Nikolay I. Kolev Multiphase Flow Dynamics 4
Nikolay I. Kolev
Multiphase Flow Dynamics 4 Nuclear Thermal Hydraulics 1st Edition
With 378 Figures
123
Nikolay Ivanov Kolev, PhD., DrSc. Möhrendorferstr. 7 91074 Herzogenaurach Germany
[email protected] ISBN: 978-3-540-92917-8
e-ISBN: 978-3-540-92918-5
Library of Congress Control Number: 2009920195 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Kirchner, Erich Printed on acid-free paper 987654321 springer.com
To my mother!
Bulgaria, Oct. 2005, Nikolay Ivanov Kolev, oil on linen
Nikolay Ivanov Kolev, PhD, DrSc Born 1.8.1951, Gabrowo, Bulgaria
The motivation to write this book
The nuclear thermal hydraulic is the science providing knowledge and mathematical tools for adequate description the process of transferring the fission heat released in materials due to nuclear reactions into its environment. Along its way to the environment the thermal energy is organized to provide useful mechanical work or useful heat. Properly arranged and controlled processes achieve this target. Not properly arranged processes or not appropriated controlled processes may lead to damages, loosing partially the investment or loosing the total investment. If power plants are so designed, that in low probable accidental processes only the investment is loosed we speak about safe nuclear power plants. Improperly designed power plants that contain the potential besides loosing the investment to destroy environment and human lives are not acceptable for the human society. The nuclear thermal hydraulics is a substantial part of the engineering discipline called nuclear reactor safety. The nuclear reactor safety is not only a technical science. It contains the relations between the society with its mature and effective control mechanisms and the technology. Only scientists and engineers can not solve the problem of the nuclear reactor safety. It is a technological and simultaneously a social problem as any problem associated with high energetic technologies. I will limit my attention in this work to the scientific part. After about 60 years research and practice we know how to build technically safe nuclear power plants. The public attitude to this subjects had its up and downs. Now the world face the problem of dramatic increasing oil and energy prises making the nuclear energy inevitable. In the same time a generation change happens and a large army of experienced nuclear engineers are retiring. The responsibility to transfer knowledge to the next generation is what drives me to write this book. I hope it will help yang scientists and engineers in their professional life of designing better facilities than those created by my generation. Herzogenaurach 22.5.2006
Summary
The nuclear thermal hydraulic is the science providing knowledge about the physical processes occurring during the transferring the fission heat released in structural materials due to nuclear reactions into its environment. Along its way to the environment the thermal energy is organized to provide useful mechanical work or useful heat or both. Chapter 1 contains introductory information about the heat release in the reactor core, the thermal power and thermal power density in the fuel, structures and moderator, the influence of the thermal power density on the coolant temperature, the spatial distribution of the thermal power density. Finally some measures are introduced for equalizing of the spatial distribution of the thermal power density. Chapter 2 gives the methods for describing of the steady and of the transient temperature fields in the fuel elements. Some information is provided regarding influence of the cladding oxidation, hydrogen diffusion and of the corrosion product deposition on the temperature fields. Didactically the nuclear thermal hydraulic needs introductions at different level of complexity by introducing step by step the new features after the previous are clearly presented. The followed two Chapters serve this purpose. Chapter 3 describes mathematically the “simple” steady boiling flow in a pipe. The steady mass-, momentum- and energy conservation equations are solved at different level of complexity by removing one after the other simplifying assumptions. First the idea of mechanical and thermodynamic equilibrium is introduced. Then the assumption of mechanical equilibrium is relaxed. Then the assumption of thermodynamic equilibrium is relaxed in addition. In all cases comparisons with experimental data gives the evidence of the level of adequacy of the different level of modeling complexity. The engineering relaxation methods are considered followed by the more sophisticated boundary layer treatment without and with variable effective bubble size. Then and introduction in the saturated flow boiling heat transfer is given and the accuracy of the methods is demonstrated by comparisons with experiments. The hybrid method of combining the asymptotic method with boundary layer treatment allowed for variable effective bubble size is also presented. Finally the idea of using the separated momentum equations and bubble dynamics is introduced and again its adequacy is demonstrated by comparison with experiments.
X
Summary
While the Chap. 3 essentially deals with the so called two-fluid model Chap. 3 demonstrates the real cases where a three fluid model is mandatory. Chapter 3 is an introduction to the “simple” steady three-fluid boiling flow in a pipe. The flow regime transition from slug to churn turbulent flow is considered in addition to the already available information from Chap. 3. The idea of the redistribution of the liquid between film and droplets is presented at two level of complexity: the instantaneous and the transient liquid redistribution in film and droplets. The transient redistribution is in fact the introduction of the ideas of droplets entrainment and deposition. The idea for the description of the mechanical interaction of the velocity fields is again presented in two level of complexity: by using drift flux correlations and by using separated momentum equation defining the forces among the fields. The next step of the sophistication is then introduced by using models for the dynamic evolution of the mean droplet size consisting of models for the droplet size stability limit, for droplet production rate due to fragmentation, for duration of the fragmentation and for collision and coalescence of droplets. Then the heat and mass transfer mechanisms in the film flow with droplet loading are introduced. Finally comparisons with experimental data demonstrate the success of the different ideas and models. To my view the reader will not understand the material of the following chapters if Chaps. 3 and 4 are not well understood. Chapter 5 describes the most powerful methods for describing the core thermal hydraulics in these days. First an introduction of the design of the reactor pressure vessels for pressurized and boiling water reactors is given. Then by using large number of experimental data sets for steady flows in heated bundles the accuracy of the modern methods is demonstrated. The experiments attracted for comparisons are the NUPEC experiment, the SIEMENS void data for the ATRIUM 10 fuel bundle, the FRIGG experiment, the THTF experiments: high pressure and low mass flow. Methods for prediction of the pressure drop for boiling flow in bundles are presented and compared with data. Then by using of experimental data sets for transient flows in heated bundles the accuracy of the modern methods is demonstrated. The experiments attracted for comparisons are the NUPEC transients in a channel simulating one subchannel of a PWR fuel assembly, the NUPEC transients in PWR 5 × 5 fuel assembly. Actually avoiding boiling crisis is the main target of a proper core design. That is why the methods for analyzing whether the critical heat flux is reached in the cores cooled by steady state flows are presented in details at different complexity level: initial 0D-guess and the 3D-CHF analysis. Several uncertainties of the physical models are identified during this process and discussed in details. New ideas for future progress in this field are presented: the large scale turbulence modeling in bundles, fine resolution analysis etc. Finally an example is given of the most complex case subject to the nuclear thermal hydraulics: the analysis of the thermal processes in a core of a boiling water reactor using the methods presented in this monograph.
Summary
XI
The stronger the driving forces for a flow processes the stable are the resulting phenomena and vise versa. Many of the processes in the nuclear thermal hydraulics are associated with low driving forces and tend to instability. This chapter presents a non-linear stability analysis on some prominent examples in the nuclear thermal hydraulics: the flow boiling and condensation stability analysis. After a state of the art review the AREVA boiling stability data for the ATRIUM 10B fuel bundle are compared with state of the art predictions using the methods presented in this monograph. The classical boiling instability analysis is accomplished with the seldom presented flow condensation stability analysis in a complex system of emergency condenser consisting of large number of 1D-condensing pipes submerged into 3D-pool. Condensation at the high pressure site leads to all flow patter for nearly horizontal pipes with all their instabilities. It is coupled with the 3Dboiling of the secondary pool site. The complex picture is very informative for what can be expected and what has to be avoided in such designs. Chapter 7 is devoted to the critical multiphase flow. It starts with the mathematical definition of the criticality condition, with the appropriate design of a numerical grid structure and numerical iteration strategy. Then the methods used in modern design are presented starting from the simple models and increasing gradually the complexity. First the single phase critical flow in pipe is considered for the case with no friction energy dissipation and constant cross section. Then the general case is presented for perfect gas. Then the same ideas are extended to simple two phase cases for pipes and nozzles: subcooled critical mass flow rate in short pipes, orifices and nozzles; frozen homogeneous non-developed flow; nonhomogeneous developed flow without mass exchange; equilibrium homogeneous flow; equilibrium non-homogeneous flow; inhomogeneous developing flow in short pipes and nuzzles with infinitely fast heat exchange and with limited interfacial mass transfer . Then the recent state of the knowledge for describing critical flow is presented by considering physical details like: bubbles origination; bubble fragmentation; bubble coalescences; droplets origination. Examples follow for application of the theory of the critical flow in real scale analysis: blow down of a closed pipe and blow down of a vessel. Chapter 8 is devoted to the basics of designing of steam generators. Chapter 9 is devoted to the basics of designing of moisture separation. Firs the importance of knowing the characteristic spectra of the moisture is underlined for proper analysis. Then some simple methods for computation of the efficiency of the separation are given for cyclone type and vane type. Different ideas based on different complexity are presented for description of the velocity field: the Kreith and Sonju solution for the decay of turbulent swirl in pipe, the potential gas flow in vanes; description of the trajectory of particles in a known continuum field; the CFD analyses of cyclones; the CFD analyses of vane separators. Then several experiments are collected from the literature for BWR cyclones, PWR steam generator cyclones, other cyclone types and vane dryers. On several cases demonstration is made for the success of different methods by comparisons with data.
XII
Summary
The nuclear power plant consists not only of large and small components but also by a forest of interconnected pipes. Chapter 10 is devoted to the estimation of the accuracy of modeling of transient processes in pipe networks by using all the methods presented in this monograph. Firs some basic definitions are introduced of pipes, axis in the space, knots, diameters of pipe sections, reductions, elbows, creating a library of pipes, creating a sub system network and discretization of pipe network for numerical treatment. Then 7 interesting experiments are simulated and a comparison with measurements is presented in order to derive conclusions about the accuracy of the methods derived in this monograph. Not only are the main systems of interest for the practicing engineer. He or she will have to handle with problems in the real life in the so called auxiliary systems. As one example of such system the high pressure reduction station is analyzed in Chapter 11. Single high pressure pipe break is analyzed and the consequences of such event are discussed. As a second example for processes in auxiliary systems an analysis of the physical and chemical processes of radiolysis gas production, air absorption, diffusion controlled gas release and transport in the coolant cleaning system of the research reactor FRM II is given. The evolution of the safety philosophy in the last 30 years leaded to the introduction of the so called passive safety systems. Such examples are the so called emergency condensers. Chapter 12 gives first a simple mathematical illustration of the operation of the system. Then the performance of the condenser as a function of the water level and pressure are analyzed with the methods introduced with this monograph. The important question of the condensate removal is discussed. Chapter 13 is devoted to the core degradation during the so called severe accidents. Chapter 14 is devoted to the melt-water interaction being important part of the moder nuclear reactor thermal hydraulics. Chapter 15 is devoted to the coolability of layers of molten nuclear reactor material. Such physics is important for designing of stabilization of spread melt in reactor compartments. After defining the problem with its boundary conditions and some simplifying assumptions the system of differential equations describing the process is presented: mass and energy conservation. The following effects are taken into account: the molten steal dropped in the melt or originating inside the melt; the gas release from a sub-layer; the viscous layer; the crust formation; the buoyancy driven convection; the film boiling; the heat conduction through the structures; oxide crust formation on colder heat conducting structures. The existence of a metallic layer is also considered. Some test cases are presented to make easy the application of the presented models: oxide over metal and oxide besides metal. A simple model for gravitational flooding of hot solid horizontal surface by water leading to hyperbolic system is also presented.
Summary XIII
Chapter 16 is devoted to the so called external cooling of reactor vessels during severe accident. It is a technology allowing to arrest the melt inside the vessel of some initial conditions are fulfilled. First the state of the art is presented. Then a brief description of the phenomenology leading to melt in the lower head is discussed: dry core melting scenario, melt relocation, wall attack, focusing effect. Brief mathematical model description is given appropriate for a set of model assumptions. The model describes: the melt pool behavior, the two-dimensional heat conduction through the vessel wall, the total heat flow from the pools into the vessel wall, the vessel wall ablation, the heat fluxes, the crust formation and the buoyancy driven convection. Solution algorithm is provided for a set of boundary conditions adequate for real situations. A summary of the state of the art regarding the critical heat flux for externally flowed lower head geometry is provided. On a several practical applications different effects are demonstrated: the effect of vessel diameter, the effect of the lower head radius, the effect of the relocation time, the effect of the mass of the internal structures. Varying some important parameters characterizing the process the difference between high powered pressurizedand boiling water reactor vessel behavior is demonstrated. Several modern aspects of the severe accident analysis can not be understood if the engineer does not have accurate information of the material properties for the participating structural materials in solid, in liquid and in some cases in gaseous states. Chapter 17 contains valuable sets of thermo-physical and transport properties for severe accident analysis for the following materials: uranium dioxide, zirconium dioxide, stainless steel, zirconium, aluminum, aluminum oxide, silicon dioxide, iron oxide, molybdenum, boron oxide, reactor corium, sodium, lead, bismuth, and lead-bismuth eutectic alloy. The emphasis is on the complete and consistent thermo dynamical sets of analytical approximations appropriate for computational analysis.
Nomenclature
Latin A A a
cross section, m² surface vector speed of sound, m / s
alw
surface of the field l wetting the wall w per unit flow volume
alσ
longing to control volume Vol (local volume interface area density of the structure w), m −1 surface of the velocity field l contacting the neighboring fields per unit
lmax
∑Vol l =1
l
be-
lmax
flow volume
∑Vol l =1
l
belonging to control volume Vol (local volume in-
terface area density of the velocity field l), m −1 al
Cui Cil c Cm Ci
lmax
total surface of the velocity field l per unit flow volume
∑Vol l =1
l
belong-
ing to control volume Vol (local volume interface area density of the velocity field l), m −1 Courant criterion corresponding to each eigenvalue, dimensionless mass concentration of the inert component i in the velocity field l coefficients, dimensionless mass concentration of the component m in the velocity field, dimensionless mass concentration of the component i in the velocity field, dimensionless
cp
specific heat at constant pressure, J / ( kgK )
c vm cd cL Dhy
virtual mass force coefficient, dimensionless drag force coefficient, dimensionless lift force coefficient, dimensionless hydraulic diameter (4 times cross-sectional area / perimeter), m
D3 E
diameter of the entrained droplets, m size of the bubbles produced after one nucleation cycle on the solid structure, bubble departure diameter, m
Dld
XVI
D1dm
Nomenclature
Dill
size of bubbles produced after one nucleation cycle on the inert solid particles of field m = 2, 3 critical size for homogeneous nucleation, m critical size in presence of dissolved gases, m most probable particle size, m characteristic length of the velocity field l, particle size in case of fragmented field, m coefficient of molecular diffusion for species i into the field l, m 2 / s
Dilt
coefficient of turbulent diffusion, m 2 / s
Dil* DCil
total diffusion coefficient, m 2 / s right-hand side of the non-conservative conservation equation for the in-
Dlch Dlcd Dl′ Dl
(
ert component, kg / sm3 D d E e F (ξ ) F , f (...
)
2
diffusivity, m / s total differential total energy, J specific internal energy, J/kg function introduced first in Eq. (42) Chapter 2 function of (...
f f Flw
force per unit flow volume, N / m3 fraction of entrained melt or water in the detonation theory surfaces separating the velocity field l from the neighboring structure
Flσ
within Vol, m 2 surfaces separating the velocity field l from the neighboring velocity field
F f im
within Vol, m 2 surface defining the control volume Vol, m 2 frequency of the nuclei generated from one activated seed on the particle
flw
belonging to the donor velocity field m, s −1 frequency of the bubble generation from one activated seed on the chan-
fl , coal
nel wall, s −1 coalescence frequency, s −1
g H h hi I i J j
acceleration due to gravity, m / s 2 height, m specific enthalpy, J/kg eigenvectors corresponding to each eigenvalue unit matrix, dimensionless unit vector along the x-axis matrix, Jacobian unit vector along the y-axis
Nomenclature
XVII
k k k kilT
unit vector along the k-axis cell number kinetic energy of turbulent pulsation, m 2 / s 2 coefficient of thermo-diffusion, dimensionless
kilp L Mi m n ΔV n nle
coefficient of baro-diffusion, dimensionless length, m kg-mole mass of the species i, kg/mole total mass, kg unit vector pointing along ΔVml , dimensionless unit vector pointing outwards from the control volume Vol, dimensionless unit surface vector pointing outwards from the control volume Vol unit interface vector pointing outwards from the velocity field l
nlσ nl
number of the particle from species i per unit flow volume, m −3 number of particles of field i per unit flow volume, particle number den-
n&coal
sity of the velocity field l, m −3 number of particles disappearing due to coalescence per unit time and
n&l , kin
unit volume, m −3 particle production rate due to nucleation during evaporation or conden-
nil
(
sation, 1/ m3s nlw′′ n&lh
) −2
number of the activated seeds on unit area of the wall, m number of the nuclei generated by homogeneous nucleation in the donor
(
velocity field per unit time and unit volume of the flow, 1/ m3s n&l , dis
number of the nuclei generated from dissolved gases in the donor velocity
(
field per unit time and unit volume of the flow, 1/ m3s n&l , sp
)
)
number of particles of the velocity field l arising due to hydrodynamic
(
disintegration per unit time and unit volume of the flow, 1/ m3s P P Per pli p q& ′′′ q&σ′′′l
)
probability irreversibly dissipated power from the viscous forces due to deformation of the local volume and time average velocities in the space, W / kg perimeter, m l = 1: partial pressure inside the velocity field l l = 2,3: pressure of the velocity field l pressure, Pa thermal power per unit flow volume introduced into the fluid, W / m 3 l = 1,2,3. Thermal power per unit flow volume introduced from the interface into the velocity field l, W / m 3
XVIII
Nomenclature
q& w′′′σ l
thermal power per unit flow volume introduced from the structure inter-
face into the velocity field l, W / m3 R mean radius of the interface curvature, m r(x,y,z) position vector, m R (with indexes) gas constant, J/(kgK) s arc length vector, m S total entropy, J/K s specific entropy, J/(kgK) Sc t turbulent Schmidt number, dimensionless tn Sc is the turbulent Schmidt number for particle diffusion, dimensionless T temperature, K Tl temperature of the velocity field l, K shear stress tensor, N / m 2 unit tangent vector dependent variables vector control volume, m3 size of the control volume, m volume available for the field l inside the control volume, m3
T t U Vol Vol1/ 3 Voll lmax
∑Vol
volume available for the flow inside the control volume, m3
V
δ iVlτ
instantaneous fluid velocity with components, u, v, w in r ,θ , and z direction, m/s instantaneous field velocity with components, ulϑ , vlτ , wlτ in r ,θ , and z direction, m/s time-averaged velocity, m/s pulsation component of the instantaneous velocity field, m/s Vl − Vm , velocity difference, disperse phase l, continuous phase m carrying l, m/s diffusion velocity, m/s
Vlτσ
interface velocity vector, m/s
Vl γ
instantaneous vector with components, ulϑ γ r , vlτ γ θ , wlτ γ z in r ,θ , and z directions, m/s specific volume, m3 / kg mass fraction, dimensionless distance between the bottom of the pipe and the center of mass of the liquid, m vector product
l =1
Vlτ Vl Vl′ ΔVlm
τ
v
x y ×
l
Nomenclature
XIX
Greek
αl α il
α l ,max γv γ r γ Δ
δ δl ∂ ε η θ κ
κ κl λ λ μlτ
part of γ vVol available to the velocity field l, local instantaneous volume fraction of the velocity field l, dimensionless the same as α l in the case of gas mixtures; in the case of mixtures consisting of liquid and macroscopic solid particles, the part of γ vVol available to the inert component i of the velocity field l, local instantaneous volume fraction of the inert component i of the velocity field l, dimensionless ≈ 0.62 , limit for the closest possible packing of particles, dimensionless the part of dVol available for the flow, volumetric porosity, dimensionless surface permeability, dimensionless directional surface permeability with components γ r , γ θ , γ z , dimensionless finite difference small deviation with respect to a given value = 1 for continuous field; = 0 for disperse field, dimensionless partial differential dissipation rate for kinetic energy from turbulent fluctuation, power irreversibly dissipated by the viscous forces due to turbulent fluctuations, W / kg dynamic viscosity, kg/(ms) θ -coordinate in the cylindrical or spherical coordinate systems, rad = 0 for Cartesian coordinates, = 1 for cylindrical coordinates isentropic exponent curvature of the surface of the velocity field l, m thermal conductivity, W/(mK) eigenvalue local volume-averaged mass transferred into the velocity field l per unit time and unit mixture flow volume, local volume-averaged instantaneous
(
mass source density of the velocity field l, kg / m3s
(
τ
)
)
μl
time average of μl , kg / m3s
μ wl
mass transport from exterior source into the velocity field l, kg / m3s
τ
μil
(
)
local volume-averaged inert mass from species i transferred into the velocity field l per unit time and unit mixture flow volume, local volumeaveraged instantaneous mass source density of the inert component i of
(
the velocity field l, kg / m3s
)
XX
Nomenclature
(
time average of μilτ , kg / m3s
μil τ
μiml
)
local volume-averaged instantaneous mass source density of the inert component i of the velocity field l due to mass transfer from field m,
(
kg / m3s
)
(
τ time average of μiml , kg / m3s
μiml τ
μilm
)
local volume-averaged instantaneous mass source density of the inert component i of the velocity field l due to mass transfer from field l into
(
) , kg / ( m s )
velocity field m, kg / m3s
μilm
time average of μilm
ν
cinematic viscosity, m 2 / s
ν
τ
t l
3
coefficient of turbulent cinematic viscosity, m 2 / s
ν ltn ξ ρ ρ ρl ρil ρl
coefficient of turbulent particles diffusion, m 2 / s angle between n lσ and ΔVlm , rad density, kg/m3 instantaneous density, density; without indexes, mixture density, kg/m3 instantaneous field density, kg/m3 instantaneous inert component density of the velocity field l, kg/m3 l
intrinsic local volume-averaged phase density, kg/m3
( ρ w )23 ( ρ w )32
(ρ V ) τ
l
l
(
entrainment mass flow rate, kg / m 2 s
(
2
deposition mass flow rate, kg / m s le
)
) (
local intrinsic surface mass flow rate, kg / m 2 s
)
σ , σ 12 surface tension between phases 1 and 2, N/m τ time, s angle giving the projection of the position of the surface point in the ϕ plane normal to ΔVlm , rad
χ
mσ l
the product of the effective heat transfer coefficient and the interfacial
(
)
area density, W / m3 K . The subscript l denotes inside the velocity field l. The superscript mσ denotes location at the interface σ dividing field m from field l. The superscript is only used if the interfacial heat transfer is associated with mass transfer. If there is heat transfer only, the linearized interaction coefficient is assigned the subscript ml only, indicating the interface at which the heat transfer takes place.
Nomenclature
XXI
Subscripts c d lm w e l i
* 0 p,v,s L R
continuous disperse from l to m or l acting on m region “outside of the flow” entrances and exits for control volume Vol velocity field l, intrinsic field average inert components inside the field l, non-condensable gases in the gas field l = 1, or microscopic particles in water in field 2 or 3 corresponding to the eigenvalue λi in Chapter 4 non-inert component mixture of entrained coolant and entrained melt debris that is in thermal and mechanical equilibrium behind the shock front from m into l from im into il maximum number of points inert component at the beginning of the time step entrainment coalescence splitting, fragmentation interface old time level new time level initial reference conditions at constant p,v,s, respectively left right
1 2 3 4 5
vapor or in front of the shock wave water or behind the shock wave melt entrained coolant behind the front – entrained coolant micro-particles after the thermal interaction – entrained melt
i M m ml iml max n 0 E coal sp
σ τ τ + Δτ
Superscripts ´ ' " "' A d e
time fluctuation saturated steam saturated liquid saturated solid phase air drag heterogeneous
XXII
Nomenclature
i imax L l le lσ m n n n+1 t vm
component (either gas or solid particles) of the velocity field maximum for the number of the components inside the velocity field lift intrinsic field average intrinsic surface average averaged over the surface of the sphere component normal old iteration new iteration turbulent, tangential virtual mass temporal, instantaneous averaging sign
τ
Operators ∇⋅ ∇ ∇n
divergence gradient normal component of the gradient tangential component of the gradient surface gradient operator, 1/m
∇t ∇l
∇2
Laplacian local volume average l le
local intrinsic volume average local intrinsic surface average
Nomenclature required for coordinate transformations
( x, y , z )
coordinates of a Cartesian, left oriented coordinate system (Euclidean space). Another notation which is simultaneously used is xi
( i = 1, 2,3) :
x1 , x2 , x3
(ξ ,η , ζ ) coordinates of the curvilinear coordinate system called transformed coordinate system. Another notation which is simultaneously used is ξ i
( i = 1, 2,3) : ξ 1 , ξ 2 , ξ 3 Vcs
g
the velocity of the curvilinear coordinate system Jacobian determinant or Jacobian of the coordinate transformation x = f ( ξ , η , ζ ) , y = g ( ξ , η , ζ ) , z = h (ξ , η , ζ )
Nomenclature
aij a
XXIII
elements of the Jacobian determinant
ij
elements of the determinant transferring the partial derivatives with respect to the transformed coordinates into partial derivatives with respect to the physical coordinates. The second superscript indicates the Cartesian components of the contravariant vectors
( a1 , a 2 , a3 )
covariant base vectors of the curvilinear coordinate system tangent
(a , a , a )
contravariant base vectors, normal to a coordinate surface on which
vectors to the three curvilinear coordinate lines represented by (ξ ,η , ζ )
1
2
3
gij
the coordinates ξ , η and ζ are constant, respectively covariant metric tensor (symmetric)
g ij
contravariant metric tensor (symmetric)
(e , e , e ) 1
2
3
unit vectors normal to a coordinate surface on which the coordinates
ξ , η and ζ are constant, respectively = ai ⋅ V , contravariant components of the vector V = ai ⋅ V , covariant components of the vector V
i
V Vi
(γ
ξ
,γ η ,γ ζ
)
permeabilities of coordinate surfaces on which the coordinates ξ , η
and ζ are constant, respectively
Greek Α, α Β, β Γ, γ Δ, δ Ε, ε Ζ, ζ Η, η Θ, ϑ
Alpha Beta Gamma Delta Epsilon Zeta Eta Theta
Ι, ι Κ,κ Λ, λ Μ, μ Ν,ν Ξ ,ξ Ο, ο Π, π Ρ, ρ
Iota Kappa Lambda Mu Nu Xi Omikron Pi Rho
Σ, σ Τ, τ Φ, ϕ Χ, χ ϒ, υ Ψ,ψ Ω, ω
Sigma Tau Phi Chi Ypsilon Psi Omega
Table of contents
1. Heat release in the reactor core ...................................................................... 1 1.1 Thermal power and thermal power density .................................................. 1 1.2 Thermal power density and fuel material .................................................... 4 1.3 Thermal power density and moderator temperature .................................... 5 1.4 Spatial distribution of the thermal power density ........................................ 6 1.5 Equalizing of the spatial distribution of the thermal power density............. 8 1.6 Nomenclature ............................................................................................. 12 References........................................................................................................ 13 2. Temperature inside the fuel elements .......................................................... 15 2.1 Steady state temperature field .................................................................... 15 2.2 Transient temperature field ........................................................................ 23 2.3 Influence of the cladding oxidation, hydrogen diffusion and of the corrosion product deposition...................................................................... 28 2.3.1 Cladding oxidation ............................................................................ 28 2.3.2 Hydrogen diffusion ........................................................................... 29 2.3.3 Deposition ......................................................................................... 29 2.4 Nomenclature ............................................................................................. 30 References........................................................................................................ 31 3. The “simple” steady boiling flow in a pipe .................................................. 33 3.1 Mass conservation...................................................................................... 35 3.2 Mixture momentum equation ..................................................................... 36 3.3 Energy conservation .................................................................................. 39 3.4 The idea of mechanical and thermodynamic equilibrium .......................... 41 3.5 Relaxing the assumption of mechanical equilibrium ................................. 42 3.6 Relaxing the assumption of thermodynamic equilibrium .......................... 43 3.7 The relaxation method ............................................................................... 45 3.8 The boundary layer treatment .................................................................... 50 3.9 The boundary layer treatment with considered variable effective bubble size ................................................................................................. 52 3.10 Saturated flow boiling heat transfer ......................................................... 56 3.11 Combining the asymptotic method with boundary layer treatment allowed for variable effective bubble size ............................................... 60
XXVI
Table of contents
3.12 Separated momentum equations and bubble dynamics ........................... 60 3.13 Nomenclature........................................................................................... 68 References ....................................................................................................... 71 Appendix 3.1: The Sani’s (1960) data for flow boiling in pipe ...................... 73 4. The “simple” steady three-fluid boiling flow in a pipe ............................... 77 4.1 Flow regime transition slug to churn turbulent flow.................................. 78 4.2 Instantaneous liquid redistribution in film and droplets ............................ 79 4.3 Relaxing the assumption for instantaneous liquid redistribution in film and droplets, entrainment and deposition ...................................... 81 4.4 Drift flux correlations ................................................................................ 84 4.5 Separated momentum equation .................................................................. 86 4.6 Dynamic evolution of the mean droplet size ............................................. 89 4.6.1 Droplet size stability limit................................................................. 89 4.6.2 Droplet production rate due to fragmentation ................................... 90 4.6.3 Duration of the fragmentation........................................................... 90 4.6.4 Collision and coalescence ................................................................. 92 4.7 Heat transfer .............................................................................................. 93 4.8 Mass transfer.............................................................................................. 95 4.9 Comparison with experiments ................................................................... 98 4.10 Nomenclature......................................................................................... 102 References ..................................................................................................... 105 5. Core thermal hydraulic............................................................................... 107 5.1 Reactor pressure vessels .......................................................................... 107 5.2 Steady state flow in heated rod bundles................................................... 114 5.2.1 The NUPEC experiment ................................................................. 114 5.2.2 The SIEMENS void data for the ATRIUM 10 fuel bundle ............ 133 5.2.3. The FRIGG experiment ................................................................. 133 5.2.4. The THTF experiments: high pressure and low mass flow .......... 139 5.3 Pressure drop for boiling flow in bundles ................................................ 144 5.4 Transient boiling ...................................................................................... 147 5.4.1 The NUPEC transients in a channel simulating one sub-channel of a PWR fuel assembly ............................................. 147 5.4.2 The NUPEC transients in PWR 5×5 fuel assembly ........................ 152 5.5 Steady state critical heat flux ................................................................... 156 5.5.1 Initial 0D-guess ............................................................................... 157 5.5.2 3D-CHF analysis ............................................................................ 162 5.5.3 Uncertainties ................................................................................... 164 5.6 Outlook – towards the large scale turbulence modeling in bundles ......... 171 5.7 Outlook – towards the fine resolution analysis ........................................ 174 5.8 Core analysis............................................................................................ 175 5.9 Nomenclature........................................................................................... 179 References ..................................................................................................... 181 Appendix 5.1: Some relevant constitutive relationship addressed in this analysis ............................................................................................... 185
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XXVII
6. Flow boiling and condensation stability analysis ...................................... 189 6.1 State of the art .......................................................................................... 189 6.2 AREVA boiling stability data for the ATRIUM 10B fuel bundle ........... 191 6.3 Flow condensation stability ..................................................................... 196 References...................................................................................................... 204 7. Critical multiphase flow .............................................................................. 207 7.1 Definition of the criticality condition....................................................... 207 7.2 Grid structure ........................................................................................... 210 7.3 Iteration strategy ...................................................................................... 212 7.4 Single phase flow in pipe ......................................................................... 212 7.4.1 No friction energy dissipation, constant cross section .................... 212 7.4.2 General case, perfect gas................................................................. 219 7.5 Simple two phase cases for pipes and nozzles ......................................... 221 7.5.1 Subcooled critical mass flow rate in short pipes, orifices and nozzles...................................................................................... 224 7.5.1 Frozen homogeneous non-developed flow ..................................... 225 7.5.2 Non-homogeneous developed flow without mass exchange .......... 228 7.5.3 Equilibrium homogeneous flow ...................................................... 229 7.5.4 Equilibrium non-homogeneous flow .............................................. 248 7.5.5 Inhomogeneous developing flow in short pipes and nuzzles with infinitely fast heat exchange and with limited interfacial mass transfer .. 261 7.6 Recent state of the knowledge for describing critical flow ...................... 269 7.6.1 Bubbles origination ......................................................................... 269 7.6.2 Bubble fragmentation...................................................................... 276 7.6.3 Bubble coalescences ....................................................................... 278 7.6.4 Droplets origination ........................................................................ 278 7.7 Examples for application of the theory of the critical flow...................... 279 7.7.1 Blow down from initially closed pipe ............................................. 279 7.7.2 Blow down from initially closed vessel .......................................... 283 7.8 Nomenclature ........................................................................................... 285 References...................................................................................................... 289 8. Steam generators ......................................................................................... 293 8.1 Introduction.............................................................................................. 293 8.2 Some popular designs of steam generators .............................................. 294 8.2.1 U-tube type ..................................................................................... 294 8.2.2 Once through type ........................................................................... 301 8.2.3 Other design types........................................................................... 301 8.3 Frequent problems ................................................................................... 301 8.4 Analytical tools ........................................................................................ 302 References......................................................................................................304
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Table of contents
9. Moisture separation .................................................................................... 307 9.1 Introduction ............................................................................................. 307 9.2 Moisture characteristics ........................................................................... 311 9.3 Simple methods for computation of the efficiency of the separation ...... 314 9.3.1 Cyclone separators .......................................................................... 315 9.3.2 Vane separators ............................................................................... 323 9.4 Velocity fields modeling in separators .................................................... 329 9.4.1 Kreith and Sonju solution for the decay of turbulent swirl in pipe ............................................................................................. 329 9.4.2 Potential gas flow in vanes ............................................................. 330 9.4.3 Trajectory of particles in a known continuum field ........................ 331 9.4.4 CFD analyses of cyclones ............................................................... 334 9.4.5 CFD analyses of vane separators .................................................... 334 9.5 Experiments ............................................................................................. 337 9.5.1 BWR cyclones, PWR steam generator cyclones............................. 337 9.5.2 Other cyclone types ........................................................................ 349 9.5.3 Vane dryers ..................................................................................... 354 9.6 Moisture separation in NPP with PWR’s analyzed by three fluid models ............................................................................................. 365 9.6.1 Separation efficiency of the specific cyclone design ...................... 367 9.6.2 Efficiency of the specific vane separators design ........................... 368 9.6.3 Uniformity of the flow passing the vane separators........................ 369 9.6.4 Efficiency of the condensate removal locally and integrally .......... 370 9.7 Nomenclature........................................................................................... 371 References ..................................................................................................... 374 10. Pipe networks............................................................................................... 377 10.1 Some basic definitions ........................................................................... 379 10.1.1 Pipes............................................................................................ 379 10.1.2 Axis in the space ......................................................................... 381 10.1.3 Diameters of pipe sections .......................................................... 382 10.1.4 Reductions .................................................................................. 383 10.1.5 Elbows ........................................................................................ 383 10.1.6 Creating a library of pipes........................................................... 384 10.1.7 Sub system network .................................................................... 384 10.1.8 Discretization of pipes ................................................................ 385 10.1.9 Knots ........................................................................................... 386 10.2 The 1983-Interatome experiments ......................................................... 388 10.2.1 Experiment 1.2 ............................................................................ 389 10.2.2 Experiment 1.3 ............................................................................ 390 10.2.3 Experiment 10.6 .......................................................................... 393 10.2.4 Experiment 11.3 .......................................................................... 394 10.2.5 Experiment 21 ............................................................................. 396 10.2.6 Experiment 5 ............................................................................... 398 10.2.7 Experiment 15 ............................................................................. 400 References ..................................................................................................... 403
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11. Some auxiliary systems ............................................................................... 405 11.1 High pressure reduction station.............................................................. 405 11.2 Gas release in research reactors piping .................................................. 408 11.2.1 Solubility of O2, N2 and H2 under 1 bar pressure ......................... 409 11.2.2 Some general remarks on the gas release- and absorption dynamics ..................................................................................... 410 11.2.3 Gas release in the siphon safety pipe .......................................... 411 11.2.4 Radiolysis gases: generation, absorption and release .................. 412 11.2.5 Mixing in the water pool ............................................................. 415 11.2.6 Computational analyses .............................................................. 415 References...................................................................................................... 421 12. Emergency condensers ................................................................................ 423 12.1 Introduction............................................................................................ 423 12.2 Simple mathematical illustration of the operation of the system ........... 424 12.3 Performance of the condenser as a function of the water level and pressure ........................................................................................... 427 12.4 Condensate removal ............................................................................... 427 13. Core degradation ......................................................................................... 429 13.1 Processes during the core degradation depending on the structure temperature ............................................................................................ 429 13.2 Analytical tools for estimation of the core degradation ......................... 430 References...................................................................................................... 431 14. Melt-coolant interaction .............................................................................. 435 14.1 Melt-coolant interaction analysis for the boiling water reactor KARENA ..................................................................................436 14.1.1 Interaction inside the guide tubes ................................................ 442 14.1.2 Melt-relocation through the lower core grid ............................... 444 14.1.3 Side melt-relocation through the core barrel ............................... 445 14.1.4 Late water injection ..................................................................... 445 14.2 Pressure increase due to the vapor generation at the surface of the melt pool ...................................................................................... 445 14.3 Conditions for water penetration into melt ............................................ 446 14.4 Vessel integrity during the core relocation phase .................................. 447 References...................................................................................................... 449 15. Coolability of layers of molten reactor material ....................................... 453 15.1. Introduction........................................................................................... 455 15.2. Problem definition ................................................................................ 455 15.3. System of differential equations describing the process ....................... 456 15.3.1 Simplifying assumptions ............................................................. 456 15.3.2 Mass conservation ....................................................................... 457 15.3.3 Gas release and gas volume faction ............................................ 459 15.3.4 Viscous layer ............................................................................... 460
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Table of contents
15.3.5 Crust formation ........................................................................... 462 15.3.6 Melt energy conservation ............................................................ 464 15.3.7 Buoyancy driven convection....................................................... 466 15.3.8 Film boiling ................................................................................ 468 15.4 Heat conducting structures..................................................................... 469 15.4.1 Heat conduction through the structures ...................................... 469 15.4.2 Boundary conditions ................................................................... 470 15.4.3 Oxide crust formation on colder heat conducting structures....... 471 15.5 Metal layer ............................................................................................. 474 15.6 Test case ................................................................................................ 474 15.6.1 Oxide over metal......................................................................... 475 15.6.2 Oxide besides metal .................................................................... 478 15.7 Gravitational flooding of hot solid horizontal surface by water ............ 479 15.7.1 Simplifying assumptions............................................................. 480 15.7.2 Conservation of mass and momentum, scaling........................... 482 15.7.3 Eigen values, eigen vectors and canonical forms........................ 485 15.7.4 Steady state ................................................................................. 489 15.8 Nomenclature......................................................................................... 491 15.9 Nomenclature to Sect. 15.7.................................................................... 493 References ..................................................................................................... 495 16. External cooling of reactor vessels during severe accident ...................... 497 16.1 Introduction ........................................................................................... 497 16.2 State of the art ........................................................................................ 498 16.3 Dry core melting scenario, melt relocation, wall attack, focusing effect ....................................................................................... 500 16.4 Model assumptions and brief model description ................................... 501 16.4.1 Molten pool behavior .................................................................. 502 16.4.2 Two dimensional heat conduction through the vessel wall ......... 503 16.4.3 Boundary conditions ................................................................... 504 16.4.4 Total heat flow from the pools into the vessel wall .................... 506 16.4.5 Vessel wall ablation .................................................................... 507 16.4.6 Heat fluxes and crust formation .................................................. 508 16.4.7 Buoyancy convection .................................................................. 509 16.5 Critical heat flux .................................................................................... 525 16.6 Application examples of the model ....................................................... 530 16.6.1 The effect of vessel diameter ...................................................... 531 16.6.2 The effect of the lower head radius ............................................. 531 16.6.3 The effect of the relocation time ................................................. 533 16.6.4 The effect of the mass of the internal structures ......................... 533 16.6.5 Some important parameters characterizing the process .............. 533 16.7 Nomenclature......................................................................................... 538 References ..................................................................................................... 540 Appendix 1: Some geometrical relations ....................................................... 544
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XXXI
17. Thermo-physical properties for severe accident analysis ........................ 549 17.1 Introduction............................................................................................ 551 17.1.1 Summary of the properties at the melting line at atmospheric pressure............................................................... 551 17.1.2 Approximation of the liquid state of melts.................................. 553 17.1.3 Nomenclature .............................................................................. 556 References...................................................................................................... 558 17.2 Uranium dioxide caloric and transport properties .................................. 559 17.2.1 Solid ............................................................................................ 560 17.2.2 Liquid .......................................................................................... 568 17.2.3 Vapor .......................................................................................... 575 References...................................................................................................... 577 17.3 Zirconium dioxide.................................................................................. 579 17.3.1 Solid ............................................................................................ 579 17.3.2 Liquid .......................................................................................... 584 References...................................................................................................... 587 17.4 Stainless steel ......................................................................................... 589 17.4.1 Solid ............................................................................................ 589 17.4.2 Liquid .......................................................................................... 596 17.4.3 Vapor .......................................................................................... 603 References...................................................................................................... 604 17.5 Zirconium .............................................................................................. 605 17.5.1 Solid ............................................................................................ 605 17.5.2 Liquid .......................................................................................... 611 References...................................................................................................... 615 17.6 Aluminum .............................................................................................. 617 17.6.1 Solid ............................................................................................ 617 17.6.2 Liquid .......................................................................................... 619 References...................................................................................................... 624 17.7 Aluminum oxide, Al2O3 ......................................................................... 627 17.7.1 Solid ............................................................................................ 627 17.7.2 Liquid .......................................................................................... 634 References...................................................................................................... 637 17.8 Silicon dioxide ....................................................................................... 639 17.8.1 Solid ............................................................................................ 639 17.8.2 Liquid .......................................................................................... 645 References...................................................................................................... 648 17.9 Iron oxide ............................................................................................... 651 17.9.1 Solid ............................................................................................ 651 17.9.2 Liquid .......................................................................................... 653 References...................................................................................................... 658 17.10 Molybdenum ........................................................................................ 659 17.10.1 Solid ........................................................................................ 659 17.10.2 Liquid...................................................................................... 663 References...................................................................................................... 666
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Table of contents
17.11 Boron oxide ......................................................................................... 667 17.11.1 Solid........................................................................................ 667 17.11.2 Liquid ..................................................................................... 669 References ..................................................................................................... 675 17.12 Reactor corium .................................................................................... 677 17.12.1 Liquid ..................................................................................... 680 17.12.2 Solid........................................................................................ 682 References ..................................................................................................... 683 17.13 Sodium ................................................................................................. 685 17.13.1 Some basic characteristics ...................................................... 686 17.13.2 Liquid ..................................................................................... 690 17.13.3 Vapor ...................................................................................... 708 References ..................................................................................................... 728 Appendix 1 .................................................................................................... 729 17.14 Lead, bismuth and lead-bismuth eutectic alloy .................................... 731 References ..................................................................................................... 737 Index ............................................................................................. ...................... 739
1. Heat release in the reactor core
Chapter 1 contains introductory information about the heat release in the reactor core, the thermal power and thermal power density in the fuel, structures and moderator, the influence of the thermal power density on the coolant temperature, the spatial distribution of the thermal power density. Finally some measures are introduced for equalizing of the spatial distribution of the thermal power density.
The nuclear energy can be used in different forms. While in the space technology some applications are known where the nuclear energy is used directly for producing electricity in all large scale technical application now the use is going trough removing and utilizing heat. Therefore the materials have to satisfy several requirements and the thermal parameters under which they operate has to be well understood for steady- and transient processes. The knowledge for the physical mechanisms for the heat release in the nuclear reactor core was already established in the 1950’s, see for instance Bonilla (1957), Etherington (1958), Alami and Ageron (1958), Petrow (1959), Grundlagen (1960), El-Wakil (1962) among many others. I will summarize only a small part of this knowledge needed to understand the basics of the safe heat removal from the nuclear reactor core. Unlike as in conventional combustion facilities where the heat is released in gas-flows in the nuclear reactor core the heat is mainly released in solid materials. If not properly removed the solid materials inevitably melt. This makes the main difference in the designing principles for the safely use of the nuclear energy compared to the conventional one.
1.1 Thermal power and thermal power density Approximate information of the available energy after each act of fission in conventional nuclear reactors is given in Table 1.1. Table 1.1. Summary of the fission energies, Etherington (1958, pp. 9–62)
Energy form
Amount in MeV
Kinetic energy of the fission products Kinetic energy of the produced neutrons Prompt γ -radiation
108 5 5
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_1, © Springer-Verlag Berlin Heidelberg 2009
1
2
1. Heat release in the reactor core
Energy form β -radiation of the fission products γ -radiation of the fission products Kinetic energy of the neutrinos Total energy 90% in form of heat
Amount in MeV 7 6 10 201 180
An averaged value of about 200 MeV per fission act is released. The total energy is split in a part called prompt energy that is immediately released, and a part called delayed decay energy associated with the farther transformation of the resulting products. The larges amount of the first group is associated with the kinetic energy of the fission product. For 235U this is 168 ± 5 MeV, for 238U it is somewhat smaller, 162 MeV and for 239Pu it is somewhat larger 172 MeV, Alami and Ageron (1958, p. 7). The high velocity fragments are decelerated in the immediate neighborhood of their origination transferring their own kinetic energy into kinetic energy of the molecules of the surrounding materials manifested in heat production inside the material. The neutrons born after each split process poses energy of 5 MeV. In thermal nuclear reactor this energy is transferred into the moderator in form of heat through deceleration of the neutrons. The γ -radiation is capable to cross larger distances. Along his way trough fuel, cladding, moderator, structure materials, radiation protection etc. it decays and transfers energy into this substance in form of heat. Therefore, in a large nuclear reactors the γ -radiation energy is completely transferred in heat within the cooled region and can completely be utilized. In fast breeder reactors there is no moderator but there is breed material layer around the core. Therefore the distribution of the γ -radiation energy a bit different: 25% of the heat is released in the breed material and 75% in the core, Etherington (1958, pp. 9–52). These considerations are valid for a developed equilibrium state of the reactor. In a fresh loaded reactor the energy production is lower for a short period of time, which is technically not so important. Much more important than this is the so called decay energy after the shut down of the reactor. Actually there is no more prompt energy after shutting down of the reactor but the decay energy is initially 6–7% of the total energy. This enforces special technical measures for cooling the reactor that is already shut down, El-Wakil (1962, p. 188). We see that about EB = 0.9 E = 180 MeV of the energy after each act of splitting is released in form of heat in the immediate neighborhood of its origin, that is inside the fuel. This amount is of basic importance for estimation of the total power of the reactor and for designing it’s safely removal from the core and its usage in the plant. Obviously in order to compute the local thermal power production per unit volume
1.1 Thermal power and thermal power density
′′′ = n&sp′′′ EB , q& Fuel
3
(1.1)
called thermal power density, we need the number of splitting acts per unit time and unit volume, n&sp′′′ = N σ f Φ .
(1.2)
Here N is the number of the atoms of the fission material per unit volume, σ f is microscopic fission cross section and Φ is the neutron flux (number of neutrons crossing a unit cross section per unit time). Therefore
′′′ = Nσ f ΦEB . q& Fuel
(1.3)
The total power of the reactor is obtained by integrating over a total fuel volume Q& = EB N
∫
σ f ΦdV .
(1.4)
VolFuel
Considering the fission cross section as some average over the core, and introducing an volume averaged neutron flux as follows Φ=
1 VolFuel
∫
ΦdV ,
(1.5)
VolFuel
results in an volume averaged thermal power density ′′′ = EB N σ f Φ , q& Fuel
(1.6)
and total power ′′′ VolFuel . Q& = q& Fuel
(1.7)
The above integration is allowed only for a homogeneous nuclear reactor core and only approximately for large cores consisting of large number of rod bundles of the same types equally distributed in space. In accordance with Eqs. (1.6) and (1.7) it follows that at constant number of fuel atoms in the core and at a constant fission cross section the averaged power density and the total power are functions of the averaged neutron flux. Because there are no limitations for the neutron flux dictated from the nuclear physics, each nuclear reactor can operate theoretically at each neutron flux, which means at each thermal power. In the practice all materials have their limitations of the properties depending on the temperature. That is the reason why as accurate as possible knowledge of the temperature field in all
4
1. Heat release in the reactor core
the core constituents is of crucial importance. Table 1.2 gives some examples for the averaged power densities in different energy engines. Obviously the nuclear industry deals with power densities which are well among those of existing conventional technologies. Table 1.2. Volume averaged thermal power density in different energy facilities
Facility
Volume averaged thermal power density in MW/m³
Fissile steam generator (forced convection) Rocket engine Pressurized water reactor Boiling water reactor Advanced gas cooled reactor Fast breeder reactor
10 ≈ 20 000 200–350 120–150 ≈ 90 400–1000
1.2 Thermal power density and fuel material The density of the fuel ρ Fuel is a material property that is obtained experimentally. Seldom fuels are pure substances. In most of the cases only C fission -part of the mass is a fuel material. The fuel mass concentration C fission is called enrichment. Therefore the number nucleus N that can be potentially split after being impacted by a neutron is N=
L M fission
C fission ρ Fuel .
(1.8)
Here M fission is the moll mass of the fuel and L is the Loschmidt number. Therefore the averaged thermal power density is ′′′ = EB Lσ f Φ q& Fuel
C fission ρ Fuel M fission
.
(1.9)
Important information dictating haw to design nuclear reactor cores is contained in this simple formula: First we realize that the power is linearly dependent on the enrichment. International conventions restrict the use of the enriched materials in power reactors. Usually 0.714–5% are used. Within this margin power increase with less then one order of magnitude is only possible. Only in some research reactors the high enrichment is used witch allows two order of magnitude increases in power. Second, technologically different chemical substances can be used as fuel e.g. uranium, uranium carbide, uranium dicarbide and uranium dioxide. In
1.3 Thermal power density and moderator temperature
5
order to obtain the same power as for 1% enrichment or metallic uranium at the same averaged neutron flux we need 1.47 % enrichment for UC, 1.81 % for UC 2, and 2.16 % for UO2, respectively. Although metallic uranium is in this regard the better material, UO2 is widely used because of its sustainability at much higher temperature.
1.3 Thermal power density and moderator temperature In the derivation of simplified Eq. (1.3) the assumption is made that all neutrons possess the same energy. In the thermal nuclear reactors the neutrons originate as fast neutrons, see Table 1.1. Then they are decelerated up to a temperature of the moderator and in the ideal case, obey Maxwell distribution controlled by the moderator temperature. In this case the fission cross section in Eq. (1.3) can be considered at best as an averaged value. The more general notation of Eq. (1.3) is then ∞
′′′ = NEB ∫ σ f ( E ) Φ ( E ) dE q& Fuel
(1.10)
0
where the integration is performed over all energies of the neutrons. Therefore the averaged fission cross section is
σf =
∞
1 σ f ( E ) Φ ( E ) dE . Φ ∫0
(1.11)
Assuming Maxwell distribution of the thermal neutrons and that the absorption cross section is proportional to the 1/v where v is the magnitude of the neutron velocity the Eq. (1.11) results in
σ f = Cσ f 0 T0 Tmod ,
(1.12)
El-Wakil (1962, p. 191), where C = 0.8862, σ f 0 = σ f (T0 ) , T0 = 293K and Tmod is the moderator temperature. With this Eq. (1.10) receives its final form ′′′ = NEB Φσ f = CEB N Φσ f 0 T0 Tmod . q& Fuel
(1.13)
Equation (1.13) indicates that the power density can also be increased if the moderator temperature is decreasing. If the moderator is used as a coolant these are contradictory requirements and therefore can not be easily realized. Therefore, the power increase is best made by increasing the enrichment. Note that the assumption of 1/v-low is only an approximation. In addition the density of the fuel changes also with temperature and therefore the influence the
6
1. Heat release in the reactor core
thermal power density itself. For fast breeder reactors the fission cross sections are not so strong functions of the temperature.
1.4 Spatial distribution of the thermal power density As already stated, the thermal power density is proportional to the local neutron flux. Therefore, the spatial distribution of the stationary neutron flux dictates the spatial distribution of the thermal power density. Analysis for real reactors is a complicated subject of the reactor physics that will not be touched here. The reader will find large number of good books in this field. We will confine our attention here only to homogeneous reactors in order to see some basic dependences. For homogeneous reactors the neutron balance is depending only on the form and the dimension of the reactor. For simple forms such dependences are listed in Table 1.3 Table 1.3. Neutron flux as a function of the spatial coordinates for simple geometries
Geometry of the core
Relative neutron flux Φ ( x, y, z ) Φ max
Parallelepiped with sizes a, b, c
cos
πx a
cos
πy
cos
πz
b c ⎛ πr ⎞ πr Sphere with radius R ⎜ sin R ⎟ R ⎝ ⎠ r⎞ πz ⎛ Cylinder with radius R and height H J 0 ⎜ 2.4048 ⎟ cos R H ⎝ ⎠
(1.14) (1.15) (1.16)
These functions can be used also as approximations for heterogeneous reactors if they consists of large number of equal fuel elements and (a) the number of the fuel elements is so large that the change of the neutron flux inside one element in radial direction is negligible and (b) the enrichment and the type of the fuel are at least in specified region of the core uniform. These functions demonstrate considerable change of the neutron flux over the space inside the core. The cylindrical form is of special interest for the practice because it is easy to realize. Therefore almost all high powered nuclear reactors possess cylindrical cores. For such cores we have
r⎞ πz ⎛ ′′′ ( r , z ) = q& Fuel ′′′ ,max J 0 ⎜ 2.4048 ⎟ cos q& Fuel , R⎠ H ⎝ where
(1.17)
1.4 Spatial distribution of the thermal power density
′′′ ,max = CEB q& Fuel
LC fission ρ Fuel M fission
σ f 0 T0 Tmod Φ max .
7
(1.18)
Here R is the outer radius and H is the height of the core. The non uniform distribution is usually quantitatively described by the so called non uniformity coefficients. The volumetric non uniformity coefficient is defined as follows kv =
′′′ ,max q& Fuel ′′′ q& Fuel
.
(1.19)
Because the influence of the radius and the axial coordinates are separated in Eq. (1.17) it is useful to introduce radial and axial non uniformity coefficients defined as follows kR =
kH =
1
r⎞ 1 ⎛ J 2.4048 ⎟ 2π rdr R⎠ π R 2 ∫0 0 ⎜⎝ R
1 1 H
⎛πz ⎞ cos ⎜ ⎟ dz ∫ ⎝H ⎠ −H / 2 H /2
=
=
1.2024 = 2.31 , J1 ( 2.405 )
π 2 = 1.57 . sin (π 2 )
(1.20)
(1.21)
Therefore kv = k R k H = 3.64 .
(1.22)
The total power of the reactor can be expressed also in terms of the non uniformity coefficients: q& ′′′ q& ′′′ ′′′ VolFuel = Fuel ,max VolFuel = Fuel ,max VolFuel . Q& = q& Fuel kv kR kH
(1.23)
Do not forget, that this equation does not take into account the energy that is released due to the neutron moderation inside the moderator. Designing the core so, as to ensure the most loaded parties with enough safety margins, means to overdesign the rest of the core. From the above relation (1.23) we see that reducing the non uniformity factors allows producing more energy at specified maximum thermal power density. This is the main idea for improving the reactor performance guiding variety of technical of measures and operational strategies.
8
1. Heat release in the reactor core
Note that the theoretical values for the non uniformity factors are not reached in praxis even for real homogeneous cores because there is always some neutron reflection back in the core. Petrow (1959, pp. 65–66) reported k R = 2 to 2.1, k H = 1.5 to 1.54 and kv = 2.86 to 3.3. During the reactor operation the fuel is burned up more intensively at those places where the fluxes are higher. Therefore the distribution changes its form with the operation time becoming more uniform. The periodical reloading and rearranging is a powerful tool for equalizing the core and exhausting more energy from the fuel.
1.5 Equalizing of the spatial distribution of the thermal power density The non uniformity does not only lower the power output from the same amount of fuel, but also reduces the burn up of the fuel. The strong non uniform thermal fields in the structure lead to thermal stresses that have to be sustained by the structure. Reff δ
R
ΔR Φ (x, y, z ) Φ max
πz r⎞ ⎛ J 0 ⎜ 2.4048 ⎟ cos R⎠ H ⎝
⎛ r J 0 ⎜ 2.4048 ⎜ R eff ⎝
⎞ πz ⎟⎟ cos H eff ⎠
r Fig. 1.1. Radial neutron flux distribution in cylindrical homogeneous core with and without reflector
Therefore, designing cores with more uniform neutron flux field increases the power exhausted from unit fuel, increases the utilization of the fuel by prolonging the time between successive reloadings (called campaigns), and reduces the thermal stresses. The most effective design feature of modern light water reactor cores leading to some equalizing of the power distribution is the so called reflector. It is a layer of moderator for instance water that reflects part of the neutrons back into the fuel region and facilitates additional power production in the boundary regions. In the last generation cores the so called heavy reflector being thick layer of
1.5 Equalizing of the spatial distribution of the thermal power density
9
steel is also implemented. Figure 1.1 demonstrates the effect of a water reflector around a cylinder. Therefore the equipping a nuclear cores with radial reflectors equalize the distribution inside the core with all the above mentioned positive consequences. The same phenomenon is acting also in axial direction. The most nuclear reactors possess naturally layers of water below and above the core therefore reflecting neutrons back in the upper and lower part in the core. The already obtained distributions for homogeneous reactors are useful approximations for cores with reflectors if appropriately modified: ⎛ r ′′′ ( r , z ) = q& Fuel ′′′ ,max J 0 ⎜ 2.4048 q& Fuel ⎜ Reff ⎝
⎞ πz , ⎟ cos ⎟ H eff ⎠
(1.24)
see Fig. 1.1. Here instead of R and H, Reff and H eff , respectively, are used, where Reff = R + ΔR ,
(1.25)
H eff = H + 2ΔH .
(1.26)
ΔR and ΔH are called extrapolation lengths. For each type of moderator there are reflector thicknesses which extension does not improve the distribution any more, Petrow (1959): light water δ = 12 cm; heavy water δ = 80 cm, graphite δ = 120 cm; beryllium δ = 50 cm. Is the reflector thickness δ ref > δ 3 so is
ΔR ≈ ΔH ≈ δ ref . Is δ ref = δ so is ΔR ≈ ΔH ≈ δ 2 . Linear extrapolation in be-
(
)
tween is useful Δ = δ 3 + δ ref − δ 3 4 . Knowing the extrapolation length and using Eq. (1.24) the non uniformity factors can be computed. The neutron flux distribution and consequently the thermal power distribution can be positively influenced by additional measures: – –
–
Changes in the fuel moderator ratio. This method is used in boiling water reactors by introducing local reflectors inside the rod bundles. An example is given in Fig. 1.2; Using of distributed neutron absorbers in the core either as burnable neutron poisons or as boric acid dissolved by the moderator. Appropriate distribution of the control rods can influence positively the neutron flux. These measures are accepted in all modern reactors; Appropriate exchange of the positions of rod bundles with different burn up combined with the use of different fuels with different enrichment. Also these measures are accepted in all modern reactors. Note that the use of rod bundles with different enrichments is more expensive and have to be balanced with the advantage in the fuel utilization.
10
1. Heat release in the reactor core
The change of the moderator temperature is complicated matter for improving the power distribution and therefore practically not used.
Fig. 1.2. Rod bundle for typical BWR: ATRIUM-10 (AREVA): Visible are the central quadratic water channel and rods with different lengths
An important instrument for influencing the moderation is the controlling the coolant mass flow at the entrance of the core. This can lead to appropriate temperature and void distribution in the core and therefore modify the moderation and the distribution of the thermal power. In modern boiling water reactors with internal pumps that have precise control of the rotation speed the overall mass flow is also changed during the campaign. At the beginning of the campaign, lower mass flow and therefore more void and increased breeding is enforced. At the end of the champagne the mass flow is increased, the void is decreased and therefore the moderation is increased. This leads to burning up of the accumulated fuel. This regime is called spectral shift. In addition, in boiling water reactors the produced two phase mixture after the core is not appropriate to be directed to the turbine. Therefore a very complex procedure is implemented to separate the large amount of the water from the steam.
1.5 Equalizing of the spatial distribution of the thermal power density
11
Fig. 1.3. Evolution of radial power distributions at beginning of cycle in a German 1300 MWe pressurized water reactor with 193 fuel assembles with 18 × 18 pins each from 1992 until 2008, Berger (2008)
12
1. Heat release in the reactor core
The so called cyclones and dryers used for this purpose operate more or less effective depending on the local water load. Therefore an appropriate designing of the water load is of great importance for the overall performance of the facility. In fast breeder reactors the blanket material is surrounding the core. Because here the emphasis is mainly on safety, changes in geometry are usually used to form the neutron flux. As already mention this chapter is only simple introduction in this complicated scientific field. In the modern nuclear core design powerful methods are developed to aggregate macroscopic data from microscopic measurements, to compute the 3D distributions in a complicated real nuclear reactor cores. An example is given in Fig. 1.3 where the evolution of the core power over 16 years of existing reactor is presented. If the reader of this book did not study nuclear reactor physics but would like to start studying nuclear reactor physics from its very basics, I recommended to start with the following books Weinberg and Wigner (1959), Henry (1975), Stacey (2001).
1.6 Nomenclature CFuel E H H eff Jo k L L M fission n N n&sp′′′
enrichment, energy, J height of the core, m effective height of the core, m Bessel-function of the I. Art, 0-order non uniformity factor Loschmidt-number length, m mol mass of the fission material, kg/kg-mole number, concentration of the fission material, -/m³ number of splitting acts per unit time and unit fuel volume, -/(m³s)
q& ′′′ Q&
power density, W/m³
power, W radius, m core radius, m effective radius of a core with reflector, m temperature, K τ time, s V volume, m³ v magnitude of the neutron velocity, m/s x coordinate, m z axial coordinate, m ΔR , ΔH extrapolation lengths, m
r R Reff T
References
δ ρ σ σf Φ
13
reflector thickness, m density, kg/m³ interaction cross section, barn fission cross section averaged over all neutron energies, barn neutron flux, the number of a neutrons crossing a unit cross section per unit time, usually in 1/(cm²s)
References Alami R and Ageron P (1958) Evacuation et récupération de la chaleur des réacteurs Nucléaires, Dunop Paris Berger H-D (September 14-19, 2008) Advances in reactor core fuel management, International Conference on the Physics of Reactors “Nuclear Power: A Sustainable Resource”, Casino-Kursaal Conference Center, Interlaken, Switzerland Bonilla CF (1957) Nuclear Engineering, McGraw-Hill Book Company Inc., New York El-Wakil MM (1962) Nuclear Power Engineering, McGraw-Hill Book Company Inc., New York Etherington H (1958) Nuclear Engineering Handbook, McGraw-Hill Book Company Inc., New York Grundlagen und Anwendung der Kerntechnik (1960) Bd. III. Energie aus Kernprozessen, VDI-Verlag, Düsseldorf Henry AH (1975) Nuclear-Reactor Analysis, The MIT Press, Cambridge, ISBN 0-262-08081-8 Petrow PA (1959) Kernenergieanlagen, VEB Verlag Technik, Berlin Stacey WM (2001) Nuclear Reactor Physics, John Wiley & Sons, New York, ISBN 0-471-39127-1 Weinberg AM and Wigner EP (1959) The Physical Theory of Neutron Chain Reactors, The University of Chicago Press, Chicago, Library of Congress Catalog Press: 58-8507
2. Temperature inside the fuel elements
Chapter 2 gives the methods for describing of the steady and of the transient temperature fields in the fuel elements. Some information is provided regarding influence of the cladding oxidation, hydrogen diffusion and of the corrosion product deposition on the temperature fields.
The fuel for the most nuclear installations used in these days is in form of small cylinders called pellets filled inside a closed tube called claddings. The closed cladding is the first barrier of the radioactivity in its way to the environment. A bundle of such tubes is called fuel element (in the east European countries) or rod bundle (in the western countries) and is prepared to be stable in transport and operation for given number of years. Groups of such bundles form the heat generating part of a nuclear reactor called core. Nuclear reactor cores are very complex in their material constitution design and control facilities. The heat is releases with different intensity at different places of the core and varies with the time depending on the regime in which the reactor is operated. In any case something is common for all places and all times: The heat is released due to the nuclear fission inside the pellets, conducted from their volume to the gas gap and then to the cladding. Finally, the coolant is removing the heat from the surface. The subject of this section is to provide the steady state and the transient solution of the Fourier equation for the specific geometry of a fuel.
2.1 Steady state temperature field Consider the geometry of the most spread type of nuclear fuel given in Fig. 2.1. The fuel radius, the gas gap thickness and the cladding thickness are RF , δ g , δ C , respectively. The heat transfer coefficient at the external wall is h . This coefficient is always related to a reference coolant temperature Tref . The heat released per unit volume of the fuel and unit time is q& ′′′ . The heat has to overcome the thermal resistance to the coolant and therefore builds a temperature profile satisfying the Fourier equation as given in Fig. 2.1. The characteristic temperatures are: The external cladding temperature Ts 3 , the internal cladding temperature Ts 2 , the
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_2, © Springer-Verlag Berlin Heidelberg 2009
15
16
2. Temperature inside the fuel elements
external fuel temperature Ts1 , and the fuel temperature at the axis of the fuel pellets Ts 0 . Problem 1: Given the volumetric heat source that is uniform along the angle and the axis, the heat transfer coefficient and the coolant reference temperature. Compute the characteristic temperatures. Solution: Text book solution is available for this task given below.
Gap
Ts0 − Ts1 =
Fuel
T s1 − Ts 2 =
q& ′′
RF
Ts2 − Ts3 =
RF + δ g
Ts 3 − Tref =
RF + δ g + δC
q&F′′′R2F 4λF
q&F′′′RF2 2λF
δ ⎞ ⎛ ln ⎜ 1 + g ⎟ RF ⎠ ⎝
2 δC q&F′′RF ⎛ ln ⎜ 1 + ⎜ 2λF ⎝ RF + δ g q&′′′RF2
(
⎞ ⎟ ⎟ ⎠
)
2 RF + δ g + δ C h
h δg
Tref δC
Cladding Fig. 2.1. Steady state temperature field in a fuel of nuclear reactor
Fuel: The Fourier equation for the cylindrical fuel is 1 d ⎛ dT λF r r dr ⎜⎝ dr
⎞ ⎟ + q& F′′′ = 0 . ⎠
(2.1)
Here λ is the thermal conductivity. Its solution is T (r ) = −
q& F′′′ 2 r + c1 ln r + c2 4λF
(2.2)
2.1 Steady state temperature field
17
with the boundary conditions dT dr
=0,
(2.3)
r =0
T ( r = 0 ) = Ts 0 ,
(2.4)
resulting in the quadratic function
T ( r ) = Ts 0 −
q& F′′′r 2 , 4λF
(2.5)
with Ts 0 − Ts1 =
q& F′′′RF2 . 4λF
(2.6)
The temperature gradient at arbitrary r inside the fuel q& ′′′r dT (r ) = − F . 2λF dr
(2.7)
will be required in a moment. Using Eq. (2.5) we can compute the volume averaged temperature between two radiuses T =
r2
1 r2
∫ rdr
∫ rT ( r ) dr = Ts 0 − r1
r1
q& F′′′ 4λF
1 r2
∫ rdr
r2
∫ r dr = T 3
s0
r1
−
q& F′′′ 2 r2 + r12 . (2.8) 8λF
(
)
r1
The point where its crosses the temperature profile is r2 =
(
)
1 2 2 r2 + r1 . 2
(2.9)
I will use this expression by constructing the transient solution. The fuel will be divided on layers of equal volumes and the position of the averaged temperature within each layer will be postulated to be defined by Eq. (2.9). Gas gap: The Fourier equation for the gas gap is
18
2. Temperature inside the fuel elements
d ⎛ dT r dr ⎜⎝ dr
2 ⎞ d T 1 dT = ⎟ dr 2 + r dr = 0 . ⎠
(2.10)
Note that there is no dependence of the heat conduction of the gap. The solution of this equation is r + c2 . RF
T ( r ) = c1 ln
(2.11)
The boundary condition at the fuel contact surface dTF dr
=
dTg
r = RF
dr
(2.12) r = RF
provides the first missing constant c1 = −
q& F′′′RF2 . 2λF
(2.13)
The second missing constant is provided by the boundary condition TF ( RF ) = Tg ( RF ) ,
(2.14)
Resulting in c2 = T1 .
(2.15)
Therefore the temperature as a function of the radius inside the gas gap is the quadratic function T ( r ) = Ts1 −
q& F′′′RF2 r ln , RF 2λF
(2.16)
with radial temperature gradient q& ′′′R 2 1 dT =− F F , 2λF r dr
(2.17)
which will be used in a moment. The temperature difference across the gas gap is therefore
2.1 Steady state temperature field
q& F′′′RF2 ⎛ δ g ⎞ ln ⎜ 1 + ⎟. 2λF ⎝ RF ⎠
Ts1 − Ts 2 =
19
(2.18)
The heat flux trough the gap can be expressed also in terms of the temperature difference
λg
′′ = − q&21 RF ln
RF + δ g
(Ts 2 − Ts1 ) .
(2.19)
RF
Note that if there is a cylinder consisting of concentric layers with left and right boundaries designated with i and i + 1 the heat transferred trough the layers is constant in the steady state conditions.
λi
Q& = −
Ri +1 Ri
Ri ln
2π Ri Δz (Ts ,i +1 − Ts ,i ) .
(2.20)
The expression can be written as Q& i ,i +1 2πΔz
=−
Ts ,i +1 − Ts ,i 1 Ri +1 ln Ri λi
(2.21)
with Rth ,i =
1
λi
ln
Ri +1 Ri
(2.22)
called thermal resistance. It is good exercise for the reader to prove that the heat transferred between the most left and the most right temperature is
Ts , right − Ts ,left Q& =− , Rth 2πΔz
(2.23)
where Rth =
left
∑
i = right
Rth,i .
(2.24)
20
2. Temperature inside the fuel elements
Cladding: The procedure for the cladding is the same as the procedure for the gas gap because it is a heat conducting cylinder without internal heat sources. The solution is again T ( r ) = c1 ln r + c2 .
(2.25)
The first constant is obtained using the boundary condition dTg dr
= r = RF + δ g
c1 = −
dTC dr
,
(2.26)
r = RF + δ g
q& F′′′RF2 . 2λF
(2.27)
The second constant is obtained from the following boundary condition
(
)
(
)
Tg RF + δ g = TC RF + δ g = Ts 2 ,
c2 = Ts 2 +
q& F′′′RF2 ln RF + δ g . 2λF
(
)
(2.28)
(2.29)
So finally the temperature as a function of the radius inside the cladding is T ( r ) = Ts 2 −
q& F′′′RF2 r ln . RF + δ g 2λF
(2.30)
The temperature difference over the cladding thickness is then Ts 2 − Ts 3 =
δC q& F′′′RF2 ⎛ ln ⎜ 1 + ⎜ 2λF ⎝ RF + δ g
⎞ ⎟. ⎟ ⎠
(2.31)
The external wall temperature is still unknown to close the solution. It is obtained attracting the Newton’s heat transfer law at the surface. Heat transfer at the surface: The heat flux at the surface is defined by
(
)
q&w′′→coolant = h Ts 3 − Tref .
(2.32)
2.1 Steady state temperature field
21
The energy conservation in steady state saying that what is generated over a fuel with axial size Δz is removed from the cladding interface over the same axial distance,
(
)
q&w′′→ coolant Δz 2π RF + δ g + δ C = q& F′′′Δzπ RF2 ,
(2.33)
provides q&w′′→coolant = q& F′′′
RF2
(
2 RF + δ g + δ C
)
.
(2.34)
Replacing in (2.32) results in Ts 3 − Tref =
(
q& ′′′RF2
)
2 RF + δ g + δ C h
.
(2.35)
A summary of the steady state solution is given in Fig. 2.1. So having a specified coolant with known temperature and velocity dictating the heat transfer coefficient all this temperatures can be computed. The relation between the heat flux and the heat inserted into unit flow is then ′′′ q&coolant = q&w′′→coolant
Heated surface 4 . = q&w′′→coolant Dheat Flow volume belonging to this heated surface (2.36)
Here the heated diameter is defined generally as follows: Dheat = 4
Flow volume belonging to this heated surface . Heated surface
(2.37)
The remarkable property of this definition is that the heated diameter does not depend on the form of the channel and on the form of the heated surface. The material temperatures are subjects to several limitations. Some of them are given in Tables 2.1 and 2.2. Table 2.1. Upper limits for the temperatures of some fuel materials, Fratzscher and Felke (1973)
Fuel U
Upper limit < Tα → β
°C 600
UO2 UC UC2
< T ′′′ < T ′′′ < Tα → β
2878 ± 20 2370 1827
22
2. Temperature inside the fuel elements
Fuel UN PuO2 ThO2
Upper limit < T ′′′ < T ′′′ < T ′′′
°C 2850 2240 3300
Table 2.2. Upper limits for temperatures of some cladding materials, Fratzscher and Felke (1973)
Cladding Aluminium Magnesium alloys Zirconium alloys Stainless steel
°C 150 200–300 500–550 300–400 350–360 400–600 800–900
H2O Air CO2 H2O H2O H2O Steam Na
The order of magnitude of some thermal conductivities is given in Table 2.3. Table 2.3. Thermal conductivity of some fuel and cladding materials, Fratzscher and Felke (1973)
Application
Material
Fuel
UO2 U 22%U+Al alloy UC Th Pu Aluminum Magnesium Magnox Steel Zirconium Graphite
Cladding
Thermal conductivity in W/(mK) at 20°C 5.4 25.1 163 29.3 37.7 33.5 at 40°C 209 147 117 167 14.6 167–251
Note that the local thermal properties are functions of the local temperatures so that the obtained solutions are idealizations. The art of designing a safe and reliable nuclear core is to select so the geometry as to guarantee in all situations the initial geometry.
2.2 Transient temperature field
23
2.2 Transient temperature field Axial non-uniformity of the heat flux and angular asymmetry makes the temperature field in a fuel really three dimensional. Frequently the asymmetry is neglected and the Fourier equations
ρcp
∂T 1 ∂ ⎛ ∂T λr − ∂τ r ∂r ⎜⎝ ∂r
⎞ ∂ ⎛ ∂T ⎟ − ∂r ⎜ λ ∂r ⎠ ⎝
⎞ ⎟ = q& ′′′ ⎠
(2.38)
is solved in two dimension numerically in order to describe the transient temperature fields in the fuel. The model used by myself since many years, see Kolev (1985, p. 220, 1986, p. 113, 1987, p. 108), is based on the following simplifying assumptions:
K r1 r2 r3 r4 = R F
δg
RF + δ g
δC
RF + δ g + δC r1
r2
r3
r4
rI ≈ RF + δ g + δC 2
2
k =1
Δz
i =1
2
3 4
I
Fig. 2.2. Definition of the geometrical sizes needed for the numerical solution of the 2DFourier equation for nuclear reactor fuel
–
The fuel is axis-symmetric: only the radial and the axial heat conduction is taken into account but not the angular;
24
2. Temperature inside the fuel elements
–
–
The gap consisting of gas does not accumulate heat because of its much lower volumetric thermal capacity ρ c p compared to the neighbouring materials; The thermal properties of the material are function of the local volume averaged temperature.
The computational grid is defined by I radial and K axial cells – see Fig. 2.2. The fuel has nF = I – 1 cells. Each of the fuel cell has the same volume resulting in
ri 2 =
RF2 i for i = 1, I – 1. nF
(2.39)
Note that
ri 2 − ri 2−1 =
RF2 . nF
(2.40)
The radius defining the position of the fuel cell averaged transient temperature ri 2 =
R 2 ( 2i − 1) 1 2 2 for i = 1, I – 1, ri + ri −1 = F 2 2nF
(
)
(2.41)
is obtained under the assumption that the radial temperature profile at each moment remains quadratic. Indication for such assumption is given by the steady solution of the Fourier equation, as already mentioned before. Using these definitions the dicretized Fourier equation reads: Radial cell 1:
(ρc )
p 1, k
dT1,k dτ
= q1,* k − λ1,k
2 T1,k − T2,k r1 r2 − r1
(2.42)
Radial cell i = 2, I - 2:
( ρc )
p i,k
dTi , k dτ
= qi*, k + λi −1, k
Ti −1, k − Ti , k ri − ri −1
(r
i
2ri −1 2
−r
2 i −1
)
− λi , k
Ti , k − Ti +1, k ri +1 − ri
(r
i
2ri 2
− ri 2−1
)
.
(2.43) Harmonic averaging of the thermal conductivities of the two neighbouring half layers is the natural choice here:
2.2 Transient temperature field
λi , k ri +1 − ri = ri
(r
= 2π ri Δz
(
2λi , k λi +1, k
2π ri − r 2
2 i −1
2λi , k λi +1, k
2
i
−r
2 i −1
)λ
i,k
(
) Δzλ
(
)
)
+ 2π ri 2+1 − ri 2 Δzλi +1, k
i ,k
+ ri 2+1 − ri 2 λi +1, k
25
.
(2.44)
Radial cell i = I - 1:
( ρc )
dTi , k dτ
p i,k
= qik* + λi −1, k
Ti −1, k − Ti , k
(r
ri − ri −1
2ri −1 2
i
−r
2 i −1
−
) (r
i
2 2
−r
2 i −1
)
Ti , k − Ti +1, k Rth , FC
.
(2.45) Here steady heat conduction is assumed in order to compute the heat flux trough the three layers between the last fuel layer temperature and the cladding averaged temperature T −T Q& FC Ts 2 − Ts1 TC − Ts 2 = − s1 i , k = − =− RF RF + δ g RF + δ g + δ C 2 1 2πΔz 1 1 ln ln ln ri λF , i , k RF RF + δ g λg λC =−
1 Rth , FC
(T
C
− Ti ,k ) ,
(2.46)
resulting in the definition of the effective thermal resistance Rth, FC =
1
λF , i , k
ln
RF + δ g RF + δ g + δ C 2 RF 1 1 . + ln + ln λg λC ri RF RF + δ g
(2.47)
Cladding i = I:
( ρc )
p i,k
dTi , k dτ
= qi*, k +
(r
2 (Ti −1, k − Ti , k )
2 C , out
−r
2 C , in
)R
th , FC
−
2 (Ti , k − Tcoolant , k )
(r
2 C , out
)
− rC2,in Rth,C
.
(2.48)
Here steady heat conduction is assumed in order to compute the heat flux trough the half of the cladding thickness
26
2. Temperature inside the fuel elements
Q& FC =− 2πΔz 1
λC
ln
Tw − TC RF + δ g + δ C
=−
(Tcoolant − Tw ) 1
RF + δ g + δ C 2i
=−
(Tcoolant − TC ) Rth,C
, (2.49)
RC ,out h
resulting in a effective thermal resistance Rth,C =
1
λC
ln
RF + δ g + δ C RF + δ g + δ C 2
+
1 RC ,out h
.
(2.50)
h is the heat transfer coefficient between the wall and the flow. Note that at any moment the heat flux at the wall is q&w′′′→coolant =
Q& FC =− RC ,out 2π RC ,out Δz
λC
(Tcoolant − TC ) ln
RF + δ g + δ C
RF + δ g + δ C
1 + 2 h
,
(2.51)
and the external wall temperature is Tw = Tcoolant + q&w′′′→coolant h .
(2.52)
Note also that in transients the heat produced inside the fuel is not necessarily the heat removed from the external wall by the coolant. q&w′′→ coolant ≠ q& F′′′
RF2
(
2 RF + δ g + δ C
)
.
(2.53)
In most of the cases it is recommendable to pool the axial conduction in the term qi*,1 = q&i′′′,1 +
qi*, k = q&i′′′, k +
λi ,1 Δz 2
(T
λi , k Δz 2
qi*, K = q&i′′′, K +
λi , K Δz 2
i ,2
− Ti ,1 ) , k = 1, adiabatic lower boundary,
(T
i , k +1
( −T
− 2Ti , k + Ti , k −1 ) , k = 2, K – 1,
i,K
(2.54)
(2.55)
+ Ti , K −1 ) , k = K, adiabatic upper boundary, (2.56)
because it is not as strong as the radial (long slabs L/D >> 1). For strong transients for instance reflood after loss of coolant accident with fast axial and temporal changes of the heat transfer coefficient in which the steam cooling is replaced fast
2.2 Transient temperature field
27
by water cooling, iterations are needed to resolve the thermal wave propagation with enough resolution. For the numerical integration of the system it is convenient to write it in the following form
E
dTk = Dk Tk + Fk E dτ
(2.57)
where E is the unit matrix, TkT = (T1, k ,..., TI , k ) , FkT = ( f1, k ,..., f I , k ) . The components of the source vectors are fi ,k =
qi*, k
( ρc )
, i = 1, I – 1,
(2.58)
+ cI , k Tcoolant , k , i = I.
(2.59)
p i ,k
f I ,k =
qI*, k
( ρc )
p I ,k
The coefficient matrix for each k is ⎛ −c1 ⎜ ⎜ b2 ⎜ 0 D=⎜ ⎜ ... ⎜ 0 ⎜ ⎜ 0 ⎝
c1 0 0 0 c2 − ( b2 + c2 ) bi − ( bi + ci ) ci ... ... ... 0 0 bI −1 0 0 0
0 0 ⎞ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟, ... ... ⎟ cI −1 ⎟ − ( bI −1 + cI −1 ) ⎟ bI − ( bI + cI ) ⎟⎠
(2.60) where the upper diagonal is ci , k =
1
( ρc )
p i ,k
cI −1, k =
λi , k 2ri , i = 1, I – 2, ri +1 − ri ri 2 − ri 2−1
(
1
( ρc )
p I −1, k
cI , k =
( ρc ) (r p I ,k
(r
i
2
2 C , out
)
2 2
−r
2 i −1
)R
)
1
,
(2.61)
(2.62)
th , FC
− rC2,in Rth ,C
,
(2.63)
28
2. Temperature inside the fuel elements
and the lower diagonal is b1, k = 0 ,
bi , k =
(2.64) 1
( ρc )
p i,k
bI , k =
λi −1, k
ri − ri −1 ri 2 − ri 2−1
( ρc ) (r p i,k
(
2ri −1
2
2 C , out
−r
2 C , in
( ρc ) ) ( ρc )
)R
th , FC
=
p i −1, k
ci −1, k , i = 2, I-1,
(2.65)
p i,k
=
( ρc ) ( ρc )
p I −1, k
cI −1, k ,
(2.66)
p i,k
Observe that the matrix is close to symmetric. It is recommendable to use explicit differentiation in the radial direction. In this case −1
n ⎞ ⎛ E ⎞ ⎛T Tk = ⎜ − Dk ⎟ ⎜ k + Fk ⎟ E . ⎝ Δτ ⎠ ⎝ Δτ ⎠
(2.67)
The axial heat fluxes can be computed explicit for slow thermal transients. For fast transients iteration is necessary to update them until a solution is obtained with prescribed accuracy. The steady state solution is easily obtained writing Δτ → ∞ ,
Tk = ( −Dk ) Fk E . −1
(2.68)
2.3 Influence of the cladding oxidation, hydrogen diffusion and of the corrosion product deposition
2.3.1 Cladding oxidation Oxidation of the cladding reduces the thickness of the cladding and therefore its strength. Rassohin et al. (1971) reported that the oxidation progression with the time depends on the cladding temperature M Zr2 O F = 10−3 exp ( − 1100 Ts 3 )(τ 3600 )
44 exp ( −1350 Ts 3 )
.
(2.69)
Here M Zr2 O in kg is the accumulated mass of zirconium oxide and F is the surface in m² on which its is build. The observations are covered with data in the region 274.15 < Ts 3 < 723.15 K. Because the critical thickness sustaining the pressure
2.3 Cladding oxidation, hydrogen diffusion and corrosion product deposition
difference
Δp
(
of the cladding is δ C ,cr = Δp RF + δ g + δ C
)σ
Zr , cr
29
where
σ Zr ,cr = 90 to 162MPa , the cladding oxidation reduces the strength of the cladding. The second effect is that the additional thickness of oxide possesses considerably lower thermal conductivity and presents additional thermal resistance that increases the surface temperature of the metal. 2.3.2 Hydrogen diffusion Hydrogen is generated in nuclear rectors due to the radiolysis of water. In some plants hydrogen is additionally introduced in the water in order to reduce the content of atomic oxygen and therefore to reduce the oxidation of the structures. Rassohin et al. (1971) reported the following kinematic expression for the reached mass concentration of hydrogen in the cladding as a function of time CH 2 in H 2 + Zr =
DH 2 → Zr (τ 3600 )
2δ C ρ Zr
4.4 exp ( −1350 TZr )
exp ( −1100 TZr )
.
(2.70)
Here DH 2 → Zr is the diffusion coefficient of hydrogen into zirconium and ρ Zr is the density of the zirconium. The effect of the hydrogen is in the reduction of σ Zr ,cr .
2.3.3 Deposition Depending on the water chemistry fuels standing long time in the core exhibits thin layer of corrosion products (Fe, Ca, Mg etc.) deposition. So for instance for CFe oxides = ( 0.05 to 0.2 ) × 10−6 kg / kg can be found in some water cooled reactors, Rassohin et al. (1973). The deposition mass flow rate was reported to be function of the wall heat flux and of the mass concentration in the coolant
( ρ w )deposits = 2.5 ×10−4 q&w′′Cdeposits
106 , 3600
(2.71)
Rassohin et al. (1973). The roughness of the deposits is depending on the time of the exposition and can take values between 0.6 to 0.65 µm up to 5 µm. The density of the deposits is reported to be very low of about 286 kg/m³. The heat conductivity for iron oxides is about 3.5–10 W/(mK) and the heat conductivity of the other oxides (Ca, Mg etc.) can be about 3.5–35 W/(mK). Therefore we have two effects of the deposits: (a) They increase the friction pressure loss because they increase the roughness; (b) They increase the thermal resistance and therefore increase the cladding temperature level.
30
2. Temperature inside the fuel elements
2.4 Nomenclature Latin b c cp
Coefficients Coefficients Specific capacity at constant pressure, J/(kgK)
DH 2 → Zr Diffusion coefficient of hydrogen into zirconium, m²/s
RC ,in
Heat transfer coefficient, W/(m²K) Fuel radius, m := RF + δ g , inside cladding radius, m
RC ,out
:= RF + δ g + δ C , outside cladding radius, m
ri
Boundary radius of layer i, m Radius at which the averaged layer temperature is defined, m Fluid temperature to which the heat transfer coefficient is related, K
h RF
ri Tref q& ′′′ Ts 0 Ts1 Ts 2 Ts 3 zk
Greek Δτ Δz
δg
δC λ ρ ρ Zr τ
Heat release per unit volume and unit time, W/m³ Fuel temperature at the axis of the fuel pellets, K External fuel temperature, K Internal cladding temperature, K External cladding temperature, K Axial coordinate of the cell boundaries, m
Time step, s Axial size of the cell, m Gap thickness, m Cladding thickness, m Thermal conductivity, W/(mK) Density, kg/m³ Density of the zirconium, kg/m³ Time, s
Subscripts F C g i k
Fuel Cladding Gap Layer i Layer k
References
31
Superscripts . ´´´
per unit time per unit volume
References Fratzscher W and Felke K (1973) Einführung in die Kernenergetik, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig Kolev NI (1985) Comparison of the RALIZA-2/02 two-phase flow model with experimental data, Nucl. Eng. Des., vol 85, pp 217–237 Kolev NI (1986) Transiente Zweiphasenströmung (Transient Two-Phase Flow), Springer Verlag, Berlin Kolev NI (1987) Numerical modelling of three-phase three-component flow, Thesis for Doctor of Science Degree, Bulgarian Academy of Science, Institute for Nuclear Research and Nuclear Energy, Discipline 02.06.04 Nuclear Reactors and Nuclear Power Plants, Sofia Rassohin NT, Gradusov GH and Gorbatych VP (1971) Korosija splava zirkonija – 1% niobija v uslovijah teploperedaci, Trudy MEI, Vyp. 83 Rassohin NT, Kabanov LP, Tevlin SA and Tepsin VA (Jan. 1973) Jelesnookisnye otlojenija na teplovydeljastich poverhnostjah I ih udalenija, Preprint B-7 doklada na mejdunarodnoj konferencii po elektrohimii jidkih rastvorov pri vysokom davlenii I temperature, England
3. The “simple” steady boiling flow in a pipe
Didactically the nuclear thermal hydraulics needs introductions at different level of complexity by introducing step by step the new features after the previous are clearly presented. The followed two Chapters serve this purpose. Chapter 3 describes mathematically the “simple” steady boiling flow in a pipe. The steady mass-, momentum- and energy conservation equations are solved at different level of complexity by removing one after the other simplifying assumptions. First the idea of mechanical and thermodynamic equilibrium is introduced. Then the assumption of mechanical equilibrium is relaxed. Then the assumption of thermodynamic equilibrium is relaxed in addition. In all cases comparisons with experimental data gives the evidence of the level of adequacy of the different level of modeling complexity. The engineering relaxation methods are considered followed by the more sophisticated boundary layer treatment without and with variable effective bubble size. Then and introduction in the saturated flow boiling heat transfer is given and the accuracy of the methods is demonstrated by comparisons with experiments. The hybrid method of combining the asymptotic method with boundary layer treatment allowed for variable effective bubble size is also presented. Finally the idea of using the separated momentum equations and bubble dynamics is introduced and again its adequacy is demonstrated by comparison with experiments.
The art in the engineering science is to find the simplest mathematical description of the investigated process. This means to make so much simplification of the problem as necessary to solve it but “without throwing the child with the water” which means without loosing substantial physical effects. We give here a description of the steady state boiling flow in a vertical pipe as an example and introduction in the nuclear thermal hydraulics. Hundreds of scientists contributed to reaching this simplicity of the model characterized with the assumptions that one of the phases is always in saturation. More sophisticated models are available now but it will be complicated to use them as an introduction to the nuclear thermal hydraulics. Although being “simple” the model presented bellow posses all characteristic features that are needed to understand the boiling flows. This understanding will be then used to introduce more sophisticated concepts. The formal mathematical description of a physical phenomenon is called usually model. A model is unperfected reflection of the nature. The engineering science distinguishes between verification and validation of a model. Verification is the check of consistency with other knowledge done by using formal logic. Part of the
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_3, © Springer-Verlag Berlin Heidelberg 2009
33
34
3. The “simple” steady boiling flow in a pipe
verification is also the formal tests of the computer program that transfers the model from the paper into a set of commands for digital computer. This mandatory step precedes the validation. The validation is a process in which the experimental data are compared with the model predictions. Conclusions are then drawn about the adequacy of the model. I also give here a simple introduction to verification and validation of physical models. The multiphase fluid dynamics is the basic science providing the tools for understanding the nuclear thermal-hydraulic processes. Cooling of heat releasing structures by boiling is a very efficient cooling mechanism. Let us consider the fundamental problem of boiling flow in vertical uniformly heated pipe with constant diameter, Fig. 3.1. Vapor
Single fluid
Droplets
Two fluid 3
Three fluid
Film + droplets
Two fluid
Vapor + liquid
Two fluid
1
Bubbles + liquid
Two fluid
&w′ Single fluid q′
Slugs + liquid
2
Liquid
m& F , T2 Fig. 3.1. Flow boiling in a vertical cylindrical heated pie
Liquid, designated with subscript 2, enters the pipe upwards with mass flow m& in kg/s. The mass flow rate is defined as mass flow per unit pipe cross section ρ w = m& F = const in kg/(m²s). It does not change in the time. The inlet temperature is T2 = const. It does not change in the time. The wall releases thermal energy
3.1 Mass conservation
35
per unit time and unit surface designated with q& w′′ in W/m². q& w′′ is called heat flux. It again, does not change with the time. At this moment we are not interested in the origin of the thermal energy source. Compute the cross section averaged parameters of the flow along the pipe axis z. The mathematicians speak about steady state “boundary value problem”, because the boundary values at the pipe entrance are specified. The heat flux is also one of the “boundary condition”. These boundary conditions and the pipe geometry control completely the processes inside the pipe. The process is called steady state because under these conditions the parameters in the flow will not change with the time or at least will repeat the same fluctuations in the time. Note that this state can not be realized at all flow conditions. I will elaborate the subject of the boiling stability in a special chapter. Although appearing “simple” this problem contains many of the characteristics of the thermal hydraulics which will be later elaborated in variety of conditions and geometries. Therefore, the reader will not understand the massage of this book if he does not understand the basic physics behind this problem. First we need the three conservation lows for mass, momentum and energy written for steady state pipe with constant cross section.
3.1 Mass conservation The mass conservation equation for steady state flow in channel with constant cross section is lmax d (α l ρl wl ) = ∑ ( μml − μlm ) . dz m =1
(3.1)
Here the subscript l means “velocity field”– a mathematical abstraction for part of the flowing fluid. We thing here, that each state of aggregate is a field. Later we will see that the field may be associated with many other properties which are of no interests for now. Convenient formalism can be obtained if we assume three velocity fields: one for the gas (1), one for the continuous liquid (2) and one for dispersed droplets (3). Let as introduce the term mass flow fraction of the velocity field l. X l = α l ρl wl G .
(3.2)
Here for each field l, α l is the cross section averaged field volumetric fraction, ρl is the “macroscopic” density and wl is the cross section averaged velocity, respectively. The term macroscopic is used for the intrinsic averaged density. We consider length scales far from the molecular size. So that the density is a function of macroscopic flow properties like pressure and temperature, ρl = ρl ( p, Tl ) .
36
3. The “simple” steady boiling flow in a pipe
This equation is called equation of state and is usually gained experimentally. The mixture mass flow rate is defined as follows lmax
G = ∑ α l ρl wl .
(3.3)
l =1
The mass conservation equation can be then rewritten as lmax d ( X l G ) = ∑ ( μml − μlm ) . dz m =1
(3.4)
Here μ ml is the mass transferred from field m into field l per unit time and unit mixture volume in kg/(m³s). The non-negative defined mass sources for each field coming from the other fields μml and μlm possess the property that their overall sum is zero. Therefore the sum of all l mass conservation equations results in the important result dG = 0 or G = const . dz
(3.5)
With other words, the mass flow rate of a steady state flow in a channel with constant cross section is a constant over the length of the channel. Now we realize the convenience of this definition. It provides also a simple notation of the mass conservation equations for each field dX l 1 lmax = ∑ ( μml − μlm ) . dz G m =1
(3.6)
Varieties of formalisms are developed for the relation α l = α l ( X l , G , p,...) in steady state flows, which will be discussed later. At this place we will only mention that either momentum equations for each field, or algebraic relations have to be used to describe the mechanical interaction between the fields.
3.2 Mixture momentum equation The fact that the sum of all forces acting on a small control volume of the fluid is equal to zero is expressed by the momentum conservation equation lmax d ⎛ lmax dp ⎛ dp ⎞ 2⎞ + g cos ϕ ∑ α l ρl + ⎜ ⎟ = 0 . ⎜ ∑ α l ρl wl ⎟ + dz ⎝ l =1 ⎝ dz ⎠ fr l =1 ⎠ dz
(3.7)
3.2 Mixture momentum equation
37
⎛ dp ⎞ Here the friction pressure drop gradient is ⎜ ⎟ . ϕ is the angle of the flow with ⎝ dz ⎠ fr respect to the upwards directed vertical. For upwards directed vertical flow cos ϕ = 1 . The mixture momentum can be rewritten as lmax
∑α l =1
lmax
l
ρl wl2 = G 2 ∑ l =1
X l2
α l ρl
(3.8)
and the momentum equation can be simplified to G2
d lmax X l2 dp ⎛ dp ⎞ + + gρ + ⎜ ⎟ = 0 , ∑ dz l =1 α l ρl dz ⎝ dz ⎠ fr
(3.9)
where lmax
ρ = ∑ α l ρl
(3.10)
l =1
is called mixture density. In many cases the simplified form of this equation is used dp ⎛ dp ⎞ + gρ + ⎜ ⎟ ≈ 0 , dz ⎝ dz ⎠ fr
(3.11)
saying that the pressure along the z-axis changes due to gravity and friction only. As an example for computation of the pressure loss due to friction we use here the Martinelli-Nelson method. In this method a friction coefficient is computed as could the total flow consists of liquid only (subscript 2o) 1 G 2 ⎛ λ fr ,20 ζ ⎞ ⎛ dp ⎞ = + ⎜ ⎟, ⎜ dz ⎟ Δz ⎠ ⎝ ⎠ fr ,20 2 ρ 2 ⎝ Dh
(3.12)
where
λ fr ,2o = λ fr ,2 o ( G Dh / η 2 , k / Dh ) .
(3.13)
Then to obtain the total mixture friction pressure loss per unit length the so computed pressure loss is multiplied by the so called Martinelli-Nelson multiplier
38
3. The “simple” steady boiling flow in a pipe
⎛ dp ⎞ 2 ⎛ dp ⎞ . ⎜ dz ⎟ = Φ 2 o ⎜ dz ⎟ ⎝ ⎠ fr ⎝ ⎠ fr ,2o
(3.14)
Here Δz is the length over which the pressure drop happens. ζ is the irreversible local pressure drop coefficient. To compute the Martinelli-Nelson multiplier we recommend the Friedel (1979) correlation valid for η2 / η1 < 1000. First additional friction coefficient is computed assuming that the flow consists of vapor λ1o = λ1o ( GDh η1 , k Dh ) only to compute λ1o λ2o . A good approximation for hydraulic smooth pipes is
λ1o λ2o ≈ (η1 η2 )
1/ 4
.
(3.15)
Then the dimensionless numbers E = (1 − X 1 ) 2 + X 12
ρ 2 λ1o , ρ1 λ2 o
(3.16)
F = X 10.78 (1 − X 1 )0.24 , ⎛ρ ⎞ H =⎜ 2 ⎟ ⎝ ρ1 ⎠
0.91
(G ρ ) Fr = gDh
⎛ η1 ⎞ ⎜ ⎟ ⎝ η2 ⎠
0.19
(3.17) 0.7
⎛ η1 ⎞ ⎜1 − ⎟ , ⎝ η2 ⎠
(3.18)
2
,
We = G 2 Dh /( ρσ )
(3.19)
(3.20)
are computed, and finally the friction multiplier Φ 220 = E + 3.24 FH /( Fr 0.045We0.035 ) .
(3.21)
This correlation approximates 25 000 experimental points for vertical upwards cocurrent flow and for horizontal flow with 30–40% standard deviation for one- and two-component flow.
3.3 Energy conservation
39
3.3 Energy conservation Neglecting the axial molecular and turbulent thermal diffusion, the irreversible energy dissipation due to velocity field deformation and turbulent fluctuation, and also the mechanical energy dissipation due to the mass exchange between the fields we obtain for the mixture energy conservation equation ⎞ ⎛ lmax ⎞ dp d ⎛ lmax w h α ρ = q&w′′′ ⎜ ∑ l l l l ⎟ − ⎜ ∑ α l wl ⎟ dz ⎝ l =1 ⎠ ⎝ l =1 ⎠ dz
(3.22)
dh dp q& w′′′ −v = , dz dz G
(3.23)
or
where lmax
h = ∑ X l hl
(3.24)
m =1
is called the mixture enthalpy and ⎛ lmax X ⎞ v = ⎜∑ l ⎟ ⎝ l =1 ρl ⎠
is called homogeneous specific volume of the mixture. In case of v
(3.25) q& ′′′ dp 70 000 − G h′′ − h′ ⎩⎪
(3.56)
Saha and Zuber (1974), where Pe2 := GDhy c p 2 / λ2 .
(3.57)
Kawara et al. (1998) reported that at low void fraction the Saha and Zuber correlation is not accurate. They proposed the modification: Nu
q&w′′ =454 or 0.88 × 107
T2* = T ′( p ) −
X 1*eq = −
q&w′′2 or λ2 399.52 × 107
Dh
Dh c p 2
λ2
q&w′′2 399.52 × 107 ( h′′ − h′ )
for Pe2 ≤ 70 000 and
St
q& w′′ = 0.0065 or 0.88 ×107
(3.58)
3.7 The relaxation method
T2* = T ′( p) −
X 1*eq = −
45
q& w′′2 or 5.72 × 104 Gc p 2
q&w′′2 , 5.72 × 104 G ( h′′ − h′ )
(3.59)
for Pe2 > 70 000, where q& w′′ Dh , * T ′( p ) − T2 λ2
(3.60)
q&w′′ . Gc p 2 ⎡⎣T ′( p ) − T2* ⎤⎦
(3.61)
Nu =:
St =:
My own analyses, Kolev (2005), based on the NUPEC (2004) data demonstrated that at low void fraction the use of the Hughes et al. (1981) model, that will be described in a moment is superior to the use of the Saha and Zuber model in combination with the Levy (1967) relaxation model. For computation of the real vapor mass flow ratio in the framework of this concept two methods are known which will be briefly presented.
3.7 The relaxation method The simplest way of modeling sub-cooled boiling is to use dX 1eq dz
=
q&w′′′ 1 ⎛ q&w′′′ dp ⎞ +v ⎟ ≈ , ⎜ h′′ − h′ ⎝ G dz ⎠ G ( h′′ − h′ )
(3.62)
to compute the equilibrium vapor mass flow rate concentration , X 1eq , and then to
use experimentally obtained relation X 1 = X 1 ( X 1*eq , X 1eq ) valid for X 1eq < 0 , and X 1 = X 1eq for X 1eq ≥ 0 e.g.
(
)
(3.63)
) , Achmad (1970).
(3.64)
X 1 = X 1eq − X 1*eq exp X 1eq X 1*eq − 1 , Levy (1967),
X1 =
(
X 1eq − X 1*eq exp X 1eq X 1*eq − 1 1− X
* 1eq
(
exp X 1eq X
* 1eq
)
−1
46
3. The “simple” steady boiling flow in a pipe
Then using the just computed real vapor mass flow ratio in the drift flux theory, the local void fraction and all other related local parameters can be computed. To demonstrate the accuracy of the relaxation method in combination with drift flux correlation I will use the systematic boiling experiments reported by Bartolomei et al. (1982). 12 mm in-diameter uniformly heated pipe of 1.5 m length is used. Fixing the boundary conditions given in Table 3.1 the void fraction was measured by gamma-ray densitometer within an error band of ± 0.04. The accuracy of the measurements of the boundary conditions was as follows: pressure 1%, mass flow rate 2%, heat flux 3% and temperature ± 1 K. All data except the points 11 and 15 has Peclet number larger then 70 000. The experimental data are presented in Fig. 3.2 as real void fraction as a function of the equilibrium vapor mass flow ratio, α1 = α1 ( X 1,eq ) . At constant heat flux the equilibrium vapor mass flow ratio is proportional to the distance from the entrance of the pipe. Observing the data we realize that there is a point where the void originates first. This point is designated in the literature as boiling initiation (bi), onset of the nucleate boiling (onb) etc. This is unstable process because the bubbles eventually completely condense in the bulk flow. There is a second point at which a stable and visible net vapor production (nvp) starts. One should distinguish between them in correlating data with models. The data demonstrate that the boiling initiation happens already in sub-cooled part of the liquid. So, it becomes clear that this can not be explained with the assumption that the vapor-liquid mixture is in thermodynamic equilibrium.
Table 3.1. Boundary conditions for the experiments performing by Bartolomei et al. (1982)
ρ w in q&w′′2 in T2 in kg/(m²s) W/m² K p ≈ const, ρ w ≈ const, T2 ≈ const, q&w′′2 → variable 1 6 810 000. 998. 440 000. 521. 2 6 890 000. 965. 780 000. 493. 3 6 840 000. 961. 1 130 000. 466. 4 6 740 000. 988. 1 700 000. 416. 5 7 010 000. 996. 1 980 000. 434. No.
p in Pa
6 7 8 9 10
14 790 000. 14 740 000. 14 750 000. 14 700 000. 14 890 000.
1878. 1847. 2123. 2014. 2012.
420 000. 770 000. 1 130 000. 1 720 000. 2 210 000.
603. 598. 583. 545. 563.
3.7 The relaxation method
47
ρ w in q& w′′2 in T2 in kg/(m²s) W/m² K p ≈ const, ρ w → variable, T2 ≈ const, q&w′′2 ≈ const 11 6 890 000. 405. 790 000. 421. 12 6 890 000. 986. 780 000. 493. 13 6 890 000. 1467. 770 000. 519. 14 6 790 000. 2024. 780 000. 520. No.
p in Pa
15 11 020 000. 503. 990 000. 494. 16 10 810 000. 966. 1 130 000. 502. 17 10 810 000. 1554. 1 160 000. 563. 18 10 840 000. 1959. 1 130 000. 563. ′′ & p → variable, ρ w ≈ const, T2 ≈ const, qw 2 ≈ const 19 3 010 000. 990. 980 000. 445. 20 4 410 000. 994. 900 000. 463. 21 6 840 000. 961. 1 130 000. 466. 22 10 810 000. 966. 1 130 000. 502. 23 14 580 000. 1000. 1 130 000. 533. 24 25 26
6 810 000. 10 840 000. 14 750 000.
2037. 1959. 2123.
1 130 000. 1 130 000. 1 130 000.
504. 563. 583.
Now I compute the void fraction using the Saha and Zuber (1974) correlation for the onset of the nucleate boiling, the relaxation formula by Levy (1967), and the drift flux parameters by Maier and Coddington (1986). The results are presented in Fig. 3.2 a-1 to f-1. We see that
(a) for variable heat flux the low heat flux cases are well predicted while the high heat flux cases are over predicted for low pressure and well predicted at high pressure; (b) for variable mass flow density at low mass flow density the data are under predicted and at high mass flow density the agreement is well; (c) for variable pressure at low pressure the data are slightly over predicted and at high pressure and high mass flow rate the agreement is good. The reason for this situation is to be seen in the simplicity of the theory for such complex processes. Slightly better results are obtained if the relaxation formula by Achmad (1970) is used as shown in Fig. 3.2 a-2 to f-2.
3. The “simple” steady boiling flow in a pipe
0,6 0,5
Void, -
0,4 0,3 0,2
0,6
1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0,5 0,4 0,3
Void, -
48
0,2 0,1
0,1
0,0
0,0 -0,2
-0,1
0,0
-0,2
0,1
0,3 0,2 0,1
a-1
a-2
0,5 0,4 0,3 0,2 0,1
6 exp. 7 exp. 8 exp. 9 exp. 10 exp. 6 th. 7 th. 8 th. 9 th. 10 th.
0,0 -0,1
X1,eq,-
0,0
-0,2
0,1
-0,1
b-1
0,3 0,2
0,5 11 exp. 12 exp. 13 exp. 14 exp. 11 th. 12 th. 13 th. 14 th.
0,4
0,1
0,3 0,2
0,0
0,1
11 exp. 12 exp. 13 exp. 14 exp. 11 th. 12 th. 13 th. 14 th.
0,1
0,1
0,0
0,0 -0,2
0,0
0,6
Void, -
Void, -
0,4
X1,eq, -
b-2
0,6 0,5
0,1
0,6 6 exp. 7 exp. 8 exp. 9 exp. 10 exp. 6 th. 7 th. 8 th. 9 th. 10 th.
0,0 -0,2
0,0 X1,eq,-
Void, -
Void, -
0,4
-0,1
X1,eq,-
0,6 0,5
1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
-0,1
0,0
0,1
-0,2
-0,1
X1,eq,-
X1,eq, -
c-1
c-2
3.7 The relaxation method
0,5
0,3
0,4
0,2
0,2 0,1
0,0
0,0
0,6 0,5 0,4 0,3 0,2
-0,1
0,0
-0,2
0,1
d-2
0,4 0,3 0,2 0,1 0,0 -0,1
0,0
-0,2
0,1
-0,1
0,0
X1,eq,-
X1,eq, -
e-1
e-2
0,1
0,6 0,5 24 exp. 25 exp. 26 exp. 24 th. 25 th. 26 th.
0,4
0,1
0,3 0,2
24 exp. 25 exp. 26 exp. 24 th. 25 th. 26 th.
0,1
0,0 -0,2
19 exp. 20 exp. 21 exp. 22 exp. 23 exp. 19 th. 20 th. 21 th. 22 th. 23 th.
0,5
Void, -
0,2
0,1
0,6
19 exp. 20 exp. 21 exp. 22 exp. 23 exp. 19 th. 20 th. 21 th. 22 th. 23 th.
0,5
0,3
0,0
d-1
0,6
0,4
-0,1 X1,eq, -
0,0 -0,2
15 exp. 16 exp. 17 exp. 18 exp. 15 th. 16 th. 17 th. 18 th.
X1,eq,-
0,1
Void, -
0,3
0,1
-0,2
Void, -
0,5
Void, -
Void, -
0,4
0,6 15 exp. 16 exp. 17 exp. 18 exp. 15 th. 16 th. 17 th. 18 th.
Void, -
0,6
49
0,0 -0,1
0,0
0,1
-0,2
-0,1
0,0
X1,eq, -
X1,eq, -
f-1
f-2
0,1
Fig. 3.2. Void fraction as a function of the homogeneous equilibrium mass fraction. Experimental data by Bartolomei et al. (1982). Prediction using: Relaxation formula by Levy (1967); Drift flux parameter by Maier and Coddington (1986); Onset of the nucleate boiling: a-1 to f-1 Saha and Zuber (1974); a-2 to f-2 Achmad (1970)
50
3. The “simple” steady boiling flow in a pipe
3.8 The boundary layer treatment The boundary layer treatment is better suited to modeling of boiling flows in nonuniformly heated pipes. The boiling boundary layer treatment needs more detailed knowledge about haw the heat flux from the wall q&w′′ = hNB (Tw − T ′) n + hconvective (T ′ − T2 )
(3.65)
is split into a part that goes directly for heating the liquid q&2′′′σ w =
4 hconvection (Tw − T2 ) Dheat
(3.66)
and a part used for evaporation. The evaporated mass per unit time and unit mixture volume is
μ21 =
4 hNB (Tw − T ′) n . Dheat h′′ − h′
(3.67)
The condensed mass per unit time and unit volume is
μ12 =
4 h2σ 1 (T ′ − T2 ) . Dheat h′′ − h′
(3.68)
Here the implicit assumption is made, that the mass transfer surface for evaporation and condensation is approximately equal to the heated surface. Later we will learn that this is much more complex process. The vapor mass conservation equation provides then the mass flow concentration change along the axis dX 1 μ21 − μ12 or = dz G
dX 1 4 ⎡ hNB (Tw − T ′) n − h2σ 1 (T ′ − T2 ) ⎤⎦ for X 1eq > X 1*eq ≈ dX 1eq Dheat q& w′′′ ⎣
(3.69)
and X 1 = 0 for X 1eq ≤ X 1*eq . The onset of nucleate boiling is then defined by
μ21 − μ12 = 0 or hNB (Tw − T ′) n = h2σ 1 (T ′ − T2,ONB ) . Eliminating the wall temperature results in
3.8 The boundary layer treatment
⎡ 2 hσ 1 hσ 1 ⎢ hconvective 2 + 4 h2σ 1 + hconvective q& w′′ − hconvective 2 hNB hNB ⎢ = T′− ⎢ σ1 2 h2 + hconvective ⎢ ⎢ ⎣
(
T2,ONB
)
(
)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
51
2
for n = 2.
(3.70) In the general case the onset of the nucleate boiling temperature is obtained by iteration starting with T2,ONB = T ′ − q& w′′
(h
σ1
2
+ hconvective
)
(3.71)
and continuing with 1/ n ⎧⎪ ⎫⎪ ⎡ hσ 1 ⎤ T2,ONB = T ′ + ⎨hconvective ⎢ 2 (T ′ − T2,ONB ) ⎥ − q& w′′ ⎬ ⎣ hNB ⎦ ⎪⎩ ⎪⎭
(h
σ1
2
)
+ hconvective . (3.72)
Already at the third iteration the value is accurate. Here again using the real vapor mass flow ratio in the drift flux theory the local void fraction and all other related local parameters can be computed. The three heat transfer coefficients required here have to be obtained empirically e.g. 0.4 hconvection = 0.023Re0.8 2 Pr2
λ2 Dheat
,
(3.73)
Dittus and Boelter (1930), one of the following h2σ 1 = 0.4 Re20.662 Pr2
λ2 Dh
,
(3.74)
Hancox and Nicoll, see in Hughes et al. (1981), h2σ 1 = 0.228 Re20.7 Pr21/ 2 (Φ 22 o / α 2 )1/ 4
λ2 Dh
, ± 30% , D1 / Dhw > 80 / Re20.7 , (3.75)
Avdeev (1986), h2σ 1 ≅
0.023Re20.8 Pr2 D1 λ2 , 1 − 1.82 Re2−0.1 Dh Dh
(3.76)
52
3. The “simple” steady boiling flow in a pipe
Labunzov (1974), and hNB = 1942 exp ( p 4.35 × 106 ) , n = 2,
(3.77)
Thom et al. (1966), or 1/ 3 1.16 ⎫ ⎧ pc 104 ) ⎛ p ⎞0.1 ⎡ ( ⎛ p ⎞ ⎤⎪ ⎪ ⎢ = ⎨238 5 / 6 1/ 6 ⎜ ⎟ 1 + 4.65 ⎜ ⎟ ⎥ ⎬ , n = 3, Tc M ⎝ pc ⎠ ⎢⎣ ⎝ pc ⎠ ⎥⎦ ⎪ ⎪⎩ ⎭ 3
hNB
(3.78)
mean error 11.2%, Borishanskii et al. (1964) for boiling on the external wall of horizontal cylinders. Here M is the molar weight in kg-mole and c indicates the critical state, the state at which the behavior of the liquid and vapor molecules of given substance can not be distinguished. D1 is the averaged local bubble diameter. The original Borishanskii et al. correlation was slightly modified by me to gain 1.5% better accuracy. The gas constant and the mol mass for waters are R = 461.526 J/(kgK) and M = 18.015257 kg/k-mol, respectively. The critical parameter for water are Tc = 647.096 K, pc = 220.64 bar, ρ c = 322 kg/m³.
3.9 The boundary layer treatment with considered variable effective bubble size Now we use the data from the previous section to check the validity of the boundary layer treatment. The results are presented in Fig. 3.3. The performance of the boundary layer model is obviously not better then the asymptotic methods. The point of the net vapor production is shifted to the smaller superheating, which means overestimation of the condensation. At higher void fractions within the subcooled region the condensation seams to be under estimated. This is associated with the not taking into account the increasing phase interface by increasing of the bubble number concentrations.
0,5
Void, -
0,4 0,3 0,2
0,6
1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0,5 0,4 Void, -
0,6
0,1
0,2 0,1
0,0 -0,2
0,3
6 exp. 7 exp. 8 exp. 9 exp. 10 exp. 6 th. 7 th. 8 th. 9 th. 10 th.
0,0
-0,1
0,0 X1,eq, -
0,1
-0,2
-0,1
0,0 X1,eq, -
0,1
3.9 The boundary layer treatment with considered variable effective bubble size 0,6
0,6 11 exp. 12 exp. 13 exp. 14 exp. 11 th. 12 th. 13 th. 14 th.
Void, -
0,4 0,3 0,2
0,5 0,4 Void, -
0,5
0,1
0,3 0,2
15 exp. 16 exp. 17 exp. 18 exp. 15 th. 16 th. 17 th. 18 th.
0,1
0,0
0,0
-0,2
-0,1
0,0
0,1
-0,2
-0,1
X1,eq, -
0,0
0,1
X1,eq,-
0,6
0,6 19 exp. 20 exp. 21 exp. 22 exp. 23 exp. 19 th. 20 th. 21 th. 22 th. 23 th.
0,4 0,3 0,2
0,5 0,4 Void, -
0,5
Void, -
53
0,3 0,2
24 exp. 25 exp. 26 exp. 24 th. 25 th. 26 th.
0,1
0,1 0,0
0,0
-0,2
-0,1
0,0
0,1
-0,2
X1,eq, -
-0,1
X1,eq,-
0,0
0,1
Fig. 3.3. Void fraction as a function of the homogeneous equilibrium mass fraction. Experimental data by Bartolomei et al. (1982). Prediction using: Drift flux parameter by Maier and Coddington (1986); Boundary layer theory Hughes et al. (1981): Condensation Hancox and Nicoll; Convection: Dittus and Boelter (1930); Evaporation: Thom et al. (1966)
One of the disadvantages of this treatment is the assumption that the condensation vapor-liquid interface is equal to the wall-flow interface. In fact bubbles are generated along the high and due several forces form a radial bubble concentration profile. The bulk bubble interface surface divided by the flow volume called bubbles interfacial area density is therefore F1 Vol flow = n1π D12 = 6α1 D1 ,
(3.79)
where the bubble number density n1 and the void volume fraction α1 are connected by the relation
n1
π 6
D13 = α1 .
(3.80)
So the interface per unit flow volume is about F1 6α 4 4 ⎛ 3 Dheat ⎞ = + 1 = ⎜1 + α1 ⎟, Vol flow Dheat D1 Dheat ⎝ 2 D1 ⎠
(3.81)
54
3. The “simple” steady boiling flow in a pipe
where D1 is the Souter mean bubble diameter. In order to take this phenomenon into account further information is necessary for the local number averaged bubble size. Gaining this information is a complicated matter as will be shown later. We will demonstrate the additional effect here by introducing the more or less arbitrary empirical relation for the effective bulk averaged bubble size in boiling processes reported by Achmad (1970, p. 31) and later modified by Kelly et al. (1981): D1 = D10 for α1 < 0.1
(3.82)
1/ 3
⎛ 9α1 ⎞ D1 = D10 ⎜ ⎟ ⎝ 1 − α1 ⎠
for α1 ≥ 0.1 ,
(3.83)
where D10 =
σ2
0.9 1 + 1.34 ⎣⎡(1 − α1 ) w2 ⎦⎤
1/ 3
ρ 2 − ρ1
.
(3.84)
For this purpose I will use a condensation correlation reported in Kolev (2007c, p. 84, Eq. (4.46)) h2σ 1 = 0.04 Re20.8 Pr20.4 (Φ 22 o / α 2 )1/ 4 λ2 Dh ,
(3.85)
with the constant 0.04 instead of 0.023. So we have finally ⎡ ⎤ ⎛ dX 1 4 1 3 Dheat ⎞ σ 1 n = ⎢ hNB (Tw − T ′ ) − ⎜ 1 + α1 ⎟ h2 (T ′ − T2 ) ⎥ , (3.86) 2 D1 ⎠ dz Dheat G ( h′′ − h′ ) ⎢⎣ ⎥⎦ ⎝
or ⎤ ⎛ dX 1 4 ⎡ 3 Dheat ⎞ σ 1 n ≈ ⎢ hNB (Tw − T ′ ) − ⎜ 1 + α1 ⎟ h2 (T ′ − T2 ) ⎥ . 2 D1 ⎠ dX 1eq Dheat q& ′′′ ⎢⎣ ⎥⎦ ⎝
(3.87)
The results of using this approach compared with the already introduced experimental data are presented in Fig. 3.4.
3.9 The boundary layer treatment with considered variable effective bubble size
Void fraction, -
0.5 0.4 0.3 0.2
0.6 1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0.5 0.4 Void, -
0.6
0.3 0.2
6 exp. 7 exp. 8 exp. 9 exp. 10 exp. 6 th. 7 th. 8 th. 9 th. 10 th.
0.1
0.1
0.0
0.0 -0.2
-0.1
0.0
-0.2
0.1
-0.1
0.6
0.3 0.2
0.5 0.4
0.1
0.3 0.2
-0.1
0.0
0.1
-0.2
-0.1
X1,eq, -
0.5 0.4 0.3 0.2
24 exp. 25 exp. 26 exp. 24 th. 25 th. 26 th.
0.1
0.1
0.0
0.0 -0.2
0.1
0.6 19 exp. 20 exp. 21 exp. 22 exp. 23 exp. 19 th. 20 th. 21 th. 22 th. 23 th.
Void, -
Void, -
0.2
0.0
X1,eq,-
0.6
0.3
0.1
0.0
-0.2
0.4
0.0
15 exp. 16 exp. 17 exp. 18 exp. 15 th. 16 th. 17 th. 18 th.
0.1
0.0
0.5
0.1
0.6 11 exp. 12 exp. 13 exp. 14 exp. 11 th. 12 th. 13 th. 14 th.
Void, -
Void, -
0.4
0.0 X1,eq, -
X1,eq, -
0.5
55
-0.1
0.0 X1,eq, -
0.1
-0.2
-0.1
X1,eq,-
Fig. 3.4. Void fraction as a function of the homogeneous equilibrium mass fraction. Experimental data by Bartolomei et al. (1982). Prediction using: Drift flux parameter by Maier and Coddington (3.1986); Boundary layer theory: Condensation Kolev (2007c); Convection: Dittus and Boelter (1930); Evaporation: Thom et al. (1966); Effective bubble size: Achmad (1970)
We realize that the overall picture improves especially regarding the point of onset of nucleate boiling. The last model indicates also the effect of the instability observed around the region of onset of nucleate boiling. That is why it requires better spatial resolution by the numerical integration. It has to be mentioned, that the bubbles over the cross section do not see the same sub-cooling. Those migrated in the center of the pipe see more sub-cooling then those being close to the heated wall. This phenomenon is not considered by this model.
56
3. The “simple” steady boiling flow in a pipe
3.10 Saturated flow boiling heat transfer At some boundary conditions the liquid in the heated pipe may become saturated. So we have flow boiling of saturated liquid in the two phase mixture. The heat transfer is then consisting of predominant convective part and nucleate boiling part. The total heat flux is then consumed for evaporation. Two successful methods can be used for computing the convective part: (a) In the first one it is assumed that the total mixture mass flows through the channel with the properties of the liquid and heat transfer coefficient hco is computed using one of the well known formulas. Then the local convective heat transfer coefficient is computed by correcting the so obtained heat transfer coefficient as follows 2 h ≈ ( Φ co )
0.35
hco .
(3.88)
This expression is very close to the theoretically obtained one in Vol. 3 of this monograph, namely 2 h ≈ ( Φ co )
0.25
hco .
(3.89)
The accuracy of this methods depends on the accuracy of the prediction of the 2 = ( dp dz )Tph ( dp dz )co . The superposition with Martinelli-Nelson multiplier Φ co the boiling part can be introduced as
(dp/dz)Tph/(dp/dz)2o
300 250
Sani's exp. Fit: mean av. error 13% mean sq. error 1.2%
200 150 100 50 0 0,0
0,5
1,0
1,5 Xtt
2,0
2,5
3,0
Fig. 3.5. Friction pressure drop gradient divided by the friction pressure drop gradient computed if the total mass is flowing through the channel with liquid properties as a function of the Martinelli parameter
3.10 Saturated flow boiling heat transfer
57
1/ 2
2 2⎫ ⎧ 2 0.35 htotal (Tw − T ′ ) ≈ ⎨ ⎡( Φ co hco (Tw − T ′ ) ⎤ + ⎣⎡ hboiling (Tw − T ′ ) ⎦⎤ ⎬ ) ⎢ ⎥ ⎣ ⎦ ⎩ ⎭
. (3.90)
This type of superposition is deduced intuitively from p. 458 of Vol. 2 3d ed. of this monograph. Reducing from the Sani’s experiment Φ 22 o = 1519.69039 exp ( − X tt 0.07719 ) + 167.21358exp ( − X tt 0.5413) , (3.91) where 0.5
⎛ ρ ′′ ⎞ ⎛ η ′ ⎞ X tt = ⎜ ⎟ ⎜ ⎟ ⎝ ρ ′ ⎠ ⎝ η ′′ ⎠
0.1
⎛ 1 − X 1,eq ⎜ ⎜ X 1,eq ⎝
0.9
⎞ ⎟ , ⎟ ⎠
see Fig. 3.5 and Appendix 3.1, and using the Borishanskii et al. correlation for nucleate boiling results in 11.3% mean averaged error compared with 259 Sani’s (1960) data for flow boiling. Simply replacing in this model the Borishanskii et al. nucleate boiling model with my model, Kolev (2007b) Vol. 2, results in mean averaged error of 11%. Applying the same model to the 697 data of Bennett et al. (1959) for annular flow channel with coaxially installed heated rod and using the hydraulic diameter as a characteristic size results in 17% mean error. (b) In the second method it is assumed that the only the continuum mass flows through the channel with the properties of the liquid and heat transfer coefficient hc is computed using one of the well known formulas. In this case the local convective heat transfer coefficient is h ≈ ( Φ c2 )
0.35
hc .
