Handbook of Phase Change: Boiling and Condensation

o o o o 0 � o oeo� �o Tc T= Tc Tj = JdzjP ( 1 .2-24) so that from Eq. ( 1 .2-23) lim cl>j = 1 ( 1 .2-25) Here, fract...

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o o o o

0

� o oeo� �o

Tc T= Tc Tj = JdzjP ( 1 .2-24) so that from Eq. ( 1 .2-23) lim cl>j = 1 ( 1 .2-25) Here, fraction of component i in a mixture. When needed to dis tinguishZi theis themolemolefraction forphase. different phases, is usedandforfugacity the vaporcoefficient or gaseousof phase and for the liquid Since the fugacity each component in a mixture are not related to the mixture fugacity and fugacity g

_

_

--

T, P,nk

T, P,nk

gi

p -+ o

p -+ o

y

x

16 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION

coefficient as partial molar properties, a carat C) is used instead of an overbar C) on

Ji and ¢ i ' As the limiting values of partial molar property, however, they become ( 1 .2-26) Ii = Ii and = = Ii / P for Zi = 1 "

0

¢i ¢i

0

"

0

1.2.4 Two-Phase Equilibrium of Mixture The Gibbs function of a multicomponent mixture in a single phase is expressed as G (T, P, n l , n2, . . . n j ), so that its differential becomes

] ]

] ]

( )

dni dG = a GT dT + a G dP + L a niG a T,P,nk a p T, n a P,n G dP G ( 1 .2-27) + L g ; dni = a T dT + a a p T,n a P,n The subscripts n in the first two terms indicate that all n's are held constant during differentiation. This implies a fixed composition, resulting in v

Therefore, Eq.

_

( aa Gp )

( 1 .2-28)

and Tn

( 1 .2-27) becomes

dG = VdP - SdT + L g;dni

or

( 1 .2-29) This expression for a mixture is the counterpart of the following equation for a single component system,

dg = vdP - sdT

( 1 .2-30)

A simple multicomponent and multi phase system is that composed of two components, " 1 " and "2," and two phases, "[" and "v," that are present at the same T and P. For each phase of the two-component mixture, Eq. ( 1 .2-29) is written at constant temperature and pressure as

dG']r,p = g� dn� + g&dn& dGv]r,p = g �dn� + g�dn�

( 1 .2-3 1 ) ( 1 .2-32)

Taking into account that the total amount of each component remaining constant, the differential of the Gibbs function for the mixture becomes

(1 .2-33)

VAPOR LIQUID EQUILIBRIUM PROPERTIES 17

Since n � and n & can vary independently, it follows from an equilibrium condi tion, i.e., dGJ T, P = 0 (as given by Eq. [ 1 . 1 - 1 ]), that the tenns in parentheses in Eq. ( 1 .2-33) must be zero.

gil

=

g¥

and

g� = g�

( 1 .2-34)

This states that for equilibrium of a two-component and two-phase system, the partial molar Gibbs function of each component is the same in each phase. The above conditions are the counterparts of Eq. ( 1 . 1 - 1 6) for the two-phase eqUilibrium of a pure substance. Equation ( 1 .2-34) is also expressed alternately in tenns of the chemical potentials and the fugacities as -/

J.J. I "/ f 1.2.5 Gibbs Phase Rule

I

-v

= J.J.I "V

=f 1

and and

-I

- v

J.J.2 = J.J.2 "/ "v f2 =f2

( 1 .2-35) ( 1 .2-36)

The above requirement for two-phase equilibrium of a two-component system can be extended with similar reasoning to an m-component and 1T -phase system, from which the Gibbs phase rule (Gibbs, 1 876) is derived. For a system involving m components, there are at most (m - 1 ) independently variable mole fractions for each phase. For the total of 1T phases, therefore, there are at most 1T (m - 1 ) independently variable mole fractions. Temperature and pressure, which are the same in each phase, are two further intensive properties, giving a maximum of [1T (m - 1 ) + 2] independently variable intensive properties for the system. Among these properties, there are m(1T - 1) equilibrium conditions describing equality of Gibbs functions in each phase. Thus, the number of intensive properties that may be arbitrarily specified, i.e., the degrees of freedom F, becomes F = L1T (m - 1 ) + 2J - m (1T - 1 ) = 2 + m - 1T

( 1 .2-37)

This is referred to as the Gibbs phase rule, which imposes important limitations on various systems. For a single component system in a single phase, m = 1 and 1T = 1 . Thus, F = 2 + 1 - 1 = 2. That is, two intensive properties, such as temperature and pressure, must be specified to fix the equilibrium state of this system. When two phases are present in a single component system, 1T = 2 and m = 1 . Hence, F = 2 + 1 - 2 = 1 . That is, the equilibrium state is determined by a single intensive property, for example, by specifying either temperature or pressure.

1.2.6 Phase Equilibrium Diagrams for Binary Mixtures The Gibbs phase rule for a two component system (m = 2) yields the maximum number of intensive variables as three for the case when a single phase (1T = 1 )


Figure 7

Pressure-temperature-composition (x I Y I ) diagram for a binary mixture. ,

exists in equilibrium. If the three intensive variables are chosen as T, P, and one of the mole fraction z, then the equilibrium state of the system is fixed. Thus, a single-phase state of liquid or vapor of binary mixtures can be represented in T - P -composition space, as shown in Figure 7. Within this space, the two phase (n = 2) equilibrium state defines surfaces because the phase rule yields two de grees of freedom, i.e., = 4 2 = 2. The upper surface in Figure 7 represents the saturated liquid state, and the lower surface represents the saturated vapor state. The vapor and liquid mixtures exist in the area enveloped by these two surfaces. For two phases in equilibrium, temperature and pressure must be equal in both phases. Thus, the compositions of equilibrium vapor-liquid mixtures are determined as the intersection of an isothermal plane with an isobaric plane. In Figure 7, an isothermal vertical plane (A-A' -A") corresponding to Ta intersects the upper and lower surfaces to form the lens-shaped envelope ALBVA, and an isobaric horizontal plane (C-C' -C") corresponding to Pa intersects both surfaces

F

-


to form the lens-shaped envelope CLOVC. The intersection of the two envelopes occurs in two places: on the upper surface for saturated liquid at point E and on the lower surface for saturated vapor at point F. These points represent values of that are in equilibrium at the the liqui d composition Xl and vapor composition temperature Ta and pressure Pa . A horizontal line connecting points E and F is called a tie line, which determines the equilibrium liquid and vapor compositions at a given temperature and pressure. Three-dimensional representation of the T - P -composition relationships illustrated in Figure 7 is usually projected on either one of the principal planes of constant pressure, constant temperature, or constant composition. Such two dimensional phase diagrams are indicated in Figures 8 through 1 0. The envelopes on the T -x- y and P-x - y diagrams will cover the entire range of mole fractions from zero to unity when mixture temperature or pressure is less than the critical values of respective pure components. Unless equilibrium near the critical region is concerned, there is little use of the P diagram shown in Figure 10. Curves CLO in Figure 8 and ALB in Figure 9 represent the state of saturated liquid, and they are referred to as the bubble-point curve. Curves CVO in Figure 8 and AVB in Figure 9 are the dew-point curves, representing the states of saturated vapor. These two curves converge to the pure-component saturation temperature or pressure at

YI

-

T

P

=

T

constant

c

o

Zl

Yl

mole fraction of more volatile component

1

Figure 8 Temperature-composition diagrams for a binary mixture at two constant pressures.


P

T = constant

B

A

o

Xl

1

mole fraction o f more volatile component

Figure 9

p

Yl

Pressure-composition diagrams for a binary mixture at two constant temperatures.

constant composition

Figure 10

T

Pressure-temperature diagrams for a binary mixture at two constant compositions.


the composition extremes of Xl = 0 and Xl = 1 . The dew-point and bubble-point curves separate the mixture state into superheated vapor, vapor-liquid two-phase, and subcooled liquid, as shown in Figures 8 and 9.

1.2.7 Tie-Line The T -x-y diagram for a constant pressure is more widely used in considering heat transfer with phase change of multicomponent substance. The tie line bd in Figure 8 that goes through the point C at an overall concentration has a useful stoichiometric property. Let n be the total number of moles of a mixture having a mole fraction for component 1 . It separates into n' moles of liquid with a mole fraction X l and n v moles of vapor with a mole fraction A balance of overall mole number gives

ZI

YI .

nl + nV

=

n

( 1 .2-38)

while a mole number balance on component 1 yields x \ nl + y \ n v

Solving these equations,

=

z\n

bc n

bd

( 1 .2-39)

( 1 .2-40)

and n

YI - ZI YI

- Xl

cd bd

( 1 .2-4 1 )

Thus, the ratio o f the numbers o f moles i n the vapor and liquid phases i s expressed by the ratio of the segment lengths of the tie line. n'

=

cd

( 1 .2-42)

This lever principle is also true for mass units if the mass fraction is substituted for the mole fraction in the above equations.

1.2.8 Phase Change at Constant Pressure If a binary mixture in a subcooled liquid state at point A in Figure 1 1 is heated in a vessel from an initial temperature Ta under a quasi-equilibrium isobaric process, it will follow a vertical line, since the overall composition X I a remains constant during this heating process. When the temperature reaches the bubble-point temperature at p oint B, the first vapor bubble is formed. This vapor has the composition

Tb

Y I b'


T

p ::; constant

vapor

Ta

A

o mole fraction of more volatile component

Figure 11

State change of a binary mixture in a heating process at constant pressure.

corresponding to point B' . It is richer in the more volatile component than liquid, so the liquid becomes depleted of the more volatile component. As heating (and evaporation) continue, the amount of vapor increases and the amount of liquid decreases, and both and YI decrease with the states of vapor and liquid following the paths B'C' and BC, respectively. Eventually, when the mixture reaches Tc at point C, the last dew of liquid with the composition X I c corresponding to point C disappears. Further heating causes the system to go through the superheated vapor region to reach point D. The cooling process of a binary mixture from a superheated vapor state to a subcooled liquid state follows the opposite path from point D to point A, shown in Figure 1 1 . The above example is enough to indicate the substantial thermo dynamic differences in the phase change process between binary mixture and pure fluid, namely that the saturation temperature and composition are not constant but are variable during the boiling and condensation processes of multicomponent mixtures. For a phase change process, such as shown in Figure 1 1 , the boiling range is defined as the temperature difference between the dew point C' and the bubble point B at the same overall composition as the mixture.

Xl

1.2.9 Azeotrope Some binary mixtures form an azeotrope at an intermediate composItIon. Figures 1 2 and 1 3 illustrate phase equilibrium diagrams for such a binary mixture.


T

P

=

constant

1

o mole fraction of more volatile component

Figure 12

P

Phase diagram at constant pressure for an azeotrope-forming binary mixture.

T = constant

o

XI , YI

1

mole fraction o f more volatile component

Figure 13

Phase diagram at constant temperature for an azeotrope-forming binary mixture.


Liquid and vapor compositions are identical at the azeotrope, and hence the fluid behaves like a pure substance. The azeotropic state is special in that it possesses onl y one degree of freedom, rather than two as required for normal two-component and two-phase equilibrium of non-azeotropic mixtures. Thus, specification of any one of temperature, pressure, or composition fixes the other two for a binary azeotrope. A maximum or minimum occurs on both the bubble-point and dew point curves at the azeotropic compositions and the respective pair of equations is satisfied:

( 1 .2-43) or

( 1 .2-44) The systems shown in Figures 1 2 and 1 3 have a minimum-temperature and a maximum-pressure azeotrope, respectively. Maximum-temperature and minimum pressure azeotropes also exist. Azeotropes may also occur in mixtures containing more than two components.

1.2.10 Property Changes on Mixing Property changes on mixing different pure substances, designated collectively by � M or � m, are often introduced to calculate properties of multicomponent mixtures. They are defined as the difference between the actual mixture properties and the mole-fraction average of the properties of pure components. The sum of any extensive property of pure components forming a mixture of a given composition at the same temperature T and pressure P as the mixture is given as

m comp = Mcomp/ n =

L z im f

( 1 .2-45)

where m f is the molar property of pure component i at T and P. The corresponding actual property of the mixture is expressed as

( 1 .2-46) Thus, the property change on mixing of pure components to fonn the mixture is defined as

( 1 .2-47)


Selecting an extensive property to be volume, internal energy, enthalpy, entropy, or Gibbs function gives the respective change of molar property on mixing: ( ( ( �v = L Zi Vi vf) , � u = L Zi Ui un , �h = L Zi iii hn ( � s L Zi Si sf) , �g = L Zi (g g f) an incomposition tegration of Eq. from the pure term (gstate The ponent i goff') Ziis =replaced to thewith mixture com Zi as gi gf = RT In (Jd It ) relation for �form g in Eq. onSubstituting mixing inthisa dimensionless as yields the change of Gibbs function �: = L Zi In (Jd It ) Similar dimensionless expressions can be derived for other property changes on mixing. (Jd It ) ] [8 In p�v = RT L Zi 8 In �RTh = L Zi [ 8 In8(JdTIt ) ] In (Ji / liO ) ] In �s / ( 3 ) [ . I' "o o O Z, 3 In T R = Z, ln I I = L Zi In ( / It ) �; all propertyor changes onderimixing canProvi be related tothethefugaci fugacittyy rati ratiooIJi //Iio and iThus, t s temperature pressure v atives. d ed that It imixture s knownproperty as a functiofogin vofentemperature, pressure, and composition, an arbi t rary composition can be obtained from the pure component properti f on the basis of the corresponding equations. For example, the molar volume eands enthalpy of a mixture are given by: (Ji / liO ) ] RT [8 In v= vcom p � v L ZiVi L Zi 8 In (Jd It ) 1 H In 3 [ h RT h = = hcom h L Zi L Zi i p � 3 In T P,Z m

-

-

-

-

=

( 1 .2-48)

i -

( 1 .2-22)

-

1

( 1 .2-49)

-

( 1 .2-48)

( 1 .2-50)

( 1 .2-5 1 )

P

T, z

_

( 1 .2-52)

P,Z

_

""""" �

, J,

-

_

""""" �

0

, PZ

( 1 .2-53)

+

m

m

v

n

n

=

+

+

=

=

°

+ -

P

°

-

P

( 1 .2-54)

T, z

( 1 .2-55)


1.2.1 1 Ideal Mixture

ideal mi!;xistureexpressed is definedoveras that for rwhich the fugacity ofaseverythe component in aAnmole mixture the enti e composition range product of its fractitemperature on Zj and theandfugaci ty ofaspurethe component the same pressure mixture. Thatit is,in the same phase and at ( 1 .2-56)

Here, 1;° is often referred to the standard-state fugacity of pure component Substitution of this relationship into Eqs. through gives /).g = RT L Z i In Z i , /). = /).h = /).s = - R L Z i In Z i Asandconsequences of anonideal mixture (though /).g and /).S are not zero), the volume enthalpy changes mixing are zero. Hence, thea mole volume,fraction enthalpy, and internal energy of an ideal mixture can be calculated as average of pure component properties: i.

( 1 .2-50)

0,

V

( 1 .2-53)

( 1 .2-57)

0,

( 1 .2-58)

1.2.12 Phase Equilibrium Equations for Mixtures

The equatibyon anforextension calculationof Eq.of n-phase equilibrium of an m-component mixturebasicis given as . , m) = I; = Ii = . . . = Ii This states that the fugacity Ii of each component in mixture must be the same in all phases. The same form of equality in each phase holds true equivalently regardi n g the Gibbs functi o n i or the fugacity coefficient ¢ i too. For equilibrium between the vapor and liquid phases, Eq. runs = . . . , m) Ii = Ii The fugacities expressed as of component i in the vapor and liquid phases are respectively ( 1 .2-36)

A[

AV

A Jf

(i

( 1 .2-59)

1 , 2, . . i

g

( 1 .2-59)

Al

Av

Therefore, Eq.

( 1 .2-47)

(i

( 1 .2-60)

1 , 2,

1/ = ¢i (Yi P ) 1/ = Yi (Xi It)

becomes

y;¢; P = x; Yi f;o

(i

= 1 , 2, . . .

( 1 .2-6 1 )

( 1 .2-62)

, m)

0 .2-63)


fugacithatty coefficient in deviation the vapor from phase,an ideal is theliquid activitymixture, coefficient i is thephase Herethe' ;Pliquid denotes the andis in and are the mole fractions in the vapor and liquid phase, respectively. fugaciandty ofit ispurewricomponent the mixture, tten formallyi inas the liquid phase at the same and as the ( ) ( �. ) ( 1 .2-64) i i Ji - J is the saturation vapor pressureis theof pure substance i corresponding to the Here, fugaci t y of pure atby Eq.the( 1 .values miof xwhich ture temperature and are equal in the liquid and vapor phases as required 1 - 1 8). Asfromshown below, the first ratio on the right-hand side of Eq. ( 1 .2-64) is evaluated the equati othen ofequati the state ofthepurestatei ininthethevapor phase, and the second ratio is calculable from o n of li q ui d phase. For a pure substance at constant temperature, Eq. ( 1 . 1 - 1 9) is written as ( 1 .2-65) ZThis equatito othen issaturati integrated in the vapor andphasethenoffrom pure substance from zero pressure o n pressure to in the liquid phase, giving Yi

Xi

Yi

T

.( 0

_

.(

( P)

pr t

p sa t

_

X

fIsa t ( pI:w t )

X

prt

dOn f )

dg RT

v -dP = RT

and

!z· ( P ) / ·(l .s a--:t) p,f t--:-(-

fr t ( pr t )

[ Pt a l exp 10 ' (Z; -

=

exp [ 1

RT

1 ) 1' dp

1 (constant

r}P;'PQI 1 (constant vj d P

pr t ,

dP P

P/ a t

pr t

=

(P)

i

= - =

ft

I�_�

/;sat ( pr t )

T,

/;sat ( p/a t ) i"' P

__

P

T)

T)

i

P

( 1 .2-66) ( 1 .2-67)

respecti viely,s thewhere Zj is the vapor-phase compressibility factortheseof pure component and i liquid-phase molar volume. Substitution of two expressions V for into Eq. ( 1 .2-63) yields i

/;0

This is a general formmixtures. of equation that should bearesatisfied foradopted the phase equilibrium of multicomponent Simplifications usually to this equation as follows.


(1)

Anleading idealtovapor mixture, i.e., and �i and an incompressible liquid, �,s t - ps t 1 Yi p XiYi psat exp [ V a ( p a ) ideal vapor i.e., Zpressure i I and � i and athenegligible liquid mo Vi mixture, larAnp volume or a small deviation from saturation pressure prt, resulting in Zj = 1

�

=

(2)

=

I

i

RT

I

( 1 .2-69)

= 1,

=

�0

1,

�

( 1 .2-70)

(3)

Ideal mixtures in bothabove, the liquid and tovapor phases. That is, Yi is added to the simplification leading = 1

(2)

( 1 .2-7 1 )

This last equation is theof statement of Raoult's law, which represents the simplest vapor-liquid behavior ideal mixtures. 1.2.13 Calculation of Phase Equilibrium for Mixture

Forliquid an idealandmixture that follows Raoult' s law,from equilibrium dia grams between the vapor phases can be detennined a knowledge of the vapor pressures of pure components as a function of temperature. For a binary mixture, for example, applying Raoult's law to each component, and gives Yl P Xl Ptat Y2P X2P;at The mole fraction in each phase totals to unity X l X2 Yl + Y2 This set ofresulting four equations compositions, in can be uniquely solved for four unknown equilibrium X l p prt BM BC AC Y1 - Xl PtPat - BM BC AM Thus, for a asgivena function pressureofP,temperature the composition Xl inthethesaturation liquid phasetemperatures is readily detennined between Ideal mixture.

1

+

=

2,

=

(T)

( 1 .2-72)

=

(T )

( 1 .2-73)

= 1,

-

= 1

prt ( T )

( 1 .2-75)

------=----

Pta t ( T)

(T )

-

(T)

-

T

( 1 .2-74)

X

( 1 .2-76)


less volatile component

more volatile

P

T sat

T=constant

o

I I I - -� - - - - - - - - - + - - - - I I I I vapor I I I I I I

1

O .-----�I----�---,r--- xl

:

I I

_ _ _ _ _ __ _ _ _

: I I

liquid

YI

I I

-----�

P=constant

I I

1 �----��----�-

Figure 14

T 1,sat

T2,sat

T

Determination of phase diagrams of an ideal mixture from the vapor-pressure curves for pure components.

Tta t (P)

and T,rt ( P) for two pure components, as shown in Figure Then the corresponding from Yl inditheagramvaporof phase Eq. Inequilibrium this way, thecomposition T phase an idealis calculated binary mixture following s lawas isa function obtainedofonlytemperature. if the vapor pressure data of pure componentstheareRaoult' available Eq. areholds true, theasliquid and vapor compositionsAt moderate of a binarypressure mixturewhere in equilibrium determined Y2 P;,a t (T) Xl = ----t YI Pt a (T) - Y2 P;, a t (T) YI ( 1 .2-76).

1 4.

-x -y

Non-ideal mixture.

( 1 .2-70)

P

-

( 1 .2-77)

( 1 .2-78)


If the activityexpressions coefficientsforinthetheexcess liquidGibbs phasefunction, are estimated (as shown below)thenusing empirical the phase diagram will be obtained. 1 .2.14 Excess Property and Activity Coefficient

An excess property (Scatchard and Raymond, of awhich mixturewould is defined as the difference between the actual mixture property and that be obtained under an assumption composition. Thus of an ideal mixture at the same temperature, pressure, and 1 938)

I1m ex = m - m id = (m - m comp) - (m i d - m comp )

=

11m - I1m i d

( 1 .2-79)

Here, denotes an excess property. If the Gibbs function is selectedtheassuperscript m, the molar excess Gibbs function is expressed as (l1hYx - T (l1sYx J (l1g)ex L Zi in ( j/i ) - L Zi in Zi � RT J L Zi In ( i 0) L Zi I n Yi z l Ii "ex"

=

=

=

( 1 .2-80)

=

Yi = J) Zi It

( 1 .2-8 1 )

Here, Yi is the activity coefficient, defined as a property useful in particular for liquid mixtures, which represents the deviation of an actualandmixture from an ideal one when (l1g ) ex Ji Zi It holds. For an ideal mixture, Yi is clear from that In Yi is related to (l1g)ex / RT as a partial molarIt property. That Eq.is, In YI - [ a [n (l1gani)eX / RT] ] - [ a [ (l1gaZi) eX / RT] ] If the excess Gibbs function (l1g ) ex of a mixture is known as a function of T, P, and composition Zi , then both the activity and fugacity coefficients are determined, making the calculation of theinmixture propertiesandpossible, asThus, exemplified by theof molar volume and enthalpy Eqs. equations (l1g)ex play the important role in mixture thermodynamics analogous to equations ofmixing state rules for pureis known substances. Needless to say,(l1g)ex if anandequation of state properti with accurate for a mi x ture, then other mixture es are directly calculable from the equation of state, as described before. This approach, ordinarily taken for pure substances, is adopted forofgasstatemixtures up to moderate pressures. For liquid mixtures, however, equations wi t h accurate mixing rules are rarely available so that empirical equations for (l1g yx are used as an alternate means to calculate the mixture properties. =

A.

= O.

= 1

( 1 .2-80)

_

_

( 1 .2-82)

T, P, z

T, P, n

( 1 .2-54)

( 1 .2-55).


Many equations have been proposed to express the composition dependence of (�gasyx. Some such equations for binary mixtures are given in the following fonn (�g yx . . (1.2-83) --- A o AI (zl - Z 2 ) A 2 (Z I - Z 2 ) 2 zIz2 RT zIz2 RT " , (l.2-84) --- A o A I (z l - Z 2 ) A 2 (z l - Z 2 ) 2 (� g Yx where parameters are functions P but not z . The two-parameter Margulesthe and van LaarA'sequations for Inof Tareandbased the above expressions, with only the first two terms being taken into account. Theon parameters in the above equations are changed in both equations as below, giving respective expressions of theTheactivity coefficient. Margules equation: (1.2-85) A O A l2 2 A2l and Al A 2 1 -2 Al 2 then (1.2-86) In zhAI 2 2Z 1 (A2l - AI 2 ) J (1.2-87) In d LA 2 l 2z2 (A1 2 - A2 dJ The van Laar (1935) equation: (1.2-88) A , - "21 ( B1l2 - B12 1 ) Ao, "21 ( B1l2 B12 1 ) then In .... [1 (B 1 2Bz1l2/B2 I Z I ) F (1.2-89) B2 (1.2-90) In Y2.... ---[1 ( B2 Iz l 1/B1 2zdF-The coefficients in the above equations must satisfy the requirements in the extremes of composition. lim1 (In 1 ) In yf (1.2-91) A1 2 , BI 2 zl� (1.2-92) A2 1 , B21 lim (In In y{ The Wilson (1964) equation: (�g) ex ( 1.2-93) RT [z n(Z G1 2 Z2 ) - z2 1n (z2 G 2 lzdl =

+

+

+ .

=

+

+

+ . . .

9i

=

+

91

92

-

-

=

----

=

+

=

+

+

�d

I

-

----

-

YI = -----------:-

+

=

+

9

=

=

-- =

-

l

l

I

+

z2 � 1

92 )

=

=

+

G_2_1 ) (1.2-94) In 91 = -In ZI G 12Z2) Z2 ( ZI GG1 212Z2 _ __ Z2 G21 Z1 G_2_1 ) (1.2-95) In 92 = -In(Z2 G21 ZI ) ZI ( Z I GG1212Z2 _ __ Z2 G21 Z1 Since each setB's,ofortheO'above equationsofforoneIn 9vapor-liquid 1 and In 92 contains only two para meters A' s , s , a minimum equilibrium data, ofT,theP, . i s sufficient to determi n e them. These data allow the calculation and YI activity coefficients 91 and 92 from Eq. (1. 2-69) or (1. 2-70) in a case when these equati o ns hold true. Thereafter, the calculated activityvcoeffici eabove nts areequations substitutedto together with the values of I into the respecti e set of and Y determine thelabletwoinparameters A's, ofB's,concentration, or O's. If manythe vapor-liquid equilibrium data are avai the full range accuracy in determining the parameters be improved. The Wilsonwillequation for m-component mixtures is given as (1.2-96) � z· ln � G I·J·z J· i=1 j=1 and then (1.2-97) In 9i = 1 - In 1=1 G ,jZj J = I GijZj k=1 G jkZk Here, O'stoarecalculate those coefficients thatcients applyin atoternary binary mixture mixtures.fromThus,O'sitdetermi shouldnbeed possible acti v i t y coeffi for the 1-2, 1-3, and 2-3, respective binary mixtures. While the Wilson equation is fficultequation to maniforpulate, been recommended ormorevandiLaar manyit hasmulticomponent systems.as superior to the Margules 32 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION

then

(

+

+

+

+

+

+

+

+

-

X I.

XI

- ( ) (t ) -t ( It ) (�g)ex

-- =

RT

m

�

m

I

�

1.2.15 Approach Based on Equations of State

For m-component mixtures, the following empirical mixing rules are employed to evaluate the constants in two-constant equationsequation, of state, such as the vanor der Waals equation, Eq. (1.1-5), the Redlich-Kwong Eq. (1.1-6), the Peng-Robinson equation, Eq. (1.1-7). (1.2-98) a = L L z iZjaij i=1 j=1 (1.2-99) m

m

m


The mole fraction of component, Zj, is replaced by Xj for the liquid phase and by j for the vapor phase. The cross coefficients in Eq. (1.2-98) are usually assumed toY follow the combination rule (De Santis et aI. , 1976). (1.2-100) Here, ai and bi are constants for pure component in the equation of state, Eqs. interactiphase on parameter, is determined so as to(1.1-5) fit thethrough equation(1.1-7). of stateTheto binary experimental equilibrium data. equationdataofofstatemixtures, followingthe thephasemixing rules is calculation able to reasonably resentIf theany P-v-T equilibrium will be rep per fonned on the basis of the original equation, Eq. (1. 2 -68), which descri b es thefu condition of phase equilibrium of mixtures. In this process, the fugaci t y and gacity coefficient of the mixture are at first evaluated from the pertinent equations. (1.2-4) In f In P + Jro - 1) dPP (1.2-5) In c/> Jro - 1) dPP Then eachphase component inactivity the mixture, the fugaci tliquid y or thephase fugacity coefficient infromthetheforvaporfollowing and the coefficient in the are determined equations. (1. 2-101) In (L ) [ a (n In f) ] i

kij '

P

=

P

=

(Z

(Z

ani [a(nlanjn c/» ] In 'Pj [ a (nanIni f) ] - ln fj ZI

=

T,P,nj

(1.2-102) (1.2-103) where fio is the fugacity of pure component at the saturated state, the values of which are equal for the saturated vapor and liquid as described by Eq. (1.1-18). Redlich-Kwong rule, Forthe thefugacity is determinequation ed as of state, Eq. (1.1-6), with the above mixing bRT - a 11 ] - RT In ( -v - b) RT In !;-Zj -bib [ -v - b ( v b) T 2 RT 2 L Z j Jaiaj b n ( v b) (1.2-104) a bTI /2 a b v T, P, nj

0

=

T. P,nj

i

=

_

__

[

+

_

i

]

I

+


For the Benedict-Webb-Rubin equation of state, Eq. (1.1-8), it becomes RT ln LZ; RT ln(RTjv) �[( v B B;)RT - 2(AA;)1/2 - 2(CC;) 1 /2 jT 2 ] =

+

+

3 [RT(b2 b;) 1 /3 - (a 2 a;) 1 /3 ] + 3 [a(a 2a;) I /3 + a(a 2a;) I /3 ] 5v 5 2v2 1 3( / + ::�, 3 {[l _ exp( _ y /v 2 )] (�) - � exp( - y /V 2 ) } _ � ( y; ) I /2 {[1 _ exp( _ y jv2 )] ( V 2 ) v2T2 y (1.2-105) - exp( - y Iv') - 2:2 exp( -y / v2 ) } +

Y

where the mixing rules are

A L L Z;ZjAij , B = L L Z;ZjAij , C L L Z;ZjCij (1.2-106) a ( L Zi a} /3) 3 , b ( L Zibil /3) 3 , C = ( L ZiCff3) 3 (1.2-107) a ( L Zi aJ /3) 3 , y ( L Ziy/ /2 ) 3 (1.2-108) =

=

=

=

=

=

1.3 THERMODYNAMIC AND TRANSPORT PROPERTIES OF PURE SUBSTANCES AND MULTI COMPONENT MIXTURES 1.3.1 Thermodynamic Properties

Thermodynami cconsistently properties offromboththeipurer respective substancesP-v-T and multicomponent mixtures are determi n ed relationship or equation of state. Calculation of some properties is based on the following equations: (l.3-1) r [p - T ( :�L} v 2: ZiU� +

u =

h

Pv s i['v YO [�v _ ( aa Tp ) ] d V + R L Zi In � RT + L ZiSf h - Ts Cp ( � aT ) = u

+

=

g =

=

Cv

=

v , n;

(� aT )

p

v

(l.3-2) (1.3-3) (1.3-4) (1.3-5) (1.3-6)


Here, uf and sf are the internal energy and entropy for pure component i , the values ofsubstituted which areinspecified a certain reference state. For pure substances, 1 is the aboveatequations. Zi

=

1.3.2 Transport Properties of Mixtures

Transport properti ehave s of pure substances andonmulti component mixturesprocesses in the vaporin and liquid phases a large influence heat and mass transfer boiling andin condensation. Whenentnecessary property data areto anyunavai lable means or not compi l ed the form of conveni use, we have to resort possible for their prediction or estimation, as done for example by Kandlikar et al. (1975a, 1975b). Reid, Prausnitz,onsand revieewed variousandpredivapors ctive methods and correlati for aPoling limited(1987) numbercritically of properti s of liquids for pure substances and their mixtures.property data are extensively measured, and For pure substances, the transport correlatingproperties equationsofforinterest those data are well established. all measurements transport are impossible for many Since mixtures, their estimationof or prediction basisxtureof thecomposition propertiessometimes of pure components area large very important. Amixture small properties, changeon thein miespecially results in changeconin diffusion coefficient, surface tension, thermal ductivity, andinviscosity. Furthermore, theproperti propertyes tovariations withthecompositions aretionnonlinear nature, causing mixture deviate from mole frac or mass fraction average values oftransport pure component properties. There areessen a few recommended methods i n predicting properti e s of liquid mixtures tial changetophenomena, solelyto inphase reference measured data.though the predicted results should be verified Thermal conductivities of organi c liquid mixtures are in general less than either the mole or mass average values of the thermalprediction conductivities for pure components. Sincetothecompensate deviationsforaretheusually small, many methods have been developed deviations from suchand anmulticomponent interpolated value. Here are two methods suitable respectively for binary mixtures. A correlation for binary mixtures (Chen et al. 1987): Thermal conductivity of liquid mixtures.

where A ofandeachA2 components, are thermal conductivity of pure components, W I and W2 are mass fraction and C is a mixture constant. An optimum value of C determiAnnedaverage separately for value 159 dioffferent mixtures, and their values are given inwasa table. constant C 0. 3 75 provides a good fit to the 90% I

=


ofextended the reference data with an error less than However, this method cannot be to multicomponent mixtures. method for multicomponent mixtures (Li, A L L ¢i ¢jAij with and where i and is a jparameter characterizi ng theaverage interactions ofabove thermalequation conductivities between components. The harmonic in the is foundi superi o r to an arithmeti c or geometric average after an extensi v e investigation. is the superficial volume fraction and Vi is the molar volume of pure components.¢ Liquid mixture viscositiescomprising are usuallytheestimated onIn the basis of the viscosities of the pure components mixture. addition to puredatacomponent data, mosttosuch interpolati ng methods requiincluded re some experimental of mixture viscosity determine adjustable constants in their correlations. Teja and ofRicenonpolar-nonpolar, include a nonpolar-polar, single parameter,polar-polar, and their method is usable i r respective and aqueous mixtures. 3%.

A

1 977):

=

j

( 1 .3-8)

Ai)

Viscosity of liquid mixtures.

( 1 98 1 )

( 1 .3-9)

Here and Ejasare parameters defined for the mixture and for the pure components respectively Ve2/3 2 E = (MTe) 1/ el / 3 1 2 Ej = (MjV�Tci) / The viscosity values for the components TI l and Tl2 are to be evaluated not at the mixture T but at the reference temperatures equal to T (Te l / Te) and T (Ted Te).temperature The mixture parameters are defined by E

( 1 .3- 1 0)

( 1 .3- 1 1 )

( 1 .3- 1 2) ( 1 .3- 1 3)


Tc

j = --Vc ( V 1 + V 1.f3 ) = 8

( l .3- 1 4)

3 /3 are molecular mass, criticalintemperature, andis ancritical molar here of pureandcomponent wvolume respectively. Eq. interaction parameter that is assumed invariantviscosity. with composition and must be determined from experimental data of mixture There isinnobinary singlemixtures correlation satisfac tory for estimating the mutual diffusion coefficient as a function of mixture concentration. The well-tested and easily-applicable Vignes method is widelyAnrecommended. empirical correlation (Vignes, Vc ij

,

C]

Cl

Mi , Tci ,

Vci

( l .3- 1 5)

l/!ij

i,

( 1 .3- 1 4)

Diffusion coefficient in liquid mixtures.

1 966) :

( 1 .3- 1 6)

where relatedphase to theasactivity, ai , or activity coefficient,is a thermodynamic defined in Eq.correctionforfactor the liquid = ( aa InIn a;' ) ( aaInInX"�i ) = 1 + ( aa InIn : ) According Gibbs-Duhem for acoefficient binary mixture, is to bedilutethe same for eachto thecomponent. DI2 isequation the diffusion of an infinitely component mediumdilution consisting essentiallyAofcorrelation Severalbymethods for an evaluationdiffusing of DI2inatainfinite are developed. Wilke and Chang is widely utilized. It gives DI2 in cm2/s as ex

Yi ,

ex

x

T, P

=

( 1 .2-62)

y'

y' I

T P .

x

( 1 .3- 1 7)

T. P

ex

1

2.

( 1 955)

( 1 .3- 1 8)

Here, is molecular3 mass in g/mol, association factor, temperature in molar volume in em /mol at its normal boiling temperature, and viscosity in cPo The composition liquid mixture on the sur face i s not always the same as that of bulk, thethoughsurface the surface is not easily amenable to measurement.theThus, tension concentrati of a mixtureon M

Surface tension of liquid mixtures.

({J

T

K, V

TJ


is unpredi cmole table fracti as a function of thefractisurface tensioofnstheof pure components. Com pared to a o n or mass o n average surface tensions of pure components, surface tension of a mi x ture is lower in many cases, whi l e it becomes higher for other mixtures. Nonlinear characteristics of surface tension against the composition variation is more pronounced in aqueous mixtures of organic sub stances. Meissner and Michaels (1949) put forth a correlation for infinitely dilute aqueous mixtures: (l. 3 -19) where isawa iconstant s surfacecharacteristic tension of water, e fractiontheof organi cofcomponent, Xo is molcomponent, and of organic values which are 4 evaluated for twenty five compounds and range from 26 10for propyl alcohol 7 for n-decanoic acid. For mixtures containing more than one mole topercent 2. 5 of10-organic compounds, Eq.(1955) (l.3-9)proposed is no longer applicable. Tamura, Kurata, and Odani a method for aqueous mixtures of organic substances: l (l.3 -20) a /4 = 1/I� (aw) I /4 + 1/1% (ao )I/4 = (l.3-21) 1/1� + 1/1% 1 where subscripts w and represent water and the organic component, and a is thethe superfici 1/Ifractions alarevolume fracti ounder n in thean assumption surface layer.of equilibrium The surfacebetween volume 1/1 � and 1/1 % determi n ed the surface and bulk phases from( the following set of equations: B == log 1/Iw) (l.3-22) 1/10 Q (1/I1/Iwa _) Q C == log _ (l.3-23) g 2 a V /3 - awV�j3 ) W == 0.441 f ( � (l. 3 -24) (1. 3 -25) C=B+W Here, is inItsK,numeri and calis avalue constant depending on number the typeofandcarbon size ofatoms organiforc component. is speci fi ed as the acids andof carbon alcohols,atomsthe tinumber ofratiocarbon atomsvolume minusofonehalogen for ketone, andve thetofattyparent number m es the of molal deri v ati fatty acid for halogen derivatives of fatty acids. 1/1 is the superficial bulk a

x

x

0

T

q


volume fraction, i.e.,

l/fw

Xw Vw = ---V + V

=

Xw w

Xo o

Xu Vu

( 1 .3-26)

=

I whereMacleod is molandal Sugden' volume.s correlation (Reid et al. , l/fo

Xw Vw

+ Xo Vo

-

l/fw

V

1 987)

( 1 .3-27)

for nonaqueous mixtures:

( 1 .3-28)

where PI and Pv are mixture densities in the liquid and vapor phases, respectively, a temperature-independent constant of purenedcomponent and called theisparachor. The value of parachor is determi from surfacei, which tensionis data, if they are available; otherwise, it is estimated from an additive method of structural contributions of the molecule. [ Pd

REFERENCES Benedict, M., Webb, G. B., and Rubin, L. C. 1 940. An Empirical Equation for Thermodynamic Pro perties of Light Hydrocarbons and Their Mixtures. J. Chem. Phys. 8:3 34-345. Chen, Z. S., Fujii, T. , Fujii, M., and Ge, X . S . 1 987. An Equation for Predicting Thermal Conductivity of Binary Liquid Mixtures. Reports of Research Institute of Industrial Science, Kyushu Univ. 82: 1 59- 1 7 1 . De Santis, R., Gironi, F. , and Marrelli, L . 1 976. Vapor-Liquid Equilibrium from a Hard-Sphere Equation of State. lnd. Eng. Chem. Fundamentals. 1 5 : 1 83- 1 89. Gibbs, 1. W. 1 876. On the Equilibrium of Heterogeneous Substances. Trans. Connecticut Acad. 3: I 08248 . Kandlikar, S . G . , Bijlani, C. A., and Sukhatme, S . P. 1 975a. Predicting the Properties o f R-22 and R- 1 2 Mixtures-Thermodynamic Properties. ASHRAE Trans. 8 1 :266-284. Kandlikar, S. G., Bijlani, C. A., and Sukhatme, S. P. 1 975b. Predicting the Properties of R-22 and R- 1 2 Mixtures-Transport Properties. ASHRAE Trans. 8 1 :285-294. Lewis, G. N. 1 90 1 . The Law of Physico-Chemical Change. Proc. Am. Acad. Arts Sci. 37 :46-69. Lewis, G. N. and Randall, M. 1 92 1 . The Activity Coefficient of Strong Electrolytes. J. Am. Chem. Soc. 43 : 1 1 1 2- 1 1 54. Li, C. C. 1 976. Thermal Conductivity of Liquid Mixtures. AlChE J. 22:927-929. Meissner, H. P. and Michaels, A. S. 1 949. Surface Tensions of Pure Liquids and Liquid Mixtures. Ind. Eng. Chem. 4 1 :2782-2787. Nishiumi, H. and Saito, S . W. 1 975. An Improved Generalized BWR Equation of State Applicable to Low Reduced Temperamres. J. Chem. Eng. Japan. 8:356-360. Peng, D. Y. and Robinson, D. B. 1 976. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundamentals. 1 5 :59-64. Redlich, O. and Kwong, J. N. S. 1 949. On the Thermodynamics of Solution. Chem. Rev. 44:233-244.


Reid, R. c., Prausnitz, J. M., and Poling, B. E. 1 987. The Properties of Gases and Liquids, 4th ed., New York: McGraw-HilI. Scatchard, G . and Raymond, C. L. 1 938. Vapor-Liquid Equilibrium. II. Chloroform-Ethanol Mixtures at 35, 45 and 55°C. J. Am. Chem. Soc. 60: 1 278- 1 287. Starling, K. E. 1 97 1 . Thermo Data Refined for LPG. Part 1 : Equation of State and Computer Prediction. Hydrocarbon Processing. 3 : 1 0 1 - 1 04. Tamura, M., Kurata, M., and Odani, H . 1 955. Practical Method for Estimating Surface Tensions of Solutions. Bull. Chem. Soc. Japan. 28: 83-88. Teja, A. S . and Rice, P. 1 98 1 . The Measurement and Prediction of the Viscosities of Some Binary Liquid Mixtures Containing n-Hexane. Chem. Eng. Sci. 36:7- 1 0. van der Waals, J. D. 1 873. Over de Continuiteit van den Gas en Vloeist of Toestand. Diss., Univ. Leiden. van Laar, J. J. 1 935. Die Thermodynamik Einheitlicher Stoffe und Biniirer Gemische. Groningen: Verlag von P. Noordhoff N.V. Vignes, A. 1 966. Diffusion in Binary Solutions. Variation of Diffusion Coefficient with Composition. Ind. Eng. Chem. Fundamentals 5: 1 89- 1 99. Wilke, C. R. and Chang, P. 1 955. Correlation of Diffusion Coefficients in Dilute Solutions. AIChE J., 1 :264-270. Wilson, G. M. 1 964. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of M ixing. J. Am. Chem. Soc. 86: 1 27-1 30.

CHAPTER

TWO REPRESENTATION OF SOLID-LIQUID-VAPOR PHASE INTERACTIONS Masahiro Shoji The University of Tokyo, 7-3- 1 Hongo, Bunkyo-ku, Tokyo 1 13-8656, Japan

Yasuhiko H. Mori Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Shigeo Maruyama The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 1 1 3-8656, Japan

2.1 INTRODUCTION OF SOLID-LIQUID-VAPOR PHASE INTERACTIONS

Solid-liquid-vapor interaction phenomena play a very important role in phase change heat transfer. Except for the cases of direct contact heat transfer, most practi cal phase-change heat-transfer involve theto thesolidsurface surfaceis apparent as a heaterin or condenser. The importance of the lisystems quid wettability dropwise the condensation, boiling heat transfer, and capillary liquid forfilmsurfaces evaporation. Actually, heat transfer coefficient of dropwise condensation with low wettability is much larger than that of film condensation. The thin liquid film onsincethethepartlythermal wet surface can contri b ute consi d erably to the overall heat transfer, resistance ofof thethe liquid film due to heat conduction is inversely proporti o nal to the thickness liqui d film. In this secti o n, fundamental liquid solid contact phenomena are considered from the tradi t i o nal macroscopic and microscopic representations. The mechanistic and thermodynamic treatments ofthe traditional macroscopic app roach yield many practical concepts and empirical correlation equations. How ever, it is sometimes essential to understand the phenomena from the microscopic point of view.condition The contact linedynamics is a singular point in thesimply macroscopic sense, since theof thenon-slip offtuid at the surface denies the movement contact line. The "monolayer liquid film" considered in some macroscopic 41


theories needs further examination. Thus,simulations a brief overview of theandmicroscopic techniques and the molecular dynamics is presented, illustrative examples of simulations for a si m pl e molecular system fol l o w in the succeeding sections. Intheboth macroscopic and microscopic representations, the approachYoung starts from liquid-vapor interaction and the surface tension. The well-known Laplace equationto the[Eq.pressure (2.2-1)]direlates thecalled curvature of liquid-vapor interface and surface tension ff erence capillary pressure. From the equa tion, it is possible to obtain therepresentation geometry ofofthetheinterface once the equation pressure term is prescribed. The microscopic Young-Laplace is used inkinetic the next sectionderiforvedthefromevaluation of the surface tension,Another whichexample should beof the the property the molecular parameters. capability ofettheor amacroscopic representation isofthegravigravitational deformatiusing on ofthea liquid dropl curved interface. The effect t y can be expressed capillary length When the systemofsizthee ismicroscopic much smallerrepresentation than the gravitational effect is negligible. No counterpart is toopossible, since the system size that the molecular dynamics method can handle is small tosmaller show than the effect, or, i n other words, the system length scale L is always much capillaryinlength. In relation representation, one of the hotthearguments molecular dynamicsto thestudymacroscopic is the determination of the condensation coefficient, whi c h wi l l be discussed here. Aswettability is well-known, thesolidcontact angleTheis iwell-known ntroduced toYoung's representequation the degree[seeofEq.the partial of a surface. (2.2-5)] relates the contact to the balance This equation can be understood from theangle mechanical balanceofofsurface forcesenergies. or the thermodynamic concept of minimizing the Helmholtz free energy. Since it is usually difficult to independently measure the surface energy between solid and vapor and between liquid and ofsolid,the Young' s equation is stillto somewhat conceptual. Furthermore, the definition contact angle seems be controversi a l if the thin li q ui d film exists the "dry" surface. The microscopic discusses thisorpoint by theover simulation of molecular dynamics method.representation There are debates whether not Young' s equation could be verified if each surface energy can be calculated. There arestructure interesti(order ng features of liofquiadmolecule) structure and nearthethespread contactoflithene: first layered liquid of the size molecular layer. Regardless of such special holds beyond the structured area. structure, it seems that Young's equation still In macroscopic treatments, theperfectly effectivesmooth contactandangledefect-free. is introduced for the practical solid surface, which is not The concep tual statisticalheterogeneous equations representing effect ofroughness (Wenzel equation) and and chemically surface arethesurveyed, and the multiplicity of the a.

a,

REPRESENTATION OF SOLID-LIQUID-VAPOR PHASE INTERACTIONS 43

contact angle (metastable angle) is described. The phase change heat transfer oc curs evaporati on orline:the thecondensation process, which usuallyor theaccompa nies thethrough movingtheofthe contact decay ofliquid by evaporation growth oftactliquid by condensation. Hence, the non-equilibrium feature of the movi n g con linetactis crucial for the heat transfer. Furthermore, reliable techniques to measure angle iinvolve the movement oftthey ofcontact line. line It is wellWeknown that con angle e contact thethadvancing s a function of the veloci the contact define the condition 0) when the contact line is moving in the direction from liquid to vapor andlimithet ofrecedin0gforcondition 0) for the opposite. It is very in teresting that the advancing condition contact angle) and that for receding conditions (receding contact(called angle)advancing do not coincide. The From contacttheangle remembers its movingstudies historysurveyed (calledhere, contact angle hysteresis). extensive macroscopic it is believed that theng movi n g contact li n e shows the range of angles between advancing and recedi due to the metastable contactheterogeneity. angle directlyHowever, related tothetherelati surfaceon conditions, such asparent roughness and chemical between the ap contact angle (dynami c contact angle) to the movi n g velocity is not well the macroscopic on technithatques.insistOnonthetheother hand, thereof areestablished reports ofin molecular dynamicscorrelati calculations reproduction theicallydynamic contact angle, even thoughontheis still surfaceopenis toperfectly and chem homogeneous. This contradicti question.smooth One possibility is that thecontact microscopic indicates a new breakthrough the concept of the dynamic angle.result However, the system size of typicalformolecular dynamics simulation ispotential so small that theAscrystal of solid molecules may be felt asdiscussions the period ically rough field. a result, some analogous or similarity torepresentation. the macroscopic contactan angle be possible in the microscopic Evendynamic though such analogymaymight help to understand the real microscopic phenomena to some extent, readers should be careful that it might simply be an analogy. Whenanglethe contact-line speed is increased for advancing condition, the dynamic contact generally increases until it reaches 1 800 • Further increase in the advancing speed beyond this critical speed induces a macroscopic saw-tooth instability of thethecontact linangle e. Theissame instability is observed for the receding condi t ion after contact decreased to 00 • It seems that the shape of the contact-line is adjusted so that the velocity component normal to the curved must be genuine kept at the and criticalno counterpart speed inSuchthe instability mCoacrntactoscline opicisphenomena, microscopic representation exists. If you force the similar condition in the microscopic molecular dynamics �imulation, you may find another kind of instability, which is never experienced the macroscopic system. V.

(V

>

V

(V

- + f3 2 where f3 is the half wedge angle. Thus, according to Eq. (4.3 .3), conical cavities present on a hydrophilic surface will not trap gas. In a more recent work, Wang and Dhir ( 1 993a) have developed the gas/vapor entrapment criterion by minimizing the Helmholtz free energy of a system involv ing the liquid-gas interface in a cavity. According to this criterion, a cavity will trap gas if ¢ > 1frmi n

(4.3-4)

where 1frmin is the minimum cavity side angle of a spherical, conical, or sinusoidal cavity. For spherical and conical cavities, 1frmin occurs at the mouth of the cavity and is equal to the cavity mouth angle, 1frm . For a sinusoidal cavity, the minimum side angle, l/Imi n , occurs at a location where the radius of the cavity is equal to half of the cavity mouth radius. It should be noted that for a conical cavity, the criterion given by Eq. (4.3-4) is the same as that given by Eq. (4.3-3) of Ward and Forest ( 1 976), and liquids with a contact angle less than 900 will not provide a stable/gas vapor embryo for nucleation. According to Eq. (4.3-4), hardly any pre-existing gas/vapor nuclei are possible for well wetting liquids, such as R-1 1 3 and FC-72. Thus, for these liquids, the wall superheat at nucleation can approach the homogeneous nucleation temperature. The observed inception superheats for these liquids, though much higher than those observed for partially wetting liquids, are much smaller (see Barthau, 1 992) than those corresponding to homogeneous nucleation temperature. It is possible that I A distinction must be made between an advancing and receding contact angle. See Chapter 2 . However, in the discussion presented here, a static contact angle, cp, is used.

NUCLEATE BOILING 81

gases dissolved in these liquids serve to initiate the nucleation and, as a result, the observed superheat is smaller than the homogeneous nucleation. After inception, the bubbles at active sites can momentarily displace the liquid from the neighboring cavities that are flooded, or the flooded cavities simply dry out during the bubble growth period. Subsequently, these latter cavities become active nucleation sites and remain so, as long as sufficient wall superheat is maintained. The sudden activation of many additional sites leads to improvement in heat transfer and, in turn, to reduction in wall temperature at a fixed heat flux (hysteresis).

4.3.2 Inception Griffith and Wallis ( 1 960) proposed that incipient superheat for boiling from pre existing nuclei corresponds to the minimum radius of curvature of the interface. The minimum radius of curvature of the interface was assumed to be equal to the radius of the cavity mouth. By replacing the pressure difference between the vapor bubble (no gas) and liquid with the liquid superheat through the use of the Clausius Clapeyron equation, they obtained an expression for inception superheat as

�T

=

4a T�at

Pvhtg Dc

(4.3-5)

where Dc is the cavity mouth diameter. It should be noted that Eg. (4.3-5) includes neither the effect of contact angle on inception superheat nor the effect of temper ature gradient that exists near a heated wall. Subsequently, Hsu ( 1 962) studied the effect of temperature profile adjacent to the heated surface on the minimum super heat needed for nucleation. In developing his model, Hsu proposed that the top of a bubble embryo should be covered with warm liquid before it can grow. Since vapor in the embryo must be at saturation temperature corresponding to the pressure of vapor in the bubble (which is higher than the pool pressure by 2a /R), the liquid surrounding the bubble must be superheated to maintain the thermal equilibrium. If the required superheat does not exist, the heat transfer to colder liquid will cause the bubble embryo to shrink. Because heat is transferred from the wall, the liquid tem perature decreases with distance from the wall, and the above criterion is satisfied everywhere around the embryo, if the temperature of the liquid at the tip of the em bryo is equal to the saturation temperature corresponding to pressure in the bubble. Figure 5 shows how the criterion can be satisfied for a bubble embryo when the wall heat flux is gradually increased. In plotting the temperature, a fictitious film thickness, 8, is defined, which with a linear temperature profile gives the same heat flux at the wall as the actual thermal layer, 8 t h • Also, it is assumed that 8 is independent of the temperature difference across the layer. During natural convection this is not a correct assumption. However, in natural convection, the dependence of film thickness on temperature difference is weak. The temperature profile traced by the solid line marks the wall heat flux at which Hsu's criterion is


y

�6.T

Tb-Tsal =Tl-Tsal

Figure 5

\\

'\

=Tw-Tsal

�

T

Hsu 's criterion for inception of a heated surface.

just satisfied. If the cavity for which the criterion is satisfied is available on the surface (i.e., is not completely filled with liquid), this cavity will be the first one to nucleate. The lowest superheat at which nucleation is possible can be determined from the expressions for the temperature distribution in the film and the liquid temperature determined from the equilibrium considerations. If the bulk liquid is saturated, the temperature profile in the liquid film is given by

Tf - Tsat = (Tw - Tsat) ( l - y / 8 )

(4.3-6)

The vapor bubble thermal equilibrium condition is written as

Tf - Tsat

=

2a Tsat RbPv h fg

(4.3-7)

If the vapor-liquid interface of the vapor bubble embryo just occupying the mouth of the cavity is considered to be part of a sphere, the interface radius of curvature and the height of the bubble tip can be written as

Rb

=

11 Rc

and

Yb

= h Rc /11

(4.3-8)

when 11 and h are functions of the contact angle, ¢. Differentiating Eqs. (4.3-6) and (4.3-7), with respect to the distance, y , normal to the heater surface, and equating the two derivatives d Tf / d y, we obtain the radius of the nucleating cavity as

Dc Rc = 2

=

[

]

1 /2 2 1?a Tsat 8 hPv h fg (Tw - Tw t )

(4.3-9)

NUCLEATE BOILING 83

If the wall superheat is replaced with wall heat flux, the expression for the cavity diameter, becomes

Dc,

TsatAe l J/2 Dc - [8fia hPvhegfJ _

(4.3- 1 0)

Upon substitution for Rb in terms of the diameter of the nucleating cavity given by Eq. (4.3- 1 0) into Eq. (4.3-7) and elimination of the liquid temperature between Eqs. (4.3-6) and (4.3-7) after y is replaced by Yb in Eq. (4.3-6), an expression for the minimum wall superheat for nucleation is obtained as

Te

Tsat �T Tw - Tsat 8ha Pvhego

h

(4.3- 1 1 )

= ---

==

equal to 1 , a comparison of wall superheats given by Eqs. (4.3-5) and (4.3- 1 1 ) suggests that the cavity diameter corresponds to half the thermal layer

For

thickness. An expression for the heat flux corresponding to the wall superheat given by Eq. (4.3- 1 1 ) is obtained as

q

=

hegPvAe 8ha Tsat (� T)2

(4.3- 1 2)

To determine the range of cavities that can nucleate if the wall superheat exceeds the minimum wall superheat for nucleation, elimination of liquid temperature be tween Eqs. (4.3-6) and (4.3-7) yields a quadratic equation in terms of the diameter of the cavity as

(4.3- 1 3) or

(4.3 - 1 4) In Figure 5, the intercepts of the dotted line representing temperature distribution in the thermal layer represent the range of cavities between De l and that can nucleate. In terms of the heat flux, Eq. (4.3- 1 3) can be written as

Dc2

� T . Dc,

8fJ2a TsatAe hPvheg D�

(4.3- 1 5)

For a given Eq. (4.3- 1 5) represents an equation in two unknowns, namely q and To solve explicitly for both, another relation between q and is needed. This relation is provided by the correlations for single phase natural convection or

�T


forced convection that may be present on the heater surface prior to inception. In terms of the wall superheat, Eq. (4.3- 1 3) can also be written as

4fl a Tsat �T =

(4.3- 1 6)

For Dc much smaller than 8, the size of a nucleating cavity varies inversely with wall superheat, as was the case for Eq. (4.3-5). As an alternative to Hsu's criterion, Mizukami ( 1 975), Nishio ( 1 985), and more recently, Wang and Dhir ( 1 993a), have proposed that the instability of vapor nuclei in a cavity detennines the inception superheat. The vapor nucleus is stable if the curvature of the interface increases with an increase in vapor volume or if it decreases with a decrease in vapor volume. Otherwise, the vapor bubble embryo or nucleus is unstable. If K is the nondimensional modified curvature of interface, the stability criterion is written as

dK d V*

-- > 0

(4.3- 1 7)

where V* is the dimensionless volume of the embryo. The minimum wall superheat required for nucleation will correspond to the value of K for which the interface is unstable. Figure 6 shows, for a spherical cavity with mouth angle of 30° , the nondimen sional modified curvature of the interface as a function of the non-dimensional volume of the vapor bubble nucleus. The plotted results are for different contact angles and in the absence of any non-condensable gas. The curvature of the liquid vapor interface is negative for V* just less than that corresponding to point A and is positive for V* larger than that for point A. The metastable position of entrapped vapor-liquid interface for a contact angle larger than the cavity mouth angle is represented by point A. The volume of trapped vapor is equal to the volume of the cavity Vc, so that V* corresponding to point A is equal to 1 . Outside of the cavity, the interface is in stable equilibrium from point A to point D because K increases with V*. The maximum value of K occurs at points B, C, and D for contact angles of 1 50° , 1 20° and for a contact angle equal to or less than 90° , respectively. The values of Kmax can be detennined from the geometric shapes shown for cases ( 1 ) and (2) i n the inset of Figure 4 and can b e expressed as

Kmax = 1 = sin ¢>

for ¢> < 90° for ¢> > 90°

(4.3- 1 8)

Nucleation occurs beyond the point where the non-dimensional modified curvature

NUCLEATE BOILING 85

1 . 5 ------�--�--__, Spherical cavity : 1m = lO·

-1

. 9---------�--------------------------... 1 .2 • 8

Figure 6 Dimensionless modified curvature as a function of dimensionless volume of vapor bubble nucleus in a spherical cavity.

has its maximum value. As such, the incipient superheat is obtained as !':::i T

=

4a Tsal Kmax h P v Dc

(4.3- 1 9)

19

The above expression of wall superheat at nucleation is obtained under the assump tion that cavity diameter, Dc, is much smaller than the thermal layer thickness. Thus, in this limit, Eq. (4.3- 1 6) suggests that II is unity for ¢ < 90° and is equal to sin ¢ for ¢ :::: 90° . Or, for non-wetted surfaces, the required superheat is smaller than that given by Eq. (4.3-5) of Griffith and Wallis. Wang and Dhir ( 1 993a), from their experiments on surfaces with different contact angles, have shown the general validity of Eq. (4.3- 1 9).

4.4 BUBBLE DYNAMICS AND HEAT TRANSFER After inception, a bubble continues to grow (in a saturated liquid) until forces causing it to detach from the surface exceed those pushing the bubble against the wall. After departure, liquid from the bulk fills the space vacated by the bubble, and the thermal layer at and around the nucleation site reforms (transient conduction). When the required superheat is attained at the tip of the vapor bubble embryo or the interface instability criterion is met, a new bubble starts to form at the same


nucleation site, and the process repeats. Wall heat transfer in nucleate boiling re sults from natural convection on the heater surface areas not occupied by bubbles and from transient conduction and evaporation at and around nucleation sites.

4.4.1 Bubble Dynamics B ubble dynamics includes the processes of bubble growth, bubble departure, and reformation of the thennal layer (waiting period). In the following, each one of these processes is described separately. 4.4.1.1 Bubble growth. Generally, two points of view with respect to the growth of a bubble on a heated surface have been put forth in the literature. One group of investigators has proposed that the growth of a bubble occurs as a result of evaporation all around the bubble interface. The energy for evaporation is supplied from the superheated liquid layer that surrounds the bubble since its inception. Bubble growth models similar to that proposed for growth of a vapor bubble in a sea of superheated liquid, such as that of PIes set and Zwick ( 1954), have been proposed. The bubble growth process on a heater surface, however, is more complex because the bubble shape continuously changes during the growth process, and superheated liquid is confined to only a thin region around the bubble. Mikic et al. ( 1 970) used a geometric factor to relate the shape of a bubble growing on a heated surface to a perfect sphere, properly accounted for the thennal energy stored in the liquid layer prior to bubble inception, and obtained an analytical solution for the bubble growth rate. Their expression for the growth rate is

dD * dt*

--

=

where

D*

[

2 t* + 1 - 0

(

t*

--

t* + t�

)

t*

_ t * I/ 2

( bPvPe�TTsath eg ) ( 1 2Ke J} ) b = Pv �TT h eg ( 1 2 J ) Pe sat Ke } 1/ 2

=

]

1/ 2 1/2

rr D

rrt

0=

and

Ja

=

(4.4- 1 )

(4.4-2) (4.4-3)

Tw - Tb �T

(4.4-4)

Pe Cpe � T Pv h eg

(4.4-5)

---

In Eq. (4.4-4), Tb is the liquid bulk temperature, and in Eqs. (4 . 4-2) and (4.4-3), b is a geometric parameter that has a value of 2 / 3 for a perfect sphere. A value of rr / 7 for b has been suggested by Mikic et al. for a bubble attached to the heater surface.

NUCLEATE BOILING 87

The second point of view is that most of the evaporation occurs at the base of the bubble, in that the micro-layer between the vapor-liquid interface and the heater surface plays an important role. Snyder and Edwards (1 956) were the first to propose this mechanism for evaporation. Subsequently, Moore and Mesler ( 1 96 1 ) deduced the existence of a microlayer under the bubble from the oscillations in the temperature measured at the bubble release site. Cooper and Lloyd ( 1 969) further confirmed the existence of the micro-layer. Using the lubrication theory for the liquid flow in the micro-layer, they deduced the average thickness of the microlayer underlying a bubble as

0 = 0. 8 00

The contribution of the micro-layer to the total heat transfer rate depends on the area occupied by the micro-layer. As such, the empirical constant in the above equation for the average micro-layer thickness may not be realistic for bubbles formed on surfaces with widely varying contact angles. A complete theoretical treatment of the evolution of the micro-layer during bubble growth still does not exist in the literature. 4.4.1.2 Bubble departure. The diameter to which a bubble grows before de parting is dictated by the balance of forces acting on the bubble. These forces are associated with the inertia of the liquid and vapor, liquid drag on the bubble, buoyancy, and surface tension. Fritz ( 1 935) correlated the bubble departure diameter by balancing buoyancy, which acts to lift the bubble from the surface, with the surface tension force, which tends to hold the bubble to the wall, so that

(4.4-6) where ¢ is the contact angle measured in degrees. Though significant deviations of the bubble diameter at departure with respect to the above equation have been reported in the literature (especially at high pressures), Eq. (4.4-6) does provide a correct length scale for the boiling process. Several other expressions that are obtained either empirically or analytically by involving various forces acting on a bubble have been reported in the literature for bubble diameter at departure. These expressions (see, e.g., Hsu and Graham, 1 976), however, are not always consistent with each other. Some investigators report an increase in bubble dia meter at departure with wall superheat, whereas others find the bubble diameter at departure to be insensitive to or decrease with an increase in wall superheat. One reason for this discrepancy could be the merger of bubbles that occurs at high heat fluxes. Cole and Rohsenow ( 1969) correlated the bubble diameter at departure at


low pressures as

Dd

=

and

Dd

=

4.65

where

X

5 /4 a 4 la* \ .5 10 J g ( Pe - Pv ) 5/4 a 4 la* 10 J X

-

for water

(4.4-7)

for other liquids

(4.4-8)

-

g ( Pe - Pv )

Ja*

=

Pee pe Tsat Pv h eg

(4.4-9)

Gorenflo et al. ( 1 986) have proposed an expression for bubble diameter at departure at high heat fluxes as

Dd

=

/

1 /3 [ ( 21C ) 1 /2]4 3 4 ) ( Ja 7 A 1+ 1+ CI --

g tg ,

-

3Ja

(4.4- 1 0)

Different values of the proportionality constant were suggested for different liquids. Knowing the growth rate and the diameter to which a bubble grows before de parting, the growth time, can be calculated. After bubble departure, cooler bulk liquid fills the space vacated by the bubble. Schlieren pictures by Hsu and Graham ( 1 976) show that an area, about two times the bubble diameter at departure, is influ enced by bubble motion. As a result, the thermal layer reforms over an area of a cir cle of diameter 2Dd surrounding the nucleation site. A new bubble at this location will now grow until the superheated liquid layer is reestablished and the inception criterion is satisfied. The time taken by the thermal layer to develop prior to incep tion is termed the Han and Griffith ( 1 965) obtained an analytical expression for the waiting period by assuming the liquid layer to be semi-infinite and by substituting the expression for the thermal layer thickness in Eq. (4.3- 1 6) as

waiting period.

[

tw 2J1CKeII [1h-Dc 4Il a 1 �TPv h eg Dc =

Tosat

]

2

(4.4- 1 1 )

It is seen from Eq. (4.4- 1 1 ) that the waiting time will first decrease and then increase with cavity size. However, it will continuously decrease as the wall superheat is increased. 4.4.1.3 Bubble release frequency. Conceivably, a theoretical evaluation of the bubble release frequency can be made from expressions for the waiting time, and the growth time, tg (the growth time can be determined by knowing the growth

tw ,

NUCLEATE BOILING 89

rate and bubble diameter at departure). In fact, such an approach meets with little success when a comparison is made with the data. Some of the reasons for the discrepancy are the following: 1 . Evaporation takes place at the base and at the surface of the bubbles, and the growth models reported in the literature generally do not account for both. 2. Generally, large cavities yield large bubbles. This in turn drastically alters the growth time from cavity to cavity. 3. Bubble activity, heat transfer, and fluid motion in the vicinity of an active site can influence the growth pattern as well as the waiting period. 4. Bubble shape continuously changes during the growth period. Thus, correlations have been reported in the literature that include both the bubble diameter at departure and the bubble release frequency. One of the most compre hensive correlations of this type is given by Malenkov ( 1 97 1 ). According to this correlation, the product of bubble release frequency, f , and diameter at departure are correlated as f Dd

where

=

Vd ----( ---'-- --------- ) --'-Jr

1 -

1

1 +

(4.4- 1 2)

Vd Pv h eg /q

(4.4- 1 3)

4.5 NUCLEATION SITE DENSITY As the wall superheat or heat flux is increased, the number density of sites that become active increases. Since adding new nucleation sites influences the rate of heat transfer from the surface, knowledge of nucleation site density as a function of wall superheat is necessary if a credible model for prediction of nucleate boiling is to be developed. In the earlier studies, listed by Hsu and Graham ( 1 976), it was noted that the density of active nucleation sites increases approximately as the square of the heat flux or as the 4th to 6th power of wall superheat. In the listed studies, no attempt was made to relate either the proportionality constant or the exponent with the surface characteristics. Kocamustafagoullari and Ishii ( 1 983) have correlated the cumulative nucle ation site density reported by various investigators for water boiling on a variety of surfaces at pressures varying from 1 - 1 98 atm as (4.5- 1 )

90 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSAnON

where

2

(4 5 - ) .

and

F(p*) 2 1 5 10-7 p * 3 2 0. 0049p*)4. 1 3 Dd (4.4-6) 0.001 2 (p*)o.9 . p* = Pe Pv- Pv =

.