(3.92)
Reducing
(Φ )
2 0.35 c
= 1.8859 + 5.72465exp ( − X tt 0.62374 )
(3.93)
from the Sani’s data results in a very good agreement with the empirically obtained Chen multiplier
3. The “simple” steady boiling flow in a pipe
NuTph/Nu2
58
9 8 7 6 5 4 3 2 1 0,0
0.35
[(dp/dz)Tph/(dp/dz)2] FChen
0,5
1,0
1,5 Xtt
2,0
2,5
3,0
Fig. 3.6. Ratio of the convective two phase flow heat transfer to the liquid only heat transfer as a function of the Martinelli parameter 0.736 FChen = max ⎡⎢ 2.35 ( X tt−1 + 0.213) , 1⎤⎥ , ⎣ ⎦
(3.94)
– see Fig. 3.6. Chen proposed a superposition of the heat transfer mechanisms htotal ≈ FChen hc + SupChen hboiling ,
(3.95)
in which the Forster and Zuber (1955) nucleate boiling model hboiling = 0.00122 λ2 c2 σ 2 / Pr
0.29 2
ρ
1/ 4 2
⎛ ρ2 c p 2 ⎞ ⎜ ⎟ ⎝ ρ1Δh ⎠
0.24
⎡⎣ p ′ (Tw ) − p ⎤⎦
0.75
(3.96)
was used. The larger the velocity of the mixture flow the smaller the influence of the nucleate boiling. This is taken into account by Chen by introducing the so called empirical suppression factor presented by Chen graphically. Bjornard and Grifith (1977) approximated it by SupChen = (1 + 0.12 Re1.14 Chen )
−1
0.78 SupChen = (1 + 0.42 ReChen )
−1
for ReChen < 32.5 ,
(3.97)
for 32.5 ≤ ReChen < 70 ,
(3.98)
SupChen = 0.0797 exp (1 − ReChen 70 ) ,
where
for 70 ≥ ReChen ,
(3.99)
3.10 Saturated flow boiling heat transfer 1.25 ReChen = 10−4 FChen (1 − X1,eq ) GDh η2 .
59
(3.100)
40
40
30
30 HTC Chen
HTC Chen
Groeneveld et al. (1989) use instead of the last expression simply SupChen = 0.1 for 70 ≤ ReChen .
20 Sani data, mean av. err. 9.35%
10 10
20 30 HTC exp.
20 Sani data, mean av. err. 9.79%
10 40
10
20 30 HTC exp.
40
Fig. 3.7. (a) Comparison between the Chen correlation with Foerster and Zuber (1955) nucleate boiling model and the data by Sani (1960); (b) Comparison between the Chen correlation with Kolev (2007b) nucleate boiling model with nucleation site density function by Wang and Dhir (1993): n1′′w = 5 × 103 ⎡⎣1 − cos (θ ) ⎤⎦
(10 D ) 5
1, cr
6
with θ the wetting angle
and D1,cr the critical bubble size
As shown in Fig. 3.7 this correlation reproduce the Sani’s data with mean averaged error of 9.35%. The correlation was validated in the region X1 = 0 to 0.71, p = (1.013 to 69.) × 105 Pa, G = 54 to 4070 kg/(m2s), q& w′′ = 44 to 2 400 kW/m², Groeneveld et al. (1989). Simply replacing in this model the Foerster and Zuber (1955) nucleate boiling model with my model, Kolev (2007b) Vol. 2, results in mean averaged error of 9.79%. Applying the Chen’s model to the 697 data of Bennett et al. (1959) for annular flow channel with coaxially installed heated rod and using the hydraulic diameter as a characteristic size results in 22% mean error. Surpassingly if the heated diameter is used as characteristic length the mean error is 9%. Many engineers working in this field possess one or other library for computing the two phase pressure drop. Depending on this the first or the second method can be used. Remember once again that the accuracy of computing the convective part of the nucleate boiling depends on the accuracy of the computing of the pressure drop.
60
3. The “simple” steady boiling flow in a pipe
3.11 Combining the asymptotic method with boundary layer treatment allowed for variable effective bubble size Kelly et al. (1981, p. 87) proposed combination of asymptotic method
μ21 =
q& w′′ ⎛ T2 − T2* 4 ⎜ Dheat h′′ − h′ ⎝ T ′ − T2*
⎞ q& w′′ ⎛ h′ − h2 4 ⎟= ⎜1 − * ⎠ Dheat h′′ − h′ ⎝ h′ − h2
⎞ ⎟, ⎠
(3.101)
and boundary layer method
μ12 =
6α1 h2σ 1 (T ′ − T2 ) ,. D1 h′′ − h′
(3.102)
Note that here the assumption that the mass transfer surface for evaporation and condensation is approximately equal to the heated surface is removed. In general, the methods presented here describe completely cross section averaged flow parameter only in a sub-cooled and saturated boiling in vertical pipes. Other regimes are not covered by the method. Local distributions over the cross section can not be recovered from the averaged values. Note that large number of void measurements in boiling channels and comparisons with varieties of theories are published in the literature in the 1960’s to the 1980’s, e.g. Pierre (1965), Bennett et al. (1967)¸ Egen et al. (1957), Levy (1967)¸ Nylund et al. (1968), Sabotinov (1974), Bartolomei et al. (1980). For details see Kolev (2007b, Ch. 26). The situation we realize on Fig. 3.2 is very much representative for the success of such approaches. The simple theory represents all important trends of the experimental observations having in some regions better coincidence then in the others. Such approaches are used for designing of more then 400 nuclear rectors worldwide.
3.12 Separated momentum equations and bubble dynamics The basic assumption in the drift flux models is the instantaneous adjustment of developed flow inside the mixture. Therefore, their use is limited to steady state flows. For transient flow separated momentum equations are generally used. The terms describing the mechanical interactions are the same for steady state and transient flow. That is why such approach is frequently tested first in comparison with steady state boiling experiments. I will give here the basic idea of haw to use the separated momentum equations. The simplified formalism here is a sub
3.12 Separated momentum equations and bubble dynamics
61
domain of the computer code IVA. The computations performed below are done with this computer code. The simplified averaged momentum equation for each field “l” is d dp α l ρl wl2 + α l + α l ρl g cos ϕ = f l d + μ ml wm − μlm wl . dz dz
(3.103)
Later I will discuss in more details what is missing of this equation. For the discussion here it is sufficient. The interaction between the bubbles and the surrounding liquid is described by the drag force per unit volume of the flow. So for a given equivalent bubble diameter Dd the drag force is f dd = −
αd πD /6 3 d
ccdd
π Dd2 1 1 3 d = −α d ρ cd ccd Δwcd Δwcd . ρcd Δwcd Δwcd 2 4 Dd 4 (3.104)
Ishii and Zuber (1978) recommended a set of correlations for describing the drag force for bubbles in non boiling flow in columns with diameter of 3–6 cm. The drag coefficients are controlled by the Reynolds number Re = Dd ρc ΔVdc / η m ,
(3.105)
where
η m = ηc (1 − α c ) .
(3.106)
This set is given below: Stokes regime: Re < 16 and
2 24 D1 / λRT < (1 − α d )0.6 Re 3
d c21 = 24 / Re .
Viscous regime: Re > 16 and
d c21 =
(3.107) (3.108)
2 24 (1 + 0.1 Re0.75 ) (3.109) D1 / λRT < (1 − α d )0.6 3 Re
24 (1 + 0.1 Re0.75 ) . Re
Distorted bubble regime: (1 − α d )0.6
(3.110) 24 2 8 (1 + 0.1Re0.75 ) ≤ D1 / λRT < (1 − α d )0.87 Re 3 3 (3.111)
62
3. The “simple” steady boiling flow in a pipe 2
d c21 =
⎛ 1 + 17.67 f 6 / 7 ⎞ 2 1.5 ( D1 / λRT ) ⎜ ⎟ , f = (1 − α d ) . 3 18.67 f ⎝ ⎠
Strongly deformed, cap bubbles:
2 8 D1 / λRT ≥ (1 − α d )0.87 . 3 3
8 d c21 = (1 − α d ) 2 . 3
(3.112)
(3.113)
(3.114)
Slug flow in a pipe d c21 = 9.8(1 − α1 )3 ,
(3.115)
where Dd ≈ 0.9 Dh ,
(3.116)
Ishii and Chawla (1979). This approach needs also information of the characteristic bubble size. The mechanism of generation of bubbles at heated surface is described in details in Vol. 2 of this monograph. I will only mention that at forced convection the bubbles generated at the heated surface inside the micro layer are extremely small. They are subject of agglomeration already in the boundary layer so that the flow receives larger babbles controlled by hydrodynamic stability. For the data comparison discussed here I will assume that the bubbles are generated with the size D1,boiling predicted by Eqs. (3.82), (3.83) and (3.84). During their transport the bubbles change their averaged size following the mechanism described by the kinetic equation for conservation of the bubble numbers: dnd = n&1,boiling + n&1, splitting − n&1, coalescence , dz
(3.117)
where n&1,boiling =
μ21 , ρ1 π D1,3boiling 6
n&1, splitting = ( n1,∞ − n1 ) Δτ br ,
(3.118)
(3.119)
3.12 Separated momentum equations and bubble dynamics
⎛π ⎞ n1, ∞ = α1 ⎜ D1,3∞ ⎟ . 6 ⎝ ⎠
63
(3.120)
The maximum stable bubble size is computed as follows: Firs the free razing bubble velocity is computed, ΔwKu = 1.41 ⎡⎣σ 21 g ( ρ 2 − ρ1 ) ρ 22 ⎤⎦
1/ 4
.
(3.121)
For smaller relative velocity, Δw12 < ΔwKu , the maximum stable size is D1, ∞ = 6λRT .
(3.122)
λRT = σ 21 ⎡⎣ g ( ρ 2 − ρ1 ) ⎤⎦ . For larger relative velocity, Δw12 ≥ ΔwKu , the maximum stable size is ⎧ 12σ 21 ⎫ D1, ∞ = max ⎨1.265λRT , ⎬. ρ 2 Δw122 ⎭ ⎩
(3.123)
In transient processes it is possible to have a transition from flow pattern with continuous gas into e.g. slug- or churn turbulent flow. It is associated with transition to highly unstable gas globules with diameter size that will be destroyed in the following time steps. For this very beginning origination of a bubble I assume 20 times larger size then the maximum possible computed by the relation derived by De Jarlais et al. (1986) 1/ 3
1/ 2 D1, ∞ = 20 × 5.66λRT ⎡(η2 ρ1σ 2 λRT ) ⎤ ⎣ ⎦
.
(3.124)
The break up time is set to the natural oscillation frequency in bubbly flow Δτ br = 0.9047 Dd Δwcd .
(3.125)
The coalescence frequency is computed for α 1 > 0.001 as follows: The collision frequency is computed following Howarth (1967) f1, col = 4.9 α1 Δw11 D1 ,
(3.126)
see Eq. (7.23) in Vol. 2. Here the relative velocity causing collisions is computed as follows Δw11 = w1, k +1 − w1, k for w1, k +1 < w1, k and Δw11 = 0 for w1, k +1 > w1, k . The physical meaning is: if the distance between two neighboring bubbles increases
64
3. The “simple” steady boiling flow in a pipe
they will never collide; if the distance decreases they collide. To this velocity difference the fluctuation velocity of the bubbles is added. If turbulence modeling is used the magnitude of the fluctuation velocity is a result of the computation. If turbulence modeling is not used the fluctuation component can be taken as being a few percent of the magnitude of the liquid velocity. The bubbles follow this magnitude. The coalescence probability is set to one P1, coal = 1 .
(3.127)
The coalescence frequency is the product f1,coal = f1, col P1, coal .
(3.128)
The time averaged coalescence rate is then for small frequencies, n&1, coalescence = n1 f1, coal 2 .
(3.129)
For large coalescence frequencies I prefer to use the time averaged expression,
(
−f Δτ n&1, coalescence = n1 1 − e 1,coal
2
)
Δτ ,
(3.130)
Eq. (7.16) in Vol. 2, and for very large coalescence frequencies n&1, coalescence = n1 Δτ .
(3.131)
So the complexity of the model increases. Let us see whether better accuracy can be obtained by this approach. I will repeat the computations performed in the previous section for boiling pipe. The set of constitutive equation used for evaporation and condensation is as follows. For modeling of the evaporation the equation 78 by Borishanskii et al. (1964) is used. Kolev (2007c, Eq. (4.46), p. 84)
h2σ 1 = 0.023Re20.8 Pr20.4 (Φ 22o / α 2 )1/ 4 λ2 Dh ,
(3.132)
multiplied by a corrector for the bulk condensation
(1 + 3α1 Dheat
D1 ) .
(3.133)
The first observation I made is that the void fractions are systematically over predicted by using the above set of drag coefficients, Fig. 3.8(a). Reasonable prediction is only achieved, if the drag coefficients as documented above are reduced by one order of magnitude, Fig. 3.8(b).
3.12 Separated momentum equations and bubble dynamics
Void fraction, -
0,5 0,4 0,3 0,2
0,6
1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0,5 Void fraction, -
0,6
0,1 0,0
0,4 0,3 0,2
65
1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0,1 0,0
-0,2
-0,1
0,0
0,1
-0,2
-0,1
X1,eq, -
0,0
0,1
X1,eq, -
(a)
(b)
Fig. 3.8. Void fraction as a function of the homogeneous equilibrium mass fraction. Experimental data by Bartolomei et al. (1982). Prediction using: Boundary layer theory: Condensation Kolev (2007c); Convection: Dittus and Boelter (1930); Evaporation: Borishanskii et al. (1964); Effective bubble size: particle conservation equation. Drag coefficients: (a) by Ishii and Zuber (1978); (b) by Ishii and Zuber (1978) multiplied by 0.1
Figure 3.9 demonstrates the final result.
Void fraction, -
0,5 0,4 0,3 0,2
0,6 1 exp. 2 exp. 3 exp. 4 exp. 5 exp. 1 th. 2 th. 3 th. 4 th. 5 th.
0,5 0,4 Void, -
0,6
0,3 0,2
11 exp. 12 exp. 13 exp. 14 exp. 11 th. 12 th. 13 th. 14 th.
0,1
0,1
0,0
0,0 -0,2
-0,1
0,0
-0,2
0,1
-0,1
0,3 0,2
6 exp. 7 exp. 8 exp. 9 exp. 10 exp. 6 th. 7 th. 8 th. 9 th. 10 th.
0,5 0,4 Void, -
Void, -
0,4
0,3 0,2
0,0
0,1
15 exp. 16 exp. 17 exp. 18 exp. 15 th. 16 th. 17 th. 18 th.
0,1
0,1
0,0
0,0 -0,2
0,1
0,6
0,6 0,5
0,0 X1,eq, -
X1,eq, -
-0,1
0,0 X1,eq, -
0,1
-0,2
-0,1 X1,eq,-
66
3. The “simple” steady boiling flow in a pipe
0,6
0,3 0,2
0,5 0,4 Void, -
0,4 Void, -
0,6
19 exp. 20 exp. 21 exp. 22 exp. 23 exp. 19 th. 20 th. 21 th. 22 th. 23 th.
0,5
0,1
0,3 0,2
24 exp. 25 exp. 26 exp. 24 th. 25 th. 26 th.
0,1
0,0
0,0
-0,2
-0,1
0,0
0,1
X1,eq, -
-0,2
-0,1
X1,eq,-
0,0
0,1
Fig. 3.9. Void fraction as a function of the homogeneous equilibrium mass fraction. Experimental data by Bartolomei et al. (1982). Prediction using: Boundary layer theory: Condensation Kolev (2007c); Convection: Kolev (2007c); Evaporation: Borishanskii et al. (1964); Effective bubble size: particle conservation equation; Drag coefficients by Ishii and Zuber (1978) multiplied by 0.1
The main problem with the drag coefficients originates from the fact that real flows possesses a enveloping spectrum of particles, velocity profiles for bubbles and liquid, and concentration profiles for bubbles over the radius of the channel. Note the interesting mathematical paradox in this case: even if the bubbles are so small that they virtually possess the same velocity as a liquid due to the profiles the difference of the cross section averaged velocities will be different from zero, Bankoff (1960). The drag force computed using this difference will be different from zero where in the nature the integral of zeros over the cross section will give zero drag force. Therefore, the drag force that has to be used in the cross section averaged momentum equations has to be the effective, cross section averaged drag force: A
f dd =
= − ρc
A
1 1 31 f dd dA′ = − ρc α d ccdd Δwcd Δwcd dA′ A ∫0 Dd 4 A ∫0
1 3 d ccd , eff α d Δwcd Δwcd Dd 4
(3.134)
with an effective drag coefficient computed as follows A
ccdd ,eff = ccdd
1 α d ,local Δwdc ,local Δwdc ,local dA′ A ∫0 . α d Δwdc Δwdc
(3.135)
For the Stokes regime where ccdd = 24 / Re with Re = Dd ρc ΔVcd / ηc the local drag force is linear function of relative velocity
3.12 Separated momentum equations and bubble dynamics
f dd =
α d ,local 18 18 ηα Δwdc ,local = 2 ηc Δwdj ,local 2 c d , local Dd Dd 1 − α d ,local
67
(3.136)
and consequently f dd =
A A 1 18 1 α d ,local Δwdj ,local d ′ = η f dA dA′ . d c A ∫0 A ∫0 1 − α d ,local Dd2
(3.137)
Here the local drift velocity is Δwdj ,local = wd ,local − jlocal = (1 − α d ,local ) Δwdc ,local . Because the Ishii and Zuber (1978) set does not take into account profiles and cross section averaging correction was necessary. The reduction with one order of magnitude seems to be arbitrary and have to be replaced with more accurate one in the feature. But it indicate, that boiling channel create kind of tunneling for family of bubbles that flows as a group locally faster then a single bubble. Such effect is observed by throwing clouds of solid sphere in water – the cloud sink faster than single sphere. This problem was already recognized by Andersen (1982) and Hassan (1987). These authors proposed some approximations based on the drift flux theory which has no general application.
Conclusions: Is the “simple” flow boiling in a vertical pipe so simple? We realize, that although the geometry and the process looks simple, the participating physical phenomena are very complex in nature: Starting with the heat surface structure and its property to generate bubbles, going through the bubble departure size entering the macroscopic boundary layer, bubble size leaving this layer and entering the main flow, bubble collision and coalescence, bubble splitting, local momentum transport, local mass condensation is sub-cooled water and many others. The most simple relaxation methods combined with drift flux models for mechanical interaction give reasonable accuracy keeping in mind on which small prize the most important phenomena are described with reasonable accuracy. Unfortunately, the drift flux model can not be used for fast transients because the velocity difference is adjusted instantaneously. That is why separated momentum equations have to be used. Here again the problem of spatial resolution arise: Either one has to resolve the space in such details in order to be allowed to use local drag coefficients for bubbles or one has to use large scale discretization e.g. one cross section – one point in the 1D description. In the second case the problem originate with the use of effective, cross section averaged drag forces, bulk condensation etc. We quickly realize that the possibility of compensating one physical phenomenon by others in combining separate models exists. This problem can be resolved in the future only with detailed measurements on the separated effects. Some works in this field are already published.
68
3. The “simple” steady boiling flow in a pipe
3.13 Nomenclature Latin
σ 3 ρc2 Archimedes number, dimensionless ηc4 g Δρ
Ar
=
Bo
cp
= ρ 2 gDh2 / σ , Bond number, dimensionless mass concentration of the inert component n inside the gas mixture, dimensionless specific heat at constant pressure, J / ( kgK )
C0
distribution parameter
Cnl
d l
drag coefficient acting on the field l, dimensionless
d d , single
c
drag coefficient for a particle in an infinite medium, dimensionless
Dhy
hydraulic diameter (4 times flow cross-sectional area / wet perimeter), m
Dheat
heated diameter (4 times flow cross-sectional area / heated perimeter), m
Eo
=
c
Fr
g Δρ 21 Dh2
σ
= ( Dh / λRT ) , pipe Eötvös number, dimensionless 2
G2 = , square of Froude number based on the liquid density, dimengDh ρ 22 sionless
( ρ w) =
2
vh2
fl d
, square of Froude number based on the mixture density, digDh mensionless drag force acting on the field l per unit mixture volume, N/m³
f l vm
virtual mass force acting on the field l per unit mixture volume, N/m³
Frh
G
g h h hNB _ T hom
= ∑ (αρ w )l , mass flow rate in the axial direction, kg/(m²s)
gravity acceleration, m/s² specific enthalpy, J/kg specific mixture enthalpy – mixture enthalpy of the non-inert components, J/kg saturated boiling heat transfer coefficient, W/(m²K)
hconvective convective heat transfer coefficient, W/(m²K)
3.13 Nomenclature
h2σ 1 j L Nu
69
heat flux coefficient by the recondensation into the bulk flow based on the heated surface, W/(m²K) volumetric flux of the mixture – equivalent to the center of volume velocity of the mixture, m/s length, m q& w′′ Dh , Nusselt number, dimensionless * T ′( p ) − T2 λ2
Nη2
= η2 / ρ 2σ 2 λRT , liquid viscous number, dimensionless
p Pe2
pressure, Pa = GDhy c p 2 / λ2 , Peclet number, dimensionless
Pr2
= c p 2η2 / λ2 , liquid Prandtl number, dimensionless
Red
= Dd ρ c Δwdc / ηc , Reynolds number for dispersed particle surrounded by continuum, dimensionless = ρ 2 w2 Dh / η2 , liquid Reynolds number, dimensionless = w1 / w2 , slip (velocity ratio), dimensionless q& w′′ , Stanton number, dimensionless ⎡ ′ c p 2 ⎣T ( p ) − T2* ⎤⎦
Re 2
S St
1/ 3
r
*
T T2*
D ⎛ ρ g Δρ ⎞ = d ⎜ c 2 ⎟ , bubble size, dimensionless 2 ⎝ ηc ⎠ temperature, K liquid temperature corresponding X 1*eq , K
q& w′′ wall heat flux, W/m² &q′′net _ evaporation heat flux at the wall causing all the net evaporation, W/m² ′′ q& NB
saturated boiling heat flux, W/m²
q&2′′σ w
convective heat flux from the wall into the bulk liquid, W/m²
q&2′′σ 1
heat flux released by the recondensation into the bulk flow, W/m²
V*
⎛ ρ c2 ⎞ = Δwdc ,∞ ⎜ ⎟ ⎝ η c g Δρ ⎠
1/ 3
, terminal velocity, dimensionless
1/ 4
VTB*
⎛ σ g Δρcd ⎞ =⎜ ⎟ , Kutateladze terminal velocity of dispersed particle in 2 ⎝ ρc ⎠ continuum, m/s = Dh g cos ϕΔρcd / ρ c , Taylor terminal velocity, m/s
Vdj*
weighted mean drift velocity, m/s
VKu
70
3. The “simple” steady boiling flow in a pipe
Wed vh w wl wd X1 Xl X 1*eq
X eq
= Dd ρc Δwdc2 / σ , Weber number for dispersed particle surrounded by continuum, dimensionless = X 1v1 + (1 − X 1 ) v2 , homogeneous mixture specific volume, m³/kg axial velocity, m/s cross section averaged axial velocity of field l, m/s weighted mean velocity, m/s αρw = 1 1 1 , gas mass flow concentration, dimensionless G mass flow rate concentration of the velocity field l inside the multi-phase mixture, dimensionless local equilibrium steam mass flow concentration in the flow, defining the initiation of the visible nucleate boiling, dimensionless equilibrium steam mass flow rate concentration in the flow, dimensionless
Greek
αl α& d α&1
α core α dm Δwcd Δwcd ∞
Δwcd , o
volume fraction of field l, dimensionless j = d , averaged volumetric flow concentration of the field d, dimensionj less 1 = , averaged volumetric flow concentration of the gas, di1 − X 1 ρ1 1+ X1 ρ2 mensionless = α 3 + α1 , core volume fraction: droplet + gas, dimensionless particle volume fraction at which the solid particles are touching each other in the control volume, dimensionless velocity difference: continuum minus disperse, m/s steady state free settling velocity for the family of solid spheres or the free rising velocity for a family of bubbles, m/s velocity difference at zero time, m/s
Δwcd ,τ →∞ steady state velocity difference, m/s Δρcd Φ 2o
ϕ
density difference: continuum minus disperse, kg/m³ two phase friction multiplier, dimensionless angle between the positive flow direction and the upwards directed vertical, rad
References
λ
thermal conductivity, W / ( mK )
λRT
=
μ21 μ12 ρ ρl ρcore ρw σ η
evaporation mass per unit time and unit mixture volume, kg/(m³s) condensation mass per unit time and unit mixture volume, kg/(m³s) density, kg/m³
σ2
g ( ρ 2 − ρ1 )
71
, Rayleigh-Taylor instability length, m
density of field l, kg/m³ = α d ρ3 + (1 − α d ) ρ1 , core density: droplet + gas, kg/m³ mixture mass flow rate, kg/(m²s) gas-liquid surface tension, N/m dynamic viscosity, kg/(ms)
Subscripts
1 2 3 c d cr core
field 1, gas field 2, liquid field 3 continuum disperse critical droplets + gas in annular flow
Superscripts
′ ′′
saturated liquid saturated vapor
References Achmad SI (1970) Raspredelenie srednomassovoj temperatury zidkosti I istinnogo obemnogo parosoderzanie vodoi obogrevaemogo kanala s nedogrevom na vchode, Teploperedaca, vol 4 translated from Axial distribution of bulk temperature and void fraction in heated channel with inlet subcooling. J. Heat Transfer, vol 92, p 595 Andersen JGM (1982) Interfacial shear for two-fluid models. Trans. ANS, vol 41, pp 669–671 Avdeev AA (1986) Growth and condensation velocity of steam bubbles in turbulent flow. Teploenergetika, in Russian, vol 1, pp 53–55 Bankoff SG (November 1960) A variable density single-fluid model for two-phase flow with particular reference to steam-water flow. J. Heat Transfer, Transactions of the ASME, vol 82, pp 265-272 Bartolomei GG, Batashova GN, Brantov VG et al. (1980) Heat and Mass Transfer IV, Izd. ITMO AN BSSR, Minsk, vol 5, p 38, in Russian
72
3. The “simple” steady boiling flow in a pipe
Bartolomei GG, Brantov VG, Molochnikov YuS, Kharitonov YuV, Solodkii VA, Batashova GN and Mikjailov VN (1982) An experimental investigation of true volumetric vapor content with subcooled boiling in tubes. Thermal Engineering, vol 29 no 3, pp 132–135 Bennett JAR, Collier JG, Pratt HRC and Thornton JD (1959) Heat Transfer to Two Phase Gas-Liquid Systems, AERE – R – 3159, Atomic Energy Research Establishment, Harwell, Berkshire Bennett AW et al. (1967) Heat transfer to steam-water mixtures flowing in uniformly heated tubes in which the critical heat flux has been exceeded, AERE-R5373 Bjornard TA and Grifith P (1977) PWR blow-down heat transfer, Thermal and Hydraulic Aspects of Nuclear Reactor Safety, American Society of Mechanical Engineers, New York, vol 1, pp 17–41 Borishanskii V, Kozyrev A and Svetlova L (1964) Heat transfer in the boiling water in a wide range of saturation pressure. High Temp., vol 2 no 1, pp 119–121 De Jarlais G, Ishii M and Linehan J (Febr. 1986) Hydrodynamic stability of inverted annular flow in an adiabatic simulation, Trans. ASME, J Heat Transfer, vol 108, pp 85–92 Dittus FV and Boelter LMK (1930) Heat transfer for automobile radiators of the tubular type, Univ. of Calif. Publ. In Engng., vol 2 no 13, p 443 Egen RA, Dingee DA and Chastain JW (1957) Vapor formation and behavior in boiling heat transfer, AEC Report BMI – 1167 Forster HK and Zuber N (1955) Dynamics of vapor bubbles and boiling heat transfer, AIChE J., vol 1 no 4 pp 531-535 Friedel L (1979) New friction pressure drop correlations for upward, horizontal, and downward two-phase pipe flow. Presented at the HTFS Symposium, Oxford, September 1979 (Hoechst AG Reference No. 372217/24 698) Groeneveld DC, Chen SC, Leung LKH, Nguyen C (1989) Computation of single and twophase heat transfer rate suitable for water-cooled tubes and subchannels, Nuclear Engineering and Design, vol 114 pp 61–77 Hassan YA (1987) Assessment of a modified interfacial drag correlation in two-fluid model codes, presented at 1987 ANS Annual Meeting, Dallas, Texas, June 1987, ANS Transaction, vol 54, pp 211–212 Hughes ED, Paulsen MP, Agee LJ (Sept. 1981) A drift-flux model of two-phase flow for RETRAN. Nucl. Technol., vol 54, pp 410–420 Howarth WJ (1967) Measurement of coalescence frequency in an agitated tank. A. I. Ch. E. J., vol 13 no 5, pp 1007–1013 Kelly JE, Kao SP and Kazimi MS (April 1981) THERMIT-2: A two-fluid model for light water reactor sub-channel transient analysis, MIT Energy Laboratory Electric Utility Program, Report No. MIT-EL-81-014 Kolev NI (2007a) Multiphase Flow Dynamics, Springer, Berlin, vol 1 Kolev NI (2007b) Multiphase Flow Dynamics, Springer, Berlin, vol 2 Kolev NI (2007c) Multiphase Flow Dynamics, Springer, Berlin, vol 3 Labunzov DA (1974) State of the art of the nuclide boiling mechanism of liquids, Heat Transfer and Physical Hydrodynamics, Moskva, Nauka, in Russian, pp 98–115 Levy S (1967) Forced convection subcooled boiling – Prediction of vapor volumetric fraction, Int. J. Heat Mass Transfer, vol 10, pp 951–965 Ishii M and Chawla TC (Dec. 1979) Local drag laws in dispersed two-phase flow, NUREG/CR-1230, ANL-79-105 Ishii M and Zuber N (1978) Relative motion and interfacial drag coefficient in dispersed two-phase flow of bubbles, drops and particles, Paper 56 a, AIChE 71st Ann. Meet., Miami Kawara Z, Kataoka I, Serizawa A, Ko YJ and Takahashi O (August 23–28, 1998) Analysis of forced convective CHF based on two-fluid and three-fluid model, Heat Transfer 19998, Proc. of the 11th IHTC, Kyongju, Korea, vol 2, pp 103–108
Appendix 3.1: The Sani’s (1960) data for flow boiling in pipe
73
Kolev NI (September 12–15, 2005) Do we have appropriate constitutive sets for sub-channel and fine-resolution 3D-analyses of two-phase flows in rod bundles? Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications Palais des Papes, Avignon, France, on CD-ROM, American Nuclear Society, LaGrange Park, IL Maier D and Coddington P (1986) Validation of RETRAN-03 against a wide range of rod bundle void fraction data, ANS Trans., vol 75, pp 372–374 Nylund D et al. (1968) Hydrodynamic and heat transfer measurements on a full-scale simulated 36-rod Marviken fuel element with uniform heat flux distribution, FRIG-2, Danish Atomic Energy Commission NUPEC (2004) OECD/NRC Benchmark based on NUPEC BWR Full-size Fine-mesh Bundle Tests (BFBT), Assembly Specifications and Benchmark Database, October 4, 2004, Incorporated Administrative Agency, Japan Nuclear Energy Safety Organization, JNES-04N-0015 Sabotinov LS (1974) Experimental investigation of void fraction in subcooled boiling for different power distribution laws along the channel. Moskva PhD Thesis in Russian Sani RleR (4 January 1960) Down flow boiling and non-boiling heat transfer in a uniformly heated tube, University of California, URL-9023, Chemistry-Gen. UC-4, TID-4500 (15th Ed.) Saha P and Zuber N (1974) Proc. Int. Heat Transfer Conf. Tokyo, Paper 134.7 Sekoguchi K, Nishikawa K, Nakasatomi M, Hirata N and Higuchi H (1972) Flow boiling in subcooled and low quality regions – heat transfer and local void fraction, B4.8, pp 180–184 St. Pierre CC (1965) Frequency-response analysis of steam voids to sinusoidal power modulation in a thin-walled boiling water coolant channel, Argon National Lab. Report, ANL-7041 Thom IRS et al. (1966) Boiling in subcooled water during up heated tubes or annuli. Proc. Instr. Mech. Engs., vol 180, 3C pp 1965–1966 Wang CH and Dhir VK (Aug. 1993) Effect of surface wettability on active nucleation site density during pool boiling of water on a vertical surface, ASME Journal of Heat Transfer, vol 115 pp 659–669
Appendix 3.1: The Sani’s (1960) data for flow boiling in pipe Although 48 years old the data by Sani (1960) are valuable. Therefore they are taken from a old and very difficult to read report, transferred in SI units. In addition the saturation temperature is added to demonstrate haw close the bulk temperature to the saturation temperature is. Test section geometry: Pipe diameter 0.01827 m Pipe length 1.728 m Flow orientation: downwards, vertical
74
3. The “simple” steady boiling flow in a pipe
Data structure: No No-test Distance from entrance in m Mass flow in kg/s Heat flux in W/m² Wall temperature in K Bulk temperature in K Saturation temperature as function of the local pressure in K Pressure in Pa Equilibrium steam mass flow ratio, dimensionless Pressure drop due to friction per unit length in Pa/m Validity region: Pressure: 1.09 to 2.13 bar Mass flow rate: 249 to 805 kg/m² Heat flux: 43 to 157 kW/m² Equilibrium steam mass flow ratio: up to 14% Experimental data by Sani (1960) in SI units 1 50 0.3048 0.21047 43533. 382.26 379.93 380.07 129139. 0.0157 2 50 0.6096 0.21047 43533. 381.82 379.43 379.61 127139. 0.0182 3 50 0.9144 0.21047 43533. 381.37 378.87 379.07 124795. 0.0208 … 259 67 1.5240 0.14334 99054. 383.21 379.93 380.36 130449. 0.0994
6040. 6831. 7804. 27394.
Computing Φ 22 o direct from the data and plotting it versus X tt results in the following picture.
(dp/dz)Tph/(dp/dz)2o
300 250
Sani's exp. Fit: mean av. error 13% mean sq. error 1.2%
200 150 100 50 0 0,0
0,5
1,0
1,5 Xtt
2,0
2,5
3,0
The best fit is the obtained by Φ 22 o = 1519.69039 exp ( − X tt 0.07719 ) + 167.21358exp ( − X tt 0.5413) .
Appendix 3.1: The Sani’s (1960) data for flow boiling in pipe
75
Similarly I proceed with Φ 22 as presented on the following picture. 400 (dp/dz)Tph/(dp/dz)2
350 300 250
Sani's exp. Fit
200 150 100 50 0 0,0
0,5
1,0
1,5 Xtt
2,0
2,5
3,0
The best fit is the obtained by Φ 22 = 2388.92798exp ( − X tt 0.07416 ) + 200.09996 exp ( − X tt 0.5024 ) .
For the heat transfer analysis ( Φ c2 )
0.35
is used. That is why I prefer to plot directly
this function and to find the best fit for it:
(Φ )
2 0.35 c
= 1.8859 + 5.72465exp ( − X tt 0.62374 ) .
4. The “simple” steady three-fluid boiling flow in a pipe
While the Chapter 3 essentially deals with the so-called two-fluid model Chapter 3 demonstrates the real cases where a three fluid model is mandatory. Chapter 3 is an introduction to the “simple” steady three-fluid boiling flow in a pipe. The flow regime transition from slug to churn turbulent flow is considered in addition to the already available information from Chap. 3. The idea of the redistribution of the liquid between film and droplets is presented at two level of complexity: the instantaneous and the transient liquid redistribution in film and droplets. The transient redistribution is in fact the introduction of the ideas of droplets entrainment and deposition. The idea for the description of the mechanical interaction of the velocity fields is again presented in two level of complexity: by using drift flux correlations and by using separated momentum equation defining the forces among the fields. The next step of the sophistication is then introduced by using models for the dynamic evolution of the mean droplet size consisting of models for the droplet size stability limit, for droplet production rate due to fragmentation, for duration of the fragmentation and for collision and coalescence of droplets. Then the heat and mass transfer mechanisms in the film flow with droplet loading are introduced. Finally comparisons with experimental data demonstrate the success of the different ideas and models.
Coming back to the flow boiling in a vertical pipe as considered in the previous chapter imagine that the heat flux is so high, that at some point the liquid mass flow is not enough to keep the bulk as a continuum any more. New structure originates in which the vapor becomes also continuum. In this structure it is thinkable that the vapor carries droplets and film continues to flow along the wall. Due to the strong friction the liquid film has much smaller velocity then the vapor and the droplet carried by the vapor. All this phenomena are experimentally observed in our vertical boiling flows. The difference in the film-droplet velocities does not allow handling mathematically both as a single velocity field. Therefore the need arise to introduce a droplet filed to describe the separate life of the droplets in the flow. Such models are called three-fluid models. So our steady state model of the one dimensional boiling flow become more complicated: evolution of droplet sizes have to be described, droplet-film, droplet-gas and film-wall interactions have to be described; droplets can be entrained into the bulk vapor flow; due to turbulence
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_4, © Springer-Verlag Berlin Heidelberg 2009
77
78
4. The “simple” steady three-fluid boiling flow in a pipe
droplets from the bulk flow can be deposed into the film; simultaneously the evaporation reduces the liquid mass flow in the film. As always in the engineering, there are many ways to do this with different degree of complexity. Here I will start with simple algebraic models, using again the drift flux approach to describe the velocity differences and combine it with a relaxation method for description of the droplet evaporation in superheated vapor. Then I will make the step to the more complex separate momentum equation model. A useful assumption regarding the thermodynamics is that the liquid is saturated and the vapor is allowed to be superheated in case of zero-wall film.
4.1 Flow regime transition slug to churn turbulent flow The first step of sophistication is the introduction of a “decision model” for the transition from two-fluid to three-fluid flow namely α1 > α1, slug to churn . In accordance with Mishima and Ishii (1984)
α1, slug to churn
⎪⎧ ( C0 − 1) j + 0.35 V12,TB ⎪⎫ = 1 − 0.813 ⎨ ⎬ j + 0.75 V12,TB b1 ⎪⎩ ⎪⎭
0.75
.
(4.1)
Here the drift flux distribution coefficient for slug flow is C0 = 1.2 ,
(4.2)
the slug (Taylor bubble) free raising velocity is V12,TB =
ρ 2 − ρ1 gDh , ρ2
(4.3)
the mixture volumetric flux is j = α1 w1 + (1 − α1 ) w2 ,
(4.4)
and 1/18
⎛ ρ − ρ1 ⎞ b1 = ⎜ 22 gDh3 ⎟ η ρ / ⎝ 2 2 ⎠
.
(4.5)
4.2 Instantaneous liquid redistribution in film and droplets
79
4.2 Instantaneous liquid redistribution in film and droplets For larger void fractions leading to film flow pattern it is important to know haw much of the liquid flow is attached to the film and how much to the bulk droplets. Two approaches are possible. Both of them are empirically obtained from steady state experiments, not very accurate, but for the time being there are no better: The first one presents the information in form of mass flow ratio for film and droplets as a function of the local parameter and the second is in form of entrainment and deposition mass flow rates. Paleev and Filippovich (1966) found the following correlation for developed annular flow E∞ =
X3 α 3 ρ3 w3 = X 2 + X 3 α 2 ρ 2 w2 + α 3 ρ3 w3
2 ⎡ α ρ w + α 3 ρ3 w3 ⎛ η 2 w1 ⎞ ⎤ = 0.985 − 0.44 log10 ⎢104 1 1 1 ⎜ σ ⎟ ⎥, ρ 2α1 ρ1 w1 ⎝ ⎠ ⎥⎦ ⎢⎣
(4.6)
α ρ w + α 3 ρ3 w3 ⎛ η 2 w1 ⎞ valid within 1 ≤ 10 1 1 1 ⎜ σ ⎟ ≤ 1000 which controls the splitting ρ 2α1 ρ1 w1 ⎝ ⎠ 2
4
of the liquid flow into droplets field indicated with subscript 3 and film attached to the wall indicated with subscript 2. Sixteen years later Nigmatulin (1982), see also Nigmatulin et al. (1995), reported information about the equilibrium film mass flow fraction of the total liquid mass flow defined as follows E∞ = 3.1Y −0.2 for Y ≤ 8000
(4.7)
E∞ = 1700Y −0.9 for Y > 8000
(4.8)
where
Y = c1c2 ( ρcore ρ 2 )(η 2 η1 )
0.3
Fr1 Eo ,
(4.9)
( gDh ) ,
(4.10)
Eo = ( Dh λRT ) ,
(4.11)
Fr1 = V12
2
80
4. The “simple” steady three-fluid boiling flow in a pipe
λRT = ⎡⎣σ ( g Δρ 21 ) ⎤⎦
1/ 2
,
(4.12)
c1 = Δρ 21 ρ1 for Δρ 21 ρ1 ≤ 10 ,
(4.13)
c1 = 1 for Δρ 21 ρ1 > 10 ,
(4.14)
c2 = 0.15 for Eo1/2 ≥ 20,
(4.15)
c2 = 1 for Eo1/2 < 20.
(4.16)
Kataoka and Ishii (1982) found that the droplet entrainment starts if the following condition is fulfilled Re2 F > Re2 Fc ,
(4.17)
where Re2F = ρ 2 w2 4δ 2 F / η2 , Re2Fc = 160 . Then the authors distinguish two entrainment regimes depending on whether the droplet field is under-entrained or over-entrained with respect to the equilibrium condition. The entrained mass flow fraction of the total liquid mass flow that defines the boundary between these two regimes is defined by the following correlation E∞ =
α 3 ρ3 w3 1/ 4 = tanh ( 7.25 × 10−7 We1.25 Ishii Re23 ) . α 2 ρ 2 w2 + α 3 ρ3 w3
(4.18)
where WeIshii =
ρ1 (α1w1 ) 2 Dh σ2
1/ 3
⎛ ρ 2 − ρ1 ⎞ ⎜ ⎟ ⎝ ρ1 ⎠
,
(4.19)
is the Weber number for the Ishii entrainment correlation, Re23 = ρ 2 (1 − α1 ) w23 Dh / η2 ,
(4.20)
is the total liquid Reynolds number, w23 =
α 2 w2 + α 3 w3 , 1 − α1
(4.21)
is the center of volume velocity of film and droplet together. E∞ defines in fact the mechanical equilibrium state. The above correlation for E∞ was verified with data
4.3 Relaxing the assumption
81
6 1/ 4 in the region 2 × 104 < We1.25 Ishii Re23 < 8 × 10 , 1< p < 4 bar, 0.0095 < Dh < 0.032 m, 320 < Re2 < 6400, α1 w1 < 100 m/s. In this region of parameters the maximum val-
ue of the dimensionless entrainment ( ρ w )23 Dh / η2 was ≈ 20 .
4.3 Relaxing the assumption for instantaneous liquid redistribution in film and droplets, entrainment and deposition Remember that the algebraic models for redistribution of the liquid • are valid for steady state and adiabatic flow; • for boiling flows such type of correlations are not reliably established; • such approach means instantaneous separation of the liquid into droplet and film while in the real flow it takes finite time to reach such state, and finally • the particle size evolution and its influence on the redistribution is not considered by this approach. Improvement in this field of knowledge is describing the mechanical transport of droplets from the-, and into the film by rate correlations. We call entrainment mass flow rate ( ρ w )23 the mass flow per unit surface of the film and unit time that is removed by the gas flow from the film into the gas core of the flow. We call deposition mass flow rate ( ρ w )32 the mass flow per unit surface of the film and unit time that is removed from the gas core and deposed into the film. Detailed review of this subject is given in Vol. 2 of this monograph, Kolev (2007b). I will give only one set of correlations for description of entrainment and deposition here in order to illustrate their use. The mass conservation of the vapor, film and droplets is d (α1 ρ1 w1 ) = μ21 + μ31 , dz
(4.22)
d (α 2 ρ2 w2 ) = μ32 − μ23 − μ21 , dz
(4.23)
d (α 3 ρ3 w3 ) = − μ32 + μ23 − μ31 . dz
(4.24)
Here the mass source of droplets due to entrainment per unit time and unit mixture volume is
82
4. The “simple” steady three-fluid boiling flow in a pipe
μ23 =
4 1 − α 2 ( ρ w )23 . Dh
(4.25)
The mass source for the film due to droplet deposition per unit time and unit mixture volume is
μ32 =
4 1 − α 2 ( ρ w )32 . Dh
(4.26)
Deposition: The Whalley deposition data are correlated by Katto (1984) as follows
( ρ w)32 = K32 ρ3 α 3 (α1 + α 3 ) ,
(4.27)
K 32 = 0.405σ 310.915 for σ 31 < 0.0383 ,
(4.28)
K 32 = 9.48 × 104 σ 314.7 for σ 31 ≥ 0.0383 .
(4.29)
Note that the deposition removes droplets with the averaged size of the droplet in the gas core D32 ≈ D3 and therefore the number of the droplets per unit time and unit mixture volume removed from the gas core due to deposition is n&32 =
4 1 − α 2 ( ρ w )32 Dh
⎛ π D323 ⎞ ⎜ ρ3 ⎟. 6 ⎠ ⎝
(4.30)
Entrainment: Kataoka and Ishii (1982) provided not only the information for the start of the entrainment and the equilibrium splitting of the liquid in the film and droplets but also for the rate of the entrainment: The under-entrained regime (entrance section and smooth injection of liquid as a film causing excess liquid in the film compared to the equilibrium condition) is defined by Re 2 > Re 2 ∞ ,
(4.31)
where Re2 = α 2 ρ 2 w2 Dh / η2 ,
(4.32)
local film Reynolds number based on the hydraulic diameter, and Re2 ∞ = Re23 (1 − E∞ )
(4.33)
4.3 Relaxing the assumption
83
is the local equilibrium film Reynolds number based on the hydraulic diameter. In this regime the entrainment mass flow rate is described by the following correlation which was published in Kataoka and Ishii in 1983
( ρ w )23
2 ⎡ E ⎞ ⎤ 0.25 ⎛ −9 1.75 ⎢ 0.72 × 10 Re23 WeIshii (1 − E∞ ) ⎜1 − ⎟ ⎥ η2 ⎢ ⎝ E∞ ⎠ ⎥ = . ⎢ 0.26 ⎥ Dh ⎢ 0.925 ⎥ 0.185 ⎛ η1 ⎞ −7 (1 − E ) ⎜ ⎟ ⎥ ⎢ +6.6 × 10 Re23WeIshii ⎝ η 2 ⎠ ⎦⎥ ⎣⎢
(
(4.34)
)
For the over-entrained regime (entrainment is caused by shearing-off of roll wave crests by gas core flow) defined by Re2 ≤ Re2∞
(4.35)
the entrainment mass flow rate is correlated by Kataoka and Ishii (1983) as follows:
( ρ w )23 =
η2 Dh
6.6 × 10
−7
( Re
WeIshii
23
) (1 − E ) 0.925
0.185
⎛ η1 ⎞ ⎜ ⎟ ⎝ η2 ⎠
0.26
.
(4.36)
The number of the droplets per unit time and unit mixture volume due to entrainment is therefore n&23 =
4 1 − α 2 ( ρ w )23 Dh
3 ⎛ π D23 ⎜ ρ3 6 ⎝
⎞ ⎟. ⎠
(4.37)
Here D23 is the averaged diameter with which the droplets are entrained from the film. The Kataoka, Ishii’s and Mishima’s model from 1983, Eq. 42, p. 237,
D23 ρ1 (α1 w1 ) σ 2 = We12 2
(4.38)
where
We12 = 0.01Re10.667 ( ρ 2 ρ1 )
1/ 3
(η1
η2 )
2/3
(4.39)
is used in the next examples. It defines the median particle size in an always observed log-normal distribution with the ratio of the maximum to median size D3,max / D23 = 4.14. The model is based on data in the region: p = 1.2 bar, 10
D3) according to the recommendation of Ishii and Chawla (1979). The effective viscosity for this case is
ηm = (1 − α d ) −2.5η1 ,
(4.57)
Roscoe (1952), Brinkman (1952), where
α d = α 3 (α1 + α 3 ) .
(4.58)
(1) The drag coefficient for the Stokes regime Re < 1, is computed as follows d c13 = 24 Re , Re = D3 ρ1ΔV13 / ηm .
(4.59)
The drag force is therefore
(
f31d = − 18α 3ηm / D32
)(w − w ) . 1
(4.60)
3
(2) The drag coefficient for the viscous regime, 1 ≤ Re < 1000, is computed as follows
(
d c13 = 24 1+0.1Re0.75
)
Re .
(4.61)
Therefore the drag force is
(
)(
f33d = − 18α 3ηm D32 1 + 0.1Re0.75
)(w − w ) . 1
3
(4.62)
(3) The drag coefficient for Newton’s regime (for single particle – Newton), Re ≥ 1000 , is computed as follows 2
d c13 =
⎛ 1 + 17.67 f 6 / 7 ⎞ 2 ( D3 / λRT ) ⎜⎜ ⎟⎟ ; 3 ⎝ 18.67 f ⎠
f = (1 − α d )3
(4.63)
88
4. The “simple” steady three-fluid boiling flow in a pipe
The drag force is therefore 1 1 f 31d = − α 3 ρ1 λRT 2
2
⎛ 1 + 17.67 f 6 / 7 ⎞ ⎜ ⎟ w1 − w3 ( w1 − w3 ) . ⎝ 18.67 f ⎠
(4.64)
Film-wall or gas-wall drag force: To describe the mechanical interaction of the three-fluid flow with the wall I recommend the use of empirical equation for pressure drop given in the previous chapter. So this force acts then at those field which is wetting the wall. For film flow it is the film and for gas-droplets flow it is the gas. Gas-film drag force: What remains to complete the description of the mechanical interaction is the gas resisting force per unit flow volume between film and gas d f12 d = −a12τ 21 = − a12 c21
=−
1 ρ1 w2 − w1 ( w2 − w1 ) 2
2 d 1 − α 2 c21 ρ1 w2 − w1 ( w2 − w1 ) , Dh
(4.65)
where a12 is the interfacial area density a12 =
4 1 − α2 . Dh
(4.66)
A critical review on how to compute the interfacial drag coefficient is given in Vol. 2 of this book, Kolev (2007b). I will introduce here only the Stephan and Mayinger (1990) correlation which was based on high pressure experiments (p = 6.7 × 105 to 13 × 105 Pa, Dh = 0.0309 m, α 2 w2 = 0.017 to 0.0035 m/s, α1w1 = 5 to 18 m/s) and which will be used in the examples provided in this chapter d c21 =
(
)
0.079 1 + 115 δ *B , Re11/ 4
(4.67)
(
)
where Re1 = ρ1w1 Dh η1 , B = 3.91 1.8 + 3 D* , δ * = δ 2 λRT , D* = Dh λRT ,
λRT = [σ /( g Δρ 21 ) ]
1/ 2
.
4.6 Dynamic evolution of the mean droplet size
89
4.6 Dynamic evolution of the mean droplet size For computing (a) the drag forces between the droplet and the carrier gas and (b) the mass transfer at the surface of a droplet the characteristic droplet size is required. I proceed as in the case of bubble flow writing the conservation of the droplet number per unit time and unit volume of the steady state flow dn3 = n&23 − n&32 + n&3, fr − n&3,coal . dz
(4.68)
Kolev (2007a), the source and the sink terms due to entrainment and depositions are already defined n&23 − n&32 =
4 6 3 ⎡( ρ w )23 D23 − ( ρ w )32 D33 ⎤⎦ . 1− α2 Dh πρ3 ⎣
(4.69)
What remains is to define droplet production due to fragmentation of unstable droplets n&3, fr , and droplet number decrease due to collisions and successful coalescence n&3,coal . Again critical review of this subject is given in Vol. 2 of this monograph. Here I will introduce models witch are addressed in computing the following examples. 4.6.1 Droplet size stability limit First the local Weber number defined as follows
ρ1 ( w1 − w3 ) We3 = σ 3 / D3
2
(4.70)
is compared with the critical Weber number. If it is larger the droplet is unstable and will fragment. Experimentally there are two limiting cases of application of the relative velocity on a drop: smoothly and abrupt. Smoothly applied relative velocity: For smoothly applied relative velocity the critical Weber number is
ρ1 ( w1 − w3 ) = = 12 , σ 3 / D3∞ 2
We3∞
(4.71)
allowing to compute the final stable droplet size D3∞ assuming that this relative velocity will remain constant.
90
4. The “simple” steady three-fluid boiling flow in a pipe
Suddenly applied relative velocity: For suddenly applied relative velocity the recommended critical Weber number is ⎛ 24 20.1807 16 ⎞ + − 2 / 3 ⎟ ⎡⎣1 + 1.077On31.64 ⎤⎦ , We3∞ = 55 ⎜ 0.615 Re3 Re3 ⎠ ⎝ Re3
(4.72)
for 200 < Re3 < 2000 and We3∞ = 5.48 ⎡⎣1 + 1.077On31.64 ⎤⎦ ,
(4.73)
for 2000 ≤ Re3 . There is no fragmentation if On3 > 4 or if Re3 < 200 . Here the Ohnesorge number is defined as follows On3 = η3
ρ3 D3σ 3 = We31/ 2 Re3 and the
Reynolds number Re3 = ρ1 D3 Δw13 / η1 .
4.6.2 Droplet production rate due to fragmentation Assuming that there is no other mass transfer processes but only the mechanical fragmentation the final number of particles will be ⎛π ⎞ n3,∞ = α 3 ⎜ D3,3 ∞ ⎟ . ⎝6 ⎠
(4.74)
The time averaged particle production rate is then
(
)
n&3, fr = ( n3,∞ − n3 ) Δτ br = n3 D33 D3,3 ∞ − 1 Δτ br .
(4.75)
Here Δτ br is the duration of the fragmentation.
4.6.3 Duration of the fragmentation The duration of the fragmentation depends on the initial Weber number because different physical fragmentation mechanisms are following depending on the departure from the stability limit. Again this is large subject discussed in details in vol. 2 of this monograph, Kolev (2007b) and here only the models addressed in the following computation will be addressed. Vibration breakup: It is expected within We3∞ < We3 ≤ We3∞ + 1 with duration
4.6 Dynamic evolution of the mean droplet size
(
)
Δτ br = 0.45 D33 ( 3ρ3 + 2 ρ1 ) 0.805 D30.225σ 3 ,
91
(4.76)
Lamb (1945). Schröder and Kitner (1965) correlated data with the group 0.805D30.225 . Droplet splitting is the result, D3∞ ≈ 0.79 D3 ,
(4.77)
which replace the critical size resulting in n&3, fr = n3 Δτ br
(4.78)
For the other regimes the review by Pilch et al. (1981) and Pilch and Erdman (1987) is used. For the data comparison the expression (4.75) is used. The dimensionless break up time is correlated to the deviation from the stability by using experimental data as explained in Vol. 2 of this monograph. Δτ br* = Δτ br Δw13
ρ1 ρ3 D3 = f (We3 − We3∞ ) .
(4.79)
Bag breakup: It is expected within We3∞ + 1 < We3 ≤ We3∞ + 6 with duration Δτ br* = 6 (We3 − We3∞ )
1/ 4
,
(4.80)
Kombaysi et al. (1964), Gelfand et al. (1976) and Fournier et al. (1955). In fact Magarvey et al. (1956) observed 30 to 100 droplets after this kind of fragmentation. Selecting 70 results in D3∞ ≈ D3 701/ 3
(4.81)
and n&3, fr = 69 n3 Δτ br
(4.82)
which is not used for the following data comparison. Bag, bag and stamen breakup: It is expected within We3∞ + 6 < We3 ≤ We3∞ + 33.12 with duration Δτ br* = 2.45 (We3 − We3∞ )
1/ 4
.
As in the previous case selecting 70 results in
(4.83)
92
4. The “simple” steady three-fluid boiling flow in a pipe
D3∞ ≈ D3 701/ 3
(4.84)
and n&3, fr = 69 n3 Δτ br
(4.85)
which is not used for the following data comparison. The following two modes are for very high Weber numbers. Bag and stamen + sheet stripping: It is expected within We3∞ + 33.12 < We3 ≤ We3∞ + 338.83 with duration Δτ br* = 14 (We3 − We3∞ )
1/ 4
.
(4.86)
Wave crest stripping followed by catastrophic breakup: It is expected within We3∞ + 338.83 < We3 ≤ We3∞ + 2658 with duration Δτ br* = 0.766 (We3 − We3∞ )
1/ 4
.
(4.87)
Catastrophic breakup: It is expected within We3∞ + 2658 < We3 with duration Δτ br* = 5.5 .
(4.88)
4.6.4 Collision and coalescence As for bubble flow the coalescence frequency is computed if α 3 > 0.001 . The collision frequency is computed following Howarth (1967) f3,col = 4.9 α 3 Δw33 D3 ,
(4.89)
see Eq. (7.23) in Vol. 2. Here the relative velocity causing collisions is computed as follows Δw33 = w3, k +1 − w3, k for w3, k +1 < w3, k and Δw33 = 0 for w3, k +1 > w3, k . The physical meaning is if the distance between two neighboring droplet increases they will never collide. If the distance decreases they collide. To this velocity difference the fluctuation velocity of the droplet is added. If turbulence modeling is used the magnitude of the fluctuation velocity is a result of the computation. If turbulence modeling is not used the fluctuation component can be taken as being a
4.7 Heat transfer
93
few percent of the magnitude of the gas velocity. Whether the droplets follow this magnitude depends on their size – see Vol. 3 of this monograph. The coalescence probability is
{
P3, coal = 0.032 1.56 σ 3 / ⎡⎣ D33 ( 3ρ3 + 2 ρ1 ) ⎤⎦ Δw33
}
1/3
.
(4.90)
Vol. 2 of this monograph, p. 267. The collision frequency is the product f3,coal = f3,col P3, coal .
(4.91)
For small frequencies the instant coalescence rate n&3,coal = n3 f 3,coal 2
(4.92)
is used. For large coalescence frequencies I prefer to use the time averaged coalescence rate
(
−f Δτ n&3,coal = n3 1 − e 3,coal
2
)
Δτ ,
(4.93)
Eq. 7.16 in Vol. 2. For very large coalescence frequencies the theoretical maximum n&3,coal =
1 n3 Δτ 2
(4.94)
is used.
4.7 Heat transfer The subject of heat transfer of boiling flow in all regimes is very complex. In the curse of this book I will come several times back to this subject. Here I will add to the previous chapter few useful heat transfer models that will be addressed in the computations. Idea for critical heat flux: Liquids boiling at heated surface is fragmented mechanically due to the bubbles generated on the surface. If the size and the frequency of the bubble generation under given local condition is so that the bubbles coalesce in the boundary layer and a stable film is formed we call this regime film boiling. If the heat generated from the surface is independent from the happenings in the flow the temperature at the surface will jump dramatically
94
4. The “simple” steady three-fluid boiling flow in a pipe
because the previous heat transfer regime, the nucleate boiling was much more effective then the film boiling. The transition from any regime at which the wall is wetted by the liquid to one by which the wall is wetted by vapor is called boiling crisis. There are more then two whys haw it happens but two of them are very prominent and important. In the first one the boundary layer bubbles coalesce. This regime transition is called departure from nucleate boiling (DNB). The second one is associated with drying out of the film attached at the wall. This regime transition is called dry out (DO). Note that the first regime is possible also in liquid film attached to the wall. Without going in any dept in this important subject at this place I will mention that one of the most reliable methods for predicting this state is experimentally elaborated in form of tables:
(
)
′′ = qCHF ′′ qCHF p, G, X 1,eq , Dhy = 0.008 ,
(
′′ = q&CHF ′′ _ 8mm 1000 Dhy 8 q&CHF
)
−1/ 2
,
(4.95) (4.96)
Groeneveld et al. (2005). The input parameters are the local flow parameters. For ′′ the heat transfer is subthe examples I will address this method. So if qw′′ < qCHF critical, else critical. I already mentioned some methods for computing the heat transfer in sub-critical heat transfer. In case of vapor flow caring droplets the heat transfer coefficient can be computed with the method derived in Vol. 3 of this book Eq. (4.46) p. 84: h1wσ = 0.2 Re10.8 Pr10.4 (Φ12o / α1 )1/ 4 λ1 Dh ,
(4.97)
Here 2 2 Φ10 ≈ Φ12 = 1 + CX LM + X LM , C = 20,
1/ 2
X LM
⎛ρ ⎞ =⎜ 1 ⎟ ⎝ ρ2 ⎠
0.1
(4.98)
0.9
⎛ η2 ⎞ ⎛ 1 − X1 ⎞ ⎜ ⎟ ⎜ ⎟ , ⎝ η1 ⎠ ⎝ X 1 ⎠
(4.99)
see Hetstroni (1982). The empirical constant 0.2 was found by comparison with experiments.
4.8 Mass transfer
95
4.8 Mass transfer For flow of liquid film attached to the wall, α 2 > 0 , in the critical heat flux regime, the heat from the wall is totally removed by the film evaporation
μ21 =
4 q&w′′ . Dh h′′ − h′
(4.100)
The heat transfer coefficient is related to the local saturation temperature. If there is no film, the heat transfer coefficient is related to the local vapor temperature. In this case the vapor receive all the heat from the wall q&w′′′1 = q&w′′ 4 Dh
(4.101)
and eventually become superheated. With knowledge of the droplet size: Droplets cared by the superheated vapor evaporate in flight. The evaporated mass per unit time and unit mixture volume is
μ31 = q&3′′′1σ
( h′′ − h′ )
(4.102)
where q&3′′′1σ = ( 6 α 3 D3 ) h13σ (T1 − T ′ )
(4.103)
is the thermal power per unit mixture volume consumed for evaporation. The heat transfer coefficient from the vapour to the droplet surface is computed using the Nigmatulin’s (1978) correlations 1 ⎛ ⎞ ⎛ 1 ⎞ h13σ = ( λ1 D3 ) ⎜ 2 + Pe0.84 ⎟ ⎜ 1 + Pe0.51 ⎟ 3 ⎝ ⎠ ⎝ 3 ⎠
(4.104)
for Pe < 1000 , else h13σ = 0.98 ( λ1 D3 ) Pe1/ 3 ,
(4.105)
where Pe = D3 w1 − w3 ρ1c p1 λ1 .
Note that using
(4.106)
96
4. The “simple” steady three-fluid boiling flow in a pipe
dX 1eq dz ≈ q&w′′′ ⎡⎣G ( h′′ − h′ ) ⎤⎦ ,
(4.107)
dX 1 dz = μ31 G ,
(4.108)
results in dX 1 dX 1, eq =
μ31 ( h′′ − h′ ) q& w′′′
=
( 6 α 3 D3 ) h13σ (T1 − T ′ ) q&3′′′1σ . =− q&w′′′ q&w′′′
(4.109)
This means that the change of the real vapor mass flow rate fraction with the equilibrium mass flow rate fraction is equal to the ratio of power density used for evaporation to those coming from the heated wall. This is intuitively expected. Such approach is used from many authors e.g. Kirilov et al. (1982) with variety of assumptions for the evolution of the droplet size. For completeness I will introduce also other methods used in the literature for computation of the droplet evaporation Without knowledge of the droplet size: If all the heat transferred from the wall is used for evaporation of the droplet the evaporation mass per unit time and unit mixture volume will be
μ31,eq = ( q& w′′1 4 Dheat ) ( h′′ − h′ ) .
(4.110)
This is called equilibrium mass source. In the reality some temperature difference is naturally necessary to have evaporation and therefore real evaporation mass density is smaller. Some authors assume
μ31 = (1 − X 1 ) μ31,eq = (1 − X 1 )( q&w′′1 4 Dheat ) ( h′′ − h′ ) .
(4.111)
This expression predicts in the limiting case of gas only no evaporation which is correct. Having in mind Eqs. (4.107) and (4.108) results in dX 1 (1 − X 1 ) = dX 1,eq or d ln (1 − X 1 ) = − dX 1, eq .
(4.112)
This equation can be integrated from the point of dry out where it can be assumed that X 1,eq = X 1 = X 1, DO . The result is X 1 = 1 − (1 − X 1, DO ) exp ⎡⎣ − ( X 1,eq − X 1, DO ) ⎤⎦ .
(4.113)
4.8 Mass transfer
97
This is Eq. (10) obtained by Barzoni and Martini (1982). It is in fact a simple relaxation method similar to those discussed for sub-cooled boiling. The difference is that here the continuum is the vapor and in the sub-cooled boiling it was the liquid and that the departure point for the integration here is the dry out point and in the sub-cooling boiling the point of net vapor production. Comparing with their experimental data the authors demonstrated that although very simple, this equation gives the right trend of void temperature and therefore of the heat transfer coefficient based on this temperature. Similar method is proposed by Hammouda et al. (1997) q&3′′′1σ =
T1 − T ′ 4 q&w′′1 , Tw − T1 Dheat
(4.114)
Resulting in dX 1 T −T′ . = 1 dX 1eq Tw − T1
(4.115)
Empirical method for computation of the droplet evaporation not taking into account the droplet size was proposed by Saha et al. (1980) and used later by Kelly et al. (1981) among other authors:
(
)
q&3′′′ = λ1 D 6300 (1 − α1 )(1 − p pc ) 1σ
μ31 = q&3′′′1σ
2 h
1/ 2
2
⎛ ρ1 w12 Dh ⎞ ⎜ ⎟ ⎝ σ ⎠
(T1 − T ′ ) ,
( h′′ − h′ ) .
(4.116)
(4.117)
Similar correlation was proposed by Webb et al. (1982)
(
)
q&3′′′1σ = λ1 Dh 1.32 (1 − α1 )
2/3
(p
pc )
⎛ X 1G ⎞ (T1 − T ′ ) . ⎜ ⎟ ρ1σ ⎝ α1 ⎠ 2
−1.1
(4.118)
Condie et al. (1984) compared the predictions of both correlations with experimental data assuming α1 = α1,hom and found that the first one systematically over predicts the data and the second inhibits large deviation.
98
4. The “simple” steady three-fluid boiling flow in a pipe
4.9 Comparison with experiments In order to validate the methods presented here I use 8 of the large number of experiments performed by Bennett et al. (1967) using 4.6576 and 5.5626 m long vertical pipes with 0.01262 m diameter. The pipes are uniformly heated. The boundary conditions are summarized in Table 4.1. Table 4.1. Boundary conditions
Author
Test No.
Pressure
Mass flow rate G / kg /(m 2 s )
p / bar
1000 950 900 850 800 750 700 650 600 550
)
Tin / K
2082 1496 1655 1317
15.95 14.65 10.34 24.13
902 1074 648 1691
18.96 18.96 25.43 11.64
850 Bennet 5253 prediction
800 Bennet 5293 prediction
750 700 650 600 550
0
1
2
3 z in m
4
5
6
850 800 700 650 600 550 0
1
2
3 z in m
1
Tw in K
Bennet 5332 prediction
750
0
1000 950 900 850 800 750 700 650 600 550 500
900
Tw in K
(
q&w′′ / kW / m 2
Temperature
900
Tw in K
Tw in K
Bennett et al. (1967), L = 4.6576 m 5442 68.93 4814.8 5407 68.93 1939.1 5424 68.93 2562.8 5456 68.93 1328.9 Bennett et al. (1967), L = 5.5626 m 5253 68.93 1356.0 5293 68.93 1979.8 5332 68.93 664.4 5380 68.93 3851.
Heat flux
4
5
6
2
3 z in m
4
5
6
4
5
6
Bennet 5380 prediction
0
1
2
3 z in m
4.9 Comparison with experiments 1100
1100
1000
1000 900
Bennet 5407 prediction
800
Tw in K
Tw in K
900
700 600 500
99
Bennet 5424 prediction
800 700 600
0
1
2 z in m
3
500
4
0
1
2 z in m
3
4
1300
1100
1200
1000
1100 900
Tw in K
Tw in K
1000 Bennet 5442 prediction
800 700
Bennet 5456 prediction
800 700 600
600 500
900
0
1
2 z in m
3
4
500
0
1
2 z in m
3
4
Fig. 4.1. Wall temperature as a function of length measured from the bottom. Comparison between the theoretical prediction and the data reported by Bennett (1967). Prediction using: Boundary layer theory: Condensation Kolev (2007c); Convection: Kolev (2007c); Evaporation: Borishanskii et al. (1964); Effective bubble size: bubble number conservation equation; Effective droplet size: droplet number conservation equation; Drag coefficients for bubbles by Ishii and Zuber (1978) multiplied by 0.1; Drag coefficients for droplets by Ishii and Chawla (1979); Deposition Katto (1984); Entrainment Kataoka and Ishii (1982); Saturated forced convection boiling is modelled using the Chen (1963) correlation. The critical heat flux is predicted by the 2005 look-up table (1996). The post-critical heat flux is computed using the correlation by Kolev (2007c) with a coefficient of 0.2; Mass transfer between the superheated gas and the droplets Nigmatulin’s (1978)
I will once again summarize the main feature of the steady state model and the constitutive relations addressed for this predictions:
• • • • • • • •
Three fluid model; Entrainment and deposition as presented in Sect. 4.3; Drag coefficients for bubbles as presented in Sect. 3.11; Drag coefficients for droplets and film as presented in Sect. 4.4; Dynamic evolution of the bubble size as presented in Sect. 3.11; Dynamic evolution of the droplet size as presented in Sect. 4.6; Heat transfer mechanism in sub-cooled boiling as presented in Chap. 3; Identification of the critical heat flux using the Groeneveld et al. 2005 look up table; • Heat transfer in film boiling as presented in Sect. 4.7;
100
4. The “simple” steady three-fluid boiling flow in a pipe
• Heat and mass transfer between the superheated gas and the droplets as presented in Sect. 4.8. The most important result of these experiments is the temperature at the wall as a function of the distance from the entrance of the heated pipe. Figure 4.1 presents the results obtained used the formalism presented here compared with the measurements.
1,0
10 9
0,6
Flow pattern ID
Voule fraction, -
0,8 Void Continuous liquid Droplets
0,4 0,2
8 7 6 5
prediction
4 0,0
0
1
2
3 z in m
4
5
6
0
1
2
3 z in m
4
5
6
Fig. 4.2. (a) Theoretical prediction of the volumetric fractions of the vapour, film and droplets as a function of length measured from the pipe entrance for the Bennet (1967) experiment 5253; (b) Flow pattern identification: 4 bubbly flow, 6 slug flow, 10 film flow with droplets in the gas core
Figure 4.2 presents details for the test 5254. Now we realize why it was necessary to introduce three velocity fields: Lumping droplets and film in a single liquid velocity field does not allow distinguishing between the droplet and the film. We also learn from this analysis, that is was useful first to model two fluid bubbly flow in Chap. 2 and then to step to the three fluid because in this way we check only the newly developed feature of the model. The following conclusions can be drawn from this comparison with data regarding the three-fluid model:
• The position of the occurrence of the critical heat flux is well predicted by the 2005 look up table; • The magnitude of the pre- and post critical heat transfer and therefore the wall temperature is also reasonably predicted by the formalism used; • Although the wall temperature is higher then the Leidenfrost temperature and even then the critical temperature and pure film boiling is the expected regime there is some heat transfer mechanism resembling transition boiling which makes the form after the jump more smooth. This is not explained by the theory so far. To account for the smooth transition to film boiling an ad hoc correction is introduced as follows
4.9 Comparison with experiments
h = 1 + δ FB − δ FB hFB
⎛1 ⎞ cos ⎜ π r ⎟ ⎝ 2 ⎠ for r > r TB ⎛1 ⎞ cos ⎜ π rTB ⎟ ⎝2 ⎠
101
(4.119)
1000 950 900 850 800 750 700 650 600 550
Bennet 5253 prediction
Tw in K
Tw in K
′′ q& w′′ rTB =0.8 , δ FB = 0.2 . The predictions with this correction are where r = q&CHF presented in Figure 4.3. The transition to film boiling is smoother. Obviously the form of Eq. (4.119) requires sophistication.
0
1
2
3 z in m
4
5
6
850
700 650 600 550 0
1
2
3 z in m
4
5
6
1100
1100
1000
1000
700 600 500
2
3 z in m
4
5
6
4
5
6
Bennet 5380 prediction
0
900
Bennet 5407 prediction
Tw in K
Tw in K
900 800
1
Tw in K
Tw in K
Bennet 5332 prediction
750
Bennet 5293 prediction
0
1000 950 900 850 800 750 700 650 600 550 500
900 800
950 900 850 800 750 700 650 600 550
1
2
3 z in m
Bennet 5424 prediction
800 700 600
0
1
2 z in m
3
4
500
0
1
2 z in m
3
4
102
4. The “simple” steady three-fluid boiling flow in a pipe 1200
1100
1100
1000
900
Tw in K
Tw in K
1000 Bennet 5442 prediction
800 700
Bennet 5456 prediction
800 700 600
600 500
900
0
1
2 z in m
3
4
500
0
1
2 z in m
3
4
Fig. 4.3. As Fig. 4.1 with corrected transition to film boiling above the Leidenfrost temperature
After going through the examples from Chaps. 2 and 3 the reader will understand the quotation marks of the titles “simple” flow boiling in a pipe: There are large numbers of complex processes controlling the heat transfer in such geometry like a cylindrical vertical uniformly heated pipe. The understanding of each of the separated mechanism is crucial for understanding haw safe heat removal can be guaranteed in nuclear installation during normal and accidental operations. In the real life the processes are even much more complicated because there are three dimensional always, transient, and the complexity of the geometry is enormous. Further in this book I well elaborate more knowledge for understanding of these processes.
4.10 Nomenclature Latin a12
c12d c1 const Dh
interfacial area density, i.e. the surface area between gas and film per unit mixture volume, m²/m³ vapor side shear stress coefficient at the liquid surface due to the gas flow, dimensionless = Δρ 21 / ρ1 constant, dimensionless hydraulic diameter, m
⎛ dp ⎞ ⎜ dz ⎟ friction pressure drop per unit length in the film, Pa/m ⎝ ⎠2 ⎛ dp ⎞ ⎜ dz ⎟ two phase friction pressure drop per unit length in the core, Pa/m ⎝ ⎠Tph
4.10 Nomenclature
E
E∞ Eo
=
103
α 3 ρ 3 w3 , mass fraction of the entrained liquid, entrainment, α 2 ρ 2 w2 + α 3 ρ 3 w3
dimensionless equilibrium mass fraction of the entrained liquid, entrainment, equilibrium entrainment, dimensionless = ( Dh / λRT ) 2 , Eötvös number, dimensionless
Re23
= V12 /( gDh ) , gas Froude number, dimensionless deposition coefficient, dimensionless frequency of the fastest growing of the unstable surface perturbation waves, dimensionless mass flow rate, kg/(m2s) = α1w1 + α 3 w3 core (gas + droplet) superficial velocity, m/s film wavelength, m pressure, Pa = α1 ρ1 w1 Dh / η1 , local gas film Reynolds number based on the hydraulic diameter, dimensionless = α 2 ρ 2 w2 Dh / η2 , local film Reynolds number based on the hydraulic diameter, dimensionless = ρ 2 w2δ 2 / η2 , local film Reynolds number based on the liquid film thickness, dimensionless = Re23 (1 − E∞ ) , local equilibrium film Reynolds number based on the hydraulic diameter, dimensionless = ρ 2 (1 − α1 ) w23 Dh / η2 , total liquid Reynolds number, dimensionless
Re2F
= ρ 2 w2 4δ 2 F / η2 , local film Reynolds number, dimensionless
Re2Fc
= 160 , critical local film Reynolds number, dimensionless
Fr1 f f m*
G jcore k2 p Re1 Re2 Re2δ Re2 ∞
Recore q& w′′ 2 u23
= (α1 ρ1w1 + α 3 ρ 3 w3 ) Dh / η1 , core Reynolds number, dimensionless
heat flux, MW/m2 interface averaged entrainment velocity, m/s 1/ 3
WeIshii
ρ (α w ) 2 Dh ⎛ ρ 2 − ρ1 ⎞ = 1 1 1 ⎜ ⎟ , Weber number for the Ishii entrainment σ2 ⎝ ρ1 ⎠ correlation, dimensionless 1/ 2
WeLopez
=
ρ1w12 Dh ⎛ ρ 2 − ρ1 ⎞ ⎜ ⎟ σ 2 ⎝ ρ1 ⎠
, Weber number for the Lopez et al. correlation,
dimensionless WeZeichik =
τ 21σ δ 2 F , Weber number for the Zeichik et al. correlation, dimensionless σ2
104
4. The “simple” steady three-fluid boiling flow in a pipe
ρ1 Dh ΔV122 , Weber number, dimensionless σ2
We12
=
We31
= ρ1 (α1V1 ) D3 E / σ , Weber number, dimensionless
w23
=
w1 w2 w3 X1
2
α 2 w2 + α 3 w3 , center of volume velocity of film and droplet together, 1 − α1
m/s axial cross section averaged gas velocity, m/s axial cross section averaged film velocity, m/s axial cross section averaged droplets velocity, m/s gas mass flow divided by the total mass flow, dimensionless
Greek
α1 α2 α3 Δw12
gas volume fraction, dimensionless film volume fraction, dimensionless droplets volume fraction, dimensionless = w1 − w2 , relative velocity, m/s
δ 2F
= Dh 1 − 1 − α 2
η1 η2 λR12 λR1
gas dynamic viscosity, kg/(ms) liquid dynamic viscosity, kg/(ms) gas-film friction coefficient, dimensionless friction coefficient, dimensionless
(
)
2 , film thickness in annular flow, m
λR 2
= λR 2 ( Re 2 , k / Dh ) , film friction coefficient, dimensionless
λRT μ23
= (σ / g Δρ 21 )1/ 2 , Rayleigh-Taylor wavelength, m
ρ1 ρ2
mass leaving the film and entering the droplet field per unit time and unit mixture volume, kg/(m³s) gas density, kg/m³ liquid density, kg/m³
ρ core
= (α1 ρ1 w1 + α 3 ρ 3 w3 ) / jcore , core density, kg/m³
ρ3c
=
α3 ρ , mass of droplets per unit volume of the gas-droplet mixture α1 + α 3 3
assuming equal velocities, kg/m³ ( ρ w) 23 entrainment mass flow rate, mass leaving the film per unit time and unit interfacial area, kg/(m²s) ( ρ w )32 deposition mass flow rate, mass leaving the droplet field per unit time and unit interfacial area and deposed into the film, kg/(m²s)
References
σ τ 21σ τ 12 θ
105
surface tension, N/m liquid side surface shear stress interfacial stress, N/m² angle with origin of the pipe axis defined between the upwards oriented vertical and the liquid-gas-wall triple point, rad
References Barzoni G and Martini R (1982) Post dry out heat transfer: An experimental study in vertical tube and a simple theoretical method for predicting thermal non-equilibrium, 7th Int. Heat Transfer Conference, Munich, Germany, pp 414–416 Bennett AW et al. (1967) Heat transfer to steam-water mixtures flowing in uniformly heated tubes in which the critical heat flux has been exceeded, AERE-R5373 Borishanskii V, Kozyrev A and Svetlova L (1964) Heat transfer in the boiling water in a wide range of saturation pressure. High Temp., vol 2 no 1, pp 119–121 Brinkman HC (1952) The viscosity of concentrated suspensions and solutions, J. Chem. Phys., vol 20 no 4 p 571 Chen JC (1963) A correlation for film boiling heat transfer to saturated fluids in convective flow, ASME Publication-63-HT-34, p2–6 Condie KG, Gottula RC, Nelson RA, Netti S, Chen JC and Sundaram RK (Aug. 5–8, 1984) Comparison of heat and mass transfer correlations with forced convective nonequilibrium post CHF experimental data, in Basic aspects of two phase flow and heat transfer, 22nd Nat. Heat Transfer Conference, Niagara Falls, New York D’Albe EMF and Hidayetulla MS (1955) The break-up of large water drops falling at terminal velocity in free air Quarterly Journal Royal Meteorological Society, vol 81 issue 350 pp 610–613 Fournier D’Albe EM and Hidayetulla MS (1955) Q J R Meteorol Soc, Kondon, vol 81, pp 610–613 Gelfand BE, Gubin SA and Kogarko SM (1976) Various forms of drop fragmentation in shock waves and their spatial characteristics. J. Eng. Phys., vol 27 Groeneveld DC, Shan JQ, Vasi AZ, Leung LKH, Durmayaz A, Yang J, Cheng SC and Tanase A (Oct. 2–6, 2005) The 2005 CHF look-up table, The 11th Int. Top. Meeting on Nuclear Thermal-Hydraulics (NURETH11) Avignon, France Hammouda N, Groeneveld DC and Cheng SC (1997) Int. J. Heat Mass Transfer, vol 40 no 11, pp 2655–2670 Hetstroni G (1982) Handbook of Multiphase Systems. Hemisphere Publ. Corp., Washington etc., McGraw-Hill Book Company, New York etc. Howarth WJ (1967) Measurement of coalescence frequency in an agitated tank, A. I. Ch. E. J., vol 13 no 5, pp 1007–1013 Imura K, Yoshida K, Kataoka I and Naitoh M (July 17–20, 2006) Subchannel analysis with mechanistic methods for thermo-hydro dynamics in BWR fuel bundles, Proc. of ICONE14, Int. Conf. on Nuclear Engineering, Miami, FA, USA Ishii M (1977) One dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. ANL-77-47 Argonne National Laboratory, Argonne Ishii M and Chawla TC (Dec.1979) Local drag laws in dispersed two-phase flow, NUREG/CR-1230, ANL-79-105
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4. The “simple” steady three-fluid boiling flow in a pipe
Ishii M and Zuber N (1978) Relative motion and interfacial drag coefficient in dispersed two-phase flow of bubbles, drops and particles, Paper 56 a, AIChE 71st Ann. Meet., Miami Kataoka I and Ishii M (July 1982) Mechanism and correlation of droplet entrainment and deposition in annular two-phase flow. NUREG/CR-2885, ANL-82-44 Kataoka I, Ishii M (March 20-24, 1983) Entrainment and deposition rates of droplets in annular two-phase flow, ASME-JSME Thermal Engineering Joint Conference Processings, Honolulu, Hawaii, vol 1 pp 69–80, Eds. Yasuo Mori, Wen-Jei Yang Kataoka I, Ishii M, Mishima K (June 1983) Generation and size distribution of droplets in annular two-phase flow Transaction of the ASME, vol 105 pp 230–238 Katto Y (1984) Prediction of critical heat flux for annular flow in tubes taking into account of the critical liquid film thickness concept. Int. J. Heat Mass Transfer, vol 27 no 6, pp 883–890 Kelly JE, Kao SP and Kazimi MS (April 1981) THERMIT-2: A two-fluid model for light water reactor sub-channel transient analysis, MIT Energy Laboratory Electric Utility Program, Report No. MIT-EL-81-014 Kirilov PL, Kokorev BV, Remizov OV and Sergeev VV (1982) Post dry-out heat transfer, 7th Int. Heat Transfer Conference, Munich, Germany, pp 414–416. Kolev NI (2007a) Multiphase Flow Dynamics, Springer, Berlin, vol 1 Kolev NI (2007b) Multiphase Flow Dynamics, Springer, Berlin, vol 2 Kolev NI (2007c) Multiphase Flow Dynamics, Springer, Berlin, vol 3 Komabayasi MT, Gonda T and Isono K (1964) Life time of water drops before breaking in size distribution fragment droplet. J. Met. Soc. Japan, vol 42 no 5, pp 330–340 Lamb MA (1945) Hydrodynamics. Cambridge, At the University Press Magarvey RH and Taylor BW (Oct.1956) Free fall breakup of large drops. J Appl. Phy., vol 27 no 10, pp 1129–1135 Mishima K and Ishii M (1984) Flow regime transition criteria for upward two-phase flow in vertical tubes. Int. J. Heat Mass Transfer, vol 27 no 5, pp 723–737 Nigmatulin BI (1982) Heat and mass transfer and force interactions in annular – dispersed two-phase flow, 7-th Int. Heat Transfer Conference Munich, pp 337–342 Nigmatulin BI, Melikhov OI and Khodjaev ID (April 3–7, 1995) Investigation of entrainment in a dispersed-annular gas-liquid flow, Proc. of the 2nd International Conference on Multiphase Flow ‘95 Kyoto, Japan, vol 3, pp P4–33 to P4–37 Nigmatulin RI (1978) Basics of the mechanics of the heterogeneous fluids, Moskva, Nauka, in Russian Paleev II and Filipovich BS (1966) Phenomena of liquid transfer in two-phase dispersed annular flow. Int. J. Heat Mass Transfer, vol 9, p 1089 Pilch M, Erdman CA and Reynolds AB (Aug.1981) Acceleration Induced Fragmentation of Liquid Drops, Department of Nucl. Eng., University of Virginia, Charlottesville, VA, NUREG/CR-2247 Pilch MM and Erdman CA (1987) Use of the breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drops. Int. J. Multiphase Flow, vol 13 no 6, pp 741–757 Roscoe R (1952) The viscosity of suspension of rigid. Spheres, Brit. J Appl. Phys., vol 3, p 267–269 Saha P, Shiralkar BS and Dix GE (1980) A post dry out heat transfer model based on actual vapor generation rate in droplet regime, ASME, 77-HT-80 Schröder RR and Kitner RC (Jan. 1965) Oscillation of drops falling in liquid field,.A.I.Ch.E. J., vol 11 no 1, pp 5–8 Stephan M and Mayinger F (1990) Countercurrent flow limitation in vertical ducts at high system pressure. In: Hetstroni G (ed), Proc. of the Ninth International Heat Transfer Conference, Jerusalem, Israel, vol 6 pp 47–52 Webb SW, Chen JC and Sundaram RK (Sept. 1982) Vapor generation rate in non equilibrium convective film boiling. 7th Int. Heat Transfer Conf., Munich, Germany, vol 4, paper FB45
5. Core thermal hydraulic
After a short review of existing nuclear reactor pressure vessels and core geometry this chapter summarizes the main ideas haw to perform a thermo-hydraulic analysis of boiling flows in nuclear reactors. Remarkable is that all known threefluid flow pattern and heat transfer mechanisms create the cooling mechanism of the core. To demonstrate the achievable accuracy using gross discretization I simulate 333-three dimensional tests in bundles with 1, 16, 25, 36 heated rods, 7 different bundles with 64 heated rods from different laboratories: 273 3D-experiments on 6 bundles for CHF, 54 3D-experiments on 7 bundles for void fraction, 2 3D-experiments in a bundles for transients and 4 1D-experiments in a sub-channel for transient. The mass flow rates and the pressure in these tests varied from 3 to 2000 kg/(m²s) and from 1 bar to 200 bar, respectively. The subcooling have been less than 140 K and the thermal power varies from some tenths of kW to 7MW. Comparison with the results of other authors was made and discussion was provided. This comprehensive analysis shows that the scale of spatial resolution calls for specific set of constitutive relations. Examples are given for the effective interfacial drag coefficients. In addition, it was demonstrated by using different spatial resolution that the turbulent void and droplets transport have to be described much more accurately in distributed parameters to provide a universal way of prediction of void and droplet dispersion with such important consequences like accurate void- and dry out prediction. Simultaneously, this chapter may also serve as a source of inspiration for improving the accuracy of the mathematical description of such processes in the future.
5.1. Reactor pressure vessels The nuclear reaction in the most nuclear power plant is organized in specially designed facilities called core inside the so-called reactor pressure vessel. The steel vessels with its internals containing the nuclear rector core are organizing the coolant in- and outflow. Two main types of nuclear reactors are world wide build with considerably difference in their pressure vessels: The boiling water reactors
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_5, © Springer-Verlag Berlin Heidelberg 2009
107
108
5. Core thermal hydraulic
(BWR’s) and the pressurized water reactors (PWR’s). Figure 5.1 shows the differences.
Fig. 5.1. RPVs of a PWR and a BWR for plants of about 1300 MW electric power. Taken from Azodi et al. (1996)
The BWR-vessels are designed for about 70 bar normal operation pressure and the PWR-vessels for about 160 bar. As the name said, in the BWR the coolant is boiled up to given moisture content depending on the power and pump mass flow. Then the moisture is separated by the so called cyclones and dryers and dry steam, < 0.1 mas% liquid content, is directed to the turbine, see Fig. 5.2. I will devote a special chapter to the cyclones and vane dryers but at this place it is important to say, that there are placed above the nuclear reactor core. This is the reason why the vessels of the BWR are about twice taller then the PWR vessels. The later organizes the cooling using almost only of single phase water. Typical RPV-sizes of the serially build NPP by Siemens in Germany are given in Fig. 5.3. Figure 5.3(b) shows the core barrel which is coaxial with the cylindrical part of the pressure vessel. It is connected with the lower grid, lower distributor plate and the sieve barrel. The core support structure of the previous design as given in Fig. 5.4 was modified as given in Fig. 5.3(b). World wide there are differences in the design of the lower head. Some providers allowed penetration for the in core instrumentation, as those presented in Figs. 5.5 and 5.6, other
5.1. Reactor pressure vessels
109
not. It is well recognized that with regards of protection against the severe accidents the solution without penetration of the lower head is better.
Fig. 5.2. Typical BWR, see Ahlinder and Tinoco (2007): 1. Upper head; 2. Dryer; 3. Feed water nozzles; 5. Core upper grid; 6. Core; 7. Down comer; 8. Internal main circulation pumps; 9. Spry system; 11. Vessel support; 12. Cyclones; 13. Reactor pressure vessel; 15. Fuel bundles; 16. Control rod blades; 17. Down comer – biological shield; 18. Control rod pipes; 20. Control rod drives
There are also other types of PWR’s the so-called pressure pipe reactors. They organize the nuclear heat release to happens in pipes: horizontal in the Canadian CANDU, and vertical in the Russian RBMK. The moderator in the CANDU is heavy water and in the RBMK – graphite.
110
5. Core thermal hydraulic
Fig. 5.3. RPV of Siemens “Konvoi”-plants (1300 to 1400 MW, 4-loop; start of operation 1988/89): (a) vessel; (b) main internal support structures. Taken from Azodi et al. (1996)
Fig. 5.4. RPV with main internal support structures of Siemens “pre-Konvoi” plants (1300 to 1400 MW, 4-loop, start of operation 1982–86). The pressure boundary is basically the same as in Fig. 5.3. Taken from Azodi et al. (1996)
5.1. Reactor pressure vessels
111
Fig. 5.5. RPV: (a) Babcock & Wilcox 900 MW 2-loop plant with its main internal support structures (TMI-2, start of Operation 1978); (b) Westinghouse plants. Sizes given are for 900MW, 3-loop plants and 1300 MW, 4-loop plants (in brackets). Taken from Azodi et al. (1996)
Fig. 5.6. RPV of a Westinghouse 1150 MW 4-loop plant with its main internal support structures (Reference plant: Sequoyah 1 & 2, start of Operation 1981/82). Taken from Azodi et al. (1996)
112
5. Core thermal hydraulic
Figure 5.6 shows where the core is placed inside the vessel – inside the so-called core barrel. The barrel may be surrounded by thermal shield. Modern reactors use instead the so-called heavy reflector, a massive steel shield having a neutron reflector functions. The core is bounded by the so-called lower- and upper core plates. The fuel rod bundles are supported by the lower core support plate and fixed by the upper core support plate. The coolant is coming from the circumferential nozzles turning 90° towards the annular space, called down comer, and serving simultaneously as a neutron reflector, and again turning 180° enters the core. The upper plenum collects the coolant with different temperatures, facilitates some mixing, and redirects it through a perforated cylindrical wall towards the exit nozzles. From the nozzles the flow goes into the steam generators.
(a)
(b)
-
Fig. 5.7. RPV of Russian VVER type reactors: (a) VVER 440, a 440 MW 6-loop plant (start of operation 1971 and later); (b) VVER 1000, a 6-loop plant with horizontal steam generators. The main difference of RPV’s of the WER 1000 plants is their larger diameter, 4 m, wall thickness of the cylindrical part 190 mm, height up to flange 11 m; 1 – control rods; 2 – reactor cover; 3 – reactor chassis; 4 – inlet and outlet nozzles; 5 – reactor vessel; 6 – active reactor zone; 7 – fuel rods
A new generation of PWR’s is started to be build with the European Pressurized Water Reactor called EPR. The plant has 1600 MW electric output, which for the time being is the largest plant using PWR reactor type.
5.1. Reactor pressure vessels
113
Fig. 5.8. (a) Nuclear Engineering International, Oct. 1997: The European Pressurized Water Reactor called EPR: A. Vessel head penetrations; B. Reactor vessel closure head; C. Control rod guide assembly; D. Upper support plate; E. Instrumentation lances; F. Inlet nozzle; G. Outlet nozzle; H. Heavy reflector; J. Reactor vessel; K. Fuel assemblies; L. Irradiation capsule basket; M. Lower support plate; N. Lower radial support system; O. Flow distribution plate. (b) European Nuclear Reactor Safety, EUR 21030, 2004: Work principles of PWR’s; (c) Reactor pressure vessel connected by primary pipes to the four steam generators. The four main circulation pipes and the pressurizer are visible
114
5. Core thermal hydraulic
The design of the reactor is given in Fig. 5.8. The design is close to the design of the previous generation – that is why the reactor is called “evolutionary”. It makes use of proven in the practice components. The effect of scale is used here – it means the higher the unit power the lower the specific prize per MWe installed power. It was designed to be competitive during the time when the barrel oil has cost 30$. The Russian type of PWR’s arrange the inlet and the outlet nozzles in two different horizontal levels, where the western type of PWR’s have a inlet and outlet nozzles in the same horizontal plate. This makes the western PWR-vessel shorter and cheaper. The first generation of the Russian PWR’s, Fig. 5.7(a) possessed additional compensation depth below the core because the control rod bundles have been connected to displacement fuel rod bundles. This had some advantages in the efficiency of the control rod bundles but was replaced later by the western solution of finger control rods penetrating the core only from the top. Variety of designs is used for homogenization of the inlet coolant flow in the lower head: sieve barrel – Fig. 5.3, perforated head parallel to the external lower head, Figs. 5.5 and 5.7 and others. Some cores possess inlet nozzles serving for profiling the mass flow entering each bundle in accordance with the specific power in order to reduce the thermal differences in the core. As the reader can imagine, the thermal non uniformity creates several technical challenges with different thermal extensions and stresses that has to be considered during the design. This is especially sensitive for fast breeder reactors.
5.2. Steady state flow in heated rod bundles Nuclear reactor cores consist of set of parallel rod bundles. Before going to the computational thermal hydraulic analysis of a nuclear reactor core I will give several examples of how to analyze processes in a single rod bundle using the computer code IVA which is based at all the methods presented in this monograph. After the main physics is understood at this level the next step of the analysis of total nuclear reactor core can be done. 5.2.1 The NUPEC experiment Valuable experimental data for boiling in 8 × 8 rod unequally heated bundles have been collected by the Japanese Nuclear Power Engineering Corporation (NUPEC) and reported by Morooka et al (1991), Yagi et al. (1992), Inoue et al. (1995a, b, c). I first concentrate my attention on the 8 × 8 bundle experiments having the geometry definition given in Table 5.1 and Fig. 5.9(a). This experimental arrangement is referred to as low burn up configuration. In addition the high burn up configuration
5.2. Steady state flow in heated rod bundles
115
as given in Fig. 5.9(b) is also used. The central rod in the last case has a diameter of 34 mm. Table 5.1. Dimensions of BWR bundle Utsuno et al. (2004) Number of fuel rods Outer diameter, mm Heated length, m Number of water rod, mm Outer diameter of water rod, mm Rod pitch, mm Width of channel box, mm Number of spacer Spacer type
62 12.3 3.7 2 15.0 16.2 132.5 7 Grid
(a)
(b)
(c) Fig. 5.9. None homogeneous power release in the bundle, Inoue et al. (1995a)
The boundary conditions for the considered experiments are given in Tables 5.2 and 5.3.
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5. Core thermal hydraulic
Table 5.2. Boundary conditions, non uniform power release, NUPEC Yagi et al.
(1992, p. 163) Exit. eq. quality 0.2415 0.2447 0.2479 0.2410 0.2489
p in MPa 7.2 7.2 7.2 7.2 7.2
Power in MW 1.143 2.313 3.509 6.458 8.219
G in kg/(m²s) 284 568 852 1562 1988
Subc. in kJ/kg (K) 50.2 (9.42) 50.2 (9.42) 50.2 (9.42) 50.2 (9.42) 50.2 (9.42)
The sub-cooling is taken from Inoue et al. (1995a, p. 393). The exit equilibrium cross section averaged quality as well as the experimental cross section void fractions are taken from Yagi et al. (1992, p.163). The power is then computed by using the internal sub-cooling, the exit quality and the mass flow rate. The axial power distribution as presented in Fig. 5.9(c) is taken from Fig. 5.3 in Inoue et al. (1995a, p. 391). The lateral power distribution is taken from Fig. 5.4(a) Inoue et al. (1995a, p. 391). The later distribution required slight renormalization to achieve strict energy conservation of the total energy release as a boundary condition. The spacers receive an irreversible pressure drop coefficient equal to 1 based on the bundle velocity due to lack of better knowledge about their actual value. Table 5.3. Boundary conditions, non uniform power release Inoue et al. (1995a)
p. 394, high burn up geometry Exit. eq. quality 0.2479 0.2410
p in MPa 7.2 7.2
Power in MW 3.509 2.313
G in kg/(m²s) 852 1562
Subc. in kJ/kg (K) 50.2 (9.43) 50.2 (9.43)
The cross section averaged void fractions are presented in the original sources Yagi et al. (1992, p. 163) and Inoue et al. (1995a, p. 394) as a function of the equilibrium quality. The equilibrium quality is computed be using the inlet sub-cooling and integrating over the height of the bundle using the axial energy distribution. The type of the channels for the low burn up bundle, their number, the hydraulic and the heated diameter and the corresponding cross section are given in Table 5.4. Table 5.4. Sub channel characteristics for the NUPEC 8 × 8 low burn up bundle Typ
nr.
γz
Dhyd
Dheat
F
1, internal 2, corner 3, periphery 4, unheated Total
22 4 36 2 64
0.5472 0.5930 0.5844 0.3266 0.5583
1.4867E-02 9.9153E-03 1.2188E-02 7.2766E-03 1.2994E-02
1.4867E-02 1.7921E-02 1.7298E-02 0 1.6291E-02
1.4362E-04 1.7313E-04 1.6711E-04 8.5725E-05 9.7575E-03
Note that Aounallah and Coddington (1999) used part of this data for verification of the two-fluid sub-channel code VIPRE-02 in which the cross flow is modeled by
5.2. Steady state flow in heated rod bundles
117
simplified momentum equations. Naitoh et al. (1999) and Utsuno et al. (2004) used part of the NUPEC data for verification of drift flux sub-channel code CAPE and TCAPE, respectively, in which the cross flow is modeled by mixing models defining cross flows, but not by momentum conservation. The idea of sub-channel: My computation in Cartesian coordinates uses 8 × 8 × 24 cells which form 64 rod-centered sub-channels, see Fig. 5.12. The term subchannel is usually used in the literature to describe a column of cells occupying only a part of the cross section of the bundle. Historically sub-channels are first considered as flow paths without exchange and later on some exchange mechanisms are introduced. Nowadays the complete 3D presentation replaces these techniques.
(flow in - flow out)/flow in in %
Obtain the steady state solution as an asymptotic of transient solution: There are two methods to obtain the steady state solution in bundles: either to solve the steady state part of the system of partial differential equations describing the flow or to solve the whole transient system for a constant boundary conditions up to the moment that the solution does not change with the time any more. While the second approach allows finding also oscillating steady state solutions the first one exclude them automatically. That is why I recommend using the second one. In this particular case I start the simulation with an arbitrary but meaningful initial state and continue the simulation until the steady state was established, controlling this by plotting the relative difference between inlet and outlet flows in %. An example is given on Fig. 5.10 for the low burn up geometry with mass flow rate 1988 kg/(m²s). For achieving an accurate steady state solution through a transient analysis the accuracy of the time integration is crucial in order not to produce artificial density waves. In this case time steps of 1 × 10–3 s or less are appropriate.
40 20
IVA, steady state
0 -20 -40 0
2
4 6 Time in s
8
10
Fig. 5.10. Non oscillating steady state for the NUPEC experiment with 1988 kg/(m²s)
Only the case with 284 kg/(m²s) resulted in oscillating but stable steady state solution. It will be discussed in a separate section. All other cases manifest non oscillating steady state solution as those characterized by Fig. 5.10.
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5. Core thermal hydraulic
Cross section averaged axial thermal power profiles: The results of the computational analyses for the low burn up cases are presented on Fig. 5.11. If nothing else is said I use the Saha and Zuber (1974) correlation for the initiation of the subcooled boiling and the Levy (1967) asymptotic model for the splitting of the heat into convection evaporation and condensation.
Cross section averaged void, -
Fine resolution versus cross section averaged resolution: The upper curve present the unchanged IVA set of drag coefficients from Vol. 2, Kolev (2007a) of this monograph, which are applicable to fine resolution analyses. The used rodcentered sub-channel discretization is by far not a fine resolution. The specifics of the flow in such sub-channels is in the velocity- and void profiles that make the application of the local drag coefficient based on local parameter not appropriate. An ad hoc modification on the drag coefficients is made as presented in Appendix 5.1. 0,9 0,8 0,7 0,6 0,5 0,4 exp IVA, drag 0,3 non modified 0,2 modified 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
Cross section averaged void, -
(a)
(b)
0,9 0,8 0,7 0,6 0,5 exp. 0,4 IVA, local drag 0,3 non modified 0,2 modified 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
Cross section averaged void, -
0,9 0,8 0,7 0,6 0,5 exp. 0,4 IVA, local drag 0,3 non modified 0,2 modified 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
0,9 0,8 0,7 0,6 0,5 Exp. 0,4 IVA, local drag 0,3 non modified 0,2 modified 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
119
(c)
(d) Fig. 5.11. NUPEC experiment with mass flow rate (a) 1998 kg/(m²s); (b) 1562 kg/(m²s); (c) 852 kg/(m²s); (d) 568 kg/(m²s). Void fraction as a function of the height. Parameter: local interfacial drag
The results with this modification are presented in each figure as a lower curve. The comparison with the data is then favorable. Figure 5.12 illustrates some details of the case with a mass flow rate 1562 kg/(m²s). The influence of the nonheated rods and of the low power rods is clearly seen. Now, let us compare the predictions with the measurements for the two high burn up cases using the modified drag coefficients. The solutions are presented in Figs. 5.13(a) and (b).
120
5. Core thermal hydraulic
Fig. 5.12. NUPEC experiment with mass flow rate 1562 kg/(m²s): (a) z = 1.3133m, X1 = 0.0596; (b) z = 1.9313m, X1 = 0.124; (c) z = 2.5492m, X1 = 0.184; (d) zexit = 3.6307m, X1,exit= 0.2489
In summary, the effect of the non uniformity of the flow profiles inside the subchannels is manifested in effectively lower drag between the liquid and the vapor. The comparison with the data using the reduced effective drag is favorable. However, it can not be expected a priori to have a unique interfacial drug reduction algorithm for all type of sub-channels. To check this, we analyze in the following sections void fraction data collected in bundles of completely different types. Regions averaged exit void fractions at different power level: Inoue et al. (1995a) reported region-averaged void fractions at the exit of bundle for the case with 1562 kg/(m²s) for different bundle power defined by equilibrium exit quality 0.05, 0.12 and 0.18, respectively.
Cross section averaged void,-
0,9 0,8 Exp. 0,7 IVA 0,6 0,5 0,4 0,3 0,2 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality,-
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
0,9 0,8 Exp. 0,7 IVA 0,6 0,5 0,4 0,3 0,2 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
121
(a)
(b) Fig. 5.13. NUPEC experiment with mass flow rate (a) 1562 kg/(m²s); (b) 852kg/(m²s). Void fraction as a function of the height
Low burn up bundle: The definitions of the five regions are given in Fig. 5.14(a). The measured region-averaged void fractions at the exit of the bundle are presented in Fig. 5.14(b). The computed void fractions are also presented for comparison in Fig. 5.14(b). High burn up bundle: The definitions of the four regions are given in Fig. 5.15(a). The measured region-averaged void fractions at the exit of the bundle are presented
122
5. Core thermal hydraulic
in Fig. 5.15(b). The computed void fractions are also presented for comparison in Fig. 5.15(b).
Region-averged void, -
0,8 0,7 0,6 0,5 0,4 X1,exit=0.05, Exp. X1,exit=0.05, IVA X1,exit=0.12, Exp. X1,exit=0.12, IVA X1,exit=0.18, Exp. X1,exit=0.18, IVA
0,3 0,2 0,1 0,0
1
2
4
5
3 4 Region Nr
5
Region averaged void, -
0,8 0,7 0,6 0,5 0,4 X1eq=0.05 exp. X1eq=0.05 IVA X1eq=0.12 exp. X1eq=0.12 IVA X1eq=0.18 exp. X1eq=0.18 IVA
0,3 0,2 0,1 0,0
1
2
3 Region Nr.
Fig. 5.14. (a) Definition of the averaging regions, Inoue et al. (1995a): PERI = 1, INR1 = 2, INR2 = 3, INR3 = 4, CNTR = 5; Region-averaged void fractions at the exit of the bundle for different bundle power corresponding to constant conditions but specified exit equilibrium quality: Resolution: (b) 8x8x24; (c) 20x20x24
For both bundles the prediction accuracy is very similar. Low burn up bundle (Fig. 5.14(b)): We see “good” agreement for the three external regions. For the central region the prediction with such gross discretization under predicts the void fraction. Comparing the cases with smaller and larger resolution, (b) and (c) in Fig. 5.13, we realize that the comparison with the data is “better” for the gross resolution. This is an indication that the void mixing computed only using the transport equations without turbulence modeling is not enough to describe appropriately this process. For low power at the high burn up bundle the computed results are higher than the measured. To my view, the reason for the discrepancy is partially in the low resolution of the sub-channel analysis. As already mention increasing the resolution alone without turbulence modeling will not improve the result. As I will show later,
5.2. Steady state flow in heated rod bundles
123
Region averaged void,-
the use of the Hughes et al. (1981) correlation for splitting of the heat fluxes instead of Saha-Zuber (1974), Levy (1967) improves prediction accuracy in the low void region, but the resolution problem will remain. 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0
Exp. X1,eq.=0.05 IVA Exp. X1,eq.=0.12 IVA Exp. X1,eq.=0.18 IVA
1
2 3 Region Nr
4
Fig. 5.15. (a) Definition of the averaging regions, Inoue et al. (1995): PERI = 1, INR1 = 2, INR2 = 3, INR3 = 4; (b) Region-averaged void fractions at the exit of the bundle for different bundle power corresponding to constant conditions but specified exit equilibrium quality
50 mm
(b) Scanning section
(a) Measuring structure (c) Scanning method
Fig. 5.16. Void fraction measurement system
Local exit void fractions at different power level: Prediction for 15 additional proprietary NUPEC experiments in a framework of an OECD/USNRC benchmark, Kolev (2005a), are compared with the outlet void fractions based on sub-channels.
124
5. Core thermal hydraulic
The 1365 experimental data are available at the exit of the bundle as a cross section averaged void fraction and as a 9 × 9 local sub-channel data, see Figs. 5.16 and 5.17.
(a) Pixel void fraction (b) Sub-channel void fraction Fig. 5.17. Spatial resolution: (a) of the experimentally measured void fractions; (b) of the region averaged void fraction subject of comparison with the predicted void fractions
The geometry of the bundles is presented on Fig. 5.18. Comparison between the computed and measured exit void fractions are presented in Tables 5.5 through 5.12 for two different models especially for the sub-cooled boiling. Tables 5.5–5.8 contain the comparison for data predicted using the Saha and Zuber (1974) correlation for the initiation of the sub-cooled boiling and the Levy (1967) asymptotic model for the splitting of the heat into convection evaporation and condensation. Tables 5.9–5.12 contain the comparison for data predicted using the Hughes et al. (1981) approach for the initiation of the sub-cooled boiling and for the splitting of the heat into convection evaporation and condensation. Table 5.5. Cross section averaged void fractions computed by IVA and measured by NUPEC. IVA heat partitioning models: Saha-Zuber (1974) and Levy (1967) Test No. z in m 3.758
0011, assembly 0-1
Exp. IVA Dev.%
0021, assembly 0-2
0031, assembly 0-3
55
58
61
16
18
21
16
18
21
0.409 0.461 +12.7
0.630 0.667 +5.9
0.794 0.797 +0.38
0.394 0.450 +14.2
0.626 0.655 +4.6
0.783 0.791 +1.0
0.393 0.447 +13.7
0.623 0.638 +2.4
0.783 0.778 +0.6
Table 5.6. Cross section averaged void fractions computed by IVA and measured by NUPEC. IVA heat partitioning models: Saha-Zuber (1974) and Levy (1967) Test No. z in m 3.758
1071, assembly 1
Exp. IVA Dev.%
55 0.430 0.453 +5.3
58 0.637 0.654 +2.7
61 0.791 0.799 +1.0
4101, assembly 4 55 0.438 0.467 +6.6
58 0.645 0.670 +3.9
61 0.807 0.803 -0.5
5.2. Steady state flow in heated rod bundles
125
Table 5.7. Local void fraction mean error at the exit of the bundle. IVA heat parti-
tioning models: Saha-Zuber (1974) and Levy (1967) Test No. Mean error%
0011, assembly 0-1 55 6.43
58 5.48
0021, assembly 0-2
61 4.61
16 7.22
18 6.31
0031, assembly 0-3
21 5.00
16 7.20
18 7.22
21 6.66
Table 5.8. Local void fraction mean error at the exit of the bundle. IVA heat parti-
tioning models: Saha-Zuber (1974) and Levy (1967) Test No. Mean error%
1071, assembly 1 55 5.38
58 4.57
4101, assembly 4 61 4.22
55 4.77
58 4.71
61 4.49
Table 5.9. Cross section averaged void fractions computed by IVA and measured by NUPEC. IVA heat partitioning models: Hughes et al. (1981) Test No. z in m 3.758
0011, assembly 0-1
Exp. IVA Dev.%
0021, assembly 0-2
0031, assembly 0-3
55
58
61
16
18
21
16
18
21
0.409 0.426 4.16
0.630 0.637 1.11
0.794 0.782 –1.51
0.394 0.415 5.33
0.626 0.627 0.16
0.783 0.777 –0.77
0.393 0.414 5.34
0.623 0.610 –2.09
0.783 0.765 –2.30
Tables 5.7 and 5.8 contain the square root mean error for the 9 × 9 sub-channel data. The total mean error is 5.62%. The data structure is presented on Fig. 5.19 where the measured void local sub-channel fractions are presented versus the computed for all 1365 data points. The spread is within +20 and –25% with the main bulk of the data grouped around the diagonal. Table 5.10. Cross section averaged void fractions computed by IVA and measured
by NUPEC. IVA heat partitioning models: Hughes et al. (1981) Test No. z in m 3.758
1071, assembly 1
Exp. IVA Dev.%
55 0.430 0.415 –3.49
58 0.637 0.629 –1.26
61 0.791 0.786 –0.63
4101, assembly 4 55 0.438 0.428 –2.28
58 0.645 0.639 –0.93
61 0.807 0.788 –2.35
Table 5.11. Local void fraction mean error at the exit of the bundle. IVA heat parti-
tioning models: Hughes et al. (1981) Test No. Mean error%
0011, assembly 0-1 55 4.97
58 4.70
61 4.53
0021, assembly 0-2 16 5.33
18 5.82
21 4.81
0031, assembly 0-3 16 5.66
18 6.87
21 6.47
126
5. Core thermal hydraulic
Assembly: 0-1, 1, 3
0-2
Assembly: 0-3
4
Fig. 5.18. IVA discretization models of the four bundles
Void fraction meas., -
100
exp./IVA +20% -25%
80 60 40 20 0
0
20
40 60 80 Void fraction comp.,-
100
Fig. 5.19. Computed versus measured void fraction at the 91 sub-channels at the exit of the bundle. Models for sub-cooled boiling: Saha and Zuber (1974) correlation for the initiation of the sub-cooled boiling and the Levy (1967) asymptotic model for the splitting of the heat into convection evaporation and condensation. The total mean error is 5.62%
5.2. Steady state flow in heated rod bundles
Void fraction meas., -
100
127
exp./IVA +20% -25%
80 60 40 20 0
0
20
40 60 80 Void fraction comp.,-
100
Fig. 5.20. Computed versus measured void fraction at the 91 sub-channels at the exit of the bundle. Models for subcooled boiling: Hughes et al. (1981) approach for the initiation of the subcooled boiling and for the splitting of the heat into convection evaporation and recondensation. The total mean error is 5.15%
Table 5.12. Local mean error of the void fraction at the exit of the bundle. IVA heat partitioning models: Hughes et al. (1981) Test No. Mean error%
1071, assembly 1 55 5.36
58 4.45
4101, assembly 4 61 3.99
55 4.55
58 4.57
61 5.21
Tables 5.9 and 5.10 contain the square root mean error for the 9 × 9 sub-channel data. The total mean error is 5.15%. The data structure is presented on Fig. 5.20 where the measured void local sub-channel fractions are presented versus the computed for all 1365 data points. The spread is within +20 and –25% with the main bulk of the data grouped around the diagonal. I recently recomputed again the discussed experiment with the last version of IVA. The total mean error is 5.07%, see Fig. 5.21. Having in mind that the reported experimental uncertainty is 3% the prediction without any turbulent mixing can be considered as very good. Comparison of the prediction of this data set with several computer codes was presented by Neykov et al. (2006) with the result given in Table 5.13 for which the standard deviation computed as follows is used
∑ (α K
σ=
k =1
exp, k
− α pred , k )
( K − 1)
2
.
128
5. Core thermal hydraulic
Void fraction meas., -
100
exp./IVA +20% -25%
80 60 40 20 0
0
20
40 60 80 Void fraction comp.,-
100
Fig. 5.21. As in Fig. 5.13: latest version of IVA. The total mean error is 5.07% Table 5.13. Standard deviation for different computer codes reported in Neykov et al. (2006) 1 2 3 01 0.18 1.7 1.2 02 0.09 1.67 0.89 03 0.39 2.01 1.25 1 1.08 0.76 0.28 4 2.11 1.02 1.09 1. MATRA (KAERI) 2. MARS (KAERI) 3. MONA (KTH) 4. CAPE Mod 1.0 (NUPEC) 5. TwoPorFlow (FZK) 6. COBRA-TF (PSU) 7. F-COBRA-TF (AREVA) 8. IVA (AREVA) 9. NEPTUNE -FLICA4 (CEA) 10. NASCA (TEPCO)
4 1.44 1.44 1.24 0.0 0.88
5 1.55 1.98
4.45
6 2.13 4.98
7 0.94 4.21 4.49 3.18 2.01
8 0.63 0.12 0.3 1.76 1.58
9 2.36 2.15 2.13 1.1 0.37
10 1.86 1.14 1.4 1 0.84
Conclusions: •
The cross section averaged void fractions predicted at the exit of the bundles with IVA depends on the used modeling approach for the sub-cooled boiling available in IVA: (a) Saha and Zuber (1974) correlation for the initiation of the sub-cooled boiling and the Levy (1967) asymptotic model for the splitting of the heat into convection evaporation and condensation; (b) Hughes et al. (1981) approach for the initiation of the sub-cooled
5.2. Steady state flow in heated rod bundles
• • •
•
• •
129
boiling and for the splitting of the heat into convection evaporation and condensation. By using the model (a) the predicted data agree excellent at high power (1%), agree well at averaged power (5.9%) and are up to 14.2% lower at 5% of the nominal power. By using the model (b) the predicted data agree excellent at high power (2.3%) and well at averaged power (2.9%) and at 5% of the nominal power (5.34%). Obviously model (b) is superior at low void fractions. The local void fractions at the exit of the bundle on a 9 × 9 sub-channel basis are predicted with an error of 5.62% using model (a) and of 5.15% using model (b). The local accuracy increases with the increasing power. Again model (b) contributes to better performance for all void fractions. As already reported in Kolev (2005b, c) comparing the cases with smaller (10 × 10 × 24) and larger (18 × 18 × 24) resolution I obtain “better” agreement with the data for the gross resolution. This is an indication that the void mixing computed using only on the transport equations is not enough to describe appropriately this process. This is confirmed by this study too. In all cases the “void diffusion” from region with higher void to region with lower void is underestimated. Without appropriate turbulence modeling the accuracy of this method regarding predicting the local void fraction can not be increased. Fine resolution is required in the future accomplished with appropriate constitutive relationships specially developed for fine resolution.
Comparison with Utsuno et al. (2004) analyses: Utsuno et al. (2004) reported analyses of the steady state void data of NUPEC. The authors used the code TCAPE. The TCAPE code uses the drift-flux formulation of all axial equations. The cross flow is modeled by algebraic models that do not satisfy lateral momentum conservation. The simplified energy conservation equation used is appropriate for slow transients only. In case of the film flow additional one-dimensional mass balance is introduced using entrainment and deposition sources. Therefore, convective transport of film and droplets across the sub-channel is not permitted in the model. The TCAPE code does not made use of dynamic fragmentation and coalescence. Even with this simplification the code predicts “reasonably” a gross scale void distribution by using flow-channel centered sub-channels discretization. As already mentioned, the apparently demonstrated good void intermixing is due to numerical diffusion and not due to appropriate turbulence description. 3D versus 1D analysis: Consider the moisture analyses of a complete boiling water reactor as reported by Kolev (2002b) for the geometry given in Fig. 5.22. The question whether the core has to be simulated in a pin-by-pin or bundle-by-bundle approach is important because the pin-by-pin resolution is still expensive. To illustrate that bundle-by-bundles analysis is accurate enough for the moisture optimization analyses I simulate one of the NUPEC experiments as presented in Fig. 5.23
130
5. Core thermal hydraulic
by using the characteristic thermo-hydraulic parameters for the overall rod bundle and compare the results with the 3D representation. As seen from Fig. 5.23 the cross section averaged void profiles are almost not distinguishable.
Fig. 5.22. (a) Typical boiling water reactor: 1. reactor pressure vessel; 2. reactor core; 3. steam-water separators (cyclones); 4. steam dryer; 5. control rod drives; 6. control assemblies; 7. feed water inlet nozzles; 8. core spray line; 9. main steam outlet nozzle; 10. forced circulation pumps; 11. annular down comer. (b) IVA 1/4th geometry model of the control rod space, core, upper plenum and stand pipes
Increasing the resolution to 20 × 20 horizontal cells gives slightly higher cross section averaged void fraction as demonstrated in Fig. 5.23(b). This allows the conclusion: For analyses of the moisture in BWR’s the core representation in a bundleby-bundle approach provides the appropriate large scale three dimensional void distributions at the exit of the core.
Cross section averaged void, -
0,9 0,8 0,7 0,6 0,5 0,4 Exp. 0,3 IVA 3D: 8x8x24 0,2 IVA 1D: 1x1x24 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
0,9 0,8 0,7 0,6 0,5 exp. 0,4 Resolution 0,3 20x20x24 cells 0,2 8x8x24 cells 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 Equilibrium quality, -
131
(a)
(b) Fig. 5.23. NUPEC experiment with mass flow rate 1562 kg/(m²s): Comparison between cross section averaged void fraction predictions using (a) 3D (rod centered channels) and 1D analysis; (b) 3D rod centered channels 8x8x24 cells and 3D separate periphery layers, one rod belongs to 4 cells, 20 × 20 × 24 cells
For approximating the bundle as a single channel for moisture analysis the averaged pressure drop coefficient for the bundle can be computed using the method proposed and verified on large data base by Rehme (1973, 1971, 1972). For the laminar regime the friction coefficient is λ fr Re = Rm , where the Rehme’s number (Rehme, 1973, 1974) is Rm = 63.172. For the turbulent regime the friction factor in accordance with Rehme (1972) is
(
)
8 λ fr = 2.55ln Re 8 λ fr − 0.255 . Here
& h ( Fη ) is based on the total mass flow, tothe bundle Reynolds number Re = mD tal flow cross section and on the effective bundle hydraulic diameter.
132
5. Core thermal hydraulic
120 100 80 60 40 20 0 -20 -40 -60 -80 -100 -120 50
(flow in - flow out)/flow in in %
(flow in - flow out)/flow in in %
The oscillating steady state for low mass fluxes: For large boiling mass flows through the bundle channels the steady state is non-oscillating. For low mass flows, however, the steady state is oscillating. As reported by Yagi et al. (1992) and Inoue et al. (1995c), oscillating void measurements are smoothed so, that the data points are not only cross section averages but also time averages. An example for stable but oscillating steady state solution is given in Figs. 5.24(a) and (b) for the NUPEC experiment with mass flow rate 284 kg/(m²s). Figure 5.25 presents the cross section averaged void fractions at different times within one cycle of oscillation. As we see they oscillate around the reported time averaged values. This is an important advantage of looking for steady state solutions as asymptotic solutions of transients. Important oscillating characteristics can be then recognized, something that is impossible by solving only the steady state part of the systems of PDE’s.
IVA, steady state 60
70 80 Time in s
90
100
120 100 80 60 40 20 0 -20 -40 -60 -80 -100 -120
IVA, steady state 101
102 Time in s
103
104
Cross section averaged void, -
Fig. 5.24. Oscillating steady state for 284 kg/(m²s) case: (a) long term steady state oscillations; (b) one oscillation cycle
0,9 0,8 0,7 0,6 0,5 Exp. 0,4 IVA, oscilating steady state in flow = out flow 0,3 in flow < out flow in flow = out flow 0,2 in flow > out flow in flow = out flow 0,1 0,0 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 z in m
Fig. 5.25. NUPEC experiment with mass flow rate 284 kg/(m²s). Void fraction as a function of the height. Parameter – time within one oscillation period
5.2. Steady state flow in heated rod bundles
133
5.2.2 The SIEMENS void data for the ATRIUM 10 fuel bundle A set of proprietary data for void fraction in ATRIUM 10XP bundles at 4 peaking patterns was obtained by Spierling (2002) at the Karlstein Thermal Hydraulic Loop in AREVA NP, Germany. At about 70 bar 12 experiments with 2.2 m/s inlet velocity, 58 kJ/kg sub-cooling and axial choked cosine power profile beam line void profiles 3.24 m from the inlet are collected and then averaged over the cross section except the water channel. Four of the experiments are at low power and 8 at high power. All the 12 test sets are computed using IVA computer code. 10 × 10 × 37 3D-discretization (10 × 10 fuel centered sub-channels with 37 axial cells) is applied. The structure of the errors is documented in Table 5.14. The maximum deviation of the predicted from the measured void was less than 8.7 and 7.55% for set (a) and (b), respectively. The averaged deviation is about 6.12 and 4.19% for set (a) and (b), respectively. Obviously, again set (b) give better performance. In summary, the void prediction for AREVA NP bundle test is with uncertainty of 4.19% is even better the prediction of the previously reported NUPEC data. Table 5.14. Measured and predicted cross section averaged void fractions at 3.24 m
from the inlet Exp no.
Lateral peaking pattern STS-66.1
Void Void Deviation Void Deviation exp. IVA, (a)* in % IVA, (b)* in % 257 0.519 0.524 0.96 0.482 –7.13 258 0.813 0.842 3.57 0.827 1.72 259 0.819 0.843 2.93 0.828 1.10 270 STS-68.3 0.468 0.519 10.2 0.476 1.71 271 0.784 0.840 7.14 0.827 5.48 272 0.779 0.840 7.83 0.828 6.29 328 STS-52.1 0.543 0.499 –8.10 0.502 –7.55 332 0.804 0.816 1.49 0.798 –0.75 339 0.839 0.861 2.62 0.843 0.48 261 STS-68.5 0.479 0.529 10.6 0.500 4.38 262 0.774 0.839 8.4 0.826 6.72 263 0.771 0.838 8.7 0.825 7.00 * Models for sub-cooled boiling: (a) Saha and Zuber (1974) correlation for the initiation of the subcooled boiling and the Levy (1967) asymptotic model for the splitting of the heat into convection evaporation and condensation; (b) Hughes et al. approach (1981) for the initiation of the sub-cooled boiling and for the splitting of the heat into convection evaporation and condensation.
5.2.3. The FRIGG experiment The FRIGG experiments are reported by Nylund et al. (1968, 1970). The geometry given in Fig. 5.26(b) is defined as follows: 36 rods with 13.8 mm diameter, heated, 1 central rod with 20 mm diameter, unheated. The heated length of the rods is 4.375 m.
134
5. Core thermal hydraulic
(a)
(b) Fig. 5.26. (a) Layout of the FRIGG loop; (b) Cross section of the FRIGG bundle
The rods are placed within a cylinder with internal diameter 159.5 mm and positioned in three circles with 21.6, 41.7, and 62.2 mm radius, respectively. The circles contain 6, 12 and 18 rods, respectively, at equal arc distance from axis to axis along the circle. The bundle is 1/6-symmetric. Eight spacers keep the rods parallel
5.2. Steady state flow in heated rod bundles
135
to each other. The spacers have an irreversible pressure drop coefficient equal to 0.6 based on the bundle velocity. Discretization used for the IVA computation: The flow in the bundle is considered as 3D axis symmetric flow in cylindrical coordinates. 1/6 segment is simulated with three radial annuli defined by the radii 0.01, 0.03165, 0.05195 and 0.07975 m, respectively. The height is discretized using 36 cells. Eight spacers are modeled by local irreversible pressure loss coefficient. Uniform power distribution: We proceed with simulation of 5 experiments for which boundary conditions are defined in Table 5.15. The lateral and axial power distribution is uniform. The results are presented in Figs. 5.27 and 5.28. Table 5.15. Boundary conditions, uniform power release FRIGG Nylund et al. (1968) No. 313009 313016 313018 313024
p in MPa 5 4.96 4.97 4.97
Power in MW 2.98 2.91 4.39 1.475
G in kg/(m²s) 1107 1208 1124 858
Subc. in K 4.4 19.3 3.7 4.2
Fig. 5.27. 1/6 of the FRIGG test section simulated by IVA (3 × 1 × 26 cells). Void fraction at different levels (1, 2, 3, 4.3142 m) as a function of the radius
As for the NUPECC data the effect of the non uniformity of the flow profiles inside the sub-channels is manifested in effectively lower drag between the liquid and the vapor. The comparison at the predictions with the data using the reduced effective drag as in the previous section is favorable.
Cross section averaged void, -
0,8
Cross section averaged void, -
5. Core thermal hydraulic
0,9 0,8 0,7 0,6 0,5 Exp. 0,4 IVA, local drag 0,3 non modified 0,2 modified 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in m
Cross section averaged void, -
136
0,9 0,8 0,7 0,6 0,5 Exp. 0,4 IVA, local drag 0,3 non modified 0,2 modified 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 Equilibrium quality, -
0,7 0,6 0,5 0,4 0,3 0,2 0,1
Exp. IVA, local drag non modified modified
0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in m
(a)
(b)
(c)
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
137
0,7 0,6 0,5 0,4 Exp. IVA, local drag non modified modified
0,3 0,2 0,1
0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in m
(d) Fig. 5.28. FRIGG experiments: (a) 3130009; (b) 3130016; (c) 3130018; (d) 3130024: Void fraction as a function of the height. Parameter: local interfacial drag
Nonlinear axial power profile: I continue the analysis with simulation of other 4 experiments which boundary conditions are defined in Table 5.16. The lateral power distribution is uniform. Smooth symmetric axial power distribution is defined by Table 5.17. I approximate the axial profile with 3% error with the Gauss function ⎧⎪ ⎡ 2 ( z − 2.812 )2 ⎤ ⎫⎪ f z = 0.62914 + 1.74672 ⎨2.47575 2 π exp ⎢ ⎥⎬ . 2 ⎢⎣ 2.47575 ⎦⎥ ⎪⎭ ⎪⎩
(5.1)
After digitizing the profile renormalization was necessary to guaranty the energy conservation.
Table 5.16. Boundary conditions, none uniform power release, FRIGG Nylund
et al. (1970) No.
p in MPa
1 10 13 19
4.88 4.87 4.99 5.03
Power in MW 1.66 5.55 5.55 4.52
G in kg/(m²s) 703 645 688 681
Subc. in K 1.5 2 22.89 26.11
Cross section averaged void, -
5. Core thermal hydraulic
1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0
Exp. IVA
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
z in m
Cross section averaged void, -
(a) 1,0 0,9 0,8 0,7 0,6 0,5 Exp. 0,4 IVA 0,3 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in z
Cross section averaged void, -
138
1,0 0,9 0,8 0,7 Exp. 0,6 IVA 0,5 0,4 0,3 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in m
(b)
(c)
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
139
1,0 0,9 0,8 0,7 0,6 Exp. 0,5 IVA 0,4 0,3 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 z in m
(d) Fig. 5.29. FRIGG experiments: (a) 1; (b) 10; (c) 13; (d) 19: Void fraction as a function of the height. The modified drag coefficient for gas in flow is used
Table 5.17. Axial power distribution z in m 0.032 0.462 0.892 1.322 1.752 2.182 2.612 3.042 3.472 3.902 4.332
Ax. distr. 0.76 0.83 0.96 1.08 1.16 1.18 1.16 1.08 0.96 0.83 0.76
The results are presented in Figs. 5.29 (a) through (d). Again we see that the predictions coincide well with the measurements. Although the comparison with the data is favorable, a physically based model for the mechanical interaction for subchannel analyses is needed. 5.2.4. The THTF experiments: high pressure and low mass flow The experiments performed in the Thermal Hydraulic Test Facility (THTF) by Anklama and Miller (1982) are characterized with such a low mass flows which in combination with appropriate low heat fluxes lead to steady states in which the bundle is partially uncovered. The two phase mixture level given in Table 5.18 is the essential parameter measured in these experiments. This is very challenging problem for transient mathematical analysis because it possesses oscillating steady
140
5. Core thermal hydraulic
state. The 8 × 8 rod bundle with quadratic arrangement is mounted in a 0.104 × 0.104 m shroud as shown in Fig. 5.30. The heated length was 3.66 m, the rod diameter 0.095 m and the axis to axis distance is 0.0127 m. Four unheated rods with diameter 0.0102 m as indicated in Fig. 5.30 are used. The boundary conditions for the experiments analyzed are given in Table 5.18. The void fraction was intended to be measured by pressure difference along a given number of segments. This method works for real steady state processes. It is questionable for oscillating steady state because the acceleration pressure drop components are not taken into account.
Fig. 5.30. Thermal hydraulic test facility rod bundle cross section, Anklama and Miller
Cross section averaged void, -
(1982)
1,0 0,9 Exp. 0,8 IVA 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 z in m 3.09.10DD
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
1,0 0,8
Exp. IVA
0,6 0,4 0,2 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
z in m
Cross section averaged void, -
3.09.10EE
1,0 0,8 0,6 0,4
Exp. IVA
0,2 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 z in m 3.09.10FF
141
Cross section averaged void, -
5. Core thermal hydraulic
1,0 0,9 0,8 0,7 0,6 0,5 0,4 Exp. 0,3 IVA 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 z in m 3.09.10I
Cross section averaged void, -
142
1,0 0,8
Exp. IVA
0,6 0,4 0,2 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 z in m 3.09.10J
Cross section averaged void, -
5.2. Steady state flow in heated rod bundles
143
1,0 0,8
Exp. IVA
0,6 0,4 0,2
0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 z in m 3.09.10K
Fig. 5.31. Snap shots of oscillating cross section averaged void fraction profiles in a bundle with partial uncover of the bundle
Table 5.18. Boundary conditions, uniform power release, THTF Anklama and Miller
(1982) No. 3.09.10DD 3.09.10EE 3.09.10FF 3.09.10I 3.09.10J 3.09.10K
p in MPa 8.1 7.7 7.5 4.5 4.2 4.0
Power in MW 0.28329 0.14055 0.07027 0.48753 0.23498 0. 7027
G in kg/(m²s) 19.7 10.9 4.8 29.3 12.9 3.1
Subc. in K 129.7 102.4 116.13 56 45.9 60.1
Mixture level in m 3.23 ± 0.04 3.47 ± 0.03 3.23 ± 0.04 2.62 ± 0.04 2.47 ± 0.4 2.13 ± 0.3
Comparing the computed with the measured two-phase mixture level presented on Fig. 5.31 we see an acceptable agreement. This demonstrates also the IVA constitutive set to cross variety of flow and heat transfer regimes with their peculiarities. Conclusions: Without special void intermixing models the prediction of the void fraction of boiling flows in bundles with IVA computer code is possible with a square mean error of about 5%. The measurement error for such experiments is 3%.
144
5. Core thermal hydraulic
The production of turbulence due to increased skin friction is much smaller then the irreversible form-induced turbulence. However, the dissipation due to skin-friction is substantial. The decay is well within the 30 hydraulic diameters as expected. The recommended approach for large scale analyses is to combine both effects: Changing of the hydraulic diameter and the cross section over the grid length and setting the irreversible friction coefficient at the end of the grid reduced by the additional grid skin friction component. In this way the effect of the increased velocity on droplet fragmentation can be taken into account. In general using correlation for the interfacial drag coefficients derived from averaged pipe flows predicts to strong cohesion between the phases. This was confirmed by all analysts using separated momentum equations worldwide. As already mentioned, the ad hoc introduced reduction here has to be further investigated in the future. Future improvements are probably possible by introducing turbulence in the boiling multiphase flow. An example is given in Sect. 5.7. The method proposed by Hughes et al. (1981) is recommended for description of the initiation of nucleate boiling in sub-cooled liquid and for the modeling of the splitting of the heat flux coming from the wall. The so called sub-channel analysis or gross discretization analysis is discretization dependent. Therefore, fine resolution analysis is required in the future to avoid this deficiency. For this purpose appropriate constitutive relationships have to be specially developed.
5.3. Pressure drop for boiling flow in bundles The check of the pressure drop in boiling bundles will be short discussed in this section. First I use the data sets for experiments with the 8 × 8 NUPEC BWR bundle tests (Fig. 5.18) as proposed in the international benchmark by Neykov et al. (2005). The single phase friction coefficient for the specific ferule grid spacer is fitted to the data with a mean error of 1%. The predicted versus the computed friction pressure drop only is given in Fig. 5.32.
5.3. Pressure drop for boiling flow in bundles
145
40 DP measured in kPa
35
NUPEC 1Ph.
30 25 20 15 10 5 0 0
5
10 15 20 25 30 DP computed in kPa
35
40
Fig. 5.32. Measured versus computed friction pressure drop (wall friction + 7 grids). Mean error 1% 160 Friedel 1979 NUPEC 2Ph +10% -10%
DP measured in kPa
140 120 100 80 60 40 20 0 0
20
40 60 80 100 120 140 160 DP computed in kPa
Fig. 5.33. Measured versus computed pressure drop for boiling channels. Two-phase friction multiplier by Friedel (1979). Mean error 7.38%
In the next step I analyze the two-phase flow data. I use the already obtained correlation for the irreversible pressure loss coefficient at the grid and two phase flow multiplier proposed by Friedel (1979). We see from Fig. 5.33 that the Friedel’s correlation predicts the data with mean error 7.38% and from Fig. 5.34 that the Baroczy correlation from 1965 modified by Chisholm (1983) predict the data with mean error 11.8%.
146
5. Core thermal hydraulic
160 Baroczy-Chiskolm (1983) NUPEC 2Ph +10% -10%
DP measured in kPa
140 120 100 80 60 40 20 0 0
20
40 60 80 100 120 140 160 DP computed in kPa
Fig. 5.34. Measured versus computed pressure drop for boiling channels, 33 experiments. Two-phase friction multiplier by Baroczy-Chiskolm (1983), mean error 11.8%
Pressure drop-measured, bar
Next I analyze 96 data points for the AREVA bundles with the FOCUS grid spacers. Again the single phase irreversible pressure loss coefficients of the spacers are fitted with accuracy of 1%. Then the total pressure loss for boiling flow is computed. The Friedel 1979 correlation for the two phase multiplier is used. The mean error of 9% is found, see Fig. 5.35.
2,5 2,0
AREVA PWR 5x5 bundle with FOCUS grid: Flow boiling with CHF Pressure drop over 2.725m from the entrance +10% -10%
1,5 1,0 0,5 0,0 0,0
0,5 1,0 1,5 2,0 2,5 Pressure drop-computed, bar
Fig. 5.35. Measured versus computed total pressure drop in a 5 × 5 bundle with FOCUS grid. 96 experiments
Conclusion: Note that the Friedel’s correlation approximates 25 000 experimental points for vertical upwards co-current flow and for horizontal flow with 30–40% standard deviation for one- and two-component flow. Therefore better accuracy
5.4. Transient boiling
147
can not be expected for this particular comparison. The two phase total pressure drop in bundles with boiling flows is best predicted by the Friedel’s correlation with a mean error between 7.38 and 9%. So I recommend the use of Friedel’s correlation for such cases.
5.4. Transient boiling 5.4.1 The NUPEC transients in a channel simulating one sub-channel of a PWR fuel assembly Hori et al. (1994, 1995) reported a series of transient experiments on a specially designed 1.5 m long heated channel – Fig. 5.36 a. The channel, Fig. 2 in Hori et al. (1995), simulates a single internal sub-channel of a PWR fuel assembly. The initial conditions for the transients are defined in Table 5.19. Only one of the four parameters in Table 5.19 is varied in each experiment as shown in the left hand side of the Figs. 5.37 through 5.40. The others are kept constant. Table 5.19. Initial conditions for the four transients, taken from Table 1 of Hori et al. (1994) No. Case 1 Case 2 Case 3 Case 4
Power in KW 55 55 55 75
Inlet temperature in °C 315 315 315 305
(a)
Pressure in MPa 15.5 15.5 15.5 15.5
Mass flow rate in ×106 kg/(m²h) 12 12 12 12
(b)
Fig. 5.36. (a) Cross section of the channel simulating one sub-channel of a PWR fuel assembly; (b) 5 × 5 experimental fuel assembly
On the right of the corresponding figures the computed and the measured mixture densities are compared as they evolve during the transients. The set of constitutive relations used is: high fidelity equation of state, models for sub-cooled boiling: Hughes et al. approach (1981) for the initiation of the sub-cooled boiling and for the splitting of the heat into convection evaporation and condensation. Note that only the temporal part of the virtual mass force is used in this code version. Changing the virtual mass coefficient from ½ for bubbly flow to the Zuber’s
148
5. Core thermal hydraulic
solution does not change this picture. The figures indicate that probably the spatial part is also important for transients like these.
5,0 4,5 winlet in m/s
4,0
case 1
3,5 3,0 2,5 2,0 1,5
26
28
30 32 34 Time in s
36
38
40
Mixture density, kg/m³
700 Exp. IVA, high fid. EOS
600 500 400 300 200
28
30
32 34 36 Time in s
38
40
Fig. 5.37. Case 1: (a) Velocity at the channel inlet as a function of time; (b) Measured and computed mixture density at the exit of the channel as a function of time
5.4. Transient boiling
149
5,0 4,5 winlet in m/s
4,0
case 2
3,5 3,0 2,5 2,0 1,5
16
18
20 22 24 Time in s
26
28
30
Mixture density, kg/m³
700 Exp. IVA high fid. EOS
600 500 400 300 200
16
18
20
22 24 Time in s
26
28
30
Fig. 5.38. Case 2: (a) Velocity at the channel inlet as a function of time; (b) Measured and computed mixture density at the exit of the channel as a function of time
150
5. Core thermal hydraulic
160 150 Exp. p in bar
140 130 120 110 100
0
50
100 Time in s
150
200
Mixture density, kg/m³
700 600
Exp. IVA high fid. EOS
500 400 300 50
100 Time in s
150
200
Fig. 5.39. Case 3: (a) Averaged pressure as a function of time; (b) Measured and computed mixture density at the exit of the channel as a function of time
5.4. Transient boiling
151
620 610
T in K
600 Exp. 590 580 570
0
50
100 Time in s
150
200
Mixture density, kg/m³
700 Exp. IVA high fid. EOS
600 500 400 300 200 0
50
100 Time in s
150
200
Fig. 5.40. Case 4: (a) Inlet water temperature as a function of time; (b) Measured and computed mixture density at the exit of the channel as a function of time
The pressure reduction in Fig. 5.39 causes evaporation and decrease of the mixture density. In this case we have some uncertainties because we actually need the pressure histories at the both ends. Fig. 5.39 (b) gives the impression that the experimental pressure and density curves are not synchronized. Similar is the impression gained from the inlet-temperature-transient presented in Fig. 5.40 (b). In any case the agreement can be considered as good having in mind that I do not resolve the fine structure of the two phase flow in the channel. The modification of
152
5. Core thermal hydraulic
the drag coefficients that has brought the improvement by the simulation of the steady state experiments in rod bundles has brought also an improvement for prediction of the transient processes. In any case we see again that the effective gas-liquid drag coefficients for rod-bundle-sub-channel analyses are smaller than those correlated for averaged parameters on pipe geometry.
5.4.2 The NUPEC transients in PWR 5
5 fuel assembly
Hori et al. (1996, 1993) and Akiyama et al. (1995, 2005) reported series of transient experiments on a 3.66 m long heated 5 × 5 rod bundle, Fig. 5.36(b), simulating a Japanese 17 × 17 PWR fuel assembly. The initial conditions for the transients are defined in Table 5.20. The axial power distribution is uniform. As reported by Hori et al. (1996, p. 802), the peripheral rods have 85% of the power of the internal rods. Averaged hydraulic characteristics of the bundle: for the laminar regime the friction coefficient is λ fr Re = 61 in accordance with Rehme’s (1973, 1971), for the turbulent regime the friction factor in accordance with Rehme (1972) is
(
)
8 λ fr = 2.5ln Re 8 λ fr − 0.17 . I selected from the 4 experiments only two
because the variable boundary conditions are specified only for them; see Fig. 8 and Fig. 9 of Akiyama et al. (1995). One of the four parameters in Table 5.20 is varied in every experiment as shown in the left hand side of the Figs. 5.41 and 5.42. The others are kept constant. Table 5.20. Initial conditions for the NUPEC 5 × 5 bundle transients, taken from Appendix 1 of Hori et al. (1996, p. 810) Variable power flow rate
Power in KW 2250 2250
Inlet temperature in °C 300 300
Pressure in MPa 15.5 15.5
Mass flow rate in ×106 kg/(m²h) 12 12
5.4. Transient boiling
153
Bundle power in MW
3,2 3,0 2,8 2,6
Exp.
2,4 2,2 30
0,5
Void, -
0,4 0,3 0,2
32
34 36 Time in s
38
34 36 Time in s
38
Exp. da mid up IVA central ch. da mid up IVA cordial av. da mid up
0,1 0,0 30
32
Fig. 5.41. (a) Bundle power as a function of time; (b) NUPEC measurements and IVA results (central sub-channel and central beam line average) for the void fraction at three different elevations as a function of time. Initial conditions: Tin = 300 °C, p = 155 bar, win = 4.588 m/s, bundle power 2250 kW
154
5. Core thermal hydraulic
5,0 Exp.
Inlet velocity in m/s
4,5 4,0 3,5 3,0 2,5 2,0
30
31
32
33
34
35
Time in s 0,6
Exp. da mid up IVA central ch. da mid up IVA cordial av. da mid up
0,5
Void, -
0,4 0,3 0,2 0,1 0,0 30
31
32
33 Time in s
34
35
Fig. 5.42. (a) Inlet velocity as a function of time; (b) NUPEC measurements and IVA results (central sub-channel and central beam line average) for the void fraction at three different elevations as a function of time. Initial conditions: Tin = 300°C, p = 155 bar, win = 4.588 m/s, bundle power 2250 kW
Measured are the beam averaged void fractions between the rod rows at three different geodetic levels. The distance between the levels is given by Hori et al. (1996, p. 811). The position of the first level is approximately taken from Fig. 1 by Akiyama et al. (2005). So sub-channel type discretization does not allow accurate computation of the beam averaged void fractions. Hori et al. (1993, p. 74) presented for the pressure transient in Fig. 14 a row cells averaged void fraction, whereas in Akiyama et al. (1995) for the power- and mass-flow-transients used for comparison the void in
5.4. Transient boiling
155
the central channel of the bundle. I put on Figs. 5.39(b) and 5.40(b) both, the central channel void and the row cells averaged void along the x-axis. As expected the central channel void is larger than the row cells averaged. Regarding the power transient: The experimental data for the upper two levels are between the predicted both void fractions. At the lowest measured position the void is over predicted, indicating that the sub-cooled boiling the real void is collected more at the wall then the smeared void computed in the sub-channel analysis. The gradient of the change reflects the experimental observations. Regarding the flow transients: There is considerable discrepancy between prediction and measurements. The moment of the intensive void formation is predicted later, the inclination of the void increase is predicted smaller, saying that the heat partitioning between evaporation and condensation is different. The predicted curves-maxima are lower than the measured. Comparison with Aounallah et al. (1999), Macian et al. (2000) and Wang et al. (2005) analyses: Analyses with the two-fluid computer code VIPRE-02 (Kelly et al. 1992) of the openly published NUPEC data which are used also in this study are already reported by Aounallah and Coddington (1999) and by Macian et al. (2000). VIPRE-02 is a two-fluid computer code using six equations for describing predominant axial flow. The momentum equations in the other two lateral directions are simplified for a normal velocity component from- or to the neighboring cannels using 1 instead of 9 non-homogeneous convection terms in a transformed coordinate system. IVA possesses a three-fluid-model with conservation equations that do not neglect any terms. Regarding selection of the drag coefficient correlations and boiling mechanisms for this particular class of processes as long as two of the fluids in IVA are addressed, the codes are similar except the use of dynamic fragmentation and coalescence in IVA but not in VIPRE-02. The observations made by Aounallah and Coddington (1999) and by Macian et al. (2000) are confirmed by my study too. In general effectively less drag between void and liquid is observed in the experiments than in the separated-momentum equation predictions with correlation gained on flows with other geometry. Macian et al. interpret the under-prediction of the void as a possible deficiency of the sub-cooled boiling instead of the effectively lower drag. This difference in the interpretation can be clarified in the future only if fine-scale analyses are done. The discrepancy with the data in Fig. 5.32 is also observed by Macian et al. The reasons remain to be clarified. Wang et al. (2005) performed a comparison between VIPRE two fluid code and 6 FRIGG experiments comparing also void fraction in lateral bundle zones. The authors come to the conclusion that addition modeling of the radial redistribution is required to better predict void in neighboring sub-channels with quite different power load. Over prediction of the axial void in the low power regions is also reported.
156
5. Core thermal hydraulic
Comparison with Hori et al. (1994): Hori et al. (1994) reported that the comparison with their transient measurements can be better reproduced in the acceleration phase by homogeneous models and in the deceleration phase by drift-flux models. This is line with my observation here and with the observation made by Aounallah and Coddington (1999) and by Macian et al. (2000). Conclusion: In general averaged measurements over a beam line have to be compared with averages of rows of fine mesh cells extracted from computational results. The larger the size of the cells the less informative is the comparison.
5.5. Steady state critical heat flux Nucleate and flow boiling is a very good cooling mechanism. Increasing heat fluxes in the wall may lead to heat transfer regimes that have considerably lower heat transfer. In such cases the frequency of the bubble production and their bubble departure size allow them to touch each other. Then film is forming worsening the heat transfer coefficient. This regime is calling departure from nucleate boiling and designated with DNB or boiling crisis of the 1. art in the Russian literature. Other important case is the boiling of a flowing liquid film which is a good heat transfer mechanism. If the liquid film dries out, the heat transfer is controlled by the steam flow which again makes the cooling much worst. This regime is called dry out and designated with DO or boiling crisis of the 2. art in the Russian literature. Both mechanisms may happen in technical facilities. The first one causes stronger temperature jump at the wall than the second. If lasting long enough both regimes can destroy the facility. We distinguish between power controlled mode- and temperature difference controlled mode of boiling. The first one is characteristic for electrical heating or nuclear heating for which the power is imposed into the system independently of the cooling conditions. The second one is typical for heat exchangers where the power is controlled by the heat transfer mechanisms at the both site of the heat transferring pipe wall. In the first mode the critical heat flux is associated with large temperature jump in increasing the power. In reducing the power the temperature difference become smaller up to the so-called minimum film boiling temperature and then jump to those characteristic for nucleate boiling. This hysteresis was first observed experimentally first by Nukiyama in 1934. The second regime is experimentally first observed by Drew and Mueller in 1937. Look-up table for critical heat flux: There are more the 400 empirical correlations and methods for prediction of the critical heat flux in pipes and rod bundles.
5.5. Steady state critical heat flux
157
The most powerful method among them is the so-called look-up table for critical heat flux. A look-up table for critical heat flux (CHF) has been developed jointly by AECL Research (Canada) and IPPE (Obninsk, Russia) Groeneveld et al. (1996) ′′ = qCHF ′′ ( p, G , X 1,eq , Dhy ) . qCHF
(5.2)
It is based on an extensive data base of CHF values obtained in tubes with a vertical upward flow of steam-water mixture. The look-up table is designed to provide CHF values for 8 mm tubes at discrete values of pressure, mass flux and dry-out quality covering the ranges 0.1 to 20.0 MPa, 0.0 to 8000 kg/(m2s) and − 0.5 to +1.0 respectively. Linear interpolation is used to determine the CHF for conditions between the tabulated values, and an empirical correction factor is introduced to extend this CHF table to tubes of diameter values other than 8 mm, ⎛ 1000 Dhy ⎞ ′′ = q&CHF ′′ _ 8mm ⎜ q&CHF ⎟ 8 ⎝ ⎠
−1/ 2
.
(5.3)
Compared against the combined AECL-IPPE world data bank (consisting of 22946 data points after excluding duplicate data and obviously erroneous data), the 1995 look-up table predicts the data with overall average and root-meansquare errors of 0.69 and 7.82%, respectively. An assessment of various CHF tables and several empirical correlations shows that the 1995 table consistently provides the best prediction accuracy and is applicable to the widest range of conditions. Groeneveld et al. (2005) issued the improved variant of their first table known as the 2005-look up table containing 20% more data. The rms error was reduced to 7.1%. 5.5.1 Initial 0D-guess Problem: Check whether it is possible to use the 2005-look up table primary designed for heated pipes also for heated rod bundles. Solution: I exploit deliberately a set of simplifying assumptions usually used in the 1960’s: 1. 2. 3.
Fictive sub-channels corresponding to the number of rods are so defined as to be equivalent in geometry; Equal mass flow rate to each sub-channel; The critical heat flux occurs at the exit of the hot channel as a function of the exit parameters; The phenomenon is similar to pipe flow; Therefore the Groeneveld’s look up table is expected to give the right dependence on mass flow rate, quality and sub-cooling;
158
4. 5.
5. Core thermal hydraulic
Uniform boundary conditions for all channels; Turbulence production due to spacer grids and its influence as well as the axial power distribution is not specifically treated.
The procedure I use is: Unless
′′ ,i ( zmax ) − qi′′( zmax ) qcrit < ε increase q with a Δq ′′ ,i ( zmax ) qcrit
starting with q0 . Here the power of the hot channel is qi = f lat ,i q nrods , where per nrods
definition the bundle power is the sum of the sub-channel powers q = ∑ qi and i =1
the lateral distribution factor obeys
1 nrods
nrods
∑f i =1
lat , i
= 1 . The exit equilibrium mass
⎞ 1 ⎛ 1 ⎛ qi q ⎞ ⎜ − Δhsub ⎟ = ⎜ flat ,i & − Δhsub ⎟ . Here by virtue of ash m Δh ⎝ m& i Δ ⎝ ⎠ ⎠ sumption 2 I use m& m& i = nrods . The local heat flux at the exit of the hot channel i,
flow quality is X 1,i =
qi′′( zmax ) = f zmax ,i
flat ,i q nrods
π Drod Lrod
, is compared with the local critical heat flux. The
critical heat flux at the exit of channel i is defined by the 2005-look-up-table ′′ ,i ( zmax ) = f ( p, G, X 1,i , Dhyd ) . Dividing this value by the lateral hot channel qcrit factor and computing the averaged flux into bundle power I obtain the final result q ′′ ( z ) q = crit ,i max π Drod Lrod nrods . Note that the last expression is based on the intuif lat ,i tive assumption that this value may represent the averaged heat flux in the hot channel. It is only a hypothesis. It is very interesting to compare the prediction of this “primitive” approach with experimental data and to see haw it works. 8 × 8 NUPEC BWR bundle tests: First I simulate three data sets for experiments with the 8 × 8 NUPEC BWR bundle tests (Fig. 5.43) as proposed in the international benchmark by Neykov et al. (2005). The first and the second tests are performed with the same axial power distribution but with different lateral power peaking pattern. They serve for analysis of the influence of the lateral power distribution. The first and the third tests are performed with the same lateral power peaking pattern but with different axial power distribution. They serve for analysis of the influence of the axial power distribution.
5.5. Steady state critical heat flux
159
CHF in MW, measured
12 NUPEC BWR bundle with ferrule spacer C2A: RMS 2.5% +5% -5%
10 8 6 4 2
2
4 6 8 10 CHF in MW, computed
12
Fig. 5.43. Geometry of the 8 × 8 NUPEC CHF bundle tests: (a) cross section; (b) ferrule spacer; (c) Measured versus computed critical bundle power for cosine profile with f z ,max = 1.4 and lateral peaking pattern characterized by f xy = 0.89 to 1.3. Data by NUPEC: p =
& = 9.98 to 65.52t/h. C2A tests, 79 5.48 to 8.67 MPa, Δh2,sub = 22.61 to 137.26 kJ/kg, m pints
CHF in MW, measured
12 NUPEC BWR bundle with ferrule spacer C2B: RMS 4.4% +5% -5%
10 8 6 4 2 2
4 6 8 10 CHF in MW, computed
12
Fig. 5.44. Measured versus computed critical bundle power for cosine profile with f z ,max = 1.4 and lateral peaking pattern characterized by f xy = 0.99 to 1.18. Data by NUPEC: p =
& = 9.93 to 64.85 t/h. C2B test, 36 7.14 to 7.20 MPa, Δh2,sub = 21.06 to 128.57 kJ/kg, m points
The measured versus the predicted bundle power is presented in Figs. 5.43(c), 5.44 and 5.45. Comparing Figs. 5.43(c) and 5.44 we see that for flatter lateral profiles this method slightly systematically under predicts the measurements. Comparing Figs. 5.44 and 5.45 we see that for the two different axial power profiles this method predicts almost similarly the measurements.
160
5. Core thermal hydraulic
CHF in MW, measured
12 NUPEC BWR bundle with ferrule spacer C3: RMS 3.4% +5% -5%
10 8 6 4 2 2
4 6 8 10 CHF in MW, computed
12
Fig. 5.45. Measured versus computed critical bundle power for inlet peak profile with f z ,max = 1.4 and lateral peaking pattern characterized by f xy = 0.89 to 1.3. Data by
& = 9.93 to 65.02 t/h. C3 NUPEC: p = 7.10 to 7.19 MPa, Δh2,sub = 21.7 to 125.74 kJ/kg, m test, 36 points
Having in mind the 2005-look up table possesses mean error of 7.1% the above predictions with mean error of 2.5, 4.4 and 3.4% can hardly be made better. AREVA ATRIUM10 BWR tests: Figures 5.46 and 5.47 presents the measured versus the predicted bundle power for a specific tests for AREVA BWR bundles ATRIUM 10.
CHF in MW, measured
12
TM
AREVA BWR bundle 17.1: RMS 4.6% +5% -5%
11 10 9 8 7 6
6
7
8 9 10 11 CHF in MW, computed
12
TM
Fig. 5.46. (a) BWR ATRIUM 10 bundle with ULTRAFLOW spacer; (b) Measured versus computed critical bundle power for cosine profile with f z ,max = 1.4 and lateral peaking pattern characterized by f xy = 0.751 to 1.255: mean error 4.6%, slight systematic
& = under prediction. Data by Fleiss et al. (1992): p = 69 bar, Δh2,sub = 23 to 184 kJ/kg, m 6.3 to 18.90 kg/s. 40 points
5.5. Steady state critical heat flux
161
There are slight differences in the geometry. For both cases we see mean error of 4.6 and 3.95%, respectively, which again lies inside the accuracy of the 2005-look up table. Again, prediction with better accuracy can hardly be achieved.
CHF in MW, measured
12 AREVA BWR bundle 17.2: RMS 3.95% +5% -5%
11 10 9 8 7 6 6
7
8 9 10 11 CHF in MW, computed
12
Fig. 5.47. Measured versus computed critical bundle power for cosine profile with f z ,max = 1.4 and lateral peaking pattern characterized by f xy = 0.731 to 1.251: mean error 3.95%, slight systematic under prediction. Data by Fleiss et al. (1992): p = 68.8 to 69.5 bar, Δh2,sub = 23 to 188 kJ/kg, m& = 6.33 to 15.7 kg/s. 37points
AREVA tests for PWR bundles with FOCUS grids: Unlike the previous four data sets this one is for pressures up to 165 bars. The bundle contains 5 × 5 rods and 5 grids. Again, the simple method predicts the CHF with mean error of 7.24 which is within the accuracy of the 2005-look up table, see Fig. 5.48.
0.0000 0.0100 0.0200 0.0300 0.0401 0.0501 0.0601 X
CHF in MW, measured
0.0000 0.0100 0.0200
Y 0.0300
0.0401
0.0501
0.0601
6 5 4 3
AREVA PWR bundle with FOCUS grid RMS 7.24% +10% –10%
2 1 1
2 3 4 5 CHF in MW, computed
6
Fig. 5.48. AREVA PWR bundle with FOCUS grid: Computed versus measured bundle power: mean error 7.5%, slight systematic under prediction. Data by Vogel et al. (1991): p = 69.3 to 166.6 bar, T2,in = 179.4 to 329.5°C, G = 1186.6 to 3629 kg/(m²s), q = 1.4643 to 5.5015 MW. 82 points
162
5. Core thermal hydraulic
Conclusions: The check whether it is possible to use the 2005-look up table primary designed for heated pipes also for heated rod bundles gives the surprising result that the bundle critical power for 5 data sets of three different bundles and different power distributions all together 273 experiments are predicted by a simple method described above using the 2005-look-up-table within the accuracy reported by the authors of this table. Therefore, unless better prediction method is developed it is good idea to use the above method as the best predictive instrument. The influence of the grid design can in addition be taken into account by empirical coefficient derived from experiment. Company designs their rod bundles with proprietary measurements usually collected for the nominal operation parameter on 1:1 geometrical models. It is of course possible to produce data fitting with a bit smaller error band in the confined parameter region. The advantage of the look-up-table is in wide validity region: pressure, mass flux and dry-out quality of 0.1 to 20.0 MPa, 0.0 to 8000 kg/(m2s) and − 0.5 to +1.0 respectively. That is why using the look up table as a scaling instrument for data on bundles with variety of spacer grids or modifications is recommendable, because of the possibility for extrapolation. 5.5.2 3D-CHF analysis
5.5.2.1 Simple preprocessor First of all I design a preprocessor and function for IVA computer code that for given mass flow, pressure, sub-cooling and geometry of the bundles which predicts the critical bundle power and parameter by the simplified procedure presented in the previous section. 5.5.2.2 3D analysis Then I perform 3 computations: (a) with the so estimated power; in (b) and (c) I vary this power with ± 5%, respectively, and for each of them compute the portion of the surface that is dry. Dry is defined as either film thickness less the 10µm in the cell or the 2005-look-up-tably predicts conditions fro CHF (DO or DNB). In most of the BWR critical heat flux cases I have had analyzed, dry out is observed. However there are cases in which the film thicknesses are large and the look-uptable identify DNB. 5.5.2.3 Importance of the spacer grid modeling Haw important the spacer grid modeling is, is demonstrated on the following example: For the case 505500 of the NUPEC experiments I consider the grid in Fig. 5.43(b) without- and with small vanes at the periphery. Figure 5.49 show the differences.
5.5. Steady state critical heat flux 16
Peripherial vanes without with
14 Dry surface in %
163
12 10 8 6 4 2
5,6 5,8 6,0 6,2 6,4 6,6 6,8 7,0 Bundle power in MW
Fig. 5.49. Portion of the dry surface as a function of the power without and with blades
It is obvious that the vanes redirect unused liquid from the periphery to the fuels and improve the wetting process. It is remarkable that such effect is possible to be taken into account in gross discretization as those shown in Fig. 5.43(a). Therefore in all next computation in this section I do consider the peripheral blades. One example of the results obtained with this procedure is given in Table 5.21. The computations to this subject are still going on and the final analysis will be provided later. Table 5.21. Computed dry out heat flux and the corresponding dry portion of the surface of the fuels in the bundle. Exp. ID
m& kg/s
p bar
ΔTsub
Q& comp
Q& exp
K
MW
MW
Fdry / Fheated % for Q& comp
Q& comp Q& comp
-5% SA505500 SA505501 SA505600 SA505800 SA505900 SA510500 SA510501 SA510600 SA510601 SA510800 SA510900 SA510901 SA512500 SA605500 SA605502 SA610503 SA610504 SA610600 SA610700 SA610701 SA610800 SA610900
5.60 5.58 5.59 5.61 5.59 15.29 15.31 15.19 15.37 15.23 15.19 15.29 18.19 5.58 5.58 15.33 15.41 15.29 15.33 15.24 15.36 15.31
54.9 54.9 55.1 55.0 54.9 54.8 55.1 55.1 55.2 55.1 55.2 55.1 55.4 71.6 71.7 71.7 71.7 71.8 71.3 72.1 72.4 72.7
–10.07 –10.15 –16.92 –26.14 –5.11 –11.17 –12.39 –19.25 –19.37 –27.30 –6.94 –6.88 –12.75 –9.46 –9.63 –11.15 –10.90 –16.97 –20.56 –21.63 –26.39 –7.00
6.216 6.203 6.323 6.504 6.121 10.129 10.159 10.413 10.431 10.720 9.895 9.920 10.702 5.667 5.661 8.956 8.966 9.217 9.445 9.397 9.634 8.654
6.13 6.13 6.23 6.39 5.98 9.72 9.81 10.09 10.19 10.20 9.560 9.660 10.410 5.770 5.730 8.850 8.910 9.200 9.370 9.380 9.520 8.660
2.65 2.37 0.63 0.57 0.63 0.61 0.66 0.54 0.61 0.56 0.54 0.61 0.56 0.21 0.19 0.00 0.00 0.02 0.00 0.00 0.00 0.00
+5% 6.49 4.87 1.86 1.83 1.72 1.64 1.69 1.38 1.46 1.39 1.38 1.46 1. 39 2.39 2.46 0.38 0.38 0.35 0.49 0.37 0.54 0.10
11.66 8.56 3.15 3.14 3.48 3.41 3.54 2.84 2.89 3.03 2.84 2.89 3.03 5.40 6.76 1.01 0.99 1.08 1.34 1.15 1.15 0.73
164
5. Core thermal hydraulic
5.5.2.4 Haw to define Dry Out in 3D-experiment and 3D-analysis? The portion of the fuel surface that is dry is an inherent property of the process. It is detected by measuring with finite number of thermocouples. It is simulated with finite number of volumes contained again finite number of portions of the fuel bundle-surface. Therefore there is a specific definition of CHF associated with the density of the thermo-couples and their distribution in each experiment. This important source of uncertainty was never considered in the literature as far I know but it has to be considered in the future by transferring experimental observation to real plant behavior. As it will be demonstrated in the next section the mean error of the 1D-prediction of the location of the dry out is about 40 cm due to the uncertainties in the involved constitutive models. They remain also in 3D. Therefore, more effort is necessary to increase the accuracy of the involved correlation, to derive complete set of source terms for generation of turbulence in all flow pattern, to accomplish the right coupling between deposition and local degree of turbulence, and finally to derive appropriate mechanistic criterion for identification of dry out. Continuation of this line of research theoretically and experimentally is recommendable. 5.5.3 Uncertainties Trying to compute dry out heat flux by using three fluid models with constitutive relations gained from adiabatic flow I found already in Kolev (1985a, b) that the uncertainty of the entrainment and deposition models does not allow very accurate prediction of the location of the dry out. Let us analyze some reasons for this situation which still does not changed very much from those days. 5.5.3.1 Interfacial drag The interfacial drag between gas and film in annular two phase flow controls the relative velocity between bulk flow and film. This relative velocity controls the entrainment of droplets. For the computation of the drag coefficients different approximations exists. All of them are reported by their authors to be based on experiments. Comparison between the predictions of different correlations, see the review in Kolev (2004b), as a function of the gas Reynolds number is given in Fig. 5.50.
5.5. Steady state critical heat flux
Stephan and Myinger 1990 Lopez and Dukler 1986 Nigmatulin 1982 Wallis 1969 Ambrosini 1991 Hewitt and Gowan 1991 Hanratty and Dukhno 1997 Hagen and Poiseuille, Blasius 1/3 Alekseenko roughness
0,5 0,4 0,3 4c21, -
165
0,2 0,1 0,0 0
5000
10000
15000
Re1, Fig. 5.50. Gas-film friction coefficient as a function of the gas Reynolds number
The correlations by Wallis, Gowan et al. and Nigmatuling do not differ from each other much. The prediction of the correlation by the Stephan and Myinger converges to the prediction of the above mentioned correlations for higher Reynolds number, but differs much for low Reynolds numbers. If one uses the formula of Alekseenko and take 1/3th of the film amplitude as effective roughness the Nikuradze formula predicts results close to those predicted by Stephan and Myinger. For comparison the prediction by the Blasius correlation for Reynolds number larger then 1187 and by the Hagen and Poiseuille correlation for lower Reynolds numbers is given also in Fig. 5.50. It is obvious that final state of the knowledge in this field is not achieved.
5.5.3.2 Entrainment The droplet entrainment in annular two phase flow influences the film thickness. For the computation of the entrainment different approximations exists. All of them are reported by their authors to be based on adiabatic experiments.
166
5. Core thermal hydraulic
400 Film thickness in µm
350 300 250
Deposition: Whalley approximation by Kato 1984 Entrainment: Kataoka and Ishii 1982 Lopez de Bertodano 1998 Nigmatulin 1982 (inc. Zeichik 1998) Sugawara 1990 Whalley 1974 Hewit and Gowan 1989
200 150 100 50 0 82 84 86 88 90 92 94 96 98 100 Cell no.
Fig. 5.51. Film thickness as a function of the distance from the entrance. Boiling water in vertical circular pipe, Bennett et al. 1967: Test nr. 5253, vertical pipe, 0.01262 m inner diameter, 5.5626 m length, uniformly heated with 199 kW, inlet water flow from the bottom: 68.93 bar and 538.90 K
Kawara et al. (1998) compared 9 correlations for entrainment obtained by different authors or teams. They found differences of six orders of magnitude at low entrainment rations and three orders of magnitude at high entrainment ratios. I give here other example. I consider one of the many vertical boiling pipe experiments reported by Bennett et al. (1967). Simulating the flow with three-fluid model (IVA) using different entrainment correlations, see the review in Kolev (2004b), I obtain the film thickness as a function of the axial coordinate as presented in Fig. 5.51. I know from the measurements that the film disappears in position 8/10 of the total pipe length. The largest entrainment, and therefore the closest to the observed result, is predicted by the Kataoka and Ishii’s correlation. None of the models takes the change of the entrainment due to the wall boiling. Therefore a final state of this research field is still not reached.
5.5.3.3 Deposition Kawara et al. (1998) compared 11 depositions coefficients obtained by different authors or teams. They found differences up to two orders of magnitude depending on the local droplet concentration.
Film thickness, µm
5.5. Steady state critical heat flux
200 180 160 140 120 100 80 60 40 20 0
167
Entrainment: Kataoka Ishii 1982 Deposition: Kataoka (Paleev modified) 1983 Paleev and Philipovich 1966 Nigmatulin 1982 Whalley approximation by Kato 1984 Owen and Hewitt 1987 Lopes and Ducler 1986 no deposition
82 84 86 88 90 92 94 96 98 100 Cell nr. Fig. 5.52. As in Fig. 5.51.
I make 7 computations for the test case of the previous section changing only the deposition models keeping the entrainment model unchanged. The results regarding the film thickness are presented on Fig. 5.52. The smallest deposition, and therefore the closest to the observed result, is predicted by the Kato’s correlation. Note that I do not take into account the deposition suppression due to boiling of the film which is obviously important. Repeating the computation with virtually no deposition gives the best results. Note that Hoyer and Stepniewski reported already in 1999 that deposition suppression due to boiling is important for dry out prediction at high void fractions boiling flows. Therefore a final state of this research field is still not reached.
5.5.3.4 Deposition and entrainment changes due to nucleate boiling: The influence of the boiling on the deposition will be next discussed. Boiling films manifest few additional phenomena controlling the film dynamics that are not present in adiabatic films. Modified deposition: Vapor produced through the film blows into the gas core and dumps the penetration of the gas pulsation into the boundary layer. Therefore there is an impact on the gas boundary layer close to the interface resembling blowing of gas from the film/gas interface. This phenomenon hinders deposition. Assuming that all generated vapor enters the bulk flow perpendicular to the wall we have for the effective gas velocity
168
5. Core thermal hydraulic
u1_ blow =
Dheat μ21 . 4 ρ1
(5.4)
If u1_ blow > V1′ , the deposition is effectively none existing. If u1_ blow > V1′ the effective gas fluctuation velocity is reduced to V1′ − u1_ blow . Doroschuk and Levitan (1971), Guguchkin et al. (1985) reported data that prove that the vapor blow from the film into the gas bulk reduce deposition. Milashenko et al. (1989) recommended ignoring the deposition in boiling flows. Modified entrainment: The steam mass flow generated at the wall surface contributes substantially to the fragmentation of the liquid by two mechanisms: At low film velocity if the bubble departures diameter is larger than the film thickness the bubble burst causes additional entrainment. At high film velocities the bubble departure diameter is small but there is a vapor net flow perpendicular to the wall crossing the film. The vapor creates two phase unstable structure that eventually break up and release the vapor into the gas core flow by entraining additionally droplets. Milashenko et al. collected in 1989 data for boiling flow in a pipe with 0.0131 m inside diameter and 0.15 and 1 m heated lengths. Setting the deposition rate μ32 equal to zero the authors correlated their data for an effective entrainment by the following correlation 1.3
μ23 _ boiling _ film + μ23 _ adiabat − μ32 =
⎛ 1.75 ρ ′′ ⎞ α 2 ρ 2 w2 ⎜ q& w′′2 10−6 ⎟ , π Dh ρ′ ⎠ ⎝
(5.5)
indicating a strong dependence on the wall heat flux. Note that the asymptotic value for zero heat flux will produce zero effective entrainment which can not be true. Nevertheless this work provides the ultimate prove that boiling process influence the effective entrainment. Probably better scaling velocity of the additional to the adiabatic entrainment is given by Eq. (5.4) which will be linearly dependent on the heat flux, w1_ blow = q&w′′2 ( ρ1Δh ) , for saturated film. This explains why Milashenko et al. are forced to introduce the constant 10–6 being of order of 1 Δh . Therefore
μ23 _ boiling _ film = f ρ 2
μ21 Dheat ρ1 4
(5.6)
with function f < 1 that remains to be found is probably the better physical basis for data correlation. Kodama and Kataoka reported in 2002 a dimensional correlation
5.5. Steady state critical heat flux
169
for the net entrainment rate due to the bubble break up with accuracy up to a constant that have to be derived from experiments
( ρ w )23 _ boiling _ film + ( ρ w )23 _ adiabat − ( ρ w )32 = const
⎡ ⎤ δ 2 τ 2w ρ2 q& w′′2 exp ⎢ − ⎥. 2.66 Δh ρ ′′ ⎣⎢ 158.7 (η ′′ η ′ ) 30 w2 ⎦⎥
(5.7)
We see here that the group w1_ blow = q& w′′2 ( ρ1Δh ) is used correctly but again for adiabatic flow the asymptotic is not correct.
140 Film thickness in µm
120 1 2 3 4
100 80 60 40 20 0 4,2
4,3
4,4
4,5 4,6 z in m
4,7
4,8
Fig. 5.53. Film thickness as a function of the axial coordinate for conditions of Bennett 5253 experiment. (1) Entrainment, deposition like in adiabatic flow; (2) Like (1) with deposition multiplied by ratio of the gas turbulent kinetic energy to the equilibrium turbulent kinetic energy; (3) Like (1) + (2) with suppression of the deposition due to the vapor blow from the film; (4) Like (1) + (2) + (3) with entrainment enhancement due to the vapor blow through the film; Experimental position of the dry out about 4.4 m.
Using correlation for entrainment and deposition gained from adiabatic experiments results obviously in over prediction of the deposition and under prediction of the entrainment. Now I demonstrate the effect of the above discussed phenomena again on the Bennett experiment as given in Fig. 5.53. Using f in of the order of 0.02 and repeating the computations for other experiments as given in Table 5.22 we see that the position of the dry out of the film can be predicted with mean error of 0.41 m.
170
5. Core thermal hydraulic
Table 5.22. Position of predicted and computed film dry out for some Bennett heated pipe experiments
Bennett exp. ID 5253 5293 5332 5380 5407 5424 5442 5456
zDO exp. ± 0.076 3.89 3.89 4.39 3.81 2.79 2.64 2.79 2.79
zDO IVA 4.31 4.42 3.37 3.7 3.02 2.94 3.27 2.94
Δ 0.42 0.53 –1.02 –0.11 0.23 0.30 0.48 0.15
Mean error 0.41 m
The mean error of the 1D-prediction of the location of the dry out is about 40 cm due to the uncertainties in the involved constitutive models. These uncertainties remain also in 3D-predictions. Therefore, as already mentioned, more effort is necessary to increase the accuracy of the involved correlation, to derive complete set of source terms for generation of turbulence in all flow pattern, to accomplish the right coupling between deposition and local degree of turbulence, and finally to derive appropriate mechanistic criterion for identification of dry out. 5.5.3.5 Residual film thickness at DO The results of such computations are always predicting some film thickness at places where the wall has to be dry. This lead some authors to introduce some critical film thickness depending on the local parameter as already mentioned in the introduction. Regarding the experimental evidence: At the dry out Milashenko et al. (1989) experimentally observed a residual film mass flow rate of 0.02 kg/s. Shiralkar and Lahey (1973) reported finite film flow rates at the location of the CHF appearance. Regarding the dry out film thickness Groeneveld (2001, Private communication) reported that in all of his dry out experiments (more then 30 years experimental research) a complete drying of the film was observed which contradicts to Milashenko et al. (1989).
5.6. Outlook – towards the large scale turbulence modeling in bundles
171
Conclusion: The influence of the boiling film on enhancing of the entrainment and suppression of the deposition has to be taken in any case into account. Then the film nucleate boiling has to be exanimate for departure from nucleate boiling. If there are no local conditions for DNB then the DO requires dry film. This logic allows finite film thickness and boiling crisis simultaneously in cases of DNB. 5.5.3.6 Amount of the liquid in the core at the onset of film flow Some authors reported that at the onset of the film flow, before DO, there is immediate amount of entrained liquid inside the gas core; see Barbosa et al. (2002). This introduces a well defined by the local parameter initial film thickness before the dry out occur. So the dry out point will be depending of this initial film thickness. This is still source of uncertainty influencing the accuracy of mechanistic dry out prediction.
5.6. Outlook – towards the large scale turbulence modeling in bundles Problem: Given rod bundle for nuclear power plant with the geometry and spatial heat release in the fuel rods specified in OECD/NRC Benchmark (2004). The horizontal cross section of the bundles is illustrated in Fig. 5.18 Assembly: 0-1, 1 and 3. Under these conditions the flow is boiling and the flow regimes are either liquid only or bubbly flow. Compute the parameters in the bundles including the turbulent kinetic energy and its dissipation in the continuous liquid as described in Vol. 3 Chap. 8 of this monograph. Solution: The lateral discretization (18 × 18 × 24 cells) used here presented also in Fig. 5.18. The geometry data input for IVA computer code is generated using the software developed by Roloff-Bock (2005). 1,6 1,4
k in m²/s²
1,2 1,0
NUPEC 8x8 bundle flow boiling middle verticle plane variation of the lateral coordinate
0,8 0,6 0,4 0,2 0,0
5
10 15 axial cell nr
20
25
172
5. Core thermal hydraulic
80 70 eps in m²/s³
60
NUPEC 8x8 bundle flow boiling middle verticle plane variation of the lateral coordinate
50 40 30 20 10 0 5
10 15 axial cell nr
20
25
Fig. 5.54. Test problem 5: Turbulence of boiling liquid in rod bundle computed with IVA computer code, Kolev (2007a, b). (a) Turbulent kinetic energy as a function of the axial coordinate; (b) Dissipation of the turbulent kinetic energy as a function of the axial coordinate
The results for a vertical plane crossing the bundle at the middle are presented in Fig. 5.54. The family of curves belongs to each vertical column of cells from the one site to the other. We see several interesting elements of the large scale averaged turbulence of the flow: (a) The distance between the spacer grids influences turbulence level. Smaller distance increases the turbulence level. Distance larger than the compete decay distance do not increase the averaged level of turbulence. (b) The boiling in the upper half of the bundle increases also the liquid velocities and therefore the production of turbulence in the wall. In addition, the bubbles increase the production of turbulence due to their relative velocity to the liquid; (c) In order to obtain smooth profiles in this case the resolution in the axial direction have to be substantially increased. Practical relevance: As I obtained this result in 2005 and reported it in Kolev (2006) I did not know any other boiling flow simulation in rod-bundles delivering the large scale averaged level of turbulence. Improving the capabilities in this field opens the door to better prediction of important safety relevant phenomena in the nuclear power plant: (a) the particles (bubble or droplets) dispersion can be better predicted; (b) the deposition of droplet influencing the dry out can be better predicted; (c) the improvement of the heat transfer in single and two phase flow behind the spacer grids.
5.6. Outlook – towards the large scale turbulence modeling in bundles
173
If such methods for prediction of the departure from the nucleate boiling (DNB) could be available that take into account the level of the local liquid turbulence the accuracy of the DNB prediction will increase. Experimentally the improved heat transfer behind the grids and increasing the margin to the critical heat flux is clearly demonstrated by Doerffer et al. (2000) on their Figs. 2 and 3 and by Groeneveld and Leung (2000) on their Fig. 3, by using innovative sliding thermocouples: 30% blockage gives 1.8 increase of the critical hat flux immediately after the grid decaying to 1 after about 100 hydraulic diameters. Empirical multiplier to take this effect into account was reported by Bobkov (2003): 1 + A exp(-0.1z/Dh),
A = 1.5ξ grid ( G /1000 ) and z being the distance of the location of the critical heat flux from the nearest flow upwards spacer grid. This important experimental observation is very much in line with the physics of the turbulence described here. 1/ 5
Problem: Consider the influence of the turbulence on heat transfer. Use the values of the local cross section averaged specific kinetic energy of the turbulence and its dissipation in order to correct the heat transfer coefficients for single phase flow obtained for developed flow. Solution: As already reported in Vol. 3 of this monograph p. 80 increasing the frequency of the turbulence with respect to the steady developed flow increases the heat transfer by following a square root function
⎛ Δτ μ e,l ,∞ ′′ q&wl =⎜ ′′ ,∞ ⎜⎝ Δτ μ e ,l q&wl
1/ 2
⎞ ⎟⎟ ⎠
.
(5.8)
Here the effective fluctuation heat transfer time constant for developed flow is 2 ⎡ ⎛1 ⎞ Δτ μ e ,l , ∞ = Dh2 ⎢π ⎜ Nul ,∞ ⎟ al ⎠ ⎣⎢ ⎝ 2
⎤ ⎥, ⎦⎥
(5.9)
(Eq. 4.26 Vol. 3) where the Nusselt number Nul , ∞ is computed by the appropriate correlation for the particular geometry for developed flow, and Δτ μ e ,l = 0.37 kl ε l
(5.10)
is the local time constant of the microscopic fluctuations. Problem: Consider the influence of the turbulence on droplet deposition. Use the values of the local cross section averaged values of the specific kinetic energy of the turbulence and its dissipation in order to correct the droplet deposition obtained for developed flow.
174
5. Core thermal hydraulic
Solution: As already reported in Vol. 3 of this monograph p. 87 increasing the turbulent kinetic energy of the continuum with respect to the steady developed flow increases the droplet deposition by following a square root function
( ρ w)32 ( ρ w )32,∞
≈
k1 . k1,∞
(5.11)
Here the local cross section averaged turbulence kinetic energy of the gas flow for developed flow is k1,∞ = ck w12 Re1−1/ 6 ,
(5.12)
with ck = 0.0306, 0.0367, 0.0368 for channels, pipes and rod bundles, respectively, in accordance with Chandesris et al. (2005).
5.7. Outlook – towards the fine resolution analysis Lumping large cross section in one sub-channel, as still practiced worldwide naturally limits the prediction accuracy. Therefore future analyses have to concentrate on much finer resolution of the processes as demonstrated in Fig. 5.55.
Fig. 5.55. Abandoning of the sub-channel analyses – a challenge for the near future
The challenges on this way are associated with the constitutive relationships. Empirical correlations that are gained by averaging values across pipe cross section are not valid for the sub-scale and have to be systematically replaced by new ones.
5.8. Core analysis
175
5.8. Core analysis After learning how to analyze boiling flow in a single pipe and in a rod bundle, knowing the accuracy of the methods after comparing them with large number of experiments, we can proceed to a thermal- and hydraulic core design. The primary information regarding the heat release in the core is coming from the neutron transport or diffusion analysis. In the past, simplified models have been used for computing hydraulic parameters and incorporate their influence on the neutron transport analysis. Since recently more institutions are coupling their complex systems for describing both: neutron transport and thermal hydraulic simultaneously. I will give here one example for analyzing the flow parameter distribution in a commercial boiling water reactor. Problem: Consider a BWR reactor pressure vessel which ¼ is discretized in Cartesian coordinates as given in Fig. 5.56(a). The lower left corner belongs to the symmetry axis. Prescribed is: the thermal power level, the 3D-power distribution, the pump mass flow, the inlet temperature, the separation efficiency of the cyclones and dryers, the geometry of the vessel, the geometry of the single bundle, and finally the bundle arrangement scheme. Compute all thermal-hydraulic characteristics inside the reactor pressure vessels, like, phase volumetric concentrations and temperatures, velocities, pressures etc. Solution: The solution is obtained using the IVA computer code. Figure 5.56 presents the horizontal cross sections at different levels containing the void distribution: (a) at the entrance of the lower head; (b) at the exit of the core; (c) inside the first upper plenum at the entrance of the chimney pipes; (d) at the exit of the chimney pipes; (e) inside the second upper plenum at the entrance of the cyclones stand pipes; (f) at the entrance of the cyclones. We realize from Fig. 5.56 haw single phase water penetrates the core (a) and evaporate at different rates in the bundles so that the void at the exit of the core is non uniform (b). Then due to the mixing in the first upper plenum some homogenization is observed (c). Further limited mixing happens in the so-called mixing chimney pipes (d). Again some mixing is observed in the second upper plenum (e). Actually this is the distribution with which the water-vapor mixture enters locally the stand pipes of the cyclones (f) which does not change much up to the entrance in the cyclones. So the characteristics of the cyclones can be used to compute the exit moisture at each cyclone exit. The next step is to use the so obtained local moisture content entering the dryers and to compute the final dryness of the steam directed to the turbine. By doing this several information are obtained: The distribution of the margins to the critical heat fluxes, the pressure losses in the reactor, the oscillation characteristics due to formation of concentration waves or due to particular mechanical designs etc. All this information is the used by one or other faculty of the design team for its own purposes: To prove the safety margins by the authorities, to check the mechanical and thermal structure sustainability etc.
176
5. Core thermal hydraulic
5.8. Core analysis
177
178
5. Core thermal hydraulic
Fig. 5.56. Steam volume fraction: (a) at the entrance of the lower head; (b) at the exit of the core; (c) in the first upper plenum at the entrance of the chimney pipes; (d) at the exit of the chimney pipes; (e) in the second upper plenum at the entrance of the cyclones stand pipes; (f) fraction at the entrance of the cyclones
5.9 Nomenclature
179
Performing transient analysis is very similar. Instead of using constant boundary conditions the changes are prescribed and the reaction of the system is then studded. In all analyzed transients or accidents the safety margins are monitored and documented. The most powerful feature of the multiphase computational fluid dynamics is to study accidents without destroying any facility: The computers do not complain. Then limited number of scenarios is selected to plane appropriate experiments and to clarify some system characteristics. The more phenomena are taken into account in the computational analysis the less surprises the engineers experiences in the practice. In the past due to the limited computational power systems are analyses separately. Now it is possible to integrate systems in virtual plant and to study its behavior in great details.
5.9 Nomenclature Latin al C0 d c21
D1 Dh Dheat Drod D3 E Fdry
temperature diffusivity, m²/s drift flux distribution coefficient, dimensionless drag coefficient due to the liquid action, dimensionless bubble diameter, m hydraulic diameter, m heated diameter, m/s fuel rod diameter, m droplet size just after entrainment, m dry heated bundle surface, m²
f lat ,i
heated bundle surface, m² lateral distribution factor for channel i, dimensionless
f zmax ,i
axial distribution factor for channel i, dimensionless
G g j k Lrod Nul , ∞
mass flow rate, kg/(m²s) gravitational acceleration, m/s² mixture volumetric flux, m³/(m²s) turbulent kinetic energy, m²/s² heated length of the fuel rod, m Nusselt number for the continuum l being in contact with the wall, dimensionless number of rods in the bundle, dimensionless mass flow trough the channel i, kg/s heat flux from the wall into the continuum l, W/m²
Fheated
nrods m& i ′′ q&wl
180
′′ ,∞ q&wl
5. Core thermal hydraulic
heat flux from the wall into the continuum l for developed flow, W/m²
′′ ,i ( zmax ) critical heat flux for channel i at the exit of the channel zmax , W/m² q&crit
q&i′′( zmax ) local heat flux for channel i at the exit of the channel zmax , W/m² ′′ q&CHF critical heat flux, W/m² ′′ _ 8 mm critical heat flux for 8 mm internal diameter heated pipes, W/m² q&CHF q
sum of the sub-channel powers power, W
q0
initial power, W power of the hot channel, W effective gas blow velocity due to film evaporation perpendicular to the heated wall, m/s velocity of field l, m/s slug or Taylor bubble free raising velocity, m/s velocity, m/s exit equilibrium mass flow quality, dimensionless
qi u1_ blow
Vl VTB w X 1,i Greek
αl local volume fractions of the fields l, dimensionless α1, B −Ch void fraction defining the transition between bubbly and slug flow, dimensionless
α1, slug to churn void fraction defining the transition between slug and churn turbulent Δh Δhsub Δq Δτ μ e ,l
flow, dimensionless specific latent heat of evaporation, J/kg sub-cooling specific enthalpy, J/kg power increase, W effective time constant for microscopic pulsation due to turbulence, s
Δτ μ e ,l ,∞ effective time constant for microscopic pulsation due to turbulence for developed flow, s power dissipated irreversibly due to turbulent pulsations in the viscous ε fluid per unit mass of the fluid (dissipation of the specific turbulent kinetic energy), m²/s³ ε small number, dimensionless irreversible friction pressure loss coefficient at the spacer grid, dimenξ grid sionless λRT Rayleigh-Taylor wavelength, m μ21 evaporated mass per unit time and unit flow volume, kg/(m³s)
References
181
μ23 _ boiling _ film entrained liquid mass per unit time and unit flow volume due to film boiling, kg/(m³s)
μ23 _ adiabat entrained liquid mass per unit time and unit flow volume for adiabatic μ32 ρ1 ρ2 ρw σ2 η1 η2
conditions only, kg/(m³s) deposed into the film droplet mass per unit time and unit flow volume, kg/(m³s) gas density, kg/m³ liquid density, kg/m³ mixture mass flow rate, kg/(m²s) viscous tension, N/m dynamic gas viscosity, kg/(ms) dynamic liquid viscosity, kg/(ms)
References Ahlinder S and Tinoco H (September 30-October 4, 2007) CFD analysis of recombination by HWC in the down comer of a BWR, The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12), Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A., Log Number: 156 Akiyama Y, Hori K, Miyazaki K, Nishioka H and Sugiyama S (23–27 April 1995) Evaluation of the bundle void fraction measurement data of PWR fuel assembly, The 3rd JSME/ASME Joint International Conference on Nuclear Engineering, Kyoto International Conference Hall, Kyoto, Japan, vol 1, 211–216 Akiyama Y, Hori K, Miyazaki K, Nishioka H and Sugiyama S (Dec. 2005) Pressurized water reactor fuel assembly subchannel void measurement. Nucl. Technol., vol 112, pp 412–421 Anglart H and Nylund O (1996) CFD application to prediction of void distribution in two phase bubbly flow in rod bundles. Nucl. Eng. Des., vol 163, pp 81–98 Anklama TM and Miller RF (1982) Void fraction under high pressure, low flow conditions in rod bundle geometry. Nucl. Eng. Des., vol 75, pp. 99-108 Aounallah Y and Coddington P (Oct. 3–8, 1999) Assessment of VIPRE-02 void fraction prediction against NUPEC experimental BWR data, 9th International Topical Meeting on Nuclear Thermal Hydraulics (NUTETH-9) San Francisco, California Azodi D, Elsert P, Gruner P, Jendrch U and Kuntze WM (Febr. 1996) Verifizierug und internationaler Vergleich von Struckturanalyse-methoden zur Beschreibung des Verhaltens des Reaktordruckbehälters bei Belastung jenseits der Auslegungsgrenzen, GRSmbH, GRS-A-2338 Barbosa JR, Hewitt GF, König G and Richardson SM (2002) Liquid entrainment, droplet concentration and pressure gradient at the onset of annular flow in vertical pipe. Int J Multiphase Flow, vol 28, pp 943–961 Bennett AW et al. (1967b) Heat transfer to steam-water mixtures flowing in uniformly heated tubes in which the critical heat flux has been exceeded, AERE-R5373 Bobkov VP (2003) The table for critical heat-flux densities in square fuel-rod assemblies. Therm.Eng., vol 50 no 11, pp 952–957
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Chandesris M, Serre G and Sagaut (2005) A macroscopic turbulence model for flow in porous media suited for channel, pipe and rod bundle flows, 4th International Conference on Computational Heat and Mass Transfer, Paris Chisholm D (1983) Two-Phase Flow in Pipelines and Heat Exchanger, George Godwin, London, p 110 Doerffer SS, Groeneveld DC, Rudzinski KF, Pioro IL and Martin JW (November 5–10, 2000) Some aspects of critical-heat-flux enhancement in tubes, Proc. of the 2000 IMECE: ASME International Mechanical Engineering Congress & Exposition, Orlando, Florida Doroschuk VE and Levitan LL (1971) Investigation of droplet deposition from annular dispersed steam-water flow core and wall liquid film. Therm. Phys. High Temp., vol 9, pp 591–596 Drew TB and Mueller C (1937) Boiling, Trans. AIChE, vol 33, p 449 Fuel Economic and Reliable, AREVA, 2004, France, see also in Advanced Nuclear Power No 3, Januar 2002, The Magazine of Framatome ANP Fleiss, Knapp, Which (1992) STS-17.2, STS-17.1: Messung der Siedeübergangsleistungen und Druckverluste am ATRIUM-10-9Q-Testbündel, e321/92/43, e321/92/44, Karlstein, proprietary Friedel L (1979) New friction pressure drop correlations for upward, horizontal, and downward two-phase pipe flow. Presented at the HTFS Symposium, Oxford, September 1979 (Hoechst AG Reference No. 372217/24 698) Groeneveld DC and Leung LKH (April 2–6, 2000) Evolution of CHF and post-CHF prediction methods for fuel bundles, Proceedings of ICONE 8: 8th International Conference on Nuclear Engineering, Baltimore, MD USA Guguchkin VV, Nigmatulin BI, Markovich EE, Vasilev NI, Arestenko YuP and Ivanovskaya VI (1985) Peculiarities in droplet movement and bubbles in wall region. Theses of a report presented at VI I All-Union Conf. an Two Phase Flow in Power Machines and Apparatus, Leningrad, vol 3, pp 316–317 Hori K, Miyazaki K, Kurosu T, Sugiyama S, Matsumoto J and Akiyama Y (1993) In bundle void fraction measurement of PWR fuel assembly, Proceedings of ICONE 2, vol 1, pp 69–76 Hori K, Akiyama Y, Miyazaki K, Kurosu T and Sugiyama S (1994) Transient void fraction measurement in a single Channel simulating one channel for a PWR Fuel Assembly, 10th Proceeding of Nuclear Thermal-Hydraulics, ANS Winter Meeting, pp 56–68 Hori K, Akiyama Y, Miyazaki K, Kurosu T and Sugiyama S (1995) Void fraction in a single channel simulating one subchannel of a PWR fuel assembly, Proceedings of the First International Symposium an Two-Phase Flow Modelling and Experimentation Hori K, Akiyama Y, Miyazaki K, Nishioka H and Takeda N (1996) Total evaluation of in bundle void fraction measurement test of PWR fuel assembly, Proceedings of ICONE-4, vol 1, Part B, pp 801–811 Hoyer N and Stepniewski M (Oct. 3–8, 1999) Dry out prediction in bundle geometry, 9th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-9), San Francisco, California Hughes ED, Paulsen MP and Agee LJ (Sept. 1981) A drift-flux model of two-phase flow for RETRAN. Nucl. Technol., vol 54, pp 410–420 Inoue A, Futakuchi M, Yagi M, Kurosu T, Mitsutake T and Morooka S (1995a) Void fraction distribution in a boiling water reactor fuel assembly and the evaluation of subchannel analysis codes. Nucl. Technol., vol 112 no 3, pp 388–400
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Inoue A, Kurosu T, Oaki T, Futaguchi M, Yagi M, Morooka S, Hoshide A, Ishizuka T, Yoshimura K, Yoshida H, Mitsutake T, Abe N, Kozima A, Nakajima T, Nakamura S and Sawa T (1995b) Void measurement of BWR fuel assembly. J. At. Energy Soc. Japan, vol 37 no 8, pp 710–720 (in Japanese) Inoue A, Kurosu T, Oaki T, Yagi M, Mitsutake T and Morooka S (1995c) Void fraction distribution in boiling water reactor fuel assembly and evaluation of subchannel code. J. Nucl. Sci. Technol., vol 32 no 7, pp 629–640 Ishii M and Chawla TC (Dec.1979) Local drag laws in dispersed two-phase flow, NUREG/CR-1230, ANL-79-105 Kataoka I, Ishii M, Mishima K (June 1983) Generation and size distribution of droplets in annular two-phase flow, Transaction of the ASME, vol 105, pp 230–238 Kataoka I and Ishii M (March 20–24, 1983) Entrainment and deposition rates of droplets in annular two-phase flow. In: Mori Y and Yang W-J eds., ASME-JSME Thermal Engineering Joint Conference Proceedings, Honolulu, Hawaii, vol 1, pp 69–80 Kawara Z, Kataoka I, Serizawa A, Ko YJ and Takahashi O (August 23–28, 1998) Analysis of forced convective CHF based on two-fluid and three-fluid model, Heat Transfer 1998, Proc. of the 11th IHTC, Kyongju, Korea, vol 2, pp 103–108 Kelly JM, Stewart CW and Cuta JM (Nov. 1992) VIPRE-02 – A two-fluid thermalhydraulics code for reactor core and vessel analysis: Mathematical modelling and solution methods. Nucl. Technol., vol 100, pp 246–259 Kodama S and Kataoka I (April 14–18, 2002) Study on analytical prediction of forced convective CHF in the wide range of quality, Proceedings of ICONE10, 10th International Conference on Nuclear Engineering Arlington, VA, USA, Paper nr ICONE10-22128 Kolev NI (1985a) Transient three-phase three-component non-equilibrium nonhomogeneous flow described by 1D-three-velocity fields slip model, Proc. Specialists Meeting on Small Break LOCA Analysis in LWRs, Pisa 23–27 June 1985, vol 3, pp 363–378 Kolev NI (August 1985b) Transiente Dreiphasen Dreikomponenten Strömung, Teil 2: Eindimensionales Schlupfmodell Vergleich Theorie-Experiment. KfK 3926 Kolev NI (April 14–18, 2002a) Simple analytical U-tube benchmarks appropriate for testing of pipe-network computer codes, Proceedings of ICONE10: Tenth International Conference on Nuclear Engineering – Arlington, Virginia (Washington, D.C.), USA ICONE10-22711 Kolev NI (27–29 November 2002b) 3D-Multiphase-Fluid-Dynamik: Reduzierung der Dampffeuchte am RDB-Austritt der Siedewasserreaktoren, Betriebsleitertagung, Geneve; in English: Controlling the moisture content in the steam by using nozzles in BWR’s, Eleventh International Conference on Nuclear Engineering, ICONE 11, Keio Plaza Inter-Continental, Shinjuku, Tokyo, Japan, April 20–23, 2003, ICONE11-36306 Kolev NI (2007a) Multiphase Flow Dynamics, Vol. 1 Fundamentals, 3rd ed., with 114 Figures and CD-ROM Springer, Berlin, New York, Tokyo Kolev NI (2007b) Multiphase Flow Dynamics, Vol. 2 Thermal and Mechanical Interactions, 3rd ed., Springer, Berlin, New York, Tokyo Kolev NI (April 25–29, 2004c) 25-Years three-fluid modeling-experience: successes and limits, Proceedings of ICONE-12 ’04 Arlington VA, USA, Paper ICONE12-49054. Also in 4th ECCOMAS (European Congress on Computational Methods in Applied Sciences and Engineering, Finland July 24–28, 2004. Also in 6th WCCM (World Congress of Computational Mechanics) Beijing Sept. 5–10 2004. In extended form in 6th International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Operations and Safety, October 4–8, 2004, Nara-Ken New Public Hall, Nara, JAPAN, NUTHOS6-N6P380
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Kolev NI (27–29 June 2005a) IVA Simulations of exercises 1 of phase 1 and 2 of the OECD/NRC benchmark based on NUPEC BWR full-size fine-mesh bundle tests (preliminary), Second Workshop on OECD/NRC Benchmark based on NUPEC BWR Full-size Fine-mesh Bundle Tests (BFBT), State College, PA, USA Kolev NI (12–15 September 2005b) Do we have appropriate constitutive sets for sub-channel and fine-resolution 3D-analyses of two-phase flows in rod bundles? Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications Palais des Papes, Avignon, France, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2005) Kolev NI (2-6 October 2005c) Flow boiling in bundles: Comparison of IVA predictions for 153 experiments for variety of bundles, flow regimes including DO, steady state and transients, The 11th International Topical Meeting on Nuclear Reactor ThermalHydraulics (NURETH-11) Log Number: 547, Popes Palace Conference Center, Avignon, France Levy S (1967) Forced convection subcooled boiling – prediction of vapor volumetric fraction, Int. J. Heat Mass Transfer, vol 10, pp 951–965 Lopez de Bertodano MA, Assad A and Beus S (1998) Entrainment rate of droplets in the ripple-annular regime for small vertical ducts, CD Proceedings of the Third International Conference on Multiphase Flow, ICMF’98, Lyon, France, June 8–12 Lucchini F and Marinelli V (1974) Experimental data on burn-out in simulated BWR fuel bundle. Nucl. Eng. Des. ,vol 31, pp 371–378 Macian R, Aounallah Y, Coddington P and Stangroom P (April 2–6, 2000) Assessment of RETRAN-3D and VIPRE-02 void prediction against experimental transient data, Proc. of ICONE 8, 8th Int. Conf. on Nuclear Engineering, Baltimore, MD USA Milashenko VI, Nigmatulin BI, Petukhov VV and Trubkin NI (1989) Burnout and distribution of liquid in evaporative channels of various lengths. Int. J. Multiphase Flow, vol 15 no 3, pp 393–402 Mishima K and Ishii M (1984) Flow regime transition criteria for upward two-phase flow in vertical tubes. Int. J. Heat Mass Transfer, vol 27 no 5, pp 723–737 Morooka S, Inoue A, Oishi M, Aoki T, Nagaoka K and Yoshida H (1991) In-bundle void measurement of BWR fuel assembly by X-ray CT Scanner, The 1st JSME/ASME Joint International Conference on Nuclear Engineering, pp 237–243 Naitoh M, Iwashita T and Ando Y (Oct. 3-8, 1999) Boiling transition analysis code for LWR fuel bundles in the IMPACT project, 9th International Topical Meeting on Nuclear Thermal Hydraulics (NUTETH-9) San Francisco, California Neykov B, Aydogan F, Hochreiter L, Ivanov K, Utsuno H, Fumio K, Sartori E and Martin M (November 2005) NUPEC BWR full-size fine-mesh bundle test (BFBT) benchmark, vol 1: Specifications, US NRC, OECD Nuclear Energy Agency, NEA/NSC/DOC (2005) 5 Neykov B, Ivanov K, Hochreiter LE and Avramova M (27–29 April 2006) Summary of comparison and analysis of submitted results for exercise I-1, OECD/NRC BFBT Benchmark, 3rd Workshop, Pisa, Italy Nukiyama S (1934) The maximum and minimum values of heat transmitted from metal to boiling water under atmospheric pressure. J. Japan Soc. Mech. Eng. Vol 37, p 367 (English translation in 1966: Int. J. Heat Mass Transfer, vol 9, p 1419) Nylund O et al. (1968) Hydrodynamic and heat transfer measurements on a full scale simulated 36-rod Marviken fuel element with uniform heat flux distribution, FRIGG-2, AB Atomenergi, Stockholm, Sweden, ASEA, VÄSTERAS, R-4-447/RTL-1007 Nylund O et al. (1970) Hydrodynamic and heat transfer measurements on a full scale simulated 36-rod BHWR fuel element with non-uniform axial and radial heat flux distribution, FRIGG-4, AB Atomenergi, Stockholm, Sweden, ASEA, VÄSTERAS,R4502/RL-1253
Appendix 5.1: Some relevant constitutive relationship addressed in this analysis
185
Paleev II and Filipovich BS (1966) Phenomena of liquid transfer in two-phase dispersed annular flow. Int. J. Heat Mass Transfer, vol 9, p 1089 Rehme K (1971) Laminar Strömung in Stabbundeln. Chemie-Ing-Tech, vol 43 ,pp 962–966 Rehme K (1972) Pressure drop performance of rod bundles in hexagonal arrangements. Int. J Heat Mass Transfer, vol 15, pp 2499–2517 Rehme K (1973) Simple method of predicting friction factors of turbulent flow in noncircular channels. Int. J. Heat Mass Transfer, vol 16, pp 933–950 Rehme K (1978) The structure of turbulent flow trough a wall sub-channel of rods bundle. Nucl. Eng. Des., vol 45, pp 311–323 Roloff-Bock I (2005) 2D-grid generator for heterogeneous porous structures in structured Cartesian coordinates, Framatome ANP, proprietary Saha P and Zuber N (1974) Proc. Int. Heat Transfer Conf. Tokyo, Paper 134.7 Schäfer H and Beisiegel A (17.03.1992) Feasibility of stability tests under natural circulation conditions in the Karlstein test loop, Siemens technical report E32/92/e14a, proprietary Shiralkar BS and Lahey RT (1973) The effect of obstacles on a liquid film. J. Heat Transfer, Trans. ASME, vol 95, pp 528–533 Spierling H (Sep. 24, 2002) Regional void fraction in ATRIUM 10XP at various peaking patterns, Framatome-ANP report, FGT, A1C-1311669-1, proprietary Utsuno H, Ishida N, Masuhara Y and Kasahara F (October 4–8, 2004) Assessment of boiling transition analysis code against data from NUPEC BWR full-size fine-mesh bundle tests, The 6th International Conference on Nuclear Thermal Hydraulics, Operations and Safety (NUTHOS-6) Nara, Japan, Paper ID. N6P041 Vogel, Bruch, Wang (19. Aug. 1991) SIEMENS Test Section 52 (DTS52) Description of experiments, KWZ BT23 1991 e 244, Erlangen, proprietary Wang G, Hochreiter LE, Sung Y and Karoutas ZE (May 16–20, 2005) VIPRE code void fraction prediction assessment using FRIGG loop data, 13th International Conference on Nuclear Engineering, Beijing, China Windecker G and Anglart H (Oct. 3–8, 1999) Phase distribution in BWR assembly and evaluation of multidimensional multi-field model, 9th International Topical Meating on Nucl. Reactor Thermohydraulics, NURETH-9, San Francisco, California Yagi M, Mitsutake T, Morooka S and Inoue A (1992) Void fraction distribution in BWR fuel assembly and the evaluation of subchannel code. In: Ninokata H and Aritomi M eds., Subchannel Analysis in Nuclear Reactors, pp 141–167, Tokyo, Japan
Appendix 5.1: Some relevant constitutive relationship addressed in this analysis IVA computer code has absorbed in the last 25 years hundreds of constitutive relationships for variety of processes as described by Kolev (2007b). We address here only those that are subject of modification influencing the void fraction prediction in bundles. Note that for the computation of the drag coefficients the procedure described in Kolev (2007b). Ch. 2 was initially used. 1. Transition between bubbly and slug flow: If D1 > 6λRT α1, B −Ch = 0.25 , else if D1 < λRT α1, B −Ch = 0.3 , else α1, B − Ch = 0.3 − 0.01( D1 / λRT − 1) , where λRT is the
Raleigh-Taylor wavelength defined as follows λRT = σ 2
( g Δρ21 ) . For the cases
186
5. Core thermal hydraulic
considered here the Mishima and Ishii (1984) correlation for transition from slug
⎪⎧ ( C − 1) j + 0.35 VTB ⎪⎫ into churn turbulent flow α1, slug to churn = 1 − 0.813 ⎨ 0 does ⎬ j + 0.75 VTB b1 ⎩⎪ ⎭⎪ predict transition from bubble to slug but not to churn turbulent flow. Here the drift flux distribution coefficient for slug flow is C0 = 1.2 , the slug (Taylor bub0.75
ble) raising velocity is VTB =
ρ 2 − ρ1 gDh , and the mixture volumetric flux is ρ2 1/18
⎛ ρ − ρ1 ⎞ j = α1V1 + (1 − α1 )V2 , and b1 = ⎜ 22 gDh3 ⎟ η ρ / ⎝ 2 2 ⎠
.
2. Drag force for distorted bubbles: None modified Ishii and Chawla (1979) 2
⎛ 1 + 17.67 f 6 / 7 ⎞ 2 1.5 c = ( D1 / λRT ) ⎜ ⎟ ; f = (1 − α d ) . 3 ⎝ 18.67 f ⎠ d 21
3. Drag force for slug flow: None modified Ishii and Chawla (1979) d c21 = 9.8(1 − α1 )3 .
4. Drag force for distorted bubbles: Modified 2
⎛ 1 + 17.67 f 6 / 7 ⎞ 2 d c21 = 0.01 ( D1 / λRT ) ⎜ ⎟ . 3 ⎝ 18.67 f ⎠
5. Drag force for slug flow modified: Here I use the expression derived for cup bubbles instead for slugs multiplied by 0.1, 8 d c21 = 0.1 (1 − α1 ) 2 . 3
6. Instead of the primarily used model by Kataoka et al. (1983), Bertodano et al. (1998) model for entrainment was used here. It was verified for low pressure p = 1.2 bar and 10 < ρ 2 w2 4δ 2 F / η2 < 9700, 2.5 < α1 ρ1V1 Dh / η1 < 170 000, and 3
1, n = 1, nmax, Δzn = Δzdiv ,1cdiv
(
Δzdiv ,1 = ( L − zcrit )(1 − cdiv ) 1 − cdiv
where mmax + nmax = kmax.
nmax +1
(7.19)
),
(7.20)
212
7. Critical multiphase flow
7.3 Iteration strategy The mathematical formalism describing the critical flow do not allow mass flow rates larger than the critical because at negative pressure the thermo-physical properties for the both phases are not defined (exception – tension state of single phase liquid only). Therefore the critical mass flow rate G* can be approached only using increasing trial values of G1 , G2 ,..., G * . The value of the denominator of the pressure gradient N1 , N 2 ,..., ε is the controlled variable. The target is the small number ε → 0 . Simple way to reach the target is if the target is becoming negative to half the step ΔG and to repeat the computation as long the ΔG is becoming small enough. Optimization strategies are also possible. I recommended in Kolev (1986) to construct a polynomial based on the couples ( Gi , N i ) then to set N = 0 and to obtain the next guess for G * . In general, the description of the critical multiphase flows is based on the tools and methods already reported in Vols. 1 and 2 of this work. Historically many authors tried to avoid computation of the generation of one phase inside the other and the interfacial heat, mass and momentum transfer by introducing simplifying assumption. The present day computers allow solving the problem in its general form but considering different simplified models is good introduction in understanding the general method. Therefore I will start with some simple cases which are also very useful for particular applications.
7.4 Single phase flow in pipe 7.4.1 No friction energy dissipation, constant cross section Frictionless flow: For frictionless isentropic flow in constant cross section we have the following simplified momentum equation κ −1 1 2 dp p κ 1 dw + = 0 or dw2 + 0 dε κ = 0 2 ρ 2 ρ0 κ − 1
(7.21)
or w12 = w02 + 2
κ −1 ⎞ p0 κ ⎛ κ − 1 ε ⎜ 10 ⎟ , ρ0 κ − 1 ⎝ ⎠
(7.22)
7.4 Single phase flow in pipe
213
where ε = p p0 . For w0 = 0 this result reduces to the one obtained by de Saint Venant and Wantzel in 1838: w12 =
κ −1 ⎞ 2κ RT0 ⎛ ⎜ 1 − ε10κ ⎟ κ −1 ⎝ ⎠
(7.23)
or in terms of the Mach number κ
⎛ κ − 1 2 ⎞1−κ ε10 = ⎜ 1 + M1 ⎟ . 2 ⎝ ⎠
(7.24)
Theoretically the maximum velocity could be reached if the exit pressure is zero: 2 w1,max =2
p0 κ . ρ0 κ − 1
(7.25)
The local velocity of sound expressed in terms of the reference state for isentropic κ
change of state, p p0 = (T T0 ) κ −1 , is ⎛ p ⎞ a 2 = κ RT = κ RT0 ⎜ ⎟ ⎝ p0 ⎠
κ −1 κ
= κ RT0ε
κ −1 κ
.
(7.26)
The criticality conditions, w = a , delivers the critical pressure ratio κ
ε =p * 10
* 1
⎛ 2 ⎞ κ −1 p0 = ⎜ ⎟ . ⎝ κ +1⎠
(7.27)
This ratio gives the critical velocity w1*2 = 2
p0 κ , ρ0 κ + 1
(7.28)
see Oswatitsch (1952), Landau and Lifshitz (1987, p. 319). Note that the critical velocity is smaller than the maximum velocity for flow into vacuum. The temperature and the density in the critical states are computed from the energy and perfect gas equation, respectively T1* = T0 −
1 w1*2 , 2 cp
(7.29)
214
7. Critical multiphase flow 1
p* ⎛ 2 ⎞ κ −1 ρ = 1 * = ρ0 ⎜ ⎟ , RT1 ⎝ κ +1⎠ * 1
(7.30)
and κ +1
( ρ w )1
*
ρ0
⎛ 2 ⎞ 2(κ −1) . =⎜ ⎟ κ RT0 ⎝ κ + 1 ⎠
(7.31)
Problem 1: Air reservoir is connected with the environment with a nozzle. The nozzle cross section divided by the volume of the reservoir is F / V . The initial pressure is p0 and the initial temperature is T0 . Compute the time history of pressure and temperature if the discharge through the nozzle is critical and the gas obeys the properties of perfect gas. Solution to problem 1: The pressure and the temperature in the vessel are controlled by the equations dp dτ = −a 2 ( ρ w )1 F V ,
(7.32)
ρ c p dT dτ − dp dτ = 0 .
(7.33)
*
1,0
1,0
analytic num., time step 1s
0,8
0,9 0,8 T/T0, -
0,6 p/p0,-
analytic num., time step 1s
0,4
0,7 0,6
0,2
0,5
0,0 0
20
40 60 Time in s
80
100
0
20
40 60 Time in s
80
100
Fig. 7.3. Air release from 1 m³ vessel through short 1cm-diam nozzle. Pressure and temperature as a function of time
After inserting the expression for the critical mass flow rate and some mathematical manipulations the analytical solution for the temperature is obtained −2
κ +1 ⎡ ⎤ 1 1F ⎛ 2 ⎞ 2(κ −1) ⎢ + κ − 1) ⎜ κ R Δτ ⎥ , T= ( ⎟ ⎢ T ⎥ 2V ⎝ κ +1 ⎠ ⎣⎢ 0 ⎦⎥
(7.34)
7.4 Single phase flow in pipe
215
κ
p = p0 (T T0 ) κ −1 ,
(7.35)
see Kolev (2007a, Chap. 6). Of course one can solve numerically the pressure equation and control the accuracy by the analytical solution. An example with air at initial conditions 10 bar and 100°C in a 1 m³-vessel and 1 cm-nozzle is given in Fig. 7.3. Both methods gives practically undistinguishable solutions. Vena contracta: Flow behind sharp edged orifices first contracts before the following expansion. The smallest jet cross section is called vena contracta (vc). The criticality condition happens in this cross section. Therefore the net mass flow is a product of the vena contracta cross section and the critical mass flow rate. The vena contracta cross section divided by the orifices cross section is called contraction coefficient. Several authors observed experimentally that the contraction coefficient decreases from Cvc ,1 = 0.84 , Perri (1949), Arnsberg (1962), to Cvc ,2 = 0.61 with the increasing pressure ratio from 0 to 1 according to Cvc = Cvc ,1 − Cvc ,2 ( p p0 ) + ( 2Cvc ,2 − Cvc ,1 ) ( p p0 ) , 2
3
(7.36)
see Salet (1984). Actually Weisbach (1872) and Freeman (1888) reported for the first time measurements for water demonstrating that the contraction coefficient is a function of the cross section ratio of the two passages before and after the abrupt contraction, γ z = Fsmall Flarge . Benedict et al. (1966) approximated the Weisbach data with Cvc ,2 = 0.61631 + 0.13318γ z2 −0.26095γ z4 + 0.51146γ z6 ≤ 1 .
(7.37)
Flow with friction: In many gas-dynamics text books like Oswatitsch (1952), Shapiro (1953), Albring (1970), an approximate solution for the critical flow in a pipe is provided. The energy conservation for adiabatic flow neglecting the gravitation and the friction component in the energy equation is used in the following form
ρ wγ z
λ fr 1 2 + λ fr d ⎛ 1 2⎞ w ylim . ⎜ h + w ⎟ + γ v ρ wg z = γ v q&l′′′ + γ v Dh 2 dz ⎝ 2 ⎠ 8
(7.38)
In this case the energy conservation for perfect fluid simplifies to cp
1 dT 1 dw dT 1 dw2 or = − (κ − 1) M 2 , =− 2 dz T dz w dz dz
and allows to write the definition of the mach number,
(7.39)
216
7. Critical multiphase flow
M2 =
w2 , κ RT
(7.40)
or after its differentiating and using the energy conservation
1 dw 1 dM 2 , = 2 2 w dz ⎡⎣ 2 + (κ − 1) M ⎤⎦ M dz
(7.41)
and the mass conservation equation 1 + (κ − 1) M 2 1 dp dM 2 . =− 2 2 p dz ⎡⎣ 2 + (κ − 1) M ⎤⎦ M dz
(7.42)
This allows in the rearranged momentum equation
1 dw2 1 dp λ fr 1 2 w =0 + + 2 dz ρ dz Dh 2
(7.43)
for separation of the variables
λ fr 1− M 2 dM 2 = dz , Dh ⎛ κ −1 2 ⎞ 4 M ⎟M κ ⎜1 + 2 ⎝ ⎠
(7.44)
and for integration it analytically for constant friction coefficient ⎛ 2 ⎞ + κ − 1⎟ ⎜ ⎠ + 1 ⎛ 1 − 1 ⎞ = λ fr Δz , ⎜ ⎟ ⎞ κ ⎝ M 12 M 22 ⎠ Dh + − 1 κ ⎜ 2 ⎟ ⎝ M1 ⎠
(κ + 1) ⎝ M 22 ln 2κ ⎛ 2
(7.45)
Albring (1970) p. 315. Later the same equation is used by Brosche (1973, p. 313). The integrated Eq. (7.41) ⎛ κ −1 2 ⎞ M1 + 1 ⎟ M1 ⎜ 2 ε 21 = ⎜ ⎟ M 2 ⎜ κ −1 M 2 +1 ⎟ ⎜ ⎟ 2 ⎝ 2 ⎠
1/ 2
,
(7.46)
7.4 Single phase flow in pipe
217
together with the momentum equation presents a system of transcendental equations with respect to the both Mach numbers for known pressure ratio. Knowing them we obtain from the integrated Eq. (7.41) w2 = w1ε 21 M 22 M 12 .
(7.47)
For critical flow at the exit M 2 = 1 the integrated momentum equation simplifies to the transcendental equation f ( M1 ) =
(κ + 1) 2κ
ln
1⎛ 1 ⎜1 − 2 2 κ M 1 ⎝ + κ −1 M 12 1+ κ
−
⎞ λ fr Δz = 0 , ⎟− ⎠ Dh
(7.48)
which is Eq. (21.48) in Albring (1970, p. 315). This equation is transcendental with respect to the Mach number at the entrance of the pipe, M1, and has to be solved by iteration. The derivative of f, df = dM 1
2 (κ + 1)
2 − , ⎛ 2 ⎞ κ M 13 κ ⎜ 2 + κ − 1⎟ ⎝ M1 ⎠
(7.49)
is useful for designing Newton iteration method M 1 = M 1,old − f old
( df
dM 1 )
(7.50)
for solving with respect to M 1 . For frictionless flow equation (7.47) is satisfied only for M 0 = 1 . Friction flow with isentropic entrance: There are many applications in the industry for which the flow path goes through isentropic flow into the pipe entrance followed by frictional pipe flow. In these cases the combination of the Eqs. (7.46) and (7.24) allows excluding the intermediate pressure p1 resulting in 1 M =− + κ −1 2 2
2
1+ κ
⎛ p0 ⎞ 2 M 12 ⎛ κ − 1 2 ⎞1−κ M1 ⎟ + ⎜ ⎟ ⎜1 + 2 2 ⎠ (κ − 1) ⎝ p2 ⎠ (κ − 1) ⎝ 1
(7.51)
Brosche (1973). Now, instead of solving the system of Eqs. (7.45) and (7.46) with respect to M 1 and M 2 , we have to solve Eqs. (7.45) and (7.51). Then Eq. (7.46)
provides ε 21 = f ( M 1 , M 2 ) , Eq. (7.24) ε10 , Eq. (7.23) w1 , Eq. (7.47) w2 , the
218
7. Critical multiphase flow
energy conservation between the points 0 and 2, T2 = T0 − w22 gas equation of state ρ 2 = p2
( RT2 )
( 2c ) , the perfect p
and finally G = ρ 2 w2 .
In the case of critical flow, M 2 = 1 the Eq. (7.48) and Eq. (7.51) 1+ κ
p2 2 ⎛ κ − 1 2 ⎞ 2(1−κ ) 1+ M1 ⎟ = M1 p0 2 κ + 1 ⎜⎝ ⎠
(7.52)
define the M 1* and p2* . Note that one has always first to ask for the criticality condition and then to compare the external pressure p2 with p2* . Then if p2 < p2* the flow is critical and the exit pressure is p2* . Otherwise, the flow is sub-critical. This approach is verified by Brosche (1973) for L/Dh = 0, 100 and 350 using some of the exhaustive measurements reported by Frössel (1936). Figure 7.4 provides the comparison of the prediction of the theory of this chapter with the measurements. Frössel (1936) reported also an interesting approximation for the data with air and almost atmospheric pressure at the inlet, G GL = 0 = 0.916 × ⎡⎣ L (10 Dh ) ⎤⎦
0.61
,
that can be used as initial value for some surge strategies. 1,0
L/Dh 0 100 350
0,8 G/GL=0, -
Exp.
0,6
100 350
0,4 0,2 0,0 0,0
0,2
0,4 0,6 p/p0, -
0,8
1,0
Fig. 7.4. Mass flow rate ratio as a function of the inlet outlet pressure ratio. Dh = 0.02 m, p0 = 1 bar, T0 = 20°C, air. Experimental data reported by Frössel (1936)
The solution summarized in this section can be used also as benchmarks for system computer codes. I give an example in Fig. 7.5 where the prediction of IVA computer code for air flow is compared with the prediction of the perfect gas theory. The slight deviations are explained with the use of real gas properties in the computer code along the tube, where the analytical solution is evaluated with perfect gas properties.
7.4 Single phase flow in pipe
219
40
Mass flow in g/s
35 30 25 20
Dh=0.02m, L/Dh=350, air, p0=1bar,T0=20°C Perf. gas theory IVA
15 10 5 0 0,0
0,2
0,4 0,6 p/p0, -
0,8
1,0
Fig. 7.5. Comparison between the prediction of the perfect gas theory and IVA computer code
Problem 2: Consider the same case as in Problem 1. Instead of short nozzle consider pipes with L/Dh = 100 and 350. Solution to problem 2: The solution is presented in Fig. 7.6. We see that the increasing friction reduces the outflow.
1,0 0,8
0,9 0,8 T/T0,-
p/p0, -
0,6 0,4 0,2 0,0
L/Dh=0 L/Dh=100 L/Dh=350
1,0
L/Dh=0 L/Dh=100 L/Dh=350
0,7 0,6
0
20
40 60 Time in s
80
100
0,5 0
20
40 60 Time in s
80
100
Fig. 7.6. Air release from 1 m³ vessel through 1 cm-diam pipes. Pressure and temperature as a function of time.
7.4.2 General case, perfect gas The dissipation of heat in long pipes due to friction with the pipe wall is important. Neglecting it results in a much stronger temperature drop then the observed in the nature. Therefore the dissipation has to be taken into account in such cases. The mass momentum and energy conservation equations for the general case of compressible single phase flow in pipes with variable geometry are:
220
7. Critical multiphase flow
d 1 dw 1 d ρ 1 dγ z + =− , ( ρ wγ z ) = 0 or dz γ z dz w dz ρ dz
(7.53)
λ fr 1 2 ⎞ γ v 1 dw2 1 dp ⎛ w ⎟ =0, + + ⎜ gz + 2 dz ρ dz ⎝ Dh 2 ⎠ γ z
(7.54)
ds γ v λ fr 1 2 + λ fr w ylim T + q& ′′′ ( ρ wγ z ) = sz . = 8 dz γ z Dh 2
(7.55)
Here γ z is the local cross section divided by some normalizing cross section Fn . The maximum cross section in the pipe is a good choice. γ v is the local flow volume over Δz divided by ΔzFn . Assuming that the flow is a perfect gas ρ =
(c
− R ) , a 2 = κ RT and s = c p ln
p , RT
T p − R ln and rewriting the above T0 p0 system in terms of temperature and pressure I obtain
κ = cp
p
1 dw 1 dp 1 dT 1 dγ z + − =− , γ z dz w dz p dz T dz
(7.56)
λ fr 1 2 ⎞ γ v ⎛ 1 dw2 RT dp w ⎟ , + = − ⎜ gz + p dz Dh 2 ⎠ γ z 2 dz ⎝
(7.57)
1 dT R dp sz − = . T dz c p p dz c p
(7.58)
Solving with respect to the derivatives I finally obtain: ⎛ sz
dp =− dz
ρ w2 ⎜⎜
⎝ cp
−
1 dγ z γ z dz
⎞ λ fr 1 2 ⎞ γ v ⎛ w ⎟ ⎟⎟ + ρ ⎜ g z + Dh 2 ⎠ γ z ⎝ ⎠ , w2 1− 2 a
(7.59)
λ fr 1 2 ⎞ γ v sz 1 d γ z 1 ⎛ w ⎟ − + 2 ⎜ gz + Dh 2 ⎠ γ z 1 dw sz 1 d γ z 1 dp c p γ z dz a ⎝ , = − − = w dz c p γ z dz κ p dz w2 1− 2 a (7.60)
7.5 Simple two phase cases for pipes and nozzles
dT T ⎛ R dp ⎞ = ⎜ sz + ⎟. dz c p ⎝ p dz ⎠
221
(7.61)
The obtained system of non-homogeneous non-linear ordinary differential equation indicates important behavior immediately. For constant pipe cross section γ z = const , the nominator of the pressure gradient is positive. For sub-critical flow, w < a , the pressure gradient is therefore negative. Consequently, the velocity gradient in the second equation is positive. If the pipe is long enough for a given pressure difference acting at the both ends and the velocity approach the sound velocity at the exit of the pipe, w → a , the pressure gradient tends to minus infinity. As already mentioned, we call such flow critical single phase flow. Problem 3: Consider the same case as in Problem 2. Simulate the process with a general purpose computer code with low spatial resolution – 20 computational cells. Solution to problem 3: The solution using IVA computer code is presented in Fig. 7.7. We see that even with very low resolution the computer code prediction is reasonable.
1,0 0,8
0,9 0,8 T/T0,-
p/p0, -
0,6 0,4 0,2 0,0
L/Dh=0 L/Dh=100 L/Dh=350 L/Dh=100 IVA L/Dh=350 IVA
1,0
L/Dh=0 L/Dh=100 L/Dh=350 L/Dh=100 IVA L/Dh=350 IVA
0,7 0,6
0
20
40 60 Time in s
80
100
0,5 0
20
40 60 Time in s
80
100
Fig. 7.7. Air release from 1 m³ vessel through 1 cm-diam pipes. Pressure and temperature as a functions of time.
Because the multi-phase flows are compressible flows they obey also such behavior in pipes. We learn on the above examples haw to proceed in multiphase flows to.
7.5 Simple two phase cases for pipes and nozzles Not considering the evolution of the particular flow pattern and the topology of the two phase flow neglects important physical information that was replaced in the past by simplifying assumptions ranging from the assumption for the so-called
222
7. Critical multiphase flow
frozen flow to the assumption for the so-called equilibrium flow. In the idealized frozen flow the participating phases are assumed not to exchange mass and energy. In the idealized thermal equilibrium flow the participating phases are assumed to exchange so much mass and energy as needed in order to keep thermodynamic equilibrium at any time and at any point. Within each of this assumption two additional assumptions are usually made: equal or non equal averaged phase velocities. The first assumption is described in the literature by the term homogeneity, which is not exact reflection of the physics, and the second as non-homogeneity. Many authors assumed also that the two phase mixture is isentropic. One should bear in mind what exactly this assumption means. To understand this let as write the entropy equation for two-fluid mixture being in steady state and neglecting the turbulent diffusion terms:
{
}
d γ z ⎡⎣α1 ρ1 s1 w1 + (1 − α1 ) ρ 2 s2 w2 ⎤⎦ dz
⎡ 2⎛ μ q& ′′′ q& ′′′ 1 μ ⎞⎤ = γ v ⎢( μ 21 − μ12 )( s12σ − s12σ ) + 1 + 2 + ( w1 − w2 ) ⎜ 21 + 12 ⎟ ⎥ T1 T2 2 T2 ⎠ ⎦⎥ ⎝ T1 ⎣⎢
1 ⎡α l ρl ( Pk ,l + δ l εη ,l + ε l′ ) + El′* ⎤ . ⎦ ⎣ T l =1 l 2
+γ v ∑
(7.62)
This equation is simplification of the exact equation given in Chap. 5 of Vol. 1 of this monograph. Here, the surface entropies within each of the phases are slmσ = sl +
hlmσ − hl − ( Δplmσ − Tl ) ρl Tl
,
(7.63)
Δplmσ is the difference between the bulk averaged pressure and the surface pressure inside of the file field l. Tl is the viscous stress at the surface of field l.
( Δp
mσ l
− Tl ) is the averaged value over all appearances of the interface
γ vα l ρl εη ,l = α leγ ⋅ ( Tη ,l : ∇ ⋅ Vl )
mσ l
.
(7.64)
is the irreversible dissipated power caused by the viscous forces due to deformation of the mean values of the velocities in the space,
γ vα l ρl ε l′ = α leγ ⋅ ( Tl′ : ∇ ⋅ Vl′
)
(7.65)
7.5 Simple two phase cases for pipes and nozzles
223
is the irreversibly dissipated power in the viscous fluid due to turbulent pulsations, and
γ vα l ρl Pk ,l = α leγ ⋅ ( Tl′ : ∇ ⋅ Vl )
(7.66)
is the power needed for production of turbulence. By modeling of the turbulence usually the last term is removed from the energy conservation equation and introduced as a generation term for the turbulent kinetic energy. Now let us answer the question what isentropic two-fluid flow means? It is a flow characterized by the equation
{
}
d γ z ⎡⎣α1 ρ1 s1 w1 + (1 − α1 ) ρ 2 s2 w2 ⎤⎦ = 0 dz
(7.67)
d ⎡ X 1 s1 + (1 − X 1 ) s2 ⎤ = 0 ⎦ dz ⎣
(7.68)
X 1 s1 + (1 − X 1 ) s2 = const ,
(7.69)
or
or
where the gas mass flow concentration is defined by X 1 = α1 ρ1 w1 G .
(7.70)
The reader immediately recognizes that isentropic flow does not exist in the nature because the irreversible energy dissipation reflected by the neglected terms always happens. In the case of steady state isentropic mixture being in thermodynamic equilibrium we have d ⎡⎣ X 1, eq s ′′ + (1 − X 1, eq ) s ′⎤⎦ −
2 μ dz 1 w1 − w2 ) 21 = 0 ( GT ′ 2
(7.71)
Using the vapor mass conservation
μ 21dz = GdX 1,eq
(7.72)
224
7. Critical multiphase flow
results in d ⎣⎡ X 1, eq s ′′ + (1 − X 1, eq ) s ′⎦⎤ −
2 dX 1, eq 1 w1 − w2 ) =0 ( T′ 2
(7.73)
or solved with respect to the pressure derivative of the equilibrium mass flow rate fraction ds′′ ds ′ + (1 − X 1, eq ) dp dp =− . 2 1 1 dp s ′′ − s ′ − w1 − w2 ) ( T′ 2
dX 1,eq
X 1,eq
(7.74)
Many authors use this equation neglecting the entropy production due to evaporation from a liquid with velocity smaller then the vapor velocity which means set2 1 1 w1 − w2 ) = 0 . ting ( T′ 2 7.5.1 Subcooled critical mass flow rate in short pipes, orifices and nozzles It was experimentally observed that outflow from vessels filled with water having temperature close to the saturation temperature through short pipes L/Dh < 12, orifices and nozzles happens slower then of the discharge flow rate is computed using the Bernoulli equation. Theories based on thermodynamic equilibrium predict discharge that is much slower than in the reality. Therefore, there is evaporation, but a delayed evaporation, in the stream. As already mentioned in Sect. 12.7 of Vol. 2 of this monograph, the most important outcome of the heterogeneous nucleation theory for the engineering design practice is the possibility to compute in a simple manner the critical mass flow rate in short pipes, orifices and nozzles for saturated and subcooled water. Probably Burnell (1947) was the first approximating critical flow in nozzles with saturated and subcooled inlet condition modifying the Beronulli equation as follows G * = 2 ρ 2 ( po − pFi )
(7.75)
where the critical pressure is set to the flashing inception pressure pFi = p ′(T2 ) − ΔpFi .
(7.76)
Jones (1982) described successfully critical mass flow rate in nozzles modifying Eq. (7.75) to
7.5 Simple two phase cases for pipes and nozzles
G * = ( 0.93 ± 0.04 ) 2 ρ 2 ( po − pFi )
225
(7.77)
The error of this approach was reported to be ± 5% for inlet conditions of p = (28 to 170) × 105 Pa and T2 = 203 to 288°C and use of the Algamir and Lienhard (1981) correlation. Fincke (1984) found for the discharge coefficient instead of 0.93 the value 0.96 and reported that this coefficient does not depend on the Reynolds number.
Gcr,comp in Mg/(m²s)
80 70
Subcooled water: Sozzi and Sutherland (1975) +20% -20%
60 50 50
60 70 Gcr,exp in Mg/(m²s)
80
Fig. 7.8. Measured versus computed critical mass flow rates for subcooled water through converging-diverging nozzle
I made an additional assessment of this approach using the data for subcooled water by Sozzi and Sutherland (1975) for converging-diverging nozzle of 12.7 mm minimum diameter. Using as a throat pressure the pressure obtained by Algamir and Lienhard (1981) results in strong under prediction of the data. Using G* = 0.93 2 ρ 2 ( po − p1 )
(7.78)
I obtain mean error of 6.8% as shown in Fig. 7.8. 7.5.1 Frozen homogeneous non-developed flow Polytrophic state change of the gas phase: Consider frictionless homogeneous flow in a pipe with constant cross section in which there is no mass transfer and the gas phase obeys polytrophic change of state for which the following relations are valid n
p p0 = (T T0 ) n −1 ,
(7.79)
1
ρ ρ0 = ( p p0 ) n .
(7.80)
226
7. Critical multiphase flow
The momentum equation is 1 2 dp dw + =0, 2 ρ
(7.81)
where the density of the mixture 1
ρ
=
X1
ρ1
+
1 − X1
ρ2
,
(7.82)
in case of no mass transfer X 1 = const ,
(7.83)
changes only due to the change of the density of the gas. The following transformation will follow exactly the already known way from the single phase fluid dynamics and will be very close to those reported by Tangren et al. (1949) and Ziklauri et al. (1975, p. 77), with the difference, that I use here instead the void fraction the mass concentration as a dependent variable for convenience because it is a constant. The density can be then rewritten as a function of the reference state at the entrance of the channel 1
ρ
=
X1
ρ0 ( p p0 )
1 n
+
1 − X1
ρ2
.
(7.84)
Replacing in the momentum equation results in ⎛ ⎞ 1 2 ⎜ A dw + 1 + B ⎟ d ε = 0 ⎜ n ⎟ 2 ⎝ε ⎠
(7.85)
where A = X 1 p0 ρ10 and B = (1 − X 1 ) p0 ρ 2 are constants. Taking the non determined integral results in ⎡ 1 2 n nn−1 ⎤ dw + d ⎢ Bε + A ε ⎥ = 0 for n ≠ 1 , n −1 2 ⎣ ⎦
(7.86)
1 2 dw + d ( Bε + A ln ε ) = 0 for n = 1 . 2
(7.87)
Integrating between entrance and the exit state results in
7.5 Simple two phase cases for pipes and nozzles
w2 = w02 + 2 B (1 − ε ) + 2 A
n −1 ⎞ n ⎛ n 1 − ε ⎜ ⎟ for n ≠ 1 , n −1 ⎝ ⎠
227
(7.88)
w2 = w02 + 2 B (1 − ε ) − 2 A ln ε for n = 1 .
(7.89)
Comparing these equations with the gas flow solution we realize that the terms containing the B-term appear due to presence of the liquid. For initial velocity and exit pressure equal to zero we have n ⎞ ⎛ 2 wmax = 2⎜ B + A ⎟ for n ≠ 1 , n −1 ⎠ ⎝
(7.90)
2 wmax = 2 ( B − A ) for n = 1 .
(7.91)
Under all assumptions introduced here the velocity of sound for any ε between the entrance and the exit of the nozzle is 2
a2 = 1
dρ dv n ρ 0 p0 = − v2 ε = dp dp X1
n +1 n
⎛ ⎞ ⎜ X1 + 1 − X1 ⎟ , 1 ⎜ ρ 2 ⎟⎟ ⎜ρ εn ⎝ 0 ⎠
(7.92)
for X 1 > 0 . The critical pressure ratio is obtained by setting the mixture velocity equal to the sound velocity in the integrated momentum equations 2
n ρ10 ε* 2 X1
n +1 n
⎛ ⎞ n −1 ⎜ X 1 + 1 − X 1 ⎟ = 1 − X 1 1 − ε * + X 1 n ⎛ 1 − ε * n ⎞ for n ≠ 1 , ( ) ⎜ ⎟ 1 ⎜ ρ 2 ⎟⎟ ρ2 ρ10 n − 1 ⎝ ⎠ ⎜ ρ ε *n ⎝ 0 ⎠ (7.93) 2
nρ0 ε* 2 X1
n +1 n
⎛ ⎞ ⎜ X 1 + 1 − X 1 ⎟ = 1 − X 1 1 − ε * − X 1 ln ε * for n = 1 . ( ) 1 ⎜ ρ 2 ⎟⎟ ρ2 ρ10 ⎜ ρ ε *n ⎝ 0 ⎠ (7.94)
These are transcendental equations that have to be solved numerically for the critical pressure ratio. Then the critical velocity is computed by using Eqs. (7.88) and (7.89) for ε * . Observe, that unlike in single phase flow, where ε * is constant, in two-phase flow ε * depends of the gas mass fraction and on the liquid properties. Discharge of mixtures of inert gases with subcooled liquids in a short nozzles is
228
7. Critical multiphase flow
well predicted by this theory provide the conditions are not very close to pure liquid conditions at the entrance of the flow. Problem 4: Consider flow of water and air at 20°C. The internal pressure and the gas mass fraction are given in the Table 7.1. Compute the critical pressure ratio using the above theory and compare it with the measurements by Tangren et al. (1949). Solution: This model gives increasing critical pressure ratio with increasing gas mass fraction up to X 1 ~ 0.2 and then almost constant values. The results are given in the following table. The model is of course not appropriate for evaporating flows. At higher pressure the interfacial heat transfer is also influencing the flow. * ε exp 0.495 0.481 0.556 0.585 0.608 0.637
p0 in bar
X1
0.162 0.213 0.330 0.456 0.576 0.668
2.05 2.25 2.39 2.81 3.36 3.77
* ε comp , n = 1.32 0.527 0.528 0.528 0.528 0.528 0.528
* ε comp ,n=1 0.605 0.606 0.606 0.606 0.606 0.606
7.5.2 Non-homogeneous developed flow without mass exchange Let as consider the following set of assumptions: Two velocity fields; S = w1 w2 = constant; the phases are adjusting the same temperature with the pressure change (instant heat exchange). These assumptions define the so called equilibrium non-homogeneous model: dp + G 2 ( dvI ) X + 1
1 λ fr ,2 o v′G 2 Φ 22 o dz = 0 , 2 Dh
(7.95)
where 1 ⎤ ⎡ ⎛ ⎞ vI = f 0 ⎡⎣ X 1v1 + S (1 − X 1 ) v2 ⎤⎦ = f 0 ⎢ X 1 ⎜ ρ10ε n ⎟ + S (1 − X 1 ) v2 ⎥ , ⎥⎦ ⎢⎣ ⎝ ⎠
f0 =
1 + ( S − 1) X 1 S
,
(7.96)
(7.97)
7.5 Simple two phase cases for pipes and nozzles
229
⎛ dv ⎞ ⎡ dv dv ⎤ 1 = − ⎜ I ⎟ = − f 0 ⎢ X 1 1 + S (1 − X 1 ) 2 ⎥ , *2 dp ⎦ G ⎝ dp ⎠ X1 ⎣ dp n +1 ⎡ ⎡ X S (1 − X 1 ) ⎤ ⎛ ⎞ 1 − X1 ⎤ = f0 ⎢ 1 + ⎥ = f 0 ⎢ X 1 ⎜ n ρ10 p0ε n ⎟ + S1 *2 ⎥ *2 G2 G2 ⎦⎥ ⎝ ⎠ ⎣⎢ n ρ1 p ⎦⎥ ⎣⎢
(7.98)
We assumed above that the polytrophic change of the gas state is described by 1
ρ1 ⎛ p ⎞ n =⎜ ⎟ . ρ10 ⎝ p0 ⎠
(7.99)
Solving the integrated momentum equation with respect to the mass flow rate results in G2 = −
Δp
( ΔvI ) X
1 λ fr ,2o + ∫ v′Φ 22o dz 2 0 Dh L
1
=
ρ10 p0 (1 − ε )
, ⎛ ⎞ L λ 1 1 ,2 fr o v′Φ 22 o dz f 0 X 1 ⎜ 1 − 1⎟ + ∫ ⎜ n ⎟ 2 0 Dh ⎝ε ⎠ (7.100)
Equalizing the both mass flow rates we obtain the criticality condition. Note that only the friction term makes possible the establishment of developed flow. Without friction the flow is non-developed and the above formalism does not apply. 7.5.3 Equilibrium homogeneous flow If the flow starts from a large vessel the initial velocity is almost zero. Therefore the flow has to be accelerated to the end steady state velocity. Such flow is called developing flow. Note that in this case the mass flow rate reaches some finite value after starting from zero. Other idealization is the flow in long pipes in which the mass flow rate is constant. In this case the inlet acceleration is neglected. Next we consider the two cases separately. 7.5.3.1 Developing flow Assuming that liquid and its vapor are always in a thermodynamic equilibrium and undergoes isentropic change of state s = X 1eq s ′′ + (1 − X 1eq ) s ′ = s0 = const
(7.101)
230
7. Critical multiphase flow
results in X 1eq =
s0 − s ′ = f ( s0 , p ) . s ′′ − s ′
(7.102)
The specific mixture volume is then
v = X 1eq v′′ + (1 − X 1eq ) v′ = s0
v′′ − v′ s′′v′ − s′v′′ + = s0 f1 + f 2 . s ′′ − s ′ s ′′ − s ′
(7.103)
Note some important differential relations for the equilibrium mixtures: ⎛ ∂v ⎞ ⎛ df df ⎞ ⎛ ∂v ⎞ dv = ⎜ ⎟ dp + ⎜ ⎟ ds = ⎜ s 1 + 2 ⎟ dp + f1ds , ⎝ ∂s ⎠ p ⎝ ∂p ⎠ s ⎝ dp dp ⎠ dρ =
dp ⎛ ∂ρ ⎞ 1 ⎛ df1 df 2 +⎜ + ⎟ ds = − 2 ⎜ s 2 a ⎝ ∂s ⎠ p v ⎝ dp dp
⎞ f1 ⎟ dp − 2 ds . v ⎠
(7.104)
(7.105)
The sound velocity under these assumptions is therefore
a2 = 1
df df ⎞ dρ 2 ⎛ = − ( s0 f1 + f 2 ) ⎜ s0 1 + 2 ⎟ . dp ⎝ dp dp ⎠
(7.106)
We recognize that the first pressure function in Eq. (7.103) is nothing else then the Clapayron’s equation from 1834: dT v ′′ − v′ = = f1 . dp s ′′ − s ′
(7.107)
Therefore two approximations only are necessary for f1 ( p ) and f 2 ( p ) to integrate the momentum equation 1 2 dw + ( s0 f1 + f 2 ) dp = 0 . 2
(7.108)
For entrance velocity equal to zero the result is p
w2 = −2 ∫ ( s0 f1 + f 2 ) dp . p0
(7.109)
7.5 Simple two phase cases for pipes and nozzles
231
From the criticality condition w
2
( p*, p0 ) = a ( p *) 2
p*
or −2 ∫ ( s0 f1 + f 2 ) dp = a*2
(7.110)
p0
the critical pressure at the exit can by calculated iteratively and then the critical velocity. To analyze in a very simple manner discharge of two phase mixture from a vessel the following information is very practical. The pressure change of equilibrium volume is computed by applying the volume conservation equation as already described in Kolev (2007a): 1 dp
ρ a 2 dτ
= μ1 v′′ + μ 2 v′ ,
(7.111)
Where μ1 and μ 2 are the mass sources per unit time and unit volume of the gas and vapour. The change of the equilibrium vapour mass fraction comes from the mass conservation:
dρ = μ w1 + μ w 2 − μ1w − μ2 w . dτ
(7.112)
Here the subscripts w indicates external site. Problem 5: Given a vessel filled with saturated water and steam at 50 bar. The water mass is 145 t, the vapour mass 5 t. A valve to a pipe with 0.299 m diameter and 1.18 m length opens at the beginning at the process and closes after 80 s. The discharge happens only from the steam space. Compute the pressure as a function of time. Solution to problem 5: I use the computational method for single phase flow to compute the critical discharge mass flow rate ( ρ w )1 . Assuming that the mixture is always in saturation the pressure change is controlled by the simple form of the volume conservation equation from Chap. 5 in Kolev (2007a) *
dp * F = −a 2 ( ρ w )1 v′′ v . dτ V
(7.113)
Here F is the cross section and V is the vessel volume. Having the pressure all saturation properties can be computed. Then from the mass conservation equation
232
7. Critical multiphase flow
dρ * F = − ( ρ w )1 , dτ V
(7.114)
and therefore v = 1 ρ and X 1eq are easily computed. For the computation of the sonic velocity for homogeneous mixture a2 = 1
⎛ df df ⎞ dρ = − v2 ⎜ s 1 + 2 ⎟ , dp ⎝ dp dp ⎠
(7.115)
I use the approximations for water f1 = a11 +
a12
− f 2 = a21 +
p
+
a22 p
a a13 df1 a , = − 123/ 2 − 132 , 2p p dp p +
(7.116)
a23 a df a , − 2 = − 223/ 2 − 232 , p 2p dp p
(7.117)
55 50
Marviken Exp. Comp.
45 p in bar
40 35 30 25 20 15 0
20
40 60 Time in s
80
Fig. 7.9. Comparison between the computed and measured pressure as a function of time for the Marviken experiment T-11, see in Grolmes et al. (1986)
70 60 GE test 1004.3 Comp.
p in bar
50 40 30 20 10 0 0
50
100
150
200
250
300
Time in s
Fig. 7.10. Comparison between the computed and measured pressure as a function of time for the General Electric experiment 1004.3, see in Hassan et al. (1985)
7.5 Simple two phase cases for pipes and nozzles
80
Vessel: 22cm-diam, 13.367m-hight Nozzle: 6mm-diam, 15cm-length Initial state: saturation p0 = 80bar, zlevel,0 = 12.3m Simple model, Cvc=0.9 Umminger et al. (2007) exp.
70
50
40 Vessel: 22cm-diam, 13.367m-hight Nozzle: 8.5mm-diam, 15cm-length Initial state: saturation p0 = 40bar, zlevel,0 = 12.19m Simple model, Cvc=0.8 Umminger et al. (2007) exp.
35 30 p in bar
p in bar
60
233
40 30 20
25 20 15 10
10 0
200
400 600 Time in s
800
1000
5 0
200
400 600 Time in s
800
1000
Fig. 7.11. Comparison between the computed and measured pressure as a function of time for the AREVA test, see in Umminger et al. (2007): (a) Initial pressure 80 bar; (b) Initial pressure 40 bar
where a11= –3.47552d-6, a12 = 0.02834, a13 = 18.95183, a21 = – 0.00658, a22 = 91.86452, a23 = 7255.25586. For the computation of the specific volume of the mixture accurate approximation for the specific volumes of the phases have to be used and not the above approximation because it will amplify the error. I use also slight contraction of 0.95 of the cross section. The result is presented in Fig. 7.9. For the simplicity of this model the agreement is surprisingly good. Problem 6: Given a vessel with volume of 0.31156 m³ filled with saturated water and steam at 69.7 bar. The water mass is 177.22 kg, the vapour mass is 2.9175 kg. A valve to a pipe with 0.00925 m diameter and 1m length opens at the beginning at the process. The discharge happens only from the steam space. Compute the pressure as a function of time. Solution: The solution is presented in Fig. 7.10. The contraction coefficient is set to 1. Again we find good prediction by this simple theoretical approach. Two additional computations are shown in Figs. 7.11(a) and 7.11(b) with geometry and initial condition given in the pictures. Using the same formalism as before I obtain the results presented in the pictures. It is obvious that this simple method of prediction of discharge from the gas space of the vessel filled initially with saturated water works well. Problem 7: Given the geometry and initial conditions as presented in Fig. 7.11. Analyze the void fraction distribution along the high over the time. Solution: This task is the so called “classical levels swell analysis”. The steam mass volumetrically generated inside the liquid with the pressure drop is removed from the liquid in forms of bubbles, slugs and continuum with finite velocity. At
234
7. Critical multiphase flow
the beginning of the process the removed vapour below the initial level is slower then the generation. This results in a two-phase level swell and then of its decrease. The phenomenon is essential for the safety function of nuclear reactors. I will perform here a very approximate analysis following the ideas reported by Grolmes et al. (1986). The mass flow of steam living the vessel is ( ρ w )1 F . The *
( ρ w )1 F *
mass flow per unit volume of the liquid
α 2 zTP Fv
is, in accordance with the equi-
librium assumption, equal to the averaged generation of steam per unit time and unit liquid volume. The local generation of steam mass per unit time and unit mixture volume is then
( ρ w )1 F * *
μ21 ( z ) = α 2 ( z )
α 2 zTP Fv
.