7

X

-

.

(4.5-3)

(1 +

In the above equations, is the bubble diameter at departure and is obtained by multiplying Eq. of Fritz ( 1 935) by The parameters and Dc are defined as

Dc = 4a [ I + ( pe/Pv ) ]/ Pe ' { exp [h f (Tv - Tsat ) / (R v Tv Tsat) ] g

p*

(4.5-4) -

I}

(4.5-5)

In Eq. (4.5-5), Tv is the temperature of vapor, PI is the liquid pressure, and R v is the gas constant for vapor. At moderate pressures, Eq. (4.5-5) reduces to Eq. (4 .3- 1 6 when cavity diameter is assumed to be much smaller than the thickness of the thermal layer. Cornwell and Brown ( 1 978) have made a systematic study of active nucleation sites on copper surfaces during the boiling of water at 1 atm pressure. Their study was limited to low heat fluxes and to surface conditions ranging from a smooth to a scratched rough surface. From their work, it was concluded that the active site density varied with wall superheat as

)

,

(4.5-6) The proportionality constant in Eq. (4.5-6) was found to increase with surface roughness, but the exponent on t1 T was independent of surface roughness. Mikic and Rohsenow ( 1 969a) have proposed that on commercial surfaces, the cumulative number of active sites per unit heater area can be assumed to vary in partial nucleate boiling as

(4.5-7) where Ds is the diameter of the largest active cavity present on the surface and m is an empirical constant. The size, Dc, of a cavity that nucleates at a wall superheat t1 T is obtained from Eq. (4.3-5). Bier et al. ( 1 978), on the other hand, have deduced an expression for active site density from heat transfer data as i n N. = i n Nm"

[

1

-

(�:) m]

(4.5 -8)

In Eq. (4.5-8), Nmax is the maximum value of Na , which occurs at Dc = O. The value of m was found to depend on the manner in which a surface was prepared.

NUCLEATE BOILING 91

With Freon 1 1 5 or Freon 1 1 boiling on a chemically etched copper surface and on a turned surface, values of 0.42 and 0.26, respectively, have been noted for m . This observation is not consistent with the conclusion of Cornwell and Brown ( 1 978). In the heat transfer experiments, the reduced pressure was varied from 0.0037 to 0.9 . It was found that to correlate the data at low and high saturation pressures, some changes in the functional form of Eq. (4.5-8) are necessary. Wang and Dhir ( 1 993a, 1 993b) have studied the effect of surface wettability during boiling of saturated water at one atm pressure on mirror finish copper surfaces. They have also provided a mechanistic approach for relating the cavities that are present on the surface to the cavities that actually nucleate. The number density of all types of cavities was found to correlate as Ns (sites/cm2 )

1 03 D; 2. 0 1 0.3 + 2.4 x 1 06 D; 5 . 2 22 1 4 + 1 .0 x 1 06 D; 5 . 4

= 9.0

x

Dc 3.5

�

�

Dc :::::

5 . 8 /Lm Dc ::::: 5 . 8 /Lm 3 .5 /Lm

(4.5-9)

In Eq. (4.5-9), Dc is measured in /Lm. From these expressions, the cumulative number density of reservoir cavities (these were cavities that were proposed to nucleate) with mouth angle, l/Im, less than 900 was correlated as (4.5- 10) Wang and Dhir ( 1 993a, 1 993b) have also noted that for spherical cavities, the number density of cavities with mouth angle less than l/I� can be approximately written as (4.5- 1 1 ) Since for cavities to trap gas/vapor, the cavity mouth angle l/Im should be less than the contact angle, the cumulative number density of cavities that become active can be written as (4.5- 1 2) where (4.5- 1 3) To obtain the cumulative number density of active sites as a function of wall Superheat, Eq. (4.5- 1 2) must be corrected to Na (¢ ,

�T)

=

Na (¢)

Kmax

(4.5- 1 4)

Where Kmax is given by Eq. (4.3- 1 8). According to Eq. (4.5- 1 0), the active nu cleation site density should vary as � T 5 .4 . Figure 7 shows a comparison of the


1 0

N E

4

1 0

u ....... II) Q)

1 0

II) '-" c z

-0 C 0

C op p e r / W a t e r ,

3

--

1 0

II) ......, c Z

-0 c 0

\/I

c z

N E

u ....... II) Q)

II) '-' c Z

-0 c 0

1/1 c Z

by Eq. (4.5-1 2) 0 : No s (lf < 9 0o )

e : No ( ep= 9 0 0 )

1 0

1

°

• ( a ) ¢= 9 0 °

1 0

u ....... II) Q)

aIm

1

2

\/I C Z

N E

: No ( ¢= 9 0° )

1 0

1 0

5

4

Co p p e r/W a t e r ,

3

---

.-------

1

0

1 0

-1

0

1 0

( b ) ¢ = 3 5°

C o p p e r /W a t e r ,

3

---

. -- - - - - -

2 1 0

1

: No s (1/Im < 9 00 )

2

aIm

: N o ( ¢= 1 8° ) by Eq. (4. 5- 1 2)

. : No

(ep= 1 8° )

0: No s (1/Im < 9 0 0 )

1

0

1 0

1 0

5

4

1 0

1

0 : No s (lfim < 9 0 o )

,

1 0

1

aIm

1

0o ) : N o s < lfm < 9 : No ( ¢= 3 5 ° ) by Eq.(4 . 5- 1 2)

. : No (ep= 3 S o )

02

2

0

-1

1 0

-2 1 0

2

Figure 7

5

1 0

C a v i t y D i a me t e r , Dc (J-I-m )

Comparison of the predicted and the measured active site density for contact angle�

35, and 1 8 ° .

NUCLEATE BOILING 93

measured and predicted number density of active sites for different contact angles. Wan g and Dhir have also found the average nearest neighbor distance, .e, to corre late with nucleation site density as

.e =

-0.84

�

(4.5- 1 5)

4.6 HEAT TRANSFER IN NUCLEATE BOILING 4.6.1 Mechanistic Models In partial nucleate boiling or the isolated bubble regime, transient conduction into liquid adjacent to the wall is probably the most important mechanism for heat removal from an upward facing horizontal surface. After bubble inception, the superheated liquid layer is pushed outward and mixes with the bulk liquid. The bubble acts like a pump in removing hot liquid from the surface and replacing it with cold liquid. This mechanism was originally proposed by Forster and Greif ( 1 959). Combining the contribution of transient conduction on and around nucleation sites, micro-layer evaporation underneath the bubbles, and natural convection on inactive areas of the heater, an expression for partial nucleate boiling heat flux is obtained as

. (A

K q. = Tf y/Jr pCp ) €: f Dd2 Na D.A T + _

Jr + aev � TNa 4 Dd2

(

K f Na Jr Dd2 1 - T

) ane- D.A T

(4.6- 1 )

Only the first two terms in the above equation were included in the original model proposed by Mikic and Rohsenow ( 1 969a). The evaporation at the bubble boundary is included in the first term, which represents the transient conduction in the liquid. The addition of the last term on the right hand side of Eq. (4.6- 1 ) was suggested by Judd and Hwang ( 1 976). This term accounts for the microlayer evaporation at the base of bubbles. For Eq. (4.6- 1 ) to serve as a predictive tool, the following must be known: the bubble diameter at departure, Dd ; bubble release frequency, f ; the proportionality constant, K 1 , for the bubble area of influence; number density, Na , of active sites; and average heat transfer coefficients, a ile and ae v , for natural convection and microlayer evaporation, respectively. Using empirical correlations for several of these parameters, Mikic and Rohsenow (1 969a) justified the validity of Eq. (4.6- 1 ) when the third term on the right hand side was not included. Judd and Hwang ( 1 976), in matching the heat fluxes predicted from Eq. (4.6- 1 ) with those observed in experiments in which dichloromethane was boiled on a glass surface, relied on the measured values of micro-layer thickness to evaluate aev and on the assumption that K had a value of 1 .8. Additionally, experimentally

f


K

�

=

1 .8

"'i: c.. 0X ;::l

1 04

f;:;

.... ell QJ

:c Natural Convection qNC

Microlayer Evap oration qME

.. � .. 103 L-..___L-.._-"-___-"--"-........ 104

Figure 8

Heat Flux qM

(W 1m2 )

Relative contribution of various mechanisms to nucleate boiling heat flux (Judd and Hwang,

1 976).

measured values of active nucleation site density and bubble release frequency were used in the model. Figure 8 shows their data and predictions. It is seen that at the total measured heat flux of 6 w/cm2 , about one third of the energy is dissipated through evaporation at the bubble base. The data plotted in Figure 8 show that at high heat fluxes or in fully developed nucleate boiling, most of the energy from the heater is removed by evaporation. A similar observation can be made from the measurement of Paul and Abdel-Khalik ( 1 983) and Ammerman et al. ( 1 996) on horizontal cylinders. The results of all of these studies are in general agreement with the findings of Gaertner ( 1 965) that after the first transition, evaporation is the dominant mode of heat transfer. According to Gaertner, most evaporation occurs at the periphery of vapor stems. Energy for the phase change is supplied by the superheated liquid layer in which the

NUCLEATE BOILING 95

stems are implanted. Thus, the boiling heat flux can be calculated if the fractional area occupied by the vapor stems and the thickness of the thermal layer are known. The h eater area fraction occupied by the vapor stems is equal to the product of the number density of stems and the area occupied by one stem. Alternatively, the heat flux can also be calculated if the vaporization rate per stem and number density of active sites are known. Lay and Dhir ( 1 994) have used the latter approach to predict the fully developed nucleate boiling heat flux. By assuming that the duration for which vapor stems exist on the heater is much larger than the time needed to form the stems, Lay and Dhir have carried out a quasi-static analysis to determine the maximum diameter of vapor stems as a function of wall superheat. The shape of the vapor stem was found to depend on the value that was chosen for the Hamaker constant. From the analysis, the vaporization rate per stem could be calculated as a function of wall superheat. Using the data of Wang and Dhir ( 1 993a, 1 993b) for den sity of active sites, they found the predicted heat fluxes to show good agreement with the data. This approach needs to be verified with data from other sources.

4.6.2 Correlations The augmented version of the model by Mikic and Rohsenow requires the use of several empirical constants, and it is not in a form that it can be readily used to predict nucleate boiling heat flux as a function of wall superheat. Instead, the earlier correlation of Rohsenow ( 1 952), which involves the use of three empirical constants, has enjoyed wide popularity. According to this correlation, the func tional dependence of partial nucleate boiling heat flux on wall superheat can be written as

qJgJleheg (PI�p")

=

C; J

( Cpe!::l.heg T ) 3 (JLecpe ) n l Al

(4.6-2)

C where s depends on heater material and fluid combination. The exponent, n I , has a value of 3 .0 for water and 5 . 1 for all other liquids. It should be noted that since gravity did not vary in the data that was used to develop Eq. (4.6-2), the parameter, should be considered as a dimensional constant. Also, in arriving C at the correlation (Eq. [4.6-2]), no attempt was made to relate the values of s to surface conditions (e.g., roughness or cleanliness), but the correlation was shown to C be applicable at different system pressures. Table 1 gives values of s for different fluid-solid combinations. From Eq. (4.6-2), it can be seen that the magnitude of heat flux at a given wall C superheat is very sensitive to the values of Cs . A factor of about two change in s can cause almost an order of magnitude change in heat flux. As such, the values given in Table 1 must be considered as a guide and, for a particular application, it is

g,


Table 1 Values of constant Cs in Rohsenow's correlation Fluid

Solid

Water Water Water Water Benzene Carbon tetrachloride Ethanol Isopropanol n-Butanol N-pentane

Nickel Platinum Copper Brass Chromium Copper Chromium Copper Copper Chromium

Cs

0.006 0.0 1 3 0.0 1 3 0.006 0.0 1 0 0.0 1 3 0.0027 0.0025 0.003 0.0 1 5

always wise t o determine the value o f constant Cs experimentally for conditions of particular interest. Although the correlation was derived for partial nucleate boiling, it has generally been successfully extended to fully developed nucleate boiling. Liaw and Dhir ( 1989) have systematically studied the effect of wettability of a polished copper surface on nucleate boiling of saturated water at one atmosphere pressure. Their data show that the empirical constant Cs is a function of the contact angle, and it increases with a decrease in contact angle. Table 2 gives the values of constant Cs for different static contact angles. Recently, Stephan and Abdelsalam ( 1 980) have developed a comprehensive correlation for saturated nucleate pool boiling of different liquids. In develop ing these correlations, they have divided the liquids into four groups: (i) water, (ii) hydrocarbons, (iii) cryogenic liquids, and (iv) refrigerants. In these correlations, dimensionless heat transfer coefficients (Nusselt numbers) were written in terms of several dimensionless parameters that depend on fluid and solid properties. The important fluid property groups were identified through regression analysis, and the values of exponents of the property groups were obtained by matching pre dictions with the data. In developing the correlations, data from different heater Table 2 Dependence of Cs on contact angle Contact angle 1 4° 27° 38° 69° 90°

Cs

0.0209 0.0202 0.0 1 94 0.0 1 86 0.0 1 72

NUCLEATE BOILING 97

geometries (such as fiat plates, horizontal cylinders, vertical cylinders, etc.) were used. Also, a mean surface roughness of mm was assumed for the heaters. Their correlations for different liquids are:

1

Water: for

Hydrocarbons:

. X i673 . Xi 1 .58 . X� ·26 . X� ·22 Nu10-=4 ::s2.46p c 10::s60.886 ¢ = 45°

(4.6-3)

Nu5 .7 = 010-.05463 ::s (X�·5c .::sX0.90.67 . X:·33¢. X=�35°·248

(4.6-4)

X

iP

and

I)

pi P Cryogenic Liquids: 24 . X�·1 17 . X�·257 . X�·374 . XiO.329 Nu4.0 = 410-.823X::s?·6plpc ::S 0.97 ¢ = 1 ° Refrigerants: Nu3.0 =x I0-2073X::s?·74pl5 . Xc �::s·581O..7X8�·533 ¢ = 35° P ± 15 X X8 for

for

and

X

(4.6-5)

and

X

for

(4.6-6)

and

All of the above equations correlate the data within The Nusselt number and various dimensionless groups

Nu =

q Dd � TA e q Dd A e T.�at K; Pe a Dd cp e Tsat D� Ke h eg D� Kt

•

•

•

-

X 1 = -X2 =

% mean absolute error. are defined as I

X5 = -PvPe

--

X3 = 2 X4 = --2-

X7 = ((pCpA)s P C p A) e X8 = Pe P-t Pv

---=----

---

Stephan and Abdelsalam have also given a generalized correlation that is applicable to all liquids, but has a larger mean absolute error:

Nu = 0.23 X?·674 . X�·297 . X�·37 1 . Xg 1.73 . X�·35 q

It should be noted that all of these correlations suggest that

'"

� T 3 to

4.

(4.6-7)


More recently, Cooper ( 1984) has proposed a much simpler correlation for saturated nucleate pool boiling. His correlation employs reduced pressure, molec ular weight, and surface roughness as the correlating parameters. This correlatio n for a flat plate can be written as

(q) I /3

-- =

�T

2 110g\O Rp ( 2-0. 1 O. ) p ( 55.0 -

Pc

.

- logl O

p- ) -O.55 . M -O.50 Pc

(4 . 6-8)

In the above equation, the roughness, R p , is measured in microns, and is the molecular weight, is measured in degrees K , and q is given in W1m 2 . Cooper suggests that for application of the correlation to horizontal cylinders, the lead constant on the right hand side should be increased to 95 . It should be noted that correlation Eq. (4.6-8) accounts for roughness but does not account for the variations in degree of surface wettability. Similarly, Eqs. (4.6-3) to (4 . 6-6) are for a specific contact angle and for an assumed value of roughness. Also, while Eq. (4 . 6-8) accounts for geometry of the heater, Eqs. (4.6-3) to (4.6-6) are independent of the heater geometry. Equation (4.6-8) is easy to use and is recommended. However, any of the equations should be used with caution, as large derivations between actual data and predictions from these equations can occur when the conditions under which the data used in developing the correlation are not duplicated.

�T

M

4.6.3 Future Research Needs The correlations described earlier are useful in providing dependence of nucle ate boiling heat fluxes on wall superheat over the range of parameters that has formed the basis of these correlations. However, the utility of these correlations diminishes as we apply them to situations involving parameters that go beyond the range used in the correlations. As a result, there is a significant impetus to develop mechanistic models for nucleate boiling. The mechanistic models developed in the literature have not progressed much in the past because they themselves have employed parameters that have an empirical basis. These models have also ex cluded the interactions that occur between simultaneously occurring processes. It is believed that significant further progress in predicting nucleate boiling heat fluxes can be made by using complete numerical simulation of bubble growth, departure and merger processes, and associated heat transfer in the vicinity of an active nucleation site. Also, mechanistic models need to be incorporated for the prediction of density of active nucleation sites. The models of physical pro cesses, such as the formation of a micro-layer underneath the bubbles, shape of advancing and receding interfaces, heat transfer around an evolving interface, and heat transfer on the unpopulated areas of the heater as influenced by the flow field created by bubble growth/departure, need to be verified with detailed expe ri mental measurements. It should be stressed that a macroscopic model need not

NUCLEATE BOILING 99

only describe macroscopic data, but it should also be verifiable at the sub-model! microsca1e lev el. 4.7 POOL BOILING IN BINARY SYSTEMS 4.7. 1 Introduction

Boiling of binary and multi-component mixtures constitutes an important process in chemical, process, air-separation, refrigeration, and many other industrial appli cations. Reboilers feeding the vapors to distillation columns and flooded evapora tors generally employ pool boiling, while the in-tube evaporation process involves flow boiling. Although the multi-component boiling is of greater interest from a process standpoint, a fundamental understanding can be obtained first through a study of pool boiling heat transfer with binary mixtures. The vapor-liquid equilibrium phase diagram introduced earlier in Chapter I provides an insight into the associated concentration gradient and changes in satu ration temperature around a growing bubble in nucleate boiling for binary mixtures. Nucleation criterion for pure components may be applied to mixtures; however, there is no work reported on this aspect in literature. As a bubble grows, the more volatile component evaporates preferentially at the liquid-vapor interface, setting up a concentration gradient in the liquid surrounding the interface. This additional mass transfer resistance along with the associated rise in interface temperature causes deterioration in heat transfer with mixtures. A summary of available liter ature and recommended model and correlations are presented in this section.

4.7.2 Heat Transfer Correlations There are a number of correlating schemes proposed in literature for predicting pool boiling heat transfer coefficient for binary mixtures. Success of a purely theoretical treatment has been limited, and many investigators have generally resorted to empirical methods. A detailed review of these correlations is provided by Kandlikar ( 1 99 8). Calus and Rice ( 1 972) incorporated the diffusion resistance term derived by Scriven ( 1 959) and Van Stralen ( 1 966, 1 967) and developed a widely known corre lation modifying the single-phase convection equation. Calus and Leonidopoulos ( 1 974) effectively employed a linear mass fraction averaged heat transfer coeffi cient of the pure components and modified it with the diffusion resistance term. Althou gh this term affects the heat transfer rate, its effect on heat transfer was not clearly derived. Calus and Leonidopoulos' equation is given below.

!:::. T

= (Xl !:::. Tl

) ( !:::. h lg ) dTdx ] [ (� D12

+ x2 !:::. T2) 1 +

�

(4.7- 1 )


Equation (4.7- 1 ) is seen to underpredict the results considerably. Stephan and Korner (1 969) recognized the importance of the tenn I Y I - X I I , representing the difference between the liquid and vapor phase mass concentrations in the redu c tion of binary heat transfer. SchItinder ( 1 982) introduced the difference between the saturation temperatures of the pure components at the same pressure as a pa rameter in his correlating scheme. He also introduced the convective mass transfer coefficient at the interface. In the absence of any reliable method to predict this coefficient, it was treated as an empirical constant specific to a system. Thome ( 1 983) recognized the need to account for the rise in the saturation temperature at the liquid-vapor interface of a bubble. He introduced the boiling range (difference between the dew point and bubble point temperatures at a given composition) as a parameter in reducing the available temperature difference. Later, Thome and Shakir ( 1 987) introduced the mass transfer correction factor proposed by SchItinder ( 1982). Wenzel et al. ( 1 995) followed a similar approach, but set out to obtain the actual value of the interface concentration by applying the mass transfer equation at the bubble boundary. This approach required knowledge of the mass transfer coefficient at the interface. It was empirically determined to be 10-4 mls. The interface concentration was then used to determine the interface temperature. Fuj ita et al. ( 1 996) developed an empirical correlation from their own ex perimental data on methanol/water, ethanol/water, methanol/ethanol, ethanol! n-butanol, methanollbenzene, benzene/n-heptane, and water/ethylene glycol sys tems. It incorporates several key features of earlier correlations, and is reproduced below.

a

=

1+

{I [ - 2 . 8 :'��2�� i:�: l } Ll::' � ;:�T2 } � � Tsat, p a2p Tdp (4.7-2)Tbp (xi / a l + X2l (2 )- 1

�----�----------------�--�----------�

----

(4.7-2)

- exp

a I , a2 , and T I , T2 are the heat transfer coefficients and wall superheat values

of the pure components, respectively, and d and Ts t. l refer to saturation temperatures of the pure components at the system pressure. Component 1 is the more volatile component with a lower boiling point. and are the dew point and bubble point temperatures, respectively. Equation predicts the parent data sets within less than 20%, but larger deviations are observed with the refrigerant mixture data from Jungnickel et al. ( 1 979) for R-221R- 1 2 and R231R- 1 2 systems. The theoretical model presented by Kandlikar ( 1 998) first defines a pseudo single-component heat transfer coefficient for mixtures. The effect of mass dif fusion and changes in saturation temperature are considered by obtaining the in terface concentration under diffusion-controlled bubble growth conditions. This approach eliminates the need for an empirical value for mass transfer coefficient in the liquid near the interface as used by the previous investigators (e.g., SchItinder,

NUCLEATE BOILING 1 9 82; Thome and Shakir, 1 987; Wenzel, is presented in the following sections.

101

1995). A detailed summary of the model

4.7.3 Pseudo-Single Component Heat Transfer Coefficient,

Qpse

Differences in pool boiling of mixtures and pure components arise due to (i) the changes in thermodynamic and thermophysical properties of the mixtures, and (ii) the presence of a mass transfer resistance to the more volatile component diffusing to the interface of a nucleating bubble. To account for the property effects before introducing the mass transfer resistance, many investigators have employed an It is calculated from a pure com ideal heat transfer coeffi c ient for the mixture, ponent pool boiling correlation with the mixture properties. Alternatively, when the pure component pool boiling data is available, a mole-fraction or mass-fraction One widely averaged value of pure component coefficients is emplyoed for used equation is the reciprocal mole fraction averaged for the pure components at the same temperature or pressure:

ai d .

a id .

a - + --_ ) -1 X2 XI aid ( --a l i p a21 p ( 1 974) aid , (1998) =

or T

(4.7-3)

or T

Calus and Leonidopoulos employed the mass-fraction average value in defining The "ideal" coefficient, does not incorporate actual mixture properties. To overcome this difficulty, Kandlikar proposed a pseudo single-component coefficient, a ps c, which utilizes an averaging equation between the pure component values, further modified by a property correction factor derived from the Stephan and Abdelsalem ( 1 980) correlation. The final form of the equation for aps c is as follows:

aid .

0.674 ( t1h LC. 0.37 1 ( 0.297 ( apse a ---0 ) t1hLC. ) ) ( ) - . 3 1 7 ( 1 ) 0 .2 4 aavg aoy• 0.5 [(X 1aI + X2a2) + G: =:f] =

where

T.wt. m

a vg

X

T.wt. avg

am

AL. m

aavg

AL.avg

m

P c. m

avg

Pc. avg

8

(4.7-4)

is obtained from the following equation,

+

=

(4.7-5)

The subscript m in the individual properties, t1hLG , PG , a , and AL, refers to the actual mixture properties, while the subscript avg refers to mass fraction averaged properties. Thus,

t1hLG.avg XI t1hLG, I + X2 t1hLG,2 =

(4.7-6)

102 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Concentration bOWldary layer thickness, B m •

Bulk liquid. XI

Liquid-vapor interface, YI,S and xl,S

Figure 9 Schematic representation of concen trations around a bubble growing in a binary mixture.

Similar equations are obtained for PG,avg , Pavg , and AL,avg . One major advan tage of using apSC is that the actual property variations are taken into account, and a sudden change in a property, such as surface tension, with the addition of the second component is reflected in the heat transfer coefficient through Eq. (4.7-4) . The effect of adding a surfactant on the heat transfer coefficient is therefore seen through the negative exponent of -0.3 1 7 in the surface tension ratio. However, if the property variations do not deviate far from a linear mass-fraction average value or as a first approximation, apSC may be assumed to be equal to aavg without applying any correction given in (4.7-4).

4.7.4 Theoretical Analysis for Pool Boiling Heat Transfer with Binary Mixtures Consider a control volume surrounding a spherical bubble (the analysis is equally applicable to a truncated bubble shape), as shown in Figure 9. Applying the mass conservation equation around the bubble within the boundary layer thickness 8m and the vapor inside, the average concentration in the boundary layer is obtained.

1 R PG X I , BL, avg = Xl - - � - (Yl,S - Xl ) 3 Urn PL

(4.7-7)

I -D analysis by Mikic and Rohsenow ( 1 969b) under the assumption of a planar The concentration gradient in the boundary layer is obtained following the

interface.

e[ z 1

Xl,z - X l ,s = if 2J D 1 2 t Xl - X l ,s

(4.7-8 )

The boundary layer thickness is estimated as 8m = (n D 12 t ) I /2 . Introducing 8m in Eq. (4.7 - 8) and integrating over the boundary layer, the average concentration is obtained as

Xl, BL, avg - Xl = 0.3 1 3 Xl.s - XI

(4.7-9)

NUCLEATE BOILING 103 Comparing Eqs.

(4.7-7) and (4.7-9),

X l, s Xl,s = XI - 1 .06 r-R -PPLG (YI,s

is obtained.

Um

-

(4.7- 10)

xd

The ratio R / Om reaches an asymptotic value and is given by Van Stralen ( 1 975) as

( K ) 1 /2

2 R (4.7- 1 1 ) J Om ; ao D I2 Combining Eqs. (4.7- 10) and (4.7- 1 1 ), the interface concentration is obtained. 2. 1 3 = XI - -- Jao (4.7- 1 2) - xd

=

( -K ) 1 /2 -PG (YI,s n D 1 2 PL

Xl, s

where

(4.7- 1 3) (4.7- 14) and

g=

X I - X I ,s

Y l,s

(4.7- 15)

- XI,s

The subscript s refers t o the interface condition. The instantaneous heat transfer rate at the interface in the absence of mass diffusion is obtained from the transient heat conduction equation

. = A d Tw -I/2Tsat) (nKt) 1 Tw - Twt q t:J.h LG t:J.h LG (n K t) 1/2

(4.7- 1 6)

q5

The resulting mass flux is obtained as

.

_

_

s

(4.7- 1 7)

m s - -- - -- --The instantaneous mass transfer rate of component similarly obtained as

1

at the interface can be

(4.7- 1 8) with

The total evaporation - XI ).

(YI ,s

ms,D ml,s [ PLD121/2 ] [XI - XI,S ] m'\', D = --YI, s - XI (n D l 2 t) YI,s - XI

at the interface is obtained by dividing Eq.

-

(4.7- 1 8) (4.7- 1 9)

104 HANOBOOK OF PHASE CHANGE: BOILING ANO CONOENSATION

The heat transfer rate may be assumed to be proportional to the evaporatio n rate at the interface. The reduction due to mass diffusion effects are then give n by a mass diffusion factor Fo, which is the ratio of the mass transfer rates given by Eqs. and Incorporating Eqs. and and simpli fy ing, we get

(4.7-19) (4.7-17).

(4.7-12) (4.7-13) I s 2 j 1/ T � !:::. � ) ) ( ( [ D m. S • FD = g m s = 0.678 1 !:::. h LG DI 2 +

(4.7-20)

The heat transfer coefficient for binary mixture is then obtained by incorporating

Fo in apSC .

a

=

apSC FD

(4.7-21)

The expression for Fo is further modified to account for the fact that when the diffusion effects are negligible, FD approaches These effects are quantified by a new volatility parameter which is obtained by simplifying the bracketed term in Eq. with Y\ , s and assuming the bubble point curve to be linear in the region.

(4.7-20)

(

VI , = YI

1.

dT (X VI = ( � � - yd ) ) ( I dX I !:::. h LG DI 2

(4. 7 -22)

After comparing with experimental data, the final form of the equation for Fo obtained.

1 1 - 64VI I FD = 2 T / Cp !:::. L [ . � 1 sj 0.678 1 + !:::. h LG ( DI 2 ) g

is

VI :::: 0.03 0.03 VI :::: 0.2 for VI 0. 2 (4.7-23) The binary pool boiling heat transfer coefficient is given by Eq. (4. 7 -21) in con junction with Eqs. (4. 7 -22) and (4.7-23). !:::. Ts and g are obtained from Eqs. (4.7-14) and (4. 7 -15), and the interface concentration is obtained from Eq. (4. 7 -12), with lao from Eq. (4. 7 -13). An iterative procedure is needed to find X I, s and a. The following procedure is recommended:

for for

1 . Calculate Tmt corresponding to the bulk liquid concentration and system pressure. 2. Assume X I . s . Determine Tsat,s and Y I , s from the thermodynamic property data for the mixture. 3. Assume a and calculate T = Tmt + q /a . 4. Calculate g, !:::. Ts, and lao from Eqs. (4.7-13)-(4.7-15). 5. Calculate X l ,s from Eq. (4.7 -12). w

NUCLEATE BOILING 105 6. Calculate VI from Eq. 7. Calculate FD from Eq. 8. Iterate steps 2 through

(4.7-22). (4.7-23) and from Eq. (4.7-21). 7 until and are converged. ex

Xl ,s

ex

The diffusion coefficients in the above equations are calculated by the Vignes

( 197 1) correlation given below. D 1 2 = ( D I 2 )X2 ( D2 1 )XJ A l

(4.7-24) This correlation was found to be very suitable for non-ideal mixtures. D ?2 and D�l in the above equation are respective diffusion coefficients for dilute solutions given by the Wilke-Chang (1955) correlation (4.7-25) D ?2 = 1.1782 10- 16 (cpM2 ) 1 /2 T /(17L, 2 vm,d A l2 is the thermodynamic factor to account for the non-ideality of the mixture. As discussed by Kandlikar et al. (1975), A 2 is assumed to be 1. 0 , since the error introduced is quite small. cp is the association factor for the solvent (2. 26 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for unassociated solvents; Taylor and Krishna, 1993), 17 L is the liquid viscosity, and is the molar specific volume of 0

0

x

l

component

1.

Vrn , l

4.7.5 Comparison with Experimental Data

10 and 1 1 show the comparison of experimental data, with the Kandlikar (1998) model given by Eqs. (4.7-22) for methanol/water data by Fujita et al. ( 1996) and R-22/R-12 by Jungnickel et al . (1979). Also shown are predictions from

Figures

E � �

1 2000

R-22/R-1 2,

i G)

8000

(I) 0 (,)

6000

'0 :e

� c:

4000

i;j

2000

� (I) J:

p=1

q=40 kW/m2

1 0000

MPa

Experimental data, Jungnickel et al. (1 979) Cal us and Leonidopoulos (1 974) Fujita et al. ( 1 996) Kandli kar ( 1 99B)

0 0

0.2

0.4

0.6

Mass Fraction of R-22

0.8

Figure 10 Comparison of lungnickel et al .'s R-22/R- 1 2 data with correlations.

106

HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION - - -

--- ----,------------

e - - Expe ri m e ntal data, Fuj ita et al. ( 1 996) - - - - - Galus and Leon idopoulos ( 1 974) --- Kan d l i kar ( 1 998) - - - - - Fujita et al. ( 1 996)

50000 �======� Methanol-Water, p= 1 00 kPa

1: � 40000 :::.::::

·u

cO)

::E

0) o

q= 500, 000 W/m 2 - - - - - - - - - -- - - - - - -

30000

'-

o

Q5 20000

U5 c (1j

�

co 0) I

1 0000 1 00, 000 o +-�--+-�-+-�--+-�-+-�-� o

Figure 11

0.2

0.4

0 .6

0 .8

Mass Fract ion of M et hanol

Comparison of methanol/water data of Fujita et al . with correlations.

Eqs_ (4.7- 1 ) and (4.7-2) by Calus and Leonidopoulos ( 1 974) and Fujita et al. ( 1 997) . An additional comparison is presented by Kandlikar ( 1 998) for different binary systems. Since there are no empirical constants introduced in the formulation, the Kandlikar ( 1 998) model is expected to be applicable to other binary systems as well . Although Eq. (4.7-2 1 ) represents the data quite well, further research is needed near the low concentration region. Extensive phase equilibrium properties are needed in applying many of the correlations in pool boiling of binary mixtures. Sources such as HYSIM ( 1 996) and NIST ( 1 995) provide such data in a computerized form and are recommended.

4.7.6 Effect of Surfactants on Pool Boiling Heat Transfer Surfactants are surface active substances that significantly alter the surface tension of a liquid even at very low concentrations. Although the surface tension is altered considerably, its effect on heat transfer is not clear. A comprehensive summary of some of the important work in this area is given in Table 3 as reported by Kandlikar and Alves ( 1 998) . They also presented results indicating a slight increase in the heat transfer coefficient with 1-2% (mass basis) solution of ethylene glycol in water.