(7.118)
The quasi steady state steam mass conservation equation for small changes of the pressure is dj1 μ21 ( z ) = . ρ ′′ dz
(7.119)
Here j1 = α1 w1 is the vapour volume flux called in the literature superficial vapour velocity. At this place of the development the following assumptions are possible: (a) the vapour rise velocity is a quasi constant, e.g. the Kutateladze large bubble free rising velocity, 1/ 4
⎛ σ g Δρ 21 ⎞ Δw12 ≈ Δw12, Ku = 2 ⎜ ⎟ 2 ⎝ ρ2 ⎠
,
(7.120)
and (b) the averaged vapour volumetric flow rate is well described by the drift flux model j1 = α1C0 ( j1 + j2 ) + α1V1*j
with j1 >> j2 and C0 ≈ 1 resulting in j1 = Δw12 dα1 2 μ 21 ( z ) = (1 − α1 ) ρ ′′Δw12 dz
or
(7.121)
α dα or dj1 = Δw12 . 2 1−α (1 − α ) (7.122)
7.5 Simple two phase cases for pipes and nozzles
dα 2
α
3 2
( ρ w )1
235
*
=−
z A z F* 1 =− , d d ′′ ρ Δw12 Fv α 2 zTP α 2 zTP
(7.123)
where
( ρ w)1
*
A=
F* . ρ ′′Δw12 Fv
(7.124)
The integration from the bottom to arbitrary level below the surface gives
⎛ A z ⎞ α 2 = ⎜1 + 2 ⎟ ⎜ α 2 zTP ⎟⎠ ⎝
−1/ 2
,
(7.125)
The high averaged liquid volume fraction is
α2 =
1 zTP
⎛ ⎜ 0⎝
zTP
1
∫ α 2 ( z ) dz = ∫ ⎜1 + 0
2A z α 2 zTP
⎞ ⎟⎟ ⎠
−1/ 2
d
z = zTP
2 1/ 2
⎡ ⎛ 2 A ⎞⎤ 1 + ⎢1 + ⎜ ⎟⎟ ⎥ ⎜ ⎣⎢ ⎝ α 2 ⎠ ⎦⎥
. (7.126)
This implicit relation allows to solve with respect to α 2 ,
α2 =
2 . 2+ A
(7.127)
Thus, the liquid volume fraction profile is uniquely defined by ⎡
α 2 ( z ) = ⎢1 + A ( 2 + A ) ⎣
z ⎤ ⎥ zTP ⎦
−1/ 2
,
(7.128)
Grolmes et al. (1986). The minimum, as intuitively expected, is at the top of the two phase mixture
α 2,min = ⎡⎣1 + A ( 2 + A ) ⎤⎦
−1/ 2
.
(7.129)
With this profile it is possible to compute the averages within any segment Δz below the surface
236
7. Critical multiphase flow 1/ 2 1/ 2 z ⎛ zTP 2 ⎡⎛ z2 ⎞ z1 ⎞ ⎤ 1 2 ⎢⎜ 1 + B = α 2 ( z ) dz = ⎟ − ⎜1 + B ⎟ ⎥ Δz B ⎢⎝ Δz z∫1 zTP ⎠ zTP ⎠ ⎥ ⎝ ⎦ ⎣
α 2, Δz
(7.130)
for z2 < zTP ,where B = A ( 2 + A) .
(7.131)
The cross check of the correctness of the derivation is achieved by setting Δz = z2 = zTP , z1 = 0 resulting in
α2 =
2 2+ A
(7.132)
which is the expected result. Note that α 2, Δz = 1 for z1 > zTP . If the two phase upper limit level is between the two taps we have
α 2, Δz =
1 z2 − z1
1/ 2 ⎧ ⎛ z1 ⎞ ⎤ ⎪⎫ 2⎡ 1/ 2 ⎪ ⎢ z z z B B 1 1 − + + − + ( ) ⎨ 2 TP TP ⎜ ⎟ ⎥⎬ B⎢ zTP ⎠ ⎥ ⎪ ⎝ ⎪⎩ ⎦⎭ ⎣
(7.133)
for z1 < zTP < z2 . The volume flow rate of vapour leaving the two phase mixture can be approximated by j1 ≈ α1,max C0 j1 + α1,maxV1*j or j1 =
α1,maxV1*j . 1 − α1,max C0
(7.134)
The change of the mixture mass below the level is then dM TP = − ρ ′′ j1 Fv . dτ
(7.135)
Having in mind that the two phase mixture mass is M TP = ⎡ ρ ′ − α1 ( ρ ′ − ρ ′′ ) ⎤ Fv zTP ⎣ ⎦
the two-phase level is controlled by the ordinary differential equation
(7.136)
7.5 Simple two phase cases for pipes and nozzles
1.0
Void fraction, -
0.8
averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
0.6 0.4 0.2 0.0 0
200
400
600
800 1000 1200
Time in s
(a) 1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0
0
200
400 600 800 1000 1200 Time in s
(b) 1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0
(c)
0
200
400 600 800 1000 1200 Time in s
237
238
7. Critical multiphase flow
1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0 0
200
400 600 800 1000 1200 Time in s
(d) Fig. 7.12. Vessel discharge starting from 80 bar. Void fractions along the high segments as a function of time: (a) Experiment by Umminger et al. (2007); Simple model with (b) Δw12 = 2Δw12, Ku ; (c) Δw12 = 3Δw12, Ku ; (d) Δw12 = 4Δw12, Ku
zTP ( ρ ′ − ρ ′′ ) d α1 ρ ′′ j1 Fv dzTP =− + . dτ ρ ′ − α1 ( ρ ′ − ρ ′′ ) ρ ′ − α1 ( ρ ′ − ρ ′′ ) dτ
(7.137)
Umminger et al. (2007) generated the experimental results presented in Figs. 7.12(a) and 13(a). The computed results are presented in Figs. 7.12(b) through 7.12(d) and Figs. 7.13(b) through 7.13(d). It is surprising haw many important features of the process are reproduced by such simple model. I drew the following conclusions from this simple analysis. (a) This simple model reproduces the qualitatively the experimentally observed process; (b) Quantitative agreement with the data is obtained if the effective vapour rise velocity is increased up to 4 times then the Kutateladze velocity. This is indication that other flow pattern, especially in the upper part of the mixture exists for which the effective velocity is much higher then the free rise velocity for large bubbles and that there is distribution of the vapour and of the relative velocity across each horizontal cross section other then the assumed in the drift flux models obtained from small diameter pipe experiments; (c) The larger the effective vapour rise velocity the smaller the local maximum void and the faster the decrease of the mixture level swell; (d) The initial inertial phase of increasing the void locally is not represented by this simple model; (e) The larger the pressure level the better the agreement with the simple model.
7.5 Simple two phase cases for pipes and nozzles
1.0
Void fraction,
0.8 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
0.6 0.4 0.2 0.0 0
200
400 600 800 1000 1200 Time in s
(a)
1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0 0
200
400 600 800 1000 1200 Time in s
(b) 1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0 0
(c)
200
400 600 800 1000 1200 Time in s
239
240
7. Critical multiphase flow
1.0 averaged over: 0.282-1.862 1.862-3.442 3.442-5.022 5.022-6.602 6.602-8.182 8.182-9.772 9.772-11.350 11.350-12.935
Void fraction,
0.8 0.6 0.4 0.2 0.0
0
200
400 600 800 1000 1200 Time in s
(d) Fig. 7.13. Vessel discharge starting from 40 bar. Void fractions along the high segments as a function of time: (a) Experiment by Umminger et al. (2007); Simple model with (b) Δw12 = 2Δw12, Ku ; (c) Δw12 = 3Δw12, Ku ; (d) Δw12 = 4Δw12, Ku
One should note that the vessel contained some test equipment inside it which was not modelled here. This is also one of the sources of the disagreement. Conclusions: 1. Regarding prediction of the pressure history: First order equilibrium methods as well as one dimensional network analysis with three fluid models predicts well the pressure history. 2. Regarding the axial void fraction averaged over specified sub-volumes: The first order analysis applied in combination with experiments for the pressure region of interests is a powerful tool for predicting the level swell. For the experiments and the real scale facility the similarity ⎡ ( ρ w )* F * ⎤ ⎡ ( ρ w )* F * ⎤ 1 1 ⎢ ⎥ =⎢ ⎥ ⎢ ρ ′′Δw12 Fv ⎥ ⎢ ρ ′′Δw12 Fv ⎥ ⎣ ⎦ exp ⎣ ⎦ real
(7.138)
is important. Without experiments, the first order analysis is not really predictive, because of the effective vapour rise velocity Δw12 is not known generally in advance. The Kutateladze vapour rise velocity of large bubbles is a good scale for the effective vapour rise velocity. In their parametric study Aounallah and Hofer (2003) reported that increasing the distribution coefficient results in faster pressure decrease and that decreasing of the effective vapor rise velocity results in slower pressure decrease. In the same
7.5 Simple two phase cases for pipes and nozzles
241
study for pressure release from 72 bar the effective vapor rise velocity have had to be increased up to 1.3 m/s in order to predict two-phase mixture level closer to the observation, which is confirmed also by my analysis. Note, that Aumiller et al. (2000) proposed to estimate the position of the two-phase mixture level by selecting the curves with transition void from continuous vapour to continuous liquid and by assuming that the mixture below the mixture level has averaged void as those in the section beneath it. Problem 9: Given the pressure and the vapour mass concentration at the inlet of a short nozzle. Compute the critical pressure ratio and the critical mass flow rate. Create plots for varying the vapour mass flow rate with the pressure as a parameter. Solution: First using the previous approximations for f1 and f2 I can derive an analytical expression for the mixture velocity as a function of the pressure difference. The result for frictionless and isentropic flow is ⎡ w2 = 2 ⎢( s0 a11 − a21 )( p0 − p ) + 2 ( s0 a12 − a22 ) ⎣
(
)
p0 − p + ( s0 a13 − a23 ) ln
p0 ⎤ . (7.139) p ⎥⎦
Decreasing the exit pressure and comparing the local sonic velocity the critical state can be detected (Fig. 7.14). p0 in bar 2 5 10 25 50 100 150 200
1,0 0,9
pcr/p0, -
0,8 0,7 0,6 0,5 0,4 0,3 0,0
0,2
0,4 0,6 X1 , -
0,8
1,0
7. Critical multiphase flow Critical mass flow rate in Mg/(m²s)
242
p0 in bar 2 5 10 25 50 100 150 200
10
1
0.0
0.2
0.4
0.6
0.8
1.0
X1, -
Fig. 7.14. Critical pressure ratio and critical mass flow rate as a function of the inlet vapor mass concentration. Parameter – inlet pressure.
We realize that increasing the inlet pressure leads to increasing critical mass flow rate. For the critical pressure ratio we see monotonic increase with the pressure up to about 100 bar. The most striking effect resulting from the assumption of the homogeneity and thermodynamic equilibrium is that at low vapour mass fraction the flow can become critical at very small driving pressure differences. Problem 10: Given the pressure and the vapour mass concentration at the inlet of a long pipe. Compute the critical pressure ratio and the critical mass flow rate. Create plots for varying the vapour mass flow rate with the pressure as a parameter. Solution: The momentum equation that has to be integrated is
1 2 1 1 λ fr ,co 2 dw + dp + v′v ( ρ w ) Φ 22 o dz = 0 . 2 2 Dh ρ
(7.140)
Therefore the integral is
w2 = f ( p ) −
L
1 2 λ fr ,co v′v ( ρ w ) Φ 22 o dz , Dh ∫0
(7.141)
where
⎡ f ( p ) = 2 ⎢( s0 a11 − a21 )( p0 − p ) + 2 ( s0 a12 − a22 ) ⎣
(
)
p0 ⎤ . p ⎥⎦ (7.142)
p0 − p + ( s0 a13 − a23 ) ln
7.5 Simple two phase cases for pipes and nozzles
243
Here Φ 22o is the two phase friction drop multiplier. Note that the friction term is a strong non linear function on the local vapor mass flow rate which for evaporating flow increases downwards the flow. Assuming linear change of the velocity from zero to w results in L 1 1 L v′ 2 λ fr ,co v′v ( ρ w ) Φ 22o dz ~ λ fr ,co w2 Φ 22o ∫ 4 Dh Dh 0 v
(7.143)
⎛ 1 L ⎞ v′ w2 ≈ f ( p ) ⎜ 1 + λ fr ,co Φ 22o ⎟ . 4 D v h ⎝ ⎠
(7.144)
or
If the formalism is used for entrance flow there is no wall friction term. 7.5.3.2 Developed flow As already mentioned, for long pipes the idealization developed flow can be used. It is expressed in the following system of ordinary differential equations for pipes with constant cross section G = const , G2
(7.145)
dv dp 1 λ fr ,2o + + v′G 2 Φ 22 o = 0 dz dz 2 Dh
(7.146)
or dp 1 λ fr ,2 o v′Φ 22 o = − G2 2 Dh dz
⎛ G2 ⎞ ⎜1 − 2 ⎟ , ⎝ Gcr ⎠
(7.147)
v = X 1, eq v ′′ + 1 − X 1,eq v′ = s0 f1 + f 2 ,
(7.148)
where
(
⎛ dv ⎞ Gcr2 = − ⎜ ⎟ ⎝ dp ⎠
)
−1
or −Gcr−2 = s0
df1 df 2 + = f ( p) , dp dp
(7.149)
are functions of the local pressure. Remember that this is much simpler method to compute the local critical mass flow rate, than those usually used in the literature which reads as follows
244
7. Critical multiphase flow
⎛ ∂v ⎛ ∂v ⎞ ⎛ ∂v ⎞ +⎜ ⎜ ⎟ =⎜ ⎟ ⎝ ∂p ⎠ s ⎝ ∂p ⎠ X1,eq ⎝⎜ ∂X 1, eq
⎞ ⎛ ∂X 1eq ⎞ ⎟ ⎜ ⎟ ⎟ ⎠ p ⎝ ∂p ⎠ s
(7.150)
with ⎛ ∂vI ⎞ dv ′′ dv′ = X 1, eq + 1 − X 1, eq ⎜ ⎟ dp dp p ∂ ⎝ ⎠ X1,eq
(7.151)
⎛ ∂v ⎜ ⎜ ∂X 1,eq ⎝
(7.152)
(
)
⎞ ⎟ = v′′ − v′ , ⎟ ⎠p
and ⎛ ∂X 1eq ⎞ ⎡ ds ′′ ds ′ ⎤ 1 + (1 − X 1eq ) ⎥ . X 1eq ⎜ ⎟ =− ⎢ ′′ ′ ∂ − dp dp ⎦ (s s ) ⎣ ⎝ p ⎠s
(7.153)
The integration of the momentum equation results in the simple expression G2 = −
Δp
1 λ fr ,2 o Δv + ∫ v′Φ 22 o dz 2 0 Dh L
.
(7.154)
So for a pipe defined with Dh and L for given pressures at the entrance and at the exit, equilibrium vapor quality at the entrance, the mass flow rate can be computed. Figure 7.15 gives an example of the prediction with this method for water.
Mass flow rate, Mg/(m²s)
60 50 40
pin in bar 2 5 10 step 10 200
30 20 10 0 0.0
0.2 0.4 0.6 0.8 1.0 Vapor mass flow rate fraction, -
7.5 Simple two phase cases for pipes and nozzles
245
Critical pressure ratio, -
0.80 pin in bar 2 5 10 step 10 200
0.75 0.70 0.65 0.60 0.0
0.2 0.4 0.6 0.8 1.0 Vapor mass flow rate fraction, -
Fig. 7.15. Critical mass flow rate and critical pressure ratio as a function of the inlet vapor mass flow fraction for a pipe with 5 cm-diameter, 1 m-length. Parameter – inlet pressure.
We realize that the critical pressure ratios are larger then those for pure gas flow. Fauske (1962) reported experimental data allowing to check the validity of this expression: His test 2 was performed for 6.83 mm-diameter pipe with 1.226 mlength, L / Dh =179.4, and his test 4 with 3.17 mm-diameter and 1.222 m-length, L / Dh =385. Measured are the pressures at six places along the pipes. The exit pressure is extrapolated using a curve fitting the six points. Reported are the measured mass flow rate and the exit equilibrium quality computed from the reference entrance equilibrium quality by using the energy conservation equation d ⎧⎡ ⎛ 1 2⎞ 1 2 ⎞⎤ ⎫ ⎛ ⎨ ⎢ X 1 ⎜ h1 + w1 ⎟ + (1 − X 1 ) ⎜ h2 + w2 ⎟ ⎥ ⎬ = 0 dz ⎩ ⎣ ⎝ 2 ⎠ 2 ⎠⎦ ⎭ ⎝
(7.155)
and the assumption of homogeneity and thermal equilibrium:
(
)
(
)
2 1 h0 = X 1, eq h′′ + 1 − X 1, eq h′ + G 2 ⎡ X 1,eq v′′ + 1 − X 1,eq v′⎤ = const . ⎣ ⎦ 2
(7.156)
For given mass flow rate, the equilibrium quality at any position in the pipe with known pressure is the solution of the above quadratic equation. So in fact we know ( p1 , X 1,eq,1 , p2 ) from the experiment which allows as to compute the mass flow rate and to compare it with the measured. The result is presented in Fig. 7.16. The mean error is 9.66%.
7. Critical multiphase flow
2
Gcomp, Mg/(m s)
246
18 16 14 12 10 8 6 4 2 0
Fauske (1962) data exp-comp HE model, friction -10% 10%
0
2
4
6 8 10 12 14 16 18 2 Gexp, Mg/(m s)
Fig. 7.16. Computed versus measured mass flow rate for 141 data points collected by Fauske (1962). Mean error 9.66%
This is a surprising result first because of the many simplification made during the deriving of the homogeneous equilibrium model. The result is also surprising secondly because the flow states are reported to be critical and the computed local critical mass flow rate at the exit of the pipe using the homogeneous equilibrium model gives 34% smaller values then the measured, see Fig. 7.17. If I impose the criticality condition at the exit defined only by the geometry and the initial conditions, Gcr ( p1 , X 1, eq ,1 ) , the mean error is 21.5%. This is a clear indication of the in-
2
Gcr,HE, Mg/(m s)
consistency of the homogeneous equilibrium model.
18 16 14 12 10 8 6 4 2 0
G=G(Δp), Gcr -34%
0
2
4 6 8 10 12 14 16 18 2 Gcomp,HE, fr., Mg/(m s)
Fig. 7.17. Critical mass flow rate at the exit of the pipe versus the p2 − p1 controlled mass
(
)
flow rate G p1 , X 1, eq ,1 , p2 computed by using the homogeneous equilibrium model
7.5 Simple two phase cases for pipes and nozzles
247
The data reported by Sozzi and Sutherland (1975) for 12.7 mm-diameter nozzles and orifices are also reproduced by the homogeneous equilibrium model as given in Table 7.2. Table 7.2. The reproduction of the Sozzi and Sutherland (1975) data by the homogeneous equilibrium model. Minimum diameter 12.7 mm
L in m 0.000 0.013 0.038 0.063 0.114 0.190 0.229 0.317 0.508 0.635 1.778 0.000 0.190 0.317 0.508 0.635
L / Dh 0 1 3 5 9 15 18 25 40 50 140 0 15 25 40 50
Error % 19 20.9 10.7 5.24 7.1 3.5 4.5 4.4 6.4 7.7 18.2 23.4 3.8 2.54 3.6 4.4
Converging-diverging nozzle Converging nozzle – pipe
Orifices with sharp entrance Pipe with sharp entrance
We realize that the data for the short converging-diverging nozzle and for the orifices with sharp entrance are under predicted by the homogeneous equilibrium model. More mass flow is flowing in this case in the reality. The short time for passing the short nozzles is not allowing recovering the complete thermodynamic equilibrium locally. This is thought to be the reason for this under prediction. The data for L / Dh = 140 are predicted with mean error of about 18%. Critical flows in pipes with L / Dh = 1 to 50 are well reproduced. In spite of the known disadvantages for very short pipes, nozzles and orifices, the homogeneous equilibrium model was found practical, and is widely used in the literature for pipe flows. It was formulated by several authors in one or other form already before the World War II but it becomes his popularity after the works published by Fred Moody (1965, 1966, 1969, 1975).
248
7. Critical multiphase flow
7.5.3.3 Entrance from a vessel followed by developed flow As in the gas flow, a combination of entrance flow accelerating from stagnation to a steady developed mass flow rate, and following flow in a pipe is possible. For this purpose the developed mass flow rate as a function of the pressure difference is G2 =
p1 − p2 . L ⎛ a ⎞ λ a a a a a a a 1 fr o ,2 2 13 13 23 23 ⎟ − 22 − v′Φ 2 o dz s0 ⎜ 12 + − 12 − + 22 + + ⎜ p p2 p2 p1 ∫0 2 Dh p1 p1 ⎟⎠ p2 p1 2 ⎝ (7.157)
The mass conservation at the intermediate point 1 gives v12 G 2 = w12 .
(7.158)
Eliminating the mass flow rate from the both equations results in a transcendental equation for the intermediate pressure p1. Remember that ⎡ w12 = 2 ⎢( s0 a11 − a21 )( p0 − p1 ) + 2 ( s0 a12 − a22 ) ⎣
(
)
p0 − p1 + ( s0 a13 − a23 ) ln
p0 ⎤ ⎥, p1 ⎦
(7.159) ⎛ a ⎞ a a a v1 = s0 ⎜ a11 + 12 + 13 ⎟ − a21 − 22 − 23 . ⎜ ⎟ p1 p1 ⎠ p1 p1 ⎝
(7.160)
7.5.4 Equilibrium non-homogeneous flow The next step of sophistication of the equilibrium model is to relax the assumption of equal velocities. In this case the coupling between the phases is due to evaporation, drag- and virtual mass force. 7.5.4.1 Developing flow Momentum exchange due to evaporation: Let as examine the behavior of a model that neglects all forces except the increased cohesion between gas and vapor due to evaporation. The mass- and the momentum conservation equations are then d (α1 ρ1 w1 ) = μ21 or μ21dz = GdX 1 . dz
(7.161)
7.5 Simple two phase cases for pipes and nozzles
d ⎡(1 − α1 ) ρ 2 w2 ⎦⎤ = − μ21 , dz ⎣
(7.162)
d dp (α1 ρ1 w1 w1 ) + α1 = μ21 w2 , dz dz
(7.163)
d dp = − μ21 w2 , ⎡(1 − α1 ) ρ 2 w2 w2 ⎤⎦ + (1 − α1 ) dz ⎣ dz
(7.164)
249
where X 1 = α1 ρ1 w1 G .
(7.165)
Using the vapor and liquid mass conservation equations the corresponding momentum equations simplify to
α1 ρ1 w1dw1 + α1dp = dz μ21 ( w2 − w1 ) or
1 2 dp G dw1 + dX = ( w2 − w1 ) ′′ ρ α1 ρ ′′ 1,eq 2 (7.166)
1 2 1 dw2 + dp = 0 . ρ2 2
(7.167)
For isentropic flow for which the local mass flow rate concentration is always defined by the initial mixture entropy and the local pressure X 1eq ( p ) =
s0 − s ′ ( p ) , s ′′ ( p ) − s ′ ( p )
(7.168)
the change of the vapor mass fraction with the pressure is dX 1eq = −
⎡ 1 ds ′′ ds ′ ⎤ + (1 − X 1eq ) ⎥ dp . X 1eq ⎢ dp dp ⎦ ( s′′ − s′ ) ⎣
(7.169)
With this the final form of the momentum equations is 1 dX 1, eq ⎫ ⎧1 2 2 w1,1 = w1,0 − 2 ∫ ⎨ + w1 ( w1 − w2 ) ⎬ dp , dp ⎭ ρ ′′ p0 ⎩
p
p1
2 2 w2,1 = w2,0 − 2∫
p0
dp . ρ′
(7.170)
(7.171)
250
7. Critical multiphase flow
Mass flow rate in t/(m²s)
60
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
50 40 30 20 10 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
1,0
(w1/w2)/(v1/v2)
1/2
0,8 0,6 0,4 0,2 0,0 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
Fig. 7.18. (a) Mass flow rate as a function of the exit/inlet pressure ratio for initial pressure 200 bar and saturated conditions; (b) Slip divided by ρ ′ ρ ′′ . Parameter: inlet mass fraction
The
system
can
be
integrated
using
the
initial
conditions: p = p0 ,
X 1eq = X 1eq ( p0 ) , w1,0 = 0 , w2,0 = 0 . At any place between the two pressures the
void fraction and the mass flow rate are −1
⎛ w 1 − X 1, eq ρ1 ⎞ α1 = ⎜1 + 1 ⎟ , ⎜ w2 X 1, eq ρ 2 ⎟ ⎝ ⎠
(7.172)
7.5 Simple two phase cases for pipes and nozzles
G = α1 ρ ′′w1 + (1 − α1 ) ρ ′w2 ,
251
(7.173)
respectively. To illustrate the behavior of this solution I compute the mass flow rate as a function of the exit/inlet pressure ratio for initial pressure 200 bar and saturated water-steam mixture. I vary the inlet mass fraction. The results are presented in Fig. 7.18. We learn that there is a maximum of the mass flow rate as a function of the pressure ratio depending on the inlet vapor mass fraction which defines the critical flow. If the two phases could be stratified and incompressible the maximum velocity ratio resulting from the two Bernulli equations would be ρ ′ ρ ′′ . Using it as a normalization variable we realize that only the cohesion due to the evaporation does not allow reaching this ratio at the critical plane as assumed by several authors in the sixteen’s. Drag force: The next step of sophistication of the thermal equilibrium model is to take into account the drag forces
f12d dz = f12d ( w1 − w2 )
2
(7.174)
where 3 ⎡ α ρ ′ d (1 − α1 ) ρ ′′ d ⎤ + f12d = sign ( w1 − w2 ) ⎢ 1 c21 c12 ⎥ dz . D2 4 ⎣ D1 ⎦
(7.175)
d In this formulation c21 is non zero only for continuum liquid and c12d is non zero only for continuum vapor. The corresponding momentum equations are then
d dp (α1 ρ1 w1 w1 ) + α1 = μ21 w2 − f12d − f wd1 , dz dz
(7.176)
d dp ⎡(1 − α1 ) ρ 2 w2 w2 ⎦⎤ + (1 − α1 ) = − μ 21 w2 + f12d − f wd2 , ⎣ dz dz
(7.177)
or after replacing the drag force with its equal and rearranging using the mass conservation equations dX 1,eq ⎤ f12d ( w1 − w2 ) v ′′ d 1 2 ⎡ dw1 + ⎢v ′′ + w1 ( w1 − w2 ) f w1 = 0 , (7.178) + ⎥ dp + v′′ dp ⎦ α1 α1 2 ⎣ 2
252
7. Critical multiphase flow
fd v′ 1 2 2 dw2 + v′dp − v ′ 12 ( w1 − w2 ) + f wd2 = 0 . 2 1 − α1 1 − α1
(7.179)
The coupling is becoming very strong and a special care is necessary to solve the system of two equations. Subtracting the second equation from the first result in dX 1, eq ⎤ ⎡ ⎛ v′′ v′ ⎞ d 1 2 d ( w12 − w22 ) + ⎢v′′ − v′ + w1 ( w1 − w2 ) ⎟ f12 ( w1 − w2 ) ⎥ dp + ⎜ + dp ⎦ 2 ⎣ ⎝ α1 1 − α1 ⎠
⎛ v′′ d ⎞ v′ +⎜ f w1 − f wd2 ⎟ dz = 0 , α α − 1 1 ⎝ 1 ⎠
(7.180)
which after discretization receives the form a ( w1 − w2 ) + b ( w1 − w2 ) + c = 0 2
(7.181)
or
w1 − w2 =
−b + b 2 − 4ac , 2a
(7.182)
where ⎛ v′′ v′ ⎞ d a = 2⎜ + ⎟ f12 + 2a ′ , ⎝ α1 1 − α1 ⎠
(7.183)
⎛ dX 1, eq ⎞ 2Δp ⎟ w1 + w2 + 2b′ , b = ⎜1 + dp ⎝ ⎠
(7.184)
c = 2 ( v ′′ − v′ ) Δp − ( w1,2 k −1 − w2,2 k −1 ) + 2c′ ,
(7.185)
Δp = pk −1 − p < 0 .
(7.186)
The subscript k indicates the integer spatial position: k–1 at the beginning of Δz and k at its end. The coefficients a and c are not dependent on the actual velocities. Only the b-coefficient is a linear function of the axial velocities. Few cycles of an iteration process are necessary to obtain the solution of this nonlinear system of two algebraic equations with respect to the velocities. This model requires knowledge about the flow pattern and about the characteristic size of the bubbles or droplets.
7.5 Simple two phase cases for pipes and nozzles
Mass flow rate in t/(m²s)
60 50 40 30 20 10 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
(w1/w2)/(v1/v2)
1/2
0,8 0,6 0,4 0,2 0,0 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
253
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
1,0
Fig. 7.19. (a) Mass flow rate as a function of the exit/inlet pressure ratio for initial pressure 200 bar and saturated conditions; (b) Slip divided by ρ ′ ρ ′′ . Parameter: inlet mass fraction. Bubble and droplet size – 1 mm, transition from bubble to droplet flow at α1,lim = 0.8
254
7. Critical multiphase flow
Mass flow rate in t/(m²s)
60 50 40 30 20 10 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
1,0
(w1/w2)/(v1/v2)
1/2
0,8 0,6 0,4 0,2 0,0 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
Fig. 7.20. (a) Mass flow rate as a function of the exit/inlet pressure ratio for initial pressure 200 bar and saturated conditions; (b) Slip divided by ρ ′ ρ ′′ . Parameter: inlet mass fraction. Bubble size – 0.1 mm, droplet size – 0.1 mm, transition from bubble to droplet flow at α1,lim = 0.8
7.5 Simple two phase cases for pipes and nozzles
255
The simplest approach is to assume bubbly flow for α1 < α1,lim and droplet flow elsewhere. In general the drag coefficient is a function of the Reynolds number e.g. Recd = ρ c Dd wc − wd ηc , ccdd =
24 (1 + 0.1Re0.75 cd ) , Re cd < 500 , Ishii and Zuber (1978). Recd
(7.187)
(7.188)
Let me evaluate this model for α1,lim = 0.8 and bubble or droplet diameter 1 and 0.1mm. The results for initial pressure of 200 bar are presented in Figs. 7.19 and 7.20. We learn from these examples that the smaller the particle sizes the smaller the velocity ratio, which is expected. We see that the slip for stratified flows and flows with large particle size due to the prehistory at the entrance of the nozzles are sensitive to the vapor mass fractions. Otherwise the slip of the critical flows is almost independent of the vapor mass fraction. The striking effect of the regime transition for relative large particle size is that under particular conditions there may be two local maxima for the critical mass flow rate for X 1, eq < 0.2 which may cause instability at the exit. Instabilities are experimentally observed around 0.1, see the discussion to Fig. 3 of Zaloudek (1961). Zaloudek reported flow instabilities in critical flow around 0.1 explaining them with the flow regimes transition. Fragmentation: Dynamic fragmentation is important in pipe flow but in nozzle flow the time is not enough for fragmentation and therefore the conditions at the inlet are controlling the flow pattern. If dynamic fragmentation could be active obeying for instance the following low Wecd = ρ c ( wd − wc ) Dd σ cd ≈ const ,
(7.189)
2 Dd = Wecd σ cd ⎡ ρc ( wd − wc ) ⎤ , ⎣ ⎦
(7.190)
2
or
the drag force between the phases will be strong function of the velocity difference
256
7. Critical multiphase flow
Mass flow rate in t/(m²s)
60 50 40 30 20 10 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
(a)
1,0
(w1/w2)/(v1/v2)
1/2
0,8 0,6 0,4 0,2 0,0 0,0
0,2
0,4 0,6 p1/p0, -
0,8
1,0
p0=200bar, X0 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.025 0.050 0.075 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999
(b) Fig. 7.21. (a) Mass flow rate as a function of the exit/inlet pressure ratio for initial pressure 200 bar and saturated conditions; (b) Slip divided by ρ ′ ρ ′′ . Parameter: inlet mass fraction. Bubble and droplet size Wecd = 12 , transition from bubble to droplet flow at
α1,lim = 0.8
7.5 Simple two phase cases for pipes and nozzles
257
2
⎛ ρ′ ⎞ 5 0.75 f = α118η ′ ⎜ ⎟ (1 + 0.1Re 21 ) ( w1 − w2 ) . ⎝ We21σ 21 ⎠ d 12
(7.191)
An illustration of the performance of such model assuming Wecd = 12 is given in Fig. 7.21. The strong coupling between steam and vapour is experimentally confirmed. For instance for low pressures the following empirical correlations for the velocity ratio are reported
(
)
w1 w2 = max 1, 0.17 X 1 ρ 2 ρ1 , Fauske (1962),
(7.192)
w1 w2 = 1 + ( 4.46 X 10.18 − 1) ( ρ 2 ρ1 ) ( ρ 2 ρ1 )1bar , Delhaye et al. (1981). (7.193) Note the weak dependence on X 1 in the empirical correlation by Delhaye et al. which is predicted for fine dispersions by the theory. 7.5.4.2 Developed flow The case in which the flow in a pipe has a constant mass flow is described in the frame work of the thermal equilibrium assumption with the following system of ordinary differential equations GA = const ,
(7.194)
vI dA 1 λ fr ,2 o − v′Φ 22 o 2 Dh dp 2 A dz , =G dz G2 1− 2 Gcr
(7.195)
s0 ≈ const ,
(7.196)
where −1
⎛ dv ⎞ Gcr2 = − ⎜ I ⎟ , ⎝ dp ⎠
(7.197)
vI = vS f 0 ,
(7.198)
(
)
vS = X 1,eq v′′ + S 1 − X 1, eq v′ ,
(7.199)
258
7. Critical multiphase flow
f0 =
1 + ( S − 1) X 1, eq S
.
(7.200)
After introducing the definitions for the two velocity non-homogeneous model G = α1 ρ1 w1 + (1 − α1 ) ρ 2 w2 ,
(7.201)
X 1 = α1 ρ1 w1 G ,
(7.202)
1 − X 1 = (1 − α1 ) ρ 2 w2 G ,
(7.203)
several useful relations can be derived as summarized by Kolev (1986, p. 18) X 1G
α1 ρ1 S S
= w2 ,
(7.204)
1 − X 1 1 − α1 ρ 2 , = X1 α1 ρ1
α1 =
1
ρ 1 − X1 1+ S 1 ρ2 X1
=
(7.205)
X 1v1 Xv = 11 , X 1v1 + S (1 − X 1 ) v2 vS
(7.206)
−1
⎛ 1 1 − α1 ρ 2 ⎞ X 1 = ⎜1 + ⎟ , ⎝ S α1 ρ1 ⎠
(7.207)
vS = X 1v1 + S (1 − X 1 ) v2 ,
(7.208)
w1 = GvS ,
(7.209)
w2 = GvS S ,
(7.210)
α1 ρ1 w12 + (1 − α1 ) ρ 2 w22 = G 2 vS
1 + ( S − 1) X 1 S
= G 2 vS f 0 = G 2 vI ,
(7.211)
which are found in one or other form in the literature up to 1962, see for instance Fauske (1962). Consequently we have
7.5 Simple two phase cases for pipes and nozzles
⎛ ∂vI ⎞ ⎡ dv′′ dv ′ ⎤ = f 0 ⎢ X 1,eq + S 1 − X 1,eq ⎜ ⎟ ⎥ dp dp ⎦ ⎝ ∂p ⎠ X1,eq ⎣
(7.212)
⎛ ∂vI ⎜ ⎜ ∂X 1, eq ⎝
(7.213)
(
)
⎞ S −1 + f 0 ( v′′ − Sv′ ) , ⎟ = vS ⎟ S ⎠p
259
and ⎛ ∂X 1,eq ⎞ ⎡ ds ′′ ds ′ ⎤ 1 X 1, eq + (1 − X 1, eq ) ⎥ , ⎜ ⎟ =− ⎢ dp dp ⎦ ( s′′ − s′ ) ⎣ ⎝ ∂p ⎠ s
(7.214)
and finally ⎛ ∂vI ⎛ ∂vI ⎞ ⎛ ∂vI ⎞ +⎜ ⎜ ⎟ =⎜ ⎟ ⎝ ∂p ⎠ s ⎝ ∂p ⎠ X1,eq ⎜⎝ ∂X 1,eq
⎞ ⎛ ∂X 1, eq ⎞ ⎟ ⎜ ⎟ . ⎟ ⎠ p ⎝ ∂p ⎠ s
(7.215)
The not resolved problem with this model is the slip. Setting the slip equal to unity results in the previously discussed homogeneous model for developed flow. To resolve this problem Isbin et al. (1957), Moy (1955), Faletti (1959), Zaloudek (1961), Fauske (1962) tried to extract from their measurements the conditions at the exit of the pipe and therefore to obtain the relation
(
)
Gcr = Gcr pcr , X 1,eq , cr .
(7.216)
The data of these authors are then used by many authors including me to check their theoretical models. Table 7.3 gives an example of such comparison for three different slip functions including S Fauske = ( ρ 2 ρ1 ) , 0.5
(7.217)
Fauske (1962) and
S Kolev
X1 ⎛ 1 − X1 = 1+ ⎜ X 1,max ⎜⎝ 1 − X 1,max
Kolev (1986, p. 106).
⎞ ⎟⎟ ⎠
1− X 1,max X 1,max
⎛ ⎛ ρ ⎞m ⎞ ⎜ ⎜ 2 ⎟ − 1⎟ , X 1,max = 0.5 , m = 0.5, ⎜ ⎝ ρ1 ⎠ ⎟ ⎝ ⎠ (7.218)
260
7. Critical multiphase flow
Table 7.3. Mean error in % of the local critical mass flow rate compared with data
Data
S =1
Isbin et al. (1957) Faletti (1959) Zaloudek (1961) Fauske (1962) Burnell (1947)
30 34.7 43 40 52.18
S Fauske 15 45 15 10 48.9
S Kolev 13 33.4 17 7.5 16.5
Unfortunately the non homogeneity and the thermal non equilibrium are not quantified by these data which let free space for hypothesis regarding the slip and the non equilibrium evolution up to the exit of the pipe. Analyzing the experimental data of several authors for short pipes, nozzles and orifices, Henry (1970) found that at low vapor mass flow fraction, X 1,eq < 0.05 , it is more important to consider the deviation from the thermodynamic equilibrium than to pay attention to the non-homogeneity. He proposed an empirical reduction of the evaporation dX 1,eq dX 1 N ⎡ ds ′′ ds ′ ⎤ =N =− ⎢ X 1, eq dp + (1 − X 1, eq ) dp ⎥ , ′′ ′ dp dp s s − ( )⎣ ⎦
(7.219)
and at the exit cross section X 1, cr = NX 1, eq , cr for L/Dh ≤ 12,
(7.220)
{
}
X 1,cr = NX 1, eq , cr 1 − exp ⎣⎡ −0.0523 ( z Dh − 12 ) ⎦⎤ for L/Dh > 12,
(7.221)
where N = 20 X 1,eq for X 1,eq < 0.05 and N = 1 for X 1,eq > 0.05 . Then he used the momentum equation for homogeneous frozen flow G2 = −
Δp = Δv
p0 (1 − ε ) ⎛v ⎞ X 1, cr ⎜ 1,01 − v2 ⎟ − X 1,0 ( v1,0 − v2 ) ⎜ n ⎟ ⎝ε ⎠
,
(7.222)
v = X 1v1 + (1 − X 1 ) v2 ,
(7.223)
Δv = X 1,cr v1, cr + (1 − X 1,cr ) v2 − X 1,0 v1,0 − (1 − X 1,0 ) v2 .
(7.224)
For constant liquid density and polytrophic change of state of the gas
7.5 Simple two phase cases for pipes and nozzles
261
1
ρ1,cr ⎛ pcr ⎞ n =⎜ ⎟ ρ10 ⎝ p0 ⎠
(7.225)
we have ⎛v ⎞ Δv = X 1,cr ⎜ 1,01 − v2 ⎟ − X 1,0 ( v1,0 − v2 ) . ⎜ n ⎟ ⎝ε ⎠
(7.226)
7.5.5 Inhomogeneous developing flow in short pipes and nuzzles with infinitely fast heat exchange and with limited interfacial mass transfer Consider the simple case of a three-field mixture having constant component mass concentration without wall interaction, without any mass transfer, and with instantaneous heat exchange that equalizes the field temperatures at any moment. The polytrophic change of state of the gas phase is derived for this case in Chap. 6 in Vol. 2. Remember that the energy conservation reduces in this case to
α le ρl wl c pl
⎤ 1 dp q& ′′′ ⎛∂h ⎞ p ⎡ 1 dTl ⎢1 − ρl ⎜ l ⎟ ⎥ − α le ρl wl = l . Tl dz p p dz Tl ρl Tl ⎢ ∂ ⎥ ⎝ ⎠Tl , all _ C ′s ⎦ ⎣
(7.227)
After dividing by the mixture mass flow rate 3
G = ∑ α l ρl wl ,
(7.228)
l =1
and summing all three equations one obtains dT R dp n − 1 dp = = , T cp p n p
(7.229)
where the mixture specific heat is 3
c p = ∑ X l c pl ,
(7.230)
l =1
and the effective analog to the gas constant is 3
R = ∑ X l Rl l =1
(7.231)
262
7. Critical multiphase flow
with the so called pseudo “gas constant” for each fluid is R=
⎤ ⎛∂h⎞ p ⎡ ⎥. ⎢1 − ρ ⎜ ⎟ ρT ⎣⎢ ⎝ ∂ p ⎠T ,all _ C′s ⎦⎥
(7.232)
Note that for the second and the third field the pseudo-gas constant is negligibly small. The polytrophic exponent is then n=
cp cp − R
≈
X 1c p1 + X 2 c p 2 + X 3 c p 3 . c p1 X1 + X 2 c p 2 + X 3c p3
(7.233)
κ1
For two velocity fields this result is obtained by Tangren et al. (1949). Integration of Eq. (7.229) yields T ⎛ p ⎞ =⎜ ⎟ T0 ⎝ p0 ⎠
n −1 n
.
(7.234)
This change of state is associated with the following change of the entropy of the gas field ds1 = c p1
⎛ 1 1 ⎞ dp dT dp , − R1 = −c p1 ⎜ − ⎟ T p ⎝ n κ1 ⎠ p
(7.235)
and for the other fields dsl = c pl
dT n − 1 dp , for l = 2, 3. = c pl T n p
(7.236)
We see that for n < κ1 this change of state is associated with entropy change of the gas phase. Useful relations are
d ρ1 ρ1 = , dp np the integrated form
(7.237)
7.5 Simple two phase cases for pipes and nozzles
263
1
ρ1 ⎛ p ⎞ n =⎜ ⎟ ρ10 ⎝ p0 ⎠
(7.238)
and d ρ1 =− ds1
ρ1
,
(7.239)
ds1 1⎛1 1 ⎞ = −c p1 ⎜ − ⎟ , dp p ⎝ n κ1 ⎠
(7.240)
⎛1 1 ⎞ nc p1 ⎜ − ⎟ ⎝ n κ1 ⎠
which is Eq. (28) in Henry and Fauske (1969). Now let as integrate the simplified momentum equation for each field without any interaction allowing a polytrophic state of change of the gas 1 2 dp dw1 + =0 2 ρ1
(7.241)
1 2 dp dwl + = 0 , l = 1, 2. 2 ρl
(7.242)
The result is w12 = w102 + 2
wl2 = wl20 + 2
n −1 ⎞ n ⎛ n 1 − ε ⎜ ⎟ ρ10 n − 1 ⎝ ⎠
p0
p0
ρl 0
(1 − ε ) , l = 1, 2.
(7.243)
(7.244)
Multiplying each equation with α l ρl and summing them I obtain
α1 ρ1 w12 + α 2 ρ 2 w22 + α 3 ρ3 w32 ≈ α1 ρ1 w102 + α 2 ρ 2 w202 + α 3 ρ3 w302 (7.245) ⎡ p n ⎛ +2 ⎢α1 ρ1 0 ⎜1 − ε ρ10 n − 1 ⎝ ⎢⎣
n −1 n
⎤ ⎞ ⎟ + (1 − α1 ) p0 (1 − ε ) ⎥ ⎥⎦ ⎠
264
7. Critical multiphase flow
Introducing the definition from Chap. 8 in Vol. 2 for the three-fluid slip model 3
3
3
l =1
l =1
l =1
Sl = wl / w , w = G / ρ , G = ∑ α l ρl wl , ρ = ∑ α l ρl , vS = ∑ X l Sl vl ,
αl =
3 X l vl Sl ⎛ 3 ⎞ X , f 0 = ∑ l , vI = ⎜ ∑ α l ρl wl2 ⎟ / G 2 = vS f 0 , wl = vS G / Sl , vS S = = 1 l l 1 ⎝ ⎠ l (7.246 –7.254)
I obtain finally the integrated mixture momentum equation 1 1 2 f 0 vS2 G 2 ≈ ( X 1 S1 w102 + X 2 S 2 w20 + X 3 S3 w302 ) 2 2
+ X 1 S1
n −1 ⎞ n ⎛ ⎜ 1 − ε n ⎟ + ( vS − X 1v1 S1 ) p0 (1 − ε ) . ρ10 n − 1 ⎝ ⎠
p0
After rearranging and introducing the vena contracta coefficient I obtain n −1 ⎡1 2 ⎞⎤ ⎛ G ⎞ p0 n ⎛ 1 f 0 vS2 ⎜ ⎜1 − ε n ⎟ ⎥ ⎟ ≈ X 1 S1 ⎢ w10 + 2 ρ10 n − 1 ⎝ ⎢⎣ 2 ⎝ Cvc ⎠ ⎠ ⎥⎦ 2
⎤ ⎡ 1 2 p0 ⎡ 1 2 p0 ⎤ + X 2 S2 ⎢ w20 + (1 − ε ) ⎥ + X 3 S3 ⎢ w30 + (1 − ε ) ⎥ . ρ2 ρ3 ⎣2 ⎦ ⎦ ⎣2
(7.255)
So for each pressure ratio ε and velocities at the entrance of the nozzle we can compute a mass flow rate at the exit of the nozzle for postulated or modeled slips as a function of the local parameter. Noting that
dvI = −
⎛ ⎞ 3 3 X l Sl ⎜ ∂ρl ∂ρl dp ⎟ f ds dC − + ∑ l il ⎟ 0∑ 2 ⎜ G *2 s C ρ ∂ ∂ l =1 l =1 l il ⎜ l ⎟ l ≠m ⎝ ⎠
3 ⎡ ⎛1 1 ⎞⎤ + ∑ ⎢ f 0 ( vl Sl − vm Sm ) + vS ⎜ − ⎟ ⎥dX l , Sl Sm ⎠ ⎦⎥ l =1 ⎣ ⎢ ⎝ l ≠m
where the local critical mass flow rate is
(7.256)
7.5 Simple two phase cases for pipes and nozzles 3 XS 1 = f 0 ∑ l*2 l , *2 G l =1 Gl
265
(7.257)
the criticality condition can be derived as follows ⎛ ⎞ 3 3 X l Sl ⎜ ∂ρl dsl ∂ρl dCil ⎟ dvI 1 1 f = − = + + ∑ 0∑ 2 ⎜ ⎟ Gcr2 dp G *2 l =1 ρ l ⎜ ∂ sl dp ll =≠1m ∂ Cil dp ⎟ ⎝ ⎠ 3 ⎡ ⎛1 1 ⎞ ⎤ dX l . −∑ ⎢ f 0 ( vl Sl − vm S m ) + vS ⎜ − ⎟⎥ Sl S m ⎠ ⎥⎦ dp l =1 ⎢ ⎝ ⎣ l ≠m
(7.258)
If we consider single component multi-phase mixture, neglect the dependences of the liquid densities on the entropies, and take into account that the gas density in its polytrophic state of change depends only on pressure we obtain for flow without any interfacial mass transfer ⎛ X S X 1 S1 X S 1 ⎜ f = + 2*2 2 + 3*2 3 0⎜ n +1 Gcr2 G G3 2 ⎜ nρ p ε n ⎝ 10 0
⎞ ⎟. ⎟ ⎟ ⎠
(7.259)
For flow with not completed evaporation required to reestablish the thermal equilibrium at the exit Henry and Fauske (1969) assumed that that the evaporation at the exit is ⎛ X 10 ds1 1 − X 10 ds ′ ⎞ dX 1 ds ⎤ 1 ⎡ ds1 X1 N =− + (1 − X 1 ) 2 ⎥ ≈ − ⎜ + ⎟ ⎢ dp s1 − s2 ⎣ dp dp ⎦ ⎝ s10 − s20 dp s ′′ − s ′ dp ⎠
≈
c p1 s10 − s20
Nc ′p dT ′ X 10 ⎛ 1 1 ⎞ , ⎜ − ⎟ − (1 − X 10 ) p0ε ⎝ n κ1 ⎠ h′′ − h′ dp
(7.260)
where use of Eq. (7.240) is made and N is the reduction factor reflecting thermodynamic non equilibrium. Note that c′p dT ′ dh′ 1 ds ′ 1 . ≈ = s ′′ − s ′ dp ( s ′′ − s ′ ) T ′ dp h′′ − h′′ dp
(7.261)
I use this idea but in a slightly different variant. I assume that only a N-part of the equilibrium vapor mass flow fraction increase over the discharge section is reached due to delayed establishing of thermal equilibrium
266
7. Critical multiphase flow
X 1 = X 10 + N ΔX 1,eq ,
dX 1 1 =− dp s1 − s2
=N
dX 1,eq dp
(7.262)
⎡ ds1 ds2 ⎤ ⎢ X 1 dp + (1 − X 1 ) dp ⎥ ⎣ ⎦
= −N
X 1,eq
ds ′′ ds ′ + (1 − X 1,eq ) dp dp , s ′′ − s ′
where
X 1,eq =
s0 − s ′ , s ′′ − s ′
(7.263)
with N being unknown function that will be derived from experiment. Replacing with the approximation for the entropy derivatives, as done by Henry and Fauske results in ⎡ X 1,eq c p1 ⎛ 1 1 ⎞ (1 − X 1,eq ) c ′p dT ′ ⎤ dX 1 ⎥. ≈N⎢ ⎜ − ⎟+ dp h′′ − h′ dp ⎥ ⎢⎣ s ′′ − s ′ p ⎝ n κ1 ⎠ ⎦
(7.264)
Assuming that the evaporation from the liquid is equally supplied by the total liquid mass flow rate dX 3 X 3 dX 1 =− , dp 1 − X 1 dp
(7.265)
I obtain for the criticality condition for the three-fluid flow ⎛ X S X 1 S1 X S 1 ⎜ = f0 ⎜ + 2*2 2 + 3*2 3 2 n +1 Gcr G2 G3 ⎜ nρ p ε n ⎝ 10 0
⎞ ⎟ ⎟ ⎟ ⎠
⎤ X3 X2 ⎪⎧ ⎡ + ⎨ f 0 ⎢ v3 S3 + v2 S 2 − v1 S1 ⎥ + vS 1 − X1 1 − X1 ⎪⎩ ⎣ ⎦
For two velocities it simplifies to
⎡ 1 X3 1 X2 1 ⎤ ⎫⎪ dX + − ⎥⎬ 1 ⎢ ⎣ S3 1 − X 1 S 2 1 − X 1 S1 ⎦ ⎪⎭ dp (7.266)
7.5 Simple two phase cases for pipes and nozzles
⎛ (1 − X 1 ) S2 X 1 S1 1 ⎜ = f0 ⎜ + 2 n +1 Gcr G2*2 ⎜ nρ p ε n ⎝ 10 0
⎞ ⎛ 1 1 ⎟ ⎡ ⎟ − ⎢ f 0 ( v1 S1 − v2 S 2 ) + vS ⎜ S − S 2 ⎝ 1 ⎟ ⎣⎢ ⎠
267
⎞ ⎤ dX 1 . ⎟⎥ ⎠ ⎦⎥ dp
(7.267) For homogeneous flow 1 = Gcr2
X1 n ρ10 p0ε X1
=
n ρ10 p0ε
n +1 n
n +1 n
+
+
1 − X1 dX − ( v1 − v2 ) 1 G2*2 dp
1 − X1 v −v ⎡ ds ′′ ds ′ ⎤ + N 1 2 ⎢ X 1, eq + 1 − X 1,eq ⎥. *2 G2 dp dp ⎦ s ′′ − s ′ ⎣
(
)
(7.268)
Compare this result with Eq. (29) by Henry and Fauske (1969), 1 = Gcr2
(
X 10 n ρ10 p0ε
n +1 n
)
⎡ 1 − X 10 X 10 c p1 ⎛ 1 1 ⎞⎤ ds ′ + ( v1 − v2 ) ⎢ − N ⎜ − ⎟ ⎥ . (7.269) dp ( s10 − s20 ) p0ε ⎝ n κ1 ⎠ ⎥ ⎢ s ′′ − s ′ ⎣ ⎦
The integrated mixture momentum equation 7.255 and the criticality condition 7.266 are valid for three velocity fields. This formalism contains obviously the inflow velocities and the flow patter information by preserving the essential part of the successful idea by by Henry and Fauske (1969) for describing flows in nozzles, orifices and very short pipes. I check the accuracy by using the data of Sozzi and Sutherland (1975). The slight non homogeneity is computed by using Eq. (7.218) with X 1,max = 0.1 and m = 0.1. The results are presented in Fig. 7.22. The mean error is about 8.6%. Therefore this extension to three velocity field of the semi-equilibrium model works and is recommended for practical use.
268
7. Critical multiphase flow
Gcr,comp in Mg/(m²s)
80 70 60
Subcooled water, two-phase mixture Sozzi and Sutherland (1975) +20% -20%
50 40 30 20 20
30
40 50 60 70 Gcr,exp in Mg/(m²s)
80
Fig. 7.22. Non-homogeneous non-developed nozzle flow with criticality conditions for equilibrium flow corrected for the non-equilibrium as proposed by Henry (1970). Mean error 8.6%. The data above 50 Mg/(m²s) are for sub-cooled water at the entrance
For the case of sub-cooled water is entering the nozzle and the exit pressure is higher than the saturation pressure we have the single phase Euler equation ⎛ p ⎞ G = Cvc 2 ρ 2 p0 ⎜ 1 − 1 ⎟ . p0 ⎠ ⎝
(7.270)
If the exit pressure is lower than the saturation pressure evaporation can be expected. In this case the entrance acceleration of sub-cooled water up to a pressure where evaporation starts is Eq. (7.270). From this point on the evaporation starts and the simplified form of Eq. (7.222) holds, ⎛ p ⎞ p1 ⎜ 1 − 1 ⎟ p2 ⎠ ⎝ . G2 = ⎛ ⎞ v1,0 X 1, cr ⎜ − v2 ⎟ 1 ⎜ ⎟ ⎜ ( p p )n ⎟ ⎝ 1 2 ⎠
(7.271)
Both equations define the intermediate pressure p1. Therefore for given pressures the mass flow rate G = f ( p0 , p2 ) can be computed. As long the mass flow rate is less then the critical one defined by Eq. (7.268) written for the states 1 and 2 the flow is sub critical. Else, again the variation of the exit pressure has to be done so, as to obtain the criticality condition at the exit. In this way bots, the exit critical
7.6 Recent state of the knowledge for describing critical flow
269
pressure and the critical mass flow rate can be computed for entrance of subcooled water in the nozzle.
7.6 Recent state of the knowledge for describing critical flow In the previous sections several simplifying assumptions are made to derive models that have limited adequacy but are easily used in the daily practice. The main ideas for such models originated mainly in the 1960’s when the computer power was very low. Now we posses fast running computers and assumptions on place of existing knowledge are no more necessary. In this section I present a mathematical description of a steady state multi-phase flow by not making assumptions regarding the mechanical and thermal interactions but using mechanistic models for each particular phenomenon. 7.6.1 Bubbles origination During the pressure drop in a discharging single phase flow the saturation temperature at this pressure may become lower then the initial liquid temperature. This creates thermodynamic instability and clusters of high energy molecules forms initially unstable and then stable bubble nucleolus. The process is called nucleation. If the nucleation happens within the volume of the fluid it is called homogeneous. If it happens at the wall of the structure it is called heterogeneous. The energy for forming a stable bubble at a solid rough surface is smaller than the energy to form a bubble within the fluid. Therefore for practical application the heterogeneous nucleation is of importance. I already provided useful information on this subject in Vol. 2 of this work. Here I will summarize few practical models which describe the constituents of the nucleation and the bubble departure process at non heated walls. 7.6.1.1 Active nucleation seeds at the surface At all the places where bubbles originate the temperature in average drops to the saturation temperature. But the creation of bubbles consumes energy that has to be supplied from the core of the flow. Therefore, the turbulent heat conduction is the mechanism transporting thermal energy from the core of the flow to the boundary layer. In this sense there is no principle difference between boiling on heated solid surfaces and spontaneous flashing at solid surfaces. Therefore the methods describing adequately developed boiling works also for flushing, Kolev (2006). As far I know there is no useful theoretical description for the activated nucleation sites per unit surface as a function of the local superheating. Empirical information for each particular surface is needed.
7. Critical multiphase flow
Act. nucl. sites density, m
-2
270
10
7
10
6
10
5
10
4
Exp. 30° 57° 80° 90° Corr. 30° 57° 80° 90°
10 Superheat, K Fig. 7.23. Active nucleation site density as a function of superheating. Saturated water at 0.1 MPa. Basu et al. (2002) data for different static contact angles: 30°, 57°, 80°, 90°. 124 < G < 886 kg/(m 2s) , 6.6 < ΔTsub < 52.5 K , 25 < q& ′′w 2 < 960 kW/m 2 . Prediction of the same data with their correlation respectively
As an example the measurements made by Basu et al. (2002) are reproduced by n1′′w = 3400(1 − cos θ )ΔT 2 = 2.183 × 10−5 (1 − cos θ ) / D12c
(7.272)
for ΔTinb < ΔT < 16.298 K , and n1′′w = 0.34(1 − cos θ ) ΔT 5.3 = 2.048 × 10−23 (1 − cos θ ) / D15.3 c
(7.273)
for 16.298K ≤ ΔT . Here ΔTinb is the superheat required for initiation of the nucleate boiling. The correlation reproduces data given Fig. 7.23 for θ = π / 2 and θ = π / 6 within ±40% . The correlation is valid in the range 124 < G < 886 kg /(m 2 s) , inlet sub-cooling 6.6 < ΔTsub < 52.5 K , heat flux 25 < q& w′′ 2 < 960 kW/m2, and wetting angle π / 6 < θ < π / 4 . These measurements demonstrated the important influence of the wetting angle on nucleation processes. Different couples of structure material and water possess different wetting angles. The wetting angle depends also on the mechanical and chemical treatment of the surface. A collection of some wetting angles for wall-water couples usually used in the technology are given in Tables 7.4 and 7.5.
7.6 Recent state of the knowledge for describing critical flow
271
Table 7.4. Static contact angles θ for distilled water at polished surfaces.
π / 3.7 π / 4.74 π / 4.76 to π / 3.83
Steel Steel, Nickel Nickel
π / 3.4 π / 3.2 π / 3.16
Siegel and Keshock (1964) Bergles and Rohsenow (1964) Tolubinsky and Ostrovsky (1966) Siegel and Keshock (1964) Arefeva and Aladev (1958) Labuntsov (1963) for p =1 to 150 bar Arefeva and Aladev (1958) Arefeva and Aladev (1958) Basu et al. (2002)
π /4 π /3 π /2 θ 25 ≈ π / 2.25
Arefeva and Aladev (1958) Gaertner and Westwater (1960) Wang and Dhir (1993) Hirose et al. (2006)
θ 25 ≈ π / 2.46 θ 25 ≈ π / 2
Hirose et al. (2006)
Nickel π / 4.74 to π / 3.83 Chrome-Nickel Steel π / 3.7 Silver π / 6 to π / 4.5 Zinc Bronze Zr-4 Note the contradictory data for copper in the literature Copper Copper Copper Stainless steel 304 (25°C) Zircaloy (25°C) Aluminum (25°C)
Hirose et al. (2006)
Table 7.5. Static contact angle θ for distilled water at thermally or chemically treated polished surfaces
Copper heated to 525 K and exposed to air one hour: Copper heated to 525 K and exposed to air two hour: Chrome-Nickel Steel chemically treated:
π / 5.14
Wang and Dhir (1993)
π /10
Wang and Dhir (1993)
π / 2.9
Arefeva and Aladev (1958)
The thermal properties of the wall and its roughness also influence the nucleation process. Benjamin and Balakrishnan (1997) reported the only correlation known to me that takes into account the thermal properties of the heated wall and its roughness as follows n1′′w = 218.8
Pr21.63
γδ w*
0.4
ΔT 3 .
(7.274)
The liquid Prandtl number, the dimensionless roughness and the dimensionless factor defining the transient interface temperature are defined as follows
272
7. Critical multiphase flow
Pr2 = c p 2η2 λ2 ,
(7.275) 0.4
δ w* = 14.5 − 4.5
δw p ⎛ δw p ⎞ +⎜ ⎟ , σ2 ⎝ σ2 ⎠
(7.276)
γ = ⎡⎣ λw ρ w c pw
( λ ρ c )⎤⎦
(7.277)
2
2 p2
1/ 2
.
δ w is the arithmetic averaged deviation of the roughness surface line from the averaged line in a vertical plane. For practical application see the summary in Table 4.7 < γ < 93 , 7.6. The correlation is valid within 1.7 < Pr2 < 5 , 0.02 < δ w < 1.17 mm, 5 < ΔT < 25 K, 13 × 10−3 < σ < 59 × 10−3 N/m, 2.2 < δ w* < 14. Table 7.6. Roughness of differently polished materials
Material
Finish
Copper Copper Copper
3/0 emery paper 4/0 emery paper 4/0 emery paper
Roughness in µm 0.14 0.07 0.07
Copper Copper Nickel Stainless steel Aluminium Aluminium Aluminium
4/0 emery paper Mirror finish 4/0 emery paper 1/0 emery paper 2/0 emery paper 3/0 emery paper 4/0 emery paper
0.07 0 .
(7.287) As already mentioned from the sixteen analytical solutions of the bubble growth problem obtained for different degrees of simplicity known to the author – see Appendix 13.1 in Vol. 2, the bubble growth model proposed by Mikic et al. (1970) is used as the most accurate one:
μ21,bulk
ρ ′′α1o = Δτ
⎡⎧ 2 ⎡ + τ +1 ⎢ ⎨1 + ⎢⎣ ⎩ 3 ⎣⎢
(
)
3/ 2
( )
− τ
+
3/ 2
3 ⎤ + ⎫ ⎤ − 1 R1o ⎬ − 1⎥ ⎦⎥ ⎭ ⎥⎦
(7.288)
where
τ + = Δτ ( B 2 A 2 ) , A2 =
2 1 ( dp / dT )sat (1 − ρ ′′ / ρ2 ) [T2 − T ′( p)] , 3 ρ2
R + = AR1 / B 2 .
(7.289)
(7.290)
(7.291)
Thus the steam generation is a superposition of the nucleation at the wall and of the bubble growth inside the bulk flow
μ21 = μ21,bulk + μ21, gen .
(7.292)
7.6.2 Bubble fragmentation A review of the existing fragmentation and coalescence models for gases and liquids was given in Chaps. 7–10 of Vol. 2. For the computational examples presented below I use the following bubble fragmentation model. The volume and time average number of generated particles per unit time and unit mixture volume due to dynamic fragmentation is
7.6 Recent state of the knowledge for describing critical flow
n&1, frag = (n1∞ − n1 ) / Δτ fr for n1∞ > n1 and Δτ ≤ Δτ fr ,
(7.293)
n&1, frag = (n1∞ − n1 ) / Δτ for n1∞ > n1 and Δτ > Δτ fr ,
(7.294)
n&1, frag = 0 for n1∞ ≤ n1 ,
(7.295)
277
where Δτ is the time step, n1 is local bubble density concentration, and n1∞ = 6α1 /(π D13∞ )
(7.296)
is the stable bubble number density concentration after fragmentation. Equation (7.294) gives in fact steady state fragmentation for large time steps. The stable bubble diameter is computed in accordance with the stability criterion either depending on the microscopic cross section averaged field velocity difference 2 D1∞ = We21, critσ 21 ⎡ ρ 2 ( w1 − w2 ) ⎤ , ⎣ ⎦
(7.297)
with critical Weber number equal to 12 or in accordance with maximum turbulence scale of the liquid, the large eddy with size D1∞ = cη k23 / 2 ε 2 ,
(7.298)
cη = 0.09 . The mechanism which dictates the smaller size is then the relevant one. The fragmentation time is set equal to the natural fluctuation period of bubble flow, 1/ 3
⎛ 6 ⎞ Δτ fr = ⎜ ⎟ ⎝π 2 ⎠
α11/ 3 Δw12* ,
(7.299)
where the effective relative velocity is a superposition of the difference between the averaged mean field velocities and the turbulent fluctuation velocity of the continuum, Δw12* = w1 − w2 + w2′ ,
(7.300)
and w2′ ≈
2 k2 . 3
(7.301)
278
7. Critical multiphase flow
For comparison the time scale of the fluctuation of eddy with size D1∞ is in such case Δτ e ,2 = 0.37 k2 ε 2 .
(7.302)
7.6.3 Bubble coalescences The volume and time average bubble disappearances per unit time and unit mixture volume due to collision and coalescence is computed in accordance with Chap. 7 of Vol. 2, n&coal = n1 [1 − 1/ exp( f coal / Δτ ) ] / Δτ ,
(7.303)
where Δτ is the time averaging period (the time step in practical computations). Here the instantaneous coalescence frequency of a single bubble is f coal = Pcoal f col ,
(7.304)
where the collision frequency is computed in accordance with Eq. (7.29) from Vol. 2,
f col = 4.5α11/ 2 Δw12* / D1
(7.305)
and the coalescence probability Pcoal is set to 1 for superheated liquid in order to approximate the effect of non-uniform particle distribution in the cross section. In case of no liquid superheating we use Eq. (7.45) from Vol. 2, Pcoal ≈ 0.032 ( Δτ col Δτ coal )
1/ 3
for Δτ col Δτ coal ≥ 1 ,
(7.306)
where 1/ 2
⎛ D1σ 2 ⎞ Δτ col Δτ coal = 1.56 ⎜ ⎟ ⎝ 3ρ1 + 2 ρ 2 ⎠
w2′ .
(7.307)
7.6.4 Droplets origination Droplets are either already available at the entrance of the considered pipe section or originate during the flashing process first by mechanical instability and transition from churn turbulent to dispersed flow.
7.7. Examples for application of the theory of the critical flow
279
7.7. Examples for application of the theory of the critical flow Next I will show two examples for application of the critical flow theory. This examples are typically used for checking of the adequacy of computer code designed to simulate loss of coolant accidents. 7.7.1 Blow down from initially closed pipe Edwards and O’Brien (1970) performed an experiment which contains all the relevant physics of the water depressurization. Figure 7.27 represents the test section: Given is a pipe with length 4.096 m and internal diameter 0.073. The pipe is filled with water at pressure 68.95 bar. The reference temperature of the water is reported to be 514.8 K, lower then the saturation temperature at the initial pressure. It is reported that there is considerable difference of the initial temperature at the top and at the bottom of about 8 K. The temperature along the pipe may vary also with about 4 K. So these are the uncertainties of the initial conditions.
Fig. 7.27. General arrangement of the blow down pipe of Edwards and O’Brien (1970)
The experiment starts with breaking of a glass wall at the right end of the pipe presumably within 0.1 ms or less. Pressures are recorded as a function of time at GS7 position as indicated in the picture. At position GS5 thermocouple inside the flow recorded also a temperature. At the same position a gamma ray station is used to measure the void fraction during the transient. The results of the computational simulation using the IVA computer code of this process will be next presented. The initial temperature was set to 510 K. The homogeneous equilibrium critical flow model is used as a boundary condition at the right end. It is assumed that 0.95 % of the glass cross section is opened after the break. The duration of the break was assumed to be 0.1 ms.
280
7. Critical multiphase flow
70
70
60
40
60 GS1 0.168m IVA Exp.
30 20 10 0 0,000
0,005 0,010 Time in s
40
0,005 0,010 Time in s
p in bar
p in bar
0,1
0,2
0,3 0,4 Time in s
GS3 1.161m IVA Exp.
40 30 20 10
0,005 0,010 Time in s
0 0,0
0,015
0,1
0,2
0,3 0,4 Time in s
0,5
0,6
70
60
60 GS4 2.072m IVA Exp.
GS4 2.072m IVA Exp.
50 p in bar
p in bar
0,6
20
50
70
30 20 10 0 0,000
0,5
30
60
10
40
0,6
70
GS3 1.161m IVA Exp.
20
50
40
0 0,0
0,015
30
0 0,000
0,5
10
70
40
0,2 0,3 0,4 Time in s
GS2 0.326m IVA Exp.
50
10
50
0,1
60
20
60
20
70 GS2 0.326m IVA Exp.
30
0 0,000
30
0 0,0
0,015
p in bar
p in bar
50
40
10
70 60
GS1 0.168m IVA Exp.
50 p in bar
p in bar
50
40 30 20 10
0,005 0,010 Time in s
0,015
0 0,0
0,1
0,2
0,3 0,4 Time in s
0,5
0,6
7.7. Examples for application of the theory of the critical flow
70
p in bar
50 40
70 60
GS5 2.627m IVA Exp.
30 20 10 0 0,000
0,005 0,010 Time in s
40
0,005 0,010 Time in s
40
0,5
0,6
0,5
0,6
30 20
0 0,0
0,015
0,1
0,2
0,3 0,4 Time in s
70 60
GS7 4.017m IVA Exp.
40 30
40 30
20
20
10
10 0,005 0,010 Time in s
GS7 4.017m IVA Exp.
50 p in bar
p in bar
0,3 0,4 Time in s
10
70
0 0,000
0,2
GS6 3.182m IVA Exp.
50
10
50
0,1
60
GS6 3.182m IVA Exp.
20
60
20
70
30
0 0,000
30
0 0,0
0,015
p in bar
p in bar
50
40
10
70 60
GS5 2.627m IVA Exp.
50 p in bar
60
281
0,015
0 0,0
0,1
0,2
0,3 0,4 Time in s
0,5
0,6
Fig. 7.28. Pressure as a function of time
The pressure histories are presented at Fig. 7.28. The void fraction 2.627 m from the left end as a function of time is presented in Fig. 7.29. The thermocouple temperature inside the flow 2.627 m from the left end as a function of time compared with the vapor and liquid temperature predicted by IVA are presented in Fig. 7.30. The force acting on the pipe as a function of time is presented in Fig. 7.31.
282
7. Critical multiphase flow
1,0
Void fraction, -
0,8 0,6 GS5 2.627m IVA Exp.
0,4 0,2 0,0 0,0
0,1
0,2 0,3 0,4 Time in s
0,5
0,6
Fig. 7.29. Void fraction 2.627 from the left end as a function of time
600 550
T in K
500 450 400 350 300 0,0
GS5 2.627m IVA Tvoid IVA Tfilm Exp.
0,1
0,2
0,3 0,4 Time n s
0,5
0,6
Fig. 7.30. Thermocouple temperature inside the flow 2.627 from the left end as a function of time compared with the vapor and liquid temperature predicted by IVA
The following observation can be made from the experimental records: - The pressure sharply drops down below the saturation. The difference between the saturation pressure at the initial liquid temperature and the reached minimum pressure value is called in the literature pressure undershoot;
7.7. Examples for application of the theory of the critical flow
283
50
Force in kN
40
IVA Exp.
30 20 10 0 -10 -20 0,0
0,1
0,2
0,3
0,4
0,5
0,6
Time in s Fig. 7.31. Force acting on the pipe as a function of time
– Then nucleation and evaporation starts recovering partially the pressure drop and keeping by evaporation a plateau of a pressure corresponding to the saturation pressure at the initial temperature; – Then a third phase is entered characterized with the final pressure reduction. The predicted character is close to the experimental results. The uncertainties in the temperature are responsible for not exactly reaching the plateau pressure. The vapor evolution in Fig. 7.29 is similar to the observed. The fluctuations are associated with the flow regime transition and slug expulsions which of course are only approximately predicted. The maximum force can not be larger then the initial pressure difference multiplied by the flow cross section which is 28.4 kN. At the plateau the maximum force at the reference temperature is 14 kN. It is not measured – an indication that effectively the initial averaged temperature was somewhat lower than the reference one. The oscillation of the measured force is not predicted by the code because no structure flow interaction is modeled. Figure 7.30 shows that the thermocouple probably senses the liquid temperature. The measured plateau liquid temperature is also somewhat smaller then the predicted which confirms the just made observation regarding the initial temperature of the liquid. 7.7.2 Blow down from initially closed vessel The Kevchishvili and Dementev small-break loss-of-coolant experiment was performed at the Moscow Energetic Institute to study strong separation effects Kevchishvili and Dementev (1985). The 36 uniformly heated rods 16 mm in diameter and 1700 mm long are mounted within a shroud 159 mm in diameter in a
284
7. Critical multiphase flow
pressure vessel 309 mm in diameter and 2100 mm high as shown in Fig. 7.32(a). We simulate an experiment with a thermal power input of 67 kW and a cold leak from a break nozzle 22 mm in diameter. The initial state of the water inside the vessel was p = 123 bar, T = 558 K, and void fraction equal to zero. The void fractions in the rod bundle in the three different locations shown in Fig. 7.33(a), (b) and (c) were recorded as a function of time.
a)
b)
Fig. 7.32. (a) IVA geometry model for the Kevchishvili and Dementev small-break loss-ofcoolant experiment. (b) Pressure at level z = 1.83 m as a function of time. Comparison of the IVA prediction with the experiment of Kevchishvili and Dementev (1985). Not heated bundle
I simulate the geometry with 2 radial, 3 angular, and 39 axial zones – 117 cells – as shown in Fig. 7.32(a). I assume that the “vena contracta” coefficient at the nozzle is 0.8. The comparison with the experimental data is shown in Figs. 7.32(b) and 7.33. From this comparison the following conclusions can be drown. The steam discharge between 7 and 11 s is stronger due to friction in the discharge pipe, which was not considered. The pressure compares well with the measurement. The end of the stratification at the three different positions of the test section is well predicted. In the upper part the stratification is well predicted.
7.8 Nomenclature
285
In the mid and lower parts the stratification is well predicted except for the region between 7 and 10 s where it happens faster in the simulation.
Fig. 7.33 The void fraction in a rod bundle on level z = 1.83, 0.975, 0.612 m as a function of time. Comparison of the IVA prediction with the experiment of Kevchishvili and Dementev (1985). Not heated bundle
7.8 Nomenclature Latin A a C C C0 c cp
coefficient matrix velocity of sound, m/s source term vector of the system of PDF describing one dimensional flow contractions coefficient, dimensionless distribution parameter in the drift flux model, dimensionless constant is controlling the grid sizes, dimensionless specific heat at constant pressure, J/(kgK)
ccdd
drag coefficient describing the action of the continuum on the dispersed particle immersed in it, dimensionless diameter, m
D
286
7. Critical multiphase flow
D1d , nc
bubble departure diameter, m bubble departure diameter for natural circulation, m
D1d , fc
bubble departure diameter for predominant forced convection, m
D1d
D1∞
F f f12d f wd1 f1 ( p ) f2 ( p ) f coal f col G G* g h j1 k k k kmax L nmax n M M
mmax N ′′ n1w n&1w n&1, frag
2 = We21, crit σ 21 ⎡ ρ 2 ( w1 − w2 ) ⎤ , stable bubble diameter, m ⎣ ⎦ cross section, m² frequency of the not dumped oscillation, 1/s drag force between field 1 and 2 per unit mixture volume (accelerating field 2), N/m³ drag force between wall w and field 1 per unit mixture volume (decelerating field 1), N/m³ dT v ′′ − v′ = = , function defining the Clapayron’s equation, function of dp s ′′ − s ′ pressure for saturated mixture, m³K/J s ′′v′ − s ′v′′ , function of pressure for saturated mixture, m³/kg = s ′′ − s ′ = Pcoal f col , instantaneous coalescence frequency of a single bubble, 1/s
collision frequency, 1/s mass flow rate, kg/(m²s) critical mass flow rate, kg/(m²s) gravitational acceleration, m/s² specific enthalpy, J/kg = α1 w1 , averaged vapour volumetric flow rate, m/s wave number, dimensionless index specific turbulent kinetic energy, m²/s² total number of cells in the Laval nozzle, dimensionless pipe length, m number of cells in the diverging part, dimensionless polytropen exponent, dimensionless mass, kg ratio of the velocity to the local sound velocity, Mach number, dimensionless number of cells in the converging part, dimensionless multiplier reducing the equilibrium evaporation, dimensionless active nucleation site density at the wall, 1/m² ⎛ π ⎞ = μ 21 ⎜ ρ1 D13d ⎟ , generated number of bubbles per unit time and mix⎝ 6 ⎠ ture volume, 1/m³ volume and time average number of generated particles per unit time and unit mixture volume due to dynamic fragmentation, 1/m³
7.8 Nomenclature
287
n1
local bubble density concentration, 1/m³
n1∞
= 6α1 /(π D13∞ ) , stable bubble number density concentration after fragmentation, 1/m³ volume and time average bubble disappearances per unit time and unit mixture volume due to collision and coalescence, 1/(m³s) power needed per unit mass of field l for production of turbulence, m²/s³ pressure, Pa coalescence probability, dimensionless = c p 2η2 λ2 , liquid Prandtl number, dimensionless
n&coal Pk ,l p Pcoal Pr2 q& ′′′ R Recd S Sl
s Tl T U U V V1*j v
heat per unit time and unit mixture volume, W/m³ gas constant, J/(kgK) = ρ c Dd wc − wd ηc , Reynolds number of dispersed particles being in continuum c, dimensionless = w1 w2 , gas to liquid velocity ratio, slip, dimensionless = wl / w , ratio of the velocity of field l to the centre of mass mixture velocity, dimensionless specific entropy, J/(kgK) viscous stress at the surface of field l, N/m² temperature, K vector of the dependent variables vector of the cross section and time time-averaged dependent variables volume, m³ drift flux velocity in the drift flux model, dimensionless specific volume, m³/kg
X1
= ρ c ( wd − wc ) Dd σ cd , Weber number of dispersed particles being in continuum c, dimensionless velocity, m/s 2 ≈ k2 , fluctuation velocity of field 2, m/s 3 = α1 ρ1 w1 G , gas mass flow concentration, dimensionless
+ ylim z
thickness of the dissipation boundary layer, dimensionless distance, m
Wecd
w w2′
2
Greek
α α1,lim γz
volumetric fraction, dimensionless void fraction defining the transition of bubble to droplet structure, dimensionless = Fsmall Flarge , cross section ratio, dimensionless
288
7. Critical multiphase flow
γz γv Δ Δplmσ
( Δp
mσ l
ratio of the flow cross section to some normalizing cross section Fn , dimensionless porosity, local flow volume over Δz divided by ΔzFn , dimensionless finite difference, increment difference between the bulk averaged pressure and the surface pressure inside of the file field l, Pa
− Tl ) averaged value over all force appearances of the interface
mσ l
, N/m²
1/ 4
Δw12,Ku
⎛ σ g Δρ 21 ⎞ = 2⎜ ⎟ 2 ⎝ ρ2 ⎠
Δτ 1d
= ⎡⎣ D1d
Δτ 1w
minimum waiting, s fragmentation time, s
Δτ fr Δτ e ,2 ∂
δw
δ 2,min Φ 22o
ε εη ,l ε l′ η κ λi λ fr λ fr ,2 o
μ21, gen ρ
( 2 B )⎤⎦
2
Kutateladze large bubble free rising velocity, m/s
, bubble departure time, s
= 0.37 k2 ε 2 , time scale of the fluctuation of eddy with size D1∞ in the liquid 2, s partial differential arithmetic averaged deviation of the roughness surface line from the averaged line in a vertical plane, m minimum of the thermal boundary layer thickness, m Martinelli-Nelson two phase friction pressure loss multiplier, dimensionless pressure ratio, dimensionless irreversible dissipated power per unit mass of field l caused by the viscous forces due to deformation of the mean values of the velocities in the space, m²/s³ irreversibly dissipated power per unit mass of field l in the viscous fluid due to turbulent pulsations, m²/s³ dynamic viscosity, kg/(ms) isentropic exponent, dimensionless propagation velocity of the harmonic oscillations, eigen value of the system of PDE’s, m/s friction coefficient, dimensionless friction coefficient computed with the total mixture mass flow rate and the liquid properties, dimensionless 4 = q&w′′2,b (h′′ − h′) , generated vapour mass per unit time and unit mixDh ture volume due to the bubble generation at the wall, kg/(m³s) density, kg/m³
References
σ τ ζ
289
surface tension, Nm time, s = z L dimensionless position
Superscripts * mσ l
′ ′′
at the critical cross section interface of field l with field m saturated liquid saturated vapor
Subscripts 0 1 1, eq 2 3 12 21 in out crit max vc h heat Fi I w wl lw Tp
initial vapour, into the vapour vapour in thermal equilibrium with the liquid liquid, into the liquid droplets from 1 to 2 from 2 to 1 inlet outlet critical maximum vena contracta hydraulic heated flushing inception derived from the momentum equation wall from the wall into field l from field l through the wall to outside the flow two phase
References Abuaf N, Wu BJC, Zimmer GA and Saha P (June 1981) A study of non equilibrium flashing of water in a converging diverging nozzle, Vol.1 Experimental, vol 2 Modeling, NUREG/CR-1864, BNL-NUREG-51317 Albring W (1970) Angewandte Strömungslehre, Verlag Theodor Steinkopf, Dresden Algamir M and Lienhard JH (1981) Correlation of pressure undershoot during hot-water depressurisation. J Heat Transfer, vol 103 no 1, p 60
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Aounallah Y and Hofer K (May 4–7, 2003) Level swell prediction with RETRAN-3D and application to a BWR steam line break analysis, Proc. of ICAP’03, Córdoba, Spain, Paper 3362 Arefeva EI and Aladev IT (July 1958) O wlijanii smatchivaemosti na teploobmen pri kipenii. Injenerno – Fizitcheskij Jurnal, vol 1 no 7, pp 11–17, in Russian Arnsberg (1962) Obsor critizeskich rashodomerof dlja izmerenija gazovyh potokov, Teoreticeskie osnovy ingenergyh raszotov, no 4 p 39, IL (in Russian) Aumiller DL, Tomlinson ET and Clarke WG (Sept. 12–14, 2000) A new assessment of Relap5-3D using a General Electric level small problem, 2000 Relap5 User Seminar, Jackson Hole, Wyoming Basu N, Warrier GR and Dhir VK (2002) Onset of nucleate boiling and active nucleation site density during subcooled flow boiling. J Heat Transfer, vol 124, pp 717–728 Benedict RP, Carlucci NA and Swetz SD (Jan. 1966) Flow losses in abrupt enlargements and contractions. Trans. ASME, J Engineering Power, vol 88, pp 73–81 Benjamin RJ and Balakrishnan AR (1997) Nucleation site density in pool boiling of saturated pure liquids: effect of surface microroughness and surface and liquid physical properties. Exp. Thermal Fluid Sci., vol 15, pp 32–42 Benjamin RJ and Balakrishnan AR (1999) Nucleate pool boiling heat transfer of binary mixtures at low to moderate heat fluxes, Trans. ASME, Journal of Heat Transfer, vol 121, pp 365–375 Bergles AE and Rohsenow WM (1964) The determination of forced convection surfaceboiling heat transfer, ASME J. Heat Transfer, vol 1, pp 365–372 Brosche D (August 1973) Berechnung kompressibler, reibungsbehafteter Rohrstroemungen mit Hilfe eines digitalen Rechenprogram, Brenst.-Waerme-Kraft, vol 25 no 8, pp 312–316 Burnell JG (Dec. 1947) Flow of boiling water through nozzles, orifices and pipes. Engineering, pp 572–576 Delhaye JM, Giot M and Rietmüller (1981) Thermodynamics of Two Phase Flow Systems for Industrial Design and Nuclear Engineering, Hemisphere Publ. Corp., McGraw-Hill Book Company, New York Edwards AR and O’Brien TP (1970) Studies of phenomena connected with the depressurization of water reactors. J. Br. Nucl. Soc., vol 9 no 1–4, pp 125–135 Faletti DW (1959) Two-phase critical flow of steam/water mixtures, Dissertation, University of Washington Fauske H (October, 1962) Contribution to the theory of two-phase, one component critical flow, ANI-6633, U. S. A.E.C. Research and Development Report, TID-4500, 18th ed. Fincke JR (August 5–8, 1984) Critical flashing flow of subcooled fluids in nozzles with contour discontinuities, Basic Aspects of Two Phase Flow and Heat Transfer, 22nd Nat. Heat Transfer Conference and Exhibition, Niagara Falls, New York, HTD – vol 34, pp 85–93 Freeman JR (1888) The discharge of water through fire nose and nozzles. Trans. ASCE, vol 2, p 303 Frössel W (1936) Strömung in glatten, geraden Rohren mit Über- und unterschalgeschwindigkeit. Forsch., vol 7 no 2, pp 75–84 Gaertner RF and Westwater JW (1960) Population of active sites in nucleate boiling heat transfer. Chem. Eng. Progr. Symp. Ser., vol 30 no 30, pp 39–48 Gaertner RF (Feb. 1965) Photographic study of nucleate pool boiling on a horizontal surface, transaction of the ASME. J. Heat Transfer, vol 87, pp 17–29 Griffith P and Wallis GB (1960) The role of the surface conditions in nucleate boiling. Chem. Eng. Prog. Symp., vol 56 pp 49–63 Grolmes M, Sharon A, Kim CS and Pauls RE (1986) Level swell analysis of the Marviken test T-11. Nucl. Eng. Des., vol 93, pp 229–239 Han CY and Griffith P (1965) The mechanism of heat transfer in nucleate pool boiling, Part I, Bubble initiation, growth and departure. Int J Heat Mass Transfer, vol 8, pp 887–904
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Hassan YA (4–7 Aug. 1985) TRAC-PF1/MOD1 prediction of the level swell data, ANS Proceedings, 1985 National Heat Transfer Conference, Denver, Colorado Henry RE and Fauske HK (1969) The two-phase critical flow of one-component mixtures in nozzles, orifices, and short tubes. J. Heat Transfer, vol 2, pp 47–56 Henry RE, Fauske HK and McComas ST (March 1970a) Two-phase critical flow at low qualities Part I: Experimental. Nucl. Sci. Eng.,vol 41, pp 79–91 Henry RE, Fauske HK and McComas ST (March 1970b) Two-phase critical flow at low qualities Part II: Analysis. Nucl. Sci. Eng., vol 41, pp 92–98 Henry RE (April 1970) The two-phase critical discharge of initially saturated or subcooled liquid. Nucl. Sci. Eng., vol 41, pp 336–342 Hirose Y, Hayashi T, Hazuku T and Takamasa T (July 17–20, 2006) Experimental study on contact angle of water droplet in high temperature conditions, Proc. of ICONE14, International Conference on Nucl. Engineering, Miami, Florida, USA Isbin HS, Moy JE and Da Cruz AJR (1957) Two-phase steam/water critical flow, A. I. Ch. E. J., vol 3 no 3, p 361 Ishii M and Zuber N (1978) Relative motion and interfacial drag coefficient in dispersed two-phase flow of bubbles, drops and particles, Paper 56 a, AIChE 71st Ann. Meet., Miami Jones OC Jr (1982) Towards a unified approach for thermal non-equilibrium in gas – liquid systems. Nucl. Eng. Des., vol 69, pp 57–73 Kevchishvili NA and Dementev BS (1985) Investigation of the influence of the decay heat on the blow down characteristics of steam-water mixtures. Teploenergetika, vol 7, p 67 Kolev NI (1986) Transiente Zweiphasenströmung, Springer, Berlin Kolev NI (1994) The influence of mutual bubble interaction on the bubble departure diameter. Exp. Therm. Fluid Sci., vol 8, pp 167–174 Kolev NI (Feb. 2006) Uniqueness of the elementary physics driving heterogeneous nucleate boiling and flashing,. Nucl. Eng. Technol., vol 38 no 1, pp 33–42, Seoul, Korea Kolev NI (2007a) Multiphase Flow Dynamics, Springer, Berlin, vol 1 Koumoutsos N, Moissis R and Spyridonos A (May 1968) A study of bubble departure in forced-convection boiling. J. Heat Transfer, Trans. ASME, vol 90, pp 223–230 Kurihara HM and Myers JE (March 1960) The effect of superheat and surface roughness on boiling coefficients, AIChE J., vol 6 no 1, pp 83–91 Labuntsov DA (1963) Approximate theory of heat transfer by developed nucleate boiling (Russ.), Izvestiya AN SSSR, Energetika i transport, no 1 Labuntsov DA, Kol'chugin VA and Golovin VS et al. (1964) Investigation by slow motion of buble growth Teplofiz. Vys. Temp., vol 3, pp 446–453 Landau LD and Lifshitz EM (1987) Course of Theoretical Physics, vol 6, Fluid Mechanics, 2nd ed., Pergamon, Oxford Mikic BB, Rohsenhow WM and Griffith P (1970) On bubble growth rates, Int. J. Heat Mass Transfer, vol 13, pp 657–666 Moody FJ (Feb. 1965) Maximum flow rate of a single component, two-phase mixture. J. Heat Transfer, vol 86, pp 134–142 Moody FJ (Aug. 1966) Maximum two-phase vessel blow down from pipes. J. Heat Transfer, vol 88 pp 285–293 Moody FJ (Aug 1969) A pressure pulse model for two-phase critical flow and sonic velocity. J. Heat Transfer, vol 91, pp 371–384 Moody FJ (1975) Maximum discharge rate of liquid-vapor mixtures from vessels, Nonequilibrium two-phase flows, pp 27–36 Moy JE (1955) Critical discharges of steam/water mixtures, MS thesis, University of Minnesota Oswatitsch K (1952) Gasdynamik, Springer-Verlag, Vienna Perri JA Jr (1949) Critical flow through sharp-edged orifices. Trans. ASME ,vol 71, p 757
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Roll JB and Mayers JC (July 1964) The effect of surface tension on factors in boiling heat transfer. A.I.Ch.E. J., pp 330–344 Salet DW (1984) Thermal hydraulic of valves for nuclear applications. Nucl. Eng. Des., vol 88, pp 220–244 Semeria RF (1962) Quelques resultats sur le mechanisme de l’ebullition, 7, J. de l’Hydraulique de la Soc. Hydrotechnique de France Shapiro AH (1953) The dynamics and thermodynamics of compressible fluid flow, The Ronald press Company, New York Siegel R and Keshock EG (July 1964) Effects of reduced gravity on nucleate boiling bubble dynamics in saturated water, AIChE J., vol 10 no 4, pp 509–551 Sozzi GL and Sutherland WA (July 1975) Critical flow of saturated and subcooled water at high pressure. General Electric, San Jose Tangren RF, Dodge CH and Seifert HS (July 1949) Compressibility effects in two-phase flow. J. Appl. Phys., vol 20 no 7, pp 673–645 Tolubinsky VI and Ostrovsky JN (1966) On the mechanism of boiling heat transfer (vapor bubbles growth rate in the process of boiling in liquids, solutions, and binary mixtures), Int. J. Heat Mass Transfer, vol 9, pp 1463–1470 Umminger K et al. (2007) Discharge experiments in the PKL test facility, AREVA proprietary van Stralen SJD, Sluyter WM and Sohal MS (1975) Bubble growth rates in nucleate boiling of water at subatmospheric pressures. Int. J. Heat Mass Transfer, vol 18, pp 655–669 Wang CH and Dhir VK (Aug. 1993) Effect of surface wettability on active nucleation site density during pool boiling of water on a vertical surface. ASME J. Heat Transfer, vol 115, pp 659–669 Weisbach J (1872) Mechanics of Engineering, translated by J. Coxe, Van Nostrand Book Company, New York, NY Wien M (2002) Numerische Simulation von kritischen und nahkritischen Zweiphasenströmungen mit thermischen and fluiddynamischen Nichtgleichgewichtseffekten, PhD Dissertation, Technischen Universität Dresden Zaloudek FR (March 1961) The low pressure critical discharge of steam/water mixtures from pipes, HW-68934 Rev. Ziklauri GB, Danilin VS and Seleznev LI (1975) Adiabatnye dvichfasnye tecenija, Atomisdat, Moskva, in Russian Zuber N (1963) Nucleate boiling: The region of isolated bubbles and the similarity with natural convection. Int. J. Heat Mass Transfer, vol 6, pp 53–79
8. Steam generators
The art in the steam generator design is by having specified primary fluid temperature, pressure and mass flow to design a vapor production with the lowest possible content on droplets at highest possible pressure and mass flow. On this way technical discoveries like introduction of economizers, redirection of separated water into the natural circulation loop using appropriate low pressure loss high efficiency separators etc. are inevitable. 8.1 Introduction The heat produced in a nuclear reactor core can be used in different ways:
Fig. 8.1. a) European Nuclear Reactor; b) Reactor pressure vessel connected by primary pipes to the four steam generators. The four main circulation pipes and the pressurizer are visible
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_8, © Springer-Verlag Berlin Heidelberg 2009
293
294
8. Steam generators
Boiling water reactors produce steam directly in the pressure vessel. The steam is then directed to turbines for producing mechanical work which then is transferred partially in electricity. In accordance with the Carnot’s low the higher the upper turbine entrance temperature is, the better the efficiency of the thermal cycle. In producing saturated steam the pressure is controlling the steam temperature. That is why the highest technically possible pressure is used. Recently operation at super critical pressures is intensively in investigation for this purpose – Pioro und Duffey (2007). Actually the super-critical technology is common in the conventional steam production. The pressurized water reactors produce steam in steam generators and direct it to the steam turbines for farther use, see Fig. 8.1. The coolant is pumped from the reactor coolant pump, through the nuclear reactor core, and through the tube side of the steam generators before returning to the pump. This cycle is called primary loop. The primary loop water with about 600 K and 150 bar flowing through the steam generator boils other water on the shell side and produce steam in the secondary loop that is delivered to the turbines to make electricity. The steam is subsequently condensed via cooled water from the tertiary loop and returned to the steam generator to be heated once again. The tertiary cooling water may be recirculated to cooling towers where it release large amount of heat into the environment before returning to condense more steam. Once through tertiary cooling may otherwise be provided by rivers, lakes, sees and oceans. This primary, secondary, tertiary cooling scheme is the most common way to extract usable energy from a controlled nuclear reaction. These three loops also have an important safety role because they constitute one of the primary barriers between the radioactive and non-radioactive sides of the plant as the primary coolant becomes radioactive from its exposure to the core. For this reason, the integrity of the tubes is essential in minimizing the leakage of water between the primary and the secondary sites of the plant.
8.2 Some popular designs of steam generators 8.2.1 U-tube type Figure 8.2 is a schematic diagram of a U-tube type vertical steam generator design.
8.2 Some popular designs of steam generators
295
Fig. 8.2. U-tube type steam generators for nuclear power plant: a) Westinghouse Model-F design, Singhal and Srikantiah (1991); b) Siemens design, Bouecke (2000)
Fig. 8.3. Pipe arrangements: quadratic: (a) inline; (b) staggered; triangle: (c) inline; (d) staggered. Typically pipes with about 19–25 mm outside diameter and pich 25–37.5 mm are used, Singhal and Srikantiah (1991)
The steam generator components are contained within a cy1indrical vessel. The primary water enters through the inlet plenum of the seam generator, flows inside the U-tubes and transfers heat to the water on the shell side. The feed water is provided at the top. This feed mixes with the water returned by the steam separator and flows down in the annular space between shell and shroud called down-comer.
296
8. Steam generators
Subsequently, this water flows upwards over the U-tubes inside the shroud, pick up heat and generates the steam. This two phase mixture rises then to steam separators.
Fig. 8.4. Antivibration bar arrangement, Singhal and Srikantiah (1991)
The tubes may have quadratic or triangular arrangements each of them may be inline or staggered as shown in Fig. 8.3. In commercial power plants steam generators can measure up to 22 m in height and weigh as much as 800 tons. Each steam generator can contain 3000–16 000 tubes, each about 19 mm in diameter. The very long tubes are exposed on flow induced oscillations and will vibrate touch each other or touch structures and erode if anti vibration spacers are not used. An example for tube support planes is given on Fig. 8.1 and for anti vibration bars in Figs. 8.2 and 8.4. Other design of U-tube steam separator is given in Fig. 8.5. In all cases a primary inlet and outlets are designed as a chambers in which mixing happens. The tubes are welded on the thick tube sheet. Several redistribution plates or other design feature improve the performance of the steam generator. The upper part of the shell has larger diameter because the two phase mixture expands in this region up to the lower deck plate. The steam quality, i.e., the ratio of the mass of steam to the mass of the mixture, varies from near zero at the bottom of the generator to 10–30% in the riser just below the lower deck plate, depending on design and operating conditions. Such wet steam can not be send to the high pressure turbine. A special technology is required to remove the liquid from the steam as good as possible. Normally it happens in two stages: the first one removing the large size liquid and the second one removing the fine moisture – see Fig. 8.6. In Chap. 9 many details of separators design are given and methods haw to analyze their performance. Here only example is given on Figs. 8.2 and 8.5 where cylindrical primary separators are contained in a deck which seals the lower portion of the generator containing the low quality steam from the upper portion containing the high quality steam.
8.2 Some popular designs of steam generators
297
The primary separators swirl the steam-water mixture so that the higher density water is thrown to the sides of the separator cylinder as a rising, rotating layer which is skimmed off. The mass carryover of water at the primary separator exit varies from 1 to 30%, again depending primarily on separator design. The water skimmed off flows from the deck into an annular down comer and is directed to the lower portion of the steam generator. There are several designs of cylindrical, centrifugal steam separators. Carson and Williams (1980) gives a compilation of various separator designs and their performance.
298
8. Steam generators
Fig. 8.5. U-tube steam generators with block separators and dryers: Combustion Engineering System-67 and System-80 design, Singhal and Srikantiah (1991)
8.2 Some popular designs of steam generators
299
(a)
(b) Fig. 8.6. Block of primary cyclone separators mounted on the lower deck plate; (a) view number large diameter cyclones, Singhal and Srikantiah (1991); (b) large number small diameter cyclones, Bussy et al. (1998)
The secondary separators which are usually of a wire mesh or parallel vane design remove most of the remainder of the moisture from the steam. Gravity separation which may be relatively significant also occurs in the inter-space between the primary and secondary separators. The exit steam quality from present design steam generators is about 0.25%, while the design specifications for the recirculation ratio, and varies from 3 to 9 depending on steam separator and generator design. In such plants higher thermal efficiency is reached because the upper temperature is considerably higher.
300
8. Steam generators
Fig. 8.7. Once through type steam generators for nuclear power plant: Babcoox and Wilcox design, Singhal and Srikantiah (1991) Besides the increase of the efficiency of the steam generators by using effective low pressure los high effective liquid separation technology the use of economizers is also important. Different type of economizers are in use: Economizers splitting the feed water flow with larger part send trough the cold part and smaller part trough the hot part. Other type of economizers called axial economizers sends the feed water trough the cold part only. Bussy et al. (1998) reported that sending the feed water trough the cold part only and redirecting 90% of the recirculation water to the hot part increases the steam pressure compared to steam generators of the design without these measures.
8.3 Frequent problems
301
8.2.2 Once through type In the once trough steam generators (OTSG) from the type presented on Fig. 8.7 the heat transfer bundle is straight. The primary fluid flows vertically downward trough straight vertical tubes, and the secondary fluid flows upwards outside the tubes. Feed water enters radially at the bottom and the superheated vapor with about 30K superheat exits radially near the top of the shell. This is the exciting feature of the OTSG – the possibility to superheat the steam. In the predominant part the use counter current flow principle provides good heat transfer efficiency. Some OTSG have either an economizer zone, other have some injection of steam in the secondary fluid about 1 to ½ of the height. This help to reach at lower elevation the saturation of the water and then to allow for superheating. These steam generators are simpler in design compared to the U-tube type. The secondary site heat transfer mechanism cross all regimes from the subcooled single phase convection to water to the single phase convection to steam. Especially endangered are the regions where the boiling crisis occurs because they naturally oscillate up and downwards around a given position causing thermal stresses. The film dry out leads to deposition of impurities at those places which change the heat transfer and accelerate corrosion.
8.2.3 Other design types Specific designs are known for nuclear power plants with different working principles then the pressurized water reactor. So for instance in pressurized heavy water reactors of the CANDU design the primary fluid is heavy water. Liquid metal cooled reactors such as the in Russian BN-600 reactor or the French Super Phoenix also use heat exchangers between primary very hot liquid metal coolant and at the secondary water coolant. 8.3 Frequent problems If a tube bursts while a plant is operating contaminated steam could escape directly to the secondary cooling loop. Thus during scheduled maintenances, outages or shutdowns, some or all of the steam generator tubes are inspected by eddy-current testing or other means. If they are hermetic they are usually plugged, Keeton et al. (1986). An example is given in Fig. 8.8. After plugging more than a prescribed number of tubes the steam generator has to be replaced. This behavior is strongly influenced by the chemical water regime and the material selection.
302
8. Steam generators
Fig. 8.8. Example of plugged tube concentration
The usual problems with the steam generators leading to degradation are associated with vibration, fretting, fatigue, stress-corrosion, cracking, pitting, denting, deposition of corrosion products at the tube sheet etc., Green (1988), Solomon et al. (1985). So depending on the design, material selection and water chemistry the good steam generators already showed 30 years operation. Others loose their reliability after 15–20 years and are replaced. The deposition at the tube sheet reduces heat transfer surface and promote corrosion of the pipes. One should design such flow conditions that do not allow for intensive sedimentation of corrosion products on the tube sheet. High pressure water jets are usually used for cleaning this parts of the steam generator. 8.4 Analytical tools Actually the method of the multi-phase flow analysis as described in this monograph is what is needed to analyze the processes inside the steam generators. Usually the porous body concept is used in which with surface permeabilities and porosities describe the secondary control volume reduction due to tubes, and flow obstacles. An example is given in Fig. 8.9(a).
8.4 Analytical tools
303
Fig. 8.9. (a) Computational model of the secondary site of a steam generator using the IVA porous body concept; (b) Void fraction at steady state as a function of space: left – cold site, right – hot site, blue – pure water, red – pure steam. Velocity of steam as a function of space
304
8. Steam generators
The primary site is described as a multiple of a representative pipe starting from the inlet plenum and ending at the outlet plenum. The thermal coupling is then arranged between corresponding primary and secondary control volumes trough transient heat conduction. It is important to have models for boiling including the boiling creases with the logic of adjustable heat fluxes at the both sites. After reaching the critical heat flux the transition boiling is entered up to the minimum film boiling temperature. Then by farther increasing of the power transition to film boiling is on place. There is no hysteresis in the boiling curve. Note the difference to nuclear heating. The modeling of the separators can happen at different level of complexity: use of experimental characteristics inside the control volumes and transmitting of the separated liquid and gas in other volumes; use in the same way analytically derived characteristics as described in Chap. 9; use of fine resolution with boundary fitted coordinates for the separators and therefore natural modeling of the entrained liquid and gas. The last method is the most expensive and still not in use. Note that the integral characteristics of the final steam are of interests. Recirculation ratio, temperature and void distribution, pressure level at given steam mass flow etc. are usually requested as output of thermo-hydraulic analysis. Figure 8.9(b) gives an illustration of numerically obtained solution for the steam generator with geometry presented on Fig. 8.2. The fascinating feature of the multi-fluid computational analysis is that it provides the local void distribution, the velocity fields, the vibration characteristics etc. General verification of multi-phase models is usually combined with prediction limited number of heat exchanger and model steam generators experiments, Fortino et al. (1980), Singhal et al. (1984), Wang and Srikantiah (1985), Lee and No (1986), Aubry et al. (1989), Keeton et al. (1990), Singhal and Srikantiah (1991), John et al. (2005). Data are reported by Gautier and Boissier (1971), Riboud and Brugeille (1987), Gouirand (1989, 1991) for heat exchanger. A numerical example given by Patankar and Spalding (1976) can be used as a benchmark for single phase flow. Tests for model steam generators are discussed by Singhal et al. (1983). Hassan and Morgan (1980) compared their analysis with a well documented simple boiler experiment. Finally the best way of confirming the design is to equip a real steam generator with measurements and to take its relevant characteristics during the operation in a real plant: Procaccia et al. (1982), Carlucci et al. (1982) compared their analysis with data collected on a real industrial steam generator of US nuclear power plant. Schwarz and Bouecke (1985) compared their analysis with data collected on a real industrial steam generator of German nuclear power plant. Bussy et al. (1998) compared their analysis with data collected on a real industrial steam generator (N4) of French nuclear power plants. References Aubry S, Cahouet J, Nicolas G and Niedergang C (1989) A finite volume approach for 3D two phase flows in tube bundles the THYC code, Proceedings of the Fourth International Topical Meeting an Nuclear Reactor Thermal – Hydraulics, pp 1247–1253 Bussy B, Dague G and Slama G (1998) Starting up of new steam generator on N4 1450 MWe plants, Proc. 3th International Conference Steam Generators and Heat Exchanger, Toronto, Ontario, Canada
References
305
Carlucci LN et al. (1982) Thermal hydraulic analysis of the Westinghouse Model 51 steam Generator, EPRI NP2683 Carson WR and Williams HK (Nov. 1980) Methods of reducing carry-over and reducing pressure drop through steam separators, EPRI Final Report NP1607 Fortino RT, Oberjohn WJ, Rice JG and Cornelius DK (1980) Thermal-Hydraulic Analyses of Once Through Steam Generators, EPRI NP-1431 Gautier D and Boissier A (1971) Les pertes de charges et le transfert thermique cote gaz dans les échangeurs tubes lisses, a circulations orthogonales. Bulletin de la Direction des Etudes et Recherches d’EDF no 2/3 Green SJ (1988) Thermal hydraulic and corrosion aspects of PWR steam generator problems. Heat Trans. Eng., vol 9, p 1 Gouirand JM (1989) CLOTAIRE Program – Thermal hydraulic test results in the straight part of the tube bundle, CEA/DTE/STRE/LGV/89/89/961 Vol 1 & 2. Gouirand JM (1991) CLOTAIRE International Program – Final report – part 1 – Thermalhydraulic, CEA/DER/SCC/LTDE/91 /012 Hassan YA and Morgan CD (1980) Steady-state and transient prediction of a 19-tube oncethrough steam generator using RELAP5/MODI. Nucl. Tech., vol 60, pp 143–150 John B, Dharne SP and Ghadge SG (October 2–6, 2005) Evolution of 434 MWth steam generator to 540 MWth, The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Paper: 332 Popes’ Palace Conference Center, Avignon, France Keeton LW, Singhal AK and Irani A (1986) A THOS3 code analysis of tube plugging effects an the thermal-hydraulic characteristics of a once-through steam generator, ASME 86-WA/NE-4 Keeton LW, Singhal AK and Srikantiah G (1990) ATHOS3: A computer program for thermal-hydraulic analysis of steam generators. Vol. 1: Mathematical und Physical Models und Method of Solution; Vol. 2: Programmer’s Manual; Vol. 3: User’s Manual. EPRI NP 4604-CCM, vol. 1-3, Revision 1 Lee JY and No HC (1986) Three-dimensional two-fluid code for U-tube steam generator thermal design analysis. Proc. 2nd International Topical Meetings an Nuclear Power Plant Thermal Hydraulics and Operations, pp. 3-21 to 3-27, Tokyo, Japan, April 1986 Patankar SV and Spalding DB (1976) A calculation procedure for the transient and steady state behavior of shell-and-tube heat exchangers, Heat Exchanger Design and Theory Source Book, Scripta, Washington, DC Pioro IL und Duffey RB (2007) Heat Transfer and Hydraulic Resistance at Supercritical Pressures in Power-Engineering Applications, Elsevier, Amsterdam Procaccia H et al. (1982) Tests of types 51A and 51M steam generators at Bugey-4 and Tricastin-1 Nuclear Power Plants, EPRI NP-2689 Riboud PM and Brugeille G (August 31–Sept. 4 1987) Validation expérimentale du calcul thermo hydraulique bidimensionnel des échangeurs tubulaires, 22rad IAHR Congres, Lausanne (Switzerland) Singhal AK, Keeton LW and Srikantiah G (1983) Thermal-Hydraulic Analysis of U-Tube and Once Through Steam Generators, AIChE Symposium Series 225, vol 79, p 331 Singhal AK, Keeton LW, Przekwas AJ and Weems JS (1984) ATHOS A Computer Program for Thermal Hydraulic Analysis of Steam Generators, vol. 4: Applications, EPRI NP-2698-CCM Singhal AK and Srikantiah G (1991) A review of thermal hydraulic analysis methodology for PWR steam generators and ATOS3 code applications, Prog. Nucl. Energy, vol 25 no 1, pp 7–70 Schwarz T and Bouecke R (1985) Utilization of the ATHOS code for split flow economizer and flow distribution plate calculations of steam generators, ASME Winter Annual Meeting Proc. HTD-vol 51, pp 57–69
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Solomon Y, Paine JPN, Steininger DA and Williams CL (1985) Principles of steam generator degradation, Steam Generator Reference Book, Ch. 5, EPRI Wang SS and Srikantiah G (1985) Numerical modeling of the phase separation processes in BWR and PWR steam separators, AIChE Symp. Series, vol 81, p 245
9. Moisture separation
Chapter 9 is devoted to the basics of designing of moisture separation. Firs the importance of knowing the characteristic spectra of the moisture is underlined for proper analysis. Then some simple methods for computation of the efficiency of the separation are given for cyclone type and vane type. Different ideas based on different complexity are presented for description of the velocity field: the Kreith and Sonju solution for the decay of turbulent swirl in pipe, the potential gas flow in vanes; description of the trajectory of particles in a known continuum field; the CFD analyses of cyclones; the CFD analyses of vane separators. Then several experiments are collected from the literature for BWR cyclones, PWR steam generator cyclones, other cyclone types and vane dryers. On several cases demonstration is made for the success of different methods by comparisons with data.
9.1 Introduction To separate straw from wheat the man used already in the ancient time the wind and the differences of the applied drag to both components. This is probably the oldest example of geodetic separation. To day in many power plants, chemical plants, air conditioning systems, petroleum pipe systems, etc. the liquid cared by the gas reduces the efficiency, facilitates erosion of the structures etc. To avoid these disadvantages already in the earlier industrial era facilities are designed for separation of the liquid or solid particles from the gas. In general I distinguish between liquid separation from the mixture and gas separation from the mixture. In the first case the dry gas flow is the whished final result of the separation and in the second the liquid without any gases. Next I will concentrate my attention to the first type of the separation.
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_9, © Springer-Verlag Berlin Heidelberg 2009
307
308
9. Moisture separation
(a)
(b)
(e)
(f)
(c)
(d)
(g)
Fig. 9.1. (a) Balke & Dürr separator for PWR’s, Powersep (2002); (b) Cyclone separator for BWR’s; Typical steam separators used in steam generators: (c) Axial cyclone separator with components: 1. axial vanes, 2. primary skimmer, 3. secondary skimmer, 4. restriction ring, 5. straightening vanes, 6. separation baffles, 7. tertiary skimmer, 8. Pre-dryer; (d) Axial vane unit for (c); (e) Separator with angular water removal: 1. riser, 2. separator down comer, 3. turbo-vane, 4. hub, 5. tangential port, 6. orifice; (f) Perforated barrel centrifugal separator: 1. inlet nozzle, 2. centrifugal vanes, 3. barrel, 4. lower perforations, 5. upper perforations, 6. discharge nozzle, 7. screen assembly; (g) as (f) with additional fine separator atop
9.1 Introduction
(a)
(b)
(c)
(d)
(e)
(h)
309
(f)
(i)
(g)
(j)
(k) Fig. 9.2. Different vane type separators: (a) “Chevron”; (b) “Babcock” (Euroform), Regehr (1967); (c) Vane type separator by Kall (1979), Patentschrift (1976); (d) Delta-P-5ft-dryer from “Burgess Minning Company, Dallas” (1973) used in Japanese BWR’s, see Nakao et al. (1999); (e) Delta-P-5ft-dryer (120°) from “Burgess Mining Company, Dallas”, (1973); (f) “Steinmüller”; (g) “Stein”; (h) Burgess Mining vertical vane separator; (i) Li et al. (2007) vane separator without hook; (j) Li et al. (2007) vane separator with hook; (k) “Peerless PX8” separator analyzed in Xiamoto et al. (2004)
Specially designed drying technology consisting mainly of cyclones, see for examples Fig. 9.1, and vane dryers, see for examples Fig. 9.2, reduces the moisture content to less then 1% depending on the power (most providers guarantee < 0.3%). Cyclones are usually used as a primary separators and vane separators as secondary separators of fine moisture. Usually inertial and centrifugal principles are exploiting for droplet separation. In the cyclone separation the rotating droplet experiences centrifugal force that is moving it towards the external wall. In the zigzag path of the so called vane separators the centrifugal force and partially the inertia of the droplet forces it to heat
310
9. Moisture separation
the wall and participate in the film formation that is then removed by gravity. For the same efficiency as a cyclone separator a zigzag vane separator needs about 10 times more flow cross section, about two times more volume and causes about 10 times less pressure drop, Gloger (1970). Therefore cyclones do not need velocity reduction. But if not properly designed they may vibrate and cause damages to the followed components of the plant. Zigzag separators, as already mentioned, works at considerably lower velocity. For this reason especially for PWR’s the high speed steam-water mixture (40–50 m/s) has to be slowed down by appropriate cross section enlargement to 1–5 m/s and then equally supplied to the zigzag separators in order to ensure proper operation. This presents considerable challenge. Problems usually originate if some of the parties receives more mass flow rate then others. In such cases unexpected large water load may cause problems with its drainage. For inlet velocities larger then the limit velocity the vane separators do not operate properly. To low mixture load at some parties reduces also the total efficiency of the separation. Therefore analysis of the spatial distribution of the velocities is necessary to judge the integral performance of the vane separators. The liquid removal in both types is as important as the separation it selves. Good designed separation but bad water removal may lead to entrainment and bad performance of the vane separator systems. There are three major performance measures associated with steam separator designs: These are carryover of water in the outlet steam flow; carry under of steam into the recirculating water flow, and a pressure drop across the steam separator. Two of the requirements are contradicting to each other: low pressure drop and high separation efficiency. Note that for NPP with 1,300 MW el. PWR’s 0.1 bar more pressure drop reduces the efficiency with 0.12 to 0.13%. The separator efficiency affects the so called recirculation ratio. This has important consequences for the operation of the overall steam generator system. The recirculation rate is the ratio of the mass flow removed by the steam separators to the generated steam flow. The larger the recirculation ratio the larger will be the flow rate through the tube or fuel bundle and the lower will be the quality of the steam in the two phase mixture surrounding the tube or rod bundle. Normally the steam leaving the mixture surface above the core of the boiling water reactor or above the steam generator bundles of pressurized water reactor contains considerable amount of liquid. Modern turbine design requires that the steam produced by the steam generator or boiling water reactors be of high quality. Two particular turbine problems are due to steam wetness: Severe erosion occurs in the blading at the low pressure end of turbines. Saturated steam of 0.27% wetness and 50–70 bar pressure entering the first high pressure stage of the turbine will produce a wetness of 12% or more at the turbine exit. In order to avoid grave erosion damage, a limitation of turbine exit wetness to about 12% is necessary: The aerodynamic efficiency of turbine stages operating in the wet steam region is considerably lower than that of dry stages. One percent of wetness present in a stage is likely to cause about a 1% decrease in efficiency. Heating the mixture up to appropriate moisture content in order to obtain the operation parameter for the next
9.2 Moisture characteristics
311
stage of the turbine is always on the account of the thermal efficiency of the plant. Therefore mechanical drying before the low pressure stage, as for the mixture after the primary steam generation, is what is needed here because it is much cheaper: Gloger (1970) reported that if the separators provide 1% more liquid separation (reduction of 2 to 1% mass content of the moisture) it increases the efficiency of the power plant with about ¼%. Balke & Dürr give guarantee for the same reduction 0.5%, Powersep (2002). Reducing the pressure drop not only improve local thermal-hydraulics in the tube or rod bundle but also decrease the tendency for chemica1 hideout, and thereby reduce the tendency for the localized corrosion that has been experienced in operating steam generators. There is a negative effect of the carry under. As already mentioned steam separators allow some steam bubbles to be drawn downward with the recirculating water into the down comer annulus – carry under. This lowers the effective density of the mixture in the down comer and, therefore, lowers the recirculation ratio. Only 1% carry under reduces the density of the recirculation mixture by approximately 20%. The pressure loss in two-phase flows is proportional to the void fraction. Therefore reducing the void fraction leads also to reduction of the pressure drop. This will bring in the steam generating system an increase in the recirculation flow rate and a reduction of the required pumping power. In a boiling water reactor system, this pressure drop decrease would lead to an increase in electrical power output without increasing vessel size or modifying the capacity of the pump system used for coolant recirculation. Although the pressure drop due to the steam separator is only a part of the total pressure losses improved steam separator design with lower pressure losses would give an immediate economic benefit also. Inlet spectra can be simply measured by exposing for a short time a oil wetted paper on the flow. Then the water droplet captured in the oil can be photographed and classified. Exit measure is usually measured by waiting of absorbing paper before and after the exposure. Of course there are modern optical methods for performing these measurements if the caring gas is optically transparent.
9.2 Moisture characteristics The prerequisite to design a good separator is to know the spectral distribution of the particles sizes in the flow to be treated. The structure of the moisture to be removed is characterized by a spectrum of the droplet size being a result of the presiding physical processes. Consider the moisture structure of mixture produced by nuclear powered installations. Figures 9.3 and 9.4 are illustrations of distribution obtained in the 0.734 m-diameter main steam line of the power plant with pressurized water reactor (Biblis) 7 m after a bend. We see, depending on the power, sizes between 0 and 400 µm.
312
9. Moisture separation
Such steam passes the high pressure stage of the turbine. The moisture increase from less than 1% before the high pressure turbine to about 13% after the high pressure stage of the turbine is due to spontaneous condensation of sub-cooled steam. Detsch and Philiphoff (1981) reported that under subcooling characteristic for high pressure turbines of nuclear power plants the originating moisture has sizes in order of 10–8 m. The collisions, coalescences, film formation on the blades and successive entrainment, secondary break up due to hitting the droplets by the rotating blades etc. lead to mean sizes of about 40 µm. So the size spectrum of the < 1% moisture entering the first stage of the turbine is substantially modified at the exit (< 13%), see the interesting discussion in Philiphoff and Povarov (1980).
Relative frequency in %
100 NPP Biblis 1300MWe Power in % 51.5 67.5 99.1
80 60 40 20 0
1
10 100 Droplet size after SG in µm
1000
Sauter mean diameter in µm
Fig. 9.3. Relative frequency of occurrence depending on the size of the droplets
100
10 0,0
NPP Biblis 1300MWe Power in % 51.5 67.5 99.1 0,2 0,4 0,6 0,8 1,0 Measurment position x/D, -
Fig. 9.4. Distribution of the Sauter mean diameter along the diameter of the pipe, Dibelius et al. (1977)
9.2 Moisture characteristics
313
Trojanovski (1978, p. 66) reported spectrums measured on the structures of turbines leading blades having maximum at 12 µm. Philiphoff and Povarov (1980, p. 72) reported mean sizes measured at the exit of experimental turbine that are depending on the rotation frequency and on the exit moisture content. They found almost linear increase of the mean size with the mass moisture content and decrease with the increase of the rotation frequency. So for instance, for 4000 rotation per minute the mean size changes between 8 and 50 µm if the moisture mass content changes from 1 to 6.5% at the low pressure exit 0.02 MPa. In KWU (1974) the droplet spectrum at the exit of the high pressure turbine was measured. The result is presented on Fig. 9.5. We realize that 50% of the moisture has a particle size less then 40 µm. The data can be extrapolated using the Boltzmann distribution model
∑
D3 < D3,i
m& 3,i
m& 3
=
a1 − a2 1− e
( D3,i − D3,0 )
ΔD
+ a2 ,
(9.1)
Accumulative mass fraction, -
where a1 = –70.40837, a2 = 92.94748, D3,0 = 10.98707 µm, ΔD = 32.66116 µm. In this formula the size of the droplet is used in µm. 100 80 60 Exp. Data model, Boltznamm
40 20 0 0
20 40 60 80 100 120 140 160 180 Diameter in µm
Fig. 9.5. Accumulative mass flow of droplets with specific size divided by the total liquid mass flow as a function of the droplet size at the exit of the high pressure turbine, KWU (1974)
Such information for specific facilities is the starting point for designing moisture separators. Using methods that allows spectral distributions of particles in a single computational cell is possible but extremely complicated. Much simple is
314
9. Moisture separation
the approach in which the separation efficiency of the facility ηi ( D3,i ) is analyzed for a particular particle size D3,i . If the mass weight of this size in the total distri-
bution is f i ( D3,i ) the total separation efficiency computed as follows imax
η = ∑ fi ( D3,i )ηi ( D3,i ) ,
(9.2)
i =1
is a good estimate for the real efficiency. Here the efficiency η is defined as the ratio of the separated liquid mass flow to the initial liquid mass flow entering the separators
η = ( m& 3,in − m& 3,out ) m& 3,in = 1 − m& 3,out m& 3,in .
(9.3)
Experiments with droplet separation require careful provisions for generating the appropriate particle size. Usually nozzles with different diameters are used for this purpose. Te droplet size is the depending on the nozzle size and on the local properties at the nozzle. An example is the correlation by Ueda (1979) ⎡ σ ⎛ ρ ⎞1.25 ⎤ D3 1 = 5.8 × 10−3 ⎢ ⎜ ⎟ ⎥ Dnoz V η ρ ⎢⎣ 1 1 ⎝ 3 ⎠ ⎥⎦
0.34
.
For completeness of this short overview I have to mention that a interesting review of the existing literature for droplet fragmentation in gas up to 1978 is given by Smith and Azzopardi (1978): For air – water system different authors reported the following: for air velocities 28 to 50 m/s particle sizes of 70 to 360 µm; for air velocities 23 to 42 m/s particle sizes of 69 to 170 µm; for air velocities 9 to 21 m/s particle sizes of 250 to 2500 µm; for air velocities 23 to 77 m/s maximum size in the distribution 600 to 2000 µm. Dependences like D3 ∝ V1−0.79 to −1.8 are reported.
9.3 Simple methods for computation of the efficiency of the separation In this section simple methods for estimation of the efficiency of cyclone and vane separators will be presented. Although not accurate these methods give the right ideas for the main influence parameters and in which direction they affect the performance. So that using them a primary design can be developed. The final performance is then estimated by experiments. In many cases the experiments are performed at different conditions than the operating conditions. The simple analysis provides the dimensionless groups that can be used for presentation of
9.3 Simple methods for computation of the efficiency of the separation
315
the experimental results and their use in other parameter regions as long as the similarity holds. 9.3.1 Cyclone separators Consider an idealized vertical cyclone separator schematically presented on Fig. 9.6 consisting of a cylinder with radius Rout and height H sep . The mixture of mono-dispersed particle and gas enters the ring cross section between Rin and Rout at the bottom. The axial gas velocity is w1 and the tangential gas velocity is v1 . The ratio v1 w1 = tan ϕ is a geometrical characteristic of the swirler imposing
the rotation. The entering particles having size D3 , follow a helical trajectory expanding in the radius. So provide the cyclone is high enough each particle will strike eventually the external wall.
z equilizer
v3 H sep
u3
swirler
Rout
r
Rin α 3 w3
Fig. 9.6. Definitions to a cyclone separator
The inlet volume fraction of the particles is α 3 so that for almost homogeneous mixture the volumetric mass concentration is C3* = α 3 ρ3 in kg droplets per m³ of the gas-droplet mixture. The target of the consideration here is to derive approximate analytical expression for the efficiency η . Approximate, because several simplifying assumptions are necessary in order to obtain analytical model:
316
9. Moisture separation
• • • • • •
Non compressible gas and particles; No heat and mass transfer; No collision, coalescences; No reflection from the external wall; Each particle striking the wall is considered as separated; The volumetric mass concentration around all undisturbed particles flowing in parallel helical trajectories remain constant and equal to the entrance one
C3* = α 3 ρ3 .
(9.4)
No entrainment considered: The steady droplet mass flow entering the cyclone
(
)
2 − Rin2 dθ ( 2π ) is trough the cross section π Rout
(
)
2 m& 3 = α 3 ρ3 w3π Rout − Rin2 dθ ( 2π ) .
(9.5)
During the time interval dτ the surface Rout dθ dz is passed by the droplet mass flow dm& 3 = −α 3 ρ3u3 Rout dθ dz .
(9.6)
The deposed mass flow dm& 3 at the wall during dτ divided by the entering mass flow m& 3 is then 2 dm& 3 u 2 Rout dz u3 2 Rout dθ . ≡ d ln m& 3 = − 3 = − 2 2 m& 3 w3 Rout v3 Rout − Rin2 − Rin2
(
)
(
)
(9.7)
The final form of the above equation is obtained by using the definitions of the velocity components Rout dθ = v3 dτ
(9.8)
and dz = w3 dτ .
(9.9)
The Eq. (9.7) is identical with the Chen’s Eq. (3), see Chen et al. (1999, p. 1607). Integrating from the entrance to the exit we obtain
9.3 Simple methods for computation of the efficiency of the separation
m& 3,out ⎛ ⎞ u 2R2 ≈ exp ⎜ − 2 out 2 3 θtotal ⎟ m& 3,in ⎝ Rout − Rin v3 ⎠
317
(9.10)
and therefore ⎛
2 ⎞ 2 Rout u3 θtotal ⎟ . 2 2 ⎝ Rout − Rin v3 ⎠
η ≈ 1 − exp ⎜ −
(9.11)
The time required for a droplet to pass the cyclone is Δτ = H sep w3
(9.12)
and therefore
θtotal =
H sep v3 H sep ≈ tan ϕ . Rout w3 Rout
(9.13)
Note that the number of completed 360° rotations of the helical trajectory is θtotal ( 2π ) . The final expression for the efficiency is then ⎛ 2 Rout H sep u3 ⎞ ⎟. 2 ⎜ Rout − Rin2 w3 ⎟⎠ ⎝
η ≈ 1 − exp ⎜ −
(9.14)
To evaluate the efficiency I need information about the particle velocity at the external wall. What I know is the gas velocity components and the particle size. Reasonable simplification is w3 = w1 .
(9.15)
The simplified radial momentum equation equalizes the centrifugal force F3c ≈ ρ3
π 6
D33
v32 v2 π ≈ ρ3 D33 1 Rout Rout 6
(9.16)
to the drag force
F3d ≈ c13d ρ1
π 8
D32 u3 u3 ,
(9.17)
318
9. Moisture separation
allowing to compute the radial particle velocity. Therefore the radial particle velocity at the external wall is u3 = v1
4 ρ3 D3 . 3c13d ρ1 Rout
(9.18)
For small particles if the Stokes flow around the particle is valid, 24η1 , ρ1 D3 u3
c13d =
(9.19)
we have
u3 =
1 ρ3 2 v12 ρ1 D3 . 18 ρ1 Rout η1
(9.20)
With this the efficiency receives the form ⎛ 1 H sep v12 ρ1 2 ρ3 ⎞ D3 ⎟ 2 2 ρ1 ⎠ ⎝ 9 Rout − Rin w1 η1
η ≈ 1 − exp ⎜ −
⎛ 1 H sep Rout ⎛ v ⎞ 2 D ρ w D ρ ⎞ 3 1 1 3 3 ⎟ 1 = 1 − exp ⎜ − ⎜ ⎟ 2 ⎜ 9 Rout − Rin2 ⎝ w1 ⎠ Rout η1 ρ1 ⎟ ⎝ ⎠ 2 2 ⎛ 1 H sep Rout tan ϕ ρ1 w1 D3 ρ3 ⎞ = 1 − exp ⎜ − ⎟. 2 ⎜ 9 Rout η1 Rout ρ1 ⎟⎠ − Rin2 ⎝
(9.21)
The geometry similarity number characterizing each particular cyclone design is Gecy =
H sep Rout tan 2 ϕ 2 Rout − Rin2
(9.22)
Here tan ϕ = v1 w1
(9.23)
is a geometrical characteristics of the swirler. The degree of swirl for a swirling flow is characterized by the swirl number. It is defined as the ratio of axial flux of angular momentum to the axial flux of axial momentum. For two phase mixture it can be extended to
9.3 Simple methods for computation of the efficiency of the separation
⎛ Rout ⎞ Sw = ⎜ ∫ (α 3 ρ3 w3 v3 + (1 − α 3 ) ρ1 w1v1 ) rdr ⎟ ⎜R ⎟ ⎝ in ⎠
319
⎛ Rout ⎞ ⎜ ∫ α 3 ρ3 w32 + (1 − α 3 ) ρ1 w12 rdr ⎟ . ⎜R ⎟ ⎝ in ⎠ (9.24)
(
)
If for single phase continuum v1 ( r ) w1 ( r ) ≈ const = tan ϕ
(9.25)
the swirl number turns to be a geometrical characteristic Sw ≈ tan ϕ .
(9.26)
Note that there are also other definitions of the swirl number. The cyclone particle Reynolds number
Re31,cy =
ρ1w1 D32 ρ1w1 D3 D3 = η1 Rout η1 Rout
(9.27)
characterizes the particle size related to the cyclone curvature and inlet axial velocity. The dependence of the efficiency on the density ratio
ρ3 ρ1 ,
(9.28)
reflects in fact the dependence on pressure. The efficiency can be rewritten in term of the dimensionless number called here cyclone number Cy =
H sep Rout tan 2 ϕ ρ1 w1 D32 ρ3 = Gecy Re31, cy ρ3 ρ1 , 2 η1 Rout ρ1 − Rin2 Rout
(9.29)
namely
η ≈ 1 − exp ( − Cy 9 ) .
(9.30)
Although simple, this expression contains all important trends. Every parameter that increases the cyclone number increases the efficiency and vice versa. So for instance increasing of the separation height increases the efficiency. Increasing the axial velocity increases the efficiency because it simultaneously increases the angular velocity component which contributes with its square. Strong
320
9. Moisture separation
effect on the efficiency has the increasing of the angular velocity which means increasing the Swirl number and of the droplet size. From great importance is the dependence on the density ratio which is in fact, as already mentioned, the dependence on the pressure. Decreasing the pressure increases the density ratio and therefore improves efficiency. All this is intuitively expected and experimentally confirmed. If model experiments have to be performed in order to have the same efficiency with the original one has to guarantee Cymodel = Cyoriginal .
(9.31)
If this is not the case one has to be careful. In such case from
ηoriginal ≈ 1 − exp ( − Cyoriginal 9 ) ,
(9.32)
η model ≈ 1 − exp ( − Cymodel 9 ) ,
(9.33)
I obtain ⎤ ⎡1 1 − ηoriginal = (1 − ηmodel ) exp ⎢ Cymodel − Cyoriginal ⎥ . ⎦ ⎣9
(
)
(9.34)
It is very convenient to perform experiments at low pressure instead of the real high pressure in the power industry. But the above result shows that the transfer of the measured efficiencies at low pressure to high pressure is associated with a non-linear dependence of the difference Cymodel − Cyoriginal . Note that this scaling
(
methodology is more general than simply setting
(F
d 3
)
)
F3c = 1 for model and
original as done for vane separators by Nakao et al. (1999). For all dimensionless characteristics equal to each other except the pressure, which means the density ratio the dependence is ⎫ ⎧1 ⎡ ( ρ 3 ρ1 ) 1 − ηoriginal = (1 − ηmodel ) exp ⎨ ( Gecy Re31,cy ) − ( ρ 3 ρ1 )original ⎤ ⎬ . model model ⎦ ⎣ ⎩9 ⎭ (9.35) It is interesting to mention that Crowe and Pratt (1974) reported the empirical relation
⎛ρ 1 − ηoriginal = (1 − ηmodel ) ⎜ 1,original ⎜ ρ ⎝ 1,model
⎞ ⎟⎟ ⎠
0.182
.
(9.36)
9.3 Simple methods for computation of the efficiency of the separation
321
for separators of solid particles from a gas flow. Entrainment considered: In cyclones for liquids, the film at the wall can become unstable and droplets can be entrained. The order of magnitude of the entrainment can be estimated by the following considerations: As discussed in Kolev (2007, vol. 2, p. 151), the entrainment velocity can be approximated by 1/ 2
⎛ λd ⎞ u23 ≈ ⎜ 1w ⎟ ⎝ 8 ⎠
( ρ1 / ρ2 )
1/ 2
V1 ( ρ1 ρ 2 )
1/ 2
V1 = 0.182
⎡⎣ ρ1V1 2 ( Rout − Rin ) η1 ⎤⎦
1/ 8
.
(9.37)
Here the friction factor is replaced with the Blasius relation
λ1w =
0.3164
( ρ1V1 Dh
η1 )
1/ 4
,
(9.38)
the hydraulic diameter Dh is two times the wall-to-wall wall distance Rout − Rin and V1 = v12 + w12 .
(9.39)
Therefore the particle mass flow effectively deposed on Rout dθ dz is now less than without considering the entrainment
(
)
dm& 3 = − ρ3 ⎡α 3u3 − 1 − f perf u23 ⎤ Rout dθ dz . ⎣ ⎦
(9.40)
Here f perf is the part of wall cross section which is perforated. Therefore dm& 3 2R ≡ d ln m& 3 = − 2 out 2 m& 3 Rout − Rin
1 − f perf ⎛ ⎞ dz u23 ⎟ , ⎜ u3 − α3 ⎝ ⎠ w3
(9.41)
or d ln m& 3 = −
2 1 − f perf ⎞ 2 Rout 1⎛ u23 ⎟ dθ . ⎜ u3 − 2 2 Rout − Rin v3 ⎝ α3 ⎠
Integrating and computing the efficiency results in
(9.42)
322
9. Moisture separation
η = 1−
⎧⎪ ⎫⎪ 1 − f perf m& 3,out ⎞ 2R2 1⎛ ≈ 1 − exp ⎨− 2 out 2 u23 ⎟ θtotal ⎬ ⎜ u3 − m& 3,in α3 ⎠ ⎩⎪ Rout − Rin v3 ⎝ ⎭⎪
(9.43)
Remembering that the total angle traveled by the most remote from the wall parH sep v3 the above expression results in ticle is θtotal = Rout w3
η = 1−
⎧⎪ Rout H sep m& 3,out ≈ 1 − exp ⎨− 2 2 m& 3,in ⎩⎪ Rout − Rin
1 − f perf u23 ⎞ ⎫⎪ ⎛ u3 −2 ⎜2 ⎟⎬ . α 3 w3 ⎠ ⎪⎭ ⎝ w3
(9.44)
Replacing the radial particle velocity and the entrainment velocity with their equals from Eqs. (9.20, 9.15, 9.23) and (9.37) I obtain
(
⎧ 2 ⎛ 1 − f perf 1 + tan ϕ ρ 1 ⎪ η = 1 − exp ⎨−Gecy ⎜ Re31,cy 3 − 0.364 ⎜⎜ 9 tan 2 ϕ ρ1 α3 ⎪ ⎝ ⎩
)
7 /16
1 Re1/1,cy8
1/ 2
⎛ ρ1 ⎞ ⎜ ⎟ ⎝ ρ2 ⎠
⎞⎫ ⎟⎪ ⎟⎟ ⎬ ⎠ ⎭⎪
(9.45) Here the vapor Reynolds number responsible for the entrainment is defined by Re1,cy = 2 ( Rout − Rin ) w1 ρ1 η1 .
(9.46)
Interesting observation is that the more perforation the external wall has the less surface is exposed on entrainment. This has a positive effect on the efficiency. No separation is possible if the entrainment could take more than the deposition supplies Re1, cy
(
2 ⎡ 1 − f perf 1 + tan ϕ > ⎢ 0.364 ⎢ tan 2 ϕ α3 ⎢⎣
)
8
7 /16
9 Re31, cy
⎛ ρ1 ⎞ ⎜ ⎟ ⎝ ρ3 ⎠
3/ 2
⎤ ⎥ . ⎥ ⎥⎦
(9.47)
One sees that increasing Re31,cy which increases the separation efficiency reduces in the same time dramatically the critical Reynolds number that prohibits separation due to entrainment. In a real cyclone there are radial non uniform profiles of the velocities. The angular velocity of the continuum at the wall has to be applied in the efficiency formula. This makes the application of the simple formula more complicated.
9.3 Simple methods for computation of the efficiency of the separation
323
9.3.2 Vane separators Consider a vane type separator as shown in Fig. 9.7 consisting of nmax-parts of a cylindrical annulus with alternating turnings. The distance between two parallel walls is δ vane . Each part of the cylindrical annulus has an angle ϕ . The external radius is Rout . n=3
n=1
Rout Rin
ϕ
n=4 H vane
n=2
δ vane G L vane
Fig. 9.7. Geometrical definitions of a vane separator
Next I will use the already derived expression for separation for cyclones without taking into account the entrainment, Eq. (9.14) ⎛ 2R2 u3 ⎞ m& 3,out , n ≈ m& 3,in , n exp ⎜ − 2 out 2 θ ⎜ R − R v ⎟⎟ 3 out in , n ⎝ ⎠
(9.48)
by taking into account that θ = nϕ and Rout = const. This implicates that after each cycle the mixture occupy the total cross section. This is then possible if there is some straight part between two subsequent turns so that the turbulence causes the dispersion to occupy the total cross section again. The ratio of the mass flow leaving the n-cycles to the entrance mass flow is then m& 3,out ,n ⎛ ⎞ u 2R2 ≈ exp ⎜ − 2 out 2 3 nϕ ⎟ , n = 1, 2,… m& 3,in R R v − 3 out in ⎝ ⎠
(9.49)
324
9. Moisture separation
and the efficiency
η = 1−
m& 3,out ,n ⎛ ⎞ u 2R2 = 1 − exp ⎜ − 2 out 2 3 nϕ ⎟ . & m3,in ⎝ Rout − Rin v3 ⎠
(9.50)
Assuming that the Stokes low is valid, Eq. (9.20),
u3 =
1 ρ3 2 v12 ρ1 , D3 18 ρ1 Rout η1
the final expression is
η = 1−
2 m& 3,out ,n ⎡ 1 ρ1v1 D3 D3 ρ 3 nϕ Rout ⎤ = 1 − exp ⎢ − . 2 2 ⎥ 9 m& 3,in R R R − η ρ 1 1 out out in ⎦ ⎣
(9.51)
The vane particle Reynolds number Re31,va =
ρ1v1 D3 D3 η1 Rout
(9.52)
characterizes the particle size related to the vane curvature and inlet velocity. The dependence of the efficiency on the density ratio
ρ3 ρ1 , reflects in fact the dependence on pressure. The efficiency can be rewritten in term of the dimensionless number called here vane number Va =
ρ1v1 D32 η1 Rout
⎛ 2 nmax 1 ⎜⎜ θ Rout ∑ 2 R Rin2 , n − n = 1 out ⎝
⎞ ρ3 = Geva Re31,va ρ3 ρ1 , ⎟⎟ ⎠ ρ1
(9.53)
and the vane geometry number Geva =
as follows
2 nϕ Rout nϕ = 2 2 Rout − Rin 1 − 1 + δ R in
(
)
−2
,
(9.54)
9.3 Simple methods for computation of the efficiency of the separation
⎛ 1 ⎝ 9
⎞
η = 1 − exp ⎜ − Geva Re31,va ρ3 ρ1 ⎟ .
325
(9.55)
⎠
This result is obtained neglecting the entrainment. Again in order to have the same efficiency in a model and original Eq. (9.31) has to be fulfilled. Problem 1: Given a vane separator with ϕ = 90° , nmax = 4, Rin = 0.01 m, Rout = 0.02 m. Compute the geometry number. Estimate the efficiency for 10, 20 and 50 µm droplets for velocities between 1 and 20 m/s. Solution: The geometry number of this vane separator is 2 Geva = nϕ Rout
(R
2 out
− Rin2
)
= 8.38.
Figure 9.8 shows the efficiency as a function of the velocity for the droplet size in question.
Separation efficiency, -
1.0 0.8 Euroform (4x90°) 10µm 20µm 50µm
0.6 0.4 0.2
0
5
10 15 Velocity in m/s
20
Fig. 9.8. Efficiency of the Regehr’s (1967) Euroform vane separator TS-4 characterized by ϕ = 90° , nmax = 4, Rin = 0.01 m, Rout = 0.02 m as a function of the velocity for air-water at atmospheric pressure. Parameter – droplet diameter. No entrainment considered
9. Moisture separation
Separation efficiency, -
326
1.0 0.8 10µm 20µm 50µm
0.6 0.4 0.2 0
50
100 150 Vane number, -
200
Fig. 9.9. Efficiency of the Regehr’s (1967) Euroform vane separator TS-4 characterized by ϕ = 90° , nmax = 4, Rin = 0.01 m, Rout = 0.02 m as a function of the vane number for airwater at atmospheric pressure. No entrainment considered
Figure 9.9 shows the same dependence as presented on Fig. 9.8 but as a function of the Vane number. Obviously for Vane numbers larger than 50 the efficiency is 100%. The universal character of the Vane number clearly demonstrated. The efficiency can be presented only as a function of this number for variety of flow characteristics, particle sizes etc. Even for unknown spectra an effective particle size can be used to fit the experimental data. Entrainment controlled efficiency of vane separators: The performance of the separation vanes for high velocities is entrainment controlled. So for instance for the Regehr’s (1967) TS-4 vane separator the velocity at which the entrainment starts for air/water mixture at atmospheric conditions is reported to be 14.1 m/s. For 100% droplet deposition at the walls (perfect separator geometry) only the rebounded liquid into the flow due to entrainment reduces the performance. Therefore the separation efficiency η is
η = 1−
ρ 2 u23 f regime H vane Lvane ρu f L = 1 − 2 23 regime vane . (1 − X 1,in ) GH vaneδ vane (1 − X 1,in ) Gδ vane
(9.56)
Here the horizontal size of the vane is Lvane , its effective height H vane and the distance between two parallel walls δ vane , see Fig. 9.7. The wetted surface portion f regime of the vane is very complex and depends on the specific design and local parameter. Having in mind that for almost homogeneous mixture
9.3 Simple methods for computation of the efficiency of the separation
ρ2
ρ2
≈
(1 − X ) G (1 − X ) ρ V 1, in
1, in
in 1
≈
1 ⎛ ρ 2 X 1,in ⎜1 + V1 ⎜⎝ ρ1 1 − X 1,in
⎞ ⎟⎟ ⎠
327
(9.57)
the efficiency receives the form
η = 1 − f regime
Lvane u23 ⎛ ρ 2 X 1,in ⎜1 + δ vane V1 ⎜⎝ ρ1 1 − X 1,in
⎞ ⎟⎟ . ⎠
(9.58)
The reader will find exhaustive review on entrainment in volume 2 of this monograph, Kolev (2007). I will estimate the entrainment using the model proposed by Hewitt and Govan (1989). For entrainment the authors recommend the Govan et al. (1988) correlation
( ρ w )23 = 0 , for
Re2 < Re2∞
(9.59)
2 u23 ρ ⎡ ρ2 ⎤ 2 η2 = 5.75 × 10−5 α1 1 ⎢( Re2 − Re2 ∞ ) ⎥ V1 Dhσ ρ12 ⎦ ρ3 ⎣
0.316
,
(9.60)
for Re2 ≥ Re2∞ ,
(9.61)
and 1 < ( Re2 − Re2∞ )
2
η22 ρ 2 < 107 , 2 Dhσ ρ1
(9.62)
where ⎛ η ρ2 ⎞ Re2 ∞ = exp ⎜ 5.8504 + 0.4249 1 ⎟ , Owen and Hewitt (1987), (9.63) ⎜ η 2 ρ1 ⎟⎠ ⎝ is the local equilibrium film Reynolds number. For example for Re2 ∞ ≈ 459
air/water flow at atmospheric pressure
( ρ w )23 /(α1 ρ1w1 ) takes values
≈ 5.75×105
to 9.37×10-3. With this entrainment correlation the separation efficiency is
η ≈ 1 − 5.75 × 10−5 f regime
Lvane ⎛ ρ 2 X 1,in ⎜1 + δ vane ⎜⎝ ρ1 1 − X 1,in
2 ⎞ ρ1 ⎡ 2 ρ 2η 2 ⎤ ⎟⎟ α1 ⎥ ⎢( Re2 − Re2 ∞ ) 2 ρ1 Dhσ ⎦ ⎠ ρ2 ⎣ (9.64)
0.316
Here Re2 = α 2 ρ 2 w2 Dh η 2 = α 2 ρ 2 w2 2δ vane η2
(9.65)
328
9. Moisture separation
is the local film Reynolds number based on the hydraulic diameter Dh which is two times the vane-to-vane wall distance δ vane . Recognizing that the efficiency multiplied by the inlet liquid mass flow is equal to the film mass flow that has to be removed from the vanes
η (1 − X 1,in ) GH vaneδ vane = α 2 ρ 2 w2 δ vane Lvane f regime
(9.66)
I realize that the film Reynolds number Re 2 = α 2 ρ 2 w2 2δ vane η2 = η
(1 − X ) G H 1, in
η2
2δ vane . Lvane f regime vane
(9.67)
is a linear function of the efficiency of the separation. Therefore the Eq. (9.64) is a implicit equation with respect to the efficiency that has to be solved by iteration. For engineering applications it is recommended to compute the thickness of a falling film δ 2 on vertical smooth walls as follows: 1/ 3
⎛ 3ν 2 ⎞ δ2 = ⎜ 2 ⎟ ⎝ g ⎠
Re1/2δ3 for Re 2δ ≤ 400 , Nusselt (1916),
(9.68)
1/ 3
⎛ν 2 ⎞ δ 2 = 0.305 ⎜ 2 ⎟ ⎝ g ⎠
Re72δ/12 for 400 > Re 2 , Alekseenko et al. (1996), (9.69)
Re 2δ = ρ 2 w2 δ 2 η 2 ,
(9.70)
α 2 = δ 2 δ vane .
(9.71)
We see immediately that the higher the pressure the higher ρ1 / ρ 2 and therefore the lower is the performance of the separator. This is experimentally confirmed by many authors. The idea to look empirically for efficiency as a function of the number
η = η ⎡⎣V1u23 ( gδ vane ) ⎤⎦ , is also used in the practice.
(9.72)
9.4 Velocity fields modeling in separators
329
In the Russian literature e.g. Glustenko et al. (1972) the idea is used to build the ratio of the limiting velocity to the Kutateladze velocity and to keep the ratio the same in model and original.
9.4 Velocity fields modeling in separators Coupled solution for multiphase-flow fields in the separators is necessary to understand the efficiency of the separators. This is just since recently possible. In the past several still useful approximate methods are used based on simplified analytical or empirical description of the continuum velocity field. Knowing the continuum velocity field the particles are inserted from the boundary and the trajectories of the particles cared by the continuum are analyzed. No feed back on the continuum is considered in this case (one-way coupling). Some of them are given below. 9.4.1 Kreith and Sonju solution for the decay of turbulent swirl in pipe Next I give the only available approximate analytical solution of the momentum equation describing a swirl in the pipe obtained by Kreith and Sonju (1965). It describes the tangential velocity as a function of the distance from the swirl inducer and from the axis of the pipe v ( r , z ) . The authors analyzed steady turbulent swirl in pipe. After making several reasonable simplifying assumptions they arrived to the following form of the tangential momentum equation w
⎛ ∂ 2v 1 ∂ v v ⎞ ∂v = (ν + ν t ) ⎜ 2 + − ⎟, r ∂ r r2 ⎠ ∂z ⎝∂r
(9.73)
or in non dimensional form
∂ v ν ⎛ ∂ 2v 1 ∂ v v ⎞ = + − ⎟ ⎜ ∂ z Re ⎝ ∂ r 2 r ∂ r r 2 ⎠
(9.74)
where v = v wmax , r = r R , v = ( v + ν t ) ν , z = z R , Re = wR ν . The authors solved this equation by separation of the variables for the following boundary conditions: v = 0 at r = 0 and r = R, v(r, 0) = f(r) at z = 0. The initial condition was gained from experimental data for the initial distribution of the tangential velocity behind a tape swirler
330
9. Moisture separation
v ( r , 0 ) = ⎡ 6.3r − 0.013 (1.1 − r ) ⎣
−2.68
⎤ Δz ts , ⎦
(9.75)
where Δz ts = Δzts R is the non dimensional pitch of the inducer tape, Δzts is the length of the tape swirler making a complete 360° rotation. The reported solution is v (r , z ) =
+
ν z ⎞ 5.26 νz⎞ ⎛ ⎛ J1 ( 3.832r ) exp ⎜ −16.7 J1 ( 7.016r ) exp ⎜ −55.7 ⎟ − ⎟ Re ⎠ Δz ts Re ⎠ Δz ts ⎝ ⎝
7.78
ν z ⎞ 3.16 νz⎞ ⎛ ⎛ J1 (10.174r ) exp ⎜ −117.9 J1 (13.324r ) exp ⎜ −203.7 ⎟ − ⎟ +… Re ⎠ Δz ts Re ⎠ Δz ts ⎝ ⎝ (9.76)
3.93
J1 is the Bessel’s function of the first kind of order one. From experimental data
the relation v = 1 + 2.03 × 10−3 Re0.86 was recommended for 4 × 104 < Re < 1× 106 . Experimental data for Re = 18 000 and 61 000 validate the approximate solution. The data indicate that the initial swirl decay to about 20% at z = 100. There are authors trying to represent the decay by a single exponential function. From the data collected by Steenberger (1995) it is visible that the decay coefficient is a decreasing function with the increasing Reynolds number as manifested by the above solution. Note the practical importance of this solution. Having the rotation introduced by twisted tapes in the cylinder the particle trajectories can be computed and therefore the efficiency of the separation devices can be judged. This theoretical result is used by Algifri and Bhardwaj (1985) for computation of the increasing convective heat transfer due to induced swirl in a pipe. The improvement is considered to be a result of the natural circulation in the strong centrifugal field. The authors proposed to extend the validity of the known relation between Nusselt and Grashoff numbers in this case by simply replacing the gravity with the centrifugal acceleration. 9.4.2 Potential gas flow in vanes Ryjkov et al. (1974) analyzed the separating properties of channel consisting of vanes that are part of cylindrical surfaces. For the computation of the gas velocity the authors used a potential solution in analytical form taking into account the influence of each discrete vortex on each subsequent point. Then the trajectory of particles with prescribed size is computed numerically as in the following section.
9.4 Velocity fields modeling in separators
331
9.4.3 Trajectory of particles in a known continuum field Consider flow of particles with very small concentrations in known gas velocity field. Compute the trajectory of a particle with mass mi taking into account only the drag force. Such task is usually solved in the past for computing trajectories of particles in cyclone and vane separators. If the trajectory ends in the particle capturing devises this class of particle size and starting coordinate is considered as entrained from the gas flow. The three simplified momentum equations are
ρ3
π D3,3 i ⎛ du3,i v3,2 i − ⎜ r 6 ⎜⎝ dτ
ρ3
π D3,3 i ⎛ dv3,i v3,i u3,i ⎞ + ⎜ ⎟ = 3πη1 D3,i ( v1 − v3,i ) , r ⎠ 6 ⎝ dτ
(9.78)
ρ3
π D3,3 i dw3,i = 3πη1 D3,i ( w1 − w3,i ) , 6 dτ
(9.79)
⎞ ⎟⎟ = 3πη1 D3,i ( u1 − u3,i ) , ⎠
(9.77)
or du3,i dτ
dv3,i dτ dw3,i dτ
=
v3,2 i r
=−
=
+
u1 − u3,i
Δτ 13
v3,i u3,i r
w1 − w3,i
Δτ 13
+
,
v1 − v3,i
Δτ 13
(9.80)
,
,
(9.81)
(9.82)
where Δτ 13 = ( ρ3 D32i ) (18η1 ) is the Stokes relaxation time constant. If the droplets are larger and the Stokes law does not hold the time constant is in general Δτ cd = ( ρ d + ρ c ccdvm ) Ddi2 ⎡⎣18ηc Ψ ( Recd ) ⎤⎦ ,
where Ψ ( Recd ) = 1 for Recd < 24 ,
0.687 for Recd ≤ 1000 , Ψ ( Recd ) = 1 + 0.15 Recd
Ψ ( Recd ) = 0.11Recd 6 for Recd > 1000 ,
(9.83)
332
9. Moisture separation V1
2ϕ
Rin
δ vane
Rout
1
2
3
4
Z
Fig. 9.10. Povarov et al. (1976): Geometry of their model of vane separator
Fig. 9.11. Povarov et al. (1976): Separation efficiency as a function of the particle size. Steam and water mixture at 1 bar. Inlet steam velocity 20 m/s. Parameter: geometrical characteristics of the channel: (1) ϕ = 90°, δ = H R1 = 0.45, Geva = 6.86; (2) ϕ = 60°, δ = 0.45, Geva = 5.68; (3) ϕ = 45°, δ = 0.45, Geva = 4.25; (4) ϕ = 30°, δ = 0.45, Geva = 2.84; (5) ϕ = 45°, δ = 0.225, Geva = 6.97; (6) ϕ = 45°, δ = 0.9, Geva = 4.92
Fig. 9.12. Povarov et al. (1976): Separation efficiency as a function of the particle size for ϕ = 45°, δ = 0.45. Parameter: Regime conditions: (1). p = 1 bar, V1 = 20 m/s; (2). p = 5 bar, V1 = 20 m/s; (3). p = 10 bar, V1 = 20 m/s; (4). p = 20 bar, V1 = 20 m/s; (5). p = 50 bar, V1 = 20 m/s; (6). p = 50 bar, V1 = 10 m/s; (7). p = 50 bar, V1 = 5 m/s; (8). p = 50 bar, V1 = 3 m/s; (9). p = 50 bar, V1 = 1 m/s
9.4 Velocity fields modeling in separators
333
Zaichik (1998). Here Re cd = ρ c ΔVcd Dd ηc is the particle Reynolds number. Here
ccdvm is the virtual mass coefficient approximately ½ (for more information to this subject see volume 1 and 2 of this monograph). For constant gas velocity and particle size the analytical solution is provided by Crowe and Pratt (1974): w3,i = w1 − ( w1 − w3,i , a ) exp ( − Δτ Δτ 13 ) ,
(9.84)
u3,i = u1 − ( u1 − u3,i , a ) exp ( − Δτ Δτ 13 ) + ⎡⎣1 − exp ( − Δτ Δτ 13 ) ⎤⎦ Δτ 13 v3,2 i r ,
(9.85) v3,i = v1 − ( v1 − v3,i , a ) exp ( − Δτ Δτ 13 ) − ⎡⎣1 − exp ( − Δτ Δτ 13 ) ⎦⎤ Δτ 13 v3,i u3,i r , (9.86) Knowing the initial position and the velocity the position after the time interval Δτ can be computed by using the Lagrange method. Similar approach was used by Gloger (1970). Povarov et al. (1976) reported application of this method for the geometry given in Fig. 9.10 for steam and water mixtures. The computed trends reproduced in Figs. 9.11 and 9.12 are very informative. We realize that (a) the larger the bend angle the higher the wall-droplet contact efficiency for a given particle size; (b) the smaller the distance between two neighboring vanes the higher the wall-droplet contact efficiency; (c) the higher the pressure the smaller the wall-droplet contact efficiency; (d) the higher the velocity the higher the wall-droplet contact efficiency. Because the authors ignored the entrainment of droplet the wall-droplet contact efficiency is not necessarily equivalent with the separation efficiency. Therefore although very informative, such analysis alone without analyzing the entrainment and the collected water removal is not recommended for practical purposes. Problem 2: Given the vane separator on Fig. 9.10 with the geometry parameter given on Fig. 9.11. Compute the geometry number and rank the cases 1, 2, 3 with respect to their separation efficiency. Perform this also for case couple 3–5 and 4–6.
(
)
−2 Solution: The geometry number is Geva = nϕ ⎡1 − 1 + δ Rin ⎤ . The following ⎥⎦ ⎢⎣ table gives the geometry numbers corresponding to the cases 1 trough 6.
334
9. Moisture separation
Case
ϕ in °
δ = H R1
Geva
1 2 3 4 5 6
90 60 45 30 45 45
0.45 0.45 0.45 0.45 0.225 0.9
6.86 5.68 4.25 2.84 6.97 4.92
Comparing the cases 1, 2 and 3 by comparing their geometry numbers I realize that the highest efficiency at constant other parameters has the case 1, followed by 2 and 3. Case 5 is better than case 3 and finally case 6 is better than case 4. Exactly this was the finding reported in Fig. 9.11 by Povarov et al. (1976). Li et al. (2007) performed experiments using the vane separators without and with hooks which geometries are presented on Fig. 9.2(i) and 9.2(j). The hooks considerably increases the pressure drop (150 to 600 Pa if the air velocity increases from 3 to 7 m/s). The vane separators without hooks manifested much lower pressure drop, less then 50 Pa up to 6 m/s and then increasing to 200 Pa for 8 m/s. Therefore if also droplets smaller the 20 µm have to be separated more efficiently the vanes with geometry as those in Fig. 9.2(d) and (e) are better suited because they provide traps but do not increase the pressure drop. 9.4.4 CFD analyses of cyclones Using two fluid computer models for flows without heat and mass exchange for analyzes of cyclones is reported by Ikeda et al. (2003), Manson et al. (1993), Chaki and Murase (2006), and Reyes et al. (2006). Note that two fluid model is not appropriate for this purpose because there are cells with continuum liquids and droplets where the three fluid model is the appropriate one. Ikeda et al. (2003) reported that changing the form of the nose of the swirler from 4c) to 4d) reduces the total irreversible pressure drop of the cyclone. Manson et al. (1993) pointed out that it is questionable to use simple algebraic steady state models in transient analyses. 9.4.5 CFD analyses of vane separators Particle tracing is usually used to analyze the performance of the vane separators.
9.4 Velocity fields modeling in separators
335
Fig. 9.13. Kim et al. (2005) analysis to the Delta-P-5ft-dryer from “Burgess Mining Company, Dallas” (1973) for fine separators in PWR’s steam generators: (a) Cross section of the dryer vanes; (b) Angular offset of the vane pocket; (c) Water droplet trajectories in a dryer vane, inlet gas velocity 1.191 m/s; (d) Droplet removal efficiency in a dryer vane
The main idea is: one consider the vane separator as a 2D flow and assume that each particle touching the wall is counted as separated. Then the size of the particle is specified. With such size, the trajectories are computed by starting at equidistant points at the entrance. Then the number of the trajectories ending in the separator divided by the total number of the investigated trajectories defines the efficiency. Example is given in Fig. 9.13. Li et al. (2007) performed performance analysis of the vane separator without and with hook which geometry is presented on Fig. 9.2(i) and 9.2(j). The authors simulated first the gas flow neglecting the influence of the droplets by using k-eps turbulence models and the exported the velocity field for external analysis of the droplet trajectories. Comparison with their pressure drop measurements with airwater flow demonstrated the correctness of this approach. The most important result of such kind of analysis is the size of particles at particular velocity that can not be separated. So analyzing the spectrum that has to be separated and waiting the contribution of each particle size group the overall efficiency is computed. Such analysis does not take into account the effect of the entrainment and can not explain the experimentally observed limitations of the performance. But for inlet velocity at which the entrainment is negligible this approach is a powerful tool for quantitative analysis.
336
9. Moisture separation
(a)
(b)
(c)
(d)
Fig. 9.14. (a) Schematic picture of typical BWR cyclone separator; (b) Function principles of cyclone separator. Mass flow definitions; (c) Conventional swirler, see Ikeda et al. (2003); (d) Swirler with low pressure losses, see Ikeda et al. (2003)
9.5 Experiments
337
9.5 Experiments 9.5.1 BWR cyclones, PWR steam generator cyclones Boiling water reactors, BWR’s, are facilities that produce two-phase mixture at about 70 bar. Ideally the steam has to be separated before supplying to the turbines. As already mentioned moisture in the steam increase the wear of the turbine blades, reduces the overall efficiency of the plant, increases the radioactivity in the turbine building and condensate system. Usually for small electric power < 70 MWe gravitational separation is possible. For larger power a separation system have to be included in the scheme. In the past such systems are made outside the vessel but the most modern systems of BWR’s operate with separations mounted inside the vessel. Usually modern boiling water systems have a set of cyclone separations followed by dryers based on the direction change between parallel plates. Here we discuss the cyclone separation. They operate with two phase mixture with void mass flow rate fraction of about 15% and if properly designed my deliver to the next stage of separation mixture with about 10–15% liquid mass flow rate fraction. Pressurized water reactors heat water close to the saturation temperature at high pressure usually around 160 bar. Then the steam is produced in a specially designed steam generators that operate again around 50 to 70 bars. Again the produced two phase mixture has to passes separation scheme before entering the turbine. Historically the performance of the cyclones and dryers was described in terms of the properties defined below: Carryover: The so-called carryover is defined as the ratio of the water to steam mass flows leaving the cyclone upwards Yco =
m& 2,cy _ co = f exp m& p , X 1, core _ out ± 20% . m& 1, cy _ co
(
)
(9.87)
Here m& p is the total mass flow produced by all main circulation pumps through the core. X 1,core _ out is the steam mass flow ratio at the core exit. The usually reported accuracy of measurements of this value is about ± 20%. With this definition the upward steam quality is computed as follows X 1,cy _ co =
m& 1,cy _ co 1 = . m& 1, cy _ co + m& 2, cy _ co 1 + Yco
(9.88)
338
9. Moisture separation
Carry under: The removed water from the main flow contains bubbles. The vapor mass concentration is characterized by the so-called carry under defined as the ratio of the vapor bubbles mass flow to the entrained water mass flow X cu =
m& 1,cy _ cu m& 2,cy _ cu
(
)
= f exp m& p , X 1, core _ out ± 10% .
(9.89)
The usually reported accuracy of measurements of this value is about ±10%. With this definition the downward steam quality is computed as follows X 1,cy _ cu =
m& 1,cy _ cu X cu = . m& 1,cy _ cu + m& 2, cy _ cu 1 + X cu
(9.90)
Problem 3: Given a nuclear reactor core cooled by a mass flow m& p . The thermal power released in the core produces a vapor mass flow concentration at the exit of the core X 1,core _ out . The total mass flow rate passes through a system of parallel cyclones having an averaged performance defined by Yco and X cu . Compute the mass flows of the liquid m& 2,cy _ co and vapor m& 1,cy _ co directed to the dryers and the mass flows of the liquid m& 2,cy _ cu and vapor m& 1,cy _ cu returned to the circulation cycle through the core. Solution: Having the nuclear reactor core outlet steam quality X 1,core _ out and the total mass flow through the core m& p , using the simple steady state mass conservation
X 1,core _ out m& p = m& 1,cy _ co + m& 1,cy _ cu ,
(9.91)
(1 − X
(9.92)
1, core _ out
) m&
p
= m& 2, cy _ co + m& 2,cy _ cu ,
and the definition Eqs. (9.87)–(9.90) we can solve with respect to the carryover and carry under mass flows. The result is m& 2,cy _ cu =
1 − (1 + Yco ) X 1,core _ out 1 − Yco X cu
m& p ,
(9.93)
⎡
1 − (1 + Yco ) X 1, core _ out ⎤ ⎥ m& p 1 − Yco X cu ⎥⎦ (9.94)
m& 1, cy _ co = X 1, core _ out m& p − X cu m& 2,cy _ cu = ⎢ X 1, core _ out − X cu
⎢⎣
9.5 Experiments
⎡ 1 − (1 + Yco ) X 1,core _ out m& 2, cy _ co = Yco m& 1, cy _ co = Yco ⎢ X 1, core _ out − X cu 1 − Yco X cu ⎣⎢ m& 1,cy _ cu = X cu m& 2, cy _ cu = X cu
1 − (1 + Yco ) X 1,core _ out 1 − Yco X cu
m& p .
339
⎤ ⎥ m& p , (9.95) ⎦⎥
(9.96)
The remaining liquid after the cyclone system m& 2,cy _ co enters the dryers. The dryer efficiency is then
η dr = m& 2, dr _ cu m& 2,cy _ co .
(9.97)
Here m& 2, dr _ cu is the mass flow of the water droplets carried into the turbine. Wolf and Moen (1973) provided valuable experimental information for the so called BWR/6 cyclones, a vertical design of the type presented in Fig. 9.14, with three pick of rings for a pressure of 73.26 bar and the following mass flows m& single_cyclon = 44.73, 51.03 and 63 kg/s. For each mass flow two skirt submergence 254 and 1016 mm are used. The skirt submergence is the distance between the end of the small diameter swirler cone of the cyclone and the water level measured upwards. So increasing positive numbers indicates increasing water level inside the vessel. The input steam quality was varied between 7 and 23%. For the case 51.03 kg/s one experimental series is performed for void quality of 14.85% and skirt submergence –254, 0, 254, 508, 762, 1016, 1270, 1524 mm. Examples for such measurements performed by Wolf and Moen (1973) are presented on Figs. 9.15, 9.13 and 9.12 together with the fits reported in Kolev (2003). The authors have found that appropriate hydraulic resistance to the drainage water can reduce the entrained bubbles with the drainage water.
340
9. Moisture separation
44.73kg/s, subm. 0.254m 1,00 0,98
X1_cy_co
0,96
A5026 A6012 fit
0,94 0,92
2
0,90
Y =0,92691+1,19392 X-5,35154 X
0,88 0,00
0,05
0,10 0,15 X1_cy_in
0,20
0,25
Fig. 9.15. Typical cyclone performance characteristics: Outlet steam mass flow concentration as a function of the inlet steam mass flow concentration, Wolf and Moen (1973). Parameters – inlet mass flow rate and skirt submergence measured from the swireler position as shown in Fig. 9.13(b). Fit by Kolev (2003)
2
3
4
Y =0,98468-0,00838 X-0,01463 X +0,09226 X -0,08151 X
1,0 0,8 A5026 A6012 fit
X1_cy_co
0,6 0,4 0,2
51.03kg/s, X1_cy_in=0.1485 0,0 0,0
0,5 1,0 1,5 Submergence in m
2,0
2,5
Fig. 9.16. Typical cyclone performance characteristics: Outlet steam mass flow concentration as a function of the skirt submergence measured from the swirler position as shown in Fig. 9.14(b), Wolf and Moen (1973). Parameters – inlet mass flow rate and inlet steam mass flow concentration. Fit by Kolev (2003)
9.5 Experiments
341
How to use this information will be demonstrated in a minute by the following Problems 4 and 5. Yoneda et al. (2003) reported air-water experiments at atmospheric conditions with typical BWR’s cyclone as shown in Fig. 9.14(a) with gas volume flow rate 5.7 m/s and liquid volume flow rate 1.9 m/s. The authors measured velocity and void profiles along the radius at different elevation useful for testing of computer code predictions. They found that the flow is developed in the stand pipe after 10 diameters. They found also that the rotating two phase flow consists in a horizontal cross section of film, transition two-phase region, and steam core. Problem 4: If a modern reactor has a 100% mass flow through the core of about m& p = 13 200 kg/s and uniform distribution over the all e.g. 279 cyclone entrances we have m& single_cyclon = m& p ncy = 13 200 279 = 47.31 kg/s, and for 120%, 56.77 kg/s which well within the Wolf and Moen (1973) experimental data. The 80% are 37.85 kg/s, which slightly below the lower data range. So for given mass flow per single cyclone, vapor mass flow and submergence the liquid mass flow at the exit of the cyclone and the vapor mass flow in the separated liquid can be estimated from the diagrams on Figs. 9.15 and 9.16 and the relations already introduced. This is a reliable method for analysis of the efficiency of the cyclones and is widely used in the industry in the last 60 years. Now let as consider a real and very complex problem of optimization of the steam dryness of an existing boiling water reactor. Problem 5: Given a nuclear reactor core as presented in Fig. 9.17(a) with the parameters specified in Problem 4. Consider the real non-equal generation of steam from the rod bundles, the multiphase flow mixing in the plenum above the core and compute the performance of each cyclone. The specific problem here is that each cyclone will receive a specific mass flow rate with a specific liquid content and therefore perform at different point in its characteristics. Compare different states over a burn up cycle characterized by different spatial power distribution. Take into account that each of the states of the burn up cycle is associated with ten 3D-power distributions with the corresponding mass flows trough the core, core thermal power levels, and a feed water temperature. Vary the number of the additional nozzles mounted at the fuel bundle support and their arrangement. With a set of such nozzles the mass flow of the corresponding bundles is reduced and the produced dryness at the exit of the bundle is increased. After particular mixture separation in each cyclone the mixture will enter the front flow surface of the dryers also with different parameters. Use the dryer characteristic for Delta-P-5ftdryer from “Burgess Mining Company, Dallas” (1973), discussed later in Sect. 9.5.3. The final performance of the cyclones-and-dryers-tract is naturally an integral over the performance of all cyclones and dryers. Compare the integral exit moisture from the reactor for each state-nozzle combination and draw conclusions.
342
9. Moisture separation
Fig. 9.17. (a) Typical boiling water reactor: 1. reactor pressure vessel; 2. reactor core; 3. steam-water separators (cyclones); 4. steam dryer; 5. control rod drives; 6. control assemblies; 7. feed water inlet nozzles; 8. core spray line; 9. main steam outlet nozzle; 10. forced circulation pumps; 11. annular down comer. (b) IVA 1/4th geometry model of the control rod space, core, upper plenum and stand pipes
9.5 Experiments
343
15 10
200 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
25 20 15 10
2000 MWd/t Add. ozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
25 20 15 10
4000 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
Water mass flow concentration, %
30
20
30 25 20 15 10
1000 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
Water mass flow concentration, %
Water mass flow concentration, %
30
25
30
Water mass flow concentration, %
Water mass flow concentration, %
30
Water mass flow concentration, %
The applied procedure described below reduces in fact the moisture and therefore radioactivity at the turbine for an existing nuclear power plant. Details of this example are given in Kolev (2003). The computer code IVA was used to simulate the 3D-distributed multi-phase flow phenomena in the space defined by Fig. 9.17b where the geometrical model used for the computation is presented. The multiphase flow is simulated starting with the subcooled single phase entrance flow. Then the spatially non equal heating and evaporation in the core followed by the mixing process in the upper mixing chamber and then followed by the stand pipes up to the entrance of the cyclones is simulated.
30
25 20 15 10
3000 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
25 20 15 10
5000 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
9. Moisture separation
30 25 20 15 10
6000 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
Water mass flow concentration, %
Water mass flow concentration, %
344
30 25 20 15 10
7330 MWd/t Add. nozzles 00 16 40 72
5 0 0.6 0.8 1.0 1.2 1.4 Cyclon skirt submergency, m
Fig. 9.18. Liquid mass flow concentration after the dryers for NPP Philipsburg 1 as a function of the cyclone skirt submergence. Parameter – the number of the nozzles. The nozzle has a irreversible friction coefficient 265 related to 0.0064516 m²
Fig. 9.19. Void fraction at the exit of the core: (a) no nozzles; (b) 72 nozzles in the second external circle of fuels
Fig. 9.20. Void fraction at the entrance of the cyclones: (a) no nozzles; (b) 72 nozzles in the second external circle of fuels
9.5 Experiments
345
I compare states over a burn up cycle – one with standard nozzle design, and one with a AREVA additional nozzles mounted at the fuel bundle support. The most important outcome of this analysis is presented in Fig. 9.18. Figures 9.19 and 9.20 illustrate the effect of the peripheral flow reduction on the void fraction at the exit of the core and at the entrance of the cyclones. I found out that the nozzles, appropriately arranged, effectively reduce the moisture content in the steam directed to the turbine. Problem 6: Investigate for case 5 haw the performance of the cyclones and the dryers influences through the recirculation ratio the averaged entrance temperature of the core. Solution: Having the core outlet steam quality X 1,core _ out and the flow through the core m& p , using the simple steady state mass conservation, see Fig. 9.21, X 1,core _ out m& p = m& 1,cy _ co + m& 1,cy _ cu ,
(9.98)
(1 − X
(9.99)
1, core _ out
) m&
p
= m& 2, cy _ co + m& 2,cy _ cu ,
and the definition
X 1_ cy _ co = m& 1,cy _ co
( m&
+ m& 2,cy _ co ,
X 1_ cy _ cu = m& 1, cy _ cu
( m&
+ m& 2, cy _ cu ,
1, cy _ co
1, cy _ cu
)
(9.100)
)
(9.101)
I solve with respect to the carryover and carry under mass flows. The result is expressed conveniently in term of the cyclone mass flow entrainment ratio
(
f cu = X 1_ cy _ co − X 1,core _ out
) (X
1_ cy _ co
)
− X 1_ cy _ cu .
(9.102)
The cyclone mass flow entrainment ratio is the portion of the total mass flow entering the cyclone entrained by the pick off rings. With this I readily obtain
m& 1,cy _ cu + m& 2, cy _ cu = f cu m& p
(9.103)
m& 1, cy _ co + m& 2, cy _ co = (1 − f cu ) m& p
(9.104)
m& 1,cy _ cu = X 1_ cy _ cu f cu m& p
(9.105)
(
)
m& 2,cy _ cu = 1 − X 1_ cy _ cu f cu m& p
(9.106)
346
9. Moisture separation
Fig. 9.21. Averaged steady state mass flows through the typical BWR with forced convection
9.5 Experiments
m& 1,cy _ co = X 1_ cy _ co (1 − f cu ) m& p
(
347
(9.107)
)
m& 2,cy _ co = 1 − X 1_ cy _ co (1 − f cu ) m& p
(9.108)
Now consider the several cyclones working in parallel, each of them designated with i, j indices – i designated the integer x-address and j designated the integer yaddress. The averaged steam mass flow carry over- and carry under concentrations will then be
X 1_ cy _ co
m& 1,cy _ co = = m& cy _ co
∑
(
)
X 1_ cy _ co,i , j 1 − f cu ,i , j m& p ,i , j
all _ i , j
∑(
,
)
(9.109)
1 − f cu ,i , j m& p ,i , j
all _ i , j
and
X 1_ cy _ cu
m& 1,cy _ cu = = m& cy _ cu
∑
X 1_ cy _ cu ,i , j f cu ,i , j m& p ,i , j
all _ i , j
.
∑
(9.110)
f cu ,i , j m& p ,i , j
all _ i , j
The upwards output of the cyclones enters the dryers, so that m& dr _ in = m& 1, cy _ co + m& 2,cy _ co = (1 − f cu ) m& p .
(9.111)
The steam mass flow does not change and therefore m& 1, dr _ in = X 1_ cy _ co (1 − f cu ) m& p = X 1_ dr _ co m& dr _ out .
(9.112)
Solving with respect to the total dryer mass flow results in m& dr _ out =
X 1_ cy _ co X 1_ dr _ co
(1 − f cu ) m& p .
(9.113)
The entrained water by the dryer is therefore ⎛ X 1_ cy _ co m& dr _ in − m& dr _ out = ⎜ 1 − ⎜ X 1_ dr _ co ⎝
⎞ ⎟⎟ (1 − f cu ) m& p . ⎠
Thus, the total recirculation mass flow is therefore
(9.114)
348
9. Moisture separation
f cu m& p + m& dr _ in − m& dr _ out ⎡ ⎛ X 1_ cy _ co = ⎢ f cu + (1 − f cu ) ⎜1 − ⎜ X 1_ dr _ co ⎢⎣ ⎝
⎞⎤ ⎟⎟ ⎥ m& p = f rec m& p . ⎠ ⎥⎦
(9.115)
Here the cyclone and dryer mass flow entrainment ratio is the portion of the total mass flow entering the cyclone that is redirected into the down comer for recirculation ⎛ X 1_ cy _ co f rec = f cu + (1 − f cu ) ⎜ 1 − ⎜ X 1_ dr _ co ⎝
⎞ ⎟⎟ . ⎠
(9.116)
With this expression Eq. (9.114) can be written in a simple form ⎛ X 1_ cy _ co m& dr _ in − m& dr _ out = ⎜ 1 − ⎜ X 1_ dr _ co ⎝
⎞ ⎟⎟ (1 − f cu ) m& p = ( f rec − f cu ) m& p ⎠
(9.117)
Now consider the several flow paths of a dryers working in parallel, each of them designated with i, j indices. The averaged steam mass flow concentrations after the dryers will then be
X 1_ dr _ co =
∑
X 1_ cy _ co ,i , j (1 − f cu ,i , j ) m& p ,i , j
∑
X 1_ cy _ co ,i , j
all _ i , j
all _ i , j
X 1_ dr _ co,i , j
(1 − fcu ,i , j ) m& p,i, j
.
(9.118)
All the above characteristics are used for the computation of the steady state mixing temperature. Starting with the simple energy balance
m& 1,cy _ cu ⎡⎣ Δh + c p1 (T ′ − Tmix ) ⎤⎦
(
)
+ m& 2,cy _ cu + m& dr _ in − m& dr _ out c p 2 (T2 − Tmix )
(
= m& dr _ out c p 2 Tmix − T feed
)
(9.119)
and solving with respect to the mixing temperature results in
Tmix = a + bT feed
(9.120)
9.5 Experiments
349
where a=
(
X 1_ cy _ cu f cu c p1
X 1_ cy _ co b=
) ( + (1 − f
)
X 1_ cy _ cu f cu Δh + c p1T ′ + f rec − f cu X 1_ cy _ cu T2 c p 2
X 1_ dr _ co
cu
)
X 1_ cy _ cu c p 2
,
(9.121)
(1 − fcu ) c p 2
(
)
X 1_ cy _ cu f cu c p1 + 1 − f cu X 1_ cy _ cu c p 2
.
(9.122)
For multiple parallel arraignments I obtain c=
∑ ⎡⎣ X
all _ ij
1_ cy _ cu , ij
f cu ,ij c p1,ij + (1 − f cu ,ij X 1_ cy _ cu ,ij ) c p 2,ij ⎤m& p ,ij , ⎦
⎡ X 1_ cy _ cu ,ij f cu ,ij ( Δhij + c p1,ij Tij′ ) ⎤ ⎢ ⎥ 1 ⎥ m& p ,ij , a= ∑ ⎢ c all _ ij ⎢ ⎥ ⎢⎣ + ( f rec ,ij − f cu ,ij X 1_ cy _ cu ,ij ) c p 2,ij T2,ij ⎥⎦
b=
X 1_ cy _ co ,ij 1 (1 − fcu ,ij ) c p 2,ij m& p,ij . ∑ c all _ ij X 1_ dr _ co ,ij
(9.123)
(9.124)
(9.125)
It is clear from this analysis that the performance of the cyclones and dryers of the BWR’s has important influence on the core inlet temperature of the coolant. This it selves influences substantially the void distribution in the core, the neutron moderation and consequently the power distribution. The power distribution over the fuel cycles influence also the burn up and therefore the economy of the power plant. 9.5.2 Other cyclone types Burkov et al. (1999) reported development of 4 types of cyclone separators. In Fig. 9.22 a cyclone design is visible together with the efficiency characteristics. The rotation device is at the left site – not presented in the picture.
350
9. Moisture separation
1,333 D
0,917D
D
0,667 D
Fig. 9.22. (a) Cyclone separator design; (b) Efficiency of separation as a function of the dimensionless gas velocity, Burkov et al. (1999), 1. 7%; 2. 15% and 3. 10% mass moisture concentration. Air-water experiments at atmospheric conditions, w1,in = 20…100 m/s, 1 − X 1 = 6…37%
The efficiency
η = f (1 − X 1,in , K1 , design similarity ) .
(9.126)
is presented as a function of the dimensionless air velocity. The air velocity is made dimensionless by the Kutateladze droplet sink velocity, K1 = w1 ρ1 ⎣⎡ gσ ( ρ 2 − ρ1 ) ⎦⎤
−1/ 4
.
(9.127)
The important observation made is that for K1 > 7 the efficiency is 35…49%. No influence of the particle size is analyzed. The idea of this presentation is to ensure similarity to liquid vapor mixtures at higher pressure. Whether the assumed similarity works is not proven. The Kutateladze droplet sink velocity is an appropriate scale for gravitational separation but not for inertial separation. Replacing the gravitational acceleration with the centrifugal acceleration v22 r is better idea.
9.5 Experiments
351
Burkov et al. (1999) reported a design which makes use of the curvature of a bend, Fig. 9.23. One sees that at high air velocities efficiencies of 60–70% are possible. Note that relaxation of the stagnation pressure in the external space is designed. This is a necessary condition to allow inflow of two phase mixture rich in liquid. The authors reported that the pressure drop coefficient ζ related to the outlet cross section could be reduced from 0.4 to 0.03 by arranging of 4…4.5% vapor release from the separation chamber, called ventilation. Inlet
.
m1, ventilation
mainsteam line
.
m
3, out
Fig. 9.23. (a) Separator design; (b) Efficiency of separation as a function of the dimensionless gas velocity, Burkov et al. (1999), 1. 10%; 2. 12% and 3. 15% mass moisture concentration. Air-water experiments at atmospheric conditions, w1,in = 20…123.7 m/s, 1 − X 1,in = 10…15%
352
9. Moisture separation
Usually such type of separators is used as the so called primary separators. The fine separation then has to happen in a special device. Other centrifugal separator design is given in Fig. 9.24. Again the authors reported that organizing ventilation the separation efficiency can be increased up to 75…93%.
d
L
1
h1 =1,25 d
Separator
Fig. 9.24. (a) Separator design using the curvature of a bent; (b) Efficiency of separation as a function of the dimensionless gas velocity, Burkov et al. (1999), 1. 4% mass moisture concentration, 0% ventilation. Airwater experiments at atmospheric conditions, w1,in = 16…75 m/s, 1 − X 1,in = 4…10%
Four different cyclone separators from the type usually used in boiling water reactors with inlet diameters 0.1 and 0.15 m are also investigated by Burkov et al. (1999), see Fig. 9.25.
9.5 Experiments
353
(a) No displacement cylinder; w1,out = 25…40 m/s, 1 − X 1,in = 8.3…21%; 1 − X 1,out = 0.1…0.2%. With ventilation 1 − X 1,out = 0.1…0.2% up to 51 m/s, ζ = 0.4 (reduced to 0.03 by introducing of 4…4.5% vapor release from the separation chamber).
2 1 5 3
4
(a)
6
(b)
(c)
(d)
Fig. 9.25. Cyclone separators: 1. shroud, 2. rotation producer, 3. equalizer, 4. pick of channel, 5. displacement cylinder, 6. venting channels. Inlet diameters: 0.1 and 0.15 m. (a) No displacement cylinder; (b) With displacement cylinder; (c) With displacement cylinder and gas recirculation inside it, provide connection to the space after the vertices generator, 0.1 m; (d) With displacement cylinder and gas recirculation inside it, 0.15 m
(b) With displacement cylinder; w1,out = 24.3…40 m/s, 1 − X 1,in = 15.6…17.3%; 1 − X 1,out = < 0.5%, w1,out = 47 m/s, 1 − X 1,out = 0.9%, ζ =3.5. With ventilation
1 − X 1,out = 0.1% up to 48.2 m/s and ζ = 3.3. (c) With displacement cylinder and gas recirculation inside it, provide connection to the space after the vertices generator; w1,out = 24.5…48.6 m/s, 1 − X 1,in = 10…21.1%; 1 − X 1,out = 0.01…0.3%, ζ = 3.7. (d) With displacement cylinder and gas recirculation inside it; w1,out = 35…50 m/s,
1 − X 1,in = 0.
(15.41)
Integrating the differential equation over the time interval Δτ and solving with respect to the new crust thickness gives 1 ⎧⎪ 2 ⎩⎪
⎡
δ c = ⎨ tanh ⎢ arctanh ⎣
2aδ c ,old + b D
+
1 Δτ 2 d
⎤ b ⎪⎫ D D⎥ − . ⎬ D ⎪⎭ a ⎦
(15.42)
The application of this equation is computationally more time consuming then its numerical solution. That is why I prefer simple procedure by discretizing implicitly Eq. (15.35) d ⎞ d ⎛ ⎞ ⎛ aδ c2 + ⎜ b + δ c,old ⎟ = 0 ⎟δ c − ⎜ c + Δτ Δτ ⎠ ⎝ ⎠ ⎝
(15.43)
and solving it with respect to the new crust film thickness. Then the interface temperature is computed by using Eq. (15.29), new heat transfer is computed using film boiling and radiation models and the procedure is repeated iteratively as many times as necessary to reach the prescribed accuracy. Finally the energy conservation is checked. Not that this procedure is unstable. It is easily made stable by averaging the new computed interface temperature with this obtained by the previous iteration. 15.3.6 Melt energy conservation Next I use the assumption (8) saying that the volumetrically generated decay heat is uniformly distributed between the liquid, crust and viscous layer. This is very strong simplification which requires improvement. Within the liquid part of the layer the generated heat in accordance with this assumption is 1 − δ cη δ part of
15.3 System of differential equations describing the process
465
the total energy generation. For M ad < M ad ,max , the energy conservation for the liquid layer in lumped parameters is ⎛ δ cη d ( ρ l c plVl Tl ) = Q⎜⎜1 − δ dτ ⎝
dM cη ⎞ dM ad ⎟ − Aq& da ′′ − Aq& up ′′ + ′ − had hl (15.44) ⎟ dτ dτ ⎠
or ⎛
ρ l c pl Aδ ⎜⎜1 − ⎝
δ cη δ
⎞ dTl ⎛ δ cη ⎟ ⎜ ⎟ dτ = Q⎜1 − δ ⎠ ⎝
⎞ dM ad ⎟ − Aq& da ′′ − Aq&up ′′ + ′ − c pl Tl had ⎟ dτ ⎠ (15.45)
(
)
or ⎛
ρ l c pl δ ⎜⎜1 − ⎝
δ cη δ
⎞ dTl Q ⎛ δ cη ⎟ ⎜ ⎟ dτ = A ⎜1 − δ ⎠ ⎝
⎞ ⎟ − q& up & ′′ f , ⎟ ′′ − qda ⎠
(15.46)
where f = 1−
′ − c pl Tl had Δhad
,
(15.47)
and f = 1 , for
dM ad =0. dτ
(15.48)
Replacing the removed energy from the liquid layer I obtain ⎛
ρ l c pl δ ⎜⎜1 − ⎝
δ cη δ
⎞ dTl Q ⎛ δ cη ⎟ ⎜ ⎟ dτ = A ⎜1 − δ ⎠ ⎝
⎞ ⎟ − hup Tl − Tη − hda f (Tl − Tbot ) , (15.49) ⎟ ⎠
(
)
or dTl Q = − ρ l c pl Aδ dτ
hup + hda f ⎛ δ ρ l c pl δ ⎜⎜1 − cη δ ⎝
⎞ ⎟⎟ ⎠
Tl +
hupTη + hda fTbot ⎛ δ ρ l c pl δ ⎜⎜1 − cη δ ⎝
⎞ ⎟⎟ ⎠
.
(15.50)
This equation is directly included in the system of ordinary differential equations describing the process. The system is then numerically integrated. An alternative
466
15. Coolability of layers of molten reactor material
approach is possible if one make use of some analytical form of the decay heat, namely, Q = Q0 a / τ n .
(15.51)
The Eq. (15.50) can be then rewritten as dTl 1 = a1 n − b1Tl + c1 dτ τ
(15.52)
where a1 =
b1 =
c1 =
Q0 a
ρ l c pl Aδ
,
(15.53)
hup + hda f ⎛ δ ρ l c pl δ ⎜⎜1 − cη δ ⎝
⎞ ⎟⎟ ⎠
hupTη + hda fTbot ⎛
ρ l c pl δ ⎜⎜1 − ⎝
δ cη δ
⎞ ⎟⎟ ⎠
,
(15.54)
.
(15.55)
Assuming constant coefficients during the time interval Δτ the implicit first order numerical solution is then ⎤ ⎡ ⎥ ⎢ Δτ ⎢ Tl = Tl ,old + a1 + c1Δτ ⎥ n ⎥ ⎢ 1 ⎛ ⎞ ⎥ ⎢ ⎜τ + Δτ ⎟ 2 ⎝ ⎠ ⎦⎥ ⎣⎢
(1 + b1Δτ ) .
(15.56)
15.3.7 Buoyancy driven convection In accordance with assumption (2) liquid layers with internal heat sources are subject of buoyancy driven convection which can be described by empirical correlations verified by experiments in the particular Rayleigh number of interest. As already mentioned review of such correlation is given by Müller and Schulenberg (1983, p. 37) and will not be repeated here. Here I give only the used correlations.
15.3 System of differential equations describing the process
467
The natural convection in layers with internal heat sources is observed by Kulacki and Goldstein (1972) to start up at Ral′ =35840,
(15.57)
where Ral′ =
gβ l q& ′′′δ l5
λl
η l λl ρ l ρ l c pl
,
δ l = δ − δ c − δη .
(15.58)
(15.59)
For lower Rayleigh numbers the heat transfer inside the layer is due to conduction only. The upward heat transfer coefficient for a volumetrically heated horizontal layer is given by Mayinger et al. (1975) as hup =
λl 0.292 Ral′0.23 Prl0.085 , δl
(15.60)
for solid surface, which is applicable in case of crust, and hup =
λl 0.368 Ral′0.23 Prl0.085 δl
(15.61)
for free liquid interface. The downwards heat transfer coefficient for volumetrically heated horizontal layer is given by the same authors hda =
λl 1.235 Ral′0.1 . δl
(15.62)
Note that the above heat transfer coefficients are averaged heat transfer coefficients. The transient values are subject of ±50% oscillations. Having the heat transfer coefficients the heat fluxes are easily computed
468
15. Coolability of layers of molten reactor material
′′ = hup (Tl − Tη ) , q&up
(15.63)
′′ = hda (Tl − Tbot ) . q& da
(15.64)
15.3.8 Film boiling The heat transfer coefficient in film boiling with negligible radiation is well described by the Berenson (1961) correlation
(
′ ,0 = c Ttop − TH′ 2O q& ′FB
)3 / 4 ,
(15.65)
where
⎡ λ ′′3 ρ ′′g (ρ ′ − ρ ′′)(h′′ − h′) ⎤ c = 0.425⎢ ⎥ η ′′λRT ⎣⎢ ⎦⎥
1/ 4
,
(15.66)
using as characteristic length scale of the process the Rayleigh–Taylor instability wave length
λRT =
σ
g (ρ ′ − ρ ′′)
.
(15.67)
The appropriate correction for radiation of the heat conduction problem is given in Kolev (1997) or in Vol. 2 of this monograph, 4 ′ = ε s k SB (Ttop q& ′rad − TH4 2O )
r=
(15.68)
′′ q& rad , ′ q& ′FB
(15.69)
1/ 4
q& ′′ ⎞ ⎛ ′ = q& ′FB ′ ,0 ⎜⎜1 + r FB ,0 ⎟⎟ q& ′FB ′ ⎠ q& ′FB ⎝
hFB =
′ q& ′FB . Ttop − TH′ 2O
,
(15.70)
(15.71)
15.4 Heat conducting structures
469
Thus the overall heat transfer coefficient is h* = hFB + ε s k SB
4 Ttop − TH4 2O
Ttop − TH′ 2 0
(15.72)
15.4 Heat conducting structures The model for heat conduction is developed by this author and documented in Kolev (1995). Although 1D model is good enough for this case we copy the 2D solution procedure as given in Kolev (1995) and use it in this case only for one column of equidistant computational cells. 15.4.1 Heat conduction through the structures The Fourier equation without heat sources governs the heat transferred from the liquid metal into the surrounding coolant. I solve this equation numerically by standard implicit method in two dimensional (r, z) space. The corresponding integer indices are (i, k), where i ∈ Z ∩ 1 ≤ i ≤ imax , k ∈ Z ∩ 1 ≤ k ≤ k max . Initially the cells with indices i = 2 to imax − 1 , and k = 2 to k max − 1 are treated as inner cells with possible contact with the melt. The cells with indices i = 1 and imax , and k = 1, and k max are auxiliary. These cells are used to impose the boundary conditions. Therefore the cell sizes are Δr = Δrmax /(imax − 2)
(15.73)
Δz = Δz max /( k max − 2)
(15.74)
I write the implicit discretized form for a computational network. For convenience of notation I omit space indices unless they differ from i, k. Subscript “a” denotes the old time level.
ρc p (T − Ta ) / Δτ = (1 / Δr )[λ2 (Ti −1 − T ) / Δr − λ1 (T − Ti +1 ) / Δr ] (15.75) + (1 / Δz )[λ6 (Tk −1 − T ) / Δz − λ5 (T − Tk +1 ) / Δz ]
Solving with respect to the temperature T I obtain
470
15. Coolability of layers of molten reactor material
T = ( ρc pTa / Δτ + b2Ti −1 + b1Ti +1 + b6Tk −1 + b5Tk +1 ) /( ρc p / Δτ + b2 + b1 + b6 + b5 )
(15.76) simple formula for applying the point Gauss–Seidel iteration method to solve the resulting system of algebraic equations. Here b2 = λ2 / Δr 2 , b1 = λ1 / Δr 2 , b6 = λ6 / Δz 2 , b5 = λ5 / Δz 2 are computed ones at the beginning of the iteration
process for λ = const. Usually up to 9 outer iterations are sufficient to obtain accurate solution. The boundary cells are treated separately as follows. 15.4.2 Boundary conditions For the inner wall I impose the heat flux as a boundary condition:
ρc p (T − Ta ) / Δτ = (1 / Δr )[q&1′′ − λ1 (T − Ti +1 ) / Δr ] (15.77) + (1 / Δz )[λ6 (Tk −1 − T ) / Δz − λ5 (T − Tk +1 ) / Δz ]
or T = ( ρc pTa / Δτ + q&1′′ / Δr + b1Ti +1 + b6Tk −1 + b5Tk +1 ) /( ρc p / Δτ + b1 + b6 + b5 ).
(15.78) Here Ti–1 is no more used, and b2 = 0. In case of crust formation into the oxide pool we have ′′ + q& ′′′δ c q&1′′ = q& da
(15.79)
For the outer wall at the coolant side we have
ρc p (T − Ta ) / Δτ = (1 / Δr ){λ2 (Ti −1 − T ) / Δr − [1 /(1 / hnc ) + 0.5Δr / λ1 )](T − Tbc )} + (1 / Δz )[λ6 (Tk −1 − T ) / Δz − λ5 (T − Tk +1 ) / Δz ]
(15.80) which modifies the main equation as follows Ti +1 = Tbc
(15.81)
15.4 Heat conducting structures
b1 = 1 /[(1 / hda ) + 0.5Δr / λ1 )Δr ]
471
(15.82)
15.4.3 Oxide crust formation on colder heat conducting structures The discussion in this section is related to modeling of crust formation on heat conducting structure. As mentioned above, the heat conduction in the structure is modeled by finite difference numerical solution of the Fourier equation. The cell being in contact with the melt posses the index number 2 and the temperature in the center of this cell is T2 . The size of the cell is Δr . In each time step, Δτ , we obtain the oxide temperature, Tl , and the temperature of the first cell of the structure contacting the melt, T2 . For non existing conditions for crust formation, δ c ,da = 0 , the heat flux at the interface in the oxide region is directly computed ′′ = hda ,eff (Tl − T2 ) , q& vi′′ = q& da
(15.83)
where the effective heat transfer coefficient from the bulk to the center of the cell is hda ,eff = 1 /(1 / hda +
1 Δr / λv ) . 2
(15.84)
The interface temperature is governed by the equality of the heat fluxes at the interface 1 2
(15.85)
1 Δr / λv . 2
(15.86)
′′ λv (Tvi − T2 ) /( Δr ) = q& da
or ′′ Tvi = T2 + q& da
For Tvi > T ′′′ , there is no crust formation. Equation (15.83) gives in this case the heat flux which is imposed as a boundary condition for the heat conduction computation in the structure. Usually Tvi ≤ T ′′′ and crust is formed. In this case the melt interface temperature can not exceed the solidification temperature T ′′′ and the heat flux from the corium into the crust is uniquely defined
472
15. Coolability of layers of molten reactor material
′′ = hda (Tl − T ′′′) . q& da
(15.87)
The heat flux leaving the crust and entering the structure wall is larger ′′ + q& ′′′δ c, da + ρ c Δhsl q& vi′′ = q& da
dδ c , da dτ
.
(15.88)
because there is volumetric decay heat generation in the crust. The structure interface temperature, Tvi , is determined by 1 2
′′ + q& ′′′δ c ,da + ρ c Δhsl λv (Tvi − T2 ) /( Δr ) = q& da
dδ c ,da dτ
(15.89)
or dδ ⎛ ′′ + q& ′′′δ c ,da + ρ c Δhsl c,da Tvi = T2 + ⎜⎜ q& da dτ ⎝
⎞1 ⎟⎟ Δr / λv . ⎠2
(15.90)
The temperature profile in the crust is governed by the solution of the Fourier equation for steady one dimensional heat conduction with uniformly distributed internal heat sources. The solution in terms of the outlet heat flux is
Tvi = q& ′′′δ c2,da /(2λc,da ) − q& vi′′ δ c,da / λc,da + T ′′′ .
(15.91)
Eliminating q&vi′′ by using Eqs. (15.88) and (15.91) I obtain dδ ⎛ ′ + ρ c Δhsl c, da q& ′′′δ c2, da /( 2λc , da ) + ⎜⎜ q& ′da dτ ⎝
⎞ ⎟⎟δ c , da / λc, da − T ′′′+ Tvi = 0 . ⎠
(15.92)
Eliminating Tvi by using the Eqs. (15.90) and (15.92) and dividing by q& ′′′ /( 2λc ,da ) I obtain
δ c2, da +
dδ 2 ⎡ 1 λ ⎤ ′′ + ρc Δhsl c , da + q& ′′′ Δr c , da ⎥ δ c , da q&da ⎢ 2 λv ⎦ q& ′′′ ⎣ dτ
15.4 Heat conducting structures
−
2λc , da q& ′′′
⎡ dδ ⎛ ′′ + ρc Δhsl c , da ⎢T ′′′− T2 − ⎜ q&da dτ ⎝ ⎣
⎤ ⎞1 ⎟ Δr / λv ⎥ = 0 ⎠2 ⎦
473
(15.93)
Solving with respect to the crust thickness and taking only the positive solutions of the quadratic equations I obtain 2 ⎡⎛ dδ 1 λ ⎞ ′′ + ρc Δhsl c , da + q& ′′′ Δr c , da ⎟ ⎢⎜ q&da 2 dτ λv ⎠ ⎢⎝ ⎢ δ c , da q& ′′′ = ⎢ ⎢ dδ ⎡ ⎛ ⎢ +2λc , da q& ′′′ ⎢T ′′′− T2 − ⎜ q&da ′′ + ρc Δhsl c , da dτ ⎢⎣ ⎝ ⎣
1/ 2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎞1 ⎟ Δr / λv ⎥ ⎥ ⎠2 ⎦ ⎥⎦
dδ ⎛ 1 λ ⎞ ′′ + ρc Δhsl c , da + q& ′′′ Δr c , da ⎟ . − ⎜ q&da 2 dτ λv ⎠ ⎝
(15.94)
Obviously there is a crust thickness larger than zero only if ′′ T ′′′> T2 + q& da
1 Δr / λv . 2
(15.95)
The effective heat conduction coefficient is ′′ + q&′′′δ c , da ) /(Tl − T2 ) = hda + q&′′′δ c , da (Tl − T2 ) . hda , eff = (q&da
(15.96)
For completeness, let me mention that the idealized task for crust formation in a liquid with a constant basement temperature Tb is approximately solved by Savino and Siegel (1968): hda (T ′′ − T ′′′ )τ
ρc Δhsl δ c , da ,∞ δ c , da ,∞ =
⎡ δ (τ ) ⎤ ⎫⎪ ⎡ c p ,c (T ′′ − T ′′′ ) ⎤ ⎧⎪ δ c , da (τ ) = ⎢1 + − ln ⎢1 − c , da ⎥⎬ , ⎥ ⎨− δ c , da ,∞ ⎥⎦ ⎪⎭ 3Δhsl ⎢⎣ ⎥⎦ ⎪⎩ δ c , da ,∞ ⎢⎣
λc (T ′′′ − Tb )
hda (T ′′ − T ′′′ )
.
The solution assumes different solidus and liquidus temperatures, T ′′′ and T ′′ , respectively.
474
15. Coolability of layers of molten reactor material
Note, that in our case the heat flux downwards into the basement is coming from the solution of the heat conduction into the basement.
15.5 Metal layer Nuclear reactor cores are usually surrounded by steel and other metals. During the core degradation radiative heat transfer may cause melting of considerable amount of steel. Steel and oxide are immiscible. I will not discuss haw the spreading behavior is influenced by the processes in the reactor pit. In any case the result after spreading may be characterized by two limiting cases: (a) metal and oxide are homogeneously spread and finally stratified one over the other and (b) metal and oxide are heterogeneously spread and occupy different surfaces. The model can be applied to the both scenarios. In the heterogeneous spreading, the area occupied by oxide and by metal is computed as to be in mechanical equilibrium A=
M Asp , M + M me
Ame =
M me Asp . M + M me
(15.97)
(15.98)
In this case the model described up to now describes the behavior of the melt, additives and structure. In the homogeneous spreading the density ratio governs whether the oxide is above or below the metal. I assume that the oxide is the lighter phase in this case. The heat transfer inside the metal layer is described in the same way as inside the oxide. Similarly I describe the interaction with additives and the heat conduction structures. Model for crust formation between oxide and melt at the interface as developed by Kolev (1995) can be applied also for the metal layer.
15.6 Test case In order to demonstrate haw the above presented models work I will construct and artificial example with extreme initial and boundary conditions. Consider a water cooled nuclear reactor with initial power of Q0 = 4.250 × 10 9 W . The decay heat follows the curve given in Fig. 15.4.
15.6 Test case
475
Relative decay power, %
2.00 Divided by the initial power
1.75 1.50 1.25 1.00 0.75 0.50
0
5 10 15 20 25 30 35 40 45 50 Time in h
Fig. 15.4. Normalized decay power as a function of time
The process starts 3 h after the shut down of the reactor and is considered up to 143 h. 193 t oxide and 100 t metal are spread over 169 m² giving collapsed oxide layer thickness 0.2 m and molten metal layer thickness 0.087 m if each of them occupy the total surface with the heavier beneath the lighter. The initial oxide temperature is assumed to be 3000 K and the initial metal temperature 2073 K. The oxide has the following mass concentrations about 62% UO2, 22% ZrO2, 8.9% SiO2 and 7.1% FeO. The 10 cm layer that can be eroded has a mass concentrations 0.42% SiO2 and 58% FeO, meting enthalpy Δhad = 2400 kJ/kg and liquidus temperature Tl , ad = 1250 + 273.15 K. The initial temperature of the cold layers is 300 K. The heat transfer coefficient of the bottom surface is 25 000 W/(m²K). The coolant has temperature of 300 K. The conducting structure at the bottom is discretized with 60 cells. The density of the released steam is 0.2 kg/m³, its viscosity 0.00004 kg/(ms) and its mass concentration in the concrete 0.0595. 15.6.1 Oxide over metal In this case the oxide is in fact considered to occupy 170 m². First I consider the case oxide over metal. Interesting information about the short time period is available in Fig. 15.5. We see that the basement interface temperature below the protective layer remains unchanged until the complete erosion of the concrete. This phase takes some 20 min. Thereafter the metal contacts the cold cast iron surface. This results in sharp drop down of the metal temperature until complete solidification of the metal. Within about the next 40 min. the metal heats up and starts to remelt. Complete melting is reached after about additional 40 min. The cast iron melts also, as shown in Fig. 15.6, and the metal and cast iron temperatures start to increase.
476
15. Coolability of layers of molten reactor material
Very similar behavior is observed by the oxide temperature. The changes are of course more inertial because of the much lower thermal conductivity of the oxide and because of the crust and viscous layer formation. The oxide/water interface temperature drops dramatically within the first h but still remains for a long time above the Leidenfrost temperature, which justify the use of film boiling regime here. The short term layers dynamics is demonstrated in Figs. 15.7 and 15.8. We see the expected behavior of decreasing sacrificial material and simultaneously increasing oxide layer thickness. After the end of the melt-concrete interaction the gas quickly abandons the both layers. After 100 h the transient heat conduction problem is reduced to a steady state one as seen from Fig. 15.9. 3000
T in K
2500 2000 Metal down averaged
1500 Oxide
down averaged up
1000 500 0.0
0.5
1.0
1.5 2.0 Time in h
2.5
3.0
Fig. 15.5. Temperatures as functions of time. Short time period
2000
T in K
1500 Cast iron top bottom Heat exchanger wall top bottom
1000 500 0.0
0.5
1.0
1.5 2.0 Time in h
2.5
3.0
Fig. 15.6. Temperature of the cast iron as function of time. 10 cm cast iron melts during the process
15.6 Test case
477
0.30
Thickness in m
0.25 0.20
Collapsed oxide layer Oxide crust - upper Oxide viscose layer - upper Oxide crust thickness - lower Collapsed molten oxide layer Sacrificial layer
0.15 0.10 0.05 0.00 0.0
0.5
1.0 1.5 2.0 Time in h
2.5
3.0
Fig. 15.7. Layer thickness as a function of time
0.35 Thickness in m
0.30 0.25
Sacrificial Two phase oxide - gas Two phase metal - gas
0.20 0.15 0.10 0.05 0.00 0.0
0.5
1.0
1.5 2.0 Time in h
2.5
3.0
Fig. 15.8. Two phase layer thickness as a function of time
Figure 15.10 shows the most important results – the temperature of the protective layer. We see that 150 h after the beginning of the transient the temperature starts to slightly decrease.
478
15. Coolability of layers of molten reactor material
30 Into
Heat flux in kW/m²
25
the protective structure the horizontal heat exchanger surface
20 15 10 5 0
0
20
40
60 80 100 120 140 Time in h
Fig. 15.9. Heat fluxes at the both side of the heat conducting structure as a function of time
2500 2000
T in K
1500
Cast iron top
1000 500 0 0
20
40
60 80 100 120 140 Time in h
Fig. 15.10. The maximum temperature of the protective layer as a function of time
15.6.2 Oxide besides metal In this case the oxide is in fact considered to occupy 112 m². One of the most important results is presented of Fig. 15.11. The interface temperature was all the time above the Lidenfrost temperature which means that the hypothesis for film boiling was true.
15.7 Gravitational flooding of hot solid horizontal surface by water
Oxide
3000
down averaged up
2500 T in K
479
2000 1500 1000 500 0
20
40
60 80 100 120 140 Time in h
3000
T in K
2500 2000 Oxide
1500
down averaged up
1000 500 0
1000
2000 Time in h
3000
4000
Fig. 15.11. Temperatures of the oxide layer and of the oxide-water interface as a function of time: (a) Short term up to 140 h; (b) Long term up to 4000 h
Complete solidification is reached after 327 days.
15.7 Gravitational flooding of hot solid horizontal surface by water Some of the postulated severe accidents in nuclear reactors with very low probability of occurrence have been receiving increasing attention in the last decade. One of the interesting physical tasks in this field is how to design effective cooling of the melt avoiding energetic steam explosions. Gravitational flooding of very hot horizontal surface by water in combination with flooding from below is an alternative.
480
15. Coolability of layers of molten reactor material
Such two phase flow possesses very interesting properties. Surprisingly for me, I did not found in the literature any contributions to this topic up to 1995. The purpose of this work in those years was to demonstrate some properties of such flows by analysing the properties of the mathematical model describing the flow, Kolev (1996). With some simplifications the problem is reduced in Sects.15.7.1 and 15.7.2 to a system of three hyperbolic partial differential equations. Eigen values, eigenvectors and canonical form are obtained in Sect. 15.7.3. The criticality condition is obtained from the system describing the steady state in Sect. 15.7.4. The system of partial differential equations is made dimensionless and physical scaling criteria are obtained. 15.7.1 Simplifying assumptions Consider horizontal hot surface temperature T3σ . In view of possible application of the model described below I choose cylindrical or Cartesian coordinates. In the following κ = 0 will stay for Cartesian and κ = 1 , for cylindrical coordinate system. The flow is considered only in r-direction. The limited pressure change of the compartment allows that the vapour and the liquid densities can be considered as a constant i.e. ρ1 ≈ const , ρ 2 = const, respectively. The vertical gas bubble velocity is the free rising Kutateladse (1951) velocity w1 = wKu = 1.41( gσΔρ / ρ 22 )1/ 4 .
(15.99)
At the bottom of the water layer film boiling is the heat transfer mechanism described e. G. with the Berenson (1961) correlation ′ ,0 q& ′FB
⎡ λ ′3 ρ ′′ΔρΔh ⎤ = 0.425⎢ g ⎥ η ′′λ RT ⎥⎦ ⎢⎣
1/ 4
(T3σ − T ′)3 / 4 ,
(15.100)
where
[
]
λ RT = σ /( gΔρ )]1 / 2 .
(15.101)
In addition the radiation ′′ ≈ ε 3κ SB (T34σ − T ′4 ) q&rad
(15.102)
may transfer considerable amount of energy because of the high surface temperatures. The film boiling heat transfer is not independent from radiation. Increasing the radiation increases the vapour thickness and reduces the heat transferred due to conduction through the film. The reduction is approximately taken into account by
15.7 Gravitational flooding of hot solid horizontal surface by water
′ = q&′FB ′ ,0 /(1 + rq&′FB ′ ,0 / q&′FB ′ )1/ m , q&′FB
481
(15.103)
see Kolev (1995b), where for the case of natural circulation m = 4 and in the case of forced convection m = 2, or ′ = q& ′FB ′ ,0 [(1 + r 2 / 4)1 / 2 − r / 2] , q& ′FB
(15.104)
and ′′ − q& ′sub ′ ) / q& ′FB ′ ,0 . r = (q& rad
(15.105) w
u
z
θ
wKu Control _ volume
r
Z Steam =1
Water = 2
T3
q&F′′B
M elt
r Fig. 15.12. Gravitational flooding of hot solid horizontal surface
The expected heat transfer mechanisms leads to the following vapour production per unit time and unit surface ′ + q& ′rad ′ − q& sub ) / Δh ( ρw) 21 = (q& ′FB
(15.106)
Note, that the heat transfer regime expected for a long time from melted corium surface is film boiling, but the flow pattern above the unstable film is bubble flow. The flow pattern bubble flow is stable in this case only if the produced steam at
482
15. Coolability of layers of molten reactor material
the surface can be removed by bubbles without to disintegrate the liquid of the two phase layer. Thus with the discussed here approach the limitation conditions for existence of continuous liquid in the two phase layer can be analysed. I assume that the difference of the horizontal phase velocities is zero and therefore u1 = u 2 = u .
(15.107)
Note that u is the radial velocity averaged over the vertical cross section of the flow. I look for description of the relationship between the two-phase layer thickness Z, the void fraction in the layer α1 and the radial velocity u.
15.7.2 Conservation of mass and momentum, scaling I introduce the distance between the hot surface and the free surface of the two phase mixture, Z, called two phase layer thickness. Consider the control volume Vol shown in Fig. 15.12. The local volume average and successively time average mass conservation equations for each phase are 1 ∂ κ ∂ (α1ρ1Z ) + κ (r Zα1ρ1u ) = ( ρw) 21 − α1ρ1wKu , ∂τ r ∂r
(15.108)
1 ∂ κ ∂ [(1 − α1 ) ρ 2 Z ] + κ [r Z (1 − α1 ) ρ 2u ] = −( ρw) 21 ∂τ r ∂r
(15.109)
Here the equations are written per unit horizontal basement surface, instead per unit mixture volume as usual, because our control volume has variable thickness Z. It is very convenient to work with the sum of the both equations and with their difference obtained after division by the corresponding densities
∂ 1 ∂ κ ( ρZ ) + κ (r Zρu ) = −α1 ρ1wKu ∂τ r ∂r
(15.110)
Δρ ∂Z 1 ∂ κ + (r Zu ) = ( ρw) 21 − α1wKu ∂τ r κ ∂r ρ1 ρ 2
(15.111)
where
ρ = α1 ρ1 + (1 − α1 ) ρ 2 , is the averaged density of the two phase flow at the point (τ , r ) .
(15.112)
15.7 Gravitational flooding of hot solid horizontal surface by water
483
The driving force for the horizontal momentum change is the gravitational pressure change. Consider two vertical surfaces being at distance Δr . The averaged gravitational pressure acting at the first surface is Z1
−g
∫
1 1 ( ρ )1 zdz = − g ( ρZ 2 )1 , Z1 2
(15.113)
0
and the corresponding pressure force is −
1 gΔθ ( ρr κ Z 2 )1 . 2
(15.114)
Similarly, the pressure force at the position r2κ is −
1 gΔθ ( ρr κ Z 2 ) 2 . 2
(15.115)
In Cylindrical coordinates the r-projection of the pressure force acting at the θ – surfaces per unit base area is −
1 gρZ 2 / r κ . 2
(15.116)
The net pressure force in radial direction divided by the base surface of the control volume is therefore Δθ ( ρr κ Z 2 ) 2 − Δθ ( ρr κ Z 2 )1 1 1 g lim − gρZ 2 / r κ = κ 2 Δr →0 2 r ΔθΔr −
1 1 ∂ κ 2 1 1 1 ∂ (r ρZ ) − g κ ρZ 2 = g ( ρZ 2 ) . g 2 r κ ∂r 2 r 2 ∂r
(15.117)
Thus the mixture momentum balance per unit base surface results in 1 ∂ κ 1 ∂ ∂ ( ρuZ ) + κ (r ρuZu ) + g ( ρZ 2 ) = −α1ρ1wKu u . r 2 ∂r ∂τ ∂ r
(15.118)
It can be further simplified by using the chain rule and comparing with the mixture mass conservation equation
484
15. Coolability of layers of molten reactor material
ρZ (
∂u ∂u 1 ∂ + u ) + g ( ρZ 2 ) = 0 2 ∂r ∂τ ∂r
(15.119)
ρZ (
∂u ∂u 1 ∂ρ ∂Z +u )+ g + gρ =0 2 ∂r ∂τ ∂r ∂r
(15.120)
or
or
1 ∂u ∂ 1 2 ∂ + ( u + gZ ) + gZ ln ρ = 0 2 ∂r ∂τ ∂r 2
(15.121)
Thus, the gravitational flooding of hot surface is completely described by the vector of dependent variables ( ρ , u, Z ) being function of the independent variables (τ , r ) . For the steady state case and when ρ = const , the momentum equation (15.121) simplifies to the form
∂ 1 2 ( u + gZ ) = 0 , ∂r 2
(15.122)
1 2 u + gZ = const , which demonstrates the well known fact 2 that the kinetic and potential energy sum of flowing layer tends to an extreme. We see that disturbance ΔZ at the surface propagates with the stationary velocity
or in integrated form
u = (2 gΔZ )1/ 2 .
(15.123)
The system can be made dimensionless by introducing the following scales: length Z m , velocity u m = ( gZ m )1/ 2 , and time Δτ m = Z m / u m . While the choice of Z m is arbitrary the choice of the velocity scale is arising from the eigen value analysis, compare with Eq. (15.132). Therefore, u m and Δτ m are scales associated with the physics of the process. Next I make dimensionless the variables u = u / u m , Z = Z / Z m , the time τ = τum / Z m , and the spatial coordinate r = r / Z m . I make the densities dimensionless using as a scale Δρ , ρ = ρ / Δρ , ρ1 = ρ1 / Δρ , and ρ 2 = ρ 2 / Δρ . In addition the dimensionless Kutateladze velocity is wKu = wKu / u m , and the dimensionless vapour generation mass flow rate is ( ρ w ) 21 = ( ρw) 21 /(Δρu m ) . Dividing the momentum equation of the system by u m2 Δρ , the mixture mass equation by u m Δρ , and the third equation by u m , I obtain
15.7 Gravitational flooding of hot solid horizontal surface by water
∂u ∂u 1 ∂ +u )+ ( ρZ 2 ) = 0 2 ∂r ∂τ ∂r
(15.124)
∂ 1 ∂ κ (ρZ ) + κ (r Z ρ u ) = −α1ρ1wKu ∂τ r ∂r
(15.125)
∂Z 1 ∂ κ 1 + (r Z u ) = ( ρ w ) 21 − α1wKu ∂τ r κ ∂r ρ1 ρ 2
(15.126)
ρZ (
485
Performing experiments with surfaces of having temperatures of order of 2000– 3000 K and water is difficult and expensive. Experiments can be performed with model liquids and much colder surfaces in laboratories providing e.g. ( ρ w ) 21, model = ( ρ w ) 21, real _ process .
15.7.3 Eigen values, eigen vectors and canonical forms Obviously, it is very convenient to choose the following set of dependent variables U T = [u, ρZ , Z ] ,
(15.127)
which completely determines the flow. The system of PDE I will analyse next is obtained after applying the chain rule for the three equations. 1 ∂Z ∂u ∂u 1 g ∂ ( ρZ ) + g +u + =0 2 ∂r ∂τ ∂r 2 ρ ∂r
(15.128)
u *2 ∂Z ∂u ∂u u *2 ∂ +u + ( ρZ ) + =0, 2 Z ∂r ∂τ ∂r 2Zρ ∂r
(15.129)
u ∂ ∂ ∂u ( ρZ ) + u ( ρZ ) + ρZ = −α1ρ1wKu − ρZ κ κ , ∂τ ∂r ∂r r
(15.130)
u ∂Z ∂Z ∂u Δρ +u +Z = ( ρw) 21 − α1wKu − Z κ κ , ∂τ ∂r ∂r ρ1ρ 2 r
(15.131)
or
where u * = ( gZ )1/ 2 ,
(15.132)
486
15. Coolability of layers of molten reactor material
as it will be demonstrated later, is the counterpart to the velocity of sound in the compressible two phase fluid mechanics. The characteristic equation of the system of PDE u−λ
ρZ Z
u *2 u *2 2 Zρ 2Z u−λ 0 = (u − λ )2 − u *2 (u − λ ) = 0 , 0 u −λ
[
]
(15.133)
gives the following eigen values
λ1 = u , λ2 = u + u * and λ3 = u − u * .
(15.134, 15.135, 15.136)
The eigen values are real and at least two of them are distinct from each other. For each of the eigen values we look for a set of linear independent eigenvectors. The result for the first vector is
0 u *2 2 Zρ u *2 2Z
or
ρZ 0 0
Z h 11 0 h12 = 0 , h13 0
[h1 ] = [0,1,− ρ ] .
(15.137)
(15.138)
The result for the second vector is
− u* u *2 2 Zρ u *2 2Z
or
ρZ − u* 0
Z h 21 0 h22 = 0 , − u*
(15.139)
h23
[h2 ] = [2 ρZ / u * ,1, ρ ].
The result for the third vector is
(15.140)
15.7 Gravitational flooding of hot solid horizontal surface by water
u* u *2 2 Zρ u *2 2Z
or
ρZ
Z h 31
u*
0 h32 = 0 ,
0
u*
487
(15.141)
h33
[h3 ] = [− 2 ρZ / u * ,1, ρ ].
(15.142)
Therefore, the system of partial differential equation describing the flow is hyperbolic. Along the characteristic curves defined by the equations dr / dτ = λi I can write the corresponding canonical equation as follows. Along dr / dτ = u
(15.143)
ρΔρ d dZ dρ ≡Z = α1 (1 − α1 )ΔρwKu − ( ρZ ) − ρ ( ρw) 21 for Z > 0 , dτ dτ dτ ρ1ρ 2 (15.144) or
dρ = −(aρ 2 + bρ + c) , dτ
(15.145)
where a = wKu /( ΔρZ ) , b = Δρ ( ρw) 21 /( ρ1ρ 2 Z ) − ( ρ1 + ρ 2 ) wKu /(ΔρZ ) ,
c = ρ1ρ 2 wKu /( ΔρZ ) .
Along dr / dτ = u + u * ,
(15.146)
2 ρZ du d dZ + ( ρZ ) + ρ =B. * dτ u dτ dτ
(15.147)
Along dr / dτ = u − u* ,
(15.148)
488
15. Coolability of layers of molten reactor material
−
2 ρZ du d dZ + ( ρZ ) + ρ =B, dτ u * dτ dτ
(15.149)
where B = −κ 2 ρZu / r κ − α1wKu ( ρ1 + ρ 2 ) + ρΔρ ( ρw) 21 /( ρ1ρ 2 ) .
(15.150)
Thus, I succeed in reducing the system of partial differential equations to a system of ordinary differential equations along the characteristic curves which is very convenient for integration e.g. by means of the method of the characteristics. The Eq. (15.147) can be integrated analytically along its characteristic curve. Knowing the value at point 1 at the characteristic curve the mixture density at the point 2 can be obtained as follows. For the trivial case α1 = 0 (consequently wKu = 0 ) and ( ρw) 21 = 0 , the solution is ρ = const . For the general case having in mind that 4ac − b 2 ≈ −( ρw) 221 /( ρ1Z ) 2 < 0 , I obtain ( ρ )2
∫(ρ )dρ /(aρ
2
+ bρ + c) = −Δτ ,
(15.151)
1
or after performing the integration ln
2a ( ρ ) 2 + b − D 2a ( ρ ) 2 + b + D
2a ( ρ )1 + b − D = −Δτ / D . 2a( ρ )1 + b + D
(15.152)
Solving with respect to ( ρ ) 2 gives ( ρ ) 2 = [E (b + D ) − b + D ] [(1 − E )2a ] ,
(15.153)
where E=
2a( ρ )1 + b − D exp(− Δτ / D ) , 2a( ρ )1 + b + D
(15.154)
(
(15.155)
D = b 2 − 4ac
)
1/ 2
.
15.7 Gravitational flooding of hot solid horizontal surface by water
489
15.7.4 Steady state The steady state is described by the following system of ordinary differential equations
ρZu
du 1 d + g ( ρZZ ) = 0 , dr 2 dr
(15.156)
1 d κ (r Zρu ) = −α1 ρ1wKu , r κ dr
(15.157)
1 d κ Δρ (r Zu ) = ( ρw) 21 − α1wKu . κ dr ρ1 ρ 2 r
(15.158)
Next I rearrange the equations in order to solve with respect to the spatial derivatives. Using the chain rule and rearranging the first equation I obtain u2
du 1 dZ 1 d + g[u + u ( ρZ )] = 0 . dr 2 dr ρ dr
(15.159)
The second and the third equation gives u
d du κ ( ρZ ) = −α1ρ1wKu − ρZ − ρuZ , dr dr r κ
(15.160)
u
dZ Δρ du κ = ( ρw) 21 − α1wKu − Z − uZ . dr ρ1 ρ 2 dr r κ
(15.161)
Replacing into the first equation I obtain ⎤ du ⎡ Δρ (ρw)21 − α1wKu (1 + ρ1 ρ ) − 2κuZ / r κ ⎥ =⎢ dr ⎣ ρ1ρ 2 ⎦
[2Z (1 − Fr )], 2
(15.162)
1 ∂ κ (r Zρu ) = −α1 ρ1wKu , r κ ∂r
(15.163)
1 ∂ κ Δρ (r Zu ) = ( ρw) 21 − α1wKu . κ ∂r ρ1 ρ 2 r
(15.164)
490
15. Coolability of layers of molten reactor material
The second and the third equations are retained in non expanded form which is very convenient for numerical integration. Here the Froud number for such kind of flows is Fr = u / u *
(15.165)
It follows from Eq. (15.162) that if Fr = 1 the velocity gradient is not defined, that is the flow is critical.
Conclusions: The canonical equations demonstrate the following properties of the flow: (i) Density waves propagate with velocity u; (ii) Density waves are caused besides the initial and boundary conditions by the volume source term α1 (1 − α1 )ΔρwKu − ρΔρ ( ρw) 21 /( ρ1ρ 2 ) , if it is different from zero. (iii) Velocity distributions are always coupled with disturbances of the density and of the mixture thickness, and (iv) propagate with velocity u + u * forwards and u − u * backwards. The following scales are used to make the system dimensionless: length Z m , velocity u m = ( gZ m )1/ 2 , time Δτ m = Z m / u m and density difference Δρ . While the choice of Z m is arbitrary the choice of the velocity scale is arising from the eigen value analysis. Therefore, u m and Δτ m are scales associated with the physics of the process. The following scaling criteria relevant to the process analysed are obtained: time τ = τu m / Z m , the spatial coordinate r = r / Z m , velocity u = u / u m , two phase layer thickness Z = Z / Z m , densities ρ = ρ / Δρ , ρ1 = ρ1 / Δρ , and ρ 2 = ρ 2 / Δρ , Kutateladze velocity wKu = wKu / u m and vapour generation mass flow rate ( ρ w ) 21 = ( ρw) 21 /(Δρu m ) . Experiments can be performed with model liquids and much colder surfaces in laboratories providing e.g. ( ρ w ) 21, mod el = ( ρ w ) 21, real _ process . From the steady state system we see that the maximum achievable flooding velocity in steady state is u*. Equations (15.157) and (15.158) manifest the following very interesting feature. Intensive evaporation at the hot surface, combined with limited steam removal
15.8 Nomenclature
491
from the two phase boundary layer leads to a dramatic increase of the boundary layer thickness and volumetric fraction of steam which leads to disintegration of the continuous liquid. Therefore in the initial state of the flooding where the surface temperature is still very high the expected cooling regime should be: (a) film boiling with bubble two phase layer above the unstable film in the neighbourhood of the water entrance region, and (b) dispersed droplets removed from-, and falling back, to the surface. Regime (a) was a subject of this analysis. The formalism described here can be used either for describing the bubbly two phase flooding process and the structure of the reached long term steady state, or for analysis the conditions which leads to local void fractions greater than e.g. 0.3 to 0.6 which leads to dispersed droplet structure above the hot surface not described by the this section. Instabilities leading to alternation of the both regimes are thinkable for given layer thickness.
15.8 Nomenclature Latin A a a,b,c,d C C0
basement surface, m2 thermal diffusivity, m2/s coefficients in Eq. (15.52) mass concentration, dimensionless concentration distribution parameter, dimensionless
Da g Gr h h hup hda kSB M Nu
= q& ′′′δ 2 /(λΔTmax ) , Dammköhler number, dimensionless gravitational acceleration, m/s2 = gβΔTmaxδ 3 /ν 2 , Rayleigh number, dimensionless g gravity acceleration, m/s2 averaged heat transfer coefficient, W/(m2K) specific enthalpy, J/kg heat transfer coefficient at the upper surface, W/(m2K) heat transfer coefficient at the basement surface, W/(m2K) Stefan - Boltzmann constant , W/(m2K4) mass, kg = h δ l / λl , averaged Nusselt number for heat transfer at the upper sur-
face, Pr Q q& ′′ q& ′′′
dimensionless (= ν / a) , Prandtl number, dimensionless power, W heat flux density, W/m2 volumetric heat flux density, W/m3
492
j Ra' Ra T V VKu z r
15. Coolability of layers of molten reactor material
superfacial velocity, m/s (= GrPrDa), modified Rayleigh number, dimensionless (= GrPr), Grashoff number, dimensionless mass averaged temperature, K volume, m3 Kutateladse velocity, m/s lateral coordinate, m vertical coordinate, m
Greek
α β
δ δc δη
volume fraction, dimensionless thermal expansion coefficient, dimensionless finite differential,latent heat of melting, J/kg latent heat of melting of the sacrificial material, J/kg maximum bulk-wall temperature difference, K layer thickness, m crust thickness, m viscous layer thickness, m
δl ρ Δτ ε ν λ ψ σ η
liquid layer thickness, m density, kg/m³ time interval, s emissivity coefficient, cinematic viscosity, m2s thermal conductivity, W/(mK) Eq. (14), dimensionless surface tension, N/m dynamic viscosity, Pas
Δ Δhsl Δhad ΔTmax
Subscripts da max min top bot up sp eff FB ad
downward in a pool maximum minimum top of the layer bottom of the layer upper surface spreading area effective film boiling additional
15.9 Nomenclature to Sect. 15.7
ml i o old rad
η
493
metal component initial old time laved radiation viscous layer
Subscripts ´´´ " ´´´
melting point per unit surface per unit volume
15.9 Nomenclature to Sect. 15.7 Latin
a
= wKu /(ΔρZ )
B
= −κ 2 ρZu / r κ − α1wKu ( ρ1 + ρ 2 ) + ρΔρ ( ρw) 21 /( ρ1ρ 2 )
b
= Δρ ( ρw) 21 /( ρ1 ρ 2 Z ) − ( ρ1 + ρ 2 ) wKu /( ΔρZ )
c g [h1 ]
= ρ1ρ 2 wKu /( ΔρZ ) gravitational acceleration, m/s2 = [0,1,− ρ ] , eigenvectors corresponding to λ1
[h2 ] [h3 ] Fr
r r ′ ,0 q& ′FB ′ q& ′FB ′′ q&rad ′ q& ′sub liquid,
[ ] = [− 2 ρZ / u ,1, ρ ], eigenvectors corresponding to λ = 2 ρZ / u * ,1, ρ , eigenvectors corresponding to λ2 *
3
= u / u , Froud number for gravitational flow with free surface, dimensionless radial coordinate, m r = r / Z m , spatial coordinate, dimensionless film boiling heat flux without radiation, W/m2 *
film boiling heat flux taking into account radiation, W/m2 radiation heat flux, W/m2 heat flux from the liquid interface at the bottom of the layer into the bulk
T′ T3σ
W/m2 saturation temperature, K surface temperature of plane spread melt layer, K
UT
= [u , ρZ , Z ] , dependent variables vector, m/s, kg/m2, m
494
15. Coolability of layers of molten reactor material
u u* u1 u2
radial surface average velocity, m/s (gZ) 2, m/s = u, horizontal gas velocity, m/s = u, horizontal liquid velocity, m/s
um
= ( gZ m )1/ 2 , velocity, m/s
u Vol w2 w1 wKu wKu Z
= u / u m , velocity, dimensionless control volume with variable thickness Z, m3 vertical liquid velocity, m/s vertical gas bubble velocity, m/s free rising Kutateladze velocity, m/s = wKu / u m , Kutateladze velocity, dimensionless distance between the melt interface and the surface of the two phase mixture, two phase layer thickness, m two phase layer thickness at position r1, m two phase layer thickness at position r2, m length, m = Z / Z m , length, dimensionless
Z1 Z2 Zm Z Greek
α1 Δρ
void fraction, dimensionless = ρ 2 − ρ1 , kg/m3
Δr Δh Δθ Δτ m
finite difference in r-direction, m = h′′ − h′ , latent heat of vaporisation, J/kg finite difference in θ direction, rad = Z m / u m , time, s differential, dimensionless dynamic vapour viscosity, kg/(ms) solid surface emissivity, e.g. ≈ 0.7 , dimensionless = 0 stays for Cartesian coordinate system, dimensionless = 1 stays for cylindrical coordinate system, dimensionless = 5.697 × 10 −8 , Steffan Boltzman constant, W/(m2K4)
∂ η ′′ ε3 κ κ
k SB
[
]
λ RT λ1
= u , eigen value, m/s
λ2
= u + u* , eigen value, m/s
λ3 λ′ ρ1
= u − u * , eigen value, m/s
= σ /( gΔρ )]1 / 2 , Rayleigh - Tailor wave length, m
saturated water thermal conductivity, W/(mK) vapour density, kg/m3
References
ρ2 ρ ρ ρ1 ρ2 ( ρw) 21 ( ρ w ) 21 ρ ′′ ρ′ σ τ τ
495
liquid density, kg/m3 = α1ρ1 + (1 − α1 ) ρ 2 , averaged density of the two phase flow at the point (τ , r ) , kg/m3 = ρ / Δρ , dimensionless = ρ1 / Δρ , dimensionless = ρ 2 / Δρ , dimensionless vapour production per unit time and unit surface, kg/(m2s) = ( ρw) 21 /( Δρu m ) vapour generation mass flow rate, dimensionless saturated water density, kg/m3 saturated steam density, kg/m3 surface tension, N/m time, s = τu m / Z m , time, dimensionless
References Alsmeier H et al. (März, 1997) COMMET-Konzept, Chap. 6.1. In B. Mühl, ed., Forschungszentrum Karlsruhe, Technik and Umwelt, Untersuchungen zu auslegungsüberschreitenden Ereignissen (Unfällen) in Leichtwasserreaktoren Berenson PJ (August, 1961) Film-boiling heat transfer from a horizontal surface. J. Heat Transfer, vol 83, pp 351–361 Fish JD, Pilch M and Arellano FE (July, 1982) Demonstration of passively-cooled particlebed core retention. Proceedings of the LMFBR Safety Topical Meeting, Lyon, Ecully France, pp III-327–336 Friedrich HJ (November 13–14, 1975) SNR-300 Tank external core retention device design and philosophy behind it. In: Coats RL ed., The Second Annual Post – Accident Heat Removal (PAHR) Information Exchange, SAND76-9008, p 333 Friedrich HJ (November 2–4, 1977) Dynamic behavior of SNR-300 core retention device in experimental support of the design concept. In Baker L and Bingle JD, eds., Proceedings of the Third Post-Accident Heat Removal Information Exchange, ANL-78-10 Gandrille P (April 4, 1997) Input data for severe accident mitigation measures. Nuclear System Supply System Part, FRA Report EPTA DC 1476, Rev. A Hübel HJ (April, 1974) The safety related criteria and design features for SNR. Proceedings Fast Reactor Safety Meeting, Beverly Hills CONF-740401-P1, pp 3–21 Kolev NI (1986) Transiente Zweiphasenströmung (Transient Two-Phase Flow), SpringerVerlag, Berlin-Heidelberg Chap. 4 pp 34–38 Kolev NI (April 20, 1995a) External Cooling of EPR 1500 Reactor Vessel under Severe Accident Conditions, Part 1. Buoyancy driven convection, metallic layer dynamics, wall ablation, KWU NA-M/95/E030, Project R&D Kolev NI (April 3–7, 1995b) IVA4 Computer code: The model for a film boiling on a sphere in subcooled, saturated and superheated water, Sub. to the Second International Conference on Multiphase Flow, '95-Kyoto
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15. Coolability of layers of molten reactor material
Kolev NI (1996) Gravitational flooding of hot solid horizontal surface by water. Kerntechnik, vol 61, pp 67–76 Kutateladse SS (1951) A hydrodynamic theory of changes in the boiling process under free convection conditions. Izv. Akad. Nauk SSSR, Otd. Tech. Nauk, vol 4, pp 529–536; AEC-tr-1991 (1954) Kolev NI (1996) External cooling Of PWR reactor vessel during severe accident. Kerntechnik, vol 61, no 2–3, pp 67–76. In abbreviated form in (March 8–12, 1996) Proceedings of ICONE-4, The Fourth International Conference on Nuclear Engineering, New Orleans, USA Kolev NI (October 6, 1997) IVA4 Layers vol 2 A computer code for analysis of coolability of molten reactor materials spread as a horizontal layers, KWU NA-M/1997/E050, Project R&D Kolev NI (May 19–21, 1997) Verification of the IVA4 film boiling model with the data base of Liu and Theofanous. Proceedings of OECD/CSNI Specialists Meeting on FuelCoolant Interactions (FCI), JAERI-Tokai Research Establishment, Japan Kolev NI (April 2–6, 2000a) Computational analysis of transient 3D-melt-water interactions. 8th International Conference on Nuclear Engineering, Baltimore, Maryland USA, ICONE-8809 Kolev NI (June 11–15, 2000b) Needs of industrial fluid dynamics applications, 2000 ASME Fluids Engineering Division Summer Meeting (FEDSM), Industry Exchange Program, Sheraton Boston Hotel, Boston, Massachusetts Kulacki FA and Goldstein RJ (1972) Thermal convection in a horizontal fluid layer with uniform volumetric energy source. J. Fluid Mech., vol 55, part 2, pp 271–287 Mayinger F, Jahn M and Reineke HH (1975) U. Steinberner, Untersuchung thermodynamischer Vorgänge sowie Wärmeaustausch in der Kernschmelze, Teil 1: Zusammenfassende Darstellung der Ergebnisse, Bundesministerium für Forschung und Technologie, Arbeitsbericht BMFT – RS 48/1 Müller U and Schulenberg T (November, 1983) Post accident heat removal research: A state of the art review, KfK 3601, Report Kernforschungszentrum Karlsruhe Richard P and Szabo I (May 26–30, 1997) In-vessel core retenuation study: Proposal for a core-catcher concept. Proceedings of the ICON 5: 5th International Conference on Nuclear Engineering, Nice, France, ICONE5-2156 Savino JM and Siegel R (1968) An analytical solution for solidification of moving warm liquid onto an isothermal cold wall. Int. J. Heat Mass Transfer, vol 12, pp 803–809 Swanson DG, Cotton I and Dhir VK (1982) A Thoria Rubble Bed for Post Accident Core Retention. In: Müller U and Günter C eds. (1983) Post-Accident Debris Cooling. Proceedings of the Fifth Post Accident Heat Removal Information Exchange Meeting, G. Braun, Karlsruhe, ISBN3 – 7650-2034-6, pp 307–312 VDI-Wärmeatlas (1991) Berechnungsblätter für den Wärmeübergang, Sechste Auflage, VDI Verlag, Düsseldorf.
Zuber N and Findlay JA (1965) Averaged volumetric concentration in the twophase flow systems. J. Heat Transfer, vol 87, p 453
16. External cooling of reactor vessels during severe accident
Chapter 16 is devoted to the so called external cooling of reactor vessels during severe accident. It is a technology allowing arresting the melt inside the vessel of some initial conditions are fulfilled. First the state of the art is presented. Then a brief description of the phenomenology leading to melt in the lower head is discussed: dry core melting scenario, melt relocation, wall attack, focusing effect. Brief mathematical model description is given appropriate for a set of model assumptions. The model describes: the melt pool behavior, the two-dimensional heat conduction through the vessel wall, the total heat flow from the pools into the vessel wall, the vessel wall ablation, the heat fluxes, the crust formation and the buoyancy driven convection. Solution algorithm is provided for a set of boundary conditions adequate for real situations. A summary of the state of the art regarding the critical heat flux for externally flowed lower head geometry is provided. On a several practical applications different effects are demonstrated: the effect of vessel diameter, the effect of the lower head radius, the effect of the relocation time, the effect of the mass of the internal structures. Varying some important parameters characterizing the process the difference between high powered pressurized- and boiling water reactor vessel behavior is demonstrated.
16.1 Introduction External cooling of rector vessel during severe accident is very attractive for arresting accident progression within the vessel. The subject of the analysis of the external cooling is to answer among others the following important questions: (a) How large are the external vessel wall heat fluxes? (b) Can the external cooling remove the resulting vessel external wall heat fluxes without leaving the nucleate boiling region? In this chapter a tool is presented for making an order of magnitude estimate of the heat fluxes at the external vessel wall of advanced light water reactor vessels for plants with 400 to 2000 MW electrical powers.
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_16, © Springer-Verlag Berlin Heidelberg 2009
497
498
16. External cooling of reactor vessels during severe accident
16.2. State of the art Review and list of references of German work performed in the field of buoyancy driven convection in enclosures with-, and without internal heat sources for the time between 1970 and 1982 is available by Sonnenkalb (1994), and will not be repeated here. Henry and Fauske (1993), Henry et al. (1993) analyzed by means of a homogeneous pool model without taking into account crust formation the external cooling of PWR-1000 rector vessel during severe accident. The authors found that there is no significant limitation to heat removal from the bottom of the rector vessel other than the thermal conduction through the reactor vessel wall, and concluded that external cooling should be a major consideration in accident management evaluations and decision-making for current plants, as well as a possible design consideration for future plants. Note that not taking into account crust formation, as done by the authors, is conservative, but neglecting the effects of the metallic layer, as also done by the authors, is of considerable concern. Crust formation in homogeneous pool is introduced by O’Brien and Hawkes (1991) for analysis of external cooling of PWR – 1000 (75% of the 69 230 kg of UO2 + 100% ZrO2 that is 17 294.5 kg resulting in 69 217 kg corium with density 7589 kg/m3).
Fig. 16.1. The focusing problem and its modeling: Two layer model, oxide pool with internal heat sources, metallic pool above it, crust formation around the molten material, natural convection inside the liquids, heat conduction inside the solid strictures, radiation emission, external nucleate boiling, heat conduction in the vessel wall, wall ablation
The feasibility of the external cooling of rector vessel during severe accident is discussed by Theofanous et al. (1994a) for the Russian VVER-440 in Loviisa,
16.2. State of the art
499
Finland and for the US design of ALWR known as AP-600. The analysis was based on two layer steady state model without taking into account the effect of the metal layer dynamics and the 2D transient heat conduction into the vessel wall. After critical estimation and appropriate selection of correlations describing the natural circulation redistribution of the heat fluxes in semi-spherical cavity filled with stratified oxide melt at the bottom and layer of steel above it, and applying them to the above two cases, the authors obtained the heat fluxes to the coolant and the remaining steel, which is the main result of their study. The experimental data from large scale experiments reported by Theofanous et al. (1994b) and in the references given there for critical heat flux, are compared with the expected values and the conclusion was drown: The results in both cases show that a failure of this severe accident management concept is so unlikely as to be considered physically unreasonable. It is worth noting, that this analysis is performed for a prescribed thickness of the steel layer without analyzing the heat fluxes in this region during the transient formation of this layer. The thicker the assumed steel layer, the smaller the resulting heat fluxes into the vessel wall. This point is of considerable concern as it will be demonstrated in this chapter. Sun (1994) introduced into the theoretical analysis the effect of molten steel relocation due to radiation and two dimensional modeling of the vessel wall heat conduction. No practical applications are reported in Sun (1994). The work of Sun provides a fast engineering methodology for estimation of the external cooling. This was in 1993 the starting point for my analysis of possible external cooling concept for an advanced light water reactors with 400 to 2000 MW electrical power. As already mentioned the purpose of the present chapter is to present a tool for order of magnitude estimation of the external coolability of such reactors. The method presented here was first developed by Kolev (1993) and carefully documented in Kolev (1995a). It was applied for analysis of the feasibility of the external cooling for VVER 640, Kolev (1993, 1995a), EPR 1500, Kolev (1995b), KKI 1, Kolev (1995c), before its external publication in Kolev (1996). Then the method was used for analysis of the severe accident management concept of the SWR 1000 Kolev (2000, 2001, 2004). Similar to Kolev’s (1993) models without dynamic formation of the layer for steady state are developed and used in US by Theofanous et al. (1996a), Rempe et al. (1997), Esmaili and Khatib-Rahbar (2004). The Russian researchers Dombrovskii et al. (1999) used slightly different approach. They compute analytically the heat transfer coefficient in the descending wall boundary layer and the effective turbulent component of the thermal conductivity. Then they use it in the 2D Fourier equation for computing the twodimensional temperature distribution in the pool.
500
16. External cooling of reactor vessels during severe accident
16.3. Dry core melting scenario, melt relocation, wall attack, focusing effect
Relative decay power, %
Chains of isotopes form after the shutdown of a nuclear reactors releasing decay heat. The initial power and the history of the burn up of the nuclear fuel dictate the level of the residual decay heat at any moment after the shutdown. Not going into the specific physical processes I refer to the American and German standards recommending haw to approximate the decay power – ANS (1971, 1973, 1979, 1994), DIN (1982, 1990). For the purpose of such studies I use an enveloping curve as presented in Fig. 16.2. This approach provides conservative results regarding safety statements.
9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 0.0
Divided by the initial power ANS-5.1 ANS73 ANS79+1σ ANS79+2σ ANS94 2σ ANS94+1σ max. actt. DIN 25463 DIN 25463+2σ
0.5
1.0 1.5 2.0 Time in h
2.5
3.0
Relative decay power, %
2.00 Divided by the initial power
1.75 1.50 1.25 1.00 0.75 0.50 0
5 10 15 20 25 30 35 40 45 50 Time in h
Fig. 16.2. Normalized decay power as a function of time: (a) The curve used in this study is compared with the used standards ANS (1971, 1973, 1979, 1994), DIN (1982, 1990); (b) Long term
16.4. Model assumptions and brief model description
501
A core under water can not melt by residual decay heat. That is why core undergoing severe damage must be dry for a specific time. First low melting point materials (cladding, stainless steel, silver, indium, cadmium, boron) melt, and run away from the hot region axially under gravity within the available open spaces. The colder parties at the outer, and at the lower part of the core facilitate freezing, plugging and therefore crust formation. What remain behind are overheated uranium dioxide pellets, which are readily reduced to rubble. Without water supply the pellets will melt and form a pool of molten ceramic mixture. The strength of the crust determines the average superheat of the melt with respect to the oxide melting point. Due to possible break or melting of the crust, the molten ceramic pool is released downwards, filling the bottom of the vessel. It may happen 1 to 8 h or more after the initiation of the accident depending on the scenario. The analysis of the processes after this moment in the course of the accident will be discussed below. First I describe the behavior described verbally below is deduced from the computational analysis performed by Kolev (1993–2004): Decay heat inside the molten pool, reduced by about 25% due to the release of noble gases and volatile fission products, causes buoyancy driven convection which considerably improves heat transfer especially at the upper corner of the melt pool. Crust formation at the internal vessel wall interface is facilitated because the vessel-steel solidification temperature is lower than the corium solidification temperature. The remarkable role of the only heat conducting crust is that it increases the heat transfer resistance because ceramic melt have an order of magnitude lower thermal conductivity than steel. This influences the redistribution of the heat flux in a way that the downwards heat transfer is further reduced. This effect is not deducible from the experiments with pure liquids. The thermal radiation from the free surface of the liquid pool is absorbed by the remaining vessel internals above the liquid corium interface. The interfaces of these internals reach the melting temperature and start to melt. The continuously melted material streams downwards into the pool, forming a metallic layer above the ceramic pool. The lower solidification temperature of the metallic material streaming into the metallic layer compared to the oxide solidification temperature reduce radiation of the top of the pool due to the cooling effect of the mixing. The metal is good heat conductor. The excess of energy which come from the oxide pool and can not be removed by radiation is forced through the vessel wall causing considerable thermal attack of the vessel. This is called focusing effect. The problem is strongly nonlinear.
16.4. Model assumptions and brief model description The following assumptions are used to develop a mathematical model of the processes governing the external cooling:
502
16. External cooling of reactor vessels during severe accident
1. The oxide and the metallic pools, as presented in Fig. 16.3, posses temperatures being functions of time but not of space due to intensive intermixing during the strong turbulent natural circulation. 2. Part of the radiation energy emitted from the free surface of the pool is consumed for melting of the surrounding metallic structures. The so called metal rain is reintroduced into the metallic layer with its saturation properties. 3. The amount of the ablated vessel steel is neglected compared to the mass of the entire molten pool. 4. The heat conduction in the vessel wall is transient and symmetric with respect to the vessel axis. That is, transient two dimensional heat conduction in a structure presented in Fig. 16.1 represents good enough the process. 5. Heat transfer coefficients adjust simultaneously to their steady state values. The only exception that the inertia processes governing the heat transfer coefficient between the metallic pool and the vessel wall is approximately taken into account. 6. The transient redistribution of the averaged heat transfer coefficient at the vessel wall is assumed to preserve the form observed experimentally in the steady state experiments. 7. The crust formation instantaneously adjusts the heat flux conditions at the melt–vessel interface. Several simple geometrical relations are required to compute different geometrical characteristics that will be used in the empirical correlations used here. A summary of them is presented in Appendix 1. 16.4.1 Molten pool behavior In accordance with assumption (1), (2) and (3) the mass and energy balances governing the mass averaged corium and steel temperatures are M u = const
(16.1)
dM s ′′ Δhs = c5 Futop q& stop dτ
(16.2)
( ρc V )
u
(16.3)
( ρc V )
s
p
p
+
dTu ′′ = Q − Quda − Qucy − Futop q&utop dτ
(
)
dTs ′′ − q&stop ′′ − Qscy = Futop q&sda dτ
dM s ′′ − c4 q&stop ′′ ) − Qscy (hs′′′− hs ) = Futop (q&sda dτ
(16.4)
16.4. Model assumptions and brief model description
503
where c4 = ⎡⎣1 + c5 c ps (Ts − Ts′′′) Δh ⎤⎦ .