.... = -...I

( 1 983)

Yang and Maa

( 1 973)

Shah and Darby

( 1 970)

Water with commercial surfactant Water with two different surfactants

Water and polymeric additives

(Continued)

Foaming results as the bubbles do not coalesce. This behavior is due to the reduction in surface tension. The localized increase in surface tension during the growth of a bubble inhibits coalescence.

Surface viscosity with high molecular weight additives retards bubble coalescence, leading to increase in heat transfer rate.

Initiation of nucleation was identified as another parameter being affected by surface tension besides the departure bubble volume. Changes in contact angle and surface viscosity are believed to affect the heat transfer rates. The complex influence of surfactants on heat transfer through bubble growth, departure size, and frequency was identified.

Volumetric study with air indicated that dynamic surface tension changed for Targitol solutions but not for aerosol solutions. Heat transfer increased by 50% with Targitol and 400% with Aerosol. Bubble frequency increased with additives; surface viscosity with higher molecular weight additives was identified as a factor. Bubble volume at departure and bubble growth rates were obtained experimentally. Bubble volume, growth time, and delay time decrease with surface tension; bubble frequency increased by an order of magnitude. Polymeric additives with long chain molecules improved heat transfer. Although the viscosity was not affected, the heat transfer was improved due to the same factors as those responsible for reducing turbulent drag for these mixtures in pipe flow. In the falling film experiments. heat transfer improved due to the increased foaming under nucleate boiling conditions. As surface tension of the mixture decreased. the heat transfer coefficient increased.

Water and Aerosol and Targitol of different concentrations Isopropanol with additives

Kochaphakdee and Williams

Rate of nucleation on the heati ng surface was affected by the presence of additives

Heat transfer increased, though the surface tension remained unchanged with additives.

Methanol with additives

Water and five surfactants

Comments

Results

Mixtures/ composition

Dunskus and Westwater ( 1 96 1 ) Roll and Myers ( 1 964)

Lowery and Westwater ( 1 957) Jontz and Myers ( 1 960)

AuthorlYear

DIble 3 Summary of some important studies on surface tension effects on boiling

""'"

Q QC

Water with surfactants

Wang and Hartnett ( 1 994) Straub ( 1 993)

Wozniak, Wozniak, and Bergelt ( 1 996) Kandlikar and Alves ( 1 998)

Malyshenko, ( 1 994) Ammerman and You ( 1 996)

Water with SLS surfactant

Tzan and Yang ( 1 990)

Water and ethylene glycol

Water under regular and microgravity

Water and FC-72 with SLS surfactant

Argon

Water and refrigerant under regular and microgravity

Mixtures/ composition

AuthorlYear

Table 3 (Continued)

Surface tension driven flow becomes important in the absence of buoyancy circulation caused by gravity. Slight increase in heat transfer observed at low concentration of ethylene glycol in water

The experimental values of superheat prior to nucleation are lower than calculated values. The heat transfer rate increased with the addition of surfactant. The convection component increased while the latent component decreased.

The photographs show the increased bubble activity with smaller bubbles as the surface tension decreased. The results are contradictory to other studies. The reasons are not clear.

Effect of contact angles and other parameters affecting bubble characteristics needs to be studied.

Photographic results under saturated conditions should be used with caution as the camera speed was low at 1 00 fpm. Presence of smaller bubbles improved heat transfer against the formation of dry spots with large bubbles. For small bubbles, the curvature effect on surface tension needs to be considered. The system is similar to Wang and Hartnett ( 1 992) for water, but the heat transfer improved with addition of surfactant. Increased nucleation with surfactants caused agitation of liquid. Role of surface tension becomes more important in micro-gravity.

Comments

Results Heat transfer improved with addition of surfactant. The bubble density also increased on the heater surface. Heat transfer with SLS and Tween surfactants in water was same or lower than water, though the surface tension was lower. The presence of bubble on a surface induced microconvection, which improved heat transfer with refrigerant but not for water. Role of surface tension through flow around bubble explained.

NUCLEATE BOILING 109

The reduction in surface tension causes the bubbles to become smaller and prevents their coalescence. The heating surface is covered by a larger number of smaller bubbles. The contact angle is also affected by the addition of surfactants, though their trends are not clearly established. The nucleate boiling heat transfer depends on the nucleation, bubble growth rates, bubble departure sizes, and contact angle, which may all be affected by the addition of surfactants. Further research is needed to establish the effect of surfactants on these parameters and the resulting influence on the pool boiling heat transfer.

4.8 HOMOGENEOUS NUCLEATION OF LIQUIDS The boiling of a pure liquid normally occurs when the liquid is in contact with its vapor and the equilibrium vapor pressure at the temperature of the liquid is equal to the ambient pressure. If, however, the extent of the vapor contact is limited to microscopic regions located within surface irregularities of the container (or in the surface structure of impurities within the liquid, e.g., motes), then the temperature at which the liquid boils increases, and the liquid can become superheated. The extent of such superheating often ranges from a few tenths of a degree to sev eral tens of degrees, depending upon the liquid, the nature of the container, the volume of liquid, the rate of heating, and the purity of the liquid, among other things. The superheated liquid is often localized in a thin layer adjacent to the heating surface. In the limiting case of no vapor (or surface) contact whatsoever, it is possible to achieve remarkably large degrees of superheat before the liquid is observed to boil-usually with explosive force. In fact, it is just this latter type of explosive boil that is believed to be responsible for a number of rather devas tating industrial accidents resulting in severe damage to property and loss of life (Reid, 1 978). The limit of superheat is the maximum temperature to which a liquid can be heated before it homogeneously nucleates (spontaneously vaporizes). The limit of superheat can be determined thermodynamically, based upon stability consi derations, as the spinodal curve for the liquid that is the locus of minima in the P- V isotherms satisfying the conditions (a p /a vh = 0 and (a 2 p /a V 2 h > O. A typical liquid spinodal is illustrated in Figure 12 as the line C-C' -Critical point. In Figure 1 2, the thermodynamic determined limit of superheat at pressure Ps is 1[. The shaded area between the liquid spinodal and the two-phase boundary (the line B Critical point) represents the metastable, superheated fluid. The calculated value of the spinodal generally depends upon the equation of state used for the analysis (Eberhart, 1 976). There have been a number of attempts to use equations of state to accurately predict the thermodynamic limit of superheat (e.g., Eberhart, [ 1 97 6] and Lienhard [ 1 976]). One example of a particularly simple relation is -

1 10 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION

Li uid spinodal q

Vapor spinodal

Stable vapor

ps

l

••"•::

Metastabl e " s perheated i Ui d q u

\ L__ \ t

_ __

\

Unstable \ region \ Metastable

_'(_u_p_er_s_atu_rate_d_v_a_po_r _

_

L-_ __ _____

Figure 12

__

___

Volume

Volume-pressure chart of fluid and the range of metastable superheated liquid_

{ (TTLSTc- Ts) }

given by Lienhard ( 1 976):

TcM

=

0.905

M

( TcTs ) + ( TcTs ) 8 Ts

_

0.095

(4.8- 1)

where i s the critical temperature, i s the saturation temperature, and the subscript denotes the value at the condition of limiting superheat. The process of boiling in liquids is analogous to the condensation of vapors. The condensation process begins with the nucleation of vapor molecules forming embryos of the condensed phase that can either grow to become macroscopic segments of the new phase or decay to molecules of the vapor. Similarly, the boiling process begins with the nucleation of vapor embryos. If the vapor embryo grows to a size sufficient to be in unstable equilibrium with the liquid (the so called critical nucleus), it can grow to macroscopic size (boil). If not, the tendency is to decay to the liquid. If an interface or foreign surface is involved, the process is called and generally occurs at low supersaturation (vapors) or smaller degrees of superheat (liquids). If not, the process is called

heterogeneous nucleation homogeneous nucleation.

We note that the homogeneous nucleation of vapors has been studied exten sively in the laboratory, and in spite of a rather long history of investigation, there is much not yet understood. This is reflected in the results of recent investigations into the effects of background gases on the nucleation process, as well as other

NUCLEATE BOILING 1 1 1

related investigations into the nucleation of vapors using thermal diffusion cloud chambers. Details of these studies are available elsewhere and will not be repro duced here (see Bertelsmann and Heist, 1 997a, 1 997b; Bertelsmann et al., 1 996; Hei st, 1 995 ; Heist and He, 1 994; and Heist et al., 1 994a, 1 994b). The limit of superheat can also be determined kinetically, based upon nu cl eation theory, as the temperature at which homogenous nucleation (sponta neous vaporization) occurs (Blander and Katz, 1 975; Carey, 1 992; Frenkel, 1 995 ; Skripov, 1 974). In this regard, we consider the rate of embryo formation from embryos of size X (containing X molecules) to size (X + 1 ) . In nucleation theory, this rate of formation is expressed as the difference between the rate of growth from size X to X + 1 (by evaporation from the embryo surface) and the rate of decay from size X + 1 to X (by condensation on the embryo surface). At steady state (constant flux of embryos through size space), the rate of embryo formation, J, per unit volume per time can expressed as (Skripov, 1974):

J=

J (A Nx ) ex

---;-J

-

-

dX

(4.8-2)

where ex is the vaporization rate from the surface of the embryo (usually taken to be the same as the condensation rate from the vapor, i.e., pv / ( 2rrmk T ) J /2 , where Pv is the vapor pressure, m is the molecular mass, and k is the Boltzmann constant), A is the interface area of the embryo, and Nx is the number density of embryos containing X molecules. The number density of embryos is generally taken to be of the form

Nx = NJ exp ( - Wx l k T )

(4.8-3)

where NJ is the number density of liquid molecules. Wx is the work required to form an embryo with X molecules and involves the work necessary to form the interface separating the two phases and the work required to vaporize the gas to a volume Vg at a pressure Pg . According to Skripov ( 1 974), Wx can be expressed as:

(4.8-4) where a is the surface tension, P I is the pressure in the superheated liquid phase, and IL is the chemical potential (on a molecule basis) given by dtt = (V IX) dp. When mechanical and chemical equilibrium apply, the embryo is the critical cluster, and the Laplace equation holds: 2a

Pg = Pv = PI + r c

Where rc is the radius of the critical cluster (also called a bubble nucleus) .

(4.8-5)

1 12

HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION

Table 4 Limit of superheat and nucleation rate in water at atmosheric pressure T (K)

J [ 1 /(cm3 sO]

560 570 575 5 80 590

2.7 8.5 5.7 4.3 4.3

x x x x x

Waiting time

1 0- 76 1 0- 20 1 0- 3 109 1023

1 2 X 1 068 yr 3.7 x 1 0 1 1 yr 1 . 8 X 1 02 s 2.3 x 1 0 - 10 s 2.3 X 1 0 - 24 S

B

Non-ideal gas effects have been included by Katz and Blander ( 1 973) in deriving the work of fonnation of the critical cluster, giving

4rrar�

where

B

Wx � -- - 4rr a ( r - rc ) 2 3

(4.8-6)

2

(4.8-7 ) for PI « Pv 3 3 The work required to form an embryo-Eq. (4.8-4) or, alternatively, Eq. (4.8-6)-increases to a maximum at r = rc and then decreases as the bubble size increases past the critical radius. Thus, bubbles with radii larger than the critical radius (continue to) grow spontaneously. With reasonable approximations, Eq. (4. 8-2) has been rewritten by Debenedetti ( 1 996) as: �

1 -

J = NM

� -

J"�B [- ��; exp

•

:� ]

82 (P

Pil 2

(4.8-8)

where Ntot is the total number density in the superheated liquid, M is the molecular mass, and

8

=

1

_

pv v

kT

(4.8-9)

where v is the specific molecular volume of the liquid. All other symbols in Eqs. (4.8-8) and (4.8-9) have already been defined. The value of J obtained using Eq. (4.8-8) varies dramatically as a function of temperature. In Table 4, the value of J obtained using Eq. (4.8-8) is shown as a function of temperature for superheated water at one atmosphere pressure (Avedisian, 1 985). The "waiting time" (essentially, the reciprocal of the rate) is also shown for comparison. At a calculated rate of J = 1 bubble/cm 3 /sec, the wait ing (expected) time for the nucleation of one bubble in one cm 3 of water is one

NUCLEATE BOILING 113

second. These data suggest that bubble nucleation is a rare event at temperatures less than 570 K under these conditions. At higher temperatures, the rate (expected 2 3 ti me) increases (decreases) dramatically-approximately 1 0 - 1 0 times for each increase in temperature of one degree. Thus, it is apparent that the kinetically detennined value of the superheat temperature is a function of the nucleation rate (actually, the embryo or bubble flux through the critical nucleus size) and increases slowly as that rate (bubble flux) increases. The thermodynamically de tennined limit, however, must be the larger value, since thermodynamics provides an upper limit on the (meta)stability of the superheated state. At the maximum possible bubble flux, the kinetically and thermodynamically determined limits of superheat should agree. It has been estimated that the maximum possible bubble flux is approximately 1 030 nuclei/cm 3 Isec 2 . In fact, there is evidence that if the thennodynamic value of the limit of superheat is taken to be the kinetic limit of superheat at a nuclei flux of 1 03 0 nuclei/cm3 /sec, the measured kinetic limit of superheat temperatures at various nuclei flux appear to be reasonably consistent with that value (Eberhart, 1 976). The limit of liquid superheat has been determined by experimental methods in which the test liquid is heated slowly with no vapor phase contact and by methods in which it is heated rapidly so as to avoid the effects of pre-existing nuclei. These methods include the isobatic heating of a small volume of liquid within a capillary tube or an immiscible liquid, the isothermal decompression technique utilizing a bubble chamber, and the rapid heating of the test liquid by a small heater immersed in the liquid. In this latter method, the heating rate is maintained sufficiently fast so as to avoid heterogeneous boiling caused by pre-existing nuclei on the heater surface. Results from these investigations indicate that the temperatures at which spontaneous boiling occur agree well with the predictions of homogeneous nucleation theory for organic liquids and low-boiling point liquids. Skripov ( 1 974) has confirmed the limit of superheat for a variety of liquids and has studied the kinetics of bubble nucleation using rapidly heated fine wires. !ida et al. ( 1 994) have immersed a small film heater (0. 1 mm x 0.25 mm) in test liquids and heated the film using a narrow pulse of current. They have been able to attain heating rates up to approximately 10 8 K/sec. Figure 1 3 shows data from these experiments for the measured onset of bubble formation in ethanol. These authors have detennined that the generation of one bubble on the heater in this system corresponds to a nucleation rate (nuclei flux) of approximately 1 0 1 4 nuclei/cm3 /sec. The data are presented by !ida et al. ( 1 993, 1 994) as the measured temperature, Tc .bi , for incipient nucleation on the heater film at various heating rates for three different pressures. The measured temperatures are also presented in reduced fonn, TC•bi / Te . Note that Te,bi increases with the heating rate and approaches a constant value for heating rates of approximately 5- 1 0 x 1 06 K/sec and above. These constant values of the nucleation temperatures shown in Figure 1 3 agree with the limit of

1 1 4 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION I

t--.tl:1

�

_

..c

0.98 0.96

�

�:E u. 230 � -1 U. � « W J:

.

20

15

--,

10 5

81 ....... :

.

• ...L/

0 10

0

82

20 =

Tw _ .

A

60

,

� II

, .

---

curve 1 (Rg. 5) A-B2: curve 2 (Rg. 5)

/ ....... 1 0 Kls

,.

20

10

60

- - - - - 81 -A:

,

30

50

- Tsat, K

.

50

� 40

�

\

40

30 L1 T

70

� �

steady-s tate • . . . . . . . heating: 1 0K/s cooling: uncontrolled

60 °C

/

"' 8 1

82

0 0

5

10

15

TIM E , Figure 9

20

25

30

S

Boiling curves for Fe 72 under steady-state conditions, and for heating and cooling tran sients with the corresponding temperature/time characteristic (Hohl et at. 1 996).

This effectobserved is involvedan inincreasing the resultscontamination shown in Figureof6.theDuring the experiments, the authors heating surface, resulting in a shift to higher heat fluxes in nucleate boiling and in the upper region ofDhi transition boilingwho(thicarrird pass). More typical datawithhavewaterbeenas presented by Bui and r (1985b), e d out experiments the deposits accumulated the heating show that an increase in deposit has anoneffect similar tosurface. that withTheimresults provedclearly wettability. Cleanliness

FILM AND TRANSITION BOILING 189

roughness is typicallinynucleate shown inboiling Figure 5.is Inimsteady Thetheeffect oftransfer heat coefficient proved ri m ents, xpe e te stwiath increasing roughness. Also CHF is higher and shifted to lower temperature The sameboiling, holdswhiforchtheis always minimumanchored film boiling point. Hence, heat ditraffnsfereernces. between these two limiting indecrtransition eases with increasing roughness. The latter alsoDhirholds(l985b) in transient poicoolints,ng. However, the effect on CHF is ambiguous. Bui and found that CHF is not significantly affected by thewithroughness, and Shoji et al. (1990) found that the CHF of water even decreases increasing roughness. Effect of heater size According to the few studies available, this effect does not seemmethanol to be significant. Abassi et al. (1989)measurements carried out transient coolingon experi montal ents with as wel l as steady-state with water a hori z heater.wavelength The diameters ofsturbance the heatersof employed were comparable to theupward-facing most dangerous for di the liquid-vapor interface. No significant effect on the heat flux in transition boiling was observed. Effect of subcooling All results presented in the open literature clearly show that subcooling has aarestrongraisedinfluence oncreasithenenti re boiling curve. Both thecular CHFis and the MHF-points with i n g subcooling. MHF in parti fted towards Hence, theas transition heat flux is shifted tshio higher valueshigher for a gisuperheats. ven wall superheat subcoolingboiling increases. Rou ghness

7.3.4 The Mechanism of Transition Boiling and Prediction Methods

Most of the modeling approaches for transition boiling follow the assumption by Berenson (1962) thatfilmtransition boilingof iswhich a combination ofexists unstable nucleate boiling and unstable boiling, each alternately at any givenis location on the heating surface. The heat flux at a given overall wall superheat consequently expressed as the sum of a component due to liquid contact and a film boiling component, (7. 2-2) where average toheatassume fluxesthatduring the liquidFurther, and vaporis time con tact, respectiandvely. areIt isthereasonable dep During the local wettingandlifetime, itFisdenotes characterithezedaverage by a statistical sequence ofp trendent. ansient conduction, nucleation, dryout. on of theFheating wall in contact withliquid liquidcontact at a given moment. Ifacroptheortiprocess is ergodic, also represents the local time-fraction cording to ql

qg

qg = q MHF .

(At / Awt )

F=

At

-

A tot

tt �ttot

q,


where atdenotes theon theliquisurface. d contact time sampled for a statistical sufficient total time a point Asn examples for modeling approaches of this type, the contributions of Pan and Li ( 1 99 1 ) and Nishio and Nagai ( 1 992) are quoted. Both models are main ly based ondiscussed the theorylabyter Katto andtext.Yokoya ( 1970) of macrolayer formation, whi ch will be in the The procedures to assumptions calculate thethatheathavefluxnotin transition boiling are still qui t e preliminary and include been verified by experiments. evant parameters boiling have been included in theMoreover, analysis.notWeallarereltherefore dependentforontransition the few experimental data for ithen thequantities inregion. Eq. (7.3-2) to provide us with at least an estimate for transition 1 0, F -factors are depicted from different experimental studies. It is notableIn Figure that regardless of the very different system conditions (see Auracher, 1992, t[ ttot

q (!!::.. Tsat )

1.0

o • {:, •

�

I U z o u

0.1

}

water.

transient

o

Shoji et 01. (1991) water. transient

o

Alem Rajabi and Winterton (1988) methanol. steady-state

�

Hohl et 01. (1997)

o

5 o ::J

Lee et 01. (1985)

u. o

F C 72.

ste a dy-state

u.

�

� 0.01 z o

I I I I I I I I

4: 0::: U.

o •

rMHF: Alem Rajabi/Winterton i MHF: Shoji e t a\. (1985)

� CHF

0 . 0 01

Nishio et at. (1997) R 1 13. transient

I I I

.

(1991 )

I L...-_'----''-----'_....I-.. --I.'-----L-_�__'__.....

a

Figure 10

2

3

5

� T sat / � T CHF'

-

6

7

8

9

Fraction of liquid contact as a function of surface superheat.

FILM AND TRANSITION BOILING 191

is belowcovered unity atbyCHF, indicating thatfurther at thenoted criticalthatboiling point, fthore desurtaifacls),e isFalready some vapor. It is F decreases rapidly with increasing wall superheat. The data in Figure 10 can be correlated by F exp[ - 2.2( � Tmr! �TCHF ) + 2 ] (7.2-3) for F in Eq. (7.3-2). Note that Eq. (7.3-3) cannot be used to to providinee an�Testimate determ MHF' availableboiinling, the literature for the quantities dataofsetstransition q ( �Tsa( ) , Taking,theandfew� TCHF the time-averaged heat flux ql during qMHF a wetting period can be determined as a function of � Tsar . The resulting correlation =

qCHF,

qCHF

(7.2-4)

0.82 ( � Tsar! � TCHF) (7.3-3) q ( � Tsa f )

in the transition region with Eq. to estimate can be=usedq together MHF is assumed. More reliable measurements are required to improve correlations (7.3-4) , respectively. An experi(7.3-3) mentalandstudy by Shoji ( 1992) attempted to verify the Katto and Yokoya ( 1970) theory of macrolayer formation, which is often the basis of mod elsublayer ing approaches to transition this theory, is assumed a liquid (macrolayer) penetratedboiling. by smallIn vapor stems isit formed on thethatheater sur face beneath a vapor bubble. The heat transfer characteristic is determi n ed by the competition of two time i n tervals, namely, the hovering time of the vapor bubble the macrolayer and assumption the evaporatithaton time t.(t)Inofhisthisstudy layer.onThesteady-state transition regiabove on is modeled under the transition ing of water,theory Shoji presents a revised version of the basic equation derived fromboilmacrolayer if qg

(r)

r >

q

=

Pt hl g Oo ( 1

- Ag / A w ) !

(7.2-5)

where denotes the liquid density, the latent heat of evaporation, the initial macro layer thickness, the area fraction of vapor stems in the macrolayer, and ! 1 / the bubble departure frequency. Shoji was able to measure, or at least estimate, the a priori unknown quantities and f, and found a good agreement with hi s transition boiling heat transfer data. It should beavai noted,lablhowever, that at present no correlations with a wider range of validity are e for the above mentioned quantities. The heat transfer models based on the macrolayer suffertofrom varitheoparametri us assumptions thatasneed to be refined, and additional data aretheoryneeded study c effects mentioned earli e r. Another phenomenological modelandto Lienhard predict transition boiling heat transfer has been recommended by Ramilison ( 1 987) based on their experi lIlents with different fluids and surface finishes (see, e.g., Figure 5). The following PI

Ag/ Aw

=

r

DO

h lg

Do, Ag / A w ,

192


correlation can be used to predict film boiling and the so-called transitional film boiling. (7.2-6) where fa = -'-- � (7.2-7) pg h \g where index h denotes: heater and is the heat flux of film boiling at the wall used butthe modified turbulent hisfilmempi boiling correlati oofn (Eq. !:::.5 )Tsbyar. KliThemauthors for(0.superheat Ja 0. enko (1981) r i c al constant 0.0086 0057is thefortemperature Freon 113 andat n-pentane, 0. 0 066 for acetone, and 0. 0 154 for benzene). whichgiventhe byfirstRamilison liquid contact occurs (MHF-point). graphi c al correlati o n for and Li e nhard is well approxi mated (see Carey, 1992) by -- TTssaart = 0.97 exp ( -0.0006e� ·8 ) (7.2-8) where ea is the advancing contact angle in degrees. denotes the homogeneous nucleation temperature, which the authors determine by (7.2-9) = 0 . 9 32 + 0. 077 ( ;,; ) 9 ] Tc is the critical temperature of the boiling liquid. Values for ea are summarized by Ramilison and Lienhard in their Table 2: at 5°C, ea = 50 for acetone on teflon-coated thesurface otherfinish ea -values ared. between 250ctioandn method 500, dependi ng ona reasonable the combinfitsurface; atitoontheof data and flui This predi provides for5) asthewelfluids already quoted above(1962). and the different surfaceAll finishes (see Fi g ure l as to data by Berenson modeling approaches suffer from effects have an influence on transition boilintheg, andfactonthattheonother,onefewhand,experinumerous menmaytal results are avai l able. Hereby, it should not be forgotten that all these effects have aspecific differentdataimforpacta giunder steady-state and lable, transieitntisconditions. Therefore, idict f noheat v en problem are avai recommended to transfer in transition boiling by using the two anchor points, namely. thesumedCHF-point and relationship the MHF-poiexists nt. Inin aa logllog-plot first approximation, be as that a linear between theif ittwocanlimiting *

Ph Cph ( Tdf b - Tw )

-

-

---

qfb

15

>

Tdjb

The

Tdj b

Tdf b Thn

Th n

[

Th n

Tc

a

pre

FILM AND TRANSITION BOILING 193 points, then In(q / qMHF )

In( /:)' TMHF / /:). Tsat ) In( /:)' TMHF / /:). TCHF )

------ = ------

In (qCHF / qMHF )

(7.2- 1 0)

With q = ex /:). Tsat , one of the quantities, q or /:). Tsat , may be replaced by the heat transfer coefficient ex. Prediction methods for the limiting quantities qCHF / /:). TCHF and qMHF / /:). TMHF are presented in chapters 6 and 7.

REFERENCES Abassi, A , Alem Rajabi, A . A , and Winterton, R. H. S . 1 989. Effect of Confined Geometry on Pool Boiling at High Temperatures. Exp. Thennal and Fluid Sci. 2 : 1 27- 1 33. Adiutori, E. F. 1 99 1 . Thermal Behavior in the Transition Region Between Nucleate and Film Boiling. Proc. ASMEIJSME Thermal Eng. Joint Con! , Volume 2. Reno, NV, pp. 5 1 -58. Alem Rajabi, A A. and Winterton, R. H. S . 1 988. Liquid-Solid Contact in Steady-State Transition Pool Boiling. Int. J. Heat Fluid Flow 9:2 1 5-2 1 9 . Andersen, J. G. M. 1 976. Low-Flow Boiling Heat Transfer o n Vertical Surfaces, Part I:Theoretical Model. AlChE Symp. Ser. 1 64:2-6. Auracher, H. 1 990. Transition Boiling. Proc. 9th Int. Heat Transfer Con! , Jerusalem, Volume I ,

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Dhir, V. K. 1 990. Nucleate and Transition Boiling Heat Transfer Under Pool and External Flow Conditions. Proc. 9th Int. Heat Transfer Conf 1 : 1 29- 1 55 . Grigoriev, V. A, Klimenko, V. v. , and Shelepen, A. G . 1 982. Pool Film Boiling from Submerged Spheres. Proc. 7th Int. Heat Transfer Conf 1 : 3 87-392. Gunnerson, F. S . and Cronenberg, A. W. 1 980. On the Minimum Film Boiling Conditions for Spherical Geometries. ASME J. Heat Transfer 1 02:335-34 1 . Hamill, T. D . and Baumeister, K. J . 1 967 . Effect of Subcooling and Radiation on Film-Boiling Heat Transfer from a Flat Plate. NASA, TN D-3925. Hendricks, R. C. and Baumeister, K. J . 1 969. Film Boiling From Submerged Spheres. NASA, TN D-5 1 24. Hohl, R., Auracher, H., Blum, J., and Marquardt, W. 1 996. Pool Boiling Heat Transfer Experiments with Controlled Wall Temperature Transients. 2nd European Thermal Science and 14th UIT Nat. Heat Transfer Conf , Rome, pp. 1 647- 1 652. Hohl, R., Auracher, H., Blum, J., and Marquardt, W. 1 997 . Identification of Liquid-Vapor Fluctuations Between Nucleate and Film Boiling In Natural Convection. Proc. Int. Engng. Foundation Conf : Convective Flow and Pool Boiling, Eds. Celata, G. P, DiMarco, P., and Mariani, A. Irsee, May 1 8-23. Pisa: Edizoni ETS. Hsu, Y. Y. and Westwater, J. W. 1 960. Approximate Theory for Film Boiling on Vertical Surface. Chern. Eng. Progr. Symp. Ser. 30: 1 5-24. Inoue, A. and Bankoff, S. G. 1 98 1 . Destabilization of Film Boiling Due to Arrival of a Pressure Shock . Part I: Experiments. ASME J. Heat Transfer 1 03(3) :459-464. Inoue, A., Ganguli, A., and Bankoff, S. G. 1 98 1 . Destabilization of Film Boiling Due to Arrival of a Pressure Shock. Part II: Analytical. ASME J. Heat Transfer 1 03(3):465-47 1 . Jackson, T. W. and Yen, H . H. 1 970. A Simplified Solution for Transient Film Boiling with Constant Heat Flux. Heat Transfer 1 970 V: B3.8. Jordan, D . P. 1 969. Film and Transition Boiling. Adv. Heat Transfer. Eds. T. F. Irvine and 1. P Hartnett, pp. 55- 1 28. New York: Academic Press. Kalinin, E. K., Berlin, I . I., and Kostyuk, V. V. 1 975. Film-Boiling Heat Transfer. Adv. Heat Transfer, Volume I I , pp. 5 1 - 1 97 . New York: Academic Press. Kalinin, E. K., Berlin, I. I . , and Kostyuk, V. V. 1 987. Transition Boiling Heat Transfer. In: Advances in Heat Transfer, Volume 1 8, pp. 24 1-323 . Katto, Y. and Yokoya, S. 1 970. Mechanism of Boiling Crisis and Transition Boiling on Pool Boiling . Proc. 4th Int. Heat Transfer Conf , Vol . B3 .2. Kim, B. J . and Corradini, M . L. 1 986. Recent Film Boiling Calculations: Implication of Fuel-Coolant Interactions. Int. J. Heat and Mass Transfer 29(8): 1 1 59-1 1 67 . Klimenko, V. V. 1 98 1 . Film Boiling o n a Horizontal Plate-New Correlation. Int. 1. Heat and Mass Transfer 24( 1 ):69-79. Lee, L. Y. w., Chen, J. c., and Nelson, R. A. 1 985. Liquid-Solid Contact Measurements Using a Surface Thermocouple Temperature Probe in Atmospheric Pool Boiling Water. Int. J. Heat Mass Transfer 28: 1 4 1 5- 1 423. Leonard, J . E., Sun, K. H., and Dix, G. E. 1 976. Low Flow Film Boiling Heat Transfer on Vertical Surface, Part II: Empirical Formulations and Application to BWR-LOCA Analysis. AIChE Symp. Ser. 1 64:7- 1 3 . Lienhard, J . H . and Wong, P. T. Y. 1 964. The Dominant Unstable Wavelength and Minimum Heat Flux During Film Boiling on a Horizontal Cylinder. ASME J. Heat Transfer 86(2) :220-226. Nikolayev, G. P. and Skripov, V. P. 1 970. Experimental Investigation of Minimum Heat Fluxes at Submerged Surface in Boiling. Heat Transfer-Soviet Research 2(3): 1 22- 1 27. Nishikawa, K. and Ito, T. 1 966. Two-Phase Boundary-Layer Treatment of Free-Convection Film B oil ing. Int. J. Heat and Mass Transfer 9(1 ) : 1 03-1 1 5 .

FILM AND TRANSITION BOILING 195 Ito, T., Kuroki, T., and Matsumoto, K. 1 972. Pool Film Boiling Heat Transfer from a Nishikawa, K., Horizontal Cylinder to Saturated Liquids. Int. J. Heat and Mass Transfer 1 5 : 853-862. Nishikawa, K., Ito, T. , and Matsumoto, K. 1 976. Investigation of Variable Thermophysical Property Problem Concerning Pool Film Boiling from Vertical Plate with Prescribed Uniform Temperature. Int. J. Heat and Mass Transfer 1 9( 1 0): 1 1 73- 1 1 8 1 . Nishio, S . 1 987. Prediction Technique for Minimum-Heat-Flux (MHF)-Point Condition of Saturated Pool Boiling. Int. J. Heat and Mass Transfer 30:2045-2057 . Nish io, S. and Nagai, N. 1 992. A Model Predicting Transition-Boiling Heat Transfer. Pool and External Flow Boiling. Eds. V. K. Dhir and A. E. Bergles, pp. 27 1 -276. New York: ASME, Engineering Foundation. Nishio, S. and Ohtake, H. 1 993. Vapor-Film-Unit Model and Heat Transfer Correlation for Natural Convection Film Boiling with Wave Motion Under Subcooled Conditions. Int. J. Heat and Mass

Transfer 36( 1 0): 254 1 -2552. Nishio, S., Gotoh, T., and Nagai, N. 1 997 . Observation of Boiling Structures. Proc. Int. Engng. Foun dation Can! , "Convective Flow and Pool Boiling. " Irsee, May 1 8-23. Pan, C. and Lin, T. L. 1 99 1 . Predictions of Parametric Effects on Transition Boiling Under Pool Boiling Conditions. Int. J. Heat Mass Transfer 34: 1 355-1 370. Payayopanakul, W. and Westwater, J . W. 1 978. Evaluation of the Unsteady-State Quenching Method for Determining Boiling Curves. Int. J. Heat and Mass Transfer 2 1 ( 1 1 ): 1 437- 1 445 . Ramilison, 1. M. and Lienhard, J. H. 1 987. Transition Boiling Heat Transfer and the Film Transition Regime. J. Heat Transfer 1 09:746--7 52. Sakurai, A. 1 990. Film Boiling Heat Transfer. Proc.9th Int. Heat Transfer Can! , Volume 1 , pp. 1 57- 1 86. Sakurai, A., Shiatsu, M., and Hata. K. 1 990a. A General Correlation for Pool Film Boiling Heat Transfer from a Horizontal Cylinder to Subcooled Liquid. Part I -A: Theoretical Pool Film Boiling Heat Transfer Model Including Radiation Contributions and Its Analytical Solution. ASME J. Heat Transfer 1 1 2(2):430-44 1 . Sakurai, A., Shiatsu, M., and Hata, K . 1 990b. A General Correlation for Pool Film Boiling Heat Transfer from a Horizontal Cylinder to Subcooled Liquid. Part 2: Experimental Data for Various Liquids and Its Correlation. ASME J. Heat Transfer 1 1 2(2):44 1- 450. Sauer, H. J., Jr. and Ragsdell, K. M. 1 97 1 . Film Pool Boiling of Nitrogen from Flat Surfaces. Adv. Cryogenic Eng. , Volume 1 6, pp. 4 1 2-4 1 5 . New York: Plenum Press. Segev, A. and Bankoff, S. G. 1 980. The Role of Adsorption in Determining the Minimum Film Boiling Temperature. Int. J. Heat and Mass Transfer 23:637-642. Shigechi, T., Kawae, N., Tokita, Y., and Yamada, T. 1 989. Film Boiling Heat Transfer from a Horizontal Circular Plate Facing Downward. JSME Int. J., Series 11 32(4): 646-65 1 . ShOji , M . 1 992. A Study o f Steady Transition Boiling of Water: Experimental Verification of Macro layer Evaporation Model. Pool and External Flow Boiling. Eds. V. K. Dhir and A. E. Bergles, pp. 237-242. New York: ASME, Engineering Foundation. ShOji, M . , Witte, L. c . , Yokoya, S., Kawakami, M., and Kuroki, H. 1 99 1 . Measurement of Liquid

Solid Contact Using Micro-Thermocouples in Pool Transition Boiling of Water on a Horizontal Copper Surface. Proc. ASMEIJSME Thermal Eng. Joint. Can! , Volume 2. Reno, NV, pp. 333338. ShOji, M., Witte, L. c., Yokoya, S., and Ohshima, M. 1 990. Liquid-Solid Contact and Effects of Surface

Roughness and Wettability in Film and Transition Boiling on a Horizontal Large Surface. Proc. 9th Int. Heat Transfer Can! 2: 1 35- 1 40. Spiegler, P., Hopenfeld, J., Silberberg, M . , Bumpus, Jr., C. F., and Norman, A. 1 963. Onset of Stable Film Boiling and the Foam Limit. Int. J. Heat and Mass Transfer 6:987-989. Srinivasan, J. and Rao, N . S . 1 984. Numerical Study of Heat Transfer in Laminar Film Boiling by the Finite-Difference Method. Int. J. Heat Mass Transfer 27( 1 ) :77-84.