(16.5)
Ts − T ′′′ is the steel layer superheat with respect to the steel melting temperature.
Here c5 gives the part of the radiation consumed for melting of vessel internals. It is very complicated function of the heat transfer conditions through- and around the reactor structures. Here it is assumed to be of order of c5 ≈ 0.2 − 0.5 .
(16.6)
Obviously during melting of internals characterized with Ts′′′< Ts the metallic layer is additionally cooled. Sun discussed in Sun (1994, p. 290) the importance of the relocation of molten steel into the melt layer as additional cooling during the thermal attack of the vessel wall. 16.4.2 Two dimensional heat conduction through the vessel wall The Fourier equation without heat sources governs the heat transferred from the liquid metal into the surrounding coolant. I solve this equation numerically by standard implicit method in two dimensional (r, z) space, see Fig. 16.3. The corresponding integer indices are (i, k), where i ∈ Z ∩ 1 ≤ i ≤ imax , k ∈ Z ∩ 1 ≤ k ≤ kmax . Initially the cells with indices i = 2 to imax − 1 , and k = 2 to kmax − 1 are treated as inner cells with possible contact with the melt. The cells with indices I = 1 and imax , and k = 1, and kmax are auxiliary. These cells are used impose the boundary conditions. The cell sizes are Δr = Δrmax /(imax − 2) , Δz = Δzmax /(kmax − 2) . I write the implicit discretized form for a computational network. For convenience of notation I omit space indices unless they differ from (i, k). Subscript “a” denotes the old time level. Solving with respect to the temperature T, I obtain T=
ρ c pTa / Δτ + b2Ti −1 + b1Ti +1 + b6Tk −1 + b5Tk +1 ρ c p / Δτ + b2 + b1 + b6 + b5
(16.7)
which is a simple formula for applying the point Gauss–Seidel iteration method to solve the resulting system of algebraic equations. Here b2 = λ2 / Δr 2 , b1 = λ1 / Δr 2 , b6 = λ6 / Δz 2 , b5 = λ5 / Δz 2 are computed at the beginning of the iteration process. Usually up to 9 outer iterations are sufficient to obtain accurate solution. The boundary cells are treated separately as follows.
504
16. External cooling of reactor vessels during severe accident 2D heat conduction thermal isolation
nz
bubbles
natural convection steel layers radiation
Tmelt
Tw,i
nucleate boiling
crust
Tw,o
T water
natural convection pool
natural two-phase convection
structures
nr
Fig. 16.3. Melt-vessel interaction during external cooling
16.4.3 Boundary conditions For the inner wall I impose the heat flux as a boundary condition: T=
ρ c pTa / Δτ + q&1′′ / Δr + b1Ti +1 + b6Tk −1 + b5Tk +1 ρ c p / Δτ + b1 + b6 + b5
(16.8)
Here Ti–1 is no more used, and b2 = 0. In case of crust formation into the oxide pool q&1′′ = q&ui′′ + q& ′′′δ u .
(16.9)
For the outer wall at the coolant side
{
}
ρ c p (T − Ta ) Δτ = (1 Δr ) λ2 (Ti −1 − T ) Δr − ⎡⎣1 (1 hnc + 0.5Δr λ1 ) ⎤⎦ (T − Tbc ) + (1 Δr ) ⎡⎣λ6 (Tk −1 − T ) Δz − λ5 (T − Tk +1 ) Δz ⎤⎦
(16.10)
16.4. Model assumptions and brief model description
505
which modified the main equation as follows Ti +1 = Tbc , and
b1 = 1 ⎡⎣(1/ hnc + 0.5Δr λ1 ) Δr ⎤⎦ . The formalism for description of the crust formation is presented in the previous chapter where I discuss the coolability of layers of molten materials. Whether the numerical solution of the Fourier equation is acceptable can be checked by comparison with the available analytical solution for the following 1D-problem. Four important analytical solutions for one-dimensional transient heat conduction in a solid are: Problem 1: constant surface temperature T ( 0,
) = Tw
for T ( z , 0 ) = T0 :
Infinite slab: T ( z ,τ ) − Tw T0 − Tw
q&w′′ =
⎛ z ⎞ = erf ⎜ ⎟, ⎝ 2 aτ ⎠
λ (T0 − Tw ) π aτ
,
Incopera and DeWitt (2002, p. 270). Finite slab: T ( z ,τ ) − T0 Tw − T0
=
z zmax
+
∞
cos ( nπ )
n =1
n
2
∑ π
⎛ an 2π 2τ ⎞ ⎛ z ⎞ sin ⎜ nπ ⎟, ⎟ exp ⎜ − 2 zmax ⎠ zmax ⎠ ⎝ ⎝
Carslaw and Jaeger (1996). Problem 2: constant surface heat flux q&w′′ = q&0′′ for T ( z , 0 ) = T0 :
λ ⎣⎡T ( z,τ ) − T ⎦⎤ 0 2q&0′′
⎛ z2 ⎞ z ⎛ z ⎞ = aτ π exp ⎜ − ⎟ − erfc ⎜ ⎟, ⎝ 2 aτ ⎠ ⎝ 4aτ ⎠ 2
Carslaw and Jaeger (1996).
506
16. External cooling of reactor vessels during severe accident
Problem 3: surface convection − λ
dT = h ⎡⎣Tcoolant − T ( 0, τ ) ⎤⎦ for T ( z , 0 ) = T0 : dz
T ( z ,τ ) − T0 Tcoolant
⎛ hz h 2 aτ ⎛ z ⎞ = erfc ⎜ − exp ⎜ + 2 ⎟ λ − T0 ⎝ 2 aτ ⎠ ⎝λ
⎛ z ⎞ h aτ ⎞ + ⎟, ⎟ erfc ⎜⎜ λ ⎟⎠ ⎠ ⎝ 2 aτ
Incopera and DeWitt (2002, p. 270). Here the Gaussian error function and the complementary error function is
(
erf (η ) = 2 π 1/ 2
η
) ∫ exp ( −u ) du , 2
0
erfc (η ) = 1 − erf (η ) ,
(
)
respectively, a = λ ρ c p is the temperature conductivity and h is the heat transfer coefficient. Incopera and DeWitt (2002) reported on p. 187 interesting two-dimensional solution of the steady Fourier equation for the following conditions: Given a rectangle in (r, z) geometry with sizes Δr and Δz . The initial temperature at the all three sites except the upper site is T0 . The upper site has the temperature T1 . The temperature inside the rectangle is then T ( r , z ,τ ) − T0 T1 − T0
=
2
∞
∑ π
( −1)
n
n
n =1
+1
sin
nπ r sinh ( nπ z Δr ) . Δr sinh ( nπ Δz Δr )
16.4.4 Total heat flow from the pools into the vessel wall The number of cells contacting oxide melt is ku = int(ΔZ u / Δz ) . Similarly the number of cells contacting the molten metal layer is k s = int(ΔZ s / Δz ) . The overall heat flow removed from the oxide pool into the wall is Quda + Qucy =
ku +1
∑ q& ′′ΔF ( z k =2
1
k
)
(16.11)
where the interface ΔF ( zk ) = 2π R ( zk )Δz . Note that, the dependence of the horizontal radius measured from the symmetry axis on the position z is different for semi-sphere and semi-toroid.
16.4. Model assumptions and brief model description
507
The heat flow removed from the metallic pool into the wall is Qscy = π Dv Δz
ku + ks +1
∑
k = ku + 2
q&1′′ .
(16.12)
The heat transferred into the cooling water is Qw =
kmax −1
∑ q& ′′ΔF ( z k =2
2
k
),
(16.13)
1 ⎛ ⎞ where q&2′′ = hw (Ti max −1 − Ti max ) and hw = 1 ⎜1/ hw, nc + Δr / λv ⎟ . Here hw, nc is the 2 ⎝ ⎠ heat transfer coefficient at the external vessel wall depending on local conditions.
16.4.5 Vessel wall ablation The melting of the vessel wall absorbs thermal energy. I take this into account simply modifying the specific heat at constant pressure within the margin, Ts′′′< T < Ts′′′+ 50 , c p = Δh / 50 , where Δh is the latent heat of solidification. The vessel wall is considered as melted if T is getting larger than Ts′′′+ 50 . In this case it is assumed that the melting front is propagating one cell into the vessel. The cell temperature is set equal to the averaged pool temperature at that level allowing the new boundary to fill the buoyancy driven convection in the pool. This technique is somewhat different to those described by Sun (1994). Thus the ablation of the vessel wall is taken into account in reducing the thermal resistance for the heat into the coolant direction. The material relocation is therefore m& = Δr Δzπ Dvessel ρ s / Δτ & psT . Neglecting this contribution to and the energy transferred into the pool is mc the mass and energy balance of the steel pool is conservative in respect of the estimation of the average temperature of the steel pool. Problem 4: Given the slab as in Problem 1. The temperature at the one surface of the slab is set to T1 at time 0. This temperature is higher then the solidus–liquidus temperature T ′′′ . Consider only one dimensional heat conduction along the axis of the slab. Compute the evolution of the temperature as a function of the time and of the axial coordinate and the propagation of the melting front. Solution: The solution to this problem is documented in Carslaw and Jaeger (1996, Eqs. (24,25,27), pp. 287–288): The position of the melting front is x (τ ) = 2γ aliqτ
508
16. External cooling of reactor vessels during severe accident
with γ being the solution of the following transcendental equation
(
exp −γ
2
erfc ( γ )
)−λ
sol
aliq
λliq
asol
aliq ⎞ ⎛ exp ⎜ −γ 2 ⎟ asol ⎠ Δhmelt T ′′′ ⎝ −γ π =0. T1 − T ′′′ c p ,liq (T1 − T ′′′ ) ⎛ aliq ⎞ erfc ⎜ γ ⎟ ⎝ asol ⎠
The evolution of the temperature profile in the remaining solid part is then ⎛ erfc ⎜ ⎜2 T ( x, τ ) ⎝ = T1 ⎛ erfc ⎜ γ ⎜ ⎝
x asolτ aliq asol
⎞ ⎟ ⎟ ⎠. ⎞ ⎟ ⎟ ⎠
Here the subscripts liq and sol means liquid and solid, respectively. T ′′′ is the solidification temperature, c p ,liq is the specific capacity at constant pressure and Δhmelt the specific meting enthalpy.
16.4.6 Heat fluxes and crust formation The heat fluxes are computed on the basic of the heat transfer coefficients and the corresponding temperature differences. The equality of the heat fluxes of the both sides at the vessel wall interface together with the temperature profiles gives the condition for estimation of the interface temperature and for possible crust formation. Similarly the interface temperature and the crust formation are estimated at the interface between oxide and metal pool. The free surface of the interface radiates heat. Again the equality of the heat fluxes of the both sides of the surface gives the interface temperature and the possible conditions for crust formation. For carbon steel and pure molten iron the emissivity is about 0.42–0.45, VDIWärmeatlas (1991), at the temperatures in question. The effect of zirconium, chrome, and nickel, as well as possible oxidation and other impurities in the layer, are presently unknown, however, it is expected that impurities would increase emissivity. Theophanous et al. (1994b) considered the value between 0.5 and 0.6 as most likely. For molten uranium oxides the emmisivity is about 0.79, VDIWärmeatlas (1991), at the temperature in questions.
16.4. Model assumptions and brief model description
509
16.4.7 Buoyancy convection Next I present a detailed discussion for the choice of the heat transfer correlation used in the model followed by a summary. 16.4.7.1 Buoyancy convection – steel layer Consider cylindrical steel layer heated from below, and cooled from the top and the side. There are no simple correlations for computing the buoyancy driven convection for this case. Direct numerical simulation is not available for Rayleigh numbers greater than 106, see the review presented by Wörner (1994), because of the required high resolution of the discretization scheme for the boundary layer. The very expensive large eddy simulation is possible only if the boundary layer is specially treated by empirical correlations for heat transfer. In other words, even in large eddy numerical simulation the empirical treatment of the boundary layer heat transfer still remains. That is why the only tractable way to estimate the problem cost effectively is to use engineering methods based on empirical correlation’s as far as possible. Vertical surface: Theofanous et al. (1994a) proposed to use the existing solution for buoyancy driven convection on vertical surface having temperature difference from those of the fluid. Text book solutions are available by Jaluria (1983, p. 324) for the averaged Nusselt numbers as a function of Rayleigh number based on the vertical wall high and bulk – wall temperature difference. For 104 < Ra < 109 Nu = 0.59 Ra1/ 4
(16.14)
and for 109 < Ra < 1013 Nu = 0.10 Ra1/ 3 .
(16.15)
For comparison see in the Russian literature, Pchelkin (1960) for 103 < Ra < 109 Nu = 0.60 Ra1/ 4 (Pr/ Prw )1/ 4 ,
(16.16)
and for Ra > 6.1010 ,
Nu = 0.15 Ra1/ 3 (Pr/ Prw )1/ 4 and Churchill and Chu (1975)
(16.17)
510
16. External cooling of reactor vessels during severe accident
⎧ ⎫ ⎪ ⎪ 0.387 Ra1/ 6 Nu = ⎨0.825 + 8 / 27 ⎬ 9 /16 ⎡1 + 0.492 Pr ) ⎤ ⎪ ⎪ ⎣ ( ⎦ ⎩ ⎭
2
(16.18)
valid for 0.1 < Ra < 1012 and any Pr-numbers. The Churchill and Chu (1975) correlation is used mostly because of the wide region of its validity. If the equation is applied for cases with constant heat flux the constant 0.492 in the denominator have to be replaced by 0.437, Baehr and Stephan (2004, p. 426). Halle et al. (1999) provided data from the COPO II experiment with frozen upper surface and recommended the Churchill and Chu (1975) correlation for vertical surface. For slightly inclined surface the Churchill and Chu (1975) gives larger values then the experimentally observed. For the narrow range of Ra < 109 the modification Nu = 0.68 +
0.670 Ra1/ 4 ⎡1 + ( 0.492 Pr )9 /16 ⎤ ⎣ ⎦
4/9
,
Churchill and Chu (1975), gives slightly better accuracy. Note the relation of the averaged to the local coefficient at z = H: for the laminar convection (4/3) times-, and for turbulent convection (5/4) times, the local value at the upper end of the plate, respectively. Theofanous et al. (1994c) used instead of Eq. (16.15) the following equation, Nu = 0.076 Ra1/ 3 ,
(16.19)
which is valid for Pr = 0.13. There is obviously a problem with this approach because buoyancy driven convection inside the cylindrical metallic pool resembles more convection in enclosures than free convection at vertical wall. Correlations are available for enclosures for the two limiting cases only: (a) heating from below, as will be discussed later, and (b) heating and cooling at the both vertical surfaces. Here we have rather combination between these two cases with the cylinder axis being adiabatic boundary. But the form of the correlation for case (b) Nu = 0.25( H / L)1/ 7 Ra 2 / 7 ,
(16.20)
which is in fact Eq. (75) by Bejan (1984, p. 193) in modified notation, reflects the limiting influence of the cylinder high in the multiplier ( H / L)1/ 7 . That is why I recommend multiplying the correlation for vertical wall by
16.4. Model assumptions and brief model description
f1 = ( H / L)1/ 7 ,
511
(16.21)
where L = R is the radius of the cylinder. Mayinger et al. (1976) reported that the profile of the heat flux in cylindrical enclosures reduces at the upper and lower part of the cylinder; see Fig. 4.52 in Mayinger et al. (1976). I take this observation into account by reducing the heat flux in the upper and lower 10% of the height up to 10% of the averaged value. The next discussion point is whether we are allowed to use steady state correlation for the time scale analyzed here which is e.g. 2 h. The thermal boundary layer thickness for laminar convection is of order of
δ T = H ( 0.59 Ra1/ 4 ) ,
(16.22)
compare with Bejan (1984). The time necessary for the penetration of the thermal boundary layer up to δ T* is obtained from the averaged analytical solution of the Fourier equation q& ′′ = 2 ⎡⎣ λρ c p
(πΔτ T )⎤⎦
1/ 2
ΔT ,
δ T* ≈ 0.886 ( aΔτ ) , 1/ 2
(16.23) (16.24)
and is
(
)
Δτ T ≈ 3.66 H 2 aRa1/ 2 .
(16.25)
Thus, the application of steady state correlation’s for transient analysis is allowed only if the characteristic time scale of the process is larger than Δτ T . This presents no restriction to the application of the correlation due to the very large Ra numbers except at the very beginning of the process. That is why I reduce the heat transfer by f 2 = δ T* / δ T
for δ T* < δ T
(16.26)
and use f 2 = 1 for δ T* ≥ δ T ,
(16.27)
512
16. External cooling of reactor vessels during severe accident
where the time τ used to compute the thermal penetration is counted since the beginning of the process considered, that is after the end of the melt relocation into the lower head. The next discussion point leading to reduction of the conservatism is the observation by Jones et al. (1976) that axis-symmetric convection in liquid metal cylinder resembles single cell “flywheel convection” which means that the establishing of the single steady state cell will take finite time. Simple inclusion of this delay time is introduced by solving the simplified momentum equation in which the difference between the buoyancy and viscous forces is the driving force for the acceleration dV / dτ = (V∞ − V ) / Δτν ,
(16.28)
where the time constant is Δτν ≈ (δ 2 /ν ) /12,
(16.29)
and the steady state velocity V∞ ≈ g βΔT Δτν .
(16.30)
The scale of δ is of order of R/2. The analytical solution for the initial condition τ = 0 , V = 0 is
(
)
f 3 = V V∞ = 1 − exp − τ Δτ ν .
(16.31)
Thus I reduce the heat transfer coefficient as discussed above by multiplying with f3 . Summary of the available correlation for heat transfer at the side wall is presented in Table 16.1.
Table 16.1. Summary of correlation for predicting the metal side wall heat transfer
Author
Nucy
Ra
Jaluria (1983)
0.59Ra1/ 4
104 < Ra < 109
0.10Ra1/ 3
109 < Ra < 1013 0.1 < Ra < 1012
Churchill and Chu (1975)
⎧ ⎫ ⎪ ⎪ 0.387 Ra1/ 6 ⎨0.825 + 8 / 27 ⎬ 9 /16 ⎡1 + ( 0.492 Pr ) ⎤ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭
2
Pr
any Pr
16.4. Model assumptions and brief model description
Author Pchelkin (1960)
1/ 4
0.60 Ra
Theofanous et al. (1994c)
Pr
Ra
Nucy 1/ 4
(Pr/ Prw )
10 < Ra < 10 3
513
9
0.15Ra1/ 3 (Pr/ Prw )1/ 4
Ra > 6.1010
0.076Ra1/ 3
109 < Ra < 1013
Pr = 0.13
Two parallel horizontal surfaces: Buoyancy convection in fluid between two parallel, horizontal surfaces with different temperatures has been studded in detail since the pioneer works by Benard (1900) and Rayleigh (1916). Jeffreys (1926a,b) found that if Ra ≥ 1108
(16.32)
the onset of cellular natural convection, called Benard convection, is possible. For low Grashoff numbers, Gr < 1700, heat conduction only transfers energy Nuup = Nuda = 1 . For larger Grashoff number and air Jacob (1949) obtained in 1949 for 104 < Gr < 4.105 Nu = 0.195Gr1/ 4 ,
(16.33)
and for 4.104 < Gr , Nu = 0.068Gr1/ 3 .
(16.34)
It was Jacob who discovered that buoyancy driven turbulent convection between two parallel plates does not depend on the distance between the planes. For very large margin of Prandtl number, 0.02 to 8750 and Ra = 3.105 to 7.109 Globe and Dopkin (1959) modified the Jacob correlation to
(
)
Nu = max 1, 0.069Gr1/ 3 Pr 0.074 .
(16.35)
Halle et al. (1999) provided data from the COPO II experiment with frozen upper surface and recommended the Globe and Dopkin (1959) modification. Summary of the available correlation for heat transfer between two parallel surfaces is presented in Table 16.2 for layer without internal heat release.
514
16. External cooling of reactor vessels during severe accident
Table 16.2. Summary of correlation for predicting the metal top and bottom wall heat transfer
Author
Nu Nuup = Nuda = 1
Ra Gr < 1700
Jacob (1949)
Nu = 0.195Gr1/ 4
104 < Gr < 4.105
Nu = 0.068Gr1/ 3 Nu = 0.069Gr1/ 3 Pr 0.074
4.104 < Gr 3 × 105 < Ra < 7 × 109
Globe and Dopkin (1959)
Pr
0.02< Pr < 8750
16.4.7.2 Buoyancy convection – cavity with internal heat sources The buoyancy convection in fluid with internal heat sources in a closed rectangular and semi-circular two dimensional slide cavity was numerically and experimentally (holographic interferometry) analyzed by Jahn and Reineke (1974). For heat transfer in rectangular cavity Gr Da = 105 to 108 and Pr = 1 to 80 the authors obtained for the upper wall Nuup = 0.78Ra ′0.2 ,
(16.36)
and for the lower wall
Nuda = 2.14 Ra ′0.1 .
(16.37)
For the heat transfer in the semi-circular (slice) cavity RaDa = 107 to 1011 and Pr = 7 the authors obtained
Nuda = 0.6 Ra ′0.2 .
(16.38)
The local Nusselt number at ϕ ≈ 45o was observed to be equal to the averaged Nusselt number. The maximum for the local Nusselt number, Nuda ,max ≈ 2 Nuda ,
(16.39)
was observed at ϕmax ≈ 90o and the minimum, Nuda ,min ≈ 0.25 Nuda at ϕ ≈ 0o . The analytical expression proposed by the authors that describes this distribution is q& ′′ = q& ′′k ⎡⎣1.07 − 0.95cos (π ϕ ϕmax ) ⎤⎦
(16.40)
where k = 1. Mayinger et al. (1976) analyzed the dependency of Eq. (16.38) on the ratio ΔZ / Req in 1976 using ΔZ / Req = 0.4, 0.6, 1. The authors found
16.4. Model assumptions and brief model description
Nuda = 0.55 Ra ′0.2
515
(16.41)
which is not dependent on the filling of the cavity. This finding was confirmed by Franz and Dhir (1992) for the same ratios. Kymäläinen et al. (1993) reported that the Mayinger’s et al. (1976) equation in the form Nuda = 0.54 Ra ′0.18 ( H R )
0.26
(16.42)
slightly under predicts their data for 1014 < Ra ′ < 1015 . Bonnet (1998) extended the validity Mayinger’s et al. (1976) equation up to 1013 < Ra ′ < 1017 by comparing with the data obtained from the BALI experiment Nuda = 0.116 Ra ′0.25 ( H R ) Nuda = 0.131Ra ′0.25 ( H R )
0.32
0.19
for 2D BALI geometry,
(16.43)
for 3D ACOPO geometry.
(16.44)
Kymäläinen et al. (1993) extended the validity of Eq. (16.41) in 1993 up to Ra' < 1.8 × 1015 and supported experimentally once again the discovered by Jahn and Reineke (1974) dependence of the local heat transfer on ϕ . Summary of the data of the both works, Kymäläinen et al. (1993) and Franz and Dhir (1992), is presented in Fig. 10 by Theofanous et al. (1994b). Note that in Theofanous et al. (1994c, pp. 5–11), for the analysis of AP-600 Eq. (16.41) was replaced as follows
Nuda = 0.048 Ra ′0.27 for 1012 < Ra ′ < 3.1013
(16.45)
and
Nuda = 0.0038Ra′0.35 for 3.1013 < Ra ′ < 7.1014
(16.46)
without any dependence on Pr number checked in the region of 2.5 < Pr < 11. Thus Theofanous et al. (1994c) considered the last correlations as the upper bound and Eq. (16.41) as the lower bound for the analytical modeling – see Fig. 5.7 in Theofanous et al. (1994c). For the same geometry Steinberner and Reineke found in 1978 that the transition from laminar to turbulent natural convection appears at Ra ≈ 108 . For the turbulent region the authors correlated their results for modified Rayleigh number 107 < Ra ′ < 7.3 × 1013 , Pr ≈ 7 as follows: For the upper surface Nuup = 0.345 Ra ′0.233
and for the vertical wall, Ra' up to 1014,
(16.47)
516
16. External cooling of reactor vessels during severe accident
Nucy = 0.85 Ra ′0.19 .
(16.48)
Experiments with Joule heated water performed by Kymäläinen, Hongisto and Pessa (1993) indicated that Eq. 47 under predicts the data and Eq. 48 provides a good agreement in the region 1014 < Ra ′ < 1015 . Bonnet (1998) extended the validity of Eq. 47 up to Ra ′ < 1017 by comparing with the BALI experiment. The slight modification introduced by Bonnet is Nuup = 0.383Ra ′0.233 .
(16.49)
Bonnet commented the difference between his 2D and 3D experiments by Theofanous et al. (1996b) ACOPO (1/2th scale), and emphasized that 3D experiments gives lower heat transfer coefficients then 2D experiments. Bonnet derived the following correlation from the 3D experiments Nuup = 1.95 Ra ′0.18 ,
(16.50)
Nuda = 0.3Ra ′0.22 .
(16.51)
Theofanous et al. (1996b) derived from the ACOPO (1/2th scale) experiment Nuup = 2.4415 Ra ′0.1772 .
(16.52)
Note that Eq. (16.47) is in fact almost the same as the correlation proposed by Kulaki and Emara (1975), Nuup = 0.34 Ra ′0.226
(16.53)
for 2.10 4 < Ra ′ < 1012 , Pr ≈ 7 . Eq. (16.45) was found to be in excellent agreement, see Theofanous et al. (1994c) App. B, in the region of Ra ′ < 1.51015 , Pr ≈ 2.5 but for Ra ′ ≥ 6.1014 under predict the measurements by 30 %. To clarify this point experimental results are provided in Theofanous et al. (1994c) App. D, showing that Eq. (16.45) can be extrapolated up to 1012 < Ra ′ < 7.1014 , 2.6 < Pr < 10.8 . Kymäläinen et al. (1993) found that the Eq. (16.45) underestimate data by approximately 10%. Mayinger et al. (1976) recognized that for isothermal boundaries the overall energy balance for the pool gives simply the driving temperature difference
(
ΔT = Q Futop hutop + Fuda huda
)
(16.54)
16.4. Model assumptions and brief model description
517
and therefore Q is partitioned as Qutop = ζ Q ,
(16.55)
and Quda = (1 − ζ )Q
(16.56)
where
ζ = Futp Nuutop
(F
utp
)
Nuutop + Fuda Nuuda .
(16.57)
Note that for the cases where no crust formation is expected and therefore the assumption isothermal boundary is violated this simple approach is not applicable. Bonnet (1998) reported from experiments with 2D geometry with H/R = ¼, ½, ¾ and 1, ζ ≈ 0.64, 0.56, 0.51 and 0.44, respectively. Summary of the available correlations for heat transfer at the upper horizontal surface of a cavity filled liquid with internal heat release is presented in Table 16.3. Table 16.3. Summary of correlation for predicting the ceramic top heat transfer
Ra ′
Pr
Author
Nuup
Jahn and Reineke (1974), rectangular cavity Kulaki and Emara (1975) Kymäläinen et al. (1993)
0.78Ra ′0.2
Gr Da = 10 to 10
0.34Ra ′0.226
2.10 4 < Ra ′ < 1012
Pr ≈ 7
Ra ′ < 1.51015
Pr ≈ 2.5
Theofanous et al. (1994c) App. B Theofanous et al. (1994c) App. B Theofanous et al. (1994c) App. D Steinberner and Reineke (1978)
Kulacki and Emara (1975), 10% under prediction Kulacki and Emara (1975) Kulacki and Emara (1975), 30% under prediction Kulacki and Emara (1975) 0.345Ra ′0.233
5
8
Pr = 1 to 80
Ra ′ ≥ 6 × 1014
1012 < Ra ′ < 7 × 1014
2.6 < Pr < 10.8
107 < Ra ′ < 7.3 × 1013 Pr ≈ 7
518
16. External cooling of reactor vessels during severe accident
Author
Nuup
Theofanous et al. (1996b) Mini ACOPO (1/8th scale): Theofanous et al. (1996b), ACOPO 1/2th scale experiment Bonnet (1998), BALI experiment Bonnet (1998), 3D experiments Kelkar et al. see in Turland et al. (1999)
0.345Ra ′
Ra ′
Pr
10 < Ra ′ < 7 × 10
0.233
12
14
2.6 < Pr < 10.8
2.4415Ra ′0.1772
0.383Ra ′0.233
Ra ′ < 1017
1.95Ra ′0.18
1.18Ra ′0.237
108 < Ra ′ < 7 × 1016
1
Summary of the available correlations for surface averaged heat transfer at the dawn words oriented surface of a cavity filled liquid with internal heat release is presented in Table 16.4. Table 16.4. Summary of correlation for predicting the ceramic bottom averaged heat transfer
Author
Nuda
Ra ′
Pr
Jahn and Reineke (1974), semi-circular slice cavity Jahn and Reineke (1974), rectangular cavity Mayinger et al. (1976) Mayinger et al. (1976) Kymäläinen et al. (1993) Theofanous et al. (1994c), AP600
0.6Ra ′0.2
RaDa = 107 to 1011
Pr = 7
2.14Ra ′0.1
Gr Da = 105 to 108
Pr = 1 to 80
0.55Ra ′0.2
7 × 106 < Ra ′ < 5 × 101
Asfa et al. (1996)
0.54 Ra ′0.18 ( H R )
0.26
0.54Ra ′0.18 ( H R )
0.26
1014 < Ra ′ < 1015
0.048Ra ′0.27
1012 < Ra ′ < 3.1013
2.5 < Pr < 11
0.0038Ra′0.35
3.1013 < Ra ′ < 7.1014
2.5 < Pr < 11
0.54 Ra ′0.2 ( H R )
0.25
2 × 1010 < Ra < 1.1× 1014 8.2 < Pr < 9.5
16.4. Model assumptions and brief model description
Author
Nuda
Ra ′
Theofanous et al. (Sept. 1996b) Mini ACOPO Theofanous et al. (1996b) ACOPO (1/2th scale): Bonnet (1998), 2D BALI geometry Bonnet (1998), 3D ACOPO geometry Bonnet (1998), 3D experiments Kelkar et al. see in Turland et al. (1999)
0.0038Ra ′0.35 ( H R )
10 < Ra < 7 × 10
0.1857Ra ′0.2304 ( H R )
0.116 Ra ′0.25 ( H R ) 0.131Ra ′0.25 ( H R )
Pr
12
0.25
0.2
519
2.6 < Pr < 10.8
14
1012 < Ra < 2 × 1016
0.32
1013 < Ra ′ < 1017
0.19
1013 < Ra ′ < 1017
0.3Ra ′0.22 1.1Ra ′0.25
108 < Ra ′ < 7 × 1016
1
Summary of the available correlations for heat transfer at the vertically oriented surface of a cavity filled liquid with internal heat release is presented in Table 16.5. Table 16.5. Summary of correlation for predicting the ceramic side averaged heat transfer
Ra ′
Pr
Author
Nucy
Steinberner and Reineke (1978) Kymäläinen et al. (1993)
0.85Ra ′0.19
Ra' < 10
Steinberner and Reineke (1978)
1014 < Ra ′ < 1015
14
16.4.7.3 Redistribution of the averaged heat flux at the lower head The Eq. (16.40) can not be applied to the reactor lower head geometry directly. Additional normalizing is necessary to ensure that 1 Fmax
Fmax
∫
q& ′′dF =q& ′′ ,
(16.58)
0
which is the definition equation for estimation of the geometry dependent factor k. For semi spherical bottom with ϕ max = 1.1344 , k = 0.786 . Experimental data of real hemispherical geometry presented in Theofanous et al. (1994c), see
520
16. External cooling of reactor vessels during severe accident
Fig. 5.8, for Freon-113 and 7 < Pr < 11, 2.1013< Ra' < 7x1014 and water 2 < Pr < 10, 1011 < Ra' < 3x1012 shows a distribution Nuda (ϕ ) Nuda = 0.1 + 1.08 (ϕ ϕmax ) − 4.5 (ϕ ϕ max ) + 8.6 (ϕ ϕmax ) 2
3
(16.59)
for 0 < ϕ / ϕ max < 0.6 , and Nuda (ϕ ) Nuda = 0.41 + 0.35 (ϕ ϕ max ) + (ϕ ϕ max )
2
(16.60)
for 0.6 < ϕ / ϕ max < 1. In the analysis performed here I use the above distribution. For lower heads having geometries different from the semi spherical it is more convenient to use instead the angular co-ordinate, the arc distance s. For elliptic semi thoroidal bottom with half axis a = 2.2675 and c = 1.317 and smax = 2.816, I obtain k = 0.694, which is characteristic for the Russian design of VVER-1000. Summary of the available correlations for the redistribution of the surface averaged heat transfer at the dawn words oriented surface of a cavity filled liquid with internal heat release is presented in Table 16.6. Table 16.6. Summary of correlation for predicting the distribution of the ceramic bottom averaged heat transfer
Nuda (ϕ ) Nuda
ϕ / ϕ max
Jahn and Reineke (1974)
⎡ ⎛ πϕ k ⎢1.07 − 0.95cos ⎜ ⎢⎣ ⎝ ϕ max
⎞⎤ ⎟⎥ , k = 1 ⎠ ⎥⎦
Theofanous et al. (1994c) Mini-ACOPO, hemispherical geometry, Freon-113, 7 < Pr < 11, 2.1013< Ra' < 7.1014 and water 2 < Pr < 10, 1011 < Ra' < 3.1012
⎛ ϕ ⎞ ⎛ ϕ ⎞ ϕ − 4.5 ⎜ 0.1 + 1.08 ⎟ + 8.6 ⎜ ⎟ ϕ max ⎝ ϕ max ⎠ ⎝ ϕ max ⎠
2
2
3
⎛ ϕ ⎞ ϕ +⎜ 0.41 + 0.35 ⎟ , ϕ max = 1.1344 ϕ max ⎝ ϕ max ⎠
0 < ϕ / ϕ max < 0.6
0.6 < ϕ / ϕ max < 1
16.4. Model assumptions and brief model description
Nuda (ϕ ) Nuda
Park and Dhir (1992)
b1 sin 2 ϕ + b2 , b1 =
521
ϕ / ϕ max 9.12 (1 − cos ϕ0 ) 8 − 9 cos ϕ0 + cos 3ϕ0
,
b2 = 0.24
Asfa and Dhir (1994)
0.25 + 0.55sin ϕ
0 < ϕ / ϕ max < 0.65
C1 sin Φ − C2 cos Φ ,
0.65 < ϕ / ϕ max < 1
C1 = 2.55 − 1.55cos Φ , C2 = 3.6 − 2.3cos Φ , Φ = 0.5π ϕ ϕ max
Asfa et al. (1996)
C1 sin 4 Φ − C2 cos Φ C1 = 1.06 − 0.31cos Φ , C2 = 1.15 + 0.24 cos Φ
0 < ϕ / ϕ max < 0.73
C1 sin Φ − C2 cos Φ
0.73 < ϕ / ϕmax < 1
C1 = 2.6 − 1.2 cos Φ , C2 = 3.6 − 2.65cos Φ , Φ = 0.5π ϕ ϕ max
Dombrovskii et al. (1998)
Nuda = BRa ′1/ 4 , B=
χ⎞ ⎛ 0.33 ⎜ sin ⎟ 2⎠ ⎝
3/ 4
, 5/ 2 χ⎞ 0.18 ⎛ 1+ ⎜ sin 2 ⎟ Pr ⎝ ⎠ χ angle between the downwards tangent and the gravity force vector 16.4.7.5 Comparison of the heat transfer coefficient used in different lumped parameter models The constitutive set of empirical heat transfer correlation is used in the same form with some variation as given in Table 16.7 from several authors Esmaili and Khatib-Rahbar (2004) reported a comparison of a prediction of heat fluxes for AP1000 by using their own set and the sets of Theofanous et al. (1996a) and Rempe et al. (1997). The results are very close to each other. Table 16.7. Heat transfer coefficient used in different lumped parameter models
1. Constitutive correlation set used by Kolev (1993) in a transient model: Metal Side wall: Jaluria (1983) multiplied by f1 f 2 f 3 : Nu = 0.59 Ra1/ 4 for 104 < Ra < 109 and
522
16. External cooling of reactor vessels during severe accident
Nu = 0.10 Ra1/ 3 for 109 < Ra < 1013 . The heat flux in the upper and lower 10% of the height is reduced linearly from the averaged value up to 10% from the averaged value at the both corners
Top and bottom: Gr < 1700: Nuup = Nuda = 1 . Jacob (1949): Nu = 0.195Gr1/ 4 for 104 < Gr < 4.105 and
Nu = 0.068Gr1/ 3
(
Nu = max 1, 0.069Gr
for 1/ 3
Pr
0.074
4.104 < Gr
)
or
Globe
and
Dopkin
(1959)
for 3 × 10 < Ra < 7 × 10 and 0.02 < Pr < 8750 . 5
9
Ceramic Top: Steinberner and Reineke (1978): Nuup = 0.345 Ra ′0.233 for 107 < Ra ′ < 7.31013 , Pr ≈ 7 .
Bottom: Mayinger et al. (1976) : Nuda = 0.55Ra ′0.2 for 7 × 106 < Ra < 5 × 1014 with Jahn and Reineke (1974), q& ′′ = q& ′′k ⎡⎣1.07 − 0.95cos (π ϕ ϕmax ) ⎤⎦ , k = 1. or Jahn and Reineke (1974): Nuda = 0.6 Ra ′0.2 for RaDa = 107 to 1011 and Pr = 7 with Theofanous et al. (1994c) Mini-ACOPO: Nuda (ϕ ) Nuda = 0.1 + 1.08 (ϕ ϕ max ) − 4.5 (ϕ ϕmax ) + 8.6 (ϕ ϕ max ) 2
0 < ϕ / ϕ max < 0.6 , and
3
Nuda (ϕ ) Nuda = 0.41 + 0.35 (ϕ ϕ max ) + (ϕ ϕ max )
for 2
for
0.6 < ϕ / ϕ max < 1 , ϕ max = 1.1344 , hemispherical geometry, Freon-113, 7 < Pr < 11, 2.1013< Ra' < 7.1014 and Water 2 < Pr < 10, 1011 < Ra' < 3.1012 .
Side wall: Steinberner and Reineke (1978): Nucy = 0.85 Ra ′0.19 for Ra' < 1014. 2. Constitutive correlation set used by Theofanous et al. (1996a) in a steady state model Metal Side wall: Churchill and Chu (1975) Nu = 0.076 Ra1/ 3 .
16.4. Model assumptions and brief model description
523
Top and bottom: Globe and Dopkin (1959) modified to Nu = 0.15Gr1/ 3 for 3 × 105 < Ra < 7 × 109 and 0.02 < Pr < 8750 . Ceramic Top: Theofanous et al. (Sept. 1996b) Mini ACOPO (1/8th scale): Nuup = 0.345 Ra ′0.233 12 14 for 10 < Ra < 7 × 10 and 2.6 < Pr < 10.8 .
Bottom: ⎛H⎞ Theofanous et al. (Sept. 1996b) Mini ACOPO: Nuda = 0.0038Ra ′0.35 ⎜ ⎟ ⎝R⎠ 12 14 10 < Ra < 7 × 10 and 2.6 < Pr < 10.8 with Mini-ACOPO distribution.
0.25
for
Side wall: – 3. Constitutive correlation set used by Rempe et al. (1997) in a steady state model Metal Side wall: Churchill
and
Chu
(1975)
⎫ ⎧ ⎪ ⎪ 0.387 Ra1/ 6 Nu = ⎨0.825 + 8 / 27 ⎬ 9 /16 ⎡1 + ( 0.492 Pr ) ⎤ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭
2
for
0.1 < Ra < 1012 and any Pr.
Top and bottom: Globe and Dopkin (1959) Nu = 0.069Gr1/ 3 Pr 0.074 for 3 × 105 < Ra < 7 × 109 and 0.02 < Pr < 8750 . Ceramic Top: Theofanous et al. (1996b) ACOPO (1/2th scale): Nuup = 2.4415 Ra ′0.1772 Bottom: ⎛H⎞ Theofanous et al. (1996b) ACOPO (1/2th scale): Nuda = 0.1857 Ra ′0.2304 ⎜ ⎟ ⎝R⎠ 12 16 for 10 < Ra < 2 × 10 with Mini-ACOPO distribution.
0.25
524
16. External cooling of reactor vessels during severe accident
Side wall: – 4. Constitutive correlation set used by Esmaili and Khatib-Rahbar (2004) in a steady state model Metal Side wall: 2
⎫ ⎧ ⎪ ⎪ 0.387 Ra1/ 6 for Churchill and Chu (1975) Nu = ⎨0.825 + 8 / 27 ⎬ 9 /16 ⎡1 + ( 0.492 Pr ) ⎤ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭ 12 0.1 < Ra < 10 and any Pr.
Top and bottom: Globe and Dopkin (1959) Nu = 0.069Gr1/ 3 Pr 0.074 for 3 × 105 < Ra < 7 × 109 and 0.02 < Pr < 8750 . Ceramic Top: Kulaki and Emara (1975): Nuup = 0.34 Ra ′0.226 for 2.10 4 < Ra ′ < 1012 , Pr ≈ 7 . Bottom: Mayinger et al. (1976) : Nuda = 0.55 Ra ′0.2 with Park and Dhir (1992)
Nuda (ϕ ) Nuda = b1 sin 2 ϕ + b2 , b1 =
9.12 (1 − cos ϕ0 )
8 − 9 cos ϕ0 + cos 3ϕ0
, b2 = 0.24 .
Side wall: – 16.4.7.4 Summary Here I summarize the results of the literature survey. Within the two liquid pools the buoyancy driven convection adjusts at the walls heat transfer corresponding to the following Nusselt numbers: For the oxide pool (a) lower head without the cylindrical part,
Nuda , nc = Eq. (16. 41) and Eqs. [16.40 or (16.59 and 16.60)], (b) lower head cylindrical wall if in contact with the melt at all
Nucy , nc = Eq. (16.48),
16.5 Critical heat flux
525
(c) top surface at the oxide pool possibly divided by crust from the metallic pool Nuup , nc = Eq. (16.47),
and for the metallic pool (a) cylindrical part Nuscy , nc = f1 f 2 f3 Eqs. (16.14 or 16.15),
where the heat flux in the upper and lower 10% of the height is reduced linearly from the averaged value up to 10% from the averaged value at the both corners. Nusup, nc = Nusda , nc = Eqs. (16.32 or 16.35)
16.5 Critical heat flux The critical heat flux database for this particular geometry without lower head intrusions is provided by the experimental investigations reported in Nishikawa et al. (1984), Gitihnji and Soberski (1963), Chen (1978), Vishnev et al. (1976), Guo and El.-Genk (1992), Kymäläinen et al. (1992), Cheung and Haddad (1994), Rouge (1995), Chu et al. (1997), Theofanous et al (1998), Liu et al. (1999), Theofanous et al. (1994a), Theofanous et al. (1994b), Schmidt et al. (2000). Small-scale experiments with lower head intrusions are available in Liu et al. (1999). Large-scale external cooling experiments with lower head intrusions are performed by Herbst and Klemm (2003). A summary is given in Table 16.8. Table 16.8. Summary of experimental findings for CHF at downwards oriented semisphere without structural penetrations, MW/m²
No penetration
Bottom KW
Guo and El.-Genk (1992) Kymäläinen et al. (1992) Cheung and Haddad (1994)
Side MW > 1.4
400–600
> 1.2 1 to 1.06 small hemisphere
Theofanous et al. (1994a) large scale test facility ZLPU-2000
500
1.4
Theofanous et al. (1994c)
700
1.7 to 1.9 depending on the isolation design
526
16. External cooling of reactor vessels during severe accident
Dinh et al. (2004)
1.7 to 1.9, 0.25 an 1 Hz 1/84 natural circulation loop, oscillation around the vessel, gap for the flow 76 to 152 mm, central stochastic in the upper part, entrance from small cross section 0.05 bar maximal amplitude, not worst if boric acid or trisodium phosphate dissolved in water
Rouge (1995), SULTAN inclined plate, forced convection steam mass concentrations 0.15 steam mass concentrations 0.7
1 for 1 bar and higher at 5 bar 0.3 for 1 bar and 0.5 at 5 bar no Ledineg instability
Sulatski et al. (1997) VVER-640 Model, 52 mm gap, ϕ = π 2 No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
p in bar 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.9 2.0 1.9 2.3 1.5 1.3
G in kg/(m²s) 280 280 267 253 248 248 246 248 262 247 250 250 242 248 246 245 575 587 511 754 50 0
Twater in °C 99 98 112 73 73 61 67 76 84 94 98 96 90 94 99 97 104 106 87 92 82 84
CHF in MW/m² 1.58 1.68 1.63 2.25 2.29 2.49 2.43 2.21 2.07 1.80 1.72 1.73 1.96 1.90 1.58 1.42 1.90 1.89 2.32 2.42 1.93 1.49
16.5 Critical heat flux
Chu et al . (1997) CYBL elliptic bottom of a vessel with 3.7 m diameter immersed in water
4.3 total power
Theofanous and Angelini et al. (2000) Jeong et al. (2003)
1.8 1.3
527
Table 16.9. Summary of experimental findings for CHF at downwards oriented semisphere with structural penetrations, MW/m²
Liu et al. (1999–2003) reported data obtained by small-scale experiments with lower head penetrations. The results are summarized as follows: Local critical heat fluxes for the heads with and without penetration Angular position 0° CHF-without 1.2 CHF-with 1.32
18° 1.37 1.3
45° 1. 1.03
60° 1.5 1.5
75° >1.5 >1.5
With penetrations T-coolant, °C 90 93 97 100 CHF - 0° 1.32 1.32 1.32 1.32 CHF - 18° >1.5 1.5 1.48 1.3 Local critical heat fluxes for the heads without and with aluminum coating at zero sub-cooling, Yang et al. (2006) Angular position 0° CHF-plane 1.37 CHF-coated 1.49
18° 1.47 1.6
45° 1.34 1.5
60° 1.61 1.73
75° 1.99 2.1
Lyon (1965), Jergel and Stevenson (1971), Vishnev (1973) proposed to relate the critical heat flux at inclined plate to those at vertical plate as follows ′′ = q&CHF ′′ ,vertical q&CHF
(190° − γ ° ) 190° .
(16.61)
Here the angle is defined between the horizontal line and the plate. Theofanous et al. (1994c) proposed the following approximation of his experimental data ′′ = 4.9 × 105 + 3.02 × 10 4 ϕ − 8.88 × 10 2 ϕ 2 + 13.5ϕ 3 − 6.65 × 10−2 ϕ 4 (16.62) q&CHF
528
16. External cooling of reactor vessels during severe accident
Rouge et al. (1998) proposed the following approximation of their experimental data for SULTAN inclined plate test facility
(
)
′′ = 106 a0 + a1 X 1 + a2 X 12 + a3 sin ϕh + a4 sin 2 ϕh m 9.7% q&CHF
(16.63)
a0 = 0.654444 − 1.2018δ ln G − 8.388 × 10−3 p *2 + 1.79 × 10 −4 G + 1.3689δ p * − 0.0774δ p *2 + 0.024967 p * ( ln G )
2
a1 = −0.0865 ( ln G ) − 4.49425δ ln G 2
a2 = 9.285δ a3 = −6.62 × 10−3 ( ln G ) + 11.62546δ p * +0.85759 X 1 ln G 2
a4 = −1.74177 p * +0.182895ln G − 1.8898 X 1 + 2.2636δ
Here p* is in MPa, δ is the gap thickness of the channel perpendicular to the heated wall in m, G is the mass flow rate in kg/(m²s), ϕ h is the angle with respect to the horizontal and X 1 vapor mass flow rate ratio. The experimental data base cover: ϕ h = 10 to 90°, δ = 0.03 to 0.15 m, p = 1 to 5 bar, ΔTsub = 0 to 50 K, G = 10 to 5000, q& ′′ = 0.1 to 2 MW/m² demineralized and degassed water. Note that for cells for which only the one site perpendicular to δ is heated δ = Dh 4 for Cartesian coordinates, and δ ≈ Dh 4 for cells close to the Cartesian cells. This allows using the correlation in system computer codes for which the heated and the hydraulic diameter in the heated cells are defined. Yang et al. (2006) continued the work by Liu et al. (1999–2003) and reported a study on vessel with 0.305 m diameter and semi-spherical lower head with 0.381 m-radius at atmospheric conditions with optimized flow path. The optimization consists in excluding recirculation zones and narrowing the annular region in order to obtain larger liquid velocities around the lower head. Two sets of data are collected, one with normal steel surface and one with surface coated with aluminum with 50 µm pores. The geometry was characterized with a narrowing of the channel at 45° which leaded to a reduction of the CHF at this place. The results are presented as follows: Plain vessel: 0 < ϕ < 0.3442 (18°) ′′ = (1.37 + 0.3501ϕ )106 q&CHF 0.3442 < ϕ < 0.7854 (45°)
(16.64)
16.5 Critical heat flux
′′ = (1.5734 − 0.2951ϕ ) ⎡1 + 0.0925 (ϕ − 0.3142 ) q&CHF ⎣
1/ 3
0.7854 < ϕ < 1.5708 (90°)
(
ΔTsub ⎤ 106 ⎦
529
(16.65)
){
}
′′ = 1.19 − 0.4393ϕ + 0.8025ϕ 2 1 + 0.0746 ⎡⎣1 − 0.573 (ϕ − 0.7854 ) ⎤⎦ ΔTsub 106 q&CHF
(16.66) Coated vessel: 0 < ϕ < 0.3442 (18°) ′′ = (1.49 + 0.3183ϕ )106 q&CHF
(16.67)
0.3442 < ϕ < 0.7854 (45°) ′′ = (1.65 − 0.191ϕ ) ⎡1 + 0.0925 (ϕ − 0.3142 ) q&CHF ⎣
1/ 3
0.7854 < ϕ < 1.5708 (90°)
(
ΔTsub ⎤ 106 ⎦
(16.68)
){
}
′′ = 1.65 − 0.9931ϕ + 1.0213ϕ 2 1 + 0.0746 ⎡⎣1 − 0.573 (ϕ − 0.7854 ) ⎤⎦ ΔTsub 106 q&CHF
(16.69) The important conclusion of Yang et al. (2006) study is: Design the flow path around the vessel as to maximize the flow velocity and avoid recirculation zones. Forming a jet towards the lowest point omits the cyclic accumulation and release of vapor. The explanation that coating improves the critical heat flux is very much in line with my theory from 1993, see Kolev (2007). Coating increases wettability manifested in decreasing wetting angle. The smaller the wetting angles at the bubbles base the smaller the heat transfer coefficient in nucleate boiling but the larger the critical heat flux. For the experimental proof see also Wang and Dhir (1993). Haw to use these correlations? Some of them are very specific to given geometry and are directly used as demonstration that given geometry removes prescribed distribution type and local maximum of heat flux. The correlation based on the SULTAN experiment is appropriate for using it in computer codes for 3D analysis. It should be noted, that appropriate discretization of the lower head can be achieved by using constant arc distances within a structure cell. This means that the structure goes trough the two opposite corners of the computational cell. The wall has to be generated computationally perfectly smooth by using appropriate surface permeabilities, volumetric porosities, heated- and hydraulic diameters or boundary fitted coordinates. The cell size which is fact the averaging scale for the conservation equations has to be about 3–10 cm, which is the scale of the thickness of the channel in the experiment. The thermal loads obtained from the melting analysis are used as boundary conditions for the thermal-hydraulic analysis. The
530
16. External cooling of reactor vessels during severe accident
heat flux profile is imposed at the external wall of the vessel. The power is then linearly increased not slower then in the real process. The boiling front propagates from the most loaded place to the others. The vapor buoyancy drives natural circulation which form is influenced by the particular geometry. Then the local velocities, void fractions, temperatures etc. are estimated. Under these local conditions the local critical fluxes are computed using appropriate correlation and compared with the actual locally imposed heat fluxes. In particular, with the local velocities, void fractions etc. the selected correlation is used in order to compute the local critical heat flux. Conclusions are then derived regarding the safety margins by judging the ratio of the local critical heat flux to the actual maximum heat flux during the transients. This ratio is called CHF ratio.
16.6 Application examples of the model Consider a typical water-cooled nuclear reactor vessel with diameter Dv = 4.42 m presented on Fig. 16.4.
Fig. 16.4. Effect of the vessel size with semispherical lower head on the external cooling for typical 3400 MWt water-cooled reactors
The semi-spherical lower head has a radius RLH = 2.21 m. The vessel is submerged in water. Consider molten debris inside the lower head with a mass of 112 t with a metallic layer of M m ,0 = 10.7 t atop the oxide layer. The relocation of this debris happens τ rel = 2 h after SCRAM of the reactor for reasons that are not important here. The initial thermal power of the reactor is assumed to be 3400 MW. Inside the debris there are no submerged structures, M m ,sub = 0 t. Above the reactor there are M m ,max = 40 t structures that can melt and move down due to absorbing radiation heat. The process starts with already-molten metal M m ,0 =10.7 t being atop 112 t oxide. The oxide height at the axis is LLH ,ox = 1.695 m. I assume that
16.6 Application examples of the model
531
c2 = 0.3 part of the energy radiated upwards from the molten pool is consumed by melting the structures as long as they are available. The other part of the radiation energy is dissipated in the vessel and also removed by external cooling. Next, I will analyze different effects on the maximum heat flux into the external water using the method described above.
16.6.1 The effect of vessel diameter Here I simply vary the diameter of the vessel, keeping all other conditions constant, in order to see the effect on the maximum external heat flux. For cases 1 to 6, the semi-spherical lower head has the same diameter as the vessel – see Table 16.10. For cases 1 to 6 the maximum heat flux is in the metal layer because of the so called focusing effect. As already mentioned, the focusing effect is a physical phenomenon based on the differences of enforced heat power from the corium into the metal and the limited radiation heat removal. The differences of the energy fluxes have to be removed by the side contact surface into the vessel wall. Therefore the smaller the metallic layer the larger the wall heat flux from the metal to the wall as illustrated in Fig. 16.1a. Conclusion 1: For the cases in which the maximum heat flux is in the metal layer (cases 1–6) there is no strong effect from the change of the size of the vessel. This changes if due to the large amount of metal atop the oxide the maximum heat flux is inside the oxide pool. 16.6.2 The effect of the lower head radius Now I introduce case 7 in which only the lower head radius is increased to the radius of the KARENA vessel – see Table 16.10 and Fig. 16.5. This increases the lower head heat transfer surface from the oxide pool and reduces the maximum heat flux by 36% with respect to case 1. Conclusion 2: Increasing the lower head radius reduces the maximum heat flux at the vessel wall and therefore influences the process positively.
532
16. External cooling of reactor vessels during severe accident
Fig. 16.5. Left: KARENA vessel without the structural penetrations with total amount of oxide debris – initial state for the analysis; Right: Typical PWR-vessel for comparison
&max Table 16.10. Effect of the vessel diameter on the maximum external heat flux q′′ No
τ rel
Dv
1
(h) 2
(m) 4.42
2
2
5
3
2
5.5
4
2
6
5
2
6.5
LLH ,ox
δ Lm
Fox ,up
Fox ,da
(m) 0.108
2
Dv 2
(m) 1.695
(m ) 14.51
(m ) 23.53
Dv 2
1.542
0.093
16.75
24.22
Dv 2
1.443
0.084
18.39
24.93
Dv 2
1.362
0.078
19.84
25.67
Dv 2
1.295
0.073
21.17
26.44
RLH (m)
2
6
2
7.12
Dv 2
1.224
0.068
22.68
27.38
7 8 9
2 8 >8
7.12 7.12 7.12
4.46 4.46 4.46
1.073 1.073 1.073
0.058 0.058 0.058
26.45 26.45 26.45
30.07 30.07 30.07
No
c1
1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 1
δ LH ,min
c2
M m ,0
M m , sub
M m ,max
&max q′′
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
(t) 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7 10.7
(t) 0 0 0 0 0 0 0 0 200
(t) 40 40 40 40 40 40 40 40 240
(MW/m ) 2.24 2.30 2.28 2.25 2.25 1.77 1.71 0.981 0.322
2
0.021 0.021 0.021 0.021 0.021 0.0278 0.0278 0.0486 No ablation
16.6 Application examples of the model
533
16.6.3 The effect of the relocation time Due to the large water inventory in the BWR’s, relocation of melt into the lower head during severe accidents happens much later than in the case of PWRs. Consider the differences between cases 7 and 8. The only difference in these two cases is that the assumed time of the melt relocation into the lower head is increased from 2 to 8 h after SCRAM. This results in about 41% lower head flux in case 8 compared to case 7. Conclusion 3: Increasing the delay time of the melt relocation reduces the maximum heat flux into the coolant – an effect that is intuitively expected. 16.6.4 The effect of the mass of the internal structures A significant difference between BWRs and PWRs of the same power level is that the first contains a considerable amount of internal structures. The KARENA has about 700 t internal structures. About 400 t of this is below the upper core plate, including the plate itself. About 200 t of structures such as control rod drives, instrumentation pipes, lower core plate, etc. are below the core region. The structures submerged into the melt first have to melt until the oxide pool starts to increase its temperature significantly above the steel melting temperature. The hanging structures which are absorbing radiation energy and are being partially melting have a similar effect and therefore increase and cool the upper metallic layer. This is manifested in the KARENA case shown in Table 16.10. It results in a significant reduction of the maximum heat flux to below 170 kW/m² – an effect again that is intuitively expected. Conclusion 4: Internal structures penetrating the debris or being closely above them during severe accidents (a) consume a significant amount of the decay energy, (b) rapidly increase the metallic layer atop the oxide pool and, therefore, significantly reduce the maximum heat flux into the coolant. 16.6.5 Some important parameters characterizing the process Now we turn our attention to the transient solutions provided by the above described model. Figure 16.6 present the oxide pool and the metallic layer averaged temperatures as a time functions for the two cases 1 and 9.
534
16. External cooling of reactor vessels during severe accident
3200
2400
3100
2300 2200 Tmetal in K
Toxid in K
3000 Case
2900
1 9
2800
2100 Case
2000
1 9
1900 1800
2700
1700 2
4
6
2
8 10 12 14 16 18 Time in h
4
6
8 10 12 14 16 18 Time in h
Fig. 16.6. Temperature as function of time for cases 1 and 9: (a) Oxide pool; (b) Metallic layer
We realize that if the maximum heat flux at the external wall is below the critical one the melt can be stabilized in the lower head because the temperatures reaches their maxima and then starts to fall. This demonstrates the sense of the transient analysis. Simulating the processes as a steady state is not necessarily conservative regarding the nuclear reactor safety. 0,14
2,0 1,5
Min. wall thickness, m
Max. heat flux in W/m²
2,5
Case 1 9
1,0 0,5 0,0 2
4
6
8 10 12 14 16 18 Time in h
0,12 0,10
Case
0,08
1 9
0,06 0,04 0,02 0,00 2
4
6
8 10 12 14 16 18 Time in h
Fig. 16.7. Heat fluxes (a) and the corresponding minimal remaining wall thickness (b) as a function of time for the two cases 1 and 9
Inspecting the heat fluxes as a function of the time and comparing their maxima with the critical heat flux is the most important part of the analysis. While for the first case the maximum heat flux is above the critical heat flux in the ninth case it is not. Therefore in the first case the assumption “nucleate boiling” was not correct and in the ninth case it was correct. The consequences for the remaining minimal wall thickness are clearly visible: In the first case the remaining wall thickness is 2 cm and in the ninth case there is no ablation of the wall. Now I recomputed the computations for the cases 1 and 9 and stop at the moment of the maximum heat flux. Then I plot the spatial distribution of the most important parameters.
0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0,00
535
0,14 Min. wall thickness in m
Crust thickness in m
16.6 Application examples of the model
Case 1 9
0
1
2 3 4 Arc distance in m
5
6
0,12 0,10
Case 1 9
0,08 0,06 0,04 0,02 0,00
0
1
2 3 4 Arc distance in m
5
6
Fig. 16.8. Crust thickness and minimal wall thickness as functions of the arc distance counted from the lowest point of the external wall for case 1 and 9
Figure 16.8 provides the dependence of the internal crust thickness attached to the vessel wall and of the minimum wall thickness on the arc distance for the cases 1 and 9. We see in the both cases a considerable crust formation in the bottom between 6 and 8cm. This crust will seal any cracks at the lower head if existing at al. We see also that the strongest ablation in the case 1 happens in the region of the metallic layer. This is the most severe consequence of the focusing effect.
Heat flux in MW/m²
2,5 2,0
Case 1 9
1,5 1,0 0,5 0,0 0
1
2 3 4 Arc distance in m
5
6
Fig. 16.9. External vessel wall heat flux as function of the arc distance counted from the lowest point of the external wall for case 1 and 9
Figure 16.9 presents the external vessel wall heat flux as function of the arc distance counted from the lowest point of the external wall for case 1 and 9. Again we see the maximum of the heat flux in the metallic layer for the first case. This is the distribution with which a subsequent hydraulic analysis for the stability of the natural circulation has to be performed.
536
16. External cooling of reactor vessels during severe accident
50
Decay heat To water Case 1 9
Power in W
40 30 20 10 0 2
4
6
8 10 12 14 16 18 Time in h
Fig. 16.10. Decay power as a function of time and its part going through the lower head into the water for the case 1 and 9
Finally the decay power as a function of time and its part going through the lower head into the water for the case 1 and 9 are presented in Fig. 16.10. Note that the total decay power has to be removed from the containment. The part not removed with the noble gasses and with the volatiles and not removed from bottom is partially dissipated in the structure and in the remaining part of the vessel and has also to be removed by natural convection. Next I give an example for the hydraulic analysis of the external cooling performed with IVA computer code as described in Vol. 1, 2 and 3 of this work. Consider case 9. The heat flux profile from Fig. 16.9 is imposed at the external wall of the vessel with the corresponding dimensions. The maximum power is imposed instead within hours within 100 s. Figure 16.10 presents the evolution of the natural circulation within 200 s. The heat transfer is controlled by two-phase phase natural circulation. As expected, void is observed in the boundary layer at the vessel wall. We see that the water supply compensating the evaporated water flows downwards around the reactor. The temperature of the water participating in the main vortex increases with the power and remains steady as the maximum of the power is reached. The most important parameter, the ratio of the local critical heat flux computed by using the SULTAN correlation to the local actual heat flux is in this case anywhere larger then 5.71. Therefore, large safety margin is proven for such systems.
16.6 Application examples of the model
537
Fig. 16.11. Water temperatures and velocities as a function of space for 0, 50, 100 and 200s
Conclusion: The computational results in this chapter clearly demonstrate that the successful realization of the external cooling for vessels of nuclear power plants strongly depend (a) on the sizes of the vessel, (b) on the type of the reactor which dictate the way of developing of the severe accident and when the melt relocations is expected to come and
538
16. External cooling of reactor vessels during severe accident
(c) on the initial power of the facility. Such analysis is obviously specific for each type of reactors. In any case, the boiling water reactors are much favorite candidates for such severe accident mitigation measure because of their much larger size, much larger water inventor in the vessel, much larger steal mass below the core etc. For the pressurized water reactor the success of this measure strongly depends on the power and has to be determinate from case to case. For high powered pressurized water reactors the melt can not be hold in the lower head with natural external circulation without special measures leading to increase the critical heat flux.
16.7 Nomenclature Latin a b’s cp c p ,liq c1, c2, c3 c4, c5 Da F f1 f2 f3 g Gr g H h hs′′′ hs hup hcy kSB k L M m&
thermal diffusivity, m2/s coefficients in the discretized Fourier equation specific heat at constant pressure, J/(kgK) specific capacity at constant pressure for the liquid, J/(kgK) heat flux distribution coefficients, dimensionless coefficients in the energy conservation equation for the metallic layers, dimensionless [= q& ′′′H 2 /(λΔTmax )] , Dammköhler number, dimensionless surface, m2 function defined by Eq. (16.21), dimensionless function defined by Eq. (16.26), dimensionless function defined by Eq. (16.13), dimensionless gravitational acceleration, m/s2 [= g βΔTmax H 3 /ν 2 ] , Grashoff number, dimensionless gravity acceleration, m/s2 thickness of the metallic layer, vertical size of the cavity, m averaged heat transfer coefficient, W/(m2K) specific liquidus enthalpy – metal, J/kg specific enthalpy – metal, J/kg heat transfer coefficient at the upper surface, W/(m2K) heat transfer coefficient at the vertical surface, W/(m2K) Steffan–Boltzman constant, VDI-Wärmeatlas (1991), W/(m2K4) form parameter, dimensionless horizontal size of the cavity, m mass, kg mass source for the metallic layer due to wall ablation, kg/s
16.7 Nomenclature Nu
Pr Q q& ′′ q& ′′′ Req Ra' Ra T T ′′′ V V V∞ Z, z
= h ΔZ / λ , averaged Nusselt number for heat transfer at the upper surface, dimensionless = ν / a , Prandtl number, dimensionless thermal power, W heat flux density, W/m2 volumetric heat flux density, W/m3 equivalent semi-sphere radius of cavity partially filled with liquid, m = GrPrDa, modified Rayleigh number, dimensionless = GrPr, Grashoff number, dimensionless mass averaged temperature, K solidification temperature, K volume, m3 spatially averaged convection velocity, m/s steady state spatially averaged convection velocity, m/s arc distance from the centre of the cavity measured in the vertical plane at the outer surface wall, m
Greek
β ϕ Δ Δhs
ΔTmax ΔZ
δT
δ T* Δhmelt Δτ T Δτν
ε ν λ τ ρ
thermal expansion coefficient, dimensionless angle, rad finite differential latent heat of melting, J/kg maximum bulk-wall temperature difference, K high of the cavity, m thermal boundary layer thickness, m thermal layer thickness due to heat conduction, m specific meting enthalpy, J/kg time necessary for the penetration of the thermal boundary layer δ T* , s viscosity time constant, s emissivity coefficient, cinematic viscosity, m2s thermal conductivity, W/(mK) time, s density, kg/m3
Subscripts cy da i
539
cylindrical part of the pool downward in a pool pool/crust interface
540
i k max min nc s T top u up v w ∞ liq sol
16. External cooling of reactor vessels during severe accident
spatial discretization index in r-direction spatial discretization index in z-direction maximum minimum natural convection steel thermal upward in the pool corium pool upper surface vessel wall steady state liquid solid
Subscripts ′′′ ′′
melting point per unit surface
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Incopera FP and DeWitt DP (2002) Fundamentals of Heat and Mass Transfer, 5th ed. John Wiley & Sons, New York Jacob M (1949) Heat Transfer. John Wiley & Sons, New York Jahn M and Reineke HH (1974) Free convection heat transfer with internal heat sources, calculation and measurements. Proceedings of 5th International Heat Transfer Conference, Tokyo, September 1974, Paper NC2.8, pp 74–78 Jaluria Y (1983) Natural Convection Heat and Mass Transfer. Pergamon Press, Oxford Jeffreys H (1926a) The stability of a layer of fluid heated below. Phil. Mag., vol 2, pp 833–844 Jeffreys H (1926b) Some cases of instability in fluid motion. Proc. R. Soc. London Ser. A, vol 118, pp 195–208 Jeong YH, Baek W-P and Chang SH (2005) CHF Experiments of the reactor vessel wall using 2-D slide test section, NURETH-10 International Topical Meeting on Nuclear Reactor Thermal Hydraulics N°10, Seoul, COREE, REPUBLIQUE DE (10/2003), vol 152, no 2, pp 162–169 Jergel M and Stevenson R (1971) Critical heat transfer to liquid Helium in open pools and narrow channels. Int. J. Heat Mass Transfer, vol 12, pp 2099–2107 Jones CA, Moore DR and Weiss NO (1976) Axis-symmetric convection in a cylinder. J. Fluid Mech. vol 73, pp 353–388 Kolev NI (1993) Sicherheitsbericht des geplanten WWER-640 (W-407) Auswertung: System für Erhaltung der Schmelze im RDB, Schmelzfänger, KWU NA-M/93/016, Project GUS-Kooperation Kolev NI (1995a) External Cooling of VVER 640 Reactor Vessel under Severe Accident Conditions, Part 1. Buoyancy driven convection, metallic layer dynamics, wall ablation, KWU NA-M/95/E029, 18.04.1995, Project WWER-640. Revision KWU NAM/95/E029r Kolev NI (1995b) External Cooling of EPR 1500 Reactor Vessel Under Severe Accident Conditions, Part 1. Buoyancy driven convection, metallic layer dynamics, wall ablation, KWU NA-M/95/E030, 20.04.1995, Project R&D Kolev NI (1995c) External Cooling of KKI 1 Reactor Vessel Under Severe Accident Conditions, Part 1. Buoyancy driven convection, metallic layer dynamics, wall ablation, KWU NA-M/95/E051, 26.07.1995, Project R&D Kolev NI (1996) External cooling of PWR reactor vessel during severe accident. Kerntechnik, vol 61 no 2–3, pp 67–76. In abbreviated form in Proceedings of ICONE-4, The Fourth International Conference on Nuclear Engineering, New Orleans, March, 8–12, 1996, USA Kolev NI (2000) SWR 1000 Severe accident control through in-vessel melt retention by external RPV cooling, SNP NDS2/00/E2515, Project SWR1000, 29-07-2000, Erlangen Kolev NI (2001) SWR 1000 Severe accident control through in-vessel melt retention by external RPV cooling. 9th International Conference on Nuclear Engineering, Nice, France, April 2–12, 2001 Kolev NI (June 13–17, 2004) External cooling – the SWR 1000 severe accident management strategy. Proceedings of ICONE-12 ’04 Arlington VA, USA, April 25–29, 2004, Paper ICONE12-49055, Presented first as SWR 1000 In-Vessel Melt Retention, STUK Meeting hold at 13.8.2003 in Helsinki, Finland; (November 17–18, 2005) European BWR Forum, 1st Seminar on SWR1000 Design Features, Framatome ANP, Offenbach, Germany; (May 10–11, 2006) European BWR Forum, 2nd Seminar on SWR1000 Design Features, Oskarshamn, Sweden Kolev NI (2002, 2004, 2007) Multiphase Flow Dynamics, vol 2 Thermal and mechanical interactions, 2nd ed. with 81 Figures, Springer, Berlin, New York, Tokyo, ISBN 3540-22107-7, see the content in http://www.springeronline.com/east/3-540-22107-7, 3rd ed., 10 March 2007
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Kulaki FA and Emara AA (1975) High Rayleigh Number Convection in Enclosed Fluid Layers with Internal Heat Sources. U.S. Nuclear Regulatory Commission Report NUREG-75/065 Kymäläinen O et al. (1992) Critical heat flux on thick walls of large natural circulation loops. ANS Proceedings National Heat Transfer Conference, San Diego, CA, August 9–11, ANS HTC, vol 6, pp 44–50 Kymäläinen O, Hongisto O and Pessa E (April, 1993) COPO experiments on heat transfer from a volumetrically heated pool, DLV1-G380-0377, Imatran Voima Oy Process Laboratory Kymäläinen O, Tuomisto H, Hongisto O and Theofanous TG (1993) Heat flux distribution from a volumetrically heated pool with high Rayleigh number, Rroc. of NURETH 6, Grenoble, France, October 5–8, 1993, pp 48–53 Liu YC, Donald SD, and Cheung FB (August 15–17, 1999) Boiling-induced flow and heat transfer in a hemispherical channel with tube intrusion. Proceedings of the 33rd National Heat Transfer Conference, Albuquerque, New Mexico Lord Rayleigh (1916) On convective currents in a horizontal layer of fluids when the higher temperature is on the under side. Philos. Mag., vol 32, pp 529–546 Lyon DN (1965) Boiling heat transfer and peak nucleate boiling fluxes in saturated liquid helium between lambda-point and critical temperature. Int. Adv. Cryog. Eng. vol 11, pp 371–379 Mayinger F, Jahn M, Reineke HH and Steinbrenner U (1976) Examination of thermohydraulic processes and heat transfer in core melt, BMFT R8 48/1. Institut für Verfahrenstechnik der TU Hanover Nishikawa K et al. (1984) Effect of surface configuration on nucleate boiling heat transfer. Int. J. Heat Mass Transfer, vol 27, no 9, pp 1559–1571 O’Brien JE and Hawkes GL (1991) Thermal analysis of a reactor lower head with core relocation and external boiling heat transfer. AIChE Symp. Ser., Heat Transfer – Minneapolis, MN, pp 159–168 Park H and Dhir VK (1992) Effect of outside cooling on the thermal behavior of a pressurized water reactor vessel lower head. Nucl. Technol., vol 100, p 331 Pchelkin IM (1960) Convective and Radiation Heat Transfer, Moscow, Publ. Academy of Science USSR, pp 56–64 (in Russian) Rempe JL et al. (1997) Potential for AP600 in-vessel retention through ex-vessel flooding, Technical evolution report, INEEL/EXT-97-0077 Rouge S (September 10–15, 1995) Large scale vessel coolability in natural convection at low pressure. NURETH 7 Conference, Saratoga Springs, NY, USA; see also in Rouge S (1997) SULTAN test facility for large scale vessel coolability in natural convection at low pressure. NED, vol 169, pp 185–195 Rouge S, Dor I and Geffraye G (March, 1998) Reactor vessel external cooling for corium retention: SULTAN Experimental program and modeling with CATHARE code. Proceedings of the Workshop on In-vessel Core Debris Retention and Coolability, Munich, NEA/CSNI/R(98), pp 351–363 Schmidt H et al. (May 29–31, 2000) Tests to investigate the RPV exterior two-phase flow behavior in the event of core melt. 38th European Two-Phase Flow Group Meeting, Karlsruhe, Paper A6 Sonnenkalb M (1994) Summary of previous German research activities and status of GRS program on in-vessel molten pool behavior and ex-vessel coolability, OECD/CSNI/NEA Workshop on large molten pool heat transfer. Nuclear Research Centre Grenoble, France, 9th–11th March 1994 Steinberner U and Reineke HH (1978) Turbulent buoyancy convection heat transfer with internal heat sources. Proceedings 6th International Heat Transfer Conference, Toronto, Canada, August 1978, NC-21, pp 305–310
544
16. External cooling of reactor vessels during severe accident
Sulatski AA, Cherny OD, Efimov VK and Granovskii VS (1997) Boiling crisis at the outer surface of VVER vessel. Proceedings of the 11th International School-Seminar of Young Scientists and Specialists: The Physics of Heat Transfer in Boiling and Condensation, May 21–24, Moskow, Russia Sun KY (June, 1994) Modeling of heat transfer to nuclear steam supply system heat sink and application to severe accident sequences. Nucl. Technol. vol 6, pp 274–291 Theofanous TG and Angelini S (August 2000) Natural convection for in-vessel retention and prototypic Rayleigh numbers, Nuclear Engineering and Design, vol 200, no 1–2, pp 1–9 Theofanous TG, Liu C, Angelini S, Kymäläinen O, Tuomisto H and Addition S (1994a) Experience from the first two integrated approaches to in-vessel retention through external cooling. OECD/CSNI/NEA Workshop on large molten pool heat transfer, Nuclear Research Centre Grenoble, France, 9th–11th March 1994 Theofanous TG, Syry S, Salmassi T, Kymäläinen O and Tuomisto H (1994b) Critical heat flux through curved, downward facing, thick walls. OECD/CSNI/NEA Workshop on large molten pool heat transfer, Nuclear Research Center Grenoble, France, 9th–11th March 1994 Theofanous TG, Liu C, Additon S, Angelini S, Kymkäläinen O and Salmassi T (November, 1994c) In-vessel coolability and retention of a core melt, DOE/ID-10460, U.S. Department of Energy Theofanous TG et al. (1996a) In vessel coolability and retention of core melt. DOE/ID10460, vol 1 Theofanous TG et al. (September, 1996b) The first results from the ACOPO experiment. Proceedings of the Topical Meeting On Probabilistic Safety Assessment (PSA ´96), Park Soty, Utah Theofanous TG et al. (June 8–12, 1998) The mechanism and prediction of the critical heat flux in inverted geometry. 3rd International Conference on Multiphase Flow, ICMF’98, Lion, France Turland BD, Dobson GP and Allen EJ (November, 1999) Models for melt-vessel interactions. AEA Technol., AEAT-4544, Issue 1 VDI-Wärmeatlas (1991) Berechnungsblätter für den Wärmeübergang, Sechste Auflage, VDI Verlag, Düsseldorf Vishnev IP (1973) Vlijanie orientatsii poverhnost nagreva v gravitationnom pole na krisis puzyrkovogo kipenija zhydkosti. Inzhenerno-Fizicheskij Zhurnal, vol 24, no 1, pp 59– 66 (in Russian) Vishnev IP et al. (1976) Study of heat transfer of boiling of helium on surfaces with various orientations. Heat Transfer-Sov. Res. vol 8, no 4, pp 104–108 Wang CH and Dhir VK (August, 1993) Effect of surface wettability on active nucleation site density during pool boiling of water on a vertical surface. ASME J. Heat Transfer, vol 115, pp 659–669 Wörner W (1994) Direkte Simulation turbulenter Rayleigh-Benard-Konvektion in flüssigem Natrium, KfK 5228, November 1994, Kernforschungszentrum Karlsruhe Yang J, Cheung FB, Rempe JL, Suh KY and Kim SB (2006) Critical heat flux for downward-facing boiling on a coated hemispherical vessel surrounded by an insulation structure. Nucl. Eng. Technol., vol 38, no 2, Special Issue on ICAPP ‘05
Appendix 1: Some geometrical relations Most of the vessels of the water cooled nuclear reactors have a spherical lower head attached to vertical cylindrical part. The collected debris at the bottom fill
Appendix 1: Some geometrical relations
545
first the spherical part and then if enough mass is relocated the cylindrical part. For proper use of the empirical correlations and for the coupling of the models describing the pool and vessel behavior some geometrical relations are needed. For convenience I summarize them below. Knowing the internal diameter of the reactor pressure vessel Dv ,in and the radius of the semi-spherical bottom rlh ,in , see Fig. 16.12, the following geometrical characteristics are easily computed: Dv ,in = 5.58
⎛ Dv ,in ⎞ ⎟⎟ ⎝ 2rlh ,in ⎠
θlh,in = arc sin ⎜⎜
rlh,in = 3.5
θlh,in l Vlh,sph
⎛1 ⎞ H lh ,in = rlh ,in − rlh2,in − ⎜ Dv ,in ⎟ ⎝2 ⎠
2
Fig. 16.12. Some vessel characteristics
The horizontal internal vessel cross section is Fv ,up =
π 4
Dv2,in .
The internal highs of the lower head that is the distance between the bottom inside and the horizontal plane at which the spherical surface crosses the cylindrical one is 2
⎛1 ⎞ H lh ,in = rlh,in − rlh2,in − ⎜ Dv ,in ⎟ . ⎝2 ⎠
The angle between the downwards directed vertical vessel axis and the vector connected the sphere centre and the cylinder-sphere common circle is ⎛D ⎞ θ lh ,in = arc sin ⎜⎜ v ,in ⎟⎟ . ⎝ 2rlh ,in ⎠
546
16. External cooling of reactor vessels during severe accident
The volume of the spherical part of the lower head is
Vlh , sph =
π 3
H lh2 ,in ( 3rlh ,in − H lh ,in ) .
For known mass and temperature of each layer the volume is then Vl = M l ρl .
The sum of the volumes from the bottom upwards is k
Vk = ∑ Vi . i =1
We have to distinguish the cases whether the group of layers ends below or above the transition of the spherical part to the cylinder, see Fig. 16.13.
Fig. 16.13. Different amount of melt in the lower head. Oxide below – metal a top
If Vk > Vlh, sph then H k = H lh,in + (Vk − Vlh, sph ) Fv ,up , rup , k = Dv ,in 2 ,
Fup , k = Fv ,up , else starting with H k = H lh,in we compute iteratively Hk =
3Vk . π ( 3rlh ,in − H k )
Less then 10 cycles are always sufficient. Then
Appendix 1: Some geometrical relations
547
rup , k = 2rlh ,in H k − H k2 , Fup , k = π rup2 , k ,
H cd , k = H k − H k −1 , see Fig. 13.
Fcy ,k = π H cd , k Dv ,in
Hcd,k
Fig. 16.14. Site interface between the metallic layer and the vessel wall and thickness of the metallic layer
θkl
rlh,in
Fup,k
θ k = arc sin ( rup ,k rlh,in )
rup,k Hk
Fup,k
θkl
rlh,in
θ k = arc sin ( rup ,k rlh,in )
rup,k Fcy,k Hk
Fig. 16.15. Angle defining the upper pool edge for spherical lower head with melt in it: (a) below the transition to the cylindrical part; (b) above the transition to the cylindrical part
The surface of the vertical cylinder site of the layer, Fcy ,l , and the angle defining the upper edge of the layer, θ k , see Figs. 16.13 and 16.14, are computed as follows:
548
16. External cooling of reactor vessels during severe accident
Bottom layer: For Vk > Vlh, sph , Fcy , k = π ( H cd , k − H lh,in ) Dv ,in , θ k = θ lh ,in else
Fcy , k = 0 , θ k = arc sin ( rup , k rlh ,in ) .
All other layers: Vk > Vlh, sph :
Vk −1 > Vlh, sph : Fcy , k = π H cd , k Dv ,in , θ k = arctan ⎡⎣ rup , k
(r
lh , in
− H k ) ⎤⎦
Vk −1 ≤ Vlh , sph : Fcy , k = π ( H k − H lh,in ) Dv ,in + 2π rlh ,in ( H lh ,in − H k −1 ) , θ k = θ lh ,in Vk ≤ Vlh , sph : Fcy , k = 2π rlh,in H cd , k , θ k = arc sin ( rup , k rlh ,in )
The arc distance from the lowest point of the vessel bottom inside to the upper edge of the k-th layer, S arc , k is computed as follows:
Vk > Vlh, sph : S arc , k = rlh,inθ k + H k − H lh,in Vk ≤ Vlh , sph : S arc , k = rlh,inθ k
The arc distance from the lowest edge of the layer to the upper edge of the k-th layer, ΔS arc ,1 is computed as follows: ΔS arc ,1 = S arc ,1 , ΔS arc , k = Sarc , k − Sarc , k −1 .
The surface of the lower layer interface, Fda ,l
Vk > Vlh, sph : Fda ,1 = 2π rlh,in H lh ,in + π Dv ,in ( Sarc ,1 − rlh ,inθlh ,in ) , Vk ≤ Vlh , sph : Fda ,1 = 2π rlh,in H1 , Fda , k = Fup , k −1 , k = 2, 3.
17. Thermo-physical properties for severe accident analysis
Several modern aspects of the severe accident analysis can not be understood if the engineer does not have accurate information of the material properties for the participating structural materials in solid, in liquid and in some cases in gaseous states. Chapter 17 contains valuable sets of thermo-physical and transport properties for severe accident analysis for the following materials: uranium dioxide, zirconium dioxide, stainless steel, zirconium, aluminum, aluminum oxide, silicon dioxide, iron oxide, molybdenum, boron oxide, reactor corium, sodium, lead, bismuth and lead-bismuth alloys. The emphasis is on the complete and consistent thermo dynamical sets of analytical approximations appropriate for computational analysis.
The mathematical description of physical processes controlling the evolution of severe accidents in nuclear power plants requires thermo-physical properties of solids, solid-liquid mixtures and liquids up to e.g. 3000 K. The generated analytical approximations that are required for state of the art multiphase fluid dynamic analysis are: For liquid and gas
ρ = ρ ( p, T )
( ∂ρ
∂p )T = f ( p, T )
Density as a function of pressure and temperature, kg/m³ Derivative of the density with respect to pressure at
( ∂ρ
∂T ) p = f ( p, T )
constant temperature, kg/(m³Pa) Derivative of the density with respect to temperature at constant pressure, kg/(m³K)
The volumetric thermal expansion coefficient, the isothermal coefficient of compressibility, isothermal bulk modulus
β = − ( ∂ρ ∂T ) p ρ , k = ( ∂ρ ∂p )T ρ = 1/ B , B = 1/ k
N.I. Kolev, Multiphase Flow Dynamics 4, DOI 10.1007/978-3-540-92918-5_17, © Springer-Verlag Berlin Heidelberg 2009
549
550
17. Thermo-physical properties for severe accident analysis
are then easily computed. h = h ( p, T )
( ∂h ∂p )T
= f ( p, T )
( ∂h ∂T ) p = c p ( p, T )
( ∂s
sure at constant temperature, J/(kgPa) Derivative of the specific enthalpy with respect to tem-
∂p )T = f ( p, T )
perature at constant pressure – specific thermal capacity at constant pressure J/(kgK) Specific entropy as a function of temperature and pressure, J/(kgK) Derivative of the specific entropy with respect to pres-
∂T ) p = f ( p, T )
sure at constant temperature, J/(kgKPa) Derivative of the specific entropy with respect to tem-
s = s ( p, T )
( ∂s
Specific enthalpy as a function of temperature and pressure, J/kg Derivative of the specific enthalpy with respect to pres-
a = a ( p, T )
λ = λ ( p, T )
ν = ν ( p, T )
perature at constant pressure, J/(kgK²) Velocity of sound, m/s Thermal conductivity, W/(mK) Cinematic viscosity, m²/s
In addition approximation of the surface tension at the liquid-gas interface as a function of temperature is given
σ = σ (T )
Surface tension at the liquid-gas interface as a function of temperature, N/m
Approximation for the saturation line is provided in two forms: T ′ = f ( p)
Saturation temperature as a function of the pressure, K
p′ = f (T )
Saturation pressure as a function of the temperature, Pa
Δh = h′′ − h′ = f (T )
Derivative of the saturation temperature with respect to pressure – the Clausius–Clapayron relation, K/Pa Latent heat of evaporation as a function of the temperature, J/kg
dT ′ dp = f (T )
The properties at the saturation line for liquid, designated with ', and for vapor, designated with '', are computed from the p-T functions using the corresponding p'-T or T'-p couples of dependent variables. Additional nomenclature is given at the end of this section.
17.1 Introduction
551
In this chapter I provide a summary of properties for the materials given in Table 17.1. Attention: In this table but also over this chapter some of the numerical constants are given with many digits after the point. This is not because they are so accurately known but because by formally construction of stable numerical algorithms the largest possible degree on consistency is necessary. Table 17.1 Materials identifiers
ID 1 2 3 4 5 6 7 8 9 10 11
Material Uranium dioxide, UO2 Reactor corium, Cor Zirconium, Zr Zirconium dioxide, ZrO2 Stainless steel, Steel Aluminium dioxide, AL2O3 Silicium dioxide, SiO2 Iron oxide, FeO+ Molybdenum, Mo Aluminum, Al Boron oxide, B2O3
17.1 Introduction Because the temperatures at which the properties of the considered materials required for severe accident analysis are very high there are considerable difficulties of measuring such properties at such high temperatures. Some times due to lack of information I am forced to make assumption or to extend in a physically based way the existing information. Here I will give an example for the liquid state of the materials considered. Before doing this let me summarize the saturation properties of the considered materials at the melting point for atmospheric pressure. 17.1.1 Summary of the properties at the melting line at atmospheric pressure In order to allow for quick comparison of the properties of each material I summarize here their properties at the melting point in Table 17.1.1. For all materials considered here I assume that the melting process happens at a single temperature called melting temperature or solidus-liquid temperature. The values are at the solidus-liquids line are computed by using the approximations collected in this chapter at the solidification temperature T ′′′ . Note that “dn” or “en” in the following text stays for 10 n .
552
17. Thermo-physical properties for severe accident analysis
Table 17.1.1. Properties of some materials at the solidus-liquid line
T ′′′ T′ ρ ′′′
UO2
Cor
Zr
ZrO2
0.3113150E+04 0.3113150E+04 0.9655299E+04
0.2920000E+04 0.2920000E+04 0.8550805E+04
0.2098000E+04 0.2098000E+04 0.6497027E+04
0.2973000E+04 0.2973000E+04 0.5707285E+04
ρ′ d ρ ′′′ dT d ρ ′ dT
0.8764894E+04
0.8105920E+04
0.6130000E+04
0.5991400E+04
–0.6835523
–0.5732749
–0.04767546
–0.2265480
–0.6448171
–0.9160000
–0.5730000
–0.9160000
h′′′
s ′ − s ′′′ c′′′p
0.1129131E+07 0.1406331E+07 0.2772000E+06 0.8016946E+03 0.8907363E+03 0.8904165E+02 0.7583083E+03
0.9974000E+06 0.1302753E+07 0.3053528E+06 0.1040234E+04 0.1135147E+04 0.9491370E+02 0.6185185E+03
0.6054876E+06 0.8308443E+06 0.2253567E+06 0.6371666E+03 0.7445816E+03 0.1074150E+03 0.3918585E+03
0.1856215E+07 0.2562215E+07 0.7060000E+06 0.1469354E+04 0.1706825E+04 0.2374706E+03 0.8150572E+03
c′p
0.5030000E+03
0.4852000E+03
0.3915546E+03
0.8150000E+03
η ′′′ η′ λ ′′′ λ′ σ′ ε′
0.4370362E-02
0.4822061E-02
0.4692419E-02
0.4170566E-02
0.4370362E-02
0.4822061E-02
0.4692419E-02
0.4170566E-02
0.3442756E+01 0.5600000E+01 0.4970000E+00 0.8700000E+00
0.3486549E+01 0.5600000E+01 0.5350000E+00 0.8700000E+00
0.3631331E+02 0.3628556E+02 0.1400000E+01 0.3500000E+00
0.1373113E+01 0.1400000E+01 0.5350000E+00 0.4000000E+00
T ′′′ T′ ρ ′′′
Steel
Al2O3
SiO2
FeO+
0.1700000E+04 0.1700000E+04 0.7255933E+04
0.2324150E+04 0.2324150E+04 0.3751342E+04
0.1993150E+04 0.1993150E+04 0.2136554E+04
0.1642000E+04 0.1642000E+04 0.5282064E+04
ρ′ d ρ ′′′ dT d ρ ′ dT
0.6979389E+04
0.3055785E+04
0.2136554E+04
0.5282070E+04
–0.5532960
–0.1266090
–0.08469055
–0.5694885
–0.5730000
–0.9650000
–0.2706760
–0.9000000
h′′′
s ′ − s ′′′ c′′′p
0.8359684E+06 0.1106298E+07 0.2703300E+06 0.9929530E+03 0.1151971E+04 0.1590176E+03 0.6901160E+03
0.2739865E+07 0.3807505E+07 0.1067640E+07 0.2548789E+04 0.3008157E+04 0.4593679E+03 0.1446692E+04
0.1960853E+07 0.1960853E+07 0.2328306E-09 0.2055283E+04 0.2055283E+04 0.0000000E+00 0.1327889E+04
0.1093364E+07 0.1428201E+07 0.3348365E+06 0.1340617E+04 0.1544537E+04 0.2039199E+03 0.8959606E+03
c′p
0.7762000E+03
0.1421713E+04
0.1327889E+04
0.9491858E+03
h′ h′ − h′′′ s ′′′
s′
h′ h′ − h′′′ s ′′′
s′
17.1 Introduction
η ′′′ η′ λ ′′′ λ′ σ′ ε′
Steel 0.6415448E-02
Al2O3
0.4661803E-01
SiO2
0.3766712E+07
0.4661803E-01
0.3766712E+07
0.5264906E-01
0.7115578E+01 0.7072828E+01 0.6698516E+00 unknown
0.2330999E+01 0.2330999E+01 0.8600000E+00 0.0660000E+01
0.3020000E+01 0.3000000E+01 0.1190000E+01 0.0950000E+01
Al
B2O3
0.2896000E+04 0.2896000E+04 0.9330013E+04
0.9332000E+03 0.9332000E+03 0.2546765E+04
0.7230000E+03 0.7230000E+03 0.2317534E+04
ρ′ d ρ ′′′ dT d ρ ′ dT
0.9330000E+04
0.2357000E+04
0.1614500E+04
–0.3649800
–0.3147714
–0.4940354
–0.5496000
–0.2330000
–0.2706760
h′′′
s ′ − s ′′′ c′′′p
0.9332872E+06 0.1323948E+07 0.3906608E+06 0.7227458E+03 0.8576425E+03 0.1348967E+03 0.5540583E+03
0.6946350E+06 0.1084635E+07 0.3900000E+06 0.1208067E+04 0.1625984E+04 0.4179169E+03 0.1282201E+04
0.5398748E+06 0.8857031E+06 0.3458282E+06 0.1084906E+04 0.1563230E+04 0.4783240E+03 0.1523124E+04
c′p
0.5715868E+03
0.1082201E+04
0.1911473E+04
η ′′′ η′ λ ′′′ λ′ σ′ ε′
0.4822061E-02
0.1100000E-02
0.2108765E+04
0.4822061E-02
0.1100000E-02
0.2108765E+04
0.8190756E+02 0.8800000E+02 0.1400000E+01 unknown
0.2350000E+03 0.1000000E+03 0.8600000E+00 0.3000000E+00
0.1972228E+01 0.2446337E+01 0.8600000E+00 unknown
s′
0.5264906E-01
0.3595500E+02 0.1798430E+02 0.1700550E+01 0.0430000E+01
Mo
h′ − h′′′ s ′′′
FeO+
0.6415448E-02
ID T ′′′ T′ ρ ′′′
h′
553
17.1.2 Approximation of the liquid state of melts Normally very limited amount of data is available for melts at very high temperatures because of the difficulties of such measurements. For thermo-hydraulic computation a strictly consistent data sets are required in order not to violate energy and mass conservation. Therefore even for very crude data, the approximations have to be strictly thermodynamically consistent in each other. Methods with different degree of complexity are discussed in Kolev (2007). Here I will summarize two of them requiring small number of constants or simple functions
554
17. Thermo-physical properties for severe accident analysis
and provide a set of this constants or functions for some materials of interests for nuclear safety. Table 17.1.2. Constants defining approximate state of melt liquids at p0 = 105 Pa
ID
T0 K
1 2 3 4 5 6 7 8 9 10 11
UO2 Cor Zr ZrO2 Steel AL2O3 SiO2 FeO+ Mo Al B2O3
3113.15 2920 2098 2973 1700 2324.15 1993.15 1642 2896 933.2 723
ρl 0 3 kg/m 8860 8105.92 6130 5991.4 6980 3055.78 2136.55 5282.07 9330 2357 1614.5
ρl 0 β kgm–3K–1 0.6448171 0.916 0.573 0.916 0.573 0.965 0.270676 0.9 0.5496 0.233 0.270676
κ Pa–1
c pl
0.7740216E-11 0.6130165E-10 0.5761464E-10 0.7080252E-10 0.1062111E-10 0.1490410E-09 0.6489103E-10 0.6135623E-10 0.4610873E-10 0.4949039E-10 0.5550999E-10
J/(kgK) 503 485.2 391.5546 815 776.2 1421.713 1327.889 949.1858 571.5868 1082.201 1911.473
Selection of such set of constants is given in Table 17.1.2. Comparison of the volume thermal expansion coefficients, 104 β l for saturated liquids estimated here with those reported by Chu et al. (1996) is given in Table 17.1.3. It seems that the uncertainties are larger then the estimated by those authors. Table 17.1.3. Comparison between the volume thermal expansion coefficients, 104 β l for saturated liquids estimated here and those estimated by Chu et al. (1996) 104 β l
UO2
ZrO2
Zr
Fe
This work Chu et al. (1996)
0.73 1.05 ± 0.1
1.53 1.05 ± 0.1
0.93 0.54 ± 0.11
0.82 1.2 ± 0.17
Given are the volumetric thermal expansion coefficient defined as follows
β l = − ( d ρl dT ) p ρl ,
(17.1.1)
and the isothermal coefficient of compressibility defined as follows kl = ( d ρl dp )T ρl .
(17.1.2)
An example are the reported by McCahan and Shepherd (1993) values for Al2O3: β l, Al2O3 = 5 × 10−5 K −1 and kl,, Al2O3 = 4.234 × 10 −11 Pa−1 . Given is also the reference density at a given temperature and pressure
17.1 Introduction
ρl 0 = ρl (T0 , p0 ) .
555
(17.1.3)
Assuming β l = const and kl = const the liquid density is approximated by Eq. (3.238) Kolev (2007) ρl = ρl 0 exp ⎡⎣ − β (T − T0 ) + κ ( p − p0 ) ⎤⎦ -
(17.1.4)
The liquid density derivative with respect to temperature is
( d ρl
dT ) p = − βρl .
(17.1.5)
The liquid density derivative with respect to pressure is
( d ρl
dp )T = κρl .
(17.1.6)
The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007),
al =
1
.
ρl κ l − T βl2 c pl
(17.1.7)
Note that if the specific capacity at constant pressure is known a function of the local temperature and pressure the corresponding local value has to be used in Eq. (17.1.7). For specific capacity at constant pressure not depending on pressure the liquid specific enthalpy is approximated by Eq. (3.243) Kolev (2007) T
hl = h′ + ∫ c pl dT − T0
1−T β
κρl 0
{
}
exp ⎣⎡ β (T − T0 ) ⎦⎤ exp ⎣⎡ −κ ( p − p0 ) ⎦⎤ − 1 ,
(17.1.8)
where h′ is the specific liquid enthalpy just at the melting temperature at atmospheric pressure. The integral T
∫ c (T ) dT = c (T − T ) + h * pl
pl ,0
0
(17.1.9)
T0
depends on the available function form for the specific capacity at constant pressure. The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (3.252) Kolev (2007)
556
17. Thermo-physical properties for severe accident analysis
⎛ ∂hl ⎞ 1−T β exp ⎡⎣ β (T − T0 ) − κ ( p − p0 ) ⎤⎦ . ⎜ ⎟ = κρl 0 ⎝ ∂p ⎠T
(17.1.10)
For specific heat at constant pressure not depending on pressure the liquid specific entropy is approximated by Eq. (3.244) Kolev (2007) T
sl = sl′ + ∫ T0
c pl (T ) T
dT +
β exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 , κρl 0
{
}
(17.1.11)
where s ′ is the specific liquid specific entropy just at the liquidus-solidus saturation line. The integral T
∫
T0
c pl (T ) T
dT = c pl ,0 ln (T T0 ) + s *
(17.1.12)
depends again on the available function form for the specific capacity at constant pressure. If one uses the entropy as a dependent variable the inversed task has to be solved. Given are the pressure and the specific liquid entropy. Find the temperature. The task is solved as follows. The temperature is computed iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ ,
(17.1.13)
where T0 and sl 0 = s ′ are the initial values. Then Eq. (3.244) Kolev (2007) is solved with respect to the temperature ⎡⎛ ⎤ ⎞ β T00 = T0 exp ⎢⎜ sl − s ′ − s * − exp ⎣⎡ β (T00 - T0 ) ⎦⎤ exp ⎣⎡ −κ ( P - P0 ) ⎦⎤ − 1 ⎟ c pl ,0 ⎥ κρ l0 ⎠ ⎣⎢⎝ ⎦⎥
{
}
(17.1.14) Less than 10 iterations are needed to reduce the error below 0.0001 K. If the specific capacity at constant pressure is not a constant, the corresponding function has to be used in Eq. (17.1.14). 17.1.3 Nomenclature T ′′′ T′ ρ ′′′ ρ′
Solidification temperature in K = T ′′′ , Liquidification temperature in K Density just complete solidified, kg/m² Density just complete liquefied, kg/m²
17.1 Introduction
557
d ρ ′′′ dT Density derivative with respect to temperature – solidus, kg/(m³K) d ρ ′ dT Density derivative with respect to temperature – liquidus, kg/(m³K) Solidus specific enthalpy, J/kg h′′′ Liquidus specific enthalpy, J/kg h′ h′ − h′′′ Melting enthalpy difference, J/kg Solidus specific entropy, J/(kgK) s ′′′ ′ Liquidus specific entropy, J/kgK) s s ′ − s ′′′ Melting entropy difference, J/(kgK) Solidus specific heat at constant pressure, J/(kgK) c′′′p Liquidus specific heat at constant pressure, J/(kgK) c′p Solidus dynamic viscosity, kg/(ms) η ′′′ Liquidus dynamic viscosity, kg/(ms) η′ Solidus thermal conductivity, W/(mK) λ ′′′ Liquidus thermal conductivity, W/(mK) λ′ Suface tension, N/m σ′ Emissivity ε′ Reference temperature, K T0 Reference pressure, Pa p0 Reference liquid density, kg/m³ ρl 0 1 ⎛ ∂v ⎞ 1 ⎛ ∂ρ ⎞ , volumetric thermal expansion coefficient, 1/K =− ⎜ v ⎜⎝ ∂T ⎟⎠ p ρ ⎝ ∂T ⎟⎠ p
β
=
κ
1 ⎛ ∂v ⎞ = − ⎜ ⎟ , isothermal compressibility, 1/Pa v ⎝ ∂p ⎠T
c pl
Liquid specific capacity at constant pressure, J/(kgK)
( d ρl ( d ρl
dT ) p Liquid density derivative with respect to temperature, kg/(m³K) dp )T Liquid density derivative with respect to pressure, kg/(m³Pa)
al
λl σl ηl hl
⎛ ∂hl ⎞ ⎜ ⎟ ⎝ ∂p ⎠T sl
ss c ps
Liquid velocity of sound, m/s Liquid thermal conductivity, W/(mK) Liquide surface tension, N/m Liquid dynamic viscosity, kg/(ms) Liquid specific enthalpy, J/kg Derivative of the liquid specific enthalpy with respect to pressure at constant temperature, J/(kgPa) Liquid specific entropy, J/(kgK) Solid specific entropy, J/(kgK) Solid specific capacity at constant pressure, J/(kgK)
558
ρs d ρs dT
λs as
17. Thermo-physical properties for severe accident analysis
Solid density, kg/m³ Derivative of the solid density with respect to the temperature, kg/(m³K) Solid thermal conductivity, W/(mK) Solid sonic velocity, m/s
References Chu CC, Sieniki JJ and Beker L Jr (October, 1996) Uncertainty analysis for thermophysical properties used in in-vessel retention analysis. In: Theophanous TG et al., eds., In-vessel Coolability and Retention of a Core Melt, DOE/ID-10460 vol 1, U.S. Department of Energy Report Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo McCahan S, Shepherd JE (January 1993) A thermodynamic model for aluminum-water interaction, Proc. Of the CSNI Specialists Meeting on Fuel-Coolant Interaction, Santa Barbara, California, NUREC/CP-0127
17.2 Uranium dioxide caloric and transport properties
559
17.2 Uranium dioxide caloric and transport properties The uranium is used as a fuel in the nuclear power plant mainly in form of uranium dioxide and in seldom cases in a metallic form. After each nuclear splitting the fragments are releasing their kinetic energy within the material in form of heat. The heat is then removed to the external cooled surface of the fuel elements. Because the temperature fields depend on the properties of the material good approximation of the thermal properties is of great importance. Moreover, during transients the accumulated thermal energy and the capability to release it are controlled by these properties. In the case of accidental condition in which the internally released heat is larger then the removed nuclear reactor cores may melt, interact with other structures, interact with the remaining coolant following variety of possible mechanisms etc. This is the motivation for the generation of a large number of careful studies to this subject. In this section I will summarize an useful set of the approximations of the thermal and transport properties for uranium dioxide. The mole mass of uranium dioxide is M = 0.2702 kg mol• 1.
(17.2.1)
Brassfield et al. (1968) reported the melting temperature of non irradiated uranium dioxide T ′′′ = 3113.15 K
(17.2.2)
and for radiated uranium dioxide T ′′′ =3113.15 – 3.2x10 FBu , –3
(17.2.3)
where FBu is the burn up in MWd/tU. Lyon and Baily (1976) found experimentally that mixtures of UO2 and PuO2 have different solidification and complete melting temperatures 2 T ′′′ = 3113.15 − 5.4195CPuO2 + 7.46839 × 10−3 CPuO − 3.2 × 10−3 FBu , 2
(17.2.4)
2 T ′ = 3113.15 − 3.21660CPuO2 − 1.448518 × 10−3 CPuO − 3.2 × 10−3 FBu . 2
(17.2.5)
Fischer (1989) computed the following critical properties of stoichiometric UO2: Tc =10 600 K, ρ c =1560 kg/m³, pc =157.873 ×106 Pa.
560
17. Thermo-physical properties for severe accident analysis
17.2.1 Solid 17.2.1.1 Specific capacity at constant pressure, specific enthalpy and specific entropy The Reymann model: The Reymann (1990) model defines the specific heat at constant pressure based on theoretical arguments as follows
c ps = c1Trel2
exp (Trel ) ⎡⎣ exp (Trel ) − 1⎤⎦
2
+ c2T +
Y 2
c3 ED , ⎛ ED ⎞ 2 RT exp ⎜ ⎟ ⎝ RT ⎠
(17.2.6)
where R θ Y ED
V Trel
= 8.3143, universal gas constant, J/(molK) Einstein temperature, K oxygen-to-metal mol ratio, = 2 for stoichiometric mixture activation energy for Frenkel defects, J/mol molar volume, m³ θ T
and the constants are given in Table 17.2.1. Table 17.2.1 contains also the constant for PuO2. Table 17.2.1. Constants of the Reymann model
Constants c1 c2 c3
θ ED
UO2 296.7 2.43e-2 8.745e7 535.825 1.577e5
PuO2 347.4 3.95e-4 3.86e7 571 1.967e5
Units J/(kgK) J/(kgK²) J/kg K J/mol
The specific heat at constant pressure for liquid UO2 and PuO2 is proposed to be c pl = 503 J/(kgK).
(17.2.7)
For mixtures of UO2 and PuO2, the specific heat at constant pressure for both solid and liquids is determination by combining the contributions from each constituent in proportion to its weighted fraction
(
)
c p = CPuO2 c p , PuO2 + 1 − CPuO2 c p ,UO2 .
(17.2.8)
17.2 Uranium dioxide caloric and transport properties
561
Actually this rule allows weighted averaging of the c-coefficients in Eq. (17.2.6) for mixtures because the form remains the same. The standard error for UO2 is 3 J/(kgK), for UO2 and PuO2 mixtures 6 to 10 J/(kgK) depending on the PuO2 fraction. For non stoichiometric oxides these uncertainties are approximately doubled. The first term in Eq. (17.2.6) represents the Einstein formulation for cvs , the second is the difference
(
c ps − cvs = VT α 2 β
)
(17.2.9)
and the third the contribution of the energy required to form Frenkel defects. The advantage of this model is that the corresponding integrals for enthalpy and entropy can be analytically obtained: T
h=
∫
Tref
−1
⎡ 1 Y ⎛ E ⎛θ ⎞ ⎤ c ps dT = Ch + c1θ ⎢ exp ⎜ ⎟ − 1⎥ + c2T 2 + c3 exp ⎜ − D 2 2 ⎝T ⎠ ⎦ ⎝ RT ⎣
⎞ ⎟, ⎠
(17.2.10)
where ⎡ ⎛ θ Ch = −c1θ ⎢exp ⎜ ⎜ Tref ⎢⎣ ⎝ T
s=
∫
Tref
+
−1
⎞ ⎤ ⎛ E 1 Y 2 ⎟ − 1⎥ − c2Tref − c3 exp ⎜ − D ⎟ ⎥ ⎜ RTref 2 2 ⎠ ⎦ ⎝
⎧⎪ ⎡ ⎛θ dT = Cs + c1 ⎨− ln ⎢ exp ⎜ T ⎝T ⎣ ⎩⎪
c ps
Yc3 R ⎛ ED ⎞ ⎛ E + 1⎟ exp ⎜ − D 2 ED ⎜⎝ RT ⎠ ⎝ RT
⎞ ⎟, ⎟ ⎠
(17.2.11)
⎞ ⎤ θ ⎛θ ⎞⎡ ⎛θ ⎟ − 1⎥ + T exp ⎜ T ⎟ ⎢exp ⎜ T ⎠ ⎦ ⎝ ⎠⎣ ⎝
⎞ ⎤ ⎟ − 1⎥ ⎠ ⎦
⎞, ⎟ ⎠
−1
⎫⎪ ⎬ + c2T ⎭⎪
(17.2.12)
where ⎧ ⎡ ⎛ θ ⎪ Cs = c1 ⎨ln ⎢exp ⎜ ⎜ ⎪⎩ ⎢⎣ ⎝ Tref
−
⎞ ⎤ θ ⎛ θ exp ⎜ ⎟ − 1⎥ − ⎟ ⎥ Tref ⎜ Tref ⎠ ⎦ ⎝
⎞ ⎛ E Yc3 R ⎛ ED + 1⎟ exp ⎜ − D ⎜ ⎜ ⎟ ⎜ RTref 2 ED ⎝ RTref ⎠ ⎝
⎞ ⎟, ⎟ ⎠
⎞⎡ ⎛ θ ⎟ ⎢exp ⎜ ⎟⎢ ⎜ Tref ⎠⎣ ⎝
⎞ ⎤ ⎟ − 1⎥ ⎟ ⎥ ⎠ ⎦
−1
⎫ ⎪ ⎬ − c2Tref ⎪⎭
(17.2.13)
by setting href (Tref ) = 0 and sref (Tref ) = 0 and selecting arbitrary Tref = 298.15 K. The equilibrium saturation solidus specific enthalpy and entropy is then
562
17. Thermo-physical properties for severe accident analysis
hs (T ′′′ ) = h′′′ = 1129131.34612 J/kg,
(17.2.14)
ss (T ′′′ ) = s ′′′ = 801.694612987 J/(kgK).
(17.2.15)
In accordance with Leibowitz et al. (1976) the melting enthalpy for urania is h′ − h′′′ = 277 200 J/kg,
(17.2.16)
the liquid saturation liquid enthalpy, the melting entropy and the saturation liquid entropy are therefore h′ = 1406331.34612 J/kg,
(17.2.17)
s ′ − s ′′′ = ( h′ − h′′′ ) 3113.15 = 89.0416459213 J/(kgK)
(17.2.18)
s ′ = 890.736258909 J/(kgK),
(17.2.19)
respectively. Therefore for s between 801.694612987 and 890.736258909 or for h between 1129131.34612 and 1406331.34612 the temperature is 3113.15 K because of melting. Figures 17.2.1, 17.2.2 and 17.2.3 demonstrate the cp, h and s functions of the temperature for constant pressure of 1 bar. The jumps are due to the phase transition at the melting point. 800
cp in J/(kgK)
700
Reymann (1990) cp model
600 500 400 300 200
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid UO2
17.2 Uranium dioxide caloric and transport properties
563
1600 1400 h in kJ/kg
1200 1000
Reymann (1990) cp model
800 600 400 200 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.2. Specific enthalpy as a function of the temperature of solid and liquid UO2. p = 1 bar
1000
s in J/(kgK)
800
Reymann (1990) cp model
600 400 200 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.3. Specific entropy as a function of the temperature of solid and liquid UO2. p = 1 bar
Several expressions modeling the specific heat for solid UO2 are reported in the literature. So for instance Fink et al. (1981a,b) selected for the melting temperature of non irradiated UO2 T ′′′ =3120 K.
In accordance with these authors for T < 2670 K we have
(17.2.20)
564
17. Thermo-physical properties for severe accident analysis
⎡ c3 k ⎢1 + 1 − Tref ⎣ c ps = c1Trel2 + c2T + 2 ⎛E ⎡⎣exp (Trel ) − 1⎤⎦ exp ⎜ D ⎝ RT
(
exp (Trel )
T
E ⎤ ) RT ⎥ ⎦ D
⎞ ⎟ ⎠
,
(17.2.21)
which differs from Eq. (17.2.6) in the third term, and for 2670 < T < 3120 K, c ps = 618.5 J/(kgK),
(17.2.22)
where the constants are given in Table 17.2.2. Table 17.2.2. Constants of the Reymann model
Constants c1 c2 c3
θ ED Tref
UO2 Reymann (1990) 296.7 2.43×10-2 8.745e7 535.825 1.577e5 298.15
k
UO2, Fink et al. (1981a) 289.67 2.8604×10-2
Units
1.259×109 516.11 1.816e5 298.15
J/(kgK) J/(kgK²) J/kg J/(kg eV) K J/mol K
8.6144e-5
eV/K
with uncertainty compared to experimental data of 0.9 % for temperatures under 1000 K and 5% for temperatures over 1000 K. The enthalpy equation is easily derived but the entropy equation turns to be more complicated containing the Euler integral. This model results in the following integral for enthalpy for T < 2670 K T
h=
∫
Tref
⎡ ⎛θ c ps dT = Ch + c1θ ⎢ exp ⎜ T ⎝ ⎣
−1
1 ⎛ ED ⎞ ⎤ 2 ⎟ − 1⎥ + 2 c2T + c3 kT exp ⎜ − RT ⎠ ⎦ ⎝
⎞ ⎟, ⎠
(17.2.23)
where ⎡ ⎛ θ Ch = −c1θ ⎢exp ⎜ ⎜ Tref ⎢⎣ ⎝
−1
⎞ ⎤ 1 ⎛ E 2 ⎟ − 1⎥ − c2Tref − c3 kTref exp ⎜ − D ⎟ ⎥ 2 ⎝ RT ⎠ ⎦
⎞, ⎟ ⎠
and in the following integral for the specific entropy for T < 2670 K
(17.2.24)
17.2 Uranium dioxide caloric and transport properties
T
s=
∫
Tref
⎧⎪ ⎡ ⎛θ dT = Cs + c1 ⎨− ln ⎢ exp ⎜ T ⎝T ⎣ ⎩⎪
c ps
⎡ ⎛E +c3 k ⎢ Ei ⎜ D ⎣ ⎝ RT
⎞ ⎛ ED ⎟ exp ⎜ RT ⎠ ⎝
⎞ ⎤ θ ⎛θ ⎞⎡ ⎛θ ⎟ − 1⎥ + T exp ⎜ T ⎟ ⎢exp ⎜ T ⎠ ⎦ ⎝ ⎠⎣ ⎝
Tref RTref ⎞ ⎟ +1− T − E ⎠ D
⎤ ⎛ ED ⎥ exp ⎜ − ⎝ RT ⎦
⎞ ⎤ ⎟ − 1⎥ ⎠ ⎦
⎞, ⎟ ⎠
−1
565
⎫⎪ ⎬ + c2T ⎭⎪
(17.2.25)
where ⎧ ⎡ ⎛ θ ⎪ Cs = c1 ⎨ln ⎢exp ⎜ ⎜ ⎪⎩ ⎢⎣ ⎝ Tref ⎡ ⎛ E −c3 k ⎢ Ei ⎜ D ⎢⎣ ⎜⎝ RTref
⎞ ⎤ θ ⎛ θ exp ⎜ ⎟ − 1⎥ − ⎟ ⎥ Tref ⎜ Tref ⎠ ⎦ ⎝
⎞ ⎛ E ⎟ exp ⎜ D ⎟ ⎜ RTref ⎠ ⎝
⎞⎡ ⎛ θ ⎟ ⎢exp ⎜ ⎟⎢ ⎜ Tref ⎠⎣ ⎝
⎞ RTref ⎤ ⎛ E ⎥ exp ⎜ − D ⎟− ⎟ ED ⎥ ⎜ RTref ⎠ ⎝ ⎦
⎞ ⎤ ⎟ − 1⎥ ⎟ ⎥ ⎠ ⎦
−1
⎫ ⎪ ⎬ − c2Tref ⎪⎭
⎞ ⎟, ⎟ ⎠
(17.2.26)
is more inconvenient then those of the Reymann (1990) model because it includes the exponential integral ∞
Ei ( x ) = 0.57721566... + ln ( x ) + ∑ 1
xn , x>0 n × n!
(17.2.27)
where ⎛ n 1 ⎞ lim ⎜ ∑ − ln ( n ) ⎟ = 0.57721566... n →∞ k ⎝ k =1 ⎠
(17.2.28)
is the Euler’s constant. This is the reason why the Reymann (1990) model is recommended. For completeness I have to mention that other data approximation in form of polynomials are also available in the literature e.g. Leibowitz et al. (1976) c ps = 194.189 + 26.277 × 10−2 T − 18.135 × 10−5 T 2 + 4.737 × 10−8 T ,
(17.2.29)
for temperatures between 298 and 3023 K. 17.2.1.2 Solid density The solid density is considered as a function of the temperature only. Fink et al. (1981a,b) reported the relation ρ s = ρ s 0 ( c1 + c2T + c3T 2 + c4T 3 )
(17.2.30)
566
17. Thermo-physical properties for severe accident analysis
where ρ s 0 = 10970 ± 1% kg/ m³, c1 = 1.0056 , c2 = −1.6324 ×10−5 , c3 = −8.3281× 10−9 , c4 = 2.0176 × 10−13 . The experimental data are fitted by these approximations with an error of 1% up to 2500 K, an error of 7% up to 3120 K and an error of 9% over the melting point. 17.2.1.3 The derivative of the solid density with respect to the temperature d ρs = ρ s 0 c2 + 2c3T + 3c4T 2 , dT
(
)
(17.2.31)
and consequently the thermal expansion coefficient is
βs = −
c2 + 2c3T + 3c4T 2 1 ⎛ d ρs ⎞ = − . ρ s ⎜⎝ dT ⎟⎠ p c1 + c2T + c3T 2 + c4T 3
(17.2.32)
17.2.1.4 Solid thermal conductivity If T < 2670 the solid thermal conductivity is computed as follows. For T < 80 then λs = ( 6.8337 × 10−2 + 1.6693 × 10−4 T + 3.1886 × 10−8 T 2 )
−1
(17.2.33)
else λs = +
( 6.8337 ×10
−2
+ 1.6693 × 10−4 T + 3.1886 × 10−8 T 2
1.2783 × 10−1 T 1.1608 ⎛ ⎞ exp ⎜ ⎟ −5 ⎝ 8.6144 ×10 T ⎠
)
−1
(17.2.34)
Fink et al. (1981). For T > 2670 λs = 4.1486 − 2.2673 × 10−4 T ,
(17.2.35)
Fink et al. (1981a,b). The standard deviation of the last two approximation compared to experimental data is reported to be 6.2%. It is also reported by these authors that the porosity decrease the thermal conductivity as follows λs = λs ,α = 0 ⎡⎣1 − ( 2.5 ± 1.5 ) α1 ⎤⎦ , 1
where α1 is the void volumetric fraction of the oxide.
(17.2.36)
17.2 Uranium dioxide caloric and transport properties
567
Approximation for the thermal conductivity of UO2 with 95% of the theoretical density is proposed by Malang (1975): λs =
3825 + 6.080109533 × 10 −11 T 3 . T + 129.4
This correlation reproduces data at 500 K with 1%, at 1000 K with2% and at 2500 K with 8%. In the Soviet literature Karim (1976) used similar form λs =
ρUO2 4000 + 3.4 × 10−14 T 4 , 10953.4 T + 130
and Kusnezov and Katkovskii (1975) λs = 7.69 − 5.78 × 10−3 T + 1.71× 10−6 T 2 .
17.2.1.5 Solid sonic velocity If the volumetric thermal expansion coefficient and the isothermal coefficient of compressibility are known the velocity of sound follows from the equation: 1 = k ρ − T β 2 cp . a2
(17.2.37)
If the elasticity modulus is known the velocity of sound is as = E ρ s .
(17.2.38)
Hagrman et al. (1990) proposed the following model for stoichiometric UO2
(
)
Es , st = 2.334 × 1011 ⎡⎣1 − 2.752 (1 − ρ s , real ρ s ,theoretical ) ⎤⎦ 1 − 1.0915 × 10−4 T N / m²
(17.2.39) valid in 450 < T < 3113 K . For non stoichiometric mixtures of UO2 and PuO2
(
)
Es = Es , st exp ( − Bx ) 1 + 0.15CPuO2 ,
(17.2.40)
where B = 1.34 for hyper-stoichiometric fuel or 1.75 for hypo-stoichiometric fuel and x is the magnitude of the deviation from the stoichiometry in MO2 ± x . With
568
17. Thermo-physical properties for severe accident analysis
this we have at the solidus and the liquids site of the saturation line approximately 3994.9 and 5396.5 m/s as shown in Fig. 17.2.4. 17.2.2 Liquid 17.2.2.1 Caloric equation of state For the velocity of sound Fink et al. (1981a) reported the following dependence on the liquid temperature within 3138 and 3196 K al (T ) = 3600 + 0.5769T = a1 + a2T .
(17.2.41)
Velocity of sound in m/s
5500 UO2 5000 4500 4000 3500
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.4. Velocity of sound of solid and liquid UO2 as a function of the temperature at 1bar pressure
Figure 17.2.4 presents the velocity of sound of solid and liquid UO2 as a function of the temperature at 1 bar pressure. From the definition of the sonic velocity, Eq. (3.202) Kolev (2007), we have ⎛ ∂ρl ⎞ 1 1 , ⎜ ⎟ = 2 = 2 ∂ p a ⎝ ⎠s ( a1 + a2T ) l
(17.2.42)
Leibowitz et al. (1976) reported the following dependence on the liquid temperature within 3043 and 3373 K for the liquid density
17.2 Uranium dioxide caloric and transport properties
ρl 0 (T , p0 ) =
r1 11.08 × 103 = . 1 + r2 (T − r3 ) 1 + 9.3 × 10−5 (T − 273)
569
(17.2.43)
Therefore r1r2 ⎛ ∂ρl ⎞ ⎛ ∂ρl 0 ⎞ = − r2 ρl20 r1 ⎜ ∂T ⎟ = ⎜ ∂T ⎟ = − 2 ⎝ ⎠p ⎝ ⎠p ⎡⎣1 + r2 (T − r3 ) ⎤⎦
(17.2.44)
or r2 1 ⎛ ∂ρ ⎞ = f ⎡⎣T , ρl (T , p ) ⎤⎦ = f (T , p ) ≈ − . ⎜ ⎟ ρl ⎝ ∂T ⎠ p 1 + r2 (T − r3 ) 5
Vol. th. expansion coefficient*10 , -
βl = −
(17.2.45)
7,4 UO2 7,3
7,2
7,1
3200
3300
3400
3500
T in K
Fig. 17.2.5. The volumetric thermal expansion coefficient of liquid UO2 as a function of the temperature at 1 bar pressure
Figure 17.2.5 shows the volumetric thermal expansion coefficient β of liquid UO2 as a function of the temperature at 1 bar pressure. From the know definition equation of the sound velocity
⎛ ∂ρ ⎞ 1 ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎜ ⎟ = 2 =⎜ ⎟ −⎜ ⎟ ∂ p a ⎝ ⎠s ⎝ ∂p ⎠T ⎝ ∂T ⎠ p
⎛ ∂h ⎞ ⎟ −1 ⎝ ∂p ⎠T
ρ⎜
ρcp
= kρ − β 2
T , cp
compare with Eq. (3.250) Kolev (2007), and using the relation
(17.2.46)
570
17. Thermo-physical properties for severe accident analysis
⎛ ∂h ⎞ 1 ⎡ T ⎛ ∂ρ ⎞ ⎤ 1 ⎛ ∂v ⎞ = ⎢1 + ⎜ ⎜ ⎟ = v −T ⎜ ⎟ ⎟ ⎥ = (1 − T β ) , ⎝ ∂T ⎠ p ρ ⎢⎣ ρ ⎝ ∂T ⎠ p ⎦⎥ ρ ⎝ ∂p ⎠T
(17.2.47)
compare with Hendricks et al. (1975, p. 54), I obtain a −2 = k ρ − β 2 T c p ,
(17.2.48)
and ⎛ d ρl ⎞ T 1 ⎛ ∂ρ ⎞ 1 β l2T 1 = 2+ ⎜ ⎟ = 2+ ⎟ 2 ⎜ dp a c a c ⎝ ⎠T pl ρ l ⎝ ∂T ⎠ p pl l l 2
= f ⎡⎣ al (T ) , β l (T , p ) , c pl (T ) ⎤⎦ = f (T , p ) .