196 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Stephan, K. 1 965 . Stabilitat beim Sieden. Brennst. - Wiirme-Kraft 1 7 :57 1 -578. Suryanarayana, N. V. and Merte, H., Jr. 1 972. Film Boiling on Vertical Surfaces. ASME 1. Heat Transfer 90(4) :377-384. Ungar, E. K. and Eichhorn, R. 1 996. Transition Boiling Curves in Saturated Pool Boiling from Hori zontal Cylinders. 1. Heat Transfer 1 1 8 :654-66 1 . Witte, L. C. and Lienhard, J . H . 1 982. On the Existence of Two Transition Boiling Curves. Int. 1. Hear Mass Transfer 25:77 1 -779. Zuber, N. 1 959. Hydrodynamic Aspects of Boiling Heat Transfer. Atomic Energy Commission Report

No.AECU-4439, Physics and Mathematics.

CHAPTER

EIGHT INTRODUCTION AND BASIC MODELS

Geofferey F. Hewitt Imperial College of Science. Technology and Medicine. London. SW7 2AZ. United Kingdom

8.1 INTRODUCTION

next fourflow. chapters of this handbook the important topic ofThetwo-phase The objective is to proviaredeconcerned a brief butwith fundamental exposition on two-phase flow that can serve as a basis for estimating flow characteristics in phase change processes, particularly evaporation (boiling) and condensation. This present chapter gives a general introduction to the topic of two-phase flow systems and Chapter the fluid mechanics of suchsuchsystems, bothTwo flow inside circular9 deals tubes with and flow in other geometries, as heatcovering exchangers. phase flows are often fluctuating in nature but, superimposed on these fluctuations, gross phenomena excursions and oscillationsin inmoreflowdetail mayinoccur due to Another system instability; these are discussed Chapter important characteristic of two-phase flows is the relatively low sound speeds (much less than those for gases and liquids) that can occur due to the combination in these systems of fluids high compressibility and high density. Thus, thesystems, maximum rate atandwhich two-phase may be discharged through, say, relief is limited, the maximum ("critical") rate is of crucial i m portance in two-phase system design. The important topic oftwo-phase critical flow is discussed in more detail in Chapter As is welexcept l known,for true predictions of even single-phase turbulent flowsfrom are nl ot possible very low Reynolds numbers, close to the transition aminar flow. In two-phase flows with deformable interfaces (i.e., gas-liquid and 1 0.

11.

197


liquid-liquid flows), the interfacial configurati on such is alsoflows usuallyare more unpredioften ctablethanin detail; when we combine this with the fact that not turbulent in nature, then it becomes obvioofus two-phase that (as inflows single-phase turbuly lent flows), fundamental analytical predictions are not readi achievable. Inandthishere,partthere of theis ahandbook, we are concerned primarily with gas liquid flows, further complication, namely, the compressibility oftwo-phase the gas phase. Thisdiscussed is of course an all important feature in the case of critical flows (as in Chapter 1 1 ). Despite the fonnidable difficulties in treating two-phase flows, theiwithr practical importance dictates that methodologies have to be developed to deal them. It should not be surprising to the reader,as however, that such methodologies contain a very large element of empiricism; aempirical consequence, extrapolatiwereondeveloped to a range iofs often conditions outside those for which the relationships uncertain at best! The maiof approaches n objective ofthatthihave s chapter isfollowed to exploreinontreati a relati vely qualitative basis theflows.types been n g two-phase gas-liquid Section 8.2 discusses the types of system (adiabatic, phase change, etc. ) that may be encountered andflowsrevimay ewsbetheobtained importantby classifying topic of equithelibtypes rium. ofMuchinterfaci betteral models for two-phase distribution c types calledis giflow terns or8.3.flowModelling regimes. Themethodsgeneralfor background itontosuchgeneri classifications ven pat in Section two-phase flow are8.5).classified in beSection 8.4, and the chapter closes with a brief overvi e w (Section It should stressed troductory9. in nature; many of the topics will be exploredthatinthemuchpresent morechapter detail iins inChapter 8.2 TWO-PHASE FLOW SYSTEMS

Much of that the work carriedwioutthouton heat gas-liquid flowsto orhasfrombeenthedone withtwo-phas adiabatice systems, is, systems transfer flowing mixture. Archetypically, suchThough experiments are flows done wihaveth thesignificant most convenient fluids, namely, air and water. adiabatic industrial importance (for instance, in the transport of oi l /natural gas mixtures in petroleum recovery systems),wherethe heat mainisapplications of the data aretheinsystem, fact in causing predictingevapdi abat i c systems, added or removed from and condensation, respectively. Such diabatic systems are the main focus oforatithison handbook. In Chapter 9, a more detailed discussion of two-phase flow in uc diabatic systems will be given. in theofcontext oflithebriupresent instrod Thecoser tory chapter, it is worth noting theHowever, importance none qui m e ff e ct . effects situationssystems in whichcan theleadapplication of adiabatic relationscanto lead phaseto change to considerable errors idata n prediandction. Non-equilibrium effects can be classified under several headings, as follows.

INTRODUCTION AND BASIC MODELS 199

8.2. 1 Hydrodynamic Non-Equilibrium

flows, theafterflowonlyparameters (particularly pressuredigradi ent) seantturbulent Inbecosinmglee i-pha ari with di s tance a short di s tance (typically a meters) nv of the flow channel. However, for gas-liquid flows, many hun froredmstheof diaentrance may behererequiareredthetodevelopment reach invariaofntbubbly conditions dequilibrium).meters Examples flows(hydrodynami (described inc in Chapter 9) as a result of bubble coalescence and the development detailflows oreannular ofBmearing as a result of entrainment ofhydrodynami the liquid cfilmequilibrium into the gasoccurcore.in in mi n d that these departures from c flows,the itflowis notratessurpri sinrespecti g that they are even more dominant in channel. diabatic flAadiabati oprime ws, where of the v e phases are changing along the example of theof dryout effectsinofevaporating departure from hydrodynami c equi libriumfilmis that of the occurrence annular flows. Here, the liquid onliquithed flow wall isdristilles out, whilein atthethechannel same time, aform largeofproporti on Inofantheadioriagbatiinalc flowing in the droplets. situation, such dryout of course would not occur. The presence of the liquid film has a dramati c effectonsonforthepressure pressuregradigradientent,underanddryit isoutclearly inappropriconditions ate to use adiabatic correlati or near-dryout (thoughcomputer this doescodes not forstoppredi suchcticorrelati from being into the mostinwidely used ng phaseonschange systems!built) . Conversely, annu lforar flow condensati o n, there is often much less entrai n ment at a gi v en quali t y the adiabatic or evaporation cases. This implies that the pressure gradiethan nt is much higher at a gi v en quali t y in condensati o n than for adi a bati c or evaporati ng flows. 10

8.2.2 Thermodynamic Non-Equilibrium

In evaporati ng and condensi nrig usystems, thetheassumpti on is often made that thereare exists t h ermodynami c equi l i b m between two phases. Thus, the phases assumedapproximation to co-exist toatassume their saturati emperature. Evenequilibrium though itappli is usually good that localon tthermodynamic es at thea vapor-liquid e phases can give rise to situatiionnsterfaces, in which,temperature on average,graditheentsflowwithin departsthe respecti grossly vfrom equilibrium. One example of such a non-equilibrium situation is that of the occurrence of vapor bubbly flowthan in thethatpresence of aon,sub-cooled liquid; ona two-phase average, theflowfluiisdoccurring. enthalpy may be less for saturati but nevertheless, This ation arisesandduehighly to sub-cooled boiling. Another example is the occurs co-exisitence ofp liqsitu uid droplets super-heated vapors such, as that which nbulk the ostdryout region. Here, droplets may persist in the flow even though the enthalpy is greater than that for the saturated vapor.


8.2.3 Component Non-Equilibrium

Manyrather of thethan phasesingle-component change situations encountered in practiceininvolve mUlti-compo_ nent fluids. For example, the process industry, reboilers and condensers associated wi t h distillation processes always (by defini tion) have to deal with mixtures of fluids, such as hydrocarbon mixtures. Again, a common assumption in calculating such thesystems is thatof thevapor-liquid equilib rium persists in a bulk sense, which dictates fractions components in the vapor and liquid phases, respectively. However, i n phase change systems, complex heat and massis nottransfer processes are taking place, which means that component equilibrium maintained. The above types of non-equilibrium effects are notflowmutually exclusive and can occur simultaneously. For instance, i n annular evaporati o n of multi component mixtures, the liquid compositions in the liquid filmon andthe theheatentrai ned droplets may be di ff erent; this can have a profound effect transfer behaviInodeali r. ng with phase change systems, it is important to be aware of these al non-equilibrium effects.forUnfortunately, largelysystems! ignored in many ofpotenti the methodologies developed prediction ofthey phasearechange 8.3 FLOW PATTERNS

Inan ainfinite gas-liquid interfaces are deformable, principlthee numberflow,ofsince ways thein which the interfaces can be there distribareutedinwithin flow. Fortunately, there are aida number of characteri of interfaci distribution, whichhowever, is a considerable in developing modelssticfortypes gas-liquid flows.al The types of interfacial distribution are termedlargely flow patterns flow regimes.andThea classification of flow patterns still depends on visualorobservation, bewildering variety ofistypes of flowdeveloping, pattern havewhibeen defined in inthemore literature. However, a consensus gradually c h is described in Chapter 9. Flow patterns can be classified into three broad types as follows.detail 8.3.1 Dispersed

Here,type elements (roughly spherical) of one bubbl of theyphases are dispersed inofthetheother.gas This of flow pattern would include flows where bubbles (vapor) areas dispersed ithen agasliquidphase.continuum. Inaredropdifficult flow,totheobtailiquidn in adiphaseabatiics dispersed droplets i n Drop flows systemsHowever, since theretheyisarea natural tendencytypeto form a wallin film characteri stic ofwhere annulthear flow. an important of flow di a bati c systems, wall film has been evaporated.

INTRODUCTION AND BASIC MODELS 201 8.3.2 Separated Flows

flowtheinregions. separateExamples regions ofherethearechannel while stillinihori nteracti ng e twoacephases Het three, inthterf between st r at i fi e dflow z ontal a which the liquid flows in a layer at the bottom of the pipe with the gas pabpoesve it,in and annul a r flow, in which the liquid flows as a film on the periphery of the channel. i

,

8.3.3 Intermittent Flows

all two-phase flows areoccurs. fluctuating ininclude nature, slthere exist several nearlywhere Thowughpatterns gross intennittency These u g (or plug) flovertical channels and slug and semi-slug flows in horizontal channels. flows The feature oftheyflowoftenpatterns inbegas-liquid flows thatabove makesthreeforcategories, additional difficulties is that cannot classified into the but rather haveis dispersed features ofintwo orgasmore ofofthethecategories. Thus, in annular flow, part liquid the core flow, making the flow simultaneously diofstheslpersed andmaking separated.the Similarly, gas bubblesintermittent are entrainedandintodispersed. the liquidFurther slugs u g flow, flow simultaneously details of such flow patterns are given in Chapter 9. in

in

8.4 MODELING TWO-PHASE FLOW SYSTEMS

the modeling of two-phase flowthesystems, thephase mostfracti important objecti ves are to predict o ns. The in-situ phase the pressure gradi e nt and i n si t u fractions are commonly known as the voidfraetion for the gas phase and the liquid hold-uthep forvolumetric the liquidphase phase.fractions In general, the inbasis.situ This phaseis fractions arevelocities different from on a flow because the of the two phases may be different (the ratio of the mean gas and liquid velocities known as the slip ratio) . In considering phase change, other hydrodynamic pextearameters, as wallnmentshearin stress, frequency characteristics of intenniHowever, ttency, nt of liqsuch ui d entrai annular flow, etc. , may be i m portant. main gradi traditional focus hasregarded been onaspressure gradithreeentcomponents, and phase fraction. The pressure e nt is usually the sum of namely frictional pressure gradients. Further detailstheof accel e rat i o nal , gravi t a t i o nal , and are given in Chapter 9. The accelerational component of the pthisressucategorization re gradient arises frompressure changesgradients in the momentum flux ofimportant the mixturein phase along ch annel. Accelerational are particularly e systems, where the vapor velocity is increasing or decreasing as a result cohfang ev apo condensation. be notedonthatsystems the accelerational pressure gradient rati canoben ornegati ve in the caseIt should of condensati where the momentum In

is

the

the


fluxreldecreases channel. Theofthe gravitational component ofthepressure gradient isFinally, atedtheto thefrialong inctiosinaltuthemean density mixture and, hence, to phase fracti on. pressure gradient is related to the wall shear stress and is nonnally negative. However, in verti c al slug flows, the mean wall shear stress may be negative, giving rise to a positive frictional pressure gradient (though not, of course, to al positive total9. pressure gradient). All zofe thethesemodels featuresthatarearediscusse d in more detai i n Chapter Here, we will categori commonly used for predicting these system variables as follows. 8.4.1 Homogeneous Model

the phases to be intimately beHere,assigned a densiarety assumed corresponding to the total mixed mass ratesuchofthatflowthedivmixture ided by canthe totalthevolume rate of flow. Thebyvelocity ofwitheth single two phases isflow. considered to be equal, and flow can be treated analogy phase The accelerati onal and gravitational components of pressure drop can be calculated directly from theOften,independent variables, and a correlation i s requi r ed only for t h e friction term. this term is calculated an appropri ate mean viscosity.on the basis of single-phase flow correlations using 8.4.2 Separated Flow Model

Here, thevelocities. two phasesHowever, are considered to be flowi ng in separate regions and wimoth different the equations describing the flow (continuity, mentum, and energy) are gradients, in a combined form. Toissolve for forthe phase accelerational and gravitational pressure a correlation required fraction (voi d fraction or li q ui d hold-up), and for fri c tional pressure gradi e nt, a separate correlation Thus,empirical while thecorrelations method is usually more accurate than the homogeneousis requi model,red.more are needed. 8.4.3 Two-Fluid Model

es th for Here, separate continuity, momentum, and energy equations are written i two phases (making 6 equations in all-hence theofname six-equatrelationshi ion modpselare often gi v en to thi s type of modeling). In thi s type modeling, required for the transfer of mass, momentum, and energy across the interneedfacfoes.r and also foristhenotwallavoishear stresses for the respective phases. Thus, theis that the empiricism d ed by using the two-fluid model; the argument empirical relationships developed formerely the interface andwall wallfriction terms areandmore generalas than those that can be achi e ved by correlating hold-Up, in the separated flow model.

INTRODUCTION AND BASIC MODELS 203

8.4 .4 Phenomenological Models

observations ofbetter the phenomena occurri ng ofin gas-liquid two-phase Efloxwsperimental sometimes gi v e a i n dication of the form model to be adopted. can ples hereas slug are thefrequency, modelingliquid of slugshedding flow, takirateng account of correlati such Exparaammeters behind slugs, etc. , oandns formodels flow, phenomenol which take odetai ledmodels accountgo further of the entrainment and deposition annular Such ofprocesses. c al than the two-fluid model in gi representing the detailed processes. However, it is possible sometimes to represent flowshereby isextending the two-fluid model to a multi-fluid representation; an theexample that of representing the droplets in annular flow as a separate "field," a further threesuitable equatioforns areintermi added,ttentmaking nine inastotal. However, thisa which pproach is often not flows such slug flow, since afundamental difficulty occurs in averaging the quantities. for

8.4.5 Computational Fluid Dynamics (CFD) Models

ngle-phase flows, is becomithere ng inarecreasiserinoglyus deficiencies prevalent toinseekmodeling solutionsof usthe ingbsimodern codes.it Though tur ulent flows that occur in most practical systems (even for single-phase flow), thecialuseapplications. of codes is now widelyflows,adopted methods For two-phase due tofortheprediction complications arisforing commer from the icannterfacial structure, is much l e ss useful for di r ect calculations. However, often be helpful in developing an understanding of the phenomena occurring.it cular challenges, when linkedTherefore, to phase two-phase change. Theflows type present of modelpartiselected would dependespecially on the desire fortechniques accuracycanandbedetail, stressed that reliedbuton ittoshould producebe accurate results.none of the current modeling In

CFD

CFD

CFD

8.5 OVERVIEW

this chapter, a brief qualitative description has been made of the general nature two-phaseflowflowsis stillanda riofchtheareawayfor inresearch, which and theyaaremorestructured and modeled.ng Two-phase detailed understandi grradually developing. When designing two-phase flow systems, it is important emember calculations are necessarily inaccurate. The design should take account of thisthatuncertainty.

In of is to

CHAPTER

NINE

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 9.1

Geofferey F. Hewitt

Imperial College of Science, Technology and Medicine, London, SW7 2AZ, United Kingdom

9.2

Masahiro Kawaji

University of Toronto, Toronto, Ontario M5S 3E5, Canada

9.1 FLOW INSIDE CIRCULAR TUBES 9.1.1 Introduction

Two-phase flow in circular tubes is found in a wide range of industrial applications, including oil/gas pipelines and systems with evaporation (such as thennosyphon reboilers inandthecondensation process industry andcondensation water-tube boilers in the powerheatgeneration industry) (in-tube in shell-and-tube exchangis ersreflected and air-cooled heat exchangers, for instance). This industrial i m portance in the vast amount of work that has been done in the area. In this chapter (for which thehopefully objectiveusefulis torelati giveonships), an introduction tobetheimpossible subject together withthisa selecti o n of it would to even list literature! Tohavefurther restrictions beenreduce imposed:the attempted coverage of the chapter, the following (1) The tubes dealt with are assumed to be straight, and two-phase flow in bends, coils, other singularities dealtewed with.by,Flows in singularities, as orifices,andbends, venturies, etc.are, arenot revi for instance, Hewitt such ( 1 984) and Chisholm and work on coiled tubes is typified by the papers of Banerjee et al. (( 11 983), 969) and Hewitt and Jayanti ( 1 992). 205


Only co-andccounter-current urrent flows willflowsbe considered. Staticthatsystems such ascondensers) bubble col umns (such as those occur in reflux are not dealt with. Typi cal(1995), of studies of two-phase flow in bubble columns isg thefilmwork of Ohnuki et al. and studies of counter-current flows in falli n systems are typified by that of Govan et al. (1991). Though only straiQuightte pipes are behavi considered, theobserved orientatinionhoriof thezontal,pipeinclined, is an impor tant parameter. different o rs are and vertical tubes, respectively. shall thefirstbasidiscuss typesons.of This interfaciis folal lstructure flow ofpatterns) then Wepresent c equati owed by(namely, a discussion predictiandon methods for pressure drop and void fraction. (2)

9.1 .2 Flow Patterns

9.1.2.1 Flow patterns in vertical upflow.

The flow regi m es i n verti c al upflow are illustrated in Figure I and are defined as follows: Bubble flow. theHere,liquid the liquid phase is continuous, and a dispersion of bubbles flows within continuum. (2) Slug or plug flow. At higher gas flows, bubble coalescence occurs, and the bubble diameter approaches thatto ofas theTaylor tube.bubbles) Large characteristically bullet shaped bubbles (often referred are fonned, which be separated by regions containing dispersions of smaller bubbles. may (3) Churn Flow. With increasing gas velocity, a breakdown of the slug flow bubbles an unstable the point ofinbreakdown ofleadsslugto flow, periodiflow c briregidgimnge called of the churnflow. pipe occurs,Nearas illustrated Figure (1)

1.

· .

-. .

Bubble Flow

Figure 1

Slug ex' Plug Flow

Churn Flow

.

. . · ·

. . .

.

..

.

.

. . . .

Annular Flow

Wispy Amular Flow

Flow regimes in vertical upftow (from Hewitt, 1 982, with permission).

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 207

However,areasswept the gasup flow rate is increased, thewithflowliquid behavifilms or is different; large waves the surface of the tube between them, which flowFlow.initially upwards andflows then onreverse in direction (Hewitt etfilmaI.,and1 985).the Annular Here, the liquid the wall of the tube as a gas phase flows in the center. Usually, some of the liquid phase is entrained as dropletsAnnular in the gasFlow.core.As the liquid flow rate is increased, the concentration of Wispy drops inthethefonnation gas core ofincreases. Ultimately, (wisps) coalescence and other processes lead to large agglomerates of liquid in the gas core,of characterizing the wispy annular flow regime. A more detailed discussion this regime is presented by Hewitt ( 1 997). be emphasized that the descriof pflow tionpatterns of flow patterns is essentially qualita Ittivshoe inuldnature, and the classification is therefore rather subjective. More detailed discussion of flow pattern determination is given by Hewitt ( 1982). earlier,ofthemoreidentification of flow patterns isData important as apatterns basisAs was forare often thediscussed development phenomenological models. for flow represented in terms ofjlo pattern maps, which are oftwo types: (1) Specifi c Flowdiameter, Patternorientation, Maps. Here,anddatafluidforphysia particular set of conditions given tube cal properties) are presented(i. ein. , terms of variables superficialmapvelocities or mass flux isandshown quality.in An example of a such speciasficphase flow pattern for vertical upflow Figure 2 (showing data for high pressure steam-water mixtures). (4)

(5)

w

4000 Wispy

Annular •

•

Burnout Limit for 3.66� 12.'J.mm-1 D Tube

Figure 2 Flow pattern data obtained by Bennett et al. ( 1 965) for steam water flow at 6.8 MPa pressure (from Hewitt, 1 982, with permission).


Annular

N

Wispy Amular

Churn

with Developing Structure

QJ)bIe Flow

Plug

Figure 3 Flow pattern map obtained by Hewitt and Roberts ( 1 969) for vertical upward two-phase upward flow (from Hewitt, 1 982, with permission).

general flowforpattern in which flow pattern transitions can be represented a widemaprangeis one of fluid physicalthe properties, tube diameters, etc. Such maps are typified by that of Hewitt and Roberts ( 1 969), as illustrated in Figure 3. Here, the flow patterns for a wi de range fluid conditions could be represented in terms of the superficial mentumof fluxes PG (UG ) 2 and pdUd 2 where PG and PL are the gas and densities, and UGfloandw rates UL are the superficial velocities of the gas and liq uid phases (volume of the phases perfor unit cross sectional area of the channel). More recent generalized maps verti c al flows are presented Taitel et al. ( 1980) and Weisman and Kang ( 1 98 1 ). Much work has been done ontothereview individual transitions between regimes. beyond the scope of this chapter these in detail, but the work is exempli tied by the following: ( 1 ) Bubble-slug transition. The transition between bubbly flow and slug flow conventionally been interpreted as a competition between bubble coalescence

(2) General Flow Pattern Maps. A

mo

liqu id by

It is

has


(Radovci Moissis,when1962)the bubbl and bubbl e free breakup due reltoaturbul eeach nce (Tai tel etcoalescence aI., 1980).ck and However, e s are to move ti v e to other, may be ratherrare. more likelywaves") explanation ithes thatbubbl regioesnsbecome ofhigh bubbl e concentration are formed ("void where closely and coalbyescence is ableandto proceed. d wave mechanipacked, sm are reported Beisheuvel Gorissen Studies (1989) andon theBourevoi(1997). (2) Slug�churn transition. The nature of the breakdown of slug flow leading ulti mately to annular flow has been a subject of considerable controversy in the literature. The three principal interpretations of the transition seem to have been as follows: (a) Chum flow isas seen an entrance regi on phenomenon. Information this view ofof chum flow, theflowregime as part of the process of the stabl e further downstream in the pipe (Dukler and Taitel; 1986; Mao slug and Dukl e r, 1993; Tai t el et aI. , 1980). Slug floinw which breakdown into chum flow is longer due to critical void This fractionclassconof (b) ditions the liquid slugs are no sustainable. models includes (1984),inwho assumed the transition occurredthatwhenof Mishima the averageandvoidIshiifraction the pipe is justthat greater than the(1986), mean voidwhofraction overthatthetheTaylortransition bubble occurred region, andwhenBrauner andd Bamea assumed the voi fraction in the liquid slugs reached a given value, namely 0.52. (c) byOnsetNicklin of flooding in the Tayl(1962) or bubbl efurther . This mechanism wasMcQuillan first proposed and Davidson and developed by and Whal l e y (1985), Govan et al. (1991), and Jayanti and Hewi t t (1992). Within the Taylor bubble, a liquid film falls downwards around the bubble, and in the flooding mechanism, breakdown of slug flow occurs when the gas velocity in the waves bubbleareis sufficient rise tofilm,flooding y, a situ ation in which formed ontothegivleiquid which (namel are transported by the gas phase, leading to the collapse of the slugs between the lupwards arge bubbles). Slug flows with short bubbles tendstateto beis rather unstable, withcircumstances the large bubbles coalescing unti l an equilibrium reached. In these (and also as the chum flow boundary is approached), the slug flow i s clearly demon strating an instability. It is important to recognize the distinction between un stable slug flow and churnflow, and this distinction has been explored in more detail byvoidHewitt and models Jayanti may (1993)applyand atCostigan andliquid Whalley (1997). The critical fraction very high mass fluxes, but itconditions. seems probable that the flooding mechanism is the governing one for most detail edifdiscussion of the transition to chum flowTheandlargeof thewaveschurnassociated flowmore regime itself given by Jayanti and Brauner (1994). with flooding appear to persist into the chum flow region, A

A

210 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION .. --- ---,-------,----, 2000 ........-

� "-

I Annular -;-Flow

I I I I I

1 500

3 N � a.

l'

c: Q)

1000

:a e (!)

�

::s en en

� a..

500

o

10

20 Gas Flu x

I m g [kg/n!s]

30

Figure 4

Data for pressure gradient in churn flow (Govan et al., 1 99 1 , with permission).

these waves being levitated by the gas phase with regions of falling film flow ing themis(Hewitt al. , 1 film 985). Thus, chum flow is similar to annular flowbetween in that there a wavyetliquid on the wall and a gas core in the center. However, as the gas flow increases, the floodingwithwaves are gradually dampedas out, leading to a reduction in pressure gradient increasing gas velocity, illustrated1 993) in Fithat gurechum 4. Due to this peculiar behavior, it is argued (Hewitt and Jayanti, flow ofshould be regarded as a specific regime rather than considering it as a subset annular flow. (3) Chum annular transition. In the light ofthe above discussion on the slug-churn transition, itinseems rational to regardgradient the transition fromincrease chum with flow increasing to annular flow as one which the pressure begins to gas mass flux, the disappearance flooding-type transition can besignaling approximately represented byof thethe condition that:waves. The (9. 1 - 0

where is the acceleration due to gravity and D the tube diameter. flosubws Though downward the are important ject of much less(particularly investigationin condensation than have vertiapplications), cal upflows. they Studieshaveofbeen flow patterns g

9.1.2.2 Flow patterns in downftow in vertical tube.

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 211 2

Annular Mist and

�

�

1 .6

g

1.2

� C)

0.8

>l:

!

..J �

U �

Annular Flow

Slug

0.4

o

0

Figure 5

0.1.

and Bubble Flow

1.2

SUPERFICIAL l I0UID VELOCITY

1 .6

U L m/s

2.0

Flow patterns in vertical downftow (from Golan and Stenning, 1 969, with pennission).

in co-current downward flow are reported by Golan and Stenning ( 1 969), Barnea et al. ( 1982), and Crawford et al. ( 1 986). The most important feature of such flows is the greater propensity to annular flow compared to upward flow systems. The flow pattern map given by Golan and Stenning is shown in Figure 5. Barnea et al. describe the annular to slug transition in terms of a critical liquid film thickness (obtained from an annular flow model) and describe the transition from slug flow to dispersed bubbly flow in terms of breakup and coalescence of bubbles in the turbulent flow field. Crawford et al. discuss both steady-state and transient flow patterns for this type of flow. 9.1.2.3 Flow patterns in horizontal tubes. The flow patterns commonly defined for horizontal flows are illustrated in Figure 6. The regimes tend to be more complex than thos e in vertical flows due to the asymmetry induced by the gravitational force acting normal to the direction of flow. The regimes shown in Figure 6 are defined as follows: ( 1) Stratified flow. Here, the gravitational separation is complete with the liquid flowing at the bottom of the tube and the gas along the top part of the tube.

212

�

Ii • •• •

HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION F low direction --

�

' • • • • • I. . ....• . I . • •• • 0 :0 0 : \0:

Dispersed b ubble f low

?

Stratified [ ,,11_ ______..::] ...» f low

-""? wavy f low

(J"' IL � ..JI _ _ __ __ � t _ _ _ __..J"'o

� co:;:>

z;::::; r fJ ""' _ ______ O

Stratif led -

Plug f low

� semislug flow . •

' : . '"

• • •

' . '

. •

Intermittent

f lows

Annular

>""":·:II: f1I! t.-·.·!= : · ·.� : ·:·O:;=:·:-=;:.:::;; · : =. - : 0-1 :':6 �:�e:rsed . · . := . ·: . ;:= Figure 6

(2)

Flow patterns in horizontal flows (from Hewitt, 1 982, with permission).

Stratified-wavy flow. At higher gas velocities, waves are formed on the gas liquid interface, giving the stratified-wavy or wavy flow regime. (3) Dispersed bubble flow. Here, the bubbles are dispersed in the liquid phase; in horizontal flow, these bubbles tend to congregate near the top of the tube, as illustrated. (4) Annular-dispersed flow. This flow regime is similar to that observed in vertical flow, except that the film thickness is non-uniform. The film is generally much thicker at the bottom of the tube than at the top due to the effect of gravity. (5) Intermittent flows. Though it is often appropriate to treat all intermittent flows as being of a single generic type, the class of intermittent flows is often con veniently divided into three subdivisions as follows: (a) Plug flow. Here, the characteristic bullet-shaped bubbles are observed, but they tend to move along in a position closer to the top of the tube, as shown in Figure 6. (b) Slug flow. Here, frothy "slugs" pass along the pipe containing entrained gas bubbles.


. � 1 000 and m > 1 00, the Chisholm ( 1 973) correlation should be used. (3) For rJL /rJ C > 1 000 and for m < 1 00, the Lockhart-Martinelli ( 1 949) correla tion should be used.

9.1.5 VOID FRACTION As was explained in Section 9. 1 .3 above, the void fraction is an important parameter in calculating total pressure drops using the separated flow model, since it appears in the terms for both accelerational and gravitational pressure drop. The last term is zero for horizontal pipes, but even for horizontal pipes, knowledge of void fraction is needed to calculate the accelerational pressure drop term (though this is often quite small, compared to the frictional pressure gradient). However, void fraction is also important in accessing the inventory of valuable and/or toxic material within pipe systems. This inventory is significant both in fiscal terms and also from a safety point of view. Again, a wide range of relationships have been proposed for void fraction, and it would be beyond the scope of the present chapter to deal with these in detail. Nevertheless, some relationships will be given that may be used over reasonably wide ranges. Again, no details are given regarding the phenomenological models (which generally predict void fraction in addition to pressure gradient); the reader wishing to pursue this (potentially more accurate) methodology should refer to the references cited in Section 9. 1 . 3 .4 above.

9.1.5.1 Homogeneous void fraction. For the homogeneous model, the void frac tion is equal to the flow volume fraction of the gas and is given by:

Pc ( l

- x) +

PLX

(9. 1 -6 3)

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 229 The void fraction approaches that calculated from the homogeneous model at high pressure and high mass flux, but in normal circumstances, there is a large overprediction of the void fraction, since the gas phase is traveling more rapidly than the liquid phase. 9. 1.5.2 Drift flux model. The drift flux model is based on the idea that effective relative motion occurs between the two phases for two reasons: ( 1 ) The gas phase at any one point in the cross section is traveling at a higher velocity than the liquid phase. Averaged over the cross section, the relative velocity is UGU . (2) Even if there is no local difference in velocity between the two phases, the weighting introduced by there being differences in the profiles of void fraction and velocity across the cross section results in an average effective slip between the phases. This can be accounted for through a distribution parameter Co. More detailed discussions of the drift flux model are given by Hewitt ( 1 982) and Wallis ( 1 969). The slip ratio and void fraction are given in terms of the parameters Co and UGU by the following equations:

-

( 1 - EG ) 1 /(Co + UGU / U) EG XP: L EG = ----Co[XPL + ( 1 x) PG] + PLPG u GU /m S =

UG UL

=

-

(9. 1 -64) (9. 1 -65)

It should be noted that for the limiting case where Co = 1 .0 and UGU = 0, the above expressions reduce to those for the homogeneous flow model. The value of Co varies with the flow regime. For instance, in fully developed bubble and/or slug flow, Co = 1 . 1-1 .2 and Co approaches 1 .0 for qualities approaching unity. For void fractions greater than 0. 1 , Rouhani ( 1 969) suggests the following expressions for the drift flux parameters:

Co UGU

= 1 + 0.2(1

( )

g DP 2 - X ) ---;L ;;r-

= 1 . 1 8 [ga ( PL

0.25

/ ] 0.25 ( 1 - x)

- PG ) PL2

(9. 1 -66) (9. 1 -67)

A more recent correlation based on the drift flux model is that of Chexal and Lellouch ( 1 986).

9.1.5.3 Correlations associated with the separated flow model. Lockhart and Martinelli ( 1 949) presented a graphical relationship between void fraction and the Martinelli parameter X (defined by Eq. 9. 1 -49), as illustrated in Figure 1 3 . This

230 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION £1 \G

1 .0

"""'"

� 0 .1

o

0.0 1

Figure 13

0.10

..... I!I....

r--"

10

1.00

PAR AM E T E R X

100

Lockhart-Martinelli ( 1 949) correlation for void fraction (from Hewitt, 1 982, with permis

sion).

relationship is fitted by the equation: ce

¢L - 1 = --¢L

(9. 1 -68)

where ¢ L may be obtained from Figure 1 2 or calculated from Eq. (9. 1 -50) using the values of C given in Table 1 . The Lockhart-Martinelli correlation for void fraction tends to overpredict the void fraction at high mass fluxes, and a number of correlations have been proposed to offset this problem. The correlation of Premoli et al. ( 1 97 1 ) covers a reasonably wide range of data and takes into account the mass flux effects. The correlation i s in terms of slip ratio S; void fraction may be calculated from S using Eq. (9. 1 - 1 0) . The Premoli e t al. correlation has the form:

S = I + EI where

(

1

y=

Y

+ yE2

- yE2

fJ 1

-

)

1 /2

fJ

( ;� )

where fJ is the volume fraction of the vapor in the flow parameters E I and E2 are given by the equations:

EI =

1 .578Re- 0 . 1 9

E2 =

0.0273 We Re- 0 . 5 1

0. 2 2

(fJ =

-0.08 ;� ( )

(9. 1 -69)

(9. 1 -70) Ve / ( Ve + Vr ) ) . The

(9. 1 -7 1 ) (9. 1 -72)

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 231 where the Reynolds and Weber numbers are defined as follows:

mD 'f/L m2D We = -apL Re = -

(9. 1 -73) (9. 1 -74)

9.1.5.4 Void fraction in subcooled boiling. The correlations given above assume a knowledge of the gas and liquid flow rates. Usually, it is assumed that these may be calculated from a knowledge of the fluid enthalpy (see Eqs. 9. 1 - 1 1-9. 1 - 1 3), al lowing the calculation of a the rmodynamic equilibrium quality from Eq. (9. 1 - 1 4), which we will define here as Xe . As was discussed earlier, void formation can occur due to subcooled boiling in the region where Xe < 0 (i .e., at neg ative thermody namic qualities). The variation of void fraction with position along a uniformly heated tube is illustrated schematically in Figure 14. The onset of nucleate boiling occurs at a distance of Xn where the quality is x (Zn ) , but significant void formation does not occur until the bubbles begin to depart from the surface at a distance Zd with a corresponding quality of x (Zd ) . Eventually, the point i s reached where Xe = 0 at Z = Zbu lk . However. at this point,

"-'-"�"'I �...---

SUBCOOL E D ____��B O I LING

F L OW I I � I X (Z n )

• (z d )

REGfON I

A

•

� Y

WALL Y OIDAGE

�

�

Q

�

o

Figure 14

n

Zd

Z INIII

Dt STANCE ALONG HEATED SUAFACE Z

Zeq

Void formation in subcooled boiling (from Hewitt, 1 982, with permission) .

232 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION vapor and subcooled liquid coexist, and it is not until distance Zeq is reached that the flow approximates to an equilibrium one. Probably the most widely used methodology for predicting void fraction in subcooled boiling is that of Saha and Zuber ( 1 974). The steps in the methodology are as follows: ( 1 ) Estimate the point of bubble departure (point of significant void generation). Saha and Zuber give the following expressions for the local thermaldynamic equilibrium quality at the point of bubble departure as follows :

Xe (Zd) =

q Dc L -0.002 --P-

for Pe < 70000 (9. 1 -75) h LC A L q (9. 1 -76) Xe (Zd) = - 1 54 -.- for Pe < 70000 mh LC where Cp L is the specific heat capacity of the liquid and A L its thermal con ductivity. The Peclet number is defined as: Pe =

rh DCp L --AL

(9. 1 -77)

(2) Calculate the actual quality x (z). Any model for the calculation of flow quality must satisfy the conditions that x (z) = 0 at x = Zd and that x (z) should approach xe (z) in the quality region, as illustrated in Figure 1 4. A relationship developed by Levy ( 1 966) that has been found to give good results when applied to the prediction of subcooled void fraction is as follows:

xe (Z ) (9. 1 -78) x (z) = xe (z) - Xe (Zd) exp -- 1 Xe ( Zd) Calculate the void fraction at position Z from the value of x (z) estimated from

]

[

(3)

Eq. (9. 1 -78). Here, any standard void fraction correlation may in principle be used, but the most popular approach is to use the drift flux model, since the constants in the model can be chosen to reflect the somewhat peculiar distribution of voids that occurs in subcooled boiling. Thus, from Eq. (9. 1 -65), it follows that:

X (Z) PL CC = ----Co {X (Z) PL + [ 1 - x (z) ]Pc } PL PC U CU /rh Appropriate values of Co and Ucu are given by Dix ( 1 97 1 )

+

(9. 1 -79) and Lahey and

Moody ( 1 977) as follows:

[

(9. 1 -80)

25 _ 2 9 ( PL - pc ) ag 0 . u cu - . 2 PL

]

(9. 1 -8 1 )

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 233 where f3 is the volumetric flow fraction of the vapor phase, which is related to the local actual flow quality x (z) by the expression: f3

=

x (z) + [ 1

x (z) - x (z)] Pc / PL

( )00 1

and where the exponent b is given by:

b = Pc PL

(9. 1 -82)

(9. 1 -83)

The generalized drift flux model of Chexal and Lellouche ( 1 986) can also be applied to the prediction of subcooled void fraction.

9.2 FLOW IN OTHER GEOMETRIES 9.2.1 Introduction Two-phase flow in non-circular channels occurs in many industrial equipment! processes, such as tube bundles of shell-and-tube heat exchangers, fuel rod bun dles of nuclear reactors, compact heat exchangers with narrow finned channels, and various types of chemical reactors and mass transfer equipment. This section describes the two-phase flow-regime, void fraction, and frictional pressure drop in various non-circular flow channels, including single channels with rectangular, triangular, and annular cross-sections, a flat channel between parallel plates, and interconnected channels found in rod bundles. The characteristics of the two-phase cross flow in tube bundles will be discussed separately in Chapter 1 2. A comprehensive list of the literature available to date and covered in this chap ter is shown in Table 2. Many similarities exist in the two-phase flow characteristics between circular pipes and non-circular channels with a hydraulic diameter, Dh , greater than about 1 0 mm, as summarized by Sadatomi et al. ( 1 982) and Sato and Sadatomi ( 1 986). However, significant differences appear in non-circular channels when the hydraulic diameter falls below 10 mm, as reviewed by Kawaji and Ali ( 1 996). Thus, in this section, the two-phase flow characteristics in non-circular channels will be discussed for relatively large channels with Dh > 10 mm and narrow channels with Dh < 10 mm. Narrow channels with internal fins are also used in various types of compact heat exchanger, and their two-phase flow characteristics have been reviewed by Westwater ( 1 983), Carey ( 1 993), and Kawaji and Ali ( 1 996). The popular fin geometries include cross-ribbed fins, offset strip fins, and round dimple fins. The effects of fins on the two-phase flow characteristics in narrow flat channels will also be summarized, based on the available data listed in Table 3 .

N (.;.I �

Rectangular 17 x 50 1 0 x 50 7 x 50 7 x 20.6 Triangular Apex-angle = 200 , height = 55 Annulus, OJ = 1 5 , Do = 30 Venkateswararao 4 x 4 rod bundle, Rod-diam. = 1 2.7 et al. ( 1 982) Pitch = 1 7 .5 Square in-line Hasan & Kabir Annulus 0.0. = 1 27, ( 1 992) 1.0. = 48, 57 , 87 Concentric and Kelessidis & Eccentric Dukler ( 1 989) Annulus 0.0. = 76.2, 1.0. = 50.8 Annulus Caetano et al. ( 1 992a, 1 992b) 0.0. = 76 1.0. = 42

Sadatomi et al . ( 1 982)

Investigators

Channel cross-section, (mm)

No

Yes

AirlWater

AirlWater

AirlWater Airl Kerosene

V-V , V-I.

V-v.

V-V .

N/A

N/A

N/A

1 8.0

79, 70, 40

34

Yes

Yes

N/A

1 5 .0 AirlWater

V-V.

N/A

Yes

Yes

No

No

Yes

Yes

Yes

No

Yes

No

Yes

Void fraction

Parameters investigated Flow Press. regime drop

1 6.3

AirlWater

Fluid mixture

2.9 5.0 7. 1 2.9

V-v.

Aspect Flow 2 orient ratio I

25.4 1 6.7 1 2.3 1 0.4

Hydr. dia., Dh (mm)

Table 2 List of previous studies on two-phase flow in non-circular channels

Void fraction measured by quick closing valves. Flow regime identification: Visual observation

Flow regime identification: PDF of conductance probe signals

Flow regimes identified visually. Bubble velocity measured by high-speed photography. Bubble rise velocity measured in stagnant liquid column.

Void fraction measured with a pair of quick shutoff valves. Flow regimes identified from slug frequency.

Notes

N 1M til

0.95 1 .3 1 2. 1 4

1 .38 3.14 7.26 9.27

0.994 1 .975 3 .90

1 .96 4.58 9.09

0.5 x 1 0 0.7 x 1 . 2 1 .2 x 1 0

0.7 x 40 1 .6 x 80 3 . 8 x 80 63.5 x 4.98

0.5 x 80 1 .0 x 80 2.0 x 80

1 .0 x 50 2.4 x 50 5.0 x 50

Fujita et al. ( 1 994)

Iida & Takahashi ( 1 976)

Lowry & Kawaji ( 1 988)

Mishima et al. ( 1 99 1 )

Jones & Zuber ( 1 975)

1 .54 2.88

0.778 x 80 1 .465 x 80

Ali et al. ( 1 993)

50 20.8 10

1 60 80 40

57. 1 50 21.1 1 2.7

20 14 8

1 02.8 54.6

V-U.

V-U.

V-U, H-H. H-H. H-H. V-U.

H-H.

V-U, V-D, H-H, H-V, I-U, I-D.

AirlWater

AirlWater

AirlWater

AirlWater

N 2 gaslWaterEthanol Solution

AirlWater

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

(Continued)

Flow regime detection: PDF3 of void fluctuations. X-ray void fraction measurement. Flow regime detection: Photographic Void fraction measurement: electrical conductance probe. Flow regime: High speed video Void fraction: Neutron Radiography (NRG) & Image processing

Flow regime detection: PDF of void fluctuations; Void fraction measurement: electrical conductance probe. Flow regime detection: Video camera; Void fraction measurement: Capacitance probe. Studied the effects of liquid viscosity and surface tension. Reported 'New' flow pattern called Fissure Flow.

N � e'I

with D h

>

10

mm.

1 0 mm


AirlWater Data I , I , BIS \ , ..;l . ...,

-- Venkateswararao et

( 1 982)

SteamlWater Data Bergles et al. ( 1 968) 6.9 MPa Williams - - - - 2.8 MPa

0.1

aI.

& Peterson ( 1 978)

--------- 8.3 MPa -.-.-

1 3 .8 MPa

.. .-&....... ...1. -----1 .. 0 . 0 11-----1.-.11.-----1-..1... 0.1 1 .0 10.0 0 . 01

Figure

16

Flow regime maps for vertical flow i n rod bundles (B -bubbly, S-slug, C-churn, F-froth,

A-annular).

slug-annular flow transition. Caetano et al . ( 1 989a, 1 989b) presented flow regime maps for a vertical annulus obtained for air/water and airlkerosene. Kelessidis and Dukler ( 1 989) have presented flow regime data for concentric and eccentric an nuli. For a rod or tube bundle geometry, a flow regime map for air/water flow in a vertical, 24-rod bundle with a subchannel hydraulic diameter, Dh = 1 8 .0 mm, was reported by Venkateswararao et al. ( 1 982), as shown in Figure 1 6. All of the above data have been obtained at near atmospheric pressures. Compared to the flow regime maps available for circular tubes, non-circular channel data show little difference if Dh > 10 mm. Typically, at low gas and high liquid flow rates, bubbly flow occurs, and at high gas flow rates and low liquid flow rates, an annular flow regime is observed. At intermediate gas and liquid flow rates, intermittent flow regimes, such as slug and churn-turbulent flows, occur. At high pressures, flow regime maps have been obtained for four-rod bundle geometries by Bergles et al. ( 1 968) and Williams and Peterson ( 1 978) by boiling water at pressures up to 1 3.8 MPa, as shown in Figure 1 6. In all of the high pressure data, transition from intermittent to annular flow occurs at much lower superficial gas velocities of about 1 mis, in comparison with 10 mls at 1 bar.

Channels with Dh < 10 mm. Flow regime maps for narrow channels, such as rectangular and flat channels, are generally similar, but the flow regime transition boundaries show some differences due possibly to the gradual nature of transition, differences in flow regime identification methods used, channel orientation, Dh , and aspect ratio. The flow regime maps obtained by Ali et al. ( 1 993) are shown in Figure 1 7 for narrow channels between two flat plates with gaps of 0.778 mm and 1 .465 mm for different channel orientations, excluding horizontal flow between Vertical plates (H-V). In the data, annular flow is observed at high gas velocities

240 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION lO �________________________________________� --- GAP = 1 .465 mm � - - - - - - - . GAP = 0.778 mm �az%Z

BUBBLY

�

./ -'/-

-

......

INTERMITTENT '� ..... H-H

- - � '-

(R-IVULE-T-�--H-H"

:=�

/

ANNULAR

0 . 1 �__�__��______��____� 20 0. 1 10 1

jG

m/s

Figure 17

Flow regimes in narrow flat channels for all orientations excluding H-V (from Ali et al . . 1 993, with permission).

(jG > 4 mls) and low to intennediate liquid velocities (k < 1 mls). Bubbly flow occurs at high liquid (jL > 1 mls) and low gas velocities. Between the annular and bubbly flow regimes, intennittent flow exists over a wide range of gas and liquid flow rates. There are some differences among the flow regime maps reported for narrow flat and rectangular channels that may be attributed to the differences in channel orientation and geometry, such as gap thickness and the channel width or aspect ratio. Overall, however, the flow regime maps reported to date for narrow channels show generally compatible results. The flow regime boundaries are also similar between small rectangular channels and small circular tubes in horizontal and possibly vertical orientations, based on a comparison of data by Wambsganss et aI. ( 1 99 1 ) with the 1-5 mm diameter circular tube data of Damianides ( 1 987) and Fukano et aI. ( 1 989). A major difference that can be readily identified in the flow regime maps between the flat narrow channels and non-circular channels with larger hydraulic diameters (Dh > 10 mm) is in the transition from intennittent to annular flow regimes. In the larger channels, this transition occurs at a superficial gas velocity, jG, greater than 1 0 mis, for all liquid flow rates. In flat narrow channels, th i s transition can be observed at j G values between 3 and 1 0 mls for low to moderate liquid flow rates ( j L < 1 mls) for all channel orientations, excluding the horizon tal flow between vertical plates (H-V) (Lowry and Kawaji, 1 987; Mishima et aI., 1 99 1 : Wambsganss et aI., 1 99 1 ; Wilmarth & Ishii, 1 994). Effects of channel orientation and fluid properties. For circular and non-circular channels, the flow regime transition boundaries are sensitive to the channel

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 241 l0 r-------, --- GAP = 1 .465 mm

- - - - GAP = O.778 mm

/

BUBBLY

/

I

INTERMITIENT

-------� STRATIFIED-WAVY

jG

m/s

Figure 18 Flow regime maps for horizontal flow between vertical plates (H- V) (from Ali et aI., 1 993, with permission). orientation. In a horizontal channel, gravity can separate gas and liquid phases, and stratified or wavy flow appears at low liquid flow rates in addition to bubbly, plug, slug and annular flow regimes. Even for narrow channels, horizontal flow between vertical plates (H-V) presents completely different flow regime maps, as shown in Figure 1 8. As in larger pipes and channels, stratified flow occurs at k below about 0.3 mls for the 1 .465 mm gap and 0.5 mls for the 0.778 mm gap over the whole range of jG values up to 1 6 mls. Annular flow was not observed even at the highest jG tested ( 1 6 mls), due possibly to the negligible interfacial area and shear. If the vertical wall of the channel is not too high « 20 mm), however, an annular flow regime can occur at jG > 1 0 mis, as evident in the horizontal flow data of Wambsganss et ai. ( 199 1 ) and Wilmarth and Ishii ( 1 994) for small rectangular channels. For narrow channels, the effects of the fluid properties have been reported to be relatively small. The effect of liquid viscosity on two-phase flow regimes has been found to be insignificant for narrow rectangular channels in vertical and horizontal orientations (Troniewski and Ulbrich, 1 984). As surface tension is decreased, the intermittent-annular transition boundary shifts to the lower jG values (Fujita et aI., 1994). This means that in low surface tension systems in narrow channels, annular flow regime would occur over a wider range of gas flow rates, compared to the air/water system at atmospheric pressure. Effect of fins. For various finned narrow channels found in compact heat exchang ers, two-phase flow regime maps have been obtained for cross-ribbed fins, offset strip fins, and round dimple fins, as listed in Table 3. The flow regime maps by Rong et al. ( 1 993) for different channel orientations shown in Figure 1 9 are qualitatively

N "'" N

0. 1

0.1

F

Rivulet

B ubbly .

Fine

Chan�1 No.

B ubbly .

JG

-

10

V

10

Wispy -An n u lar

Orientation

(m/s)

Intermittent

D

H - H

Wispy-Ann ulor

Orientation

0. 1

0.1

Fine Bubbly

C han nel No.

B ubbly .

Fine Bubbly

C hannel No.

IG

- U

- V

10 H

(m/s)

10

Wispy-An n ular

Orientation

Interm ittent

I ntermitten t

V

Wispy-Ann u lar

Orientation

Flow regime maps for a finned channel with round dimple fins in different channel orientations (H-H: Horizontal flow between horizontal plates. V-D: Vertical downward flow. V-U: Vertical upward flow. H-V: Horizontal flow between vertical plates) (from Rong et aI . , J 99.3 . with permission).

Figure 19

� I-)

"-'"

a

..-.. 00

�

"-'"

a

..-.. 00

Fine

Cha nnel No.

FLUID MECHANICS ASPECTS OF TWO-PHASE FLOW 243 similar to those obtained for narrow smooth channels; however, several significant differences are found due to the enhanced turbulence levels in the two-phase mix ture. For example, a fine-bubbly flow regime occurred for all channel orientations when the superficial liquid velocity ( jd exceeded 1 .3 mls or so, rivulet flow was not found in horizontal flow, and annular flow occurred at higher gas velocities for all four orientations. The above differences can be attributed to the fact that the liquid is slowed down by the fins, and some liquid films are broken up into large droplets. Also, some gas bubbles are trapped in the wake of the fins due to the surface tension force and may not move even in the presence of a moderately strong liquid flow (Carey, 1 993). Flow regime transition correlations. In order to predict the flow regime transi tions, various theories have been proposed for circular pipes with diameters greater than 1 0 mm. Widely used theories have been proposed by Taitel and Dukler ( 1 976) for horizontal flow, Taite1 et al. ( 1 980) and Mishima and Ishii ( 1 984) for vertical flow, and Bamea ( 1 987) for inclined pipes, among others. These models and other correlations developed for circular pipes can also be used for non-circular channels if the hydraulic diameter is greater than about 1 0 mm. For flat narrow channels, the applicability of the existing correlations is less certain, and correlations specific to small channels have been developed, as sum marized in Table 4. For slug-annular transition in a narrow rectangular channel in vertical upward flow, the correlation by Jones and Zubers ( 1 979) can predict well their own data obtained in a 5 mm x 63 .5 mm rectangular channel and that of Mishima et al. ( 1 99 1 ) obtained in 1 mm and 2.4 mm gap channels. For bubbly-slug flow transition in a narrow, vertical rectangular channel, the models of Taitel et al. ( 1 980) and Mishima and Ishii ( 1 984) are recommended. For the latter's correla tion, a drift flux model (described later in this section) is used to determine the values of k and jo that would give 0.3. For intermittent-annular flow transition in a horizontal rectangular channel in H V orientation, Wilmarth and Ishii ( 1 994) found that the modified model of Bamea et al. ( 1 983) is applicable, where the height of the gas phase, ho, must be predicted using a void fraction correlation.

E=

-

Table 4 Flow regime transition correlations for narrow channels Flow regime transition

Channel orientation and flow direction

Recommended correlation

Reference

Slug-Annular Vertical Upward

k

Jones & Zuber ( 1 979)

Bubbly-Slug

Vertical Upward

Slug A nnu lar

Horizontal (H- V)

jL = 3.0jG - 1 . 1 5 (gtJ.pa j pl)O.75 E = 0.3 hG < (rrj4) J [aj PLg( l - rr j4)]

-

= 0.25jG - Vg j where Vg j = (0.23 + 0. 1 38jW) J( tJ.pgWj pd

Taitel et al. ( 1 980) Mishima & Ishii ( 1 984) Bamea et al. ( 1 983)


9.2.3 Void Fraction The void fraction or liquid holdup in the flow channel is another important parame ter in two-phase flow, as it affects the flow regime, pressure drop, and heat transfer characteristics. In this section, the experimental data available and methods of prediction will be discussed first, followed by the effects of channel geometry, orientation, size, and fluid properties. Mean void fraction. The channel average void fraction data for non-circular chan nels have been successfully correlated in terms of the Lockhart-Martinelli param eter, described in the previous section or in the form of a drift flux model (Zuber and Findlay, 1 965). For example, the void fraction data for flat channels obtained by Ali et al. ( 1 993) are shown in Figures 20 and 2 1 for horizontal flow. They are fairly well correlated in terms of the Lockhart-Martinelli parameter, X2 , which for both phases turbulent is given by (9.2- 1 ) where x i s the mass quality (Lockhart and Martinelli, 1 949). A widely used correlation proposed by Chisholm and Laird ( 1 958) to predict the mean void fraction in pipes and channels with Dh > 1 0 mm is given by (9.2-2) The value of C ranges from 5 to 2 1 , depending on whether each phase is laminar or turbulent. This correlation is indicated by solid curves for several values of C in

HORIZONTAL FLOW BETWEEN HORIZONTAL 0.8

--

= 0

GAP

=

1 .465

Equation 9.2-2

-.: 0 . 6 CJ = I..

o �

.... o

0.2

0

0. 1

Figure 20

jL m/s o

� "0 '0 0 . 4 �

10

X

1 00

PLATES

mm

0. 1 9 0.5 1 .0

1 .9

1 0 00

Void fraction data for horizontal flow between plates (H-H) for 1 .465 mm gap channel (from Ali et aI., 1 993, with permission).


HORIZONTAL FlOW BETWEEN HORIZONTAL PLATES GAP

0.8

t�

c �

..

=

0.778

mm

Equation 9.2-2 iL m/s

0.6

o

� "0 0 . 4 ·S ;;>

o t:. • o

0.2 0 0. 1

X

10

Figure 21

0.30 0 . 48

0.96 1 .9

3. 1

1 000

1 00

Void fraction data for horizontal flow between horizontal plates (H-H) for a 0.778 mm gap channel (from Ali et aI . , 1 993, with permission).

Figures 20 and 2 1 . The mean void fraction data in Figure 21 obtained by Ali et al. ( 1 993) can be predicted reasonably well with C � 20 for > 0.3 and C � 10 for E < 0.3 . Another widely used approach i s the drift flux model (Wallis, 1 969; Zuber and Findlay, 1 965), given by

E

UG

= jG/E = Co ( j) + Vgj

(9.2-3)

where UG is the weighted mean gas velocity, (j) is the mixture mean velocity ( = j G + jd, Co is the distribution parameter, and Vgj is the mean drift velocity, which represents the difference between the gas velocity and the mixture mean velocity and is usually considered to be a function of the terminal rise velocity of a bubble in a stagnant liquid. The void fraction data for vertical upward flow in triangular and rectan gular channels (Sadatomi et al., 1 982), narrow rectangular channels (Mishima et al., 1 988, 1 99 1 ), and a 4 x 4 rod bundle (Morooka et al., 1 987) are shown in Figures 22-24. The distribution parameter, Co, which accounts for the differences in velocity and void fraction profiles in the channel cross section, varies with the channel geometry, ranging from 1 .068 for a rod bundle to 1 .34 for a triangular channel. For non-circular channels with Dh > 10 mm, Sadatomi et al. ( 1 982) rec ommended the values given in Table 5. For narrow rectangular channels, Mishima et al. found the correlation by Ishii ( 1 977) to be adequate for rectangular channels, Co

=

1 .35 - 0.35 ,J( PG / pd

(9.2-4)

Ali et al. ( 1 993) found Co = 1 .25 for a narrow flat channel. In extremely narrow channels with gap thicknesses less than 0. 1 mm, Moriyama and Inoue ( 1 99 1 )

246 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Table 5 Distribution parameter values for non-circular channels with Dh > 10 mm (from Sadatomi et aI., 1982) Shape

Dimension (mm)

Rectangular,

17 x 50 10 x 50 7 x 50 7 x 20.6 20° , h = 55

A is defined in subcooled region,

not valid for small Nsub , verified by experiments and extended by Saha( l 974) to include subcooled boiling

Am = 2AI is recommended.

Remarks

}"'L/ (2d). qv: volume heat flux, f}.hsub: subcooled enthaply, }... : friction factor,

uniform heat flux, subcooled, boiling, & superheated vapor regions

uniform heat flux, subcooled & boiling region

Condition

Validity

frictional pressure drop dominant, constant pressure drop between plenums

frictional pressure drop dominant. constant pressure drop between plenuns

System

Table 2 Simplified stability criterion for density wave oscillation

N -...J ="

Heat flux

not considered not considered not included included

homogeneous model

not included

not considered or considered in BWR model

Channels. Internal Report of Dept. Nuclear Eng. & Eng. Physics, Rensselaer Polytechnic Institute,

1990.

5 : See Takenaka et al. ( 1 99 1 ), which is based on Numerical Study on Non-Linear Two-Phase Flow Dynamics with Neutron Feedback and Fuel Heat Transfer in a Natural Circulation Loop and Parallel

4: Lumped-parameter analysis with several nodes. dynamics is expressed by ordinary differential equations, applied to discussion on non· linear dynamics. including chaos.

3 : Applicable to SG in FBR, various extended versions exit.

2: Detailed discussion is found in this handbook.

not considered

not considered included slip flow model not considered

considered

uniform or heat exchanger mode constant in space but changable in time considered or not considered

not considered included

slip flow model

not considered not considered

included

not included

not included included

not considered

Nuclear feedback

included

Superheated region

considered

considered considered

homogeneous model homogeneous model drift-flux model drift-flux model, homogeneous if superheated region exist slip flow model slip flow model

Two-phase model

considered

considered considered

not considered

considered in S.R. but not in B .R. constant heat input considered in S.R. but not in B.R. considered not considered

not considered

Subcooled boiling

not included

Heater dynamics

arbitrary

arbitrary arbitrary

I: Typical linearized stability analysis.

.

Non-linear and time domain HYDNA code (Currin et al. 1 96 1 ) DEW code (Takitani & Sakano, 1 979) Lahey et a1 4 ( 1 989, 1 99 1 5 , 1 992)

STABLE (Jones, 1 96 1 ) DYNAM code3 (Efferding, 1 968)

Linearized andfrequency domain uniform LOOP code I (Davis & Potter, 1 967) arbitrary in S .R., NUFREQ code (Lahey & uniform in B.R. Yadigaloglu, 1 974) uniform Saha ( 1 974) uniform Nakanishi et al . 2 ( 1 978)

Code or Authors

Table 3 Codes and analysis for flow stability

1WO-PHASE FLOW INSTABILITIES 277

k,=1

C\I

2

+

\-.l

\-.l

-

>-

2

C\I

+

-

>-

0

(a )

0

y( r )

(b)

y( r )

C\I +

Figure 9

(c)

y( r )

Pseudo-phase plane traj ectories of pressure drop oscillation (a) Steady limit cycle of pressure drop oscillation. (b) Torus. (c) Chaos.

Non-linear dynamics canis bemodeled well understood, but only ifofthefreedom boilingsystem. channelForor two-phase flow system as a limited-number relevance to the readers, one of the examples is shown in Figure 9. This figure showsadiabatic pseudo-phase planeflowtrajectori es obtained by the numerical simulation ofof the two-phase system with two compressible volumes upstream theandchannelrepresent of negative resistance (Ozawa 1996). The parameters the dimensionless velocity Umekawa, and time, respectively, similar to Eq. (10-7), while the equation system is slightly different from the equation, owing y

&

T


Thebetween dimensionless parameter krvolumes represents thetheto thetwo-phase restriadditional ction channel. tocompressible controlThethelarge involume. teracti o n two compressible and value of kr gives a simple limit cycle oscillation (a), i. e . ,then the pressure drop oscillation, but the smalltendency value ofhaskrbeen realizeswelltheveritorus (b)by experiment, and chaotic behavior (c). The qualitative fied whi l e the cri t i c al revi e w on such a non-linear approach may give rise to ofa serious question as toand/or whatvarious kind ofdimensions physical model is reconstructed by means the strange attractor of chaos. Further research is strongly encouraged in this new field of flow instability. REFERENCES Boure, J. A, Bergles, A E., and Tong, L. S. 1 97 1 . Review of Two-Phase Flow Instability. ASME paper, 7 1 -HT-42. Currin, H. B., Hunin, C. M . , Rivlin, L., and Tong, L. S. 1 96 1 . HYDNA-Digital Computer Program for Hydrodynamic Transients in a Pressure Tube Reactor or a Closed Channel Core. CVNA-77, Westinghouse Electric Co. Davis, A. L. and Potter, R. 1 967. Hydraulic Stability: An Analysis of the Causes of Unstable Flow in Parallel Channels. Proc. Symp. on Two Phase Flow Dynamics, Volume 2, Session 9, pp. 1 2251 266. Eindhoven: Euratom. Efferding, L. E. 1 968. DYNAM, A Digital Computer Program for Study of the Dynamic Sta bility of Once-Through Boiling Flow with Steam Superheat. GAMD-8656, Gulf General Atomic. Ishii, M. and Zuber, N. 1 970. Thermally Induced Flow Instabilities in Two-Phase Mixtures. Proc. 4th Int. Heat Transfer Conference, Paris, Paper No. B 5 . 1 1 . Jones, A . B . 1 96 1 . Hydrodynamic Stability of a Boiling Channel. KAPL-2 1 70, KAPL-2208( l 96 1 ), KAPL-2290( 1 963), KAPL-3070( 1 964). Lahey, Jr., R. T, ed. 1 992. Boiling Heat Transfer. Amsterdam: Elsevier Sci. Pub. B . v. Lahey, Jr., R. T. and Yadigaroglu, G. 1 973. NUFREQ, A Computer Program to Investigate Thermo Hydrodynamic Stability. NEDO- 1 3344, G.E. Lahey, Jr., R. T, Clause, A, and DiMarco, P. 1 989. Chaos and Non-Linear Dynamics of Density-Wave Instabilities in a Boiling Channel. AIChE Symp. Ser. 85-269:25fr.26 1 . Nakanishi, S., Ishigai, S . , Ozaw, M., and Mizuta, Y. 1 978. Analytical Investigation of Density Wave Oscillation. Osaka University Technology Report, 28- 1 42 1 : 243-25 1 . Nakanishi, S., Kaji, M., and Yamauchi, S . 1 986. An Approximation Method for Construction ofStability Map of Density-Wave Oscillation. Nuclear Eng. and Design 95:55-64. Ozawa, M. and Umekawa, H. 1 996. Non-Linear Flow Oscillation in Two-Phase Flow System. Proc. Japan-U.S. Seminar on Two-Phase Flow Dynamics, Fukuoka, pp. 273-278. Ozawa, M., Nakanishi, S., Ishigai, S., Mizuta, Y., and Tarui, H. 1 979. Flow Instabilities in Boiling Channels, Part I . Pressure Drop Oscillation. Bulletin of the JSME, 22- 1 70: 1 1 1 3- 1 1 1 8. Saha, P. 1 974. Thermally Induced Two-Phase Flow Instabilities, Including the Effect of Thermal Non Equilibrium Between the Phases. Ph.D. Thesis, Georgia Inst. Tech., Atlanta, GA Takenaka, N., Lahey, Jr., R. T, and Podowski, M. Z. 1 99 1 . The Analysis of Chaotic Density-Wave Oscillations. Trans. ANS 63 : 1 97 198. Takitani, K . and Sakano, K. 1 979. Density Wave Instability in Once-Through Boiling Flow System (lII)-Distributed Parameter Model. J. Nucl. Sci. & Tech. 1 6: 1 6-29. -

CHAPTER

ELEVEN

CRITICAL lWO PHASE FLOW Hideki Nariai University of Tsukuba. Tsukuba. /baragi. Japan

11.1 INTRODUCTION

When a tube vapor-liquid twoincrease phase ofmixture flows through a nozzle or a uniform cross section with the the pressure difference between upstream and downstream of the nozzle, the flow rate i s determi n ed onl y by the upstream pres sure, and it is not affected by the downstream pressure of the nozzle or the tube. The flow is named as the critical two phase flow, and the flow rate becomes maximum at factor the condition. Historically, the critical tworatephase flowboiler has been anvalves. important in determining the discharge flow from the vent For the past thi r ty years, it has drawn considerable interest in the evalua tion of the postulated Loss of Coolant Accident (LOCA) of water cooled nuclear reactors. The criticalequations flow condition is determitwoned phase by the flow. mass,The momentum, andonenergy conservation of vapor-liquid conservati equa tions and the constitutive equations have to be formulated as strictly as possible. However, since thengtwois indispensable. phase flow is, Forin general, of verytwocomplicated structure, simplified modeli steady state phase flow, servation law in differential equation form is written as n basic equationsthewithconn 279


variables:

,

Y I , Y 2 . . . Yn .

etc.

a i i d Y I + a l 2 d Y2 + a 2 1 dY2 + a 22dY2 +

.. .. . .. . .. + a l n dYn = b l a2n dYn = b2 . .