(17.2.49)
With this relation the density is then function of temperature and pressure ⎛ d ρl ⎞ ⎟ ( p − p0 ) . ⎝ dp ⎠T
ρl (T , p ) ≈ ρl 0 (T ) + ⎜
(17.2.50)
Note that the derivative in the above equation is function of temperature and week function of pressure. Figure 17.2.6 presents the density for solid and liquid UO2 as a function of the temperature at 1 bar pressure with the approximation documented here. 11000
Density in kg/m³
10500
UO2
10000 9500 9000 8500
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.6. Density of solid and liquid UO2 as a function of the temperature at 1 bar pressure
17.2 Uranium dioxide caloric and transport properties
571
Having the results for β l (T ) and ρl (T , p ) the isothermal coefficient of compressibility and derivative of the specific enthalpy with respect to the pressure are computed kl =
1 ⎛ 1 βl2T ⎜ + ρl ⎜⎝ al2 c pl
⎞ ⎟, ⎟ ⎠
(17.2.51)
(17.2.52)
8.0
12
Is. coef. of compressibility*10 , -
⎛ ∂hl ⎞ ⎜ ⎟ = (1 − T β l ) ρl . ⎝ ∂p ⎠T
UO2
7.9 7.8 7.7 7.6
3200
3300 T in K
3400
3500
Fig. 17.2.7. The isothermal coefficient of compressibility of liquid urania as a function of the temperature at 1 bar pressure
Figure 17.2.7 shows the isothermal coefficient of compressibility k of liquid urania as a function of the temperature at 1 bar pressure. The specific liquid enthalpy and entropy are then computed as follows. Integrating the differential form of the enthalpy equation ⎛ ∂h ⎞ dh = c p dT + ⎜ ⎟ dp , ⎝ ∂p ⎠T
(17.2.53)
having in mind that ( ∂h ∂p )T is function of the temperature and only week temperature of pressure results in
572
17. Thermo-physical properties for severe accident analysis
⎛ ∂h ⎞ h ≈ href + c p T − Tref + ⎜ ⎟ p − pref ⎝ ∂p ⎠T
(
)
(
)
(17.2.54)
with href = h′ = 1406331.34612 J/kg at Tref = T ′′′ = 3113.15 K, pref = 105 Pa and c p =503 J/(kgK). Similarly integrating ds = c p
dT 1 ⎛ ∂h ⎞ + ⎜ ⎟ dp T T ⎝ ∂p ⎠T
(17.2.55)
results in
(
)
s = sref + c p ln T Tref +
1 ⎛ ∂h ⎞ ⎜ ⎟ p − pref T ⎝ ∂p ⎠T
(
)
(17.2.56)
with sref = s ′ = 890.736258909 J/(kgK). The inversed computation of the temperature of the liquid for known specific entropy and pressure is then ⎫⎪ ⎤ 1 ⎛ ∂h ⎞ ⎪⎧ ⎡ T = Tref exp ⎨ ⎢ s − sref − ⎜ ⎟ p − pref ⎥ c p ⎬ . T ⎝ ∂p ⎠T ⎪⎭ ⎦⎥ ⎩⎪ ⎣⎢
(
)
(17.2.57)
At maximum 5 iterations are needed to solve the above equation with respect to the unknown temperature starting with T = T ′′′ . Figures 17.2.1, 17.2.2 and 17.2.3 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as a temperature functions at 1bar pressure for solid and liquid UO2. For completeness, the Fischer’s (1992) approximations of the saturation liquid density as functions of the temperature are given below in implicate form: T ′ = 3120 + 1.092 ( 8860 − ρ ′ ) − 1.7 × 10−6 ( 8860 − ρ ′ ) , T ′′′ < T ′ < 9951.66 K, 2
T ′ = 10600 − 427.13 × 10 −6 ( ρ ′ − 1560 ) − 1120 × 10 −9 ( ρ ′ − 1560 ) 2
−1242 × 10−12 ( ρ ′ − 1560 ) − 365.1× 10 −15 ( ρ ′ − 1560 ) 4
5
3
, 9951.66 < T ′ < Tc K,
The transport properties for liquid UO2 will be discussed below.
17.2 Uranium dioxide caloric and transport properties
573
17.2.2.2 Transport properties Liquid thermal conductivity: For the thermal conductivity of liquid UO2 Fink et al. (1981a,b) give the value 11 W/(mK) measured by Kim et al. (1977). In the overview given in Gmelin (1986) this high value is explained by systematic errors and as correct thermal conductivity of liquid UO2 a value of 2.2 W/(mK) is suggested. In 1985 an theoretical analysis of the three only existing experiments is done by Fink and Leibowitz (1985). This work is not taken into account by the author of the overview in Gmelin (1986). Fink and Leibowitz (1985) estimated the systematic deviation of the three different experiments. Summarizing these investigations a thermal conductivity of
λl = 5.6 W/(mK)
(17.2.58)
Thermal conductivity in W/(mK)
with an uncertainty of 1.2 W/(mK) is given. This value is recommended. In an uncertainty estimation done by Chu et al. (1996) this thermal conductivity is confirmed.
8 7 UO2
6 5 4 3 2
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.2.8. Thermal conductivity of solid and liquid UO2 as a function of the temperature at 1 bar pressure
Figure 17.2.8 presents the thermal conductivity as a function of the temperature at 1bar pressure for solid and liquid UO2. Remarkably the lower conductivity of the solid compared with the liquid in a large region. Liquid surface tension: Chawla et al. (1981) reported for the liquid surface tension σ l = 0.497 − 0.00019 (T - T0 )
(17.2.59)
574
17. Thermo-physical properties for severe accident analysis
valid for 3120 < T < 3225 K. If the temperature is higher than 3225 K, a constant value is assumed.
Surface tension N/m
0,500 0,495 0,490
UO2
0,485 0,480 0,475 0,470
3200
3300 T in K
3400
3500
Fig. 17.2.9. Surface tension of liquid UO2 as a function of the temperature at 1bar pressure
Dynamic viscosity /1000 in kg/(ms)
Figure 17.2.9 presents the surface tension of liquid UO2 as a function of the temperature at 1 bar pressure. No data are known to me for T > 3225 K. 4,4 4,3
UO2
4,2 4,1 4,0 3,9 3,8 3,7
3200
3300 T in K
3400
3500
Fig. 17.2.10. Dynamic viscosity of liquid UO2 as a function of the temperature at 1 bar pressure
Liquid dynamic viscosity: Woodly cited by Fink et al. (1981a,b) proposed for the liquid viscosity ηl = 0.000988exp ( 4629 T ) , T > T0
(17.2.60)
17.2 Uranium dioxide caloric and transport properties
575
with an error of 2.5 %. Figure 17.2.10 presents the dynamic viscosity of liquid UO2 as a function of the temperature at 1 bar pressure. Liquid emissivity: Fink et al. (1982) recommended for the emmisivity the following averaged value ε ( λ = 0.63μm ) = 0.87 .
(17.2.61)
In Hohorst (1990) this value is recommended for T < 1273.15 K . The authors recommended ε = 1.311 − 4.404 × 10 −4 T
(17.2.62)
for 1273.15 < T < 2323.15 K and ε = 0.408
(17.2.63)
for T > 2323.15 K . For more accurate analysis at high temperatures Fink et al. (1982) proposed the approximation ε ( λ = 0.63μm ) = 1 −0.16096 exp ⎡⎣ −3.7897 ×10−4 ΔT − 3.2718 ×10−7 ΔT 2 ⎤⎦ ,
(17.2.64)
where ΔT = T − 3120 , reproducing experimental data within 3120–4200 K with uncertainty of 0.14%. 17.2.3 Vapor The boiling pressure of UO2 liquid depends on pressure, Bober et al. (1978): p ′ = 3.09227 × 1012 T −0.265 exp ( − 54871 T ) .
(17.2.65)
As discussed by Chawla et al. (1981, p. 60) the uncertainties in this field remain large. The above relation gives a boiling temperature at 1 bar of 3606.5 K. The critical temperature estimated by Chawla et al. (1981, p. 61) using the Watson’s formula is Tc = 8816 K .
(17.2.66)
576
17. Thermo-physical properties for severe accident analysis
The Clapeyron’s equation dp ′ h′′ − h′ 54871 T − 0.265 , = T′ = 3.09227 × 1012 1.265 dT ′ v′′ − v′ exp ( 54871 T ) T
(17.2.67)
allows to estimate the vapor specific volume knowing the liquid specific volume at the saturation line v′′ (T ′ ) = v′ +
dT ′ T ′ ( h′′ − h′ ) . dp
(17.2.68)
Karow (1977) noted that the use of the Clapeyron’s equation for computing the specific volume of the saturated vapor is questionable because of the ionization. Breitung and Reil (1985) reported approximation for the saturation pressure of large number of data with
(
)
ln 10−6 p ′ = 23.7989 − 29605.5 T − 4.75783ln T .
(17.2.69)
This relation gives 3817 K for normal boiling point. The specific heat at constant pressure for saturated uranium oxide by the same authors as c′′p = 553.3 + 2.189 × 10−2 T ,
(17.2.70)
and therefore h′′ − hg ( 298 K ) = −221150 + 553.3T + 1.0945 ×10−2 T 2 .
(17.2.71)
These relations are valid for temperatures within 3120 to 8500 K, pressures up to 540 bar for pure UO2, reactor grade UO2 and reactor grade (U0.77Pu0.23)O2. Fischer (1989) reported the following approximation for the total pressure including oxygen
(
)
log10 10−6 pU′ + O2 = 47.287 − 36269 T − 4.8665ln T + 0.3615 × 10−3 T
and for the saturation pressure
(
)
log10 10−6 p ′ = 39.187 − 34715 T − 3.8571ln T + 0.1921× 10−3 T .
Both equations are differs 7000 K within the uncertainty band. For larger temperatures the last equation is recommended.
References
577
Fischer (1992) introduced a modification into the well known Redlich–Kwong equation (1949) of state for pure vapors: p=
f (T ) RT − , v − a1 v ( v + a3 )
f (T ) = a2 (T Tc )
{
a4
for T < Tc ,
}
f (T ) = a2 1 + a4 ⎡⎣(T Tc ) − 1⎤⎦ for T ≥ Tc ,
The parameters a1 , a2 , and a3 , are determined from the critical constants and the fact that the critical isotherm an a pressure–volume p–v diagram has an inflection point at the critical point. Consequently, the first and second derivatives of the pressure with respect to specific volume are zero. The method to determine a4 is material dependent. It can be used to fit the slope of the vapor–pressure curve at the critical temperature, Riedel (1954), or to obtain as best fit as possible to the vapor density. The equation reduces to the van der Waals equation if a3 , = 0 and a = const, and to the original Redlich–Kwong equation Redlich and Kwong (1949) if a1 = a3 and a4 = –1/2. The modification Redlich–Kwong equation reproduces the evaluated UO2 vapor data with a1 = 1.41301×10−4 , a2 = 2.94299 × 102 , a3 = 2.85846 × 10−4 , a4 = 0.2 rather well, Fischer (1992).
References Brassfield HC et al. (April, 1968) Recommended property and reactor kinetics data for use in evaluating a light-water-coolant reactor loss-of-coolant incident involving Zircaloy4 or 304-SS-clad UO2, GEMP-482 Breitung W and Reil KO (August, 1985) In-pile vapor pressure measurements on UO2 and (U, Pu)O2, Kernforschungszentrum Karlsruhe, KfK 3939 Bober M and Singer J (1987) Vapor pressure determination of liquid UO2 using a boiling point technique, Nucl. Sci. Eng. vol 97 pp 344–352 Chawla TC et al. (1981) Thermophysical properties of mixed oxide fuel and stainless steel type 316 for use in transition phase analysis, Nuclear Engineering and Design, vol 67 pp 57–74 Chu CC, Sieniki JJ and Beker L Jr (1996) Uncertainty analysis for thermophysical properties used in in-vessel retention analysis, in Teofanous TG et al. (Oct. 1996) Invessel coolability and retention of a core melt, DOE/ID-10460 vol 1 Gmelin (1986) Handbook of Inorganic Chemistry, 8th ed. Uranium, Supplement Volume C5, Chap. 4.8.6: Thermal conductivity, Springer-Verlag, Berlin Fink JK, Chasanov MG and Leibowitz L (April, 1981a) Thermodynamic properties of uranium dioxide, ANL-CEN-RSD-80-3 Fink JK, Chasanov MG and Leibowitz L (1981b) Thermo-physical properties of uranium dioxide. J. Nucl. Mater., vol 102, pp 17–25
578
17. Thermo-physical properties for severe accident analysis
Fink JK, Chasanov MG and Leibowitz L (1982) Properties for safety analysis, ANL-CENRSD-82-2 Fink JK and Leibowitz L (1985) An analysis of measurements of the thermal conductivity of liquid urania. High Temperatures-High Pressures, vol 17, pp 17–26 Fischer EA (1989) A new evaluation of the urania equation of state based on recent vapor pressure. Nucl. Sci. Eng., vol 101, pp 97–116 Fischer EA (May, 1992) Fuel evaluation of state data for use in fast reactor accident analysis codes, KfK 4889, Kernforschungszentrum Karlsruhe Hagrman DL, Laats ET and Olsen CS (1990) In Hohorst JK ed. SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-waterreactor accident analysis, NUREG/CR-5273, EGG-2555 Hendricks RC, Baron AK and Peller CP (February, 1975) GASP – A computer code for calculating the thermodynamic and transport properties for ten fluids: Parahydrogen, helium, neon, methane, nitrogen, carbon monoxide, oxygen, fluorine, argon, and carbon dioxide, NASA technical note NASA TN D-7806, Washington, DC Hohorst JK ed. (1990) SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR5273, EGG-2555 Karim S (1976) PhD Thesis, MEI Moscow Karow HU (February, 1977) Thermodynamic state, specific heat and enthalpy function of saturated UO2 vapor between 3000K and 5000K, Kernforschungszentrum Karlsruhe, KfK 2390 Kim et al. (May 12–12, 1977) Measurements of thermal diffusivity of molten UO2. Proceedings of the 7th Symposium on Thermophysical Properties at the National Bureau of Standards, Gaithersberg, MD, CONF 770537-3, pp 338–343 Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo Kusnezov VD and Katkovskii EA (1975) Teplovoj I gidravlicheskij raschet na EVM reaktorov s vodoj pod davleniem, MEI Moscow Leibowitz L et al. (April, 1976) Properties for LMFBR safety analysis, ANL-CEN-RSD76-1 Lyon WF and Baily WE (1976) The solid-liquid diagram of the UO2-PO2 system. J. Nucl. Mater, vol 22, p 332 Malang S (1975) Simulation of nuclear fuel rods by using process-computer controlled power for indirect electrically heated rods, Oak Ridge National Laboratory, ORNLTM-4712 (GEMP-482) Redlich O and Kwong JNS (1949) On the thermodynamics of solutions. V, An equation of state. Fugacities of gaseous solutions. Chem. Rev. vol 44, pp 233–244 Reymann GA (1990) Specific heat capacity and enthalpy. In: Hohorst JK ed. SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR-5273, EGG-2555 Riedel L (1954) Eine neue universelle Dampfdruckformel. Chem.-Ing. Tech., vol 26, pp 83–89
17.3 Zirconium dioxide
579
17.3 Zirconium dioxide Zirconium is the main constituent of the cladding of nuclear fuels for water cooled nuclear reactors. During accidents leading to overheating the cladding may oxidize in oxygen or water environment. The produced zirconium oxide is then a part of the materials participating in the following events, relocation, melt-water interactions, freezing etc. It is obvious that thermal and transport properties also of zirconium oxide are of primary importance for understanding the accidental processes in the nuclear engineering. The subject of this section is to provide a useful set of approximation describing properties of solid and liquid zirconium oxide. The solidus and liquidus temperatures of ZrO2 are reported to be different and function of the content of oxygen. For stoichiometric oxide the melting temperature is T ′′′ = 2973 K,
(17.3.1)
Hagrman (1990). 17.3.1 Solid For increasing temperatures at T12 = 1478 K the monoclinic zirconium dioxide changes to tetragonal zirconium dioxide, at T23 = 2000 K tetragonal zirconium dioxide starts to coexists with cubic zirconium dioxide. After T23 = 2558 K only cubic zirconium dioxide exists, and finally above T ′′′ = 2973 K the zirconium dioxide is molten. I introduce identifiers of the region as follows: 300 < T ≤ T12 T12 < T ≤ T23 T23 < T ≤ T34 T34 < T ≤ T ′′′
i = 1, monoclinic, i = 2, tetragonal only, i = 3, tetragonal and cubic, i = 4, cubic.
17.3.1.1 Solid specific capacity at constant pressure The approximation proposed by Hammer (1967) for the solid specific heat at constant pressure is c ps = ai1 + ai 2T + ai 3 T 2 ,
where
(17.3.2)
580
17. Thermo-physical properties for severe accident analysis
⎛ 565 6.11×10 −2 ⎜ 604.5 0 a=⎜ ⎜ 171.7 0.2164 ⎜⎜ ⎝ 171.7 0.2164
− 1.14 × 107 ⎞ ⎟ 0 ⎟. ⎟ 0 ⎟⎟ 0 ⎠
For T > 2973 K the ZO2 is in liquid state and c pl = 815 J/(kgK).
(17.3.3)
The transition from one structures into the other are associated with the following enthalpies of formation Δh12 ( at T12 ) = 48 200 J/kg,
(17.3.4)
Δh34 ( at T34 ) = 102 000 J/kg.
(17.3.5)
The heat of fusion is h′ − h′′′ = 706 000 J/kg.
(17.3.6)
The specific enthalpy and the specific entropies are obtained after the integration setting href (Tref ) = 0 and sref (Tref ) = 0 and selecting arbitrary Tref = 298.15 K:
(
)
hs ,1 (T ) = a11 T − Tref +
(
)
(
1 a12 T 2 − Tref2 − a13 1 T − 1 Tref 2
) for T ≤ T
,
(17.3.7)
12
hs ,2 = hs ,1 (T12 ) + Δh12 + a21 (T − T12 ) for T12 < T ≤ T23, hs ,3 (T ) = hs ,2 (T23 ) + a31 (T − T23 ) +
(
1 a32 T 2 − T232 2
hs ,4 (T ) = hs ,3 (T34 ) + Δh34 + a41 (T − T34 ) +
(
)
(17.3.8)
for T23 < T ≤ T34 ,
(17.3.9)
)
1 a42 T 2 − T342 for T34 < T ≤ T ′′′ , 2
(17.3.10)
(
)
(
)
(
1 ss ,1 (T ) = a11 ln T Tref + a12 T − Tref − a13 1 T 2 − 1 Tref2 2
) for T ≤ T
ss ,2 (T ) = ss ,1 (T12 ) + Δh12 T12 + a21 ln (T T12 ) for T12 < T ≤ T23,
, (17.3.11)
12
(17.3.12)
17.3 Zirconium dioxide
ss ,3 (T ) = ss ,2 (T23 ) + a31 ln (T T23 ) + a32 (T − T23 ) for T23 < T ≤ T34 ,
581
(17.3.13)
ss ,4 (T ) = ss ,3 (T34 ) + Δh34 T34 + a41 ln (T T34 ) + a42 (T − T34 ) for T34 < T ≤ T ′′′ ,
(17.3.14) 900 Hagrman (1990) cp model
cp in J/(kgK)
800 700 600 500 400
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.3.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid ZrO2
3000
h in kJ/kg
2500 2000
Hagrman (1990) cp model
1500 1000 500 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.3.2. Specific enthalpy as a function of the temperature of solid and liquid ZrO2. p = 1 bar
with this we have h′′′ = 1856215.45843 J/kg,
(17.3.15)
582
17. Thermo-physical properties for severe accident analysis
h′ = 2562215.45843 J/kg, s ′′′ = 1469.35394164 J/(kgK), s ′ = 1706.82451009 J/(kgK), h′ − h′′′ = 706000.000000 J/kg, melt enthalpy, s ′ − s ′′′ = 237.470568449 J/(kgK), melt entropy.
(17.3.16) (17.3.17) (17.3.18) (17.3.19) (17.3.20)
Figures 17.3.1, 17.3.2 and 17.3.3 demonstrate the cp, h and s functions of the temperature for constant pressure of 1 bar. The jumps are due to the phase transitions and due to the melting. 2000 Hagrman (1990) cp model
s in J/(kgK)
1500 1000 500 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.3.3. Specific entropy as a function of the temperature of solid and liquid ZrO2. p = 1 bar
The standard deviation error for specific heat capacity and enthalpy is given by the authors with 0.1. Slightly different approximation of the specific enthalpy based on large data set is reported by Cordfunke and Konings (1990, pp. 471–472). 17.3.1.2 Solid density The solid density is considered as a function of the temperature only. The reference solid density of black oxide at 300 K is reported by Gilchrist (1976) to be ρ s 0 = ρ s ( 300 ) = 5800 kg/m³.
(17.3.21)
Hammer (1967) reported ρ s = ρ s 0 (1 − 3ε 0 ) ± 300 kg/m³.
(17.3.22)
17.3 Zirconium dioxide
583
The linear thermal strain of zirconium oxide in m/m is ε = c11 + c12T within 300 < T ≤ T12,
(17.3.23)
for monoclinic zirconium oxide and ε = c21 + c22T within T21 < T ≤ T ′′′ .
(17.3.24)
for tetragonal only and tetragonal and cubic zirconium oxide. The constants are −2.34 × 10 − 3 ⎞ ⎛ 7.8 × 10 − 6 c=⎜ ⎟. ⎝1.302 × 10 − 5 −3.338 × 10 − 2 ⎠
(17.3.25)
Note the 7.7% decrease of the volume across 1478 K. The liquid zirconium dioxide is reported to have 5% reduction in volume if it melts. 17.3.1.3 The derivative of the solid density with respect to the temperature After the differentiation of the density function with respect to the temperature we have d ρs = − ρ s 0 3ci ,2 for i = 1, 2. dT
(17.3.26)
17.3.1.4 Solid thermal conductivity Hammer (1967) reported the following relation for the thermal conductivity of the solid ZrO2 λs = 0.835 + 1.81× 10−4 T ± 0.75 W/(mK).
(17.3.27)
17.3.1.5 Solid sonic velocity If the elasticity modulus is known the velocity of sound is as = E ρ s .
(17.3.28)
Hammer (1967) proposed the following model for stoichiometric zirconium oxide Es , st = 1.637 × 1011 − 3.77 × 107 T for 300 < T < T12 ,
(17.3.29)
Es , st = 2.255 × 1011 − 8.024 × 107 T for T12 < T < T ′′′ .
(17.3.30)
584
17. Thermo-physical properties for severe accident analysis
Velocity of sound in m/s
5500 ZO2
5000 4500 4000 3500 3000 2500 2000 1500
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.3.4. Velocity of sound of solid and liquid UO2 as a function of the temperature at 1bar pressure
The expected error is ± 20%. In Fig. 17.3.4 the velocity of sound as function of the temperature is presented using only the elasticity modulus for the lower temperature region because the dependence in the second region gives unphysical results for temperature larger then 1800 K. 17.3.1.6 Solid emmisivity Hammer (1967) reproduced data by several authors with the following approximations for the emmisivity of the solid: ε = 0.325 + 0.1246 × 106 δ ± 0.1 for δ ≤ 3.88 × 10 −6 m
(17.3.31)
ε = ( 0.808642 − 50δ ) exp ⎡⎣(T − 1500 ) 300 ⎤⎦ for δ > 3.88 × 10−6 m ,
(17.3.32)
where δ is the oxide layer thickness in m. 17.3.2 Liquid Specific capacity at constant pressure: Hohorst (1990) proposed to use for the specific capacity at constant pressure the following constant c pl = 815 J/(kgK).
(17.3.33)
17.3 Zirconium dioxide
585
For computation of the liquid specific enthalpy, the derivative of the liquid specific enthalpy with respect to pressure at constant temperature, the liquid specific entropy and the liquid temperature as a function of pressure and specific entropy I use the results from Ch. 17.1 with the following reference values. The liquid specific enthalpy is h′ = 2562215.45843 J/kg. The liquid specific entropy is s ′ = 1706.82451009 J/(kgK). Density: The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105Pa, T0 = 2973 K, ρl 0 = 5991.4 kg/m³, the volumetric thermal expansion coefficient is β = 0.916/ ρl 0 K–1, and the isothermal compressibility is κ = 4.234×10–11 Pa–1.
6100
Density in kg/m³
6000
ZO2
5900 5800 5700 5600 5500
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.3.5. Density of the solid and liquid ZrO2 as a function of the temperature
Figure 17.3.5 shows the density of the solid and liquid ZrO2 as a function of the temperature. The first jump is due to the phase transition in the solid state and the second due to melting. The liquid density derivative with respect to temperature, the liquid density derivative with respect to pressure and the liquid velocity of sound are computed as shown in Chap. 17.1. Liquid thermal conductivity: Hammer (1967) reported for the liquid thermal conductivity the constant
λl = 1.4.
(17.3.34)
17. Thermo-physical properties for severe accident analysis
Thermal conductivity in W/(mK)
586
1,5 1,4 1,3
ZO2
1,2 1,1 1,0 0,9 0,8
500 1000 1500 2000 2500 3000 3500 T in K
Dynamic viscosity /1000 in kg/(ms)
Fig. 17.3.6. Thermal conductivity of solid and liquid ZrO2 as a function of the temperature at 1 bar pressure
4,5 4,0
ZO2
3,5 3,0 2,5
3000
3100
3200 3300 T in K
3400
3500
Fig. 17.3.7. Dynamic viscosity of liquid ZrO2 as a function of the temperature at 1 bar pressure
Figure 17.3.6 shows the thermal conductivity of solid and liquid ZrO2 as a function of the temperature at 1bar pressure. Liquid surface tension: The liquid surface tension for ZrO2 is not known to me. For analysis I use the value σ l = 0.535 N/m.
(17.3.35)
References
587
which has to be improved in the future. Liquid dynamic viscosity: The liquid dynamic viscosity is approximated in Hohorst (1990) with ηl = 0.000122 exp (10500 T )
(17.3.36)
with the limit of ηl = 0.004170566344 for T < T0 . Figure 17.3.7 visualizes this dependence. Note that Chu et al. (1996) estimated the uncertainty of such type relations, in this particular case ηl = (1.5 ± 0.5 ) × 10-4 exp (10 430 T )
to be 33%. References Cordfunke EHP and Konings RJM (1990) Thermochemical Data for Reactor Materials and Fussion Products. North-Holland, Amsterdam Chu CC, Sieniki JJ and Beker L Jr (October, 1996) Uncertainty analysis for thermophysical properties used in in-vessel retention analysis. In: Teofanous TG et al., eds., In-Vessel Coolability and Retention of a Core Melt, DOE/ID-10460, vol 1, U.S. Department of Energy Report Gilchrist KE (1976) Thermal property measurements on Zirkaloy-2 and associated oxide layers. J. Nucl. Mater., vol 62, pp 257–264 Hagrman DL (1990) In: Hohorst JK ed., SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR-5273, EGG-2555 Hammer RR (September, 1967) Zircaloy-4, uranium dioxide and materials formed by their interaction, A literature review with extrapolation of physical properties to high temperatures, IN-1093 Hohorst JK ed. (1990) SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR5273, EGG-2555
17.4 Stainless steel
589
17.4 Stainless steel Stainless steel is an alloy of Fe, Cr, Ni, Mo and a small amount of C. On the average it may contain by weight 69% Fe, 17% Cr, 12% Ni and 2% Mo with mol-mass M = 0.05593756 kg mol• 1,
(17.4.1)
Chawla et al. (1981, p. 62). The melting temperature is T ′′′ = 1700 K .
(17.4.2)
Using the critical temperature Tc =9600 K evaluated by Fortov et al. (1975), Fischer (1989) computed the following critical properties of stoichiometric UO2: ρc =1143 kg/m³, pc =456.76 ×106 Pa.
17.4.1 Solid
17.4.1.1 Solid specific capacity at constant pressure Perry and Green (1985) reported the following heat capacities at constant pressure for pure iron c ps = 308.91 + 0.4772 T ± 3% within 273 and 1041 K, 457.76 + 0.2513T ± 3% within 1041 and 1179 K, 628.3 ± 5% within 1179 and 1674 K, 747.98 ± 5% within 1674 and 1803 K, and 609.6 ± 5% for T >1803 K. Leibowitz et al. (1976) obtained from enthalpy data the following correlation for stainless steel type 316 c ps = c1 + c2T , T ≤ 1700 K ,
(17.4.3)
where c1 = 462.656 , c2 = 0.1338 . The specific melt enthalpy is reported to be h′′′ − h′ = 270330 J/kg.
(17.4.4)
The specific melt entropy is therefore s ′′′ − s ′ = ( h′′′ − h′ ) T ′′′ = 159.01764705882354. J/(kgK)
(17.4.5)
Different steels possess different dependences of the specific capacity at constant pressure on temperature. Two additional examples are given for the two type of steels used in designing of the German nuclear reactor vessels taken from Richter (1983). For ferrite steel 22NiMoCr 37 the specific capacity at constant pressure is presented in Fig. 17.4.1(b) and fitted by the following polynomial:
590
17. Thermo-physical properties for severe accident analysis
c ps = 569.69603 − 0.77608T + 0.00174T 2 − 1.22599 ×10 −6 T 3 + 2.83233 × 10−10 T 4
(17.4.6) 700
cp in J/(kgK)
650 600 550 500
Leibowitz et al. (1976) cp model
450 500
1000 T in K
700
700 22 NiMoCr 37 Fit
650 cp in J/(kgK)
650 cp in J/(kgK)
1500
600 550 500
X10 10 CrNiNb 18 9 Fit
600 550 500 450
450 400 600 800 1000 1200 1400 1600 T in K
400 600 800 1000 1200 1400 1600 T in K
Fig. 17.4.1. Specific capacity at constant pressure as a function of temperature: (a) US type 316; (b) German ferrite steel 22NiMoCr 37; (c) German austenite steel X10 10 CrNiNb 18 9
For austenite steel X10 10 CrNiNb 18 9 the specific capacity at constant pressure is presented in Fig. 17.4.1c and fitted by the following polynomial: c ps = 47.67669 + 0.10214T − 2.48663 × 10−5 T 2 + 3.95267 × 10−8 T 3 − 1.74743 × 10−11 T 4
(17.4.7) 17.4.1.2 Solid specific enthalpy The specific solid enthalpy is then
(
)
(
)
1 hs = c1 T − Tref + c2 T 2 − Tref2 , 2
(17.4.8)
with Tref = 298.15 K and hs (Tref ) = 0 . For h between h′′′ = 835968.35178193124 J/kg and h′ = 1106298.3517819312 J/kg the temperature is T ′′′ = 1700 K because of melting.
17.4 Stainless steel
591
17.4.1.3 Solid specific entropy The specific solid entropy is 800
cp in J/(kgK)
750 700 650 600 550 500
Leibowitz et al. (1976) cp model
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.2. Specific capacity at constant pressure as a function of the temperature of solid and liquid steel
3000 Leibowitz et al. (1976) cp model
h in kJ/kg
2500 2000 1500 1000 500 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.3. Specific enthalpy as a function of the temperature of solid and liquid steel. p = 1 bar
(
)
(
)
ss = c1 ln T Tref + c2 T − Tref ,
(17.4.9)
with ss (Tref ) = 0 . For s between s ′′′ = 992.95300457091207 J/(kgK) and s′ = 1151.9706516297356 J/(kgK) the temperature is 1700 K because of melting.
592
17. Thermo-physical properties for severe accident analysis
2000 Leibowitz et al. (1976) cp model
s in J/(kgK)
1500 1000 500 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.4. Specific entropy as a function of the temperature of solid and liquid steel. p = 1 bar
Figures 17.4.2, 17.4.3 and 17.4.4 presents the specific heat, the specific enthalpy and the specific entropy as a function of temperatures for 1 bar pressure. 17.4.1.4 Solid density The solid density is considered as a function of the temperature only: ρ s = c0 + c1T + c2T 2
(17.4.10)
Density in kg/m³
with c0 = 8084, c1 = −4.209 × 10−1 , c2 = −3.894 ×10−5 , Leibowitz et al. (1976).
8000 7900 7800 7700 7600 7500 7400 7300 7200 7100
Iron
400 600 800 1000 1200 1400 1600 T in K
8000 7900 7800 7700 7600 7500 7400 7300 7200 7100
22 NiMoCr 37 Fit
Density in kg/m³
Density in kg/m³
17.4 Stainless steel
400 600 800 1000 1200 1400 1600 T in K
8000 7900 7800 7700 7600 7500 7400 7300 7200 7100
593
X10 10 CrNiNb 18 9 Fit
400 600 800 1000 1200 1400 1600 T in K
Fig. 17.4.5. Density as a function of temperature: (a) iron; (b) ferrite steel 22NiMoCr 37; (c) austenite steel X10 10 CrNiNb 18 9
Different steels possess different density dependences on temperature. Two examples are given for the two type of steels used in designing of the German nuclear reactor vessels taken from Richter (1983). For ferrite steel 22NiMoCr 37 the density is presented in Fig. 17.4.5(b) and fitted by the following polynomial: ρ s = 7915.4444 − 0.22109T − 1.00051× 10−4 T 2.
(17.4.11)
For austenite steel X10 10 CrNiNb 18 9 the density is presented in Fig. 17.4.5(c) and fitted by the following polynomial: ρ s = 8031.05773 − 0.42608T − 2.13034 × 10−5 T 2 .
(17.4.12)
17.4.1.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is then d ρs = c1 + c2T . dT
(17.4.13)
17.4.1.6 Solid thermal conductivity Leibowitz et al. (1976) reported the following relation for the thermal conductivity of the solid steel λs = 9.248 + 1.57 × 10−2 T .
(17.4.14)
Different steels possess different thermal conductivity dependences on temperature. Two examples are given for the two type of steels used in designing of the German nuclear reactor vessels taken from Richter (1983). For ferrite steel
594
17. Thermo-physical properties for severe accident analysis
22NiMoCr 37 the thermal conductivity is presented in Fig. 17.4.6(b) and fitted by the following polynomial: λs = 30.32317 + 0.0951T − 1.92767 × 10−4 T 2 + 1.22827 × 10−7 T 3 − 2.53805 × 10−11 * T 4. (17.4.15) For austenite steel X10 10 CrNiNb 18 9 the thermal conductivity is presented in Fig. 17.4.6(c) and fitted by the following polynomial:
Iron
45 40 35 30 25 20 15 10
500
50 22 NiMoCr 37 Fit
45 40 35 30 25 20 15 10
(17.4.16)
50
400 600 800 1000 1200 1400 1600 T in K
1000 T in K
Thermal conductivity in W/(mK)
Thermal conductivity in W/(mK)
Thermal conductivity in W/(mK)
λs = 8.97849 + 0.02014T − 5.8917 × 10−6 T 2.
1500
50 X10 10 CrNiNb 18 9 Fit
45 40 35 30 25 20 15 10
400 600 800 1000 1200 1400 1600 T in K
Fig. 17.4.6. Thermal conductivity as a function of temperature: (a) iron; (b) ferrite steel 22NiMoCr 37; (c) austenite steel X10 10 CrNiNb 18 9
17.4.1.7 Solid sonic velocity If the elasticity modulus is known the velocity of sound is as = E ρ s .
(17.4.17)
In DIN 17240 the Es = f (T ) N / m² is given for stainless steel, see Fig. 17.4.7. The dependence is approximated here with
17.4 Stainless steel
(
)
E = b0 + b1T + b2T 2 + b3T 3 109
595
(17.4.18)
where b0 = 197.8549 , b1 = 0.07275 b2 = −1.884 ×10−4 b3 = 5.95263 × 10−8 . The reader may find thermo-physical and mechanical properties of 52 German steels in Richter (1983). Different steels possess different elasticity modulus dependences on temperature. Two examples are given for the two type of steels used in designing of the German nuclear reactor vessels taken from Richter (1983). For ferrite steel 22NiMoCr 37 the elasticity modulus is presented in Fig. 17.4.7(b) and fitted by the following polynomial: Elastic modulus in GN/m²
250 200 150 100 Steel, DIN 17240 Fit
50 0
500
1000 Temperature in K
250 22 NiMoCr 37 Fit
150 100 50 0
400 600 800 1000 1200 1400 1600 T in K
Elastic modulus in GN/m²
Elastic modulus in GN/m²
250 200
1500
200
X10 10 CrNiNb 18 9 Fit
150 100 50 0
400 600 800 1000 1200 1400 1600 T in K
Fig. 17.4.7. Elastic modulus as a function of the temperature: (a) SA-533B based on the German DIN 17240 Standard; (b) German ferrite steel 22NiMoCr 37; (c) German austenite steel X10 10 CrNiNb 18 9
⎛ 363.05196 − 1.12811T + 0.00307T 2 − 3.95421×10−6 T 3 ⎜ E =⎜ ⎜ +2.17705 ×10 −9 T 4 − 4.27926 × 10−13 T 5 ⎝
⎞ ⎟ 9 ⎟10 ⎟ ⎠
(17.4.19)
For austenite steel X10 10 CrNiNb 18 9 the elasticity modulus is presented in Fig. 17.4.7(c) and fitted by the following polynomial:
596
17. Thermo-physical properties for severe accident analysis
⎛ −57.73687 + 2.11813T − 0.00635T 2 + 8.0966 ×10 −6 T 3 − 3.2556 ×10−9 T 4 ⎞ ⎜ ⎟ 9 E =⎜ ⎟10 . ⎜ −2.09638 × 10 −12 T 5 + 2.20172 × 10 −15 T 6 − 5.15042 ×10−19 T 7 ⎟ ⎝ ⎠ (17.4.20)
17.4.1.8 Emissivity Text books give for polished steel with temperature between 273.15 and 1273.15 K emissivity of 0.07 to 0.17, for polished cast iron with C ≈ 4%, 0.2 to 0.25, for oxidized surfaces 0.55 to 0.6 and for rusted surfaces 0.6 to 0.8. For wrought iron with C ≈ 0.5% for polished surfaces 0.3 to 0.35 and for oxidized surfaces 0.9 to 0.95. 17.4.2 Liquid 17.4.2.1 Thermal properties For the velocity of sound Kurz and Lux (1969) reported the following dependence on the liquid temperature within 1773 and 1923 K al = 5838.2 − 1.02T = a1 + a2T .
(17.4.21)
Velocity of sound in m/s
5500 5000 4500 4000 3500 3000 2500
steel
2000 1500
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.8. Velocity of sound of solid and liquid steel as a function of the temperature at 1 bar pressure
Figure 17.4.8 presents the velocity of sound of solid and liquid steel as a function of the temperature at 1 bar pressure.
17.4 Stainless steel
597
From the definition of the sonic velocity, Eq. (3.202) Kolev (2007), we have ⎛ ∂ρ ⎞ 1 1 . ⎜ ⎟ = 2 = 2 ⎝ ∂p ⎠ s a ( a1 + a2T )
(17.4.22)
Leibowitz et al. (1976) reported the following dependence on the liquid temperature for the liquid density
ρl = 7.433 × 103 + 3.934 ×10−2 T − 1.801× 10 −4 T 2 = r1 + r2T + r3T 2 . (17.4.23)
8000 steel
Density in kg/m³
7500 7000 6500 6000 5500 5000
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.9. Density of solid and liquid steel as a function of the temperature at 1 bar pressure
Figure 17.4.9 presents the density for solid and liquid steel as a function of the temperature at 1 bar pressure with the approximation documented here. The derivative with respect to the temperature is computed as follows: ⎛ ∂ρ ⎞ ⎜ ∂T ⎟ = r2 + 2r3T ⎝ ⎠p
(17.4.24)
The volumetric thermal expansion coefficient is therefore
β =−
r2 + 2r3T 1 ⎛ ∂ρ ⎞ =− . ⎜ ⎟ ρl ⎝ ∂T ⎠ p r1 + r2T + r3T 2
(17.4.25)
17. Thermo-physical properties for severe accident analysis
5
Vol. th. expansion coefficient*10 , -
598
24 22 20 18 16 14 12 10 8 6
steel
2000
2500
3000
3500
T in K Fig. 17.4.10. The volumetric thermal expansion coefficient of liquid steel as a function of the temperature at 1 bar pressure
Figure 17.4.10 shows the volumetric thermal expansion coefficient of liquid steel as a function of the temperature at 1 bar pressure. The specific capacity at constant pressure is reported by Hohorst (1990) to be c pl = 776.2 J/(kgK).
(17.4.26)
Note that Chase et al. (1988) reported the value 835 ± 25 J/(kgK). The derivative of the density with respect to the pressure is then ⎛ d ρl ⎞ T 1 ⎛ ∂ρ ⎞ 1 β l2T 1 = 2+ ⎜ ⎟ = 2+ c pl al c pl ρl2 ⎜⎝ ∂T ⎟⎠ p ⎝ dp ⎠T al 2
= f ⎡⎣ al (T ) , β l (T , p ) , c pl (T ) ⎤⎦ = f (T , p ) .
(17.4.27)
With this relation the density is then function of temperature and pressure ⎛ d ρl ⎞ ⎟ ( p − p0 ) . ⎝ dp ⎠T
ρl (T , p ) ≈ ρl 0 (T ) + ⎜
(17.4.28)
Note that the derivative in the above equation is function of temperature and week function of pressure.
17.4 Stainless steel
599
The isothermal coefficient of compressibility is then 1 ⎛ 1 β 2T ⎜ + cp ρ ⎜⎝ a 2 11
Is. coef. of compressibility*10 , -
k=
⎞ ⎟. ⎟ ⎠
(17.4.29)
8,0 7,0
steel
6,0 5,0 4,0 3,0 2,0 1,0 0,0
2000
2500 T in K
3000
3500
Fig. 17.4.11. The isothermal coefficient of compressibility of liquid steel as a function of the temperature at 1 bar pressure
Figure 17.4.11 shows the isothermal coefficient of compressibility of liquid steel as a function of the temperature at 1 bar pressure. The specific liquid enthalpy and entropy are then computed as follows. Integrating the differential form of the enthalpy equation ⎛ ∂h ⎞ dh = c p dT + ⎜ ⎟ dp , ⎝ ∂p ⎠T
(17.4.30)
having in mind that ( ∂h ∂p )T is function of the temperature only results in ⎛ ∂h ⎞ h = href + c p T − Tref + ⎜ ⎟ p − pref ⎝ ∂p ⎠T
(
)
(
)
(17.4.31)
with href = h′ = 1106298.3517819312 J/kg at Tref = T ′′′ = 1700 K and pref = 105 Pa Similarly integrating
600
17. Thermo-physical properties for severe accident analysis
ds = c p
dT 1 ⎛ ∂h ⎞ + ⎜ ⎟ dp T T ⎝ ∂p ⎠T
(17.4.32)
results in
(
)
s = sref + c p ln T Tref +
1 ⎛ ∂h ⎞ ⎜ ⎟ p − pref T ⎝ ∂p ⎠T
(
)
(17.4.33)
with sref = s ′ = 1151.9706516297356 J/(kgK). The inversed computation of the temperature of the liquid for known specific entropy and pressure is then ⎤ 1 ⎛ ∂h ⎞ ⎪⎧ ⎡ ⎪⎫ T = Tref exp ⎨ ⎢ s − sref − ⎜ ⎟ p − pref ⎥ c p ⎬ . T ⎝ ∂p ⎠T ⎪⎩ ⎣⎢ ⎪⎭ ⎦⎥
(
)
(17.4.34)
At maximum 5 iterations are needed to solve the above equation with respect to the unknown temperature starting with T = T ′′′ . As already mentioned Fig. 17.4.2, 17.4.3 and 17.4.4 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as a temperature functions at 1 bar pressure for solid and liquid steel. The transport properties will be discussed below.
17.4.2.2 Transport properties Liquid thermal conductivity: Correlation for liquid thermal conductivity of stainless steel 304 is reported by Leibowitz et al. (1976):
λl = 12.41 + 3.279 × 10−3 T .
(17.4.35)
Figure 17.4.12 shows the thermal conductivity for solid and liquid steel as a function of the temperature. Liquid surface tension: The liquid surface tension for pure iron (not for stainless steel) is reported by Fraser et al. (1971) to be σ l = 0.773 + 0.65 (T − 273)10−3 ,
(17.4.36)
17.4 Stainless steel
601
Thermal conductivity in W/(mK)
valid in 1700 < T < 3000 K. Figure 17.4.13 illustrates this dependence. It is surpassing that the surface tension increase with temperature. Usually for liquids it decreases to zero for temperatures approaching the critical point. 40 steel
35 30 25 20 15 10
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.4.12. Thermal conductivity of stainless steel as function of temperature
Surface tension N/m
3.0 2.8 2.6 2.4 2.2 2.0
iron
1.8 1.6
2000
2500 T in K
3000
3500
Fig. 17.4.13. The surface tension as a function of temperature for pure iron
The steel consists of several components having different surface tensions at the steel boiling point: < 0.08 mass% C (3.7 N/m), < 2 mass% Mn (1.09 N/m), < 0.045 P (0.05 N/m), < 0.03 mass% S (0.06 N/m), < 1 mass% Si (0.87), 16 to 18 mass% Cr (1.7 N/m), 10 to 14 Ni (1.78 N/m), 2 to 3 mass% Mo (2.25 N/m). This leads Ostensen et al. (1977) to recommend the following very uncertain (as these authors mentioned) dependence:
602
17. Thermo-physical properties for severe accident analysis
σ l = 2.08 − 4 (T − 273)10−4 ,
which is a decreasing function with temperature. Liquid dynamic viscosity: Liquid dynamic viscosity for pure iron is reported by Leibowitz et al. (1976) to be ηl = 0.0002536 exp ( 5492.2 T ) ,
(17.4.37)
Dynamic viscosity /1000 in kg/(ms)
covered by data for T < 2000 K. This dependence is illustrated on Fig. 17.4.14. 6,5 6,0 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5
steel
2000
2500 T in K
3000
3500
Fig. 17.4.14. Dynamic viscosity of stainless steel as function of temperature
For pure iron slightly different correlation ηl = ( 3.699 ± 0.33) × 10-4 exp ( 4979.3 T )
(17.4.38)
is reported by Brandes (1983). Chu et al. (1996) reported ηl = ( 2.21 ± 0.33) × 10-4 exp ( 5776 T ) ,
containing the uncertainty estimate. Liquid emissivity: In VDI-Wärmeatlas (1991) we find ε ( 2043.15 K) = 0.45 . Teofanous et al. (1996) reported the value 0.43.
17.4 Stainless steel
603
17.4.3 Vapor The boiling pressure of stainless steel liquid depends on pressure, Leibowitz et al. (1976): ln p ′ = 25.601 −43446 T .
(17.4.39)
This equation gives a boiling temperature at 1 bar of 3086.75 K. Comparing with Kolev (2007, p. 147) we see that this relation is derived for constant evaporation enthalpy and for low pressure. The critical temperature can be computed using the Watson’s formula relating the critical temperature to the boiling temperature at 1 bar pressure. It gives Tc = 9230 K ,
(17.4.40)
Chawla et al. (1981). The latent heat of evaporation is estimated to be h′′ − h′ = 7.46417 × 106 J / kg ,
(17.4.41)
Leibowitz et al. (1976). The Clapeyron’s equation dT ′ v′′ − v′ 43446 , = T′ = ′′ ′ dp h − h ( 25.601 − ln p ′ ) 2 p ′
(17.4.42)
allows to estimate the vapor specific volume knowing the liquid specific volume at the saturation line v′′ (T ′ ) = v′ +
1 dT ′ ( h′′ − h′ ) . T ′ dp
(17.4.43)
Letter Fischer (1989) approximated the saturation pressure as follows log10 ( p ′ ) = 23.47496 − 22027.61 T − 1.4359 ln T + 67.2678 × 10 −6 T .
Fischer (1992) introduced a modification into the well known Redlich–Kwong equation (1949) of state for pure vapors: p=
f (T ) RT − , v − a1 v ( v + a3 )
f (T ) = a2 (T Tc )
{
a4
for T < Tc ,
}
f (T ) = a2 1 + a4 ⎡⎣(T Tc ) − 1⎤⎦ for T ≥ Tc ,
604
17. Thermo-physical properties for severe accident analysis
For stainless steel he found the constants a1 = 1.51243 × 10−4 , a2 = 2.02244 × 103 , a3 = 6.50753 × 10−4 , a4 = 0.257346 .
References Brandes EA ed. (1983) Smithels metals reference book (sixth ed.), Butterworth and co., London Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Nonograph No. 9, American Chemical Sociaty Chawla TC et al. (1981) Thermophysical properties of mixed oxide fuel and stainless steel type 316 for use in transition phase analysis, Nuclear Engineering and Design, vol 67 pp 57–74 Chu CC, Sieniki JJ and Beker L Jr (October, 1996) Uncertainty analysis for thermophysical properties used in in-vessel retention analysis. In: Teofanous TG et al., eds., In-Vessel Coolability and Retention of a Core Melt, DOE/ID-10460 vol 1, U.S. Department of Energy Report Fischer EA (1989) A new evaluation of the urania equation of state based on recent vapor pressure. Nucl. Sci. Eng., vol 101, pp 97–116 Fischer EA (May, 1992) Fuel evaluation of state data for use in fast reactor accident analysis codes, KfK 4889, Kernforschungszentrum Karlsruhe Fortov VE, Dremin AN and Leont’ev AA (1975) Evaluation of the parameters of the critical point. High Temp., vol 13, pp 984–992 Fraser ME, Lu WK, Hamielec AE and R. Murarka R (März 1971) Surface tension measurements on pure liquid iron and nickel by an oscillating drop technique, Metallurgical and Materials Transactions B, vol 2 no 3 pp 817–823 Hohorst JK ed. (1990) SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR5273, ECG-2555 Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo Kurz W and Lux B (1969) The sound velocity of iron and iron alloys in solid and fluid states (in German). High Temp. – High Pressure, vol 1, pp 387–399 Leibowitz L et al. (April, 1976) Properties for LMFBR safety analysis, ANL-CEN-RSD76-1 Ostensen RW, Murphy WF, Wrona BJ, Dietrich LW and Florek JC (1977) Intrusion of molten steel into cracks in solid fuel in transient-undercooling accident in liquid-metal fast breeder reactor. Nucl. Technol, vol 36, pp 200–214 Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGraw-Hill, New-York, pp 3–285 Redlich O and Kwong JNS (1949) On the thermodynamics of solutions. V, An equation of state. Fugacities of gaseous solutions. Chem. Rev., vol 44, pp 233–244 Richter F (1983) Die wichtigsten physikalischen Eigenschaften von 52 Eisenwerkstoffen Stahleisen-Sonderberischte Heft 8, Verlag Stahleisen M.B.H.-Düsseldorf Teofanous TG et al. (October, 1996) In-vessel coolability and retention of a core melt, DOE/ID-10460 vol 1, U.S. Department of Energy Report VDI-Wärmeatlas (1991) Berechnungsblätter für den Wärmeübergang, 6. Aufl., VDI-Verlag, Düsseldorf
17.5 Zirconium
605
17.5 Zirconium Due to its low absorption capability for thermal neutrons Zirconium is massively used in nuclear reactors for designing of nuclear fuel containing tubes (claddings) and nuclear core internal structures. Usually alloys are used with small amount of other materials. Note that the alloys have somewhat different properties then the pure zirconium. Dissolved hydrogen has a strong impact on the thermal properties of zirconium. The mole-mass of Zirconium is M = 0.091224 kg.
(17.5.1)
The melting temperature used here is T ′′′ = 2098 K.
(17.5.2)
Cordfunke and Konings (1990) gives for the melting temperature T ′′′ = 2128 ± 5 K and Chase (1998) 2125 K. The boiling temperature at atmospheric pressure is reported to be T ′ = 4702.633 K,
(17.5.3)
Chase (1998). 17.5.1 Solid With increasing temperature within the solid state at a temperature Ts ,α → β = 1139 ± 5K
(17.5.4)
there is a phase transition from hexagonal structure called alpha-zirconium to bccstructure called beta-zirconium responsible for abrupt change of the properties, Cordfunke and Konings (1990), e.g. the density changes from 6506 to 6445 kg/m³. These authors reported phase-transition specific enthalpy Δhs ,α → β = 4106 J/g-mole.
(17.5.5)
Compare with 4017 ± 0.3 J/g-mole reported by Chase (1998). 17.5.1.1 Solid specific capacity at constant pressure Cordfunke and Konings (1990, p. 466) reported for T < Ts ,α → β for α -zirconium
606
17. Thermo-physical properties for severe accident analysis
c ps M = c10 + c11T + c12T 2 + c13T 3 + c14 T 2
(17.5.6)
c10 = 24.161800, c11 = 8.75682 × 10−3 , c12 = 0, c13 = 0, c14 = −69942 . For β -zirconium
I use the approximation c ps M = c20 + c21T + c22T 2 + c23T 3 + c24 T 2 .
where
c20 = 25.607406 ,
c23 = 9.16714728 × 10
c21 = 6.80168 × 10−4 ,
(17.5.7) c22 = 5.837384 × 10−8 ,
−10
, c24 = −50466 . Note that Cordfunke and Konings (1990, p. 466) reported the following constants for the β -zirconium: c20 = 43.246 , c21 = −18.5806 × 10−3 , c22 = 7.438 ×10−6 , c23 = 0 , c24 = −5173470 which provide almost the same approximation. The specific melt enthalpy is reported to be h′′′ − h′ = 225356.7 J/kg.
(17.5.8)
The specific melt entropy is therefore s ′′′ − s ′ = ( h′′′ − h′ ) T ′′′ = 107.41501429933270 J/(kgK).
(17.5.9)
Note that Cordfunke and Konings (1990) reported slightly larger melting specific enthalpy of 210 00 J/g-mole = 230 213 J/kg, and Chase (1998) 20 920 J/g-mole = 229 336 J/kg. For fuel claddings in U.S. zirconium alloy called zircaloy is used having the following property: ρ s c ps = 1.673 ×106 + 721.6T for 300 < T < 1090 K , ρ s c ps = 5.346 ×106 + 3.608 ×104 T − 1170 for 1090 ≤ T ≤ 1254 K , ρ s c ps = 2.316 ×106 for T > 1254 K ,
with 5% error for 300 < T < 1090 K , Kelly et al. (1981). 17.5.1.2 Solid specific enthalpy For h between 605487.59408074501 J/kg and 830844.29408074496 J/kg the temperature is 2098 K because of melting. For T < 1139 K we have
17.5 Zirconium
(
)
hs M = c10 T − Tref +
(
−c14 1/ T − 1/ Tref
(
)
(
)
(
1 1 1 c11 T 2 − Tref2 + c12 T 3 − Tref3 + c13 T 4 − Tref4 2 3 4
)
607
) (17.5.10)
else 1 1 ⎡ 2 2 3 3 ⎢c20 T − Tref + 2 c21 T − Tref + 3 c22 T − Tref 1 ⎢ hs = h0 + ⎢ M ⎢ 1 ⎢ + c23 T 4 − Tref4 − c24 1/ T − 1/ Tref ⎣⎢ 4
(
(
)
(
)
)
(
(
)
)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥
(17.5.11)
with h0 = 278823.29312710464 J/kg and reference temperature Tref = 298.15 K .
(17.5.12)
17.5.1.3 Solid specific entropy For s between 637.16662880936565 and 744.58164310869836 J/(kgK) the temperature is 2098 K because of melting. For T < 1139 K we have
(
)
(
(
)
)
(
1 1 ss M = c10 ln T Tref + c11 T − Tref + c12 T 2 − Tref2 + c13 T 3 − Tref3 2 3
(
1 − c14 1/ T 2 − 1/ Tref2 2
)
) (17.5.13)
else 1 ⎡ 2 2 ⎢c20 ln T Tref + c21 T − Tref + 2 c22 T − Tref 1 ⎢ ss = s0 + ⎢ M ⎢ 1 1 ⎢ + c23 T 3 − Tref3 − c24 1/ T 2 − 1/ Tref2 2 ⎣⎢ 3
(
)
(
(
)
)
(
with s0 = 4.3171463184809460 × 102 J/(kgK).
(
)
)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥
(17.5.14)
608
17. Thermo-physical properties for severe accident analysis
400
cp in J/(kgK)
380 360 340 320 300 Cordfunke and Konings (1990)
280
500
1000
1500 T in K
2000
2500
Fig. 17.5.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid Zr
1200
h in kJ/kg
1000
Cordfunke and Konings (1990) cp model
800 600 400 200 0
500
1000
1500 T in K
2000
2500
Fig. 17.5.2. Specific enthalpy as a function of the temperature of solid and liquid Zr. p = 1 bar
17.5 Zirconium
609
1000 Cordfunke and Konings (1990) cp model
s in J/(kgK)
800 600 400 200 0
500
1000
1500 T in K
2000
2500
Fig. 17.5.3. Specific entropy as a function of the temperature of solid and liquid Zr. p = 1 bar
17.5.1.4 Solid density The solid density is considered as a function of the temperature only ρ s = ρ s 0 (1 + ε )
3
(17.5.15)
where for T < 1139 K ε = c10 + c11T + c12T 2 + c13T 3
(17.5.16)
else ε = c20 + c21T + c22T 2 + c23T 3
(17.5.17)
and ρ s 0 = 6570, c10 = −1.11× 10−3 , c11 = 2.325 × 10−6 , c12 = 5.595 ×10−9 , c13 = −1.768 × 10−12 , c20 = −7.59 × 10−3 , c21 = 1.474 × 10−6 , c22 = 5.140 × 10−9 , c23 = −1.559 × 10−12.
This relation is valid for T > 298.15 K and T < 2098 K.
610
17. Thermo-physical properties for severe accident analysis
17.5.1.5 The derivative of the solid density with respect to the temperature d ρs 4 = −3ε T ρ s 0 (1 + ε ) dT
(17.5.18)
where for T < 1139 K ε = c10 + c11T + c12T 2 + c13T 3 ,
(17.5.19)
ε T = c11 + 2c12T + 3c13T 2 ,
(17.5.20)
else ε = c20 + c21T + c22T 2 + c23T 3 ,
(17.5.21)
ε T = c21 + 2c22T + 3c23T 2 .
(17.5.22)
17.5.1.6 Solid thermal conductivity Fink and Leibowitz (1995) performed a statistical analysis of thermal conductivity and thermal diffusivity measurements from 1951 to 1995 and reported the following correlation: λs = c0 + c1T + c2T 2 + c3 T
(17.5.23)
with c0 = 8.8527, c1 = 7.082 × 10−3 , c2 = 2.5392 × 10−6 , c3 = 2.9918 × 103. The standard deviation error of the fit to the data range from 5 to 9.5%, depending on temperature. Note that in a previous work published by Hohorst et al. (1990) much higher values for the thermal conductivity are suggested. For the zirconium alloy used in USSR called zircaloy-4 Rassohin et al. (17.5.1971) used in their work c0 = 1.96 , c1 = −2.41× 10−2 , c2 = 4.63 ×10−7 , c3 = −1.95 × 10−10 . For the zirconium alloy used in US for producing a fuel claddings called zircaloy the thermal conductivity is reported by Kelly et al. (1981) to be λs = c0 + c1T + c2T 2 + c3T 3 for 300 < T < 1800 K ,
where c0 = 7.51, c1 = 9.09 × 10−2 , c2 = −1.45 × 10−5 , c3 = 7.67 × 10−9 with 20% error.
17.5 Zirconium
611
17.5.1.7 Solid sonic velocity With the Youngs modulus (elasticity modulus) of E = 68000000000Pa, the velocity of sound is as = E ρ s ,
(17.5.24)
which is in order of 3217 m/s. 17.5.2 Liquid 17.5.2.1 Density The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105Pa, T0 = 2098 K, ρl 0 = 6130 kg/m³, the volumetric thermal expansion coefficient is β = 0.573/ ρl 0 K–1, and the isothermal compressibility is κ = 4.234×10–11Pa–1. 6800
Density in kg/m³
6600 6400 6200
Zr
6000 5800
500
1000
1500 T in K
2000
2500
Fig. 17.5.4. Density of solid and liquid Zr as a function of the temperature at 1 bar pressure
Figure 17.5.4 presents the density of solid and liquid Zr as a function of the temperature at 1 bar pressure. 17.5.2.2 Liquid density derivative with respect to temperature Equation (17.1.5).
612
17. Thermo-physical properties for severe accident analysis
17.5.2.3 Liquid density derivative with respect to pressure Equation (17.1.6). 17.5.2.4 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007), Eq. (17.1.7).
Velocity of sound in m/s
3500 3000 Zr
2500 2000 1500
500
1000
1500 T in K
2000
2500
Fig. 17.5.5. Velocity of sound of solid and liquid Zr as a function of the temperature at 1 bar pressure
Figure 17.5.5 presents the velocity of sound of solid and liquid Zr as a function of the temperature at 1 bar pressure. 17.5.2.5 Specific capacity at constant pressure I use the constant value of c pl = 391.5545933J/(kgK).
Figures 17.5.1, 17.5.2 and 17.5.3 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as a temperature functions at 1bar pressure for solid and liquid Zr. Note that Bonnell (1972) and Chase (1998) estimated larger value c pl = 461.98 J/(kgK) and 458.67 ± 3% J/(kgK), respectively. 17.5.2.6 Liquid thermal conductivity Fink and Leibowitz (1995) reported
17.5 Zirconium
613
Thermal conductivity in W/(mK)
λl = 36.28556W/(mK) 40 35
Zr
30 25 20 15
500
1000
1500 T in K
2000
2500
Fig. 17.5.6. Thermal conductivity of solid and liquid Zr as a function of the temperature at 1 bar pressure
Figure 17.5.6 presents the thermal conductivity as a function of the temperature at 1 bar pressure for solid and liquid Zr. 17.5.2.7 Liquid surface tension The surface tension is reported in TAPP (1994) to be σ l = 1.4 .
17.5.2.8 Liquid emissivity The emissivity reported by Krishnan et al. (1993) for high temperatures is ε = 0.35 .
17.5.2.9 Liquid dynamic viscosity Hohorst (1990) proposed the following correlation for the liquid dynamic viscosity of zirconium ηl = 1.9 × 10−4 exp ( 6500 T ) which is very similar to the one I use ηl = ( 2 ± 0.2 ) × 10−4 exp ( 6620 T ) .
17. Thermo-physical properties for severe accident analysis
Dynamic viscosity /1000 in kg/(ms)
614
5,0 4,5
Zr
4,0 3,5 3,0
2100
2200
2300 T in K
2400
2500
Fig. 17.5.7. Dynamic viscosity of liquid Zr as a function of the temperature at 1 bar pressure
Figure 17.5.7 presents the dynamic viscosity of liquid Zr as a function of the temperature at 1 bar pressure. 17.5.2.10 Liquid specific enthalpy For hl between 605487.59408074501 and 830844.29408074496 J/kg, T is = 2098 K because of melting. The melt enthalpy is h′ − h′′′ = 225356.7 J/kg. The liquid specific enthalpy is approximated by Eq. (17.1.8) with h′ = 830844.29408074496 J/kg. 17.5.2.11 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10). 17.5.2.12 Liquid specific entropy The liquid specific entropy is approximated by Eq. (17.1.11) with s ′ = 744.58164310869836 J/(kg/K). 17.5.2.13 Liquid temperature as a function of pressure and specific entropy Given are the pressure and the specific liquid entropy. The temperature is computed iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , where sl 0 = s ′ = 744.58164310869836 J/(kgK) and using Eq. (17.1.11) solved with respect to the
References
615
temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. References Bonnell DW (1972) Property measurements at high temp., levitation calorimetry studies of liquid metals. Ph.D. Thesis Rice University, Houston, TX. Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Nonograph No. 9, American Chemical Sociaty Cordfunke EHP and Konings RJM (1990) Thermochemical Data for Reactor Materials and Fussion Products, North-Holland, Amsterdam Fink JK and Leibowitz L (1995) Thermal conductivity of zirconium. J. Nucl. Mater., vol 226, pp 44–50 Hohorst JK ed. (1990) SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR5273, EGG-2555 Kelly JE, Kao SP and Kazimi MS (April, 1981) THERMIT-2: A two fluid model for light water reactor subchannel transient analysis, MIT Energy Laboratory Electric Utility Program, Report No. MIT-EL.81-014 Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo Krishnan S, Weber JKR, Anderson CD and Nordine PC (1993) Spectral emissivity and optical properties at l=632.8 nm for liquid uranium and zirconium at high temperature. J. Nucl. Mater., vol 203, pp 112–121 Rassohin NT, Gradusov GH and Gorbatych VP (1971) Korosija splava zirkonija – 1% niobija v uslovijah teploperedaci, Trudy MEI, Vyp. 83 TAPP (1994) A Database of Thermo-Chemical and Physical Properties. ES Microware, Hamilton, Ohio
17.6 Aluminum
617
17.6 Aluminum In accordance with Chase (1998) the mole-mass of aluminum is M = 0.02698154 kg,
(17.6.1)
the melting temperature T ′′′ = 933.35 K and the boiling temperature at atmospheric pressure is T ′ = 2790.812 K.
(17.6.2)
Note that Perry and Green (1985, pp. 2–7) reported much lower boiling point 2329.15 K. Touloukian and DeWitt (1972) reported T ′′′ = 933.2 K,
(17.6.3)
which value will be used here. Perry and Green (1985, pp. 2–7) reported 933.15 K.
17.6.1 Solid 17.6.1.1 Solid specific capacity at constant pressure The temperature dependence of the heat capacity is approximated using the values given by Touloukian and DeWitt (1972): c p (T = 301.6 K) = 907.5 J/(kg K) , and c p (T = 923.0 K) = 1276.15 J/(kg K) , resulting in c ps = c1 + c2 (T − 301.6 ) ,
(17.6.4)
where c1 = 907.5 , c2 = 0.593257161 . The relation is valid for 298.15 < T < 933.2 K. The relation reported by Perry and Green (1985) c ps = 744.33 + 0.5T ± 1% is very close the above relation. 17.6.1.2 Solid specific enthalpy Integrating between the reference temperature Tref = 298.15 K and the actual temperature results in
(
)
1 hs = ( c1 − c2 301.6 )(T − T1 ) + c2 T 2 − Tref2 . 2
(17.6.5)
618
17. Thermo-physical properties for severe accident analysis
For h between 694634.9905662248 J/kg and 1084634.990566225 J/kg the temperature is 933.2 K because of melting. The specific melt enthalpy is reported by Kammer (1995) to be h′′′ − h′ = 390000 J/kg.
(17.6.6)
The specific melt entropy is therefore s ′′′ − s ′ = ( h′′′ − h′ ) T ′′′ = 417.916845263609 J/(kgK).
(17.6.7)
17.6.1.3 Solid specific entropy The solid specific entropy is therefore
(
)
(
)
ss = ( c1 − c2 301.6 ) ln T Tref + c2 T − Tref .
(17.6.8)
For s between 1208.067112995626 and 1625.983958259235 J/(kgK) the temperature is 933.2 K because of melting. 17.6.1.4 Solid density The solid density considered as a function of the temperature only is approximated by Kammer (1995) with ρ s = ρ s 0 ⎡1 − c0 ( c1 + c2 ΔT + c3 ΔT 2 + c4 ΔT 3 ) ⎤ , ρ s 0 = 2700 , −7 c3 = 4.164 × 10 , c4 = 8.27 × 10−10 .
where
ΔT = T − 300 ,
(17.6.9)
⎦
⎣
c0 = 0.03 ,
c1 = 0.018 ,
c2 = 2.364 × 10−3 ,
17.6.1.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is then d ρs = − ρ s 0 c0 ( c2 + 2c3 ΔT + 3c4 ΔT ) . dT
(17.6.10)
17.6.1.6 Solid thermal conductivity The solid thermal conductivity is approximated with the constant λs = 235 W/(mK),
(17.6.11)
17.6 Aluminum
619
Touloukian and DeWitt (1972), Kammer (1995). 17.6.1.7 Solid sonic velocity With the Youngs modulus (elasticity modulus) of E = 70000000000Pa, the velocity of sound is as = E ρ s ,
(17.6.12)
which is in order of 5092 m/s. 17.6.1.8 Emissivity Text books give emissivities within 273.15 and 673.15 K of 0.04 to 0.06 for polished, 0.07 to 0.09 for commercial and 0.2 to 0.3 for oxidized aluminum. 17.6.2 Liquid
17.6.2.1 Specific capacity at constant pressure Kammer (1995) reported a constant value for the specific capacity at constant pressure c pl = 1082.201222 J/(kgK).
Perry and Green (1985) reported c pl = 1085.48 ± 5% J/(kgK). 17.6.2.2 Liquid specific enthalpy For h between 694634.9905662248 and 1084634.990566225 J/kg the temperature is = 933.2 K because of melting. The melt enthalpy is 390 000 J/kg. The liquid specific enthalpy is approximated by Eq. (17.1.8) with h′ = 1084634.990566225 J/kg. 17.6.2.3 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10).
620
17. Thermo-physical properties for severe accident analysis
1300
cp in J/(kgK)
1200 1100 1000 Touloukian and DeWitt (1972), Kammer (1995)
900 800
500
1000
1500 T in K
2000
2500
Fig. 17.6.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid Al
2500
Touloukian and DeWitt (1972), Kammer (1995), cp model
h in kJ/kg
2000 1500 1000 500 0
500
1000
1500 T in K
2000
2500
Fig. 17.6.2. Specific enthalpy as a function of the temperature of solid and liquid Al. p = 1 bar
17.6 Aluminum
3000
621
Touloukian and DeWitt (1972), Kammer (1995), cp model
s in J/(kgK)
2500 2000 1500 1000 500 0
500
1000
1500 T in K
2000
2500
Fig. 17.6.3. Specific entropy as a function of the temperature of solid and liquid Al. p = 1 bar
17.6.2.4 Liquid specific entropy For s between 1208.067112995626 and 1625.983958259235 J/(kgK) the temperature is 933.2 K because of melting. The melt entropy is 417.916845263609. The liquid specific entropy is approximated by Eq. (17.1.11) with s ′ = 1625.983958259235 J/(kgK). Figures 17.6.1, 17.6.2 and 17.6.3 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as a temperature functions at 1-bar pressure for solid and liquid Al. 17.6.2.5 Density The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105Pa, T0 = 933.2 K, ρl 0 = 2357 kg/m³, the volumetric thermal expansion coefficient is β = 0.233/ ρl 0 K–1, and the isothermal compressibility is –11 –1 κ = 4.234×10 Pa .
622
17. Thermo-physical properties for severe accident analysis
2700 Al
Density in kg/m³
2600 2500 2400 2300 2200 2100 2000
500
1000
1500 T in K
2000
2500
Fig. 17.6.4. Density of solid and liquid Al as a function of the temperature at 1 bar pressure
Velocity of sound in m/s
5500 5000 Al
4500 4000 3500 3000
500
1000
1500 T in K
2000
2500
Fig. 17.6.5. Velocity of sound of solid and liquid Al as a function of the temperature at 1 bar pressure
Figure 17.6.4 presents the density of solid and liquid Al as a function of the temperature at 1 bar pressure 17.6.2.6 Liquid density derivative with respect to temperature Equation (17.1.5). 17.6.2.7 Liquid density derivative with respect to pressure Equation (17.1.6).
17.6 Aluminum
623
17.6.2.8 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007), Eq. (17.1.7). Figure 17.6.5 presents the velocity of sound of solid and liquid Al as a function of the temperature at 1bar pressure 17.6.2.9 Liquid thermal conductivity The liquid thermal conductivity is approximated with the constant λl = 100 W/(mK),
Thermal conductivity in W/(mK)
Touloukian and DeWitt (1972), Kammer (1995). 240 220 Al
200 180 160 140 120 100 80
500
1000
1500 T in K
2000
2500
Fig. 17.6.6. Thermal conductivity of solid and liquid Al as a function of the temperature at 1bar pressure
Figure 17.6.6 presents the thermal conductivity as a function of the temperature at 1bar pressure for solid and liquid Al. 17.6.2.10 Liquid surface tension σ l = 0.86 .
17.6.2.11 Liquid dynamic viscosity Kammer (1995) give for the viscosity of the liquid aluminum
624
17. Thermo-physical properties for severe accident analysis
Dynamic viscosity /1000 in kg/(ms)
ηl = 1.1× 10−3 Pas. 1,5 1,4 1,3 1,2 1,1 1,0 0,9 0,8 0,7 0,6 0,5
Al
1000
1500 2000 T in K
2500
Fig. 17.6.7. Dynamic viscosity of liquid Al as a function of the temperature at 1 bar pressure
Figure 17.6.7 presents the dynamic viscosity of liquid Al as a function of the temperature at 1 bar pressure. 17.6.2.12 Emissivity The emissivity of aluminum depends strongly on the structure of the surface, Kammer (1995). We use the value:
ε = 0.3 . 17.6.2.13 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , where s ′ = 1625.983958259235 J/(kgK) and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. References Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Nonograph No. 9, American Chemical Sociaty Hohorst JK edt. (1990) SCDAP/RELAP5/M002 Code Manual, Volume 4: MATPRO – A library of material properties for light-water-reactor accident analysis, NUREG/CR5273, ECG-2555
References
625
Kammer C (1995) Aluminium-Taschenbuch, Aluminium-Verlag, Düsseldorf, 15. Auflage Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo Touloukian YS and DeWitt DP (1972) Thermo-Physical Properties of Matter. Plenum Press, New York Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New-York.
17.7 Aluminum oxide, Al2 O3
627
17.7 Aluminum oxide, Al2O3 Usually experiments for severe accident analysis are performed instead of radioactive materials with stimulants. So for instance melt water interaction of molten core materials with water is simulated some times by ejecting thermit and injecting the produced aluminum oxide into water. To understand the processes accurate modeling of the thermo-physical and transport properties is part of the investigation. Therefore there is a need to summarize thermal and transport properties for aluminum oxide. In accordance with Perry and Green (1985) the mole-mass of alumina is M = 0.10194 kg,
the melting temperature T ′′′ = 2272.15 to 2305.15 K and the boiling temperature at atmospheric pressure is T ′ = 2483.15 K.
Barin and Knacke (1973) reported for the melting point T ′′′ = 2324.15 K
which will be used here. 17.7.1 Solid 17.7.1.1 Solid specific capacity at constant pressure Data for specific capacity at constant pressure for solid alumina are reported by Samsonov (1982), Touloukian and Buyco (1970), Shpil’rain et al. (1973), Barin and Knacke (1973) and critically evaluated by Turnay (1985). Turnay (1985) finally recommended the following approximations: c ps = c1T 3 + c2T 4 for T < 38 K ,
(17.7.1)
c ps = c3 + c4T + c5T 2 + c6T 3 + c7T 4 for 38 K < T ≤ 90 K ,
(17.7.2)
c ps = c8 + c9T + c10T 2 + c11T 3 + c12T 4 for 90 K < T ≤ 300 K ,
(17.7.3)
c ps = c13 + c14T + c15 T 2 for 300 K < T ≤ 1273K ,
(17.7.4)
c ps = c16 + c17T + c18 T 2 for 1273K < T ≤ 2324.15K .
(17.7.5)
628
17. Thermo-physical properties for severe accident analysis
with c1 = 1.24695 × 10−4 , c2 = 4.39873 × 10−7 , c3 = −0.185615, c4 = 1075.46 × 10−4 , c5 = −82.5507 × 10−4 , c6 = 3.39221× 10−4 , c7 = −1.359170 × 10−6 , c8 = 119.7012, c9 = −53910.2 × 10 −4 , c10 = 812.132 × 10−4 , c11 = −2.94665 ×10 −4 , c12 = 3.60813 × 10−7 , c13 = 1126.3493, c14 = 0.125652, c15 = −3.47801× 107 , c16 = 1046.2771, c17 = 0.174515, c18 = −2.80047 × 107 , s1 = −6650.57125817941, s2 = −5969.21115510038, T1 = 298.15, T2 = 300, T3 = 1273.
Perry and Green (1985) reported for the fourth region 273 to 1974 K a relation c ps = 906.25 + 0.3682T − 2.144536 × 107 T 2 ± 3% which is very close to those of Turnay (1985). The phase change by T = 1273 K is found by Barin and Knacke (1973) with transformation specific enthalpy 215 170 J/kg and the melting point T ′′′ = 2324.15 K
(17.7.6)
with melting specific enthalpy of 1 067 640 J/kg. 17.7.1.2 Solid specific enthalpy For h between 1104917.5485113175 J/kg and 1320087.5485113175 J/kg the temperature is 1273 K because of phase transition. The transformation specific enthalpy is 215 170 J/kg, Barin and Knacke (1973). For h between 2739865.3242982719 J/kg and 3807505.3242982719 J/kg the temperature is 2324.15 K because of melting, Shpil’rain (1973). The melt enthalpy is h′ − h′′′ = 1 067 640 J/kg,
(17.7.7)
Barin and Knacke (1973). The specific enthalpy is then: If T < 300 K then hs = h0 + c8T +
1 1 1 1 c9T 2 + c10 T 3 + c11T 4 + c12T 5 2 3 4 5
(17.7.8)
else if T < 1273 K then hs = h1 + h13T +
1 h14T 2 − h15 T 2
(17.7.9)
17.7 . Aluminum oxide,, Al2 O3
629
else hs = h2 + c16T +
1 c17T 2 − c18 T , 2
(17.7.10)
with h0 = −101456.733166016 , h1 = −458057.7597819015 , h2 = −390395.487737848 + 0.21517 × 106 , h1 = 298.15 T2 = 300 T3 = 1273 . 17.7.1.3 Solid specific entropy For s between 1572.5337976134472 J/(kgK) and 1741.5597206299437 J/(kgK) the temperature is 1478 K because of phase transition. For s between 2548.7886432044293 J/(kgK) and 3008.1565841720949 J/(kgK) the temperature is 2324.15 K because of melting. The melt entropy is 459.36794096766556 J/(kgK). The solid specific entropy is: If T < 300 K then ss = s0 + c8 ln T + c9T +
1 1 1 c10T 2 + c11T 3 + c12T 4 2 3 4
(17.7.11)
else if T < 1273 then 1 ss = s1 + c13 ln T + c14T − c15 T 2 2
(17.7.12)
else 1 ss = s2 + c16 ln T + c17T − c18 T 2 2
(17.7.13)
with c1 = 1.24695 × 10−4 , c2 = 4.39873 × 10−7 , c3 = −0.185615, c4 = 1075.46 × 10−4 , c5 = −82.5507 × 10−4 , c6 = 3.39221× 10−4 , c7 = −1.359170 × 10−6 , c8 = 119.7012, c9 = −53910.2 × 10 −4 , c10 = 812.132 × 10−4 , c11 = −2.94665 × 10 −4 , c12 = 3.60813 × 10−7 , c13 = 1126.3493, c14 = 0.125652, c15 = −3.47801× 107 , c16 = 1046.2771, c17 = 0.174515, c18 = −2.80047 × 107 , s1 = −6650.57125817941, s2 = −5969.21115510038, T1 = 298.15, T2 = 300, T3 = 1273.
The relation is valid for T > 298.15 and T < 2324.15 K.
630
17. Thermo-physical properties for severe accident analysis
1500 1400 cp in J/(kgK)
1300 1200 1100 1000
Thurnay (1985)
900 800 700
500 1000 1500 2000 2500 3000 3500 T in K
h in kJ/kg
Fig. 17.7.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid alumina
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0
Thurnay (1985) cp model
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.7.2. Specific enthalpy as a function of the temperature of solid and liquid alumina. p = 1 bar
17.7 Aluminum oxide, Al2 O3
631
4000 3500 s in J/(kgK)
3000 2500 2000 1500 1000 500 0
Thurnay (1985) cp model 500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.7.3. Specific entropy as a function of the temperature of solid and liquid alumina. p = 1 bar
Figures 17.7.1, 17.7.2 and 17.7.3 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as functions of the temperature for atmospheric pressure for solid and liquid alumina. The first jump is due to phase change and the second one due to melting. 17.7.1.4 Solid density The data for solid density 0 < T ≤ 2324.15 K by Touloukian (1966) are approximated as a function of the temperature only ρ s = c0 + c1T ,
(17.7.14)
With c0 = 4045.6, c1 = −1.26609 × 10-1 by Turnay (1985). Figure 17.7.4 present the density of the solid and liquid alumina as a function of temperature at atmospheric pressure. 17.7.1.5 The derivative of the solid density with respect to the temperature Therefore the derivative of the solid density with respect to the temperature is d ρs = c1 . dT
(17.7.15)
632
17. Thermo-physical properties for severe accident analysis
4500
Density in kg/m³
4000 3500 Al2O3
3000 2500 2000
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.7.4. Density of the solid and liquid alumina as a function of temperature at atmospheric pressure
17.7.1.6 Solid thermal conductivity Data for solid thermal conductivity of alumina are reported by Samsonov (1982), Touloukian (1966), Touloukian et al. (1970). After critical review of the data Turnay (1985) recommended the following approximations. The solid thermal conductivity is approximated with six different polynomials for six different temperature intervals defined as follows: T < 22.9 K , i = 1; 22.9 K < T ≤ 35.224 K , i = 2; 35.224 K < T ≤ 79.9 K , i = 3; 79.9 K < T ≤ 290 K , i = 4; 290 K < T ≤ 1050 K i = 5, 1050 < T ≤ 2324.15 K , i = 6.
The approximation provided by Turnay (1985) is λs = 1× 10−5 ( c0,i + c1,iT + c2,iT 2 +c3,iT 3 +c4,iT 4 ) ,
where
(17.7.16)
17.7 Aluminum oxide, Al2 O3
633
Thermal conductivity in W/(mK)
c0 = (0.0d0, 0.0d0, 42.9399d6, 52.3554d6, 9.02681d6, 2.26552d6), c1 = (0.0d0, –12619.0d0, –3.83439d6, –702786.0d0, –27384.7d0, –2372.3d0), c2 = ( 2590.18d0, 6307.41d0, 133297.0d0, 4107.05d0, 37.7291d0, 0.993531d0), c3 = (529.814d0, 224.444d0, –1711.70d0, –11.1004d0, –25.371d-3, –112.082d-6), c4 = (–12.0454d0, –4.74074d0, 7.42942d0, 11.3772d-3, 6.7842d-6, 0.0d0). 40 35 30
Al2O3
25 20 15 10 5 0
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.7.5. Thermal conductivity of solid and liquid alumina as a function of temperature
Figure 17.7.5 presents the thermal conductivity of solid and liquid alumina as a function of temperature. 17.7.1.7 Solid sonic velocity With the Youngs modulus (elasticity modulus) of E = ( 431.10594 − 0.05257T )109 ,
(17.7.17)
approximating values taken from Ceramic Nist (2007) valid within 293.15 and 1773.15 K the velocity of sound is as = E ρ s .
(17.7.18)
This corresponds very well to the properties β = 0.00005 K–1 and κ = 2.8×10–12Pa–1 reported by McCahan and Shepherd (1993).
634
17. Thermo-physical properties for severe accident analysis
17.7.2 Liquid 17.7.2.1 Liquid specific capacity at constant pressure Turnay (1985) recommended the constant value for the specific thermal capacity at constant pressure c pl = 1421.7131 J/(kgK) for T > 2324.15 K .
(17.7.19)
17.7.2.2 Liquid specific enthalpy As already mentioned for h between 2739865.3242982719 J/kg and 3807505.3242982719 J/kg the temperature is 2324.15 K, Shpil’rain (1973), because of melting. The melt enthalpy is 1067640 J/kg. The liquid specific enthalpy is approximated by Eq. (17.1.8) with h′ = 3807505.3242982719 J/kg. 17.7.2.3 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10). 17.7.2.4 Liquid specific entropy As already mentioned for s between 2548.7886432044293 J/(kgK) and 3008.1565841720949 J/(kgK) the temperature is 2324.15 K because of melting. The melt entropy is 459.36794096766556 J/(kgK). The liquid specific entropy is approximated by Eq. (17.1.11) with s ′ = 3008.1565841720949 J/(kgK). 17.7.2.5 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed where s′ = iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , 3008.1565841720949 and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. 17.7.2.6 Liquid density The liquid density data reported by Shpil’rain et al. (1973) for T > 2324.15 K are approximated with
17.7 Aluminum oxide, Al2 O3
ρl = c0 + c1T ,
635
(17.7.20)
where c0 = 5298.59, c1 = −0.965 by Turnay (1985). Note that ρl 0 = ρl ( 2324.15 K ) = 3055.78 kg/m³. The liquid density as a function of temperature and pressure is then approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105Pa, T0 = 2324.15K, ρl 0 = 3055.78525kg/m³. The volumetric thermal expansion coefficient is taken to be β = 0.965/ ρl 0 K-1, and the isothermal compressibility is -12 -1 κ = 2.8x10 Pa in accordance with McCahan and Shepherd (1993). 17.7.2.7 Specific capacity at constant pressure c pl = 1421.7131J/(kgK), Turnay (1985).
(17.7.21)
17.7.2.8 Liquid density derivative with respect to temperature Equation (17.1.5). 17.7.2.9 Liquid density derivative with respect to pressure Equation (17.1.6). 17.7.2.10 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007),
Velocity of sound in m/s
Eq. (17.1.7). 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000
Al2O3
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.7.6. Velocity of sound for solid and liquid alumina as a function of the temperature at atmospheric conditions
636
17. Thermo-physical properties for severe accident analysis
Figure 17.7.6 shows the velocity of sound for solid and liquid alumina as a function of the temperature at atmospheric conditions. 17.7.2.11 Liquid thermal conductivity We assume
λl = 7.072827583108203 W/(mK),
(17.7.22)
due to lack of better information. 17.7.2.12 Liquide surface tension The surface tension is approximated by 2 σ l = ⎡1560-522.2 × 10-3 (T − 273.15 ) + 43 × 10-6 (T − 273.15) ⎤ × 10−3 N/m (17.7.23)
⎣
⎦
which goes through the points σ l ( 2500 K) = 0.612 N/m and σ l ( 3000 K) = 0.46 N/m, see Kuhn et al. (1983), Peppler et al. (1983). Figure 17.7.7 presents this dependence.
Surface tension N/m
0.70 0.65 Al2O3
0.60 0.55 0.50 0.45 0.40 0.35 0.30
2400 2600 2800 3000 3200 3400 T in K
Fig. 17.7.7. Surface tension of a liquid alumina as a function of the temperature
17.7.2.13 Liquid dynamic viscosity Data for liquid viscosity are published by Touloukian (1966), Bates et al. (1971), Biomquist et al. (1978), Elyutin et al. (1972) and Urbain (1982). Turnay (1985) selected the Urbain (1982) data and approximated them by
References
Dynamic viscosity /1000 in kg/(ms)
ηl = 0.1× 10( 6156.52 T − 2.98038) .
637
(17.7.24)
50 Al2O3
40 30 20 10 2500
3000 T in K
3500
Fig. 17.7.8. Liquid dynamic viscosity of alumina as a function of the temperature
Figure 17.7.8 presents this dependence. Interestingly, Samsonov (1973) reported the dynamic viscosity values for solid alumina. Turnay (1985) approximated them by η s = 0.1× 10(13227.5 T + 3.77669) .
(17.7.25)
References Barin I and Knacke O (1973) Thermo-Chemical Properties of Inorganic Substances. Springer-Verlag, Berlin Heidelberg New York Bates JL, McNeilly CE and Rasmussen JJ (1971) Properties of Molten Ceramics. Batelle Memorial Institute, Richland, Washington BNWL-SA-3529 Biomquist RA, Fink JK and Leibowitz L (1978) Viscosity of molten alumina. Ceram. Bull., vol 5, p 522 Ceramic Nist (2007) http://www.ceramics.nist.gov/srd/summary/scdaos.htm Elyutin VP, Mitin BC and Nagibin YuA (1972) Properties of liquid aluminum oxide. Fiz. Aerodispersnykh Syst., vol 7, pp 104–109 Kolev NI (2007) Multiphase flow dynamics, vol 1, Springer, Berlin Kuhn D, Moschke M and Werle H (Okt., 1983) Freezing of aluminum oxide and iron flowing upward in circular quartz glass tubes, KfK 3592 McCahan S and Shepherd JE (January, 1993) A thermodynamic model for aluminum-water interaction. Proceedings of the CSNI Specialists Meeting on Fuel-Coolant Interaction, Santa Barbara, California, NUREC/CP-0127 Peppler W, Menzenhauer P and Will H (July, 1983) Simulated fuel melt movement and relocation in two seven-pin-bundle geometries, KfK 3591
638
17. Thermo-physical properties for severe accident analysis
Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New-York, pp 3–285 Samsonov GV ed. (1973) The Oxide Handbook, IFI/PLENUM, New York Samsonov GV ed. (1982) The Oxide Handbook, 2nd ed. IFI/PLENUM, New York Shpil’rain EE, Yakimovich KA and Tsitsarkin AF (1973) Experimental study of the density of liquid alumina up to 2750°C. High Temp. – High Pressures, vol 5, pp 191–198 Touloukian YS ed. (February, 1966) Recommended values of the thermo-physical properties of eight alloys, major constituents and their oxides thermo-physical properties research center, Purdue University, Lafayette, Indiana Touloukian YS and Buyco EH (1970) Thermo-Physical Properties of Matter, vol 5, Specific Heat/Nonmetallic Solids. DATA TABLE NO. 62-A, IFI/PLENUM, New York Touloukian YS, Powell RW, Ho CY and Klemens PG (1970) Thermo-Physical Properties of Matter, vol 2, Thermal Conductivity/Nonmetallic Solids. DATA TABLE no 32, IFI/PLENUM, New York Turnay K (Mai 1985) Thermophysicalische Eigenschaften des Aluminiumoxides und Quarzglases, Research Center Karlsruhe Urbain G (1982) Viscosite de 1’alumine liquide. Rev. Int. Hautes Temper. Refract, vol 19, pp 55–57
17.8 Silicon dioxide
639
17.8 Silicon dioxide Silicon dioxide or silica is one of the most commonly encountered substances in daily life. Crystalline silicon dioxide is an important constituent of a great many minerals and gemstones, both in pure form and mixed with related oxides. Beach sand is mostly silica. In the nature transport of sand by wind or water is of great ecological importance. Therefore the properties of sand are of importance for proper mathematical description of such processes. During severe accident the interaction of molten materials with the power plant structures is important. This is the reason for the interest to the thermal and transport properties of sand in the nuclear technology. Silicon dioxide is known in the nature as a quartz glass being transparent or non transparent; see Touloukian and Ho (1976). Quartz glass is a subcooled liquid of SiO2 with molar mass of M = 0.0606 kg/kmole, Perry and Green (1985), a density between 2200 and 2600 kg/m². The melting happens within 1950 and 2000 K. 1983 K is reported by Perry and Green (1985). In Samsonov (1982) some of the naturally known minerals density and melting points are reported: quartz, 2651 kg/m³, 1883 K, tridymite, 2260 kg/m³, 1953 K and cristobalite, 2320 kg/m³ and T ′′′ = 1993 K.
(17.8.1)
For comparison in Wikipedia T ′′′ = 1923.15 ± 75 K is given. The normal boiling point is T ′ = 2503.15 K, Perry and Green (1985). We consider here the properties of cristobalite as representative for SiO2. 17.8.1 Solid 17.8.1.1 Solid specific capacity at constant pressure Data for the specific capacity at constant pressure reported by Samsonov (1982) and Touloukian and Buyco (1970) are critically reviewed by Turnay (1985) and approximated as follows c ps = c1 + c2T + c3 T 2 ,
(17.8.2)
where c1 = 1035.9088, c2 = 0.150342, c3 = −3.0487 × 107 . This approximation will be used here. The relation is valid for 298.15 < T < 1993.15 K. For completeness we give the approximation provided by Perry and Green (1985): c ps = 750.5 + 0.6015T − 1.6653×107 T 2 m 1%
640
17. Thermo-physical properties for severe accident analysis
valid for 273 < T < 848 K α -quartz, c ps = 756.02 + 0.3797 T m 3.5%
valid for 848 < T < 1873 K β -quartz, c ps = 252 + 1.657 T m 2.5%
valid for 273 < T < 523 K α -cristobalite, c ps = 1179.94 + 0.031345 T− 6.1945293 ×107 T 2 m 2%
valid for 523 < T < 1973 K α -cristobalite and c ps = 883.75 + 0.30862 T− 2.0850957 ×107 T 2 m 3.5%
valid for 273 < T < 1973 K glas. 17.8.1.2 Solid specific enthalpy The specific enthalpy for the solid cristobalite is therefore
(
)
hs = c1 T − Tref +
(
)
(
)
1 c2 T 2 - Tref2 − c3 1 T − 1 Tref , 2
(17.8.3)
where Tref = 298.15 K. The relation is valid for 298.15 K < T < 1993.15 K. 17.8.1.3 Solid specific entropy The specific entropy for the solid cristobalite is
(
)
(
)
(
)
1 ss = c1 ln T Tref + c2 T − Tref − c3 1 T 2 − 1 Tref2 . 2
The relation is valid for 298.15 K < T < 1993.15 K.
(17.8.4)
17.8 Silicon dioxide
641
1500 1400 cp in J/(kgK)
1300 1200 1100 1000
Thurnay (1985)
900 800 700
500
1000
1500 T in K
2000
2500
Fig. 17.8.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid SiO2
3500 3000 h in kJ/kg
2500
Thurnay (1985) cp model
2000 1500 1000 500 0
500
1000
1500 T in K
2000
2500
Fig. 17.8.2. Specific enthalpy as a function of the temperature of solid and liquid SiO2. p = 1 bar
642
17. Thermo-physical properties for severe accident analysis
3000
s in J/(kgK)
2500
Thurnay (1985) cp model
2000 1500 1000 500 0
500
1000
1500 T in K
2000
2500
Fig. 17.8.3. Specific entropy as a function of the temperature of solid and liquid SiO2. p = 1 bar
Figures 17.8.1, 17.8.2 and 17.8.3 presents the caloric properties of cristobalit for solid and liquid states. Note that there is smooth transition between solid and liquid because the melt entropy is zero. 17.8.1.4 Solid density Data reported for solid density by Touloukian (1966) for Cristobalite are approximated by Turnay (1985) as a function of the temperature only: ρ s = c10 + c11T + c12T 2 for T < 523 K for α - cristobalite,
(17.8.5)
ρ s = c20 + c21T + c22T 2 for 523 K < T < 1993.15 K for β -cristobalite, (17.8.6)
with c10 = 2338.57, c11 = 5 × 10−2 , c12 = −3.57143 × 10−4 , c20 = 2193.09, c21 = 2.79599 × 10−2 , c22 = −2.82594 × 10−4 .
17.8 Silicon dioxide
643
Density in kg/m³
2400 2300
SiO2
2200 2100 2000
500
1000
1500 T in K
2000
2500
Fig. 17.8.4. Density of solid and liquid cristobalite as a function of the temperature
Figure 17.8.4 presents the density of solid and liquid cristobalite as a function of the temperature. The first jump in density is due to phase transition. Note again that there is smooth transition between solid and liquid. 17.8.1.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is therefore d ρs = c11 + 2c12T for T < 523 K for α -cristobalit, dT
(17.8.7)
d ρs = c21 + 2c22T for 523 K < T < 1993.15 K for β -cristobalit. dT
(17.8.8)
17.8.1.6 Solid thermal conductivity The solid thermal conductivity data are provided by Samsonov (1982), Touloukian et al. (1970), Krzhizhanovskii and Shtern (1973), Men’ and Chechel’nitskii (1973), Bityukov et al. (1984), Bityukov et al. (2000). Turnay (1985) critically reviewed these data and proposed the following formalism of approximation. First four temperature regions are defined as follows T < 2.3 K , i = 1,
2.3 K < T ≤ 450 K , i = 2, 450 K < T ≤ 1395.1 K , i = 3,
644
17. Thermo-physical properties for severe accident analysis
1395.1 K < T , i = 4.
In each region the thermal conductivity is then λs = 10−5 ( ci ,0 + ci ,1T + ci ,2T 2 +ci ,3T 3 ) .
(17.8.9)
Here c10 = 0, c11 = 0, c12 = 143.951, c13 = 0, c20 = −1103.87, c21 = 815.255, c22 = −1.45043, c23 = 1.02794 × 10−3 , c30 = 52426.40, c31 = 366.975, c32 = −0.289129, c33 = 8.09922 × 10−5 ,
Thermal conductivity in W/(mK)
c40 = 172657, c41 = 52.8927, c42 = −0.0161725, c43 = 2.43335 × 10−6.
2.6 2.4 2.2 2.0 1.8
SiO2
1.6 1.4 1.2
500
1000
1500 T in K
2000
2500
Fig. 17.8.5. Thermal conductivity for solid and liquid cristobalite
Figure 17.8.5 presents the thermal conductivity for solid and liquid cristobalite. 17.8.1.7 Solid sonic velocity With the Youngs modulus (elasticity modulus) of E = 46 to 75GPa, Memsnet (2007) the velocity of sound is as = E ρ s .
(17.8.10)
17.8 Silicon dioxide
645
17.8.1.8 Emissivity Text books give emissivities within 273.15 and 673.15 K of 0.94 to 0.66 for glass (window). 0.79 is reported by Perry and Green (1985, pp. 2–337). 17.8.2 Liquid 17.8.2.1 Liquid density Data reported for liquid density by Touloukian (1966) for Cristobalit are approximated by Turnay (1985) as a function of the temperature only: ρl = 267567 − 0.270676T .
(17.8.11)
The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105Pa, T0 = 1993.15 K, ρl 0 = 2136.5536562433035 kg/m³, the volumetric thermal expansion coefficient is β = 0.270676/ ρl 0 K–1, and the isothermal compressibility is κ = 4.234×10–11Pa–1. 17.8.2.2 Specific capacity at constant pressure The function used for the solid region is extended also into the liquid region. 17.8.2.3 Liquid density derivative with respect to temperature Equation (17.1.5) 17.8.2.4 Liquid density derivative with respect to pressure Equation (17.1.6). 17.8.2.5 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007), Equation (17.1.7). Note that here c pl is a function of the local temperature. The velocity of sound as a function of the temperature at 1 bar is presented in Fig. 17.8.6.
646
17. Thermo-physical properties for severe accident analysis
Velocity of sound in m/s
6000 5000 SiO2
4000 3000 2000
500
1000
1500 T in K
2000
2500
Fig. 17.8.6. Velocity of sound for solid and liquid silicon oxide
17.8.2.6 Liquid thermal conductivity The liquid thermal conductivity is approximated by Turnay (1985) as a continuation of the fourth temperature interval of the solid thermal conductivity
λl = 1× 10−5 ( c40 + c41T + c42T 2 + c43T 3 ) for T > 1993.15K ,
(17.8.12)
with c40 = 172657, c41 = 52.8927, c42 = −0.0161725, c43 = 2.43335 × 10 −6 . 17.8.2.7 Liquid surface tension The liquid surface tension is assumed to be σ l = 0.86 N/m
(17.8.13)
due to lack of better knowledge. 17.8.2.8 Liquid dynamic viscosity The data for the liquid dynamic viscosity published by Samsonov (1982) and Touloukian (1966) are correlated by Turnay (1985) as follows ηl = 0.1× 10( 30530 T − 7.7415) .
(17.8.14)
17.8 Silicon dioxide
647
Dynamic viscosity in kg/(ms)
7
10
SiO2
6
10
5
10
4
10
3
10 2000
2100
2200 2300 T in K
2400
2500
Fig. 17.8.7. The dynamic viscosity of liquid silicon dioxide as a function of the temperature
Figure 17.8.7 represents this dependence. 17.8.2.9 Liquid specific enthalpy There is smooth transition between solid and liquid because the melt enthalpy is 0. The liquid specific enthalpy is approximated by Eq. (3.243) Kolev (2007a) hl = h′ + c1 (T − T0 ) + −
1−T β
κρl 0
⎛ 1 1⎞ 1 c2 T 2 − T02 + c3 ⎜ − ⎟ 2 ⎝ T0 T ⎠
(
)
{
}
exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 ,
(17.8.15)
where h′ = 1960853.0397222259 J/kg. 17.8.2.10 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.8.11).
17.8.2.11 Liquid specific entropy There is smooth transition between solid and liquid because the melt entropy is 0. The liquid specific entropy is approximated by Eq. (3.244) Kolev (2007a)
648
17. Thermo-physical properties for severe accident analysis
1 ⎛ 1 1 ⎞ sl = sl′ + c1 ln (T T0 ) + c2 (T − T0 ) − c3 ⎜ 2 − 2 ⎟ 2 ⎝T T0 ⎠ +
β exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 , κρl 0
{
}
(17.8.16)
where s ′ = 2055.2832670628213 J/(kgK). 17.8.2.12 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed where s′ = iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c1 ⎤⎦ , 2055.2832670628213 J/(kgK) and using Eq. (3.244) Kolev (2007a) solved with respect to the temperature ⎡⎛ β ⎢⎜ sl − s ′ − κρ exp ⎡⎣ β (T00 - T0 ) ⎤⎦ exp ⎡⎣ −κ ( P - P0 ) ⎤⎦ − 1 l0 T00 = T0 exp ⎢⎢⎜ ⎜ 1 ⎛ 1 1 ⎞ ⎢⎜ −c2 (T − T0 ) + c3 ⎜ 2 − 2 ⎟ ⎜ 2 ⎝T T0 ⎠ ⎢⎣⎝
{
} ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎤ ⎥ c1 ⎥⎥ ⎥ ⎥⎦
(17.8.17) Less than 10 iterations are needed to reduce the error below 0.0001 K. References Bityukov VK and Petrov VA (März 2000) Optical quartz glass as a reference substance for the thermal conductivity coefficient of partially transparent materials, High Temperature. vol 38 no 2 pp 293–299 Bityukov VK, Petrov VA and Stepanov SV (1984) Determination of the coefficient of thermal conductivity of silica glass by the contactless flat-plate method at 950-1500K. High Temp., vol 21, no 6, p 840 (Teplofiz. Vys. Temp. (1983) vol 21, no 6, p 1099) Kolev NI (2007a) Multiphase flow dynamics, vol 1, Springer, Berlin Krzhizhanovskii RE and Shtern ZYu (1973) Thermo-physical properties of nonmetallic materials, Energija, Leningrad Men’ AA and Chechel’nitskii AZ (1973) Teplofis. Vys. Temp., vol 11, no 6, p 1309 Memsnet (2007) www.memsnet.org/material/silicondioxidesio2film/ Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New-York Samsonov GV ed. (1982) The Oxide Handbook, 2nd ed. IFI/PLENUM, New York Touloukian YS ed. (February, 1966) Recommended values of the thermo-physical properties of eight alloys, major constituents and their oxides thermo-physical properties research center, Purdue University, Lafayette, Indiana
References
649
Touloukian YS and Buyco EH (1970) Thermo-Physical Properties of Matter, vol 5, Specific Heat/Nonmetallic Solids. DATA TABLE NO. 62-A, IFI/PLENUM, New York Touloukian YS, Powell RW, Ho CY and Klemens PG (1970) Thermo-Physical Properties of Matter, vol 2, Thermal Conductivity/Nonmetallic Solids. DATA TABLE no 32, IFI/PLENUM, New York Touloukian YS and Ho CY, eds. (1976) Thermo-physical properties of selected aerospace materials, Part I, Thermo-physical and Electronic Properties Information Center, CINDAS – Purdue University Turnay K (Mai, 1985) Thermophysicalische Eigenschaften des Aluminiumoxides und Quarzglases, Research Center Karlsruhe
17.9 Iron oxide
651
17.9 Iron oxide As a substance the iron monoxide is reported to be stable for temperatures larger than 850 K, Massalski (1992). It has a mole mass M = 0.0718464 kg/kmol.
(17.9.1)
Perry and Green (1985) give 0.07185 kg/mol. The melting temperature is reported by Lide and Frederikse (1997) to be T ′′′ = 1642 K.
(17.9.2)
Massalski (1992) reported a melting temperature 1650 K and in Wikipedia I find 1643.15 K which is close to the accepted here. Perry and Green (1985) reported 1693.15 K. The boiling point is 3687.15 K. 17.9.1 Solid 17.9.1.1 Solid specific capacity at constant pressure Barin and Knacke (1973) reported the following approximation for the heat capacity at constant pressure for solid iron oxide c ps = c0 + c1T + c2T 2 + c3T 3 ,
(17.9.3)
where c0 = 606.8240695652054 , c1 = 0.340551519681837 , c2 = −1.531627089846903 ×10−4 , c3 = 3.227906855636743 × 10−8 . For comparison Perry and Green (1985) reported c ps = 734.93 + 0.086887T − 4.437533 × 106 T 2 ± 2%
valid within 273 and 1173 K. 17.9.1.2 Solid specific enthalpy For h between 1200971.868 J/kg and 1535110.7568999999 J/kg the temperature is 1642 K because of melting. The melt enthalpy is 334836.4649 J/kg, see Barin and Knacke (1973).
(
)
hs = c0 T − Tref +
(
)
(
)
(
)
1 1 1 c1 T 2 − Tref2 + c3 T 3 − Tref3 + c4 T 4 − Tref4 , 2 3 3
(17.9.4)
652
17. Thermo-physical properties for severe accident analysis
where Tref = 298.15 K . The relation is valid for 298.15 < T < 1642 K. 17.9.1.3 Solid specific entropy For s between 1340.617421893374 J/(kgK) and 1544.537315293374 J/(kgK) the temperature is 1642 K because of melting. The melt entropy is 203.9198934 J/(kgK). The solid specific entropy is
(
)
(
)
ss = c0 ln T Tref + c1 T − Tref +
(
)
(
1 1 c2 T 2 − Tref2 + c3 T 3 − Tref3 2 3
)
(17.9.5)
The relation is valid for 298.15 < T < 1642 K. 17.9.1.4 Solid density The solid density is approximated as a function of the temperature only by Touloukian et al. (1977) ρ s = c0 ⎡⎣1 − 3 ( c11 + c12T + c13T 2 + c14T 3 ) ⎤⎦ ± 5%
(17.9.6)
with c0 = 5702.133168, c11 = −0.409 × 10−2 , c12 = 1.602 × 10−5 , c13 = −7.913 × 10−9 , c14 = 5.348 × 10−12.
17.9.1.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is therefore d ρs = −c0 3 c12 + 2c13T + 3c14T 2 . dT
(
)
(17.9.7)
17.9.1.6 Solid thermal conductivity The solid thermal conductivity data by Touloukian et al. (1970) are approximated in MATPRO (1990) with the expression λs = 4.6851 + 100T (−3.3292 × 10−7 − 2.5618 × 10−8 T ) .
(17.9.8)
The data are valid for T < 796.15 K only. For larger temperature the constant
17.9 Iron oxide
λs = 3.02
653
(17.9.9)
is assumed due to lack of better knowledge. 17.9.1.7 Solid sonic velocity The solid velocity of sound is unknown to me. 17.9.1.8 Emissivity As already mentioned text books give for polished steel with temperature between 273.15 and 1273.15 K emisivities for cast iron with C ≈ 4% for rusted surfaces the value 0.6 to 0.8. For wrought iron with C ≈ 0.5% with oxidized surfaces the emissivity is frequently reported to be 0.9 to 0.95. These values can be taken as orientation for the emissivity of the solid iron oxide. 17.9.2 Liquid 17.9.2.1 Liquid specific capacity at constant pressure Barin and Knacke (1973) reported a constant value for the heat capacity at constant pressure for liquid iron oxide c pl = 949.1858038 J/(kgK).
(17.9.10)
17.9.2.2 Liquid specific enthalpy For h between 1093364.142077756 J/kg and 1428200.606977756 J/kg the temperature is 1642 K because of melting. The melt enthalpy is 334836.4649 J/kg, see Barin and Knacke (1973). The liquid specific enthalpy is approximated by Eq. (17.1.8) with h ′ = 1428200.606977756 J/kg. 17.9.2.3 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10). 17.9.2.4 Liquid specific entropy For s between 1340.617421893374 J/(kgK) and 1544.537315293374 J/(kgK) the temperature is 1642 K because of melting. The melt entropy is 203.9198934
654
17. Thermo-physical properties for severe accident analysis
J/(kgK). The liquid specific entropy is approximated by Eq. (17.1.11) with s′ = 1544.537315293374 J/(kgK). 1000
Barin and Knacke (1973)
cp in J/(kgK)
900 800 700 600
500
1000
1500 T in K
2000
2500
Fig. 17.9.1. The specific capacity at constant pressure as a function of the temperature for solid and liquid iron mono oxide
2500
h in kJ/kg
2000
Barin and Knacke (1973) cp model
1500 1000 500 0
500
1000
1500 T in K
2000
2500
Fig. 17.9.2. The specific enthalpy as a function of the temperature for solid and liquid iron mono oxide at atmospheric pressure
17.9 Iron oxide
655
2000
s in J/(kgK)
1500 1000 500 0
Barin and Knacke (1973) cp model 500
1000
1500 T in K
2000
2500
Fig. 17.9.3. The specific entropy as a function of the temperature for solid and liquid iron mono oxide at atmospheric pressure
Figures 17.9.1, 17.9.2 and 17.9.3 present the specific capacity at constant pressure, the specific enthalpy and the specific entropy as a function of the temperature for solid and liquid iron mono oxide at atmospheric pressure. 17.9.2.5 Liquid temperature as a function of pressure and specific entropy Given the pressure and the specific liquid entropy. The temperature is computed where s′ = iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , 1544.537315293374 J/(kgK) and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. 17.9.2.6 Liquid density Reimann and Stiefel (1989) proposed the following approximation for the liquid density ρl =
5613 . 1 + 1.6 ×10 −4 (T − 1673)
(17.9.11)
We derive from this equation for the volumetric thermal expansion coefficient –1 β ≈ 0.9/ ρl 0 K . Then we approximate the liquid density by Eq. (17.1.4) using as reference state defined by p0 = 105 Pa, T0 = 1642 K, ρl 0 = 5282.070148697579 kg/m³ which is the solid density at the melting point and the isothermal compressibility is κ = 4.234×10–11 Pa–1.
656
17. Thermo-physical properties for severe accident analysis
6000
Density in kg/m³
FeO 5500 5000 4500 4000
500
1000
1500 T in K
2000
2500
Fig. 17.9.4. The density as a function of the temperature for solid and liquid iron mono oxide at atmospheric pressure
Figure 17.9.4 presents the density as a function of the temperature for solid and liquid iron mono oxide at atmospheric pressure. 17.9.2.7 Liquid density derivative with respect to temperature Equation. (17.1.5). 17.9.2.8 Liquid density derivative with respect to pressure Equation. (17.1.6). 17.9.2.9 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007), Eq. (17.1.7). 17.9.2.10 Liquid thermal conductivity I assume λl = 3 W/(mK) due to lack of better knowledge.
Thermal conductivity in W/(mK)
17.9 Iron oxide
657
5
4
FeO
3
2
500
1000
1500 T in K
2000
2500
Dynamic viscosity in kg/(ms)
Fig. 17.9.5. The thermal conductivity as a function of the temperature for solid and liquid iron mono oxide
0.06 0.05
FeO
0.04 0.03 0.02 0.01 0.00
1800
2000 2200 T in K
2400
Fig. 17.9.6. The dynamic viscosity as a function of the temperature for liquid iron mono oxide
Figure 17.9.5 presents the thermal conductivity as a function of the temperature for solid and liquid iron mono oxide. 17.9.2.11 Liquid surface tension No information is available. I use σ l = 1.19 N/m due to lack of better knowledge.
658
17. Thermo-physical properties for severe accident analysis
17.9.2.12 Liquid dynamic viscosity Powers et al. (1986) proposed the following expression for modeling of the dynamic viscosity ηl = 10-5 exp (14070 T ) .
(17.9.12)
Figure 17.9.6 presents the dynamic viscosity as a function of the temperature for liquid iron mono oxide. References Barin I and Knacke O (1973) Thermochemical Properties of Inorganic Substances. Springer-Verlag, New York Kolev NI (2007a) Multiphase flow dynamics, vol 1, Springer, Berlin Lide DR and Frederikse HPR eds. (1997) CRC Handbook of Chemistry and Physics, 78th ed. CRC Press, New York Massalski TB ed. (1992) Binary Alloy Phase Diagrams ASM International, 2nd ed. ASM International, Materials Park, Ohio MATPRO (1990) SCDAP/RELAP5/MOD2 Code Manual, vol 4: MATPRO – A Library of Materials Properties for Light- Water-Reactor Accident Analysis, NUREG/CR-5273 Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New-York, pp 3–285 Powers DA, Brockmann JE and Shiver AW (July, 1986) VANESA: A mechanistic model of radionuclide release and aerosol generation during core debris interactions with concrete, NUREG/CR-4308 Reimann M and Stiefel S (June, 1989) The WECHSL-Mod2 Code: A computer program for the interaction of a core melt with concrete including the long term behavior – Model Description and User’s Manual, KfK 4477 Touloukian YS, Powell RW, Ho CY and Klemens PG (1970) Thermophysical Properties of Matter, vol 2, Thermal Conductivity – Nonmetallic Solids. IFI/PLENUM, New York, Washington Touloukian YS and DeWitt DP (1972) Thermophysical Properties of Matter. Plenum Press, New York Touloukian YS, Kirby RK, Taylor RE and Lee TYR (1977) Thermophysical Properties of Matter – vol 13, Thermal Expansion – Nonmetallic Solids. IFI/PLENUM, New York, Washington
17.10 Molybdenum
659
17.10 Molybdenum Molybdenum posses a density and melting point close to the uranium mixtures and therefore is used some times for experimental investigation instead of uranium mixtures. This is the reason why we give a summary of its properties here. The mole-mass of molybdenum is M = 0.09595 kg/kmol,
(17.10.1)
Perry and Green (1985). The melting temperature used here is T ′′′ = 2896 ± 10 K,
(17.10.2)
in accordance with Chase (1998), Cordfunke and Konings (1990) and T ′′′ = 2893.15 ± 10 K in accordance with Perry and Green (1985), Touloukian and DeWitt (1972). The boiling temperature at atmospheric pressure is reported to be T ′ = 4951.969 K,
(17.10.3)
by Chase (1998) and 3973.15 K by Perry and Green (1985). The heat of vaporization at atmospheric conditions is given in Wikipedia to be 617 kJ/mol. The heat of atomization of 659 kJ/mol is also reported. 17.10.1 Solid 17.10.1.1 Solid specific capacity at constant pressure Cordfunke and Konings (1990) reported for the specific heat at constant pressure Mc ps = a0 + a1T + a2T 2 + a3T 3 + a4 T 2
(17.10.4)
where a0 = 23.56414 , a1 = 6.8868 × 10−3 , a2 = −3.39771×10−6 , a3 = 15.7112 ×10−10 , a4 = −1.31625 × 105 , and for the specific melt enthalpy
h′′′ − h′ = 390660.8297 J/kg.
(17.10.5)
The specific melt entropy is therefore s ′′′ − s ′ = ( h′′′ − h′ ) T ′′′ = 134.8966953 J/(kgK).
(17.10.6)
For completeness let as mention that Touloukian and DeWitt (1972) reported for the specific heat at constant pressure c ps = 440 J/(kgK), and for the specific melt
660
17. Thermo-physical properties for severe accident analysis
enthalpy h′′′ − h′ = 305352.7941 J/kg. c3 = 3.227906855636743 × 10−8 . Perry and Green (1985) reported c ps = 428.12 + 0.081979T − 2.193384 ×106 T 2 ± 5% valid within 273 and 1773 K. Chase (1998) reported h′′′ − h′ = 375 234 J/kg. 17.10.1.2 Solid specific enthalpy With the Cordfunke and Konings (1990) approximation the specific solid enthalpy is obtained after integration
(
)
Mhs = a0 T − Tref +
(
)
(
)
(
)
(
a a1 2 a T − Tref2 + 2 T 3 − Tref3 + 3 T 4 − Tref4 − a4 1 T − 1 Tref 2 3 4
)
(17.10.7) Here the reference temperature is Tref = 298.15 K . The relation is valid for 298.15 K < T < 2896 K. For h between 933 287 J/kg and 1 323 948 J/kg the temperature is 2896 K because of melting. 17.10.1.3 Solid specific entropy The solid specific entropy is therefore
(
)
(
)
Mss = a0 ln T Tref + a1 T − Tref +
(
)
(
)
(
a a2 2 a T − Tref2 + 3 T 3 − Tref3 − 4 1 T 2 − 1 Tref2 2 3 2
)
(17.10.8) 800
cps-Cordfunke and Konings (1990), cpl-Chase (1998)
cp in J/(kgK)
700 600 500 400 300 200
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.10.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid Mo
17.10 Molybdenum
661
h in kJ/kg
The relation is valid for 298.15 K < T < 2896 K. For s between 722.746 J/(kgK) and 857.642 J/(kgK) the temperature is 2896 K because of melting.
1800 1600 1400 1200 1000 800 600 400 200 0
cps-Cordfunke and Konings (1990), cpl-Chase (1998)
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.10.2. Specific enthalpy as a function of the temperature of solid and liquid Mo. p = 1 bar
1000
s in J/(kgK)
800 600 400 200 0
cps-Cordfunke and Konings (1990), cpl-Chase (1998)
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.10.3. Specific entropy as a function of the temperature of solid and liquid Mo. p = 1 bar
In Figs. 17.10.1, 17.10.2 and 17.10.3 cp, h and s functions of the temperature for constant pressure of 1 bar for the solid molybdenum are visible.
662
17. Thermo-physical properties for severe accident analysis
17.10.1.4 Solid density The solid density is considered as a linear function of the temperature only: ρ s = 10387 − 0.365 T,
(17.10.9)
providing at 293.15 K 10 280 kg/m³ and at 2896 K, 9330 kg/m³. 17.10.1.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is therefore d ρs = −0.365 kg/(m³K). dT
(17.10.10)
17.10.1.6 Solid thermal conductivity The solid thermal conductivity is approximated with the constant given in Wikipedia λs ( 300 K) = 138 Wm• 1K• 1. Touloukian et al. (1970) proposed the following correlation
Thermal conductivity in W/(mK)
λs = 152.547 − 5.02038 ×10−2 T + 8.91289 × 10−6 T 2 .
(17.10.11)
140 130
Mo
120 110 100 90 80
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.10.4. Thermal conductivity of solid and liquid molybdenum
Figure 17.10.4 shows the above dependence.
17.10 Molybdenum
663
17.10.1.7 Solid sonic velocity The Young’s modulus of elasticity reported in Molybdenum (2007) is approximated here with Es = 3.48447 × 1011 − 1.19958 × 108 T + 45144.52381T 2
(17.10.12)
within 293.15 and 1273.15 K. The velocity of sound is then as = Es ρ s .
(17.10.13)
17.10.1.8 Emissivity Text books give emissivities within 273.15 and 673.15 K of 0.06 to 0.1 for polished, molybdenum. Other sources like Molybdenium (2007) reported that the spectral emissivity at 1000°C, ~0.65 µ for wavelength is 0.37, the total emissivity at 1500°C is 0.19, and the total emissivity at 2000°C is 0.24. 17.10.2 Liquid
17.10.2.1 Specific capacity at constant pressure Chase (1998) reported for the specific heat of liquid at constant atmospheric pressure data that can be approximated as follows Mc ps = a5 + a6T
(17.10.14)
where a5 = –43.62596, a6 = 0.034. For comparison Cordfunke and Konings (1990) reported for the specific heat at constant pressure c pl = 444.424 J/(kgK). 17.10.2.2 Liquid specific enthalpy As already mentioned for h between 933 287 J/kg and 1 323 948 J/kg the temperature is 2896 K because of melting. The liquid specific enthalpy is approximated by Eq. (17.1.8) with h′ = 1 323 948 J/kg. hl = h′ +
−
1−T β
κρl 0
1 ⎡ 1 ⎤ a5 (T − T ′′ ) + a6 T 2 − T ′′2 ⎥ M ⎢⎣ 2 ⎦
(
{
)
}
exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 ,
(17.10.15)
664
17. Thermo-physical properties for severe accident analysis
17.10.2.3 Liquid specific entropy As already mentioned for s between 722.746 J/(kgK) and 857.642 J/(kgK) the temperature is 2896 K because of melting. The liquid specific entropy is approxmated by the modified Eq. (17.1.11) with s ′ = 857.642 J/(kgK). sl = sl′ +
1 β ⎡ a5 ln (T T0 ) + a6 (T − T0 ) ⎤⎦ + exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 , M ⎣ κρl 0
{
}
(17.10.16) Figures 17.10.1, 17.10.2 and 17.10.3 demonstrate the cp, h and s functions of the temperature for constant pressure of 1 bar. 17.10.2.4 Liquid density The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105 Pa, T0 = 2893.15 K, ρl 0 = 9330 kg/m³ (Wikipedia), the volumetric thermal expansion coefficient is β = 0.5496/ ρl 0 K-1, and the isothermal compressibility is κ = 4.234×10-11 Pa-1.
10400 Density in kg/m³
10200
Mo
10000 9800 9600 9400 9200 9000
500 1000 1500 2000 2500 3000 3500 T in K
Fig. 17.10.5. Density of solid and liquid molybdenum as a function of temperature
17.10.2.5 Liquid density derivative with respect to temperature Equation (17.1.5). 17.10.2.6 Liquid density derivative with respect to pressure Equation (17.1.6).
17.10 Molybdenum
665
17.10.2.7 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007a), Eq. (17.1.7). 17.10.2.8 Liquid thermal conductivity The liquid thermal conductivity of liquid molybdenum is not known to me. I assume λl = 88 . 17.10.2.9 Liquid surface tension Elyutin et al. (1970) reported σ l = 0.213 for pure molybdenum. A dependence of dissolved Carbon impurities is reported also. 17.10.2.10 Liquid dynamic viscosity The dynamic viscosity of liquid molybdenum is not known to me. I assume ηl = 0.004822061 . 17.10.2.11 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10). 17.10.2.12 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , where sl 0 = s ′′ = 857.642 J/(kgK) and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. 17.10.2.13 Saturation temperature For pressures less than 1 bar the saturation temperature as a function of the pressure given in Wikipedia can be approximated by T´(p)= 2588.57143+ 182.82557lnp
666
17. Thermo-physical properties for severe accident analysis
References Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Nonograph No. 9, American Chemical Sociaty Cordfunke EHP and Konings RJM (1990) Thermochemical Data for Reactor Materials and Fussion Products. North-Holland, Amsterdam Elyutin VP, Kostikov VI and Pen’kov IA (September, 1970) Effect of carbon on the surface tension and density of liquid vanadium, niobium, and molybdenum. Poroshovaya Metallurgya, vol 93, no 9, pp 46–51 Kolev NI (2007) Multiphase Flow Dynamics, vol 1 Fundamentals. Springer, Berlin, New York, Tokyo Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New York, pp 3–285 Touloukian YS, Powell RW, Ho CY, and Klemens PG (1970) Thermophysical Properties of Matter – vol 1, Thermal Conductivity – Metallic Elements and Alloys. IFI/PLENUM, New York, Washington Touloukian YS and DeWitt DP (1972) Thermo-Physical Properties of Matter. Plenum Press, New York
17.11 Boron oxide
667
17.11 Boron oxide The substance boron is a strong absorber of thermal neutrons. This is the reason for its wide use in the nuclear energy mainly as boron oxide dissolved in water. Tracing the concentrations of boron oxide is an important safety issue. In any transients and accidents the local concentration in the primary circuit is of interests. That is why the properties of boron oxide are of importance for its use in nuclear safety analysis. The molar mass of the boron oxide, B2O3 is M = 0.0696202 kg/kmol.
(17.11.1)
In Wikipedia the molar mass is reported to be 0.0696182 kg/kmol and in Perry and Green (1985) 0.06964kg/kmol. Its melting temperature is relatively low T ′′′ = 723 ± 1K,
(17.11.2)
see in Kracek, Morey and Merwin (1938), Schmidt (1966). Perry and Green (1985) reported 850.15 K. The normal boiling temperature is reported in Wikipedia to be T ′ = 1953.15 K,
(17.11.3)
and in Perry and Green (1985) > 1773.15 K. 17.11.1 Solid 17.11.1.1 Solid specific capacity at constant pressure Experimental data for boron oxide reported by Kelly (1941) and Kerr et al. (1950), Glushko et al. (1981), Cox et al. (1989) are approximated by Cordfunke and Konings (1990) with Mc ps = c1 + c2T +
c3 T2
(17.11.4)
where c1 = 64.141, c2 =0.064643, c3= –18.359d5. Kruh and Stern (1956) proposed the polynomial expression with the same accuracy c ps = c1 + c2T + c3T 2 ,
(17.11.5)
where c1 = 95.944103, c2 = 3.2659831, c3 = −1.7870174 × 10−3 which is used here. Note that at the melting point c ps = 1523.12406582J/(kgK).
668
17. Thermo-physical properties for severe accident analysis
17.11.1.2 Solid specific enthalpy For h between 539 875 J/kg and 885 703 J/kg the temperature is 723 K because of melting. The melt enthalpy is 345 828 J/kg. Note that Schmidt (1966) reported 352 771 ± 2154 J/kg.
(
)
hs = c1 T - Tref +
(
)
(
)
1 1 c2 T 2 − Tref2 + c3 T 3 − Tref3 . 2 3
(17.11.6)
where Tref = 298.15 K. The relation is valid for 298.15 < T < 723 K. 17.11.1.3 Solid specific entropy For s between 1084.9 J/(kgK) and 1563.2 J/(kgK) the temperature is 723 K because of melting. The melt entropy is 478.32 J/(kgK). The solid specific entropy is
(
)
(
)
ss = c1 ln T Tref + c2 T − Tref +
(
1 c3 T 2 − Tref2 2
)
(17.11.7)
The relation is valid for 298.15 < T < 723 K. 17.11.2.4 Solid density For 298.15 K the density is 2460 kg/m³, Alfa Aesar (1999). Solid densities at higher temperature are not known to me. Using the thermal expansion coefficient of SiO2 I construct the following polynomial so that the above reference density is obtained for 298.15 K. The result is ρ s = c10 + c11T + c12T 2 ,
(17.11.8)
where c10 = 2.477 × 103 , c11 = 5.296 × 10−2 , c12 = −3.783 × 10−4 . 17.11.2.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is therefore d ρs = c11 + 2c12T . dT
(17.11.9)
17.11.2.6 Solid thermal conductivity No information is known to me for the solid thermal conductivity of the boron oxide. For approximate analysis I use the approximations for silica oxide.
17.11 Boron oxide
669
17.11.2.7 Solid sonic velocity No information is known to me. 17.11.2 Liquid 17.11.2.1 Specific capacity at constant pressure The data by Glushko et al. (1981) and Cox et al. (1989) are approximated by Cordfunke and Konings (1990) with Mc pl = c1 −
c2 T2
(17.11.10)
where c1 = 127.074, c2 = –31.380e5. This equations gives at the melting point the liquid specific capacity at constant pressure is c ps = 1911.472685 J/(kgK). Because the dependence on the temperature is low the value at the melting point can be used as a good approximation also for the liquid state. Note that at the melting point the solid specific capacity at constant pressure is lower. 17.11.2.2 Liquid specific enthalpy For h between 539874.811303 J/kg and 885703.061303 J/kg the temperature is 723 K because of melting. The melting enthalpy is 345828.25643036142 J/kg. The liquid specific enthalpy is then by hl = h′ +
−
1−T β
κρl 0
⎛1 1 ⎡ 1 ⎢ c1 T − Tref + c2 ⎜ − ⎜ M ⎢ ⎝ T Tref ⎣
(
)
{
⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦
}
exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 .
(17.11.11)
with h′ = 885703.061303J/kg. 17.11.2.3 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10).
670
17. Thermo-physical properties for severe accident analysis
17.11.2.4 Liquid specific entropy For s between 1084.90554354 J/(kgK) and 1563.22954077 J/(kgK) the temperature is 723 K because of melting. The melt entropy is 478.32400612774745 J/(kgK). The liquid specific entropy is approximated by Eq. (17.11.12)
2000
cp in J/(kgK)
1800 1600 1400 1200
Cordfunke and Konings (1990)
1000 800
500
1000 T in K
1500
2000
Fig. 17.11.1. Specific capacity at constant pressure as a function of the temperature of solid and liquid B2O3
4000 3500 h in kJ/kg
3000
Cordfunke and Konings (1990) cp model
2500 2000 1500 1000 500 0
500
1000 T in K
1500
2000
Fig. 17.11.2. Specific enthalpy as a function of the temperature of solid and liquid B2O3. p = 1 bar
sl = sl′ +
1 M
⎡ 1 ⎛ 1 1 ⎞⎤ ⎢ c1 ln (T T0 ) + c2 ⎜ 2 − 2 ⎟ ⎥ 2 ⎝T T0 ⎠ ⎦⎥ ⎣⎢
17.11 Boron oxide
+
β exp ⎡⎣ β (T − T0 ) ⎤⎦ exp ⎡⎣ −κ ( p − p0 ) ⎤⎦ − 1 , κρl 0
{
}
671
(17.11.12)
with s ′ = 1563.22954077 J/(kgK).
4000
Cordfunke and Konings (1990) cp model
s in J/(kgK)
3000 2000 1000 0
500
1000 T in K
1500
2000
Fig. 17.11.3. Specific entropy as a function of the temperature of solid and liquid B2O3. p = 1 bar
Figures 17.11.1, 17.11.2 and 17.11.3 presents the caloric properties of boron oxide for solid and liquid state. The jump is due to melting. 17.11.2.5 Liquid density Kruh and Stern (1956) reported the following approximation for the liquid boron oxide ρl = 1751.87 − 0.19T .
(17.11.13)
Therefore the volumetric thermal expansion coefficient is β = 0.19/ ρl 0 K–1. The liquid density is then approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105 Pa, T0 = 723 K, ρl 0 = 1614.5 kg/m³, and the isothermal compressibility is κ = 4.234×10–11 Pa–1. Figure 17.11.4 shows the density as a function of the temperature for solid and liquid boron oxide at atmospheric conditions.
672
17. Thermo-physical properties for severe accident analysis
2400 Density in kg/m³
2200 2000 B2O3
1800 1600 1400 1200 1000
500
1000 T in K
1500
2000
Fig. 17.11.4. Density as a function of the temperature for solid and liquid boron oxide at atmospheric conditions
17.11.2.6 Liquid density derivative with respect to temperature Equation. (17.1.5). 17.11.2.7 Liquid density derivative with respect to pressure Equation. (17.1.6). 17.11.2.8 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007), Eq. (17.1.7).
17.11 Boron oxide
673
Velocity of sound in m/s
5000
4000
B2O3
3000
2000
1000
1500 T in K
2000
Fig. 17.11.5. Velocity of sound for liquid boron oxide
With the assumed parameters the velocity of sound is presented in Fig. 17.11.5. 17.11.2.9 Liquid thermal conductivity Touloukian et al. (1970) proposed to use a linear fit between measurements for two temperatures as follows
Thermal conductivity in W/(mK)
λl = −2.49231 + 6.83077 ×10−3 T .
(17.11.14)
12 10
B2O3
8 6 4 2 0
500
1000 T in K
1500
2000
Fig. 17.11.6. Thermal conductivity for boron oxide as a function of the temperature for solid and liquid state
674
17. Thermo-physical properties for severe accident analysis
Figure 17.11.6 shows the thermal conductivity for boron oxide as a function of the temperature for solid and liquid state. 17.11.2.10 Liquid surface tension Due to not better knowledge I assume the surface tension to be σ l = 0.86 N/m. 17.11.2.11 Liquid dynamic viscosity Kruh and Stern (1956) proposed the following approximation for the liquid dynamic viscosity ηl = max ⎡⎣1, 1.71465 × 107 exp (1.24529 × 10−2 T ) ⎤⎦ ,
(17.11.15)
valid within 873.15 and 1173.15 K.
Dynamic viscosity in kg/(ms)
10000 1000
B2O3
100 10 1 0.1
1000
1500 T in K
2000
Fig. 17.11.7. Dynamic viscosity of liquid boron oxide as a function of the temperature
Figure 17.11.7 illustrates this dependence. 12.2.12 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , where s ′ = 2338.5012753994474 J/(kgK) and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.19). Less than 10 iterations are needed to reduce the error below 0.0001 K.
References
675
References Alfa Aesar (1999) Bestellkatalog Forschungschemikalien. Metalle and Materialien 1999– 2000 Cordfunke EHP and Konings RJM (1990) Thermochemical Data for Reactor Materials and Fussion Products North-Holland, Amsterdam Cox JD, Wagman DD and Medvedev VA (1989) CODATA Key Values for Thermodynamics. Hemisphere Publ. Corp., New York Glushko VP, Gurvich LV, Bergman GA, Veyts IV, Medvedev VA, Kachkuruzov GA and Yungman VS (1981) Termodinamicheskie Svoistva Individual’nykh. Tom III, Nauka, Moskva Kelly KK (1941) J. Am. Chem. Soc., vol 63, p 1137 Kerr EC, Hersh N and Johnston HL (1950) J. Am. Chem. Soc., vol 72, p 4738 Kolev NI (2007) Multiphase flow dynamics, vol 1, Springer, Berlin Kracek FC, Morey GW and Merwin HE (1938) Am. J. Sci., vol 35, p 143 Kruh R and Stern KH (1956) The effect of solutes on the properties and structure of liquid boric oxide. J. Am. Chem. Soc., vol 78, pp 278–281 Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New-York, pp 3–285 Schmidt NE (1966) Zh. Neorg. Khim., vol 11, p 441 (Russ. J. Inorg. Chem., vol 11, p 241) Touloukian YS, Powell RW, Ho CY and Klemens PG (1970) Thermophysical Properties of Matter, vol 2, Thermal Conductivity – Nonmetallic Solids. IFI/PLENUM, New York, Washington
17.12 Reactor corium
677
17.12 Reactor corium The nuclear reactor core contains different materials. The melting temperatures of some of them are given in Table 17.12.1. Table 17.12.1. Independent material species in a debris bed, Hodge and Ott (1997)
Material
T-melting in K
T-boiling in K
Fe Cr Ni Zr B4C FeO Fe3O4 NiO Cr2O3 B2O3 ZrO2 UO2
1808 2130 1728 2125 2728 1650 1839 2244 2572 2728 2978 3011
3008 2750 3003
In combination some of them form eutectic mixtures with lower melting temperatures as sown in Table 17.12.2. Table 17.12.2. Some eutectic mixture compositions, Hodge and Ott (1997)
Mixture
Mole fractions
Melting temperature in K
Zr–Steel Fe–Cr–Ni Zr–SS*UO2 ZrO2–UO2
0.1000-0.900 0.733-0.192-0.075 0.367-0.605-0.028 0.868-0.132
1523 1700 1873 2573
* stays for stainless steel The predominant amount of materials of nuclear reactor core is UO2, Zr and steel. So for instance boiling water reactors in the scale of 440–840 MW electrical power have cores consisting mainly of 84 to 160 t UO2, 37 to 71 t Zr and 5 to 10 t steel. During the melting process part of the Zr burns in water atmosphere so that part of the Zr is found as a ZrO2 in the oxide mixture. The mass of the ZrO2 is then 1.3516 times of the mass of the oxidized Zr, M ZrO2 = 1.3516 M oxidized Zr . Presence of the chemically very active Zr-metal inside the oxide mixture reduces the melting temperature as seen from all pseudo-binary phase diagrams presented in Fig. 17.12.1.
678
17. Thermo-physical properties for severe accident analysis
(a)
(b)
(c)
(d)
17.12 Reactor corium
679
(e) Fig. 17.12.1. Schematic pseudo-boundary phase diagram of: (a) UO2–ZrO2, Bottomley and Cooquerelle (1989); (b) Zr–UO2, Juenke and Whitte (1969); (c) ZrO0.43–UO2, Skokan (1984); (d) ZrO0.51–UO2, Politis (1975); (e) Hayward and George (1996): ZrO0.54–UO2–x constructed from Zry/25%O solubility data with Zr representing Zry
Additional information of haw Zirconium dissolves urania (UO2) is available in Romberger et al. (1967) and Tim and Olander (1988). So the core melt called corium has specific mixture properties depending on the concentrations of these main constituents, melting scenarios etc. In any case it starts its relocation from its initial position at temperatures less then the melting temperatures of the participating oxides. Some times this temperature is called mobility temperature. Following Chong et al. (1971, p. 2017), the measured viscosity of such mixtures of liquids and solids with particles 5 to 435 µm can be approximated as a function of the volumetric ratio of the solid volume fraction to the maximum packing solid fraction, α solid α max-packing , ⎛ ⎜ ⎝
ηeff = η ′′ (T ) ⎜ 1 + 0.75
α solid α max-packing 1 − α solid α max-packing
2
⎞ ⎟ for α solid α max-packing < 1 , ⎟ ⎠
(17.12.1)
α max-packing = 0.605 . The Young’s elasticity modulus exhibits the same dependence as the ratio ηeff η ′′ (T ) . Moshev and Ivanov (1990) summarized several approximations and confirmed the Chong et al. result. State of the art review and more information to this subject is available in Ramacciotti et al. (1999). See also Spindler and Vateau (2006a,b). Therefore mobility is possible also in presence of solid phase. Such information is important for analysis of the relocation of
680
17. Thermo-physical properties for severe accident analysis
the melt, for the analysis of the spreading of the melt in case of failure of the lower head etc. For the time being the description of mixtures at any temperatures consisting of all materials from Table 17.12.1 seems hopeless. Nevertheless approximations are needed for finding controllable configurations after severe accidents. One possible approach is to postulate limited number of characteristic constituents with a single combination of their concentration and to derive approximate properties from the most dominant constituent by taking into account the changed melting temperature and reference density. As an example the properties of a corium melt consisting of 80% UO2 and 20% ZrO2 are approximated here by the relations of UO2. The only changes done in comparison to the UO2-properties are the melt temperature which is now set to 2830 K and the reference density of corium at 298.15 K is set to 9550 kg/m³. 17.12.1 Liquid 17.12.1.1 Liquid density The liquid density is approximated by Eq. (17.1.4) where the reference state is defined by p0 = 105 Pa, T0 = 2920 K, ρl 0 = 8105.92 kg/m³, volumetric thermal expansion coefficient is β = 0.916/ ρl 0 K–1, and isothermal compressibility κ = 4.234×10–11Pa–1. 17.12.1.2 Specific capacity at constant pressure The specific capacity at constant pressure is set to c pl = 485.2 J/(kgK),
(17.12.2)
which is close to those used for liquid urania 503 J/(kgK), 17.12.1.3 Liquid density derivative with respect to temperature The liquid density derivative with respect to temperature is computed using Eq. (17.1.5). 17.12.1.4 Liquid density derivative with respect to pressure The liquid density derivative with respect to pressure is computed using Eq. (17.1.6).
17.12 Reactor corium
681
17.12.1.5 Liquid velocity of sound The liquid velocity of sound is approximated by Eq. (3.250) Kolev (2007) which is Eq. (17.1.7) in this chapter. 17.12.1.6 Liquid thermal conductivity The liquid thermal conductivity of this mixture is set to the urania liquid thermal conductivity
λl = 5.6 W/(mK).
(17.12.3)
17.12.1.7 Liquide surface tension Due to the uncertainty a constant value for the liquid surface tension of σ l = 0.535
(17.12.4)
is used which is close to this of urania, 17.12.1.8 Liquid dynamic viscosity For the liquid dynamic viscosity I usually use ηl = 0.004356 for T < T0 ,
(17.12.5)
and ηl = 0.000988exp ( 4629 T ) ,
(17.12.6)
as for UO2 taken from Woodly cited by Fink et al. (1981a,b). 17.12.1.9 Liquid specific enthalpy The liquid specific enthalpy is approximated by Eq. (17.1.8) with h′ = 1302752.7941 J/kg. 17.12.1.10 The derivative of the liquid specific enthalpy with respect to pressure at constant temperature The derivative of the liquid specific enthalpy with respect to pressure at constant temperature is approximated by Eq. (17.1.10).
682
17. Thermo-physical properties for severe accident analysis
17.12.1.11 Liquid specific entropy The liquid specific entropy is approximated by Eq. (17.1.11) with where s ′ = 1135.1472193464033 J/(kgK). 17.12.1.12 Liquid temperature as a function of pressure and specific entropy Given is the pressure and the specific liquid entropy. The temperature is computed where s′ = iteratively by starting with T00 = T0 exp ⎡⎣( sl − sl 0 ) c pl ⎤⎦ , 1135.1472193464033 J/(kgK) and using Eq. (17.1.11) solved with respect to the temperature resulting in Eq. (17.1.14). Less than 10 iterations are needed to reduce the error below 0.0001 K. 17.12.2 Solid 17.12.2.1 Solid specific entropy For s between 1040.2335207162664 J/(kgK) and 1135.1472193464033 J/(kgK) the temperature is 2920 K because of melting. The melting entropy is s ′ − s ′′′ = 94.913698630136992 J/(kgK). 17.12.2.2 Solid specific enthalpy The solid specific entropy is computed as for UO2. 17.12.2.3 Solid specific capacity at constant pressure The solid specific capacity at constant pressure is computed as for UO2. 17.12.2.4 Solid density The solid density is considered as a function of the temperature only. Here the Fink et al. (1981a,b) function for UO2 multiplied by factor 0.874 is used ρ s = ρ s 0 ( c1 + c2T + c3T 2 + c4T 3 )
(17.12.7)
where ρ s 0 = 0.874 × 10970 kg/m³ , c1 = 1.0056 , c2 = −1.6324 × 10−5 , c3 = −8.3281× 10−9 , c4 = 2.0176 × 10−13 .
References
683
2.2.5 The derivative of the solid density with respect to the temperature The derivative of the solid density with respect to the temperature is taken from Eq. (17.12.7), d ρs = 0.874c0 c1 + c2 + 2c3T + 3c4T 2 . dT
(
)
(17.12.8)
17.12.2.6 Solid thermal conductivity The solid thermal conductivity is taken as for UO2. 17.12.2.7 Solid sonic velocity No measurements of the elasticity modulus and of the sonic velocity for such mixtures are known to me. For practical analysis I use the Eq. (17.2.38) strictly valid for UO2 only as an approximation.
References Bottomley PD and Coquerelle (Aug. 1989) Metallurgical examination of bore samples from the three mile island unit 2 reactor core, Nuclear Technology vol 87 pp 120-136 Chong JS, Chrisiansen EB and Baer AD (1971) Rheology of concentrated suspensions, J. Appl. Polym., vol 15 p 2007-2021 Fink JK, Chasanov MG, Leibowitz L (April 1981a) Thermodynamic properties of uranium dioxide, ANL-CEN-RSD-80-3 Fink JK, Chasanov MG, Leibowitz L (1981b) Thermo-physical properties of uranium dioxide, J. of Nuclear Materials, vol 102, pp 17-25 Hayward PJ and George IM (1966) Dissolution of UO2 in Zirkaloy-4 Part-4: Phase evolution during dissolution and cooling of 2000 to 2500°C specimen (ZrO0.54-UO2), vol 232 pp 13-22 Hodge SA and Ott LJ (June 1-5, 1997) Interpretation of the XR2-1 experiment and characteristics of the BWR lower plenum debris bed, Proc. of the Int. Top. Meeting on Advanced Reactor Safety, vol 1 Orlando, Florida Juenke EF and Whitte JF (1969) Zr-UO2, Report GEMP-731 Kolev NI (2007a) Multiphase flow dynamics, vol 1, Springer, Berlin Moshev VV and Ivanov VA (1990) Rheological behavior of concentrated non-newtonian suspensions, Nauka, Moskva Politis C (Oct. 1975) Untersuchungen im Dreistoffsystem Uran-Zirkon-Sauerstoff, ZrO0.51-UO2, Kernforschungszentrum Karlsruhe, KfK Report 2167 Ramacciotti M, Journeau C, Sudreau F and Cognet G (Oct. 3-8 1999) Viscosity models for corium melts, Ninth Int. Top. Meeting on Nuclear Thermal Hydraulics (NURETH-9) San Francisco Romberger KA, Baes CF Jr and Stone HH (1967) Phase equilibrium studies in the UO2-ZrO2 system, J. Inorg. Nucl. Chem., vol 29 pp 1619-1630 Skokan A (Sept. 9-13, 1984) High temperature phase relations in the U-Zr-O system, ZrO0.43UO2, 5th Int. Meeting on Thermal Nuclear Reactor Safety, Karlsruhe pp 1035-1042
684
17. Thermo-physical properties for severe accident analysis
Spindler B and Vateau JM (2006a) The simulation of melt spreading with THEMA code Part: 1 Model, assessment strategy and assessment against analytical and numerical solutions, Nuclear Engineering and Design, vol 236 pp 415-424 Spindler B and Vateau JM (2006b) The simulation of melt spreading with THEMA code Part: 2 Assessment against spreading experiments, Nuclear Engineering and Design, vol 236 pp 425-441 Tim KT Olander DR (1988) Dissolution of uranium dioxide by molten zitcaloy, Journal of Nuclear Materials, vol 154 pp. 85-101
17.13 Sodium
685
17.13 Sodium The purpose of this chapter is to review the openly available sources of information and to attempt to derive from them a consistent set of thermodynamic, caloric and transport functions for sodium. Even being forced to accept the uncertainty of the available data I will explicitly document them. The so obtained functions are then recommended for use in computer codes for consistent multi-phase dynamic analysis of fast breeder nuclear reactors which uses sodium as a coolant. The interest in the power engineering to the alkali metals as a coolant for high powered fast breeder reactors arises in the 60s. With the increasing the energy prizes worldwide there is again increasing interests to the fast breeder technology. The interest on sodium as coolant is based on its specific thermal and transport properties: − Even for high temperatures, the primary circuits can be designed as a lowpressure system; the steam conditions of conventional power plants can be approximately achieved. Therefore high overall efficiency is possible which improves use of the nuclear fuel; − Under normal operating conditions sodium is at a low pressure, and there is no danger of sudden rupturing of large pipes and vessels. Moreover, fractured pipes or vessels do not cause high containment pressures, so that the containment can be designed for low pressure. − The considerable difference between the operating temperature and the boiling point makes it possible for the coolant to absorb excess heat under accidental conditions before reaching boiling, so that accidents remain controllable; − The high heat conduction of the liquid sodium warrant heat removal within permissible temperature gradients between the fuel rods and the sodium, in spite of the high power density in the core. This also reduces the heat exchanger surface and therefore their sizes and their costs. Besides the advantages the following has to be taken into account: − The low specific heat capacity necessitates a large coolant temperature rise for about the square rate of coolant circulation as in pressurized water reactors; − The high operating temperatures demand extensive measures to provide adequate thermal expansion compensation. All components must be designed for high temperatures;
686
17. Thermo-physical properties for severe accident analysis
− The favourable heat transfer properties, coupled with high operating temperatures and a large coolant temperature rise, requires extensive thermal stress analyses for designing system with maximum thermal shock resistance. − Because the solidification temperature is higher then the environmental temperature all sodium-filled components and piping require electric or hot-gas trace heating for commissioning and for keeping them ready for operation. The purpose of this chapter is to review the openly available sources of information and to attempt to derive from them a consistent set of thermodynamic, caloric and transport functions for sodium. Even being forced to accept the uncertainty of the available data I will explicitly document them. The so obtained functions are then recommended for use in computer codes for consistent multi-phase dynamic analysis of fast breeder nuclear reactors which uses sodium as a coolant. 17.13.1. Some basic characteristics 17.13.1.1 Moll mass and gas constant The moll mass for mono-atomic sodium is reported to be •1 M1 = 22.997 g mol Na,
(17.13.1)
Perry and Green (1985), compare with 22.98976928(2) g mol• 1 Na, Wikipedia (2007). In the literature slightly different values are found e. g. 22.991 g/mol Fink et al. (1982), 22.98977 g/mol Vargaftic et al. (1996, p. 203), 22.98922 g/mol Chase (1998, p. 1642). For practical engineering calculation the 22.99 g/mol is sufficient. The universal gas constant is Ru = 8314.41 ± 0.26 J/(kmol K),
(17.13.2)
Gurvich et al. (1985, p. 577). Therefore the gas constant for mono-atomic sodium vapor is R1 = Ru / M 1 = 365.52 J/(kgK).
(17.13.3)
The gas state of sodium is possible at such a high temperatures so that the association and ionization of the sodium atoms has to be taken into account. This change the value of the effective gas constant as it will be shown in a moment.
17.13 Sodium
687
17.13.1.2 Melting temperature and enthalpy at atmospheric conditions The melting temperature at 1 bar is reported as follows: 370.98 K, Golden and Tokar (1967), 371.01 K Cordfunke and Konings (1990), T ′′′ (1bar) = 371.02 ± 0.03 K,
(17.13.4)
Borishanskij et al. (1976), Chase (1998), compare also with Perry and Green (1985) 370.65 K. The melting enthalpy is reported to be h′ − h′′′ = 113.18 kJ/kg,
(17.13.5)
by Fink et al. (1982), 113.6 kJ/kg by Chase (1998), 113 ± 0.2 kJ/kg by Cordfunke and Konings (1990). The corresponding melting entropy used here is s ′ − s ′′′ = ( h′ − h′′′ ) T ′′′ = 305.06 J/(kgK).
(17.13.6)
17.13.1.3 Enthalpy of sublimation from T = 0 K to mono-atomic Na 107.763 kJ/g-mol Na, Gurvich et al. (1985, p. 597), 107.76 ± 0.7 kJ/g-mol Na, Bystrov et al. (1988). 17.13.1.4 Ionization potential of to mono-atomic Na for Na = Na+ + e 495.845 ± 0.001 kJ/g-mol Na, Gurvich et al. (1985, p. 578). 17.13.1.5 Dissociation energy for Na2 = 2Na 2 Na of the reaction Na2=2Na is given as The dissociation energy or standard heat ΔhNa 2
follows: 76.6038 kJ/g-mol Na2, Ewing et al. (1967, p. 476); 71.09 ± 0.25 kJ/g-mol Na2, Gurvich et al. (1985, p. 583); 71.380 ± 0.85 kJ/g-mol Na2, Vargaftic and Voljak (1985, p. 536), 70.492 kJ/g-mol Na2, Makansi et al. (1960). I will use here the value reported by Golden and Tokar (1967) 76.62054 kJ/g-mol Na2. 17.13.1.6 Dissociation energy for Na4 = 4Na 4 Na of the reaction Na4=4Na is given as The dissociation energy or standard heat ΔhNa follows: 173.63528 kJ/g-mol Na4, Ewing et al. (1967, p. 476); I will use here the value reported by Golden and Tokar (1967) 173.6269 kJ/g-mol Na4. 4
688
17. Thermo-physical properties for severe accident analysis
17.13.1.7 Critical point The critical point is not exactly known. Borishanskij et al. (1976) summarized the estimations and the measurements of different authors and showed that the reported critical temperature varies between 1648 and 2800 K and the critical pressure varies between 116 and 510 atm. Fink et al. (1982) reported the following values for the critical temperature, pressure and density Tc = 2509.46 m 0.96% K, pc = 256.4 m 0.96% bar,
ρ c = 214.1 m 0.4% kg/m³.
(17.13.7) (17.13.8) (17.13.9)
For comparison see Petiot and Seiler (1982) Tc = 2630 ± 50 K , pc = 340 ± 40 bar , ρ c = 205 kg/m³ adapted also by Vargaftic et al. (1996). I recommend for practical analysis to keep always the critical density in accordance with the used set of equations of state and the critical pressure in accordance with the used critical temperature and saturation line approximation. Note that the main difficulty is to find whether there is a point for chemically unstable gases that satisfy the criticality conditions. The similarity theory used usually for chemically stable substances is not applicable for sodium. 17.13.1.8 Specific capacity at constant pressure for solid sodium
cp in kJ/kg
Data for the specific capacity at constant pressure for solid sodium are reviewed by Chase (1998) and presented in Fig. 17.13.1.
1.4 1.2 1.0 0.8 0.6 0.4 collected by Chase (1998) 0.2 Fit 0.0 -0.2 -50 0 50 100 150 200 250 300 350 400 T in K
Fig. 17.13.1. Specific capacity at constant pressure for solid sodium
They can be approximated by the following cubic polynomial
17.13 Sodium
c ps = b0 + b1T + b2T 2 + b3T 3 ,
689
(17.13.10)
where b0 = 71.15653, b1 = 17.38527, b2 = –0.07733, b3 = 1.08436e-4. The resulting expressions for the specific enthalpy and for the specific entropy are then 1 1 1 hs − hs , ref = b1 (T 2 − Tref2 ) + b2 (T 3 − Tref3 ) + b3 (T 4 − Tref4 ) , 2 3 4
ss − sref = b0 ln
(17.13.11)
T 1 1 + b1 (T - Tref ) + b2 (T 2 - Tref2 ) + b3 (T 3 − Tref3 ) , (17.13.12) Tref 2 3
respectively. We select arbitrarily hs , ref = 0 and ss , ref = 0 at Tref = 293.15 K and pref = 105 Pa . The solidus and the liquidus specific enthalpies and entropies are then: h′′′ = 95917.4514452 J/kg, h′ = 209097.451445 J/kg, s ′′′ = 306.23033 J/(kgK), s ′ = 611.297716903 J/(kgK).
(17.13.13) (17.13.14) (17.13.15) (17.13.16)
For comparison see Bystrov et al. (1988, p. 42). Their correlation c ps = ( −23.346 + 1.385 × 106 T −2 + 120.736 ×10−3 T ) 0.02299
(17.13.17)
is valid for 298.15 < T < 371.02 K and Perry and Green (1985) c ps = 911.5 + 0.975T m1.5% valid for 273 < T < 371 K. 17.13.1.9 Density of solid sodium Cordfunke and Konings (1990) reported the value 966 kg/m³. With the linear thermal expansion coefficient from Wikipedia (2007) recomputed in volumetric thermal expansion coefficient we obtain
ρ s = 966 − 2.13 × 10−4 (T − 298.1) kg/m³.
(17.13.18)
17.13.1.10 Sonic velocity in solid sodium If the elasticity modulus is known, E = 10 GPa,
(17.13.19)
690
17. Thermo-physical properties for severe accident analysis
Wikipedia (2007), the velocity of sound is as = E ρ s = 3217 m/s.
(17.13.20)
17.13.1.11 Thermal conductivity of solid sodium The thermal conductivity of solid sodium is reported to be
λs =142 W/(mK), Wikipedia (2007).
(17.13.21)
17.13.2. Liquid 17.13.2.1 Velocity of sound of liquid sodium The velocity of sound for liquid sodium at 1 bar is reported by Trelin et al. (1960)
Sound velocity in m/s
al (1 bar) = 2694 – 0.577T m/s.
2800 2600 2400 2200 2000 1800 1600 1400 1200 1000
(17.13.22)
Trelin et al. (1960) Vargaftic et al. (1996) Fink et al. (1982)
500
1000
1500 T in K
2000
2500
Fig. 17.13.2. Velocity of sound for liquid sodium at 1bar pressure. Comparison between the prediction of the Trelin et al. (1960), Vargaftic et al. (1996) and Fink et al. (1982) correlations
This correlation gives at the boiling point 2028 m/s. Borishanskij et al. (1976) demonstrated in their Fig. 1.7, p. 31 a very good representation of the experimental data within 263 and 973 K. Vargaftic et al. (1996) recommended the approximation of the Novikov et al. (1981) data
17.13 Sodium
al (1bar ) = 2746 - 0.5673T m/s,
691
(17.13.23)
valid for 400 < T < 1650 K with an error between 10 and 26%. The error increases from 12 to 26% from 1100 to 1650K. In the Bystrov et al. (1988, p. 35) correlation the constant is 2747. Fink et al. (1982) reported al (1bar ) = 2660.7 – 0.37667 T – 0.90356e-5 T m/s. 2
(17.13.24)
The comparison between the three correlations is given in Fig. 17.13.2. The Fink’s et al. correlation gives close to the boiling point about 12% higher values. The other two correlations have larger error approaching the critical point. Therefore the Fink’s et al. correlation is recommendable. 17.13.2.2 Density of liquid sodium From the Table 1.6 by Borishanskij et al. (1976) the following correlation can be derived 950 Borishanskij et al. (1976) Fit
ρ in kg/m³
900 850 800 750 700 200
400
600
800 1000 1200 1400 T in K
Fig. 17.13.3. Density for liquid sodium as a function of temperature at 1 bar
⎛ ∂ρ ⎞ (17.13.25) ρl (1bar ) = rl 0 + rl1T , ⎜ l ⎟ = rl1 , ⎝ ∂T ⎠ p where rl 0 = 1015.87425 , rl1 = −0.23773 . The quality of the fit is visible from Fig. 17.13.3. Golden and Tokar (1967) reported the following approximation for the density at atmospheric pressure ⎛ ∂ρ ⎞ (17.13.26) ρl (1bar ) = rl 0 + rl1t + rl 2 t 2 + rl 3t 3 , ⎜ l ⎟ = rl1 +2rl 2 t + 3rl 3t 2 ⎝ ∂T ⎠ p
692
17. Thermo-physical properties for severe accident analysis
where t = T − 273.15 , rl 0 = 950.076 ,
rl1 = −0.22976 , rl 2 = −1.46049 × 10−5 ,
rl 3 =5.63788 × 10 −9 . Bystrov et al. (1988, p. 23) reproduced their data with the very practical correlation p = a00 + a01 ρ + a02 ρ 2 + ( a10 + a11 ρ ) T ± 0.5% ,
(17.13.27)
where a00 = 0.75985e10, a01 = –0.21766e8, a02 = 0.14068e5, a10 = –0.17318e7,
a11 = 0.32803e4, valid for p ′ (T ) < p < 500 bar , 500 < T < 1000 K. This correla-
tion allows computing analytically the derivatives in the equation of state
dρ =
( ∂p ∂T )ρ ⎛ ∂ρ ⎞ dp ⎛ ∂ρ ⎞ dT = ⎜ − ⎟ dp + ⎜ ⎟ dT , ( ∂p ∂ρ )T ( ∂p ∂ρ )T ⎝ ∂T ⎠ p ⎝ ∂p ⎠T
(17.13.28)
⎛ ∂ρ ⎞ 1 , ⎜ ⎟ = p p ∂ ∂ ∂ρ )T ( ⎝ ⎠T
(17.13.29)
( ∂p ∂T )ρ ⎛ ∂ρ ⎞ , ⎜ ⎟ =− ( ∂p ∂ρ )T ⎝ ∂T ⎠ p
(17.13.30)
with
⎛ ∂p ⎞ ⎜ ⎟ = a10 + a11 ρ , ⎝ ∂T ⎠ ρ
(17.13.31)
⎛ ∂p ⎞ ⎜ ⎟ = a01 + 2a02 ρ + a11T . ⎝ ∂ρ ⎠T
(17.13.32)
Density in kg/m³
17.13 Sodium
1000 950 900 850 800 750 700 650 600 550
693
Borishanskij et al. (1976) Golden and Tokar (1967) Bystrov et al. (1988)
500
1000 T in K
1500
Fig. 17.13.4. Density of the liquid sodium at 1 bar pressure as a function of the temperature. Comparison of the correlations proposed by Borishanskij et al. (1976), Golden and Tokar (1967) and Bystrov et al. (1988)
The density as a function of temperature and pressure is then the solution of the quadratic equation a02 ρ 2 + ( a01 + a11T ) ρ + a00 + a10T − p = 0
(
ρ = −b + b 2 − 4ac
) ( 2a )
(17.13.33)
where a = a02 , b = ( a01 + a11T ) , c = a00 + a10T − p . Comparison of the predictions of the three correlations for atmospheric pressure is presented in Fig. 17.13.4. We see that all correlations are very close to each other except the Bystrov et al. (1988) correlation for high temperatures. 17.13.2.3 Density of the saturated liquid sodium Golden and Tokar (1967) reported the following expression for the density of the saturated liquid sodium
ρ ′ (T ) = r0 + r1 (T − 273.15 ) + r2 (T − 273.15 ) + r3 (T − 273.15 ) , 2
3
(17.13.34)
where r0 = 949.44, rls1 = –0.2296058, rls2 = –1.4595e-5, rls3 = 5.6341e-9. Bystrov et al. (1988, p. 28) reported an expression for the density of the saturated liquid sodium
ρ ′ (T ) = rls 0 + rls1T * + rls 2T *2 + rls 3T *3 + rls 4T *4 + rls 5T *5 + rls 6T *6 ,
(17.13.35)
694
17. Thermo-physical properties for severe accident analysis
valid for Tmelt < T < Tc where T * = T /1000 , rls0 = 896.42, rls1 = 517.11, rls2 = – 1831.7, rls3 = 2203.58, rls4 = –1398.56, rls5 = 448.92, rls6 = –57.99. The density of saturated liquid sodium reported by Stone et al. (1965) is
ρ ′ (T ) = rls 0 + rls1T + rls 2T 2 + rls 3T 3
(17.13.36)
for T < 1644.26 K with rls0 = 1011.8, rls1 = –0.22054, rls2 = –1.9226e-5, rls3 = 5.6371e-9, and by Fink et al. (1982)
ρ ′ (T ) = ρc ⎡1 + 2.3709 (1 − T Tc )
0.31645
⎣
2 + 2.8467 × 10−7 (Tc − T ) ⎤ ⎦
(17.13.37)
Saturated liquid density in kg/m³
for T > 1644.26 K with Tc = 2509.46 K and ρ c = 214.1 kg/m³. The advantage of this correlation is that it gives exactly the critical density at the critical point. The saturation liquid sodium density predicted by the three correlations is presented in Fig. 17.13.5. We see that high order approximations are better close to the critical point.
Golden and Tokar (1967) Bystrov et al. (1988) Stone et al. (1965) T < 1644.26 K, Fink et al. (1982) T > 644.26 K
1100 1000 900 800 700 600 500 400 300 200 500
1000
1500 T in K
2000
2500
Fig. 17.13.5. Saturation liquid sodium density as a function of temperature
17.13.2.4 Boiling temperature at 1 bar The boiling temperature at 1 bar is reported to be T ′ (1 bar) = 1154.6 K
by Borishanskij et al. (1976).
(17.13.38)
17.13 Sodium
695
17.13.2.5 Saturation pressure as a function of temperature Golden and Tokar (1967, p. 77) recommended the following correlation for the saturation pressure as a function of temperature. For T < 1144.28 K p ′ (T ) = pa1Tr0.5 exp ( − pb1Tr ) ,
(17.13.39)
dp ′ = p ′Tr ( pb1Tr − 0.5 ) , dT
(17.13.40)
where Tr = 1 T , pa1 = 2.2904 × 1011 , pb1 = 12818.5 else p ′ (T ) = pa 2Tr0.61344 exp ( − pb 2Tr ) ,
(17.13.41)
dp ′ = p ′Tr ( pb 2Tr − 0.61344 ) , dT
(17.13.42)
where pa 2 = 4.8621× 1011 , pb 2 = 12767.756 . If the critical pressure is selected to be 256.4 bar Golden and Tokar (1967) equations delivers a critical temperature of Tc = 2531.83 K. Fink et al. (1982) reported the Bonilla’s least squares fit to the experimental vapor pressure data of four authors 1 ⎛ ⎞ p ′ (T ) = 0.981× 105 exp ⎜18.832 − 13113 − 1.0948ln T + 1.9777 × 10 −4 T ⎟ . T ⎝ ⎠ (17.13.43) Comparison between the correlations proposed by Golden and Tokar (1967), and those by Bonilla in Fink et al. (1982) is shown in Fig. 17.13.6. We realize that both correlations give values close to each other. Both equations are reliable but the last contains potential for a generalization that will be revealed below. Note that if the latent heat of evaporation is a quadratic function of the temperature Δh = a1 + a2T + a3T 2,
(17.13.44)
and if the specific liquid volume at the saturation line is negligible compared to the vapor specific volume which simplifies the Clausius–Clapayeron relation dT v′′ − v′ , =T dp Δh
to
(17.13.45)
696
17. Thermo-physical properties for severe accident analysis
dT Tv ′′ . ≈ dp Δh
(17.13.46)
and assuming that the vapor is a perfect gas we have after integration dp 1 ⎛ a1 a2 ⎞ = ⎜ + + a3 ⎟ dT , p R ⎝T2 T ⎠
(17.13.47)
see Kolev (2007, p. 148). Comparing this relation with the Bonilla’s relation we obtain for R = 365.52 J/(kgK) the following constants: a1 = 4793063.76, a2 = – 400.17, a3 = 0.072288890 which uniquely defines the evaporation specific enthalpy. Integrating the above equation and setting as reference point the critical point we obtain ⎤ ⎪⎫ T ⎪⎧ 1 ⎡ ⎛ 1 1 ⎞ p = pc exp ⎨ ⎢ a1 ⎜ − ⎟ + a2 ln + a3 (T − Tc ) ⎥ ⎬ . Tc ⎦ ⎭⎪ ⎩⎪ R ⎣ ⎝ Tc T ⎠
(17.13.48)
Comparison between the two preceding correlations and the last one is presented on Fig. 17.13.6. In the same picture for verification the measurements by Petiot and Seiler (1982) are given.
250
p' in bar
200
Bonilla in Fink et al. (1982) Golden and Tokar (1967) Kolev (2007) Mozgovoi et al. (1988) Exp. Petiot and Seiler (1982)
150 100 50 0 1000
1500
2000
2500
T in K Fig. 17.13.6. Saturation vapor pressure as a function of the temperature. Comparison between the correlations proposed by Golden and Tokar (1967), Bonilla in Fink et al. (1982) and the new correlation strictly related to the critical parameters and the evaporation enthalpy
We realize that this equation reproduce closely the Golden and Tokar (1967) equation by having the following advantages. It is
17.13 Sodium
697
(a) strictly consistent with the Clausius–Clapayron equation, (b) with the definition of the critical point, (c) with the definition of the evaporation enthalpy as a function of temperature, and (d) with the assumption that the vapor is a perfect monomer (which is not the case as we will see later but compensated by the empirical constants). Mozgovoi et al. (1988, p. 206) reported the following correlation for the saturation pressure of sodium ln ( p ′ 106 ) = c ln T * + ∑ aiT *i 5
(17.13.49)
i =−1
where T * = T /1000 , c = –2.494631, a–1 = –13.29055, a0 = 7.844046, a1 = 1.709349, a2 = –0.171569, a3 = –0.008757, a4 = –0.009092, a5 = 0.002906. The equation is based on the following values: Tc = 2503 K and pc = 25.64 MPa. Vargaftic et al. (1996) commented on this that due to the large uncertainty of the critical parameter ( ΔTc = ±50 K, Δpc = ±1.5 MPa ), which was demonstrated by improved data, which are Tc = 2630 ± 50 K , Δpc = 34 ± 4 MPa , the prediction of saturated pressure values close to the critical point should be considered as tentative. The accuracy of the data is estimated by Vargaftic et al. (1996) as follows: 2% for T < 700 K, 1.5% for T between 700 and 1000 K, 1% for T between 1000 and 1500 K, 1.5% for T between 1500 and 2000, 3% for T between 2000 and 2200 K and 5% above 2200 K. Figure 17.13.6 gives the predictions of the all four reviewed correlations. It is very interesting to note that Eqs. (17.13.48) and (17.13.49) predicts values very close to each other so that the error estimate by Vargaftic et al. (1996) can be applied also to both equations. 17.13.2.6 Specific capacity at constant pressure of liquid sodium at atmospheric conditions Borishanskij et al. (1976) data are approximated by the following correlation c p (1bar ) = cl 0 + cl1T + cl 2T 2 ,
(17.13.50)
where cl 0 = 1613.56767 , cl1 = −0.80563 , cl 2 = 4.43624 × 10−4 . The data representation is presented on Fig. 17.13.7. Note that the third order polynomial does not give better approximation.
698
17. Thermo-physical properties for severe accident analysis
1380
Borishanskij et al. (1976) Fit
cp in J/(kgK)
1360 1340 1320 1300 1280 1260 1240 200
400
600
800
1000 1200 1400
T in K
Fig. 17.13.7. Specific capacity at constant pressure for liquid sodium as a function of temperature at 1 bar
Bystrov et al. (1988, p. 42) approximated their data by
c ps = ( b0 + b1T *− 2 + b2T * + b3T *2 ) 0.02298977
(17.13.51)
valid for 371.02 < 2300 K where T * = T /1000 , b0 = 38.121, b1 = –0.069, b2 = – 19.493, b3 = 10.24. 1450 Borishanskij et al. (1976) Bystrov et al. (1988) Golden and Tokar (1967)
cp in J/(kgK)
1400 1350 1300 1250 1200 400
600
800 1000 1200 1400 T in K
Fig. 17.13.8. Specific capacity of liquid sodium at atmospheric pressure
17.13 Sodium
699
17.13.2.7 Specific capacity at constant pressure of saturated liquid sodium at atmospheric conditions From the derived by Golden and Tokar (1967) enthalpy function at saturation we have c p (1bar ) = cl 0 + cl1T + cl 2T 2 ,
(17.13.52)
where cl0 = 1630.13942, cl1 = – 0.83344, cl2 = 4.62816e-4. Comparing this equation with the Borishanskij et al. (1976) equation we do not see substantial difference. Therefore the accuracy of the both correlations is practically the same, see Fig. 17.13.8. The three above introduced correlations for specific capacity at constant pressure gives similar predictions as demonstrated on Fig. 17.13.8. 17.13.2.8 The pressure dependence of the specific liquid capacity at constant pressure The specific enthalpy can be approximated by using the following approach ⎛ ∂h ⎞ hl = hpref (T ) + ⎜ l ⎟ ( p − pref ) , ⎝ ∂p ⎠T assuming that ⎛ ∂hl ⎞ ⎛ ∂v ⎞ ⎜ ⎟ = v −T ⎜ ⎟ ∂ p ⎝ ∂T ⎠ p ⎝ ⎠T
(17.13.53)
(17.13.54)
is not a pressure function. Note that ⎡ ∂ ⎛ ∂h ⎞ ⎤ c pl = c pl (T , pref ) + ⎢ ⎜ l ⎟ ⎥ ( p − pref ) , ⎣⎢ ∂T ⎝ ∂p ⎠T ⎦⎥ p
(17.13.55)
⎡ ∂ ⎛ ∂hl ⎞ ⎤ ⎛ ∂2v ⎞ T = − ⎢ ⎜ ⎥ ⎜ 2⎟ . ⎟ ⎢⎣ ∂T ⎝ ∂p ⎠T ⎥⎦ p ⎝ ∂T ⎠ p
(17.13.56)
Obviously, it is very convenient to have at the reference pressure the dependence of the specific volume on temperature. I exercise the development of the specific capacity, enthalpy and entropy function using the Golden and Tokar (1967) approximations for h (T − 273.15 ) and ρ (T − 273.15 ) for pref = 1 bar. Approximat-
ing the first function with h (T ) , writing the second as v (T ) = 1 ρ (T − 273.15 ) , and approximating it I obtain
700
17. Thermo-physical properties for severe accident analysis
h′ (T ) = hl 0 + hl1T + hl 2T 2 + hl 3T 3
(17.13.57)
c p = cl 0 + cl1T + cl 2T 2 = hl1 + 2hl 2T + 3hl 3T 2 ,
(17.13.58)
v = bl 0 + bl1T + bl 2T 2 + bl 3T 3 ,
(17.13.59)
where hl 0 = –67500.61329, hl1 = 1630.13942, hl 2 = –0.41672, hl 3 = 1.54272E-4 and bl0 = 9.71621E-4, bl1 = 2.85914E-7, bl2 = –1.93508E-11 and bl3 = 4.90723E-10. The success of these approximations is demonstrated on Fig. 17.13.9. Then the following enthalpy derivatives are only function of the temperature ⎛ ∂hl ⎞ ⎛ ∂v ⎞ 2 3 ⎜ ⎟ = v −T ⎜ ⎟ = bl 0 − bl 2T − 2bl 3T , p T ∂ ∂ ⎝ ⎠p ⎝ ⎠T
(17.13.60)
⎡ ∂ ⎛ ∂hl ⎞ ⎤ ⎛ ∂2v ⎞ 2 ⎢ ⎜ ⎟ ⎥ = −T ⎜ 2 ⎟ = − ( 2bl 2T + 6bl 3T ) . ⎢⎣ ∂T ⎝ ∂p ⎠T ⎥⎦ p ⎝ ∂T ⎠ p
(17.13.61)
Therefore
c pl = cl 0 + cl1T + cl 2T 2 − ( 2bl 2T + 6bl 3T 2 ) ( p − pref ) .
(17.13.62)
3
v'*10 in m³/kg
h' in MJ/kg
17.13 Sodium
4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0
2,4 2,2 2,0 1,8 1,6 1,4 1,2 1,0 0,8
701
f(T-273.15) f(T) fit
500
1000
1500 T in K
2000
2500
1/ρ'(T-273.15) v'(T)
500
1000
1500 T in K
2000
2500
Fig. 17.13.9. (a) Specific enthalpy of saturated liquid as a function of temperature; (b) Specific saturated liquid volume as a function of temperature
17.13.2.9 Specific liquid enthalpy The liquid specific enthalpy is obtained after integration first between T ′′′ and T for p = pref and the between pref and p along T = const. The result is h = h′′′ + hl1 (T − T ′′′ ) + hl 2 (T 2 − T ′′′2 ) + hl 3 (T 3 − T ′′′3 )
− ( 2bl 2T + 6bl 3T 2 ) ( p − pref ) .
(17.13.63)
702
17. Thermo-physical properties for severe accident analysis
17.13.2.10 Specific liquid entropy The liquid specific entropy is obtained after integration first between T ′′′ and T for p = pref and then between pref and p along T = const. The result is s = s ′′′ +
p 1 1 ⎡ ∂ ⎛ ∂hl ⎞ ⎤ , + c T p dT ( ) ref ∫T ′′′ T pl ∫p T ⎢⎢ ∂T ⎜⎝ ∂p ⎟⎠ ⎥⎥ dp ⎣ T ⎦p ref T
1 = s ′′′ + cl 0 ln (T T ′′′ ) + cl1 (T − T ′′′ ) + cl 2 (T 2 − T ′′′2 ) − ( 2bl 2 + 6bl 3T ) ( p − pref ) . 2 (17.13.64)
17.13.2.11 The pressure dependence of the liquid density Once there is observed that the velocity of sound in the liquid is finite we know that the liquid is compressible. A consistent equation of state of the liquid density has to contain this dependence. The density ca be approximated by the following linearization ⎛ d ρl ⎞ ⎟ ( p − p0 ) , ⎝ dp ⎠T
ρl (T , p ) ≈ ρl 0 (T ) + ⎜
(17.13.65)
where
⎛ d ρl ⎞ 1 β l2T 1 1 T 1 ⎛ ∂ρ ⎞ T 1 ⎛ ∂vl ⎞ = 2+ = 2+ . (17.13.66) ⎜ ⎟ = 2+ ⎟ 2 ⎜ c pl al c pl ρl ⎝ ∂T ⎠ p al c pl vv2 ⎝⎜ ∂T ⎠⎟ p ⎝ dp ⎠T al 2
2
Having in minded that ⎛ d ρl ⎞ ⎜ ⎟ = f ⎡⎣ al (T ) , β l (T , p ) , c pl (T , p ) ⎤⎦ = f (T , p ) ⎝ dp ⎠T
(17.13.67)
we realize that the above equation is implicit with respect to the density and therefore have to be solved by iteration. Because the pressure dependence is much weaker then the temperature dependence two cycles are enough to obtain converged solution. If there is no addition treatment of such correlation the coincidence of de vapor and liquid densities in the critical point not necessarily occur. Some authors like Miller et al. (1967) make use of Eq. (17.13.65) by taking its derivative with respect to the temperature, dividing by the density and taking as reference state the saturation line
17.13 Sodium
1 ∂ρ 1 ∂ρ ′ 1 ⎛ d ρ ′ ⎞ dp ′ ≈ + ⎜ ⎟ ρ ∂T ρ ′ ∂T ρ ′ ⎝ dp ⎠T dT
703
(17.13.68)
which is in fact
β ≈ β′+κ′
dp ′ . dT
(17.13.69)
Then the isothermal coefficient of compressibility is computed from Eq. (17.13.66) at the saturation line. 17.13.2.12 Thermal conductivity or liquid sodium Borishanskij et al. (1976) reported data for the thermal conductivity at atmospheric conditions that can be approximated by
λl (1 bar) = 103.9169 − 0.04654T ,
(17.13.70)
λ in W/(mK)
as given in Fig. 17.13.10. Borishanskij et al. (1976) Fit
90 85 80 75 70 65 60 55 50 400
600
800 T in K
1000
1200
Fig. 17.13.10. Thermal conductivity or liquid sodium as a function of temperature at 1 bar
Bystrov et al. (1988, p. 66) reported similar correlation
λl (1 bar) = 99.5 − 0.0391T
(17.13.71)
valid for T < 2400 K with error 5% at the melting point and 15% at T > 1400 K. Golden and Tokar (1967) proposed
704
17. Thermo-physical properties for severe accident analysis
λl (1 bar) = 92.948 − 0.05809 (T − 273.15 ) + 1.1727 ×10 −5 (T − 273.15 ) . 2
Thermal conductivity in W/(mK)
(17.13.72) 100
Borishanskij et al. (1976) Bystrov et al. (1988) Golden and Tokar (1967)
90 80 70 60 50 40 30 400
600
800 1000 1200 1400 T in K
Fig. 17.13.11. Thermal conductivity or liquid sodium as a function of temperature at 1 bar
Figure 17.13.11 shows the comparison between the predictions of the three correlations. Up to 1200 K the prediction by Borishanskij et al. (1976) correlations is very close to the prediction of the Golden and Tokar (1967) correlation. 17.13.2.13 Dynamic viscosity or liquid sodium Borishanskij et al. (1976) reported data for the dynamic viscosity at atmospheric conditions that can be approximated by
ηl (1bar ) = 1.07656 ×10−4 +
10.41994 0.00116 + , (17.13.73) exp (T 36.08604 ) exp (T 361.22725 )
as shown in Fig. 17.13.12. Bystrov et al. (1988, p. 68) reported ln (ηl ) = −6.4406 − 0.3958ln T + 556.835T −1
(17.13.74)
valid for T < 2300 K. Shpilrain et al. cited in Vargaftic et al. (1996) estimated the error of this correlation as follows: 3 to 5% for T < 1500 K and 5 to 10% for 1500 < T < 2300 K. Golden and Tokar (1967) reported
4
Dynamic viscosity*10 in kg/(ms)
17.13 Sodium
10 9 8 7 6 5 4 3 2 1 200
705
Borishanskij et al. (1976) Fit
400
600
800 1000 1200 1400 T in K
4
Dyn. viscosity*10 in kg/(ms)
Fig. 17.13.12. Dynamic viscosity of liquid sodium as a function of time
10 9 8 7 6 5 4 3 2 1
Borishanskij et al. (1976) Bystrov et al. (1988) Golden and Tokar (1967)
400
600
800 1000 1200 1400 T in K
Fig. 17.13.13. Dynamic viscosity of liquid sodium as a function of time – comparison of different correlations
ηl = 3.241903 × 10−3 T −0.4925 exp( 508.07 / T ) .
(17.13.75)
Figure 17.13.13 demonstrates that above 400 K these correlations predict values that are very close to each other. 17.13.2.14 Prandtl number of liquid sodium The Prandtl number is used for analysis of heat transfer. It is interesting to know its behavior as a function of temperature at atmospheric conditions. The Borishanskij et al. (1976) estimate is plotted in Fig. 17.13.14. Therefore, un-
706
17. Thermo-physical properties for severe accident analysis
like water, liquid sodium has a two to three order of magnitude lower Prandtl number dictating the need for special heat transfer correlations that are valid in this region. 0,016 0,014
Borishanskij et al. (1976)
Pr
0,012 0,010 0,008 0,006 0,004 400
600 800 T in K
1000
1200
Fig. 17.13.14. Prandtl number for liquid sodium as a function of temperature at 1 bar
17.13.2.15 Surface tension of liquid sodium Kiriyanenko et al. (1965) correlated their own measurements within 467.15 and 1205.15 K with σ = 0.202 − (T − 371.15 ) 10989 .
Surface tension in N/m
0.22 Bystrov et al. (1988) Golden and Tokar (1967)
0.20 0.18 0.16 0.14 0.12 0.10 0.08 400
600
800 1000 1200 1400 T in K
Fig. 17.13.15. Surface tension for liquid sodium as a function of temperature
Golden and Tokar (1967) proposed similar correlation
17.13 Sodium
707
σ = 0.2067 − (T − 273.15 ) 10000 (17.13.76) valid in extended temperature region of 371.15 and 1623.15 K. Bystrov et al. (1988, p. 58) reproduced their own data with
σ = 10−3 ( 247 − 142.3 ×10−3 T + 50.33 × 10−6 T 2 − 16.62 ×10−9 T 3 ) ± 5% . (17.13.77) Both correlation deliver values very close to each other as demonstrated in Fig. 17.13.15. 17.13.2.16 Wetting angle The wetting angle is important characteristic influencing the internal characteristics of surface boiling. Brandhurst and Buchanan (1961) reported for the contact angle of sodium on uranium oxide values of about 160 to 105deg for temperatures between 293 to 425°C. With increasing oxygen concentrations the contact angle decreases. For sodium saturated with oxygen the contact angle at 300°C tends to zero. Ivanovskij, Sorokin and Subbotin (1976) reported wetting angle for sodium-stainless steel for temperatures between 645.15 and 923.15K of 40°. For polished nickel the same authors reported 162 – 0.197(T – 273.15) valid for 473.15 to 773.15 K. 17.13.2.17 Solubility of argon and helium in sodium In most of the fast breeder reactors argon or helium is used as a cover gas in different components. Therefore the sodium can be saturated with argon or helium at a given temperature and pressure and at other conditions the gas can be released and influence the working processes. That is why knowledge of the solubility of argon in sodium is important. Thormeier (1970) reported experimental data for solubility of argon and helium in the region of about 300 to 600°C and about 0.5 to 4 bar which are approximated by the Henry coefficient k Ar → Na = 10
( 7.252 + 4365 Tliquid Na ) ,
(17.13.78a)
( 7.332 + 3595 Tliquid Na ) .
(17.13.79a)
k He → Na = 10
Similarly Veleckis et al. (1971) reported k Ar → Na = 10
( 7.59 + 4221 Tliquid Na )
±16% ,
(17.13.78b)
(8.161+ 2833 Tliquid Na )
±4%
(17.13.79b)
k He → Na = 10
valid for 330–530°C. Knowing the Henry’s coefficient the molar concentration of the saturated solution is then for instance
708
17. Thermo-physical properties for severe accident analysis
(
)
YAr in liquid Na , ∞ = p Ar in gas k Ar → Na Tliquid Na ,
(17.13.80)
where p Ar in gas is the partial pressure of argon in the gas being in contact with the liquid. Note that the solubility of these gases in sodium is four orders of magnitude less than their solubility in water. 17.13.3. Vapor
17.13.3.1 Constituents of the sodium vapor Water vapor at boiling point consists 100% of water molecules. In contrasts at the boiling point of sodium 11% of the matter is not in a mono-atomic state – the normal state at high temperatures, Cordfunke and Konings (1990). The sodium vapor is therefore a mixture of mono-atomic (monomer), diatomic (dimmer), probably four-atomic (tetramer) components with ion and electron components in addition, Ewing et al. (1967). This makes the analysis of processes like film boiling for instance more complicated because of the additional association and dissociation in the boundary layer, Dwyer (1976). The general way haw to compute the constituency of the sodium vapor is associated with the classical approach of the chemical thermodynamic for computing the equilibrium mole concentrations. The interested reader may consult Chapter 4.2.4.3 of Kolev (2007a). There are differences in the literature in the implementation of this method by considering different number of components and the corresponding reactions. As an example I give here the approach clearly stated by Golden and Tokar (1967), based on the important findings of Ewing et al. (1967): The Stone et al. (1966) pvT-data collected for temperatures up to 1673 K and 30 bar are reproduced best by the assumption that the sodium vapor consists of mono-atomic (monomer), diatomic (dimmer) and four-atomic (tetramer) components. Given the temperature T and the total pressure p of a mixture of jmax = 3 compounds that may react in a number imax = 2 of chemical reactions, jmax
∑ n Symb
= 0 , for i = 1, imax ,
(17.13.81)
−2 Na + Na2 = 0 , reaction Nr. 1,
(17.13.82)
−4 Na + Na4 = 0 , reaction Nr. 2.
(17.13.83)
j =1
ij
j
or in particular
17.13 Sodium
709
Here nij is the stoichiometric coefficient, < 0 for reactants and > 0 for products. I look for a solution of 3 molar concentrations YT = (Y1 , Y2 , Y4 ) ,
(17.13.84)
for which the system is in chemical equilibrium. I know from Dalton’s law that the system pressure is the sum of the partial pressures p = p1 + p2 + p4 , Yi =
pj p
,
Y1 + Y2 + Y4 = 1 .
(17.13.85)
(17.13.86)
(17.13.97)
For each chemical reaction i we have the condition enforcing chemical equilibrium, see Kolev (2007a) Eq. (3.174), ⎡ 1 ⎤ K p 2 = exp ⎢ − ⎡ −2 ( h10 − Ts10 ) M 1 + ( h20 − Ts20 ) M 2 ⎦⎤ ⎥ ⎣ ⎣ TR ⎦ ⎡ 2 ( h10 − h20 ) R1 ⎤ = exp ⎢ −2 ( s10 − s20 ) R1 + ⎥ = exp ( ea1 + ea 2 T ) , T ⎣ ⎦
(17.13.88)
⎡ 1 ⎤ K p 4 = exp ⎢ − ⎡⎣ −4 ( h10 − Ts10 ) M 1 + ( h40 − Ts40 ) M 4 ⎤⎦ ⎥ ⎣ TR ⎦ ⎡ 4 ( h10 − h40 ) R1 ⎤ = exp ⎢ −4 ( s10 − s40 ) R1 + ⎥ = exp ( eb1 + eb 2 T ) , T ⎣ ⎦
(17.13.89)
with the chemical equilibrium factors defined as follows K p2 =
p0 Y2 , p Y12
(17.13.90)
K p4 =
p03 Y4 . p 3 Y14
(17.13.91)
710
17. Thermo-physical properties for severe accident analysis
Using the molar enthalpy of demerization and tetramerization given above recomputed pro kg 2 Na h20 − h10 = ΔhNa 2
( 2M 1 ) = −1.666315 ×106 J / kg ,
(17.13.92)
4 Na h40 − h20 = ΔhNa 4
( 4M 1 ) = −1.887987 ×106 J / kg ,
(17.13.93)
results in ea 2 = 2 ( h10 − h20 ) R1 = 9117.50383 K,
(17.13.94)
eb 2 = 4 ( h10 − h40 ) R1 = 20660.83388 K,
(17.13.95)
which are the values used here. Stone et al. (1965) used values for the enthalpy and entropy changes of the mixture due to each chemical reaction derived from his p-v-T data. They come to ea2 = 9217.72 K and eb2 = 20772.05 K which are slightly different. For the entropy terms this authors come to ea1 = −2 ( s10 − s20 ) R1 = -9.95845,
(17.13.96)
eb1 = −4 ( s10 − s40 ) R1 = -24.59115,
(17.13.97)
related to reference pressure p0 = 1 atm which means s20 − s10 = –1820.006322 J/(kgK), s40 − s10 = –2247.139287 J/(kgK). These values are used by Hame (1986). Recomputed to p0 = 1 Pa results in ea1 = –21.4845, eb1 = –59.1694 which means s20 − s10 = −3926.50722 J/(kgK), s40 − s10 = −5406.899772 J/(kgK). Therefore for given pressure and temperature the algebraic system of Eqs. (17.13.87), (17.13.90) and (17.13.91) completely defines the mole concentrations. Replacing in Eq. (17.13.87) the mole concentrations Y2 = K p 2 pY12 ,
(17.13.98)
Y4 = K p 4 p 3Y14
(17.13.99)
results in the fourth order equation
K p4 or
p3 4 p 2 Y + K p2 Y1 + Y1 − 1 = 0 3 1 p0 p0
(17.13.100)
17.13 Sodium
a4Y14 + a2Y12 + Y1 − 1 = 0
711
(17.13.101)
where a2 = K p 2 p = p exp ( ea1 + ea 2 T ) ,
(17.13.102)
a4 = K p 4 p 3 = p 3 exp ( eb1 + eb 2 T ) .
(17.13.103)
The solution of the fourth order equation is performed by iteration using the Newton method starting with Y1 = 0.8 and iterating
f = Y1 + a2Y12 + a4Y14 − 1 ,
(17.13.104)
df dY1 = 1 + 2a2Y1 + 4a4Y13 ,
(17.13.105)
ΔY1 = f
( df
dY1 ) ,
Y1 = Y1 − ΔY1 ,
(17.13.106) (17.13.107)
until ΔY1 < ε Y1 where ε = 10–6. So at each given temperature and pressure the mole concentrations are uniquely defined. The mixture moll mass is usually computed by assuming instantaneous adjustment of chemical equilibrium for each pressure and temperature. Therefore M ( p, T ) = Y1 M 1 + Y2 M 2 + Y4 M 4 = M 1 (Y1 + 2Y2 + 4Y4 ) = M 1 ( 4 − 3Y1 − 2a2Y12 ) .
(17.13.108) Knowing the mole concentrations, the mixture mole mass the mass concentrations are then Ci = Yi M i M .
(17.13.109)
The method presented here for computation of the components of the gas mixture is called some times in the literature quasi-chemical method. 17.13.3.2 Sodium vapor density The equation of state for sodium vapor is then not for a single component but for a mixture of components. Assuming the mixture as a perfect mixture of perfect gases we have for the equation of state
ρTRu p = M ,
(17.13.110)
712
17. Thermo-physical properties for severe accident analysis
or
ρTRu p = M 1 ( 4 − 3Y1 − 2a2Y12 )
(17.13.111)
or finally
ρTR1 p = A ,
(17.13.112)
where
Density in kg/m³
A = 4 − 3Y1 − 2a2Y12 . 1000 900 800 700 600 500 400 300 200 100 0
(17.13.113)
Na
500
1000
1500 T in K
2000
2500
Density in kg/m³
0.30 Na vapor
0.25 0.20 0.15 0.10
1500
2000 T in K
2500
Fig. 17.13.16. Sodium density as a function of the temperature at atmospheric pressure: (a) Solid, liquid and vapor region; (b) Vapor region only
17.13 Sodium
713
Other set of reactions will generate slightly different mole mass dependence on pressure and temperature; see for instance Vargaftic and Voljak (1985, p. 536). Fink et al. (1982) commented that no rigorous error analysis can be made to determine the accuracy of the density thermo-chemical function since no experimental measurements exist, but based on the standard deviation of the values calculated using the quasi-chemical method for saturated sodium of 5%, the standard deviation for the values of the density of superheated sodium can be estimated as 10%. Figures 17.13.16 (a) and (b) present the sodium density as a function of the temperature at atmospheric pressure. The jumps in Fig. 17.13.16(a) are due to the phase transitions. 17.13.3.3 Density of the saturated sodium vapor The sodium vapor density at the saturation line can be computed by using the quasi-chemical method as described above by setting the pressure equal to the saturation pressure at the given temperature
ρ ′′ (T ) = ρ ⎡⎣ p ′ (T ) , T ⎤⎦ .
(17.13.114)
There are also pure empirical methods. Miller et al. (1967) proposed for liquid and vapor sodium density at saturation the following approximation
ρ ′ (T ) T b′ = 1 + a ′ (1 − T Tc ) + d c (1 − T Tc ) , ρc ρc
(17.13.115)
ρ ′′ (T ) T b ′′ = 1 − a ′′ (1 − T Tc ) + d c (1 − T Tc ) , ρc ρc
(17.13.116)
where a´ = 4.097, b´ = 0.682, a´´ = 1.422, b´´ = 0.389, d = 0.1194 kg/(m³K) . The vapor density equation is valid for T > 1644.26 K with Tc = 2780 K and ρ c = 180 kg/m³. The advantage of this correlation is that it gives exactly the critical density at the critical point. 17.13.3.4 Evaporation enthalpy – using Clausius–Clapayron equation For known pressure, saturation temperature, saturation specific vapor and liquid volumes at this pressure the latent heat of vaporization can be computed from the Clausius–Clapayron equation Δh = T ′ ( p )
v ′′ − v′ . dT ′ dp
(17.13.117)
714
17. Thermo-physical properties for severe accident analysis
The latent entropy of vaporization is then Δs = Δh T ′ ( p ) .
(17.13.118)
Having expressions defining the liquid specific enthalpy and entropy, the vapor saturation enthalpy and entropy are then
h′′ = h′ + Δh = hl ⎡⎣ p, T ′ ( p ) ⎤⎦ + Δh ,
(17.13.119)
s ′′ = s ′ + Δs = sl ⎡⎣ p, T ′ ( p ) ⎤⎦ + Δs .
(17.13.120)
Alternatively, approximation for Δh can be used and the saturation temperature gradient can be defined with dT ′ dp = T ′ ( p )( v ′′ − v ′ ) Δh . 17.13.3.5 Specific enthalpy of evaporation–empirical approximations Golden and Tokar (1967) proposed
Δh = 1.4482 ×106 (1 − T Tc ) + 3.4849 × 106 (1 − T Tc )
0.2
(17.13.121)
reproducing data up to 1644 K with 2% mean square error. For higher temperature data are not available. Miller et al. (1967) proposed the correlation
Δh = 1887.3Tc (1 − T Tc )
0.32227
,
(17.13.122)
which was later recommended by Fink et al. (1982) for use for T > 1644 K. The prediction of those correlations is presented in Fig. 17.13.17. Note that the combination of both correlations gives a non-smooth transition.
Heat of evaporation in MJ/kg
17.13 Sodium
715
5 4 3 2 1
Eq.44 For low pressure Eq.121 for T < 1644 else Eq.122 Eq.121 Golden and Tokar (1967) Eq.122 Miller et al. (1967) Golden and Tokar (1967), quasi-chemical method
0 500
1000
1500 T in K
2000
2500
Fig. 17.13.17. Specific enthalpy of evaporation as a function of the temperature
From Fig. 17.13.17 is also obvious that for higher pressure neglecting the diminishing density difference of vapor and liquid as done in Eq. (17.13.46) results in unphysical evaporation enthalpy and therefore is not justified. Nevertheless, as already mentioned Eq. (17.13.48) reproduces successfully the experimental data for saturated pressure because the deviation is compensated by the empirical constants. 17.13.3.6 Specific enthalpy of evaporation – quasi-chemical approach The quasi-chemical method allows also computing the heat of evaporation as proposed by Golden and Tokar (1967, p. 34). It needs expression for computing the evaporation enthalpies of the constituents of the mixture at the saturation line ⎣⎡ p ′ (T ) , T ⎦⎤ . The evaporation specific molar enthalpy for monomers is Δh1m = 1.087552 × 108 − 16675.4T + 9.57899T 2 − 3.546196 × 10−3 T 3 J/(kgmol Na). (17.13.123) Transferring liquid into dimers requires less enthalpy, namely
Δh2m = 2Δh1m − 7.662054 × 107 J/(kgmol Na2),
(17.13.124)
2 Na where the averaged heat of dimerization is ΔhNa = −7.662054 × 107 J/(kgmol 2
Na2). Transferring liquid into tetramers requires also less enthalpy, namely
Δh4m = 4Δh1m − 1.736269 × 108 J/(kgmol Na4),
(17.13.125)
716
17. Thermo-physical properties for severe accident analysis
4 Na where the averaged heat of tetramerization is ΔhNa = −1.736269 × 108 J/(kgmol 4
Heat of evaporation in MJ/kg
Na4). 5 4 3 Chemical method Approximation Golden and Tokar (1967)
2 1
500
1000
1500 T in K
2000
2500
Fig. 17.13.18. Specific enthalpy of evaporation as a function of the temperature
Then the mole- and mass concentrations of the saturated vapor, (Y1′′, Y2′′, Y4′′) ,
( C1′′, C2′′, C4′′) , respectively are computed at
⎣⎡ p ′ (T ) , T ⎦⎤ . The evaporation enthalpy
plus the components for association is
(
Δh = Y1′′ Δh1 + Y2′′ Δh2 + Y4′′Δh4
) M ′′ = C ′′ ΔMh + C ′′ 2ΔMh 1
1
2
2
1
1
+ C4′′
Δh4 . 4M1
2 Na 4 Na ΔhNa ΔhNa Δh Δh Δh 2 4 = C1′′ 1 + C2′′ 1 + C4′′ 1 + C2′′ + C4′′ M1 M1 2M 1 4M 1 M1
= Δh10 + C2′′Δh20 + C4′′Δh40 ,
(17.13.126)
where Δh10 = Δh1 M 1 ,
(17.13.127)
2 Na Δh20 = ΔhNa 2
( 2M 1 ) = −1.666315 ×106 J / kg ,
(17.13.128)
4 Na Δh40 = ΔhNa 4
( 4M 1 ) = −1.887987 ×106 J / kg .
(17.13.129)
17.13 Sodium
717
The prediction of the evaporation heat by the quasi-chemical method is also presented in Fig. 17.13.18. It is approximated by Golden and Tokar (1967) which is also presented in Fig. 17.13.18 as well on Fig. 17.13.17 for comparison with other methods. Remarkably, at the critical point due to association the evaporation enthalpy is not zero but substantial. 17.13.3.7 Specific entropy of evaporation The specific entropy of evaporation is Δs = Δh T ′ ( p ) .
(17.13.130)
17.13.3.8 Velocity of sound of sodium vapor The general expression for the velocity of sound for compressible medium is a=1
⎤ ⎛ ∂ρ ⎞ 1 ⎛ ∂ρ ⎞ ⎡ ⎛ ∂h ⎞ ⎜ ⎟ − ⎜ ⎟ ⎢ ρ ⎜ ⎟ − 1⎥ . ⎝ ∂p ⎠T ρ c p ⎝ ∂T ⎠ p ⎣ ⎝ ∂p ⎠T ⎦
(17.13.131)
It is interesting to compare it with the velocity of sound for perfect gas mixture is a=
cp cp − R
RT ,
(17.13.132)
where
Velocity of sound in m/s
R = Ru M = R1 A .
(17.13.136)
3500 Na
3000 2500 2000 1500 1000 500
500
1000
1500 T in K
2000
2500
718
17. Thermo-physical properties for severe accident analysis
Velocity of sound in m/s
3500 Na
3000 2500 2000 1500 1000 500
500
1000
1500 T in K
2000
2500
Fig. 17.13.19. Velocity of sound as a function of temperature at atmospheric pressure: (a) General expression; (b) Perfect gas mixture
Figure 17.13.19 presents the velocity of sound as a function of the temperature at atmospheric conditions. The jumps are associated with the melting and the evaporation. The differences in the vapor region between the general expression and Eq. (17.13.132) are visible only close to the saturation. For high temperature the differences disappears. 17.13.3.9 The density derivative with respect to the temperature at constant pressure The density derivative with respect to the temperature at constant pressure is computed as follows. First the derivatives of the coefficients containing the equilibrium constants are ea 2 a2 ⎛ ∂a2 ⎞ ⎜ ∂T ⎟ = − T 2 , ⎝ ⎠p
(17.13.134)
eb 2 a4 ⎛ ∂a4 ⎞ ⎜ ∂T ⎟ = − T 2 . ⎝ ⎠p
(17.13.135)
Differentiating the equation for the mono-atomic molar concentration with respect to the temperature at constant pressure and solving with respect to dY1 dT results in
17.13 Sodium
⎡⎛ ∂a ⎞ ⎛ ∂a ⎞ ⎤ Y12 ⎢⎜ 2 ⎟ + Y12 ⎜ 4 ⎟ ⎥ 2 2 ⎝ ∂T ⎠ p ⎦⎥ Y1 ( ea 2 a2 + Y1 eb 2 a4 ) ⎛ ∂Y1 ⎞ ⎣⎢⎝ ∂T ⎠ p = − = . ⎜ ∂T ⎟ T 2 df dY1 1 + 2a2Y1 + 4a4Y13 ⎝ ⎠p
719
(17.13.136)
The density derivative with respect to the temperature at constant pressure is then p ⎛ ∂ρ ⎞ ⎜ ⎟ = 2 ⎝ ∂T ⎠ p R1T
⎡ ⎛ ∂A ⎞ ⎤ ⎢T ⎜ ⎟ − A⎥ , ⎥⎦ ⎣⎢ ⎝ ∂T ⎠ p
(17.13.137)
where ⎡ e a ⎤ ⎛ ∂Y1 ⎞ ⎛ ∂A ⎞ − 2Y12 a 2 2 2 ⎥ . ⎜ ⎟ = − ⎢( 3 + 4a2Y1 ) ⎜ ⎟ T ⎦⎥ ⎝ ∂T ⎠ p ⎝ ∂T ⎠ p ⎣⎢
(17.13.138)
17.13.3.10 The density derivative with respect to the pressure at constant temperature The density derivative with respect to the pressure at constant temperature is computed as follows. First the derivatives of the coefficients containing the equilibrium constants are ⎛ ∂a2 ⎞ ⎜ ⎟ = exp ( ea1 + ea 2 T ) = a2 p , ⎝ ∂p ⎠T
(17.13.139)
⎛ ∂a4 ⎞ 2 ⎜ ⎟ = 3 p exp ( eb1 + eb 2 T ) = 3 a4 p . p ∂ ⎝ ⎠T
(17.13.140)
Differentiating the equation for the monomer molar concentration with respect to the pressure at constant temperature and solving with respect to ( ∂Y1 ∂p )T results in ⎛ ∂Y1 ⎞ Y12 a2 + 3a4Y12 . ⎜ ⎟ =− p df dY1 ⎝ ∂p ⎠T
(17.13.141)
The density derivative with respect to the pressure at constant temperature is then ⎛ ∂ρ ⎞ 1 ⎡ ⎢A+ ⎜ ⎟ = ⎝ ∂p ⎠T TR1 ⎢⎣
⎛ ∂A ⎞ ⎤ p⎜ ⎟ ⎥ , ⎝ ∂p ⎠T ⎦⎥
(17.13.142)
720
17. Thermo-physical properties for severe accident analysis
where ⎡ ⎤ ⎛ ∂A ⎞ ⎛ ∂Y1 ⎞ 2 ⎜ ⎟ = − ⎢( 3 + 4a2Y1 ) ⎜ ⎟ + 2Y1 a2 p ⎥ . ⎝ ∂p ⎠T ⎝ ∂p ⎠T ⎣⎢ ⎦⎥
(17.13.143)
17.13.3.11 Specific enthalpy Golden and Tokar (1967) proposed to compute the vapor enthalpy by first adding to the evaporation enthalpy Δh = Δh10 + C2′′Δh20 + C4′′Δh40 the amount C2′′Δh20 + C4′′Δh40 h* = h′ ⎣⎡ p ′ (T ) , T ⎦⎤ + Δh − C2′′Δh20 − C4′′Δh40 = h′ ⎣⎡ p ′ (T ) , T ⎦⎤ + Δh10
(17.13.144)
and then reducing the amount C2 Δh20 + C4 Δh40 (remember that the demerization and tetramerization enthalpies are negative numbers) resulting in h ( p, T ) = h * +C2 Δh20 + C4 Δh40 = h′ ⎡⎣ p ′ (T ) , T ⎤⎦ + Δh − ( C2′′ − C2 ) Δh20 − ( C4′′ − C4 ) Δh40
(17.13.145) or
h ( p, T ) = h′ ⎡⎣ p ′ (T ) , T ⎤⎦ + Δh10 + C2 Δh20 + C4 Δh40 .
(17.13.146)
Expressing the concentrations in terms of the mole monomer concentration, C2 = 2 Y2 A and C4 = 4 Y4 A results in h ( p, T ) = h′ ⎡⎣ p ′ (T ) , T ⎤⎦ + Δh10 +
1 ⎡ 4Δh40 (1 − Y1 ) + ( Δh20 − 2Δh40 ) 2 K p 2 pY12 ⎤ ⎦ A⎣
(17.13.147) or
h ( p, T ) = h1 (T ) + Δhas0 A
(17.13.148)
where h1 (T ) = h′ ⎡⎣ p ′ (T ) , T ⎤⎦ + Δh10 ,
(17.13.149)
Δh = ak 4 (1 − Y1 ) + ak 2 2 K p 2 pY , 0 as
2 1
(17.13.150)
ak 4 = 4Δh = −7.551948 × 10 J/kg , 0 4
6
ak 2 = Δh − 2Δh = 2.109659 ×10 J/kg . 0 2
0 4
6
(17.13.151) (17.13.152)
If other expression for the saturated liquid enthalpy h′′ ( p ) is used the adjustment of the vapor enthalpy is easily done hg (T , p ) = h′′ ( p ) − h ⎡⎣T ′ ( p ) , p ⎤⎦ + h (T , p )
(17.13.153)
17.13 Sodium
721
to obtain h ⎡⎣T ′ ( p ) , p ⎤⎦ = h′′ ( p ) . Figure 17.13.20 presents the specific sodium enthalpy as a function of temperature at atmospheric pressure. The two jumps are associated with the melting and evaporation with partial association.
p in bar 1 10 50 100 150 200 250 300 350 400 450 500 600 700
h in MJ/kg
8 6 4 2 0 1000
2000 T in K
3000
4000
Fig. 17.13.20. Specific enthalpy for solid, liquid, and gaseous sodium as a function of temperature at atmospheric pressure
Padilla (1978) reported that the quasi-chemical approach is a valid approximation only below 1644 K. As already mentioned Fink et al. (1982) commented that no rigorous error analysis can be made to determine the accuracy since no experimental measurements exist, but based on the standard deviation of the values calculated using the quasi-chemical method for saturated sodium, the standard deviation for the values of the enthalpy of superheated sodium can be estimated as 10%. 17.13.3.12 Specific heat at constant pressure This expression allows computing the specific capacity at constant pressure by differentiating this equation with respect to the temperature. Taking for the evaporation enthalpy of the monomer Δh1 = 4730337.958 − 725.3012048T + 0.416640859T 2 − 1.542427907 × 10−4 T 3 (17.13.154) and for the saturation enthalpy h′ (T ) = 1630.14T − 0.416751T 2 + 1.542719 × 10−4 T 3 − 67500.5 ,
Golden and Tokar (1967), results in
(17.13.155)
722
17. Thermo-physical properties for severe accident analysis
c p = c p 0 + c p1T + c p 2T 2 +
⎤ 1 ⎡ ⎛ ∂Δhas0 ⎞ 0 ⎛ ∂A ⎞ ⎢ A⎜ ⎟ − Δhas ⎜ ⎟ ⎥, 2 A ⎢ ⎝ ∂T ⎠ p ⎝ ∂T ⎠ p ⎥⎦ ⎣
(17.13.156)
with cp0 = 904.8387952, cp1 = –2.20282d-4, cp2 = 8.73279d-8.
(17.13.157)
I use for my analysis these constants. Note that the recomputed constants from the original dependence on T °C into T K provided by Golden and Tokar (1967) are cp0 = 904.698412984, cp1 = 1.022404919687d-06, cp2 = –9.795711218086d-10. Because the first term is much larger then the others the agreement is very close. Figure 17.13.21 gives the specific capacity as a function of temperature for different pressures computed by this method and compared with the tabulated data by Vargaftic et al. (1996). It is obvious that using only a single constant e.g. c p = 904.16 J/(kgK), Chase (1998, p. 1642) is not correct.
Vargaftic et al. (1996) 1bar 50bar 100bar Golden and Tokar (1967) 1bar 50bar 100bar
cp in J/(kgK)
3000 2500 2000 1500 1000 1000
1500
2000 2500 T in K
3000
Fig. 17.13.21. Specific capacity at constant pressure of sodium vapor as a function of temperature. Comparison between the Golden and Tokar (1967) correlation and the data table by Vargaftic et al. (1996)
17.13 Sodium
Na Solid state: Chase (1998); liquid and vapor state: Golden and Tokar (1967).
3000 cp in J/(kgK)
723
2500 2000 1500 1000 500
1000
1500 T in K
2000
2500
Fig. 17.13.22. Specific capacity at constant pressure of sodium as a function of temperature after Chase (1998) for the solid state, Golden and Tokar (1967) for the liquid and vapor state
Figure 17.13.22 presents the specific sodium capacity at constant atmospheric pressure as a function of temperature for solid, liquid and vapor. One clearly sees the phase transitions. 17.13.3.13 Specific entropy Let as remember one basic expression for mixtures of perfect gases. If the specific imax
entropy at an initial state defined with ( p0 , T0 ) , s0 = ∑ Ci 0 si 0 is known the specifi =1
ic entropy at ( p, T ) is imax
T
i =1
T0
s − s0 = ∑ Ci si = T
=
∫
T0
c p (T ) T
∫
dT − R ln
c p (T ) T
imax
dT − ∑ Ci Ri ln ( pi p0 )
p + Δsmix . p0
i =1
(17.13.158)
Here imax
Δsmix = − R ∑ Yi ln Yi > 0 i =1
(17.13.159)
724
17. Thermo-physical properties for severe accident analysis
is the change of the specific entropy due to mixing if the gas components are previously separated in the same volume under the same total pressure and temperature. For c p = const we have ⎛ p imax ⎞ (17.13.160) s − s0 = c p ln (T T0 ) − R ⎜ ln + Yi ln Yi ⎟ . ⎜ p ∑ ⎟ i =1 0 ⎝ ⎠ The Golden and Tokar (1967) algorithm for computation of the vapor entropy is based on an idea proposed by Ewing et al. (1965): It is similar as the algorithm for computation of the vapor specific enthalpy
s* = s ⎡⎣ p ′ (T ) , T ⎤⎦ + +
Δh −C2′′ ( s20 − s10 ) −C4′′ ( s40 − s10 ) T
{
}
Ru ln ⎣⎡ p ′ (T ) p0 ⎦⎤ + Y1′′ln Y1′′+ Y2′′ln Y2′′+Y4′′ln Y4′′ M ′′
s = s * +C2 ( s20 − s10 ) +C4 ( s40 − s10 ) −
Ru ⎡ ln ( p p0 ) + Y1 ln Y1 + Y2 ln Y2 +Y4 ln Y4 ⎤⎦ . M⎣
(17.13.161)
Remember that Ru h −h ln K p 2 = s20 − s10 − 20 10 , T 2M 1
(17.13.162)
Ru h −h ln K p 4 = s40 − s10 − 40 10 , T 4M 1
(17.13.163)
and therefore s20 − s10 =
h20 − h10 R R Δh 0 + u ln K p 2 = 2 + u ln K p 2 , 2M 1 2M 1 T T
h40 − h10 R R Δh 0 + u ln K p 4 = 4 + u ln K p 4 , 4M 1 4M 1 T T so that finally s40 − s10 =
s* = s ⎡⎣ p ′ (T ) , T ⎤⎦ + +
{
(17.13.164)
(17.13.165)
Δh ⎛ Δh 0 ⎞ ⎛ Δh 0 ⎞ R R −C2′′ ⎜ 2 + u ln K p 2 ⎟ −C4′′ ⎜ 4 + u ln K p 4 ⎟ T 2M 1 4M 1 ⎝ T ⎠ ⎝ T ⎠
}
Ru ln ⎡⎣ p ′ (T ) p0 ⎤⎦ + Y1′′ln Y1′′+ Y2′′ln Y2′′+Y4′′ln Y4′′ M ′′
(17.13.166)
17.13 Sodium
725
⎛ Δh20 ⎞ ⎛ Δh 0 ⎞ R R + u ln K p 2 ⎟ +C4 ⎜ 4 + u ln K p 4 ⎟ 2M 1 4M 1 ⎝ T ⎠ ⎝ T ⎠
s = s * + C2 ⎜ −
Ru ⎡ ln ( p p0 ) + Y1 ln Y1 + Y2 ln Y2 +Y4 ln Y4 ⎤⎦ . M⎣
(17.13.167)
p in bar 1 10 50 100 150 200 250 300 350 400 450 500 600 700
7000 6000 s in J/(kgK)
5000 4000 3000 2000 1000 0 1000
2000 T in K
3000
4000
Fig. 17.13.23. Specific entropy as a function of temperature for solid, liquid, gas state. Parameter: pressure. Computed using the Golden and Tokar (1967) algorithm
Again if other expression for the saturation liquid entropy is used the adjustment sg ( p, T ) = s ′′ ( p ) − s ⎡⎣ p, T ′ ( p ) ⎤⎦ + s ( p, T ) .
(17.13.168)
An illustration of the results delivered by this algorithm is given in Fig. 17.13.23. The two jumps are associated with the melting and evaporation with partial association. 17.13.3.14 Thermal conductivity or sodium vapor Golden and Tokar (1967) reported the following expression for the thermal conductivity of the sodium vapor:
λg = a0 + a1 (T − 273.15 ) + a2 (T − 273.15 ) , 2
(17.13.169)
where a0 = 5.02209152760992039e-3, a1 = 1.21963001333615982e-4, a2 =– 5.43767168358e-8. Note that this correlation can not be extrapolated for T > 1600 K. Vargaftic et al. (1996, p. 1047) reported
λg = 4.056 × 10−4 T 0.675
(17.13.170)
726
17. Thermo-physical properties for severe accident analysis
valid for 500 to 6000 K with an error 1% up to 2500 K and 5% up to 6000 K. The lower range of validity is extended by
λg = ( 7.8T 0.3 + 4.75 T − 30.4 + 1.14 × 10−2 T ) × 10−3
(17.13.171)
Thermal conductivity, W/(mK)
Thermal conductivity in W/(mK)
valid in 90 to 6000 K. 160 140
Na
120 100 80 60 40 20 0
500
0.08
1000
1500 T in K
2000
2500
Na, Vargaftic, Vonogradov and Yargin (1996)
0.07 0.06 0.05 0.04 1000
1500
2000
2500
T in K Fig. 17.13.24. Thermal conductivity for liquid and vapor sodium at atmospheric conditions
The thermal conductivity for liquid and vapor sodium at atmospheric conditions is presented in Fig. 17.13.24.
17.13 Sodium
727
17.13.3.15 Dynamic viscosity or sodium vapor Golden and Tokar (1967) reported the following expression for the dynamic viscosity of sodium vapor:
η g = 1.797 × 10−5 + 6.0836 ×10−9 T .
(17.13.172)
7
4
4
Dynamic viscosity*10 in kg/(ms)
7
Dynamic viscosity*10 in kg/(ms)
The dynamic viscosity for liquid and vapor at atmospheric conditions is presented in Fig. 17.13.25.
Na, Golden and Tokar (1967)
6 5 4 3 2 1 0
500
1000
1500 T in K
2000
2500
Na, Vargaftic, Vonogradov and Yargin (1996)
6 5 4 3 2 1 0
500
1000
1500 T in K
2000
2500
Fig. 17.13.25. Dynamic viscosity for liquid and vapor at atmospheric conditions. Vapor approximations: (a) Golden and Tokar (1967); (b) Vargaftic et al. (1996)
Vargaftic et al. (1996) reported
728
17. Thermo-physical properties for severe accident analysis
η g = 5.196 × 10−7 T 0.675 ,
(17.13.173)
valid in 500 to 6000 K which gives higher values, compare Figs. 17.13.25 (a) and (b). References Borishanskij VM, Kutateladze SS, Novikov II and Fedynskij OS (1976) Jidkometaliceskie teplonositeli (Liquid metal coolants), Atomisdat Brandhurst DH and Buchanan AS (1961) Surface properties of liquid sodium and sodiumpotassium alloy in contact with metal oxide surfaces. Aust. J. Chem.. vol 14, no 3, pp 397–408 Bystrov PI, Kagan DN, Krechetova GA and Shpilrain EE (1988) Zhidkometallicheskie teplonositeli dlya teplovyh trub i energeticheskih ustanovok (Liquid Metal Heat-Carriers for Heat Pipes and Power Facilities). Nauka Press, Moscow Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Monograph No. 9 Cordfunke EHP and Konings RJM eds. (1990) Thermochemical Data for Reactor Materials and Fusion Products. Nord Holland, Amsterdam Dwyer OE (1976) Boiling Liquid Metal Heat Transfer. American Nuclear Society, Hinsdale, IL Ewing CT, Stone JP, Spann JR, Steinkuller EW, Williams DD and Miller RR (September, 1965) High-Temperature Properties of Potassium, NRL-6233. Naval Research Laboratory, Washington, DC Ewing CT, Stone JP, Spann JR and Miller RR (February, 1967) Molecular association in Sodium, Potassium and Cesium vapors at high temperature. J. Phys. Chem., vol 71, no 3, pp 473–477 Fink JK, Chasanov MG and Leibowitz L (1982) Properties for Safety Analysis, ANL-CENRSD-82-2 Golden GH and Tokar JV (August, 1967) Thermophysical Properties of Sodium. ANL7323, Argonne National Laboratory, Argonne, IL Gurvich LV, Yorish VS, Khandamirova NE and Yungman VS (1985) Ideal gaseous state. In: Ohse RW ed., Handbook of Thermodynamic and Transport Properties of Alkali Metals. Blackwell Scientific Publishing, Oxford Hame W (Dez., 1986) Aufbereitung der Stoffunktionen für Natrium; Einsatz in COMIXReferenzversion KfK auf M7890 und Vektorrechner, PTF report delivered to KfK Ivanovskij MN, Sorokin VP and Subbotin VI (1976) Isparenie I kondensazija metalov. Atomizdat, Moskva Kolev NI (2007a) Multiphase flow dynamics, vol 1, Springer, Berlin Kiriyanenko AA, Makarova OP, Romanov VD and Solov’ev AN (1965) Experimental investigation of the surface tension of liquid sodium. J. Appl. Mech. Tech. Phys., vol 59 no 4, pp 121–122 Makansi M, Selke WA and Bonilla CF (October, 1960) Thermodynamic properties of sodium. J. Chem. Eng. Data, vol 5, no 4, pp 441–452 Miller D, Cohen AB and Dickerman CE (September, 1967) Estimation of vapor and liquid density and heat of vaporization of alkali metal to the critical point. International Conference of Safety of Fast Breeder Reactors, Aix-en Provence, France Mozgovoi AG, Roshchupkin VV, Pokrasin MA, Fokin LR and Handomirova NE (1988) Lithium, sodium, potassium, rubidium, cesium. Saturation Vapor Pressure at High Temperature. GSSSD 112-87, Standards Press, Moscow
Appendix 1
729
Novikov II, Roshchupkin VV, Trelin YuS, Tsiganova TA and Mozgovoi AG (1981) Review series an thermophysical properties of substances, no 6 (32), Institute of Hi, Temperatures Acad. Sci. USSR, Moscow, p 65 Padilla A Jr (February, 1978) High-Temperature Thermodynamic Properties of Sodium, HEDL-TME 77-27. Hanford Engineering Development Laboratory, Richland, WA Perry RH and Green D (1985) Perry’s Chemical Engineer’s Handbook, 6th ed. McGrawHill, New York, pp 3–285 Petiot F and Seiler J-M (1982) Physical properties of sodium. A contribution to the estimation of the critical coordinates. 10th Liquid Metal Boiling Working Group, Katlsruhe, October 1982 Shpilrain EE, Yakimovich KA, Fomin VA, Skovorod’ko SN and Mozgovoi AG (1994). Handbook of Thermodynamic and Transport Properties of Alkali Metals. Blackwell Scientific Publications, Oxford, p. 753 Stone JP, Ewing CT, Spann JR, Steinkuller EW, Williams DD and Miller RR (September, 1965) High-Temperature Properties of Sodium. NRL-6241, Naval Research Laboratory, Washington, DC Stone JP, Ewing CT, Spann JR, Steinkuller EW, Williams DD and Miller RR (1966) High-temperature properties of sodium. J. Chem. Eng. Data, vol 11, p 309 Thormeier K (1970) Solubility of noble gases in liquid sodium. Nucl. Eng. Design, vol 14, p 69 Trelin JuS, Vasiljev II and Rostchupkin BB (1960) Atomaja Energia, vol 9, no 5, p 410 (in Russian) Vargaftic NB, Vonogradov YK and Yargin VS (1996) Handbook of Physical Properties of Liquids and Gases, 3rd Augmented and revised edition. Begel House, New York Vargaftic NB and Voljak LD (1985) Thermodynamic properties of alkali metal vapors at low pressures. In: Ohse RW ed., Handbook of Thermodynamic and Transport Properties of Alkali Metals. Blackwell Scientific Publishing, Oxford Veleckis E, Dhar SK,Cafasso FR and Feder HM (1971) Solubility of helium and argon in liquid sodium. J. Phys. Chem., vol 75, no 18, pp 2832–2838 Wikipedia (2007) http://en.wikipedia.org/wiki/Sodium
Appendix 1 ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎟ dp + ⎜ ⎟ dT , ⎝ ∂T ⎠ p ⎝ ∂p ⎠T
ρ = ρ ( p, T ) , d ρ = ⎜
⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎟ dp + ⎜ ⎟ dh , ∂ p ⎝ ∂h ⎠ p ⎝ ⎠h
ρ = ρ ( p, h ) , d ρ = ⎜
⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎜ ⎟ ⎜ ⎟ −⎜ ⎟ ⎛ ∂h ⎞ ⎝ ∂T ⎠ p ⎝ ∂p ⎠T ⎝ ∂p ⎠ h dh = dp + dT = ⎜ ⎟ dp + c p dT ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎝ ∂p ⎠T ⎜ ⎟ ⎜ ⎟ ⎝ ∂h ⎠ p ⎝ ∂h ⎠ p
730
17. Thermo-physical properties for severe accident analysis
⎡⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎤ ⎛ ∂ρ ⎞ ⎛ ∂h ⎞ ⎜ ⎟ = ⎢⎜ ⎟ −⎜ ⎟ ⎥ ⎜ ⎟ ⎝ ∂p ⎠T ⎣⎝ ∂p ⎠T ⎝ ∂p ⎠ h ⎦ ⎝ ∂h ⎠ p ⎛ ∂ρ ⎞ ⎛ ∂ρ ⎞ ⎛ ∂h ⎞ ⎛ ∂ρ ⎞ ⎜ ⎟ =⎜ ⎟ +⎜ ⎟ ⎜ ⎟ ⎝ ∂p ⎠T ⎝ ∂p ⎠ h ⎝ ∂p ⎠T ⎝ ∂h ⎠ p
⎛ ∂ρ ⎞ cp = ⎜ ⎟ ⎝ ∂T ⎠ p
⎛ ∂ρ ⎞ ⎜ ⎟ ⎝ ∂h ⎠ p
17.14 Lead, bismuth and lead-bismuth eutectic alloy
731
17.14 Lead, bismuth and lead-bismuth eutectic alloy
During the last 30 years Soviet scientists invented a fast breeder reactor for submarines cooled by liquid-bismuth eutectic alloys. Being much heavier as a coolant it shield radiation better and makes possible to safe intermediate heat exchanger circuit or even to directly produce steam. In the recent years these interesting properties attracted the attention of many countries to investigate this kind of reactor for future use. A OECD sponsored activities lead in 2008 to the Lead-Bismuth Handbook (2007) that contains a state of the art collection of sources, approximations and reasoning why to prefer one or not other of this approximations. I will give here only a brief summary of the properties part in Tables 17.14.1, 17.14.2 and 17.14.3. The interested reader should consult this book for more information. Table 17.14.1. Main thermodynamic properties of molten lead at atmospheric conditions recommended by Lead-Bismuth Handbook (2007)
Property
Correlation
T-range in K
Error ±
Atomic mass in kg/mol Critical temperature in K Critical pressure in Pa Critical density in kg/m³ Melting temperature in K Latent heat of melting in J/kg Boiling temperature in K Latent heat of boiling in J/kg Saturated vapor pressure in Pa
207.2, Chase (1998)
T ′′′ = 600.6
–
0.1
h′ − h′′′ = 23 800
–
0.7
T ′ = 2016
–
10
h′′ − h′ = 858 200
–
1.9
p ′ = 6.5715 × 109 exp ( −22247 T )
610– 2016
15%
Tc = 4870 pc = 1000 ×105
ρ c = 2490
732
17. Thermo-physical properties for severe accident analysis
Property Liquid Surface tension in N/m Density in kg/m³ Sound velocity in m/s Modulus of elasticity in Pa Specific heat at constant pressure in J/(kgK) Dynamic viscosity in Pa s Electrical resistivity in Ωm Thermal conductivity in W/(mK) Solid Specific heat at constant pressure in J/(kgK)
Correlation
T-range in K
Error ±
σ l = 0.519 − 1.13 × 10−4 T
601– 1200 601– 1900 601– 2000 601– 2000
5%
601– 1300
7%?
ηl = 4.55 × 10−4 exp (1069 T )
601– 1470
4%
Rl = ( 66.6 + 0.0479T ) × 10−8
601– 1300
1%
λl = 9.2 + 0.011T
601– 1300
10%
c p , Pb = a1, Pb + a2, PbT + a3, PbT 2
0–600.6
ρl = 11367 − 1.1944T al = 1951.75 − 0.3423T + 7.635 × 10 −5 T 2 El = ρl al2
(
)
= 42.15 − 1.652 × 10−2 T + 3.273 × 10 −6 T 2 × 109 c p ,l = 175.1 − 4.961× 10−2 T + 1.985 × 10−6 T 2 −2.099 × 10 −9 T 3 − 1.524 × 106 T −2
0.7% 0.05%? –
a1, Pb = 111.74544 , a2, Pb = 0.06866 ,
a3, Pb = −3.15488 × 10−5 , approximation of the Chase (1998) data Vapor: Chase (1998) recommended considering the lead vapor as a perfect monatomic gas. Table 17.14.2. Main thermodynamic properties of molten bismuth (Bi) at atmospheric conditions recommended by Lead-Bismuth Handbook (2007)
Property
Correlation
Atomic mass in kg/mol Critical temperature in K
208.9804, Wikipedia
Tc = 3990
T-range in K
Error ±
17.14 Lead, bismuth and lead-bismuth eutectic alloy
Property Critical pressure in Pa Critical density in kg/m³ Melting temperature in K Latent heat of melting in J/kg Boiling temperature in K Latent heat of boiling in J/kg Saturated vapor pressure in Pa Liquid Surface tension in N/m Density in kg/m³ Sound velocity in m/s Modulus of elasticity in Pa Specific heat at constant pressure in J/(kgK) Dynamic viscosity in Pa s
Correlation
733
T-range in K
Error ±
T ′′′ = 544.4
–
0.3
h′ − h′′′ = 52 600
–
1.7
T ′ = 1806
–
51
h′′ − h′ = 857000
–
2
p ′ = 2.4723 ×1010 exp ( −22 858 T )
600– 1806
70%
σ l = 0.4255 − 8 ×10−5 T
545– 1300
5%
ρl = 10726 − 1.2208T
0.4%
al = 2111.3 − 0.7971T
545– 1300 545–603
El = ρl al2 = ( 44.67 − 0.0299T ) × 109
545–603
–
c p ,l = 118.2 − 5.934 × 10−3 T + 71.83 × 105 T −2
545– 1300
7%?
ηl = 4.458 × 10−4 exp ( 775.8 T )
545– 1300
5%
pc = 1300 ×105
ρ c = 2890
0.1%?
734
17. Thermo-physical properties for severe accident analysis
Property
Correlation
T-range in K 545– 1420
Error ± 0.8%
Electrical resistivity in Ω m Thermal conductivity in W/(mK) Solid Specific heat at constant pressure in J/(kgK)
Rl = ( 97.1 + 0.05534T ) × 10−8
λl = 6.55 + 0.01T
545– 1000
10%
c p , Bi = a1, Bi , a1, Bi = 122.117
25°C
Table 17.14.3. Main thermodynamic properties of molten lead-bismuth eutectic alloy (44.29 at.% Pb and 55.71 at.% Bi) at atmospheric conditions recommended by LeadBismuth Handbook (2007)
Property
Correlation
T-range in K
Error ±
Atomic mass in kg/mol Critical temperature in K Critical pressure in Pa Critical density in kg/m³ Melting temperature in K Latent heat of melting in J/kg Boiling temperature in K Latent heat of boiling in J/kg
208
T ′′′ = 397.7
–
0.6
h′ − h′′′ = 38600
–
200
T ′ = 1943
–
10
h′′ − h′ = 854 000
–
2000
Tc = 4890 pc = 880 ×105
ρ c = 2170
17.14 Lead, bismuth and lead-bismuth eutectic alloy
Property Liquid Saturated vapor pressure in Pa Surface tension in N/m Density in kg/m³ Sound velocity in m/s Modulus of elasticity in Pa Specific heat at constant pressure in J/(kgK) Dynamic viscosity in Pa s Electrical resistivity in Ωm Thermal conductivity in W/(mK) Solid Specific heat at constant pressure in J/(kgK)
Correlation
735
T-range in K
Error ±
p ′ = 11.1× 109 exp ( −22 552 T )
508– 1943
50%
σ l = 0.4371 − 6.6 ×10−5 T
423– 1400 403– 1300 403– 1300
5%
430– 605
0.05%
430– 605
7%?
ηl = 4.94 × 10−4 exp ( 754.1 T )
400– 1100
5%
Rl = ( 86.334 + 0.0511T ) × 10 −8
403– 1100
6%
λl = 3.61 + 1.517 ×10−2 T − 1.741×10−6 T 2
403– 1100
5%
ρl = 11096 − 1.3236T al = 1773 + 0.1049T − 2.873 ×10−4 T 2 El = ρl al2
(
)
= 35.18 − 1.541× 10 −3 T − 9.191× 10 −6 T 2 × 109 −2
−6
c p ,l = 159 − 2.72 × 10 T + 7.12 × 10 T
(
2
0.8% –
)
c p , PbBi = CPb a1, Pb + a2, PbT + a3, PbT 2 + (1 − CPb ) a1, B 0– 600.6
In general the specific heat at constant pressure for liquid is approximated by the expression
736
17. Thermo-physical properties for severe accident analysis
⎡ ∂ ⎛ ∂h ⎞ ⎤ c p ,l T , pref = c1 + c2T + c3T 2 + c4T 3 + c5T −2 + ⎢ ⎜ l ⎟ ⎥ p − pref , ⎣⎢ ∂T ⎝ ∂p ⎠T ⎦⎥ p
(
)
(
)
(14.14.1) and the liquid density by
ρl = a1 + a2T .
(14.14.2)
Therefore ⎧⎪ 1 ∂ 2 ρ 2 ⎛ ∂ρ ⎞ 2 ⎫⎪ ⎡ ∂ ⎛ ∂hl ⎞ ⎤ ⎛ ∂2v ⎞ 2T = − = − 4⎜ T T ⎢ ⎜ ⎨ 2 ⎜ 2⎟ ⎟ ⎥ ⎟ ⎬ =− 4 2 ρ ⎝ ∂T ⎠ ⎪⎭ ρ ⎝ ∂T ⎠ p ⎪⎩ ρ ∂T ⎣⎢ ∂T ⎝ ∂p ⎠T ⎦⎥ p
⎛ ∂ρ ⎞ ⎜ ⎟ , ⎝ ∂T ⎠ 2
(14.14.3) ⎛ ∂hl ⎞ 1 T ⎛ ∂ρ ⎞ ⎛ ∂v ⎞ . ⎜ ⎟ = v −T ⎜ ⎟ = + 2⎜ ∂ p ∂ T ρ ρ ⎝ ∂T ⎟⎠ p ⎝ ⎠p ⎝ ⎠T
(14.14.4)
The liquid components of the specific enthalpy and entropy are then computed as follows
(
)
(
1 1 hl (T ) = h′ + c1 (T − T ′ ) + c2 T 2 − T ′2 + c3 T 3 − T ′3 2 3
)
⎛ ∂h ⎞ 1 + c4 T 4 − T ′4 + c5 T ′−1 − T −1 + ⎜ l ⎟ p − pref , 4 ⎝ ∂p ⎠T
(
sl = s′ + c1 ln
+
)
(
)
(
(
)
)
(14.14.5)
(
)
(
T 1 1 1 + c2 (T − T ′ ) + c3 T 2 − T ′2 + c4 T 3 − T ′3 + c5 T ′−2 − T −2 ′ T 2 3 2
1 ⎡ ∂ ⎛ ∂hl ⎞ ⎤ ⎢ ⎜ ⎟ ⎥ ( p − pref T ⎢⎣ ∂T ⎝ ∂p ⎠T ⎥⎦ p
)
)
(14.14.6)
Proper design of steam properties for Pb, Bi and the eutectic alloy of them is still not available. It is expected as in the case of Na that there are dimmers in the vapor. The solid components of the specific enthalpy and entropy are
(
1 hPbBi (T ) = ⎡⎣CPb a1, Pb + (1 − CPb ) a1, Bi ⎤⎦ (T − T0 ) + CPb a2, Pb T 2 − T02 2
)
References
(
)
1 + CPb a3, Pb T 3 − T03 , 3
737
(14.14.7)
sPbBi (T ) = ⎡⎣CPb a1, Pb + (1 − CPb ) a1, Bi ⎤⎦ ln (T / T0 ) + CPb a2, Pb (T − T0 )
(
)
1 + CPb a3, Pb T 2 − T02 . 2
(14.14.8)
Here CPb is the mass concentration of the lead in the alloy. The selection of the reference temperature is arbitrary, e.g. T0 = 293.15 K at which the specific enthalpy and entropies are set to zero. The specific enthalpies and entropies at the melting points are then ′′′ = hPbBi (T ′′′ ) , hPbBi
(14.14.9)
′′′ (T ) = sPbBi (T ′′′ ) . sPbBi
(14.14.10)
References Chase MW Jr (1998) NIST-JANAF Thermochemical Tables, 4th ed., part II, Cr-Zr. J. Phys. Chem. reference data, Monograph No. 9 Lead-Bismuth Handbook (2007) Handbook on Lead-Bismuth Eutectic Alloy and Lead Properties, Materials Compatibility, Thermal-Hydraulics and Technology. Nuclear Energy Agency, NEA No. 6195, ISBN 978-92-64-99002-9
Index
absolute coordinates 381 ACOPO 515 act of fission 1 Active nucleation seeds at the surface 269 ALPHA 447 alpha-zirconium 605 Aluminum 617 Aluminum oxide 627 annular flow 79 Antivibration bar 296 asymptotic method 60 ATRIUM 10XP bundles 133 austenite steel 593, 594 bag and stamen breakup 91 Bag breakup 91 BALI 515 Baroczy correlation 145 BERDA 448 beta-zirconium 605 bismuth 732 Bismuth 731 Blow down 279 Blow down from initially closed vessel 283 boiling crisis 94 boiling flow 33, 34, 61 boiling initiation 46 boiling on heated solid surfaces 269 boiling on the external wall of horizontal cylinders 52 boiling pressure 575 boiling pressure of stainless steel 603 boiling temperature 667, 694 boiling water reactor 203 boiling water reactors 10, 107 Boron oxide 667 boundary layer treatment 50, 52, 60 boundary value problem 35 Bubble coalescences 278 Bubble departure diameter 272
bubble departure diameter as a function of pressure 273 bubble departure diameter as a function of superheat 273 bubble departure diameter as function of mass flow rate 274 bubble disappearance 278, 287 bubble dynamics 60 Bubble fragmentation 276 Bubble growth in the bulk 275 bubble growth model 276 bubble number density 53 bubble numbers 62 bubble size 54, 62 bubbles interfacial area density 53 building condensers 203 Buoyancy convection 509 Buoyancy driven convection 466 BWR cyclones 337 BWR reactor pressure vessel 175 BWR stability analysis 190 BWR-vessels 108 canonical forms 485 cap bubbles 62 carry under 310, 338 carryover 310, 337 catastrophic breakup 92 Catastrophic breakup 92 cavitation 378, 405 CFD analyses of cyclones 334 CFD analyses of vane separators 334 characteristic points 379, 380 chemical equilibrium 709 chemical equilibrium factors 709 Chernobyl 435 CHF analysis 162 churn turbulent flow 63 Cladding oxidation 28 claddings 15 coalescence 92
740
Index
coalescence frequency 63, 64, 72, 278, 286 coalescence probability 64, 93, 278, 287 coalescence rate 64, 93 Collision 92 collision frequency 63, 93, 278, 286 condensation induced instability benchmarks 196 control rod bundles 114 converging-diverging nozzles 209 Coolability of layers 453 core 15 Core analysis 175 core barrel 108, 112 Core degradation 429 core flooding pool 436 core plates 112 corium 439, 677, 679 corrosion products 29 cracking 302 critical flow 208 critical heat flux 93, 525 critical mass flow rate 264 critical mass flow rate in nozzles 224 Critical multiphase flow 207 critical pressure ratio 213 critical velocity 213 critical Weber number 89, 277 criticality condition 207, 265 criticality condition for the three-fluid flow 266 crust formation 471, 508 Crust formation 462, 501 cubic zirconium dioxide 579 cyclone mass flow entrainment ratio 345 cyclone number 319 cyclone particle Reynolds number 319 Cyclone separators 315 cyclones 299 decay energy 2 decay heat 500 deformation of the mean values of the velocities 222, 288 delayed decay energy 2 density 582, 592, 609, 618, 621, 631, 634, 642, 645, 652, 655, 656, 662, 664, 668, 671, 680, 682, 694, 702
Density 570, 585, 593, 597, 611, 622, 632, 664, 672, 689, 691, 693 density wave oscillations 196 denting 302 deposition 81, 167 Deposition 82, 166 deposition mass flow rate 81 deposition of corrosion products 302 deposition rate 187 depressurization system 437 Developed flow 243 deviation angle 381 dimmer 708 Dissociation energy 687 distorted bubble regime 62 Distorted bubble regime 61 distribution parameter 42 distribution parameter for annular dispersed 86 distribution parameter for annular flow 85 down comer 112 drag coefficient 87 drag force 61, 66, 88 drag forces 251 Drift flux 84 drift velocity 42, 67, 84, 85, 86 droplet evaporation 96 Droplet production rate 90 droplet separation 309 Droplet size stability limit 89 Droplet-gas drag force 87 droplets-gas system 87 dry out 94 dryer efficiency 339 Duration of the fragmentation 90 dynamic viscosity 587, 602, 623, 636, 637, 646, 657, 658, 665, 674, 681, 704, 727 Dynamic viscosity 614, 624, 674, 705, 727 effect of scale 114 effective analog to the gas constant 261 efficiency 314 eigen values 208 Eigen values 485 eigen vectors 208, 485 Elastic modulus 595
Index elasticity modulus 594, 611, 619, 633, 679, 689 Elbows 383 emergency condensers 203 Emergency condensers 423 Emisivity 619 emissivity 602, 613 Emissivity 596, 624, 645, 653, 663 emmisivity 584 Energy conservation 39 enrichment 4 Enthalpy of sublimation 687 entrainment 80, 81, 168, 327 Entrainment 82, 165 Entrainment controlled efficiency of vane separators 326 entrainment mass flow rate 81 entropy equation 39 entropy of vaporization 714 EPR 112, 453 equation of state 36 Equilibrium homogeneous flow 229 Equilibrium non-homogeneous flow 248 European Nuclear Reactor 293 European Pressurized Water Reactor 112 eutectic mixtures 677 evaporation 50 External cooling 497 extrapolation lengths 9, 12 Ex-vessel core catchers 455 FARO 440 FARO-FAT 440 fatigue 302 ferrite steel 593, 594 film boiling 93 Film boiling 468 film thickness 170 fine resolution analysis 174 fission cross section 3, 5, 13 fission energies 1 flashing inception pressure 224 flow boiling heat transfer 56 Flow condensation stability 196 Flow regime transition 78 Flow with friction 215 fluctuation velocity 64 flux distribution 8
741
focusing effect 501 focusing problem 498 Fourier equation 15, 16, 17, 23, 24, 469, 503 fragmentation 255 free razing bubble velocity 63 frequency of the not dumped oscillation 208 fretting 302 Friction flow with isentropic entrance 217 frictionless and isentropic flow 241 frictionless isentropic flow 212 Friedel 38 FRIGG experiment 133 Frozen homogeneous non-developed flow 225 fuel element 15 fuel moderator ratio 9 gas separation 307 Gas-film drag force 88 Gravitational flooding 479 grid sizes 210 harmonic oscillations 208 Heat conduction 469 heat flux 35 heat of vaporization 713 Heat transfer 93 heated diameter 21 heated pipe 102 heated rod bundles 114 heavy reflector 9, 112 Henry’s coefficient 707 High burn up bundle 121 High pressure reduction station 405 homogeneous specific volume 39 Hydrogen diffusion 29 hyperbolic 208 inclination angle 381 instabilities in boiling systems 189 Integrity of the penetrations 449 interfacial area density 88 Interfacial drag 164 In-vessel core catchers 455 in-vessel retention 436 Ionization potential 687 iron 594 Iron oxide 651
742
Index
irreversible dissipated power 222, 288 isentropic 40 isothermal coefficient of compressibility 571, 599, 703 IVR 436 KARENA 423, 436 KATHY loop 191 knots 386 KROTOS 442 Kutateldse droplet free falling velocity 84 Landau and Lifshitz 213 Late water injection 445 Laval nozzles 211 lead 731 Lead 731 lead-bismuth 734 lead-bismuth eutectic alloy 731 levels swell analysis 233 Linear stability analysis using homogeneous equilibrium models 190 Linear stability analysis using slip or drift flux equilibrium models 190 linear thermal strain 583 Liquid dynamic viscosity 574 Liquid emmisivity 575 liquid redistribution in film and droplets 79 liquid separation 307 liquid slug energy 448 Liquid surface tension 573 Liquid thermal conductivity 573 look-up table for critical heat flux 156, 157 Loschmidt number 4 Low burn up bundle 121 lower distributor plate 108 lower grid 108 Mach number 213 Martinelli-Nelson method 37 Martinelli-Nelson multiplier 37 Mass conservation 35 mass flow 34 mass flow fraction 35 mass flow rate 34 Mass transfer 95 MATPRO 652 Maxwell distribution 5
MCI 435 mean droplet size 89 mechanical and thermodynamic equilibrium 41 Melt relocation 444 melt spreading 454 melt-coolant interaction 435 Melt-coolant interaction 435 melting enthalpy 687 melting entropy 687 melting temperature 559, 579, 589, 605, 617, 627, 651, 659, 667, 687 melting temperatures 677 Melt-vessel interaction 504 melt-water interaction 435 Metal layer 474 metallic layer 501 mixture density 37 mixture enthalpy 39 mixture entropy 40 mixture entropy equation 40 mixture mass flow rate 36 Mixture momentum equation 36 mixture specific heat 261 moderator 109 moderator temperature 5 modulus of elasticity 663 Moisture characteristics 311 molar mass 667 mole mass 559, 651 mole-mass 617, 627, 659 mole-mass of Zirconium 605 moll mass 686 mol-mass 589 Molten pool behavior 502 Molybdenum 659 Momentum exchange due to evaporation 248 monoclinic zirconium dioxide 579 monomer 708 natural oscillation frequency in bubbly flow 63 net vapor production 46 neutron absorbers 9 neutron flux 3 Neutron flux 6 neutron velocity 5 Newton iteration method 217 Newton's regime 87
Index noble gases 501 non uniformity coefficients 7 non-damped oscillations 195 non-linear stability analysis of boiling bundles in loops 190 non-uniformly heated pipes 50 nozzles mounted at the fuel bundle support 345 nucleation 269 Nukiyama 156 NUPEC experiment 114 Ohnesorge number 90 once trough steam generators 301 onset of film flow 171 onset of the nucleate boiling 44, 46 oops heated by external condensation 190 over-entrained regime 83 passive safety 437 pellets 15 phase diagram 679 phase-transition specific enthalpy 605 pipe 379 Pipe arrangements 295 pipe library 384 pipe networks 379 Pipe networks 377 pipe section 382 pitting 302 plugged tube 302 politropic exponent 262 Polytrophic state change of the gas phase 225 Potential gas flow in vanes 330 power controlled mode 156 Prandtl number 705, 706 PREMIX 441 pressure drop 310 Pressure drop for boiling flow in bundles 144 pressurized water reactors 108 pressurizer 293 primary cyclone separators 299 prompt energy 2 propagation velocity of the harmonic oscillations 208 PWR steam generator cyclones 337
743
quasi-chemical method 711 QUEOS 439 radial power distributions 11 radiolysis gases 413 Raleigh-Taylor wavelength 185 reactor cores 114 recirculation mass flow 347 recirculation rate 310 reduction 383 reflector 9 relative coordinates 381 relaxation method 45 rod bundle 15, 171 Rod bundle 10 Saint Venant and Wantzel 213 saturation pressure 695 Saturation vapor pressure 696 self-triggered thermal detonation 439 Separated momentum equation 86 Separated momentum equations 60 separation efficiency 314 separator designs 297 severe accident management concept 437 sheet stripping 92 sieve barrel 108 Silicon dioxide 639 slug 63 slug flow 62 small deformation limit 448 Sodium 685 Sodium density 712 sodium vapor 708 Sodium vapor density 711 Solid density 565 Solubility 707 sonic velocity 567, 594, 611, 619, 633, 644, 653, 663, 669, 683 Sonic velocity 689 spacer grid 162 specific capacity 589, 605, 617, 627, 634, 639, 651, 653, 654, 659, 667 Specific capacity 560, 562, 581, 584, 590, 591, 608, 612, 619, 620, 630, 635, 641, 645, 660, 663, 669, 670, 680, 688, 697, 698, 699, 722, 723 specific enthalpy 560, 580, 606, 614, 617, 619, 628, 634, 640, 647, 651,
744
Index
653, 654, 660, 663, 668, 669, 681, 682, 699, 701 Specific enthalpy 563, 581, 591, 608, 620, 630, 641, 661, 670, 701, 720, 721 Specific enthalpy of evaporation 714, 715, 716 specific entropies 580 specific entropy 560, 607, 618, 621, 629, 640, 647, 652, 653, 655, 660, 664, 668, 670, 682, 702 Specific entropy 563, 582, 592, 609, 621, 631, 642, 661, 671, 723, 725 specific entropy of evaporation 717 specific heat 579 Specific heat 721 specific melt enthalpy 606, 659 specific melt entropy 606, 659 specific solid enthalpy 590 specific solid entropy 591 spectral shift 10 spontaneous flashing 269 stability of small heating reactors 190 Stainless steel 589 steam explosion 435 Steam generators 293 Stokes regime 61, 66, 87 stress-corrosion 302 sub system network 384 sub-channel 117 sub-channels 117 sub-cooled boiling 43 suddenly applied relative velocity 90 SULTAN 526, 528 surface tension 601, 613, 623, 636, 646, 657, 665, 674 Surface tension 636, 706 surface tension for pure iron 600 swirl decay 330 swirl number 319 Taylor bubble 78, 180, 186 team generator design 293 temperature difference controlled mode 156 tetragonal zirconium dioxide 579 tetramer 708 thermal conductivity 566, 583, 585, 593, 600, 610, 612, 618, 623, 632,
636, 643, 644, 646, 652, 656, 657, 662, 665, 668, 673, 681, 683, 703 Thermal conductivity 594, 613, 623, 633, 644, 662, 673, 690, 703, 704, 725 Thermal conductivity of stainless steel 601 thermal expansion coefficient 566 thermal power density 1, 3, 4, 6, 8 Thermal power density 4, 5 thermal power profiles 118 thermal shield 112 Thermo-physical properties for severe accident analysis 549 three velocity fields 100 three-field mixture 261 three-fluid boiling flow 77 three-fluid models 77 Three-Mile Island 435 THTF experiments 139 total power 3 Transient boiling 147 transition to film boiling 100 turbulence 172, 173 turbulence modeling in bundles 171 turbulent fluctuation velocity 277 turbulent pulsations 223, 288 under-entrained regime 82 universal gas constant 686 Uranium dioxide caloric and transport properties 559 U-tube type vertical steam generator 294 vacuum breaker 353 validation 34 vane number 324 vane particle Reynolds number 324 vane separators 309 Vane separators 323 vapor Reynolds number 322 vapor specific volume 576, 603 Velocity fields modeling in separators 329 velocity of sound 213, 583, 596, 611, 619, 633, 635, 644, 645, 656, 663, 665, 672, 681, 690, 717 Velocity of sound 568, 584, 596, 612, 622, 635, 673, 690, 718
Index vena contracta 215, 264, 284 Verification 34 Vessel discharge 240 Vessel integrity 447 Vessel wall ablation 507 vibration 302 viscous regime 87 Viscous regime 61 void mixing 122 volatile fission products 501 volumetric fraction 35 volumetric thermal expansion coefficient 597, 598, 655
745
wall-droplet contact efficiency 333 water hammer 378 water penetration into melt 446 water removal 333 water with dissolved gases 378 Wave crest stripping 92 wave number 208 Weber number 89 wetting angle 707
Zirconium 605 Zirconium dioxide 579