1

ann dYn = bn �n dYn an2dY2 Here aij (i, j 1, 2, . . . n) and bi (i = 1, 2, . . . n) are the coefficients in each equation. In the assystem writtentwo inequations: equation (11.1-1), the critical flow condition is determined following I

+

+

+......+

(11.1-1)

=

a l l , a 1 2 , . . . . . . a ln a2 1 , a22 , . . . . . . a2n

and

b l , a l 2 , . . . . . . a ln b2 , a22 , · · · · · · a2n

=

0

(1 1.1-2)

=

0

(1 1.1-3)

Equation (11.1-2) givesflowtheoccurrence critical flowin therate,flowandpassage. equationWhen(11.1-3) gives the location of the critical the cri t i c al flow occurs at the(11.1-2). exit of theThis flowcritical passage,flothew principle critical flisowapplicable condition tois the determi nedphase only byflow,equation single The principle developed to the(1975), vapor-liquid phase flow bytoo.Ogasawara (1965),wasKatto (1968),andSudoapplied and Katto and Bouretwo(1977). Innon-equilibrium two phase flow,statewhenarethewellslipmodeled, betweenandthe basic liquidequations and the vapor, thettentherin mal are wri the form of Eq.It is,(1 1.1-1), thenverythe difficult critical flow ratete theis dericomplicated ved by Eqs.flow(11.1-2) and (11.1-3). however, to wri in eand accurate form of Eq. (11.1-1)-to overcome thi s difficulty, several simplifi models have been proposed. We typically have two kinds of models: the equi librium critical flow model for two phase equilibrium flow through the nozzle orat thethe entrance tube, andofthethenon-e critical flow model for subcooled condition tubequilibrium or the nozzle followed by flashing in the tube or the nozzle. The equilibrium critical fl o w model assumes thephases thermal equilibrium state forrationbothtemperature. liquid and Onvaporthephases; that means both have the same satu other hand, thethenon-equilibrium criticatal least flowformodel assumes the non-equilibrium state, such as superheated liquid, one phase.

CRITICAL lWO PHASE FLOW 281

Though the principle of two-phase critical flow has been made clear, the simplified been used for the real applications. Typical of them will bemethods presentedhavein widely the following sections. 1 1.2 EQUILIBRIUM CRITICAL FLOW 1 1 .2.1 Homogeneous Equilibrium Model

The vaporandandtemperatures. the liquid areTheassumed to a beforhomogeneous mixturemixture with equal ve locities mass velocity the homogeneous through the flow passage is given by the following equation. m. = {2[ho - (I1 --xx)hl x- Xhg]} 1/2 ( 1 1 .2- 1 )

-PI

+-

Pg

The toquality x is given by assuming isentropic flow from upstream reference loca tion the location oftubecritical flow condition,crosswhich is attube. the smallest crossvelocity sectionat ofcritical the nozzle or the exit of uniform section The mass flow condition by changing the downstream pressure until the mass velocity in equationis derived ( 1 1 . 2- 1 ) reaches a maximum value. 1 1 .2.2 Slip Flow Model

The slipsliflow model assumesasthethedifferent velocities for the toliquid and thevelocity. vapor, and the p rati o S is defined rati o of the vapor velocity the liquid The mass velocity for slip flow is given as follows: m. = yl2(ho - ( 1 - x)hl - xh g ) ( 1 1 .2-2) [ SO - x) + �l 2 (x �2 ) S Pg PI Fauske ( 1 962) gave the slip ratio at critical flow condition as Eq. ( 1 1 .2-3), assuming the maximum pressure gradient at tube exit: s= 0 1 .2-3) ( ;: f' Moody ( 1 965) gave the slip ratio at the critical flow condition, assuming the maximum kinetic energy as follows: S = ( PgPI ) 1/3 0 1 .2-4) +


The qualitymassx invelocity Eq. 01.is deri 2-2)vedis from detennined by2-2)assuming isentropic flow. The maximum Eq. (11. by decreasing the downstream pressure. 11.3 NON-EQUILIBRIUM CRITICAL FLOW

When the subcooled liquid entersliquid into a atnozzle or the uniform crossflashing sectionoccurs tube, theduringliquid flows as superheated the entrance, and the the flow ineffect the innozzle or the tube. In this case, we have to consider the non-equilibrium the model. 1 1.3.1 Zaloudek Model

When the subcooled liquid enters into The a nozzle orflashes a shortin tube, the flowandisthea single phase liquid flow at the entrance. liquid the nozzle, two phaseobserved criticalbothflowupstream conditionchoke occursat theat theentrance nozzleandexit.downstream In this case,choke Zaloudek (1961) at the exit. The mass velocity at upstream choking is given by the following equation, based on the superheated liquid flow at the entrance. (11. 3-1) where is an experimentally a sharpCedged entrance. determined constant and has a value of about 0.6 for 11.3.2 Henry-Fauske Model

Henry and Fauske (1971) developed aincrinozzles tical flowand model, takingThey intoassumed account thethe retard non-equilibrium effect in the flow short tubes. of vaporization at small quality region and gave the relation between the true quality as follows: at non-equilibrium condition Xa and the thennal equilibrium quality x dx dXa - = N (11. 3 -2) dP dP where N is the experimental parameter, given as follows: N = -- for X ::; 0.14 0.14 (11.3-3) N 1. 0 for x > 0.14 Thei that theequilibrium change ofquality actual quality is veryr assumption small at theis small region.with Whenthethedecrease qualityofx pressure is larger x

=

CRITICAL 1WO PHASE FLOW 283

than 0.14, the non-equilibrium effect disappears, and thermal equilibrium holds between liquid and vapor phases. 1 1.4 SUMMARY AND RECOMMENDATION

Moody (1975) showed that for the saturated liquid blowdown from a vessel, ho mogeneous equilibrium model wasto develop rather applicable at the entrance offlow the flow passage. The flow is accelerated the vapor-liquid slip in the pas sage. At the tube exit, the slip model is applicable. For nozzles and short tubes, the Henry-Fauske generally holds. For subcoolednon-equilibrium liquid discharge,model the phenomena Zaloudek (1961) showed has todownstream be taken intochoking account.at tube The exit upstream choking by superheated has to be considered. liquid flow and the REFERENCES Boure, J. A. 1 977. The Critical Flow Phenomenon with Reference to Two-Phase Flow and Nuclear Reactor Systems. Thermal Hydraulic Aspects ofNuclear Reactor Safety, Volume I : Light Water Reactors. New York: ASME, pp. 1 95-2 1 6 . Fauske, H. K. 1 962. Contribution t o the Theory o f Two-Phase, One-Component Critical Flow. ANL6633, ANL. Henry, R. E. and Fauske, H. K. 1 97 1 . The Two-Phase Critical Flow of One-Component Mixtures in Nozzles, Orifices, and Short Tubes. Trans. ASME, Series C, 1. Heat Transfer 93: 1 79- 1 87. Katto, Y. 1 968. Dynamics of Compressible Saturated Two-Phase Flow. Trans. lSME 34(260):73 1 - 742 (in Japanese). Moody, F. J. 1 965. Maximum Flow Rate of a Single-Component, Two-Phase Mixture. Trans AS ME Series C. 1. Heat Transfer 87( 1 ) : 1 34 - 1 42. Moody, F. J. 1 975. Maximum Discharge Rate of Liquid-Vapor Mixtures from Vessels. Non-Equilibrium Two-Phase Flows. Eds. Lahey, R. T., Jr. and Wallis, G. B., pp.27 36. New York: ASME. Ogasawaara, H. 1 965. Theory of Two-Phase Critical Flow. Trans. lSME 3 1 (225):75 1 -768 (in Japanese). Sudo, Y. and Katto, Y. 1 975. Study on Compressible Two-Phase Critical Flow. Trans. lSME 4 1 (342):624 - 655 (in Japanese). Zaloudek, F. R. 1 96 1 . The Low Pressure Critical Discharge of Steam-Water Mixtures from Pipes. HW-68934 Rev. -

CHAPTER

TWELVE TWO-PHASE FLOW AND B OILING HEAT TRANSFER IN TUBE BUNDLES Ramin Dowlati Owens-Corning Science and Technology Center, Granville, OH 43023-1200, USA

Masahiro Kawaji University of Toronto, Toronto, Ontario M5S 3£5, Canada

12.1 INTRODUCTION

Aindustries significantare porti oton boil of allpureheatorexchangers employed inofthetheseprocess and power used liquid mixtures. Many heat exchangers involve boiling heat transfer, andbundles. henceIntwo-phase (vapor-liquid) cross-flow on the shell-side of horizontal tube the chemical and petroleum industries, such equipment is commonly referred to as 'reboilers,' or by the more general ' Ultimately, term, 'process vaporizers. lers addcharge energyfortovapor-phase distillation columns, remove energy in refrigerationthese cycles,reboi or prepare reactions. reboi lers Baffled (Figure heat 1), forexchangers, example, such are commonly found at thesteambasegenerators ofKettle-type distillation towers. as i n verted U-tube commonly employed in nuclear power reactors, also experience two-phase cross-flow in theatopsummary U-bend ofregion. This chapter presents vertical upward two-phase cross-flow and boiling heat transfer data obtained to date onparameters, the shell-sidesuchof ashorizontal tube bundles. The focus will be on important physical void fraction (also known as vapor or liquid hold-up), two-phase frictional pressure drop, and boiling heatflowtransfer coefficient. The first section will alsoInreview the status of two-phase regime maps for horizontal tube bundles. the present chapter, thetwo-phase term "reboi ler" willandbeshell-side used to refer cross-flow boiling.to horizontal tube bundles subjected to 285

286 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSAnON PISTILLAUQN

�

"EffiE RE50ILER

+

HEATING FLUID

Figure 1

1

eorrOMS PRODUCT

5AFFLElSUf'PORT PLATES

Illustration of a kettle reboiler.

Whileperformance the single-phase heatheattransfer correlations availableaccurate for predicting thermal in tubular exchangers are typically to withinis ±30%, the accuracy becomes worse for reboilers, as two-phase heat transfer often verythedifficult toperformance estimate, especially with hasmixtures of several components. Although overall of reboilers been extensively analyzed in thephasepast,flowrelatively few studies have been conducted to determine the local two and boiling heat transfer conditions onoccurring the shell-side ofreboiler a tube bundle. This is mainly due to the complex phenomena in the under two-phase flow and the associated di ffi culty in gatheri n g experimental data, even withAssophianstiexample, cated instrumentation. the circulation pattern in a kettle reboiler is illustrated in Figure 2. At the bottom ofoftheeach tubetube, bundle,andtheas itsubcooled liquid comesof tubes, into contact with the hot outer surface flows past each row more sensi b le heat is absorbed until the boi l i n g poi n t is reached and vaporization begins totheoccur. The two-phase (vapor-liquid) mixture continues to remove energy from(x, tube bundle i n the form of latent heat, causing the flow or mass quality defined as the ratio of the vapor mass flow rate to the total flow rate) to increase. Local conditions, suchtheasbundle. liquid andAs vapor velocitiesmixture and voidreaches fraction,thearetopexpected totubevarybundle, throughout the two-phase of vtheen the vapor phase is separated, and any remaining liquid phase i s dri away from the tube bundle towards the shell wall. The difference between the

TWO-PHASE FLOW AND BOILING HEAT TRANSFER IN TUBE BUNDLES 287

l '" /0'0 ' 0'0\ Y) \ 000 0 0 ) o q o o O\c:O 0 r;J0 0 0 0'.10 0000 0000 o o oto o o 00 00 � 'Figure 2

Circulation patterns in a kettle reboiler.

two-phase mixture density in thecreates core ofthethestatitubec head, bundlewhichandisthatthe main of thedriving liquid phase outside of the tube bundle force for recirculating the liquid phase. According to a visualexperi studyences by Cornwell of a thinflow, slice and of a reboi ler, the core of the bundle nearly a etvertiaI. c(1980) al two-phase the void fraction is the highest near thetransverse top of theflowbundle (Figure 3).ofInthethebundle outer region ofWittheh thebundle, a significant from the sides exists. presence of other physical effects, such as boundary layer separati o n and two-phase wake i n teractions, it is quite clear that such complex thermal-hydraulic behavior does data not lend itself easily to instrumentation for the purpose of Also, gatheritheng experi m ental and hence developi n g sound mechanistic models. bundlePresent geometry is expected toforhavereboia signifi cantknown effecttoonbetheveryflowconservati characteristics. design methods l ers are ve (by ascostsmuch(Bell,as 1983; 300 percent), hence leading to over-design and additional capital Grant etallaI.studies , 1983; onNiels, 1979;cross-flow Schuller,boiling 1982; Smith, 1985; Westwater, 1969). Nearly vertical in horizontal tube bundles to dateandhaveveryfocused on thedirmeasurement andtheprediction ofhydrody the heat transfer coefficient, few have ectly addressed two-phase namics, which hasclear an important effect onofthetubeoverall heatperformance transfer rate.must Frominvolve these studies, it is quite that the analysis bundle consideration of a strong interaction between heat transfer and the two-phase flow

288 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Heat Flux

=

20 kW/m2

L--�==--'

Liquid Level

Figure 3

Streamlines in kettle reboiler (from Leong and Cornwell, 1 979, with permission).

parameters, suchas theas void fraction, liquid and vaporcorrelations velocities,developed two-phasetopressure drop, as well flow regime. Many of the date to predi c t the heat transfer coefficient al s o requi r e some knowledge of the two-phase flow parameters. Furthermore, knowledge of these parameters and (CHF) two-phase flow regimes is an important factor in the prediction of critical heat flux in a tube bundle in order to avoid dryout conditions that can severely degrade heat transfer. The3)observation by region Cornwellof aetbundle al. (1980) of a nearly vertiresults cal crossflow (Figure in the central suggested that the fromler,exis periments in which a horizontal tube bundle, simulating the core of a reboi subjectedcould to forced vertito cdevelop al two-phase flow under controlleddropmasscorrelations velocity andfor quality, be used heat transfer and pressure designbeenpurposes. Hence,onmany of the recent advances in the study of reboitube lers have made based experiments conducted in horizontal, rectangular bundles, simulating the core of aitreboi ler beunderusefulcontrolled flowsome conditions. Before proceeding further, would to explain of thegenerally termi nology associated with tube bundle geometry. Tube rows in a bundle are categorized as4).being in-line (aligned) or staggered in the direction of oncoming flow (Figure The bundle geometry is further zed bywith the outer diameter, D, and by the pitch-to-diameter (PID)characteri ratios defined eithertube the

TWO-PHASE FLOW AND BOILING HEAT TRANSFER IN TUBE BUNDLES 289 In-Line

Staggered

i

i

i

FLOW Figure 4 Tube

i

layout in bundles.

i i

F LOW

transverse pitch, S T , or longitudinal pitch, SL, measured between the tube centers. Typical P /0 ratios used in industry vary between and (Rubin,

1.25 2.5

1984).

12.2 FLOW REGIMES Several two-phase flow regime maps have been reported for upward air/water flow across horizontal tube bundles. Grant and Chisholm visually observed bubbly, intermittent, and spray (or annular-dispersed) flow regimes in a segmen tally baffled model heat exchanger. The bundle had an equilateral triangular layout of mm 0.0. tubes with a P/O ratio of Ulbrich and Mewes also visually detected the flow regimes in an in-line bundle consisting of rows of tubes with mm 0.0. and a P /0 ratio of They detected all the three flow regimes as reported by Grant and Chisholm for L < mis, but only bubbly and annular-dispersed flow regimes at k > m/s. Flow regimes have been studied in tube bundles using non-visual techniques as well. Hahne et al. used a fiber optics probe to measure local void fraction distributions in in-line and staggered tube bundles containing finned tubes with P/0 ratios of and The working fluid, R- l l , was boiled in the heated tube bundle to create natural circulation driven two-phase cross flow. Probability density histograms of local void fractions and their higher moments were used to identify the flow regimes inside the bundles. Only bubbly flow regime was detected, and no slug or annular flow regimes were observed within the tube bundle. Although the flow conditions could not be determined under the pool boiling conditions, the detection of only the bubbly flow regime may have been due to the limitation in flow conditions or small gaps between the finned tubes, which could prevent formation of large vapor slugs. Ueno et al. employed double-sensor probes to measure the local void fraction and bubble velocities for vertical upward flow of

(1979)

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290 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION 1 .2

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Figure 5 Two-phase flow regime maps for in-line tube bundles (-Noghrehkar et al. ( 1 995, 1 999), P/D = 1 .47 ; ---- Ulbrich and Mewes ( 1 994), P/D = 1 .5 , (B) = bubbly flow, (I) = Intermittent flow, (A)

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20

an air-water mixture across an in-line tube bundle containing rows of two half rods with a P jD ratio of Slug flow was identified by large bubbles, which had crowds of small bubbles in their tail, flowing mainly in the gap region. They also noted that gas slugs do not have a bullet shape, which is often observed in circular pipes. Recently, Noghrehkar and Noghrehkar et al. have re ported flow regime maps for both in-line and staggered bundles consisting of and rows, respectively, of five mm O.D. rods per row arranged with a P jD ratio of They analyzed the probability density function of void fraction fluc tuations, which were measured with an electrical resistivity probe inserted deep into the bundle. The flow regime maps of Noghrehkar et al. and Ulbrich & Mewes are shown in Figures and for in-line and staggered bundles, respectively. For the in-line bundle, the flow regime transition boundaries reported are consistent m1s). However, Noghrehkar between the two maps at low liquid flow rates ( k < et al. detected intermittent flow at higher liquid flow rates ( j L > m1s), unlike the results of Ulbrich and Mewes The difference between the

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TWO-PHASE FLOW AND BOILING HEAT TRANSFER IN TUBE BUNDLES 291 1 . 2 ,-------,--,---,

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306 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION z

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boiling is expected to be the dominant heat transfer mode in a tube bundle, where a simple power law relationship given by Eq. ( 1 2.5-5) is capable of predicting the overall heat transfer coefficient with reasonable accuracy. Bergles and Rohsenow ( 1 964) also reported regions on the boiling curve for single horizontal tubes where either nucleate boiling or forced-convective boiling will dominate, and hence there would be no need to form an additive correlation such as Eq. ( 1 2.5- 1 ). This mechanism appears to hold true for boiling in tube bundles as well as single tubes. More recently, Cornwell ( 1 990a, 1 990b) and Houston and Cornwell ( 1 996) have introduced an additional heat transfer mechanism based on the effect of sliding bubbles over a tube surface. The mechanism involves the process of bubbles sliding on the tube surface and enhancing heat transfer either by disruption of the liquid boundary layer or by direct evaporation. They concluded that for low quality flow regimes, the boiling heat transfer coefficient in the upper region of a tube bundle can be determined by the effect of sliding bubbles and that nucleation is important only at the lowest tubes where quality approaches zero. This new approach holds some promise; however, there are currently no viable correlations based on this mechanism that can be recommended. The effect of tube bundle geometry on two-phase heat transfer is not yet fully clear. Cornwell et al. ( 1986) and Palen et al. ( 1 972) reported that tube arrangement

TWO-PHASE FLOW AND BOILING HEAT TRANSFER IN TUBE BUNDLES 307 had little or no effect on the boiling heat transfer characteristics in horizontal tube bundles. On the other hand, as the P /D ratio was increased from 1 .3 to 1 .7, Hsu and Jensen ( 1 988) reported an increase in heat transfer coefficient of nearly 20 percent for an in-line tube bundle at low heat fluxes (q " < 10 kW/m2 ). Reinke and Jensen ( 1 987) also reported higher heat transfer coefficients in a staggered tube bundle when compared to an in-line bundle with the same P /D ratio at low to moderate heat fluxes. Intuitively, one would expect higher heat transfer coefficients in staggered tube bundles due to the additional turbulence and mixing created by the staggered tube arrangement. Also, a staggered layout can provide more surface area for a given shell diameter. Finally, there are some tube bundle heat transfer data reported with finned tubes. Hahne and Muller ( 1 983), Windisch et al. ( 1 985), and Fujita et al. ( 1 986) presented heat transfer data and correlations for finned tube bundles under natural circulation-driven two-phase cross-flow of R- l 1 and R- 1 1 3 . Compared to single finned tubes, finned tube bundles also provided higher heat transfer coefficient.

REFERENCES Bell, K. J. 1 983. Heat Exchangers with Phase Change. Heat Exchangers-Theory and Practice. Eds. Taborek, J., Hewitt, G. F., and Afghan, N. Washington DC : Hemisphere Publishing Co. Bennet, D. L., Davis, M. w., and Hertzler, B. L. 1 980. Boiling by Forced Convective Flow. AIChE Symp. Ser. 76( 1 99):9 1 - \ 03 . Bergles, A. E. and Rohsenow, W. M . 1 964. The Determination o f Forced-Convection Surface-Boiling Heat Transfer. 1. Heat Transfer 86: 365-372. Brisbane, T. W. c., Grant, I. D. R., and Whalley, P. B. 1 980. A Prediction Method for Kettle Reboiler Performance. ASME Paper No. 80-HT-42. Butterworth, D. 1 975. A Comparison of Some Void-Fraction Relationships for Co-Current Gas-Liquid Flow. Int. 1. Multiphase Flow 1 :845-850. Carlucci, L. N., Gapin, P. F., and Brown, 1. D. 1 984. Numerical Predictions of Shellside Heat Ex changer Flows. A Reappraisal of Sheliside Flow in Heat Exchangers. Eds. W. J. Marner and 1. M. Chenoweth, HTD-Vol. 36, pp. 1 9-26. New York: ASME. Chan, A. M. C. and Shoukri, M. 1 987. Boiling Characteristics of Small Multitube Bundles. 1. Heat Trans. \ 09:753-760. Chen, J. C. 1 966. A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow. Industrial and Eng. Chemistry Process Design and Development 5(3):322-329. Chisholm, D. and Laird, A . D. K. 1 95 8 . Two-Phase Flow in Rough Tubes. ASME Trans. 80:276--2 86. Collier, J . G. 1 972. Convective Boiling and Condensation, New York: McGraw-Hill. Cornwell, K. 1 990a. The Influence of Bubbly Flow on Boiling from a Tube in a Bundle. Int. 1. Heat Mass Transfer 33( 1 2) :2579-2584. Cornwell, K. 1 990b. The Role of Sliding Bubbles in Boiling on Tube Bundles. Inti. Heat Transfer Conj , Jerusalem, Paper 9-TPF- l l . Cornwell, K., Duffin, H . w. , and Schuller, R . B . 1 980. An Experimental Study of the Effects of Fluid Flow on Boiling Within a Kettle Reboiler Tube Bundle. ASME Paper No. 80-HT-45. Cornwell, K., Einarsson, 1. G., and Andrews, P. R. 1 986. Studies on Boiling Tube Bundles. 8th IntI. Heat Transfer Conj, San Francisco, Paper FB-03 .

308


Dowlati, R. 1 992. Hydrodynamics of Two-Phase Cross-Flow and Boiling Heat Transfer in Horizontal Tube Bundles. Ph.D. Thesis, University of Toronto. Dowlati, R., Chan, A. M. c., and Kawaji, M. 1 992b. Hydrodynamics of Two-Phase Flow Across Horizontal In-Line and Staggered Rod Bundle. ASME J. Fluids Eng. 1 1 4(3) :450--456. Dowlati, R., Kawaji, M . , and Chan, A. M. C. 1 990. Pitch-to-Diameter Effect on Two-Phase Flow Across an In-Line Tube Bundle. AlChE J. 36(5):765-772. Dowlati, R., Kawaji, M., and Chan, A. M. C. 1 996. Two-Phase Crossftow and Boiling Heat Transfer in Horizontal Tube Bundles. J. Heat Transfer 1 1 8( 1 ) : 1 24- 1 3 1 . Dowlati, R., Kawaji, M., Chisholm, D., and Chan, A . M . C . 1 992a. Void Fraction Prediction i n Two Phase Flow Across a Tube Bundle. A.I. Ch.E. J. 3 8(4) : 6 1 9-622. ESDU. 1 973. Convective Heat Transfer During Crossftow of Fluids Over Plain Tube Banks. Engineer ing Sciences Data Unit, No. 7303 1 , London. ESDU. 1 979. Crossftow Pressure Loss over Banks of Plain Tubes in Square and Triangular Arrays Including Effects of Flow Direction. Engineering Sciences Data Unit, No. 79034, London. Fair, J. R., and Klip, A. 1 983. Thermal Design of Horizontal Reboilers. Chern. Eng. Progress 79(8):8696. Fujita, Y., Ohta, H., Hidaka, S., and Nishikawa, K. 1 986. Nucleat Boiling Heat Transfer on Horizontal Tubes in Bundles. Proc. Int. Heat Transfer Can! , Volume 5, pp. 2 1 3 1-2 1 36, San Francisco. Grant, I. D. R. and Chisholm, D. 1 979. Two-Phase Flow on the Shell Side of a Segmentally Baffled Shell-and-Tube Heat Exchanger. J. Heat Transfer 1 0 1 :38-42. Grant. 1. D. R., Cotchin, C. D., and Henry, 1. A. R. 1 983. Tests on a Small Kettle Reboiler. Heat Exchangersfor Two-Phase Applications. Eds. J. B. Kitto and J. M. Robertson, HTD-Vol. 27 , pp. 4 1 - 45 . New York: ASME. Hahne, E. and Muller, J. 1 983. Boiling on a Finned Tube and a Finned Tube Bundle. Int. J. Heat Mass Transfer 26(6):849-859. Hahne, E., Spindler, K., Chen, Q., and Windisch, R. 1 990. Local Void Fraction Measurements in Finned Tube Bundles. Proc. Ninth Int. Con! on Heat Transfer, Volume 6. Jerusalem, Israel, pp. 4 1 -45. Houston, S . D. and Cornwell, K. 1 996, Heat Transfer to Sliding Bubbles on a Tube Under Evaporating and Non-Evaporating Conditions. Int. J. Heat Mass Transfer 39( 1 ): 2 1 1 -2 14. Hsu, J. T. 1 987. A Parametric Study of Boiling Heat Transfer in Horizontal Tube Bundles. Ph.D. Thesis, Univ. of Wisconsin, Milwaukee. Hsu, J. T. and Jensen, M. K. 1 988. Effect of Pitch-to-Diameter Ratio on Crossflow Boiling Heat Transfer in an Inline Tube Bundle. HTD-Vol. 104, pp. 239-245. New York: ASME. Hwang, T. H . and Yao, S. C. 1 986. Forced Convective Boiling in Horizontal Tube Bundles. Int. J. Heat Mass Transfer 29(5) :785-795. Ishihara, K., Palen, J. W., and Taborek, J. J. 1 979. Critical Review of Correlations for Predicting Two-Phase Pressure Drop Across Tube Banks. Heat Trans. Eng. 1 (3): 1-8. Ishii, M. 1 977. One-Dimensional Drift-Flux Model and Constitutive Equations for Relative Motion Between Phases in Various Two-Phase Flow Regimes. Argonne Nat. Lab. Report, ANL-77-47, October 1 977. Jensen, M . K. and Hsu, J. T. 1 987. A Parametric Study of Boiling Heat Transfer in a Tube Bundle. 2nd ASMEIJSME Thermal Engineering Joint Can! 3: 1 32- 140. Kondo, M . and Nakaj ima, K. 1 980. Experimental Investigation of Air-Water Two-Phase Upftow Across Horizontal Tube Bundles, Part I: Flow Pattern and Void Fraction. Bulletin JSME 23( 1 77) :385393. Leong, L. S . and Cornwell, K. 1 979. Heat Transfer Coefficients in a Reboiler Tube Bundle. The Chemical Engineer 343 (April 1 979) : 2 1 9-22 1 .

TWO-PHASE FLOW AND BOILING HEAT TRANSFER IN TUBE BUNDLES 309 Lian, H. Y., Chan, A. M. C., and Kawaji, M. 1 992. Effects of Void Fraction on Vibration of Tubes in Tube Bundles Under Two-Phase Cross Flow. Proc. 3rd International Symposium on Flow-Induced Vibration and Noise, Volume 1 , pp. 1 09- 1 1 8. New York: ASME. Lockhart, R W. and Martinelli, R C. 1 949. Proposed Correlation of Data for Isothermal Two-Phase Two-Component Flow in Pipes. Chem. Eng. Prog. 45( 1 ) : 39-48. Martinelli, R. C. and Nelson, D. B . 1 948. Prediction of Pressure Drop During Forced-Circulation Boiling of Water. Trans. ASME 65:695-702. Mostinski, J. L. 1 963. Application of the Rule of Corresponding States for the Calculation of Heat Transfer and Critical Heat Flux. Teploenergetika 4:66. Niels, G. H. 1 979. Some Boiling Aspects in Kettle Reboilers. Boiling Phenomena: Physicochemical and Engineering Fundamentals and Applications, Volume 2, Chapter 29. New York: Hemisphere Publishing Corp. Noghrehkar, R 1 996. Investigation of Local Two-Phase Flow Parameters in Cross Flow-Induced Vibration of Tubes in Tube Bundles. Ph.D. Thesis, University of Toronto. Noghrehkar, R, Kawaji, M., and Chan, A. M. C. 1 995. An Experimental Study of Local Two-Phase Parameters in Cross Flow Induced Vibration in Tube Bundles. Proc. Sixth Int. Con! On Flow Induced Vibrations. Ed. Bearman, P. W., pp. 373-382. Rotterdam: Balkema. Noghrehkar, R., Kawaji, M., and Chan, A. M. C. 1 999. An Experimental Study of Two-Phase Flow Regimes In-Line and Staggered Tube Bundles Under Cross-Flow Conditions. Int. 1. Multiphase Flow, submitted. Palen, J. W and Taborek, J. J. 1 962. Refinery Kettle Reboilers-Proposed Method for Design and Optimization. Chem. Eng. Progress 58(7) : 37-46. Palen, J. W and Yang, C. C. 1 983. Circulation Boiling Model for Analysis of Kettle and Internal Reboiler Performance. Heat Exchangers for Two-Phase Applications. Eds. J. B. Kitto and J. M. Robertson, HTD-Vol. 27, pp. 55-6 1 . New York: ASME. Palen, J. W , Yarden, A., and Taborek, J. J. 1 972. Characteristics of Boiling Outside Large-Scale Horizontal Multitube Bundles. AlChE Symp. Ser. 68( 1 1 8) :50-6 1 . Payvar, P. 1 985. Analysis of Performance of Full Bundle Submerged Boilers. Two-Phase Heat Ex changer Symposium, HTD-Vol. 44, pp. 1 1 - 1 8. New York: ASME. Reinke, M. J. and Jensen, M. K. 1 987. Comparison of Boiling Heat Transfer and Two-Phase Pressure Drop Between an In-Line and Staggered Tube Bundle. HTD-Vol. 85, pp. 4 1-50. New York: ASME. Rohsenow, W M. 1 952. A Method of Correlating Heat-Transfer Data for S urface Boiling of Liquids. Trans. ASME 74:969-976. Rubin, F. L. 1 984. Heat Transfer Equipment. Perry's Chemical Engineers ' Handbook, 6th edition, Section I I . Eds. R H . Perry, D. W. Green, and J. O. Maloney. New York: McGraw-Hill. Schrage, D. S., Hsu, J. T, and Jensen, M. K. 1 987. Void Fraction and Two-Phase Friction Multipliers in a Horizontal Tube Bundle. A.l. Ch.E. Symp. Ser. 83(257) : 1 -8. Schrage, D. S . , Hsu, J. T, and Jensen, M. K. 1 988. Two-Phase Pressure Drop i n Vertical Crossftow Across a Horizontal Tube B undle. A.l. Ch.E. 1. 34( 1 ): 1 07- 1 1 5 . Schuller, R B. 1 982. Boiling on Horizontal Tube Bundles. Ph.D. Thesis, Heriot-Watt University, Edinburgh, U.K. Smith, B . 1 985. Update on Shellside Two-Phase Heat Transfer. The Chemical Engineer 414 (May 1 985): 1 6- 1 9 . Stephan, K. and Abdelsalam, M . 1 980. Heat-Transfer Correlations for Natural Convection Boiling. Int. 1. Heat Mass Transfer 23:73-87. Ueno, T, Leung, W H., and Ishii, M. 1 995. Local Measurement in Two-Phase Flow Across a Horizontal Tube Bundle. Proc. 2nd Int. Con! on Multiphase Flow, Kyoto, Japan, April 3-7, 1 995, pp. 89-95.

310 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Ulbrich, R. and Mewes, D. 1 994. Vertical, Upward Gas-Liquid Two-Phase Flow Across a Tube Bundle. Int. J. Multiphase Flow 20(2):249-272. Wallis, G. B. 1 969. One-Dimensional Two-Phase Flow. New York: McGraw-Hill. Westwater, J . W. 1 969. Nucleate Pool Boiling. Advanced Heat Transfer. Ed. B . T. Chao, pp. 2 1 7-232. Urbana, IL: Univ. of Illinois Press. Whalley, P. B. and Butterworth, D. 1 983. A Simple Method for Calculating the Recirculating Flow in Vertical Therrnosyphon and Kettle Reboilers. Heat Exchangers for Two-Phase Applications, HTD-Vol. 27, pp. 47-53. New York: AS ME. Windisch, R., Hahne, E., and Kiss, V. 1 985 . Heat Transfer for Boiling on Finned Tube Bundles. Int. Commun. Heat Mass Transfer 1 2(4):355-368. Zuber, N. and Findlay, J. A . 1 965. Average Volumetric Concentration in Two-Phase Flow Systems. J. Heat Transfer 87:453-468. Zukauskas, A. 1 972. Heat Transfer from Tubes in Crossflow. Advances in Heat Transfer 8 :93- 1 60.

CHAPTER

THIRTEEN

EXTERNAL FLOW FILM BOILING Larry C. Witte University of Houston. Houston. TX 77204-4792. USA

13.1 INTRODUCTION The theory of flow film boiling across bluff bodies that induce flow separation and wake formation is still not complete, especially for subcooled systems, even though it has been studied for many years. More than 40 years ago, Bromley et al. ( 1 953) conducted the first systematic theoretical and experimental studies for heated cylinders experiencing flow film boiling and subcooled flow film boil ing (Motte and Bromley, 1 957) as extensions of his seminal work on pool film boiling. In the intervening years, research has been directed at refining the basic models proposed by Bromley and his co-workers, especially in the areas of finding the influence of liquid velocity and subcooling on the behavior of the vapor film that surrounds the body. This section will be directed toward summarizing the ad vances made in recent years to understand the basic nature of flow film boiling from bodies like cylinders and spheres, that is, bodies over which flow separation will occur.

13.2 PHYSICALIMATHEMATICAL MODELS FOR FLOW FILM BOILING Figure 1 shows the vapor envelope that surrounds a bluff body during flow film boiling. The picture shows an upward flow of liquid, though the orientation is 311


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i

liquid flow Vc.> , Tc.>

Physical model of flow film boiling across a cylinder (Chou and Witte, 1 995).

insignificant as long as the flows are strong enough so that buoyancy effects are not important. A thin, smooth vapor layer over the forward part of the body feeds into a wake region beyond the flow separation point. This is the basic model that was proposed by Bromley et aI., though they did not attempt to analyze the processes that occur in the wake region. Indeed, only recently have efforts been undertaken at predicting the heat transfer in the wake region. In Figure 1 , Part One refers to the region where a local similarity (boundary-layer-like) analysis can be justified, while Part Two refers to the wake region for which only crude models for heat transfer have been explored. The key to predicting heat transfer is being able to predict how the vapor layer develops around the body. Doing this requires knowledge of the vapor velocity and temperature profiles. Once the vapor thickness is known, it is a straightforward matter to integrate for the total heat transfer from the body. Analyses usually fall within two categories, depending on the boundary con dition that is imposed at the heater surface. The most common boundary condition is a uniform temperature, though some studies have been done using a uniform heat flux condition at the surface. Other conditions that are commonly assumed are: •

the liquid-vapor interface is smooth and is at the local saturation temperature

EXTERNAL FLOW FILM BOILING 313 • • •

•

the liquid free stream velocity and temperature are uniform. the vapor layer and the adjacent liquid boundary layer are thin and laminar. viscous dissipation and liquid compression work are negligible. Usually, radia tion heat transfer is considered negligible as well, even though surface tempera ture might be relatively high. If needed, a correction to include radiation can be performed following the method proposed by Bromley et a1. a potential flow exists in the far-field of the liquid.

Steady state conditions are usually assumed, and if thermophysical properties are variable, they depend only on temperature. Several researchers have performed analyses that generally follow the as sumptions stated above. Brief summaries of some of them follow. Witte ( 1 968) studied saturated film boiling from a sphere, assuming that the vapor velocity is a simple linear function, going from zero at the wall to the potential flow velocity at the liquid-vapor interface. However, the assumption of linear velocity in the vapor cannot be physically justified. Indeed, as recognized by Kobayashi ( 1 965), Epstein and Hauser ( 1 980), and Witte and Orozco ( 1 984), the imposition of the liquid pressure gradient caused by the flow around the body causes a non-linear velocity profile. Epstein and Hauser ( 1 980) modeled only the processes near the stagnation region of the body. They then drew inferences about the nature of the film boiling process, assuming basically a vapor film of uniform thickness around the body. Wilson ( 1 979) used an integral technique to describe the behavior of the vapor boundary layer. A result similar to that of Epstein and Hauser was found. Witte and Orozco ( 1 984) ignored inertial effects in the vapor film while re taining the assumption of potential flow in the liquid overiding the vapor film. They found that if a non-dimensional velocity is formed using the velocity at the edge of the vapor layer as the reference, one can write

Uv U8

- =

Y

-

8

+

3 V - PI oo cos� (y8 - y 2 ) --

4 J.lv R

( 1 3.2- 1 )

Figure 2 shows Eq. ( 1 3.2- 1 ) plotted for one particular case of subcooled film boiling over a sphere. The velocity profiles appear to be "squeezed" into the vapor thickness over the forward portion of the body, going to a linear distribution at the 90-degree point, and actually demonstrating flow reversal (separation) at some point past the 90-degree point. The shape of the profile is a response to the liquid pressure profile, which is represented by the last term on the right side of Eq. ( 1 3 .2- 1 ). If Eq. ( 1 3 .2- 1 ) is differentiated and set to zero at y = 0, a relationship for the separation angle in terms of 8r at �sp can be obtained as

COS�s p

=

( 1 3 .2-2)


1.0

0.75

y/8

0. 5

0. 2 5

o -0. 2

0

0.4

0.8

1.2

1.6

Figure 2 Tw

=

Vapor velocity profiles for subcooled water flowing over a 6. 35-mm sphere at 4.26 m1s: 400DC, b.Tsub 40° C (Witte and Orozco, 1 984). =

Equation ( 1 3 .2-2) does not predict {sp; rather, it gives the relationship between {sp and Dr at separation for a given boiling case. Recently, Liu and Theofanous ( 1 995a) analyzed flow film boiling from spheres using essentially the same laminar model as Witte and Orozco ( 1 984) up to the separation point. They found that for saturated liquids, the heat transfer in the wake could be adequately accounted for by assuming a constant thickness of the vapor film in the wake equal to the vapor film thickness at the separation point. For subcooled systems, however, they found this to be inadequate. Conse quently, they invoked a heat transfer version of a turbulent eddy model developed by Theofanous et al. ( 1 976) for mass transfer at a free gas-liquid interface. Additional theoretical work related to flow film boiling around spheres was done by Fodemski and Hall ( 1 982) and Fodemski ( 1 992). Chappidi et al. ( 1 99 1 ) and Walsh and Wilson ( 1 979) perfonned boundary-layer like analyses for film boiling on wedges. Chou and Witte ( 1 995) perfonned a much more rigorous analysis of film boiling around a cylinder, relaxing the potentially-restrictive assumptions of Witte and Orozco. While the analysis is lengthy and complicated, some of the details are given below to illustrate the nature of recent analytical research.

EXTERNAL FLOW FILM BOILING 315

13.2.1 Governing Equations

The governing equations, including mass, momentum, and energy conservation, in the vapor layer = v) and the liquid layer = are ( 1 3.2-3) ax (pu)j + ay (pv)j = 0 au. ) = 2PLoov� sin (2X) + g(PLoo - Pj ) sin (X ) pJ. ( uJ· _Jaxu + v·_J R J ay R ( 32 ) + -y (wJ uyJ ) (2X) 1 + - (A · - ) - ) = [ 2PLOOV� (p c ) ( U · sin R R ( 2- ) whereThetheboundary index denotes whether thethevapor or(without the liquidsubscript)-liquid region is being described. conditions and vapor (with sub script interface conditions are ( 3 .2- ) y = O; u = O, v = O, T = Tw, y = 8; u = y = ay ' T = TL = Tsi(X), ( 3 2- ) P (v :: ) = PL ( :: ) L - p ( d8 ) h jg 3 .2- ) -A- + qr = -AL -ay y= = 2Voo sin ( i} TL = Too ( 3 2- ) To simplifytransfonnation-a Eqs. (13.2-3) to (well 3 2Chou and Wi t te developed a local-similarity knownbodies. technique for computing thetheheat transfer oinn are the given non-separated region ontte bluff The detailed steps of transformati by Chou and Wi only and a summary is presented here.canStream functions forby two the vapor and liquid are defined, it is assumed that they be separated single-variable functions ({) and f(17) so that ( 3 2 0) [f(�n = [M:({) 1 ' (j

(j

a

a

a .

R

a

a

P

L)

a . a

1 . -4

a Tj a Tj + v)· )ax ay

-

u

a

j

ay

a Tj ) ay

1 3. 5

j

L)

1

UL ,

-

aT

J.L

au a

J.L L

aU L

V L - UL

U

aT

UL

13.2.1.1 Local-similarity transformations.

9),

v - u-

dx

ay

D + DL ;

( 1 995)

6

1 . 7

(1

8

1 . 9 1 .

( 1 995);

1 . -1


Ps )1(1;) loy = (�)

where l; and h are new independent variables, defined as 17j

= Nj

-R

O (j

8( j

or

=

v)

L)

j

dy

( 1 3.2- 1 1 )

where ( 1 3.2- 1 2) The following variables for the vapor and the liquid layers are introduced: () _

T - Ts

___

fj. T sat

'

( 1 3.2- 1 3)

where qrp is the radiation heat flux between two parallel plates. The transfonned equations become a set of ordinary differential equations locally for any given location l; that yields a local-similarity solution. By assigning a succession of l; values, the streamwise dependence of velocity and temperature fields can be determined.

13.2.1.2 Selected results of the model. The equations were solved using a finite difference method. Once f is obtained, 1/1 can be found, which in tum gives both the u- and v-components of velocity everywhere. Energy is balanced at the vapor liquid interface to detennine 8 . The temperature profiles i n the vapor are virtually linear, suggesting that conduction dominates heat transfer in the vapor film. In the liquid, however, the T-distributions are in general non-linear, indicating that convection plays a role in the total heat transfer in the liquid. The effects of bulk liquid velocity, superheat, cylinder diameter, and system pressure were also studied (see Chou and Witte, 1 995). Results showed that the velocities of the vapor and liquid layers are increased with bulk liquid velocity, as expected. 8 is reduced with increased bulk liquid velocity. For a given angle l; , 8 increases with the superheat fj. Tsat and the cylinder diameter, D. The effect of fj. Tsat on 8 is nearly linear. For a given angle l; , increasing the system pressure p has only a small effect on 8 in the p-range explored by Chou. 13.2.2 Uniform Heat Flux Models Shoji and Montasser ( 1 993) used an extension of Witte and Orozco's model to study the influence of a uniform qw condition on flow film boiling from cylinders.

EXTERNAL FLOW FILM BOILING 317 In this analysis, 8 and Tw must take on values necessary to accommodate the imposed q. They showed that local Nu's for the uniform-q condition were about 25% higher than those for the uniform-Tw model at any given location.

13.2.3 The Wake Region Until recenly, most analyses simply omitted the heat transfer in the wake region. Chou and Witte ( 1 995), however, attempted to include the wake region in their heat transfer analysis from a uniform Tw cylinder. The wake begins at the separation points predicted by the analysis described in the previous sections. The vapor was assumed to form a continuous, smooth, and steady film layer (vapor bubble) in the wake. The 8 of the wake region must match the 8 at the same point on the front part of the body. Based on these assumptions, Chou performed energy and mass balances as follows.

13.2.3.1 Energy balance. Considering the right half of a symmetric vapor bub ble, as shown in Figure 3, for steady-state, the heat flow-in from the front part at the separation point, Qsp , and the heat conducted-in from the hot cylinder wall, Qw , should balance the net heat flow-out across the vapor-liquid interface, Q,s , i.e., ( 1 3 .2- 1 4) y

Figure 3

Two-dimensional wake model (Chou and Witte, 1 995).

318 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION where

18.fP (A-axaT) dY 18JP oX 0 aT) w = 1 sp ( -A a y dX

Q sp Q

=

+

sp

o

( puh)sp dY

w

13.2.3.2 Mass balance. For a steady-state vapor bubble, the mass flow-in from the front part at the separation point, msp , and across the lower part of the bubble interface, min , should balance the mass flow-out across the upper part of the bubble interface, mOUI> i.e.,

msp + min where

msp

=

mout

= tsp ( pu)sp dY

and IDo", - ID;n =

io

1x• p,Vo � dX

1X• p, V" � dX = 1'· (puj,p

is the net mass flux across the interface. Therefore, we have

1Xsp (Aw -aT) ay

( 1 3 .2- 1 5 )

dY

Xsp ( aT) 1 w dX 0 As-an � dX = 8sp = r [A aT ] dY io ax

Combining energy and mass balance equations ( 1 3 .2- 1 4) and ( 1 3 .2- 1 5) gives

where

o

Q :p

-

+

p u (h - hs )

sp

Q:p

( 1 3 .2- 1 6)

Q�p is evaluated at the separation section (X = Xsp) and thus can be obtained using the numerical results on the front of the cylinder at the separation point.

EXTERNAL FLOW FILM BOILING 319 By applying e = (T - Ts ) / (Tw - Ts) , r = X/Xsp and w fiat interface results, and Eq. ( 1 3 .2- 1 6) can be transfonned into

t{ ( )

A w ae � Tsat Xsp io 8 aw =

w

=

o

+

As

[ X; ( 1

p

d8

ae

dr

ar

-

)

= Y /8, a regular

8" dr aw 1 d8

ae

Q�p

� ae

]

8 aw

}

dr

w= l

( 1 3 . 2- 1 7)

To solve Eq. ( 1 3 . 2- 1 7), the following additional assumptions were made: • •

e = 1 - w (linear temperature distribution across the vapor film) = A[ 1 + cos(rr r ) ] the interface shape)

8 (r)

+ 8sp,

( 1 3 .2- 1 8)

where A = [8 (0) - 8spl/2; cosine function for ( 1 3 .2- 1 9)

The cosine shape gives a closed curve that satisfies the boundary conditions: 8 = 8sp at X = Xsp ; 8' = 0 at X = 0; and 8 = 8max at X = O. By apply ing the assumed temperature and interface profiles and integrating Eq. ( 1 3.2- 1 7), we get

Equation ( 1 3 .2-20) yields a unique solution for the variable 'A' that detennines a cosine shape given in Eq. ( 1 3 . 2- 1 9). The average a is detennined by integrating the local ex over the entire surface. The integration includes the front and the wake of the cylinder. In other words, the a of a cylinder can be calculated by using the following equation :

-

ex =

Q 2rr R (Tw - Ts ) ,

10K a (n /H,a, R

( 1 3 .2-2 1 )

where Q i s the total heat transfer over the entire cylinder surface given by

Q

=

dl;

( 1 3 .2-22)

The average Nu for a cylinder can then be calculated as:

Nu =

aD A '

-

( 1 3 .2-23)

where A is evaluated at (Tw + Ts) /2. The numerical solutions for Nu as a function of liquid subcooling at one set of system variables are shown in Figure 4. It is clear that Nu is a strong function of liquid subcooling.

320 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION 600 500 400 Nu

300 200 100

Figure 4

The effect of liquid subcooling on average Nusselt number for a 3 .2-mm cylinder for water at 2.95 rn/s: Ew 0. 1 2, li Tsub = 5 1 5°C, P = 0.3 1 3 MPa, Pry = 0.85, ReL = 77, 400, JaL = 0.03 to 0. 1 (Chou and Witte, 1 995). =

The relative magnitude (Qwake /Qfront) of wake and front heat transfer is in creased with increased subcooling, as shown in Figure 5. The magnitude is up to about 20% at high subcooling and therefore should not be neglected, especially at high subcooling. This result helps explain why previous flow-film-boiling models that neglected the wake underestimated experimental data by a significant amount.

13.3 DATA CORRELATION AND COMPARISON TO THEORY 13.3.1 Cylinders Bromley et al. ( 1 953) found that the results of their analysis of flow film boiling did not correlate data very well. Consequently, they turned to dimensional reasoning to develop a correlation. For saturated boiling, Bromley found separate correlations that fit different velocities according to the value of the parameter V / (gD) 1/2. where

Voo r;:n y gD

ygi5

( 1 3 .3-3)

The factor Z is defined as

( 1 3.3-4) and

f3 =

322 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATlON Liu et al. ( 1 992) found that Epstein and Hauser's correlation agreed reason ably well with their experimental data for larger cylinders but not for smaHer ones; thus, Liu concluded that a more general correlation was needed. Yilmaz and Westwater ( 1 980) found that they could correlate their data for flow film boi ling of R- 1 1 3 in a manner similar to Bromley et al. ( 1 953): it

= 3 .68

[ Pv VooAv h ] 1 /2 D � Tsa t

fg

the only exception being the lead constant, 3 .68, compared to 2.7 for Bromley's correlation. Chou et al. ( 1 995) compared her solution (described in previous sections) that includes the Qwake to the data of Motte and Bromley ( 1 953), Yilmaz and Westwater ( 1 980), Chang and Witte ( 1 990), Sankaran and Witte ( 1 990), and Liu et al. ( 1 992). Figure 6 shows a comparison between available experimental data, the correlation, Data of Motte & Bromley ( 1 957) Data of Yilmaz & Westwater (1 980) Data of Chang ( 1 987) + • Data of Sankaran & White (1990) Data of Liu, Shiotsu & Sakurai (1992) o - . - - - Correlation of Liu, Shiotsu & Sakurai (1 992) - - - • • - - • • Bromley, LeRoy & Robbers' Model ( 1 953) - - - - - Witte & Orozco's Model ( 1984) --- Authors' Model (1 994)

•

1 000

Nu 1 00

10

o

Figure 6

0 . 05

0. 1

J'\.

0. 1 5

0.2

Comparison o f various models to experimental data and correlations (Chou et aI., 1 995).

EXTERNAL FLOW FILM BOILING 323 and the models of Chou and Witte, Witte and Orozco, and Bromley, Leroy, and Robbers. The data are plotted as average vapor Nu vs. liquid Ja, which represents the liquid subcooling. Three different Re's are used that represent as close a match as possible to experimental conditions. The model by Bromley et al. ( 1 953) did not consider the liquid subcooling effect; therefore, it can be used only for very small subcooling cases. As shown in Figure 6, Motte and Bromley's data are scattered in their sub cooling range. Both the ChoulWitte and the Witte/Orozco models predict lower heat transfer than the data. It should be pointed out that Motte and Bromley's data overlap the JaL range where the ChoulWitte model begins to lose its applicability. The applicable range of the ChoulWitte model is for Jal > 0.04, and non-zero liquid velocity. The data of Chang and Yilmaz and Westwater fall at JaL values less than the applicable range of the ChoulWitte model. Hence, comparative calculations were not performed for those data. Figure 6 shows that the ChoulWitte model is in better agreement with the subcooled-ftow-film boiling data than the previous models. The merit of the ChoulWitte model is much more obvious at high Ja than at low Ja. The model of Bromley et al. is significantly lower than the data, especially with increasing Ja. The Witte/Orozco model is better than Bromley's model; however, it diverges from the data when subcooling is increased. The ChoulWitte model yields higher q results than the previous models. The difference between the ChoulWitte model and the Witte/Orozco model is greater at higher Ja, and the two models are closer at lower subcooling, since Qwake is less important at lower Ja. The ChoulWitte model gives closer predictions to experimental data than previous models, especially at high subcooling. Figure 6 also shows selected data from Liu et al. ( 1 992), along with the correlation developed by Liu. Their correlation indeed fits their data better in the low JaL range than does the ChoulWitte model. However, at the higher JaL end of Liu's data, the ChoulWitte model results appear to coincide well with the experimental data. Because Liu's correlation was based on data limited to relatively low JaL and ReL ranges, it was deemed inappropriate to extend it into the range where it could be compared to Sankaran's data. Montasser ( 1 994) compared his data for cylinders to those of Motte and Bromley ( 1 957), Yilmaz and Westwater ( 1 980), Sankaran and Witte ( 1 990), and Chang and Witte ( 1 990). The result is shown in Figure 7. The data are plotted in terms of modified Nu and Ja (as shown below). Also shown on the figure is the correlation

Nu ** = O A I (Ja; * ) 1 .3 8

( 1 3 .3-5)


t

> �

Z :::J

•

Bromley et al. [ 1 953]

Present data

0

Y1lmaz and Westwater [ 1 980]

6-

0

Sankaran and Witte [ 1 990]

¢

V

Chang and Witte [ 1 990]

Tw_const Qin - const

Ja

**

Figure 7

where

I

Comparison of data for flow film boiling from cylindrical heaters (Montasser, 1 994).

1 **v ) nSUb NU ** - NU * (Ja PI Ja[ -

Jai * = _----'P_v'__ ' Pr[ -v J.i-I

R

and

nsub

= 0.356

'

Nu *

=

VRe,

Nu

--,===

PI ' Pv !!i... Ja v � Ja ** v = Pr v

( ) R Jal PI P

�

Pn v J.i-l

0.24

EXTERNAL FLOW FILM BOILING 325 The figure shows that Montasser's data are in good agreement with those of Sankaran and Witte. Motte and Bromley's data are widely scattered within the range of J ai* . The data are correlated to within ±40% by Eq. ( 1 3.3-5). Liu et al. ( 1 992) obtained an extensive set of data for different fluids over a fairly wide range of velocities, pressures, diameters, and subcoolings. They developed correlations for saturated liquids as well as for subcooled liquids. For saturated conditions, their correlation took the form of

NUv 2 + Nu

(I J

=

H (Fr, D ' ) K (D' ) M 1/4

( 1 3.3-6)

V2 / gD.

acoD/ A,

where Fr is a Froude number, The Nu is defined as Nuv = where is the convective part of the heat transfer only. Other parameters are defined as follows

aco

and

H (Fr, D') The parameter

=

( 1 + 0.68Fro.625 ) O .4 + 0.45tanh[0.4(D'

-

1 .3)Fr]

K(D') depends upon the value of D' as

K (D') K (D , ) K (D') K ( D')

0.44 ( D , ) 1/4 for D' < 0. 14 0.75 0. 14 < D , < 1 .25 (1 + 0.28D') 2 1 D' 1 . 25 � D' :::: 6.6 ( 1 + 3.0D') = 0.4 1 5 ( D, ) 1/4 D' :::: 6.6 =

-

_

_

The factor M is one that Liu et al. developed for pool film boiling and is defined as

M

=

[ 1

E3 Gr (RPr[Sp) -2 Sp 1 + E SpPn --

where Sp is the nondimensional superheat

Cp/:). Tsat Sp = ----'h f g Prv

326 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION and

E

=

�

-271 SC* -4 Sc*2 (A

+

cv'B)) 1 / 3 + (A - Cv'B) 1 / 3 + Sc*

1 1 - R 2 SpPrI + _ R 2 Sp2 p?,I 3 1 2 2 32 B SpR 2 Pn + 4 Sp2 PI[2 + Sc*3 / R 2 + SpPnSc 3 27 27 27 Cpi fj. T.w b 1 Sc* = 0.93Pr�· 22Sc C = - R 2 SpPrl , Sc h fg 2 A

=

=

3+

*

=

4

,

Sc is a nondimensional subcooling while Sc * is a modified subcooling parameter. The Grashof number also appears in M and is defined below, along with the modified latent heat and R.

,

Gr

=

g (pl - p, ) D J Pv v�

h lg = h + O .5cp , !1 T,a' . R

•

=

JPP,I fi.MI,

Liu found that he could use the same basic form of Eq. ( 1 3.3-6) to correlate subcooled data if a modification of Sc is performed. He redefined Sc as Sc* = Ksc Sc where Ksc must take on appropriate values to fit experimental data. Liu found he could fit an equation for the modified Ksc, shown below

Ksc = 0.92Pr?·22 + 0. 1 8(FrPrl )o . 22 For purposes of data correlation, Liu recast Eq. ( 1 3.3-6) into the form ( 1 3.3-7) N Nuv /(1 + 2/Nuv)/ K (D')/ H (Fr, D ') M I /4 Figure 8 shows Liu's data for water at various pressures compared to Eq. ( 1 3.3-7). The agreement is excellent. Comparisons to other data for water and R- 1 1 3 ( 1 992) =

=

show equally good agreement between the correlating equation and the data. Liu also compared his correlation to the data of Shigechi ( 1983). Liu's correla tion, accounting for radiation, predicts Shigechi's data to within - 15% to +20%, which is a reasonably good fit for boiling data.

13.3.2 Spheres Liu and Theofanous ( 1995b) found that they could correlate their data for spheres in single phase flow with separate expressions for saturated and subcooled conditions. For saturated water, they found

Nus

=

0.5Ref/2 Ml

/-Lv

( -K R4 ) 1 /4 Sp

( 1 3.3-8)

!j

!M

I

10

10'

•

G- I O K

0.47m1.

IO'

M

IO' 10'

G-20K

0.39mIJ

G-20K G-20K

• O.l6m1s

Subcoolln, G-20K

a 0.32 mIJ

Flow � o Oml.

Diameter: 3 mm Near atmospheric pressure a Superheat: 200-800 K a

Y"MO.lS

Water

loa

7.

�

10

10' IO' M

Diameter. 3 mm Pressure: 294 K SuperbeaC 2»-800 K

MO.lj

10'

IO'

Row � Subcoolinl o-lOK o O ml. • 0.06 mi. G-30K a O.24m1s O-30 K o-2O K a 0.39 m1. a 0.47 m1. 0-20 K OK • 0.55 m1. • 0.7 1 m1. OK • 0.78m1. OK

�

I

10

10'

IO'

Diameter. 3 mm

Pressure: 490 K Superheat: 200-800

M

K

•

a a a · .

10'

0.32m1. 0.47m1. 0.63 mi. 0.78m1. 0.86m1. 1.03m1s

G-20 K G-20 K OK OK OK OK

lOS

Flow nle SubcoolinS ()....4() K o O ml, �K • 0. 16 mi•

Figure 8 (a), (b), (c) Comparison of flow film boiling correlation with experimental data on 3-mm dia. cylinders in subcooled water at various pressures (Liu et aI ., 1 992).

�

100 .

(c)


)(

]

+

1.5

o 0

D=6.35mm D=9.53mm D= 12.7mm D= 1 9. 1mm

�

� -iG)

1 . 0 1-115-5-....p,....__ �iWI��.

0.5

/(gD) 112 U

0.5

Figure 9

2

Comparison of the data of Liu and Theofanous ( 1 995b) to their correlation, ( 1 3.3- 1 0).

R,

correlated their data well, where K, and Sp are the same as used by Liu et al. ( 1 992) for film boiling from cylinders. For subcooled flow, they found the correlation Sc Nusc = Nu + 0.07ReIo. 77 PrIo.s f-tl Sp

/1v

S

( 1 3 .3-9)

worked well, where Sc is the subcooling parameter defined by Liu et al. They found a general correlation by combining Eqs. ( 1 3 .3-8) and ( 1 3 .3-9), which is

where

Nu = [Nu� + (F(Fr)Nusc » S ] l iS Nup = 0.64(Ar/Sp) 3

( 1 3 .3- 1 0)

1 /4

( 1 3.3- 1 1 )

Ar is the Archimedes number, and F(Fr) is an empirical function

Ar = g (PI - Pv) D Pv v;

F(Fr) = 1

_

I Fro . s - 1 1 0.2

1 +

Figure 9 shows the data o f Liu and Theofanous plotted i n terms of the Fr. I t shows that the data can be correlated in a band within ± 1 5 % .

13.4 CONCLUDING REMARKS In this article, a review of the status of present-day understanding of external flow film boiling has been undertaken. The concentration was on flow film boiling over

EXTERNAL FLOW FILM BOILING 329 bodies that have wakes behind them-primarily cylinders and spheres. A brief review of a typical analytical method of treating the phenomenon was presented. Analyses can be characterized as being laminar boundary-layer-like on the forward portions of the bodies, with few attempts at modelling the heat transfer in the wake until just recently. Emphasis was placed on the correlations that have been developed to predict film boiling heat transfer. It is clear that our understanding of flow film boiling is incomplete, especially as it relates to treating the behavior of film boiling wakes properly.

REFERENCES Bromley, L. A., LeRoy, N. R., and Robbers, J. A. 1 953. Heat Transfer in Forced Convection Film Boiling. Ind. and Eng. Chern. 45( 1 2) :2639-2646. Chang, K. H. and Witte, L. C. 1 990. Liquid-Solid Contact During Flor Film Boiling of Subcooled Freon- 1 1 . 1. Ht. Trans. 1 1 2 :465-47 1 . Chappidi, P. R., Gunnerson, F. S., and Pasamehmetoglu, K. O. 1 99 1 . Subcooled Forced Convection Film Boiling Drag and Heat Transfer of a Wedge. 1. Therm. and Ht. Trans. 5(3):355-365. Chou, X. S . and Witte, L. C. 1 995. Subcooled Flow Film Boiling Across a Horizontal Cylinder, Part I: Analytical Model. 1. Ht Trans. 1 1 7( 1 ) : 1 67-1 74. Chou, X. S., Witte, L. c., and Sankaran, S . 1 995 . Subcooled Flow Film Boiling Across a Horizontal Cylinder, Part II: Comparison to Experimental Data. 1. Ht Trans. 1 1 7( 1 ): 1 75-178. Epstein, M . and Hauser, G. 1 980. Subcooled Forced Convection Film Boiling in the Forward Stagnation Region of a Sphere or Cylinder. Int. 1. Ht. Mass Trans. 23 : 1 79-1 89. Fodemski, T. R. 1 992. Forced Convection Film Boiling in the Stagnation Region of a Molten Drop and Its Application to Vapor Explosions. Int. 1. Ht. Mass Trans. 35(8):2005-20 1 6. Fodemski, T. R. and Hall, W. B. 1 982. Forced Convection Boiling on a Sphere in a Subcooled or Superheated Liquid. Proc. 7th Int. Ht. Trans. Con! Volume 4, pp. 375-379, Munich. Huang, L., and L. C. Witte, 1 996. An Experimental Investigation of the Effects of Subcooling and Velocity on Boiling of Freon- 1 1 3 , 1. Ht. Trans, 1 1 8(2):436-44 1 . Kobayashi, K . 1 965. Film Boiling Heat Transfer Around a Sphere in Forced Convection. 1. Nucl. Sci. and Tech. 2(2) :62--67 . Liu, C, and Theofanous, T. G. 1 995a. Film Boiling on Spheres in Single- and Two-Phase, Part II: A Theoretical Study. Proc. of 1 995 National Heat Transfer Conference (ANS), pp. 48--6 1 . Liu, C. and Theofanous, T. G . 1 995b. Film Boiling on Spheres in Single- and Two-Phase, Part I : Experimental Studies. Proc. of 1 995 National Heat Transfer Conference (ANS), pp. 34-47. Liu, Q. S., Shiotsu, M., and Sakurai, A. 1 992. A Correlation for Forced Convection Film Boiling Heat Transfer from a Horizontal Cylinder Under Subcooled Conditions. Fundamentals of Subcooled Flow Boiling. ASME 1992, HTD-Vol. 2 1 7 :2 1 -32. Montasser, O. 1 994. Effect of Wall Heat Flux Conditions on Cross Flow Film Boiling Heat Transfer. Ph.D. Dissertation, University of Tokyo. Motte, E. A. and Bromley, L. A. 1 957. Film Boiling of Flowing Subcooled Liquids. Ind. and Eng. Chern. 49( 1 1 ) : 1 92 1 - 1 928. Sankaran, S . and Witte, L. C. 1 990. Highly Subcooled Flow Boiling of Freon- l 1 3 over Cylinders. Presented at the AIAAIASME Heat Transfer and Thermophysics Conference, Seattle; ASME HTD Vol. 1 36:29-34.

330 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION Shigechi, T. 1 983. Studies on External Forced Convection Film Boiling Heat Transfer. Doctoral Thesis, Kyushu University, Japan (in Japanese). Shoji, M. and Montasser, O. 1 993. Theoretical Analysis of Uniform Wall Heat Flux and Uniform Wall Temperature Forced Convection Film Boiling Heat Transfer. 1. Faculty Engineering, The University a/ Tokyo (8) XLII, ( 1 ):4 1 -56. Theofanous, T. G., Houze, R. N., and Brumfield, L. K. 1 976. Turbulent Mass Transfer at Free, Gas Liquid Interface, with Applications to Open-Channel, Bubble, and Jet Flows. Int. 1. Ht. Mass Trans. 1 9 : 6 1 3-624. Walsh, S. and Wilson, S. D. R. 1 979. Boundary-Layer Flow in Forced Convection Film Boiling on a Wedge. Int. 1. Ht. Mass Trans. 22:596 -574. Wilson, S. D. R. 1 979. Steady and Transient Film Boiling on a Sphere in Forced Convection. Int. 1. Ht. Mass Trans. 22: 207-2 1 8 . Witte, L. C. 1 968. Film Boiling From a Sphere. I. E. C. -Fundamentals 7(3):5 1 7 . Witte, L . C. and Orozco, J . 1 984. The Effect o f Vapor Velocity Profile Shape o n Flow Film Boiling from Submerged Bodies. 1. Ht. Trans. 1 06: 1 9 1 - 1 97. Yilmaz, S . and Westwater, J. W. 1 980. Effect of Velocity on Heat Transfer to Boiling Freon- 1 1 3 . 1. Ht. Trans. 1 02:26 -3 1 .

CHAPTER

FOURTEEN

AUGMENTATION TECHNIQUES AND EXTERNAL FLOW B OILING 14.1

�asanori �onde

Saga University, 1 Honjo Saga, 840 Japan

14.2

Tatsuhiro Ueda

University of Tokyo, Tokyo, 1 1 3 Japan

14.2

Yasuo Koizumi

Kogakuin University, Hachioji, Tokyo, 192 Japan

14.3

�asanori �onde and Vijay K. Dhir

University of California, Los Angeles, CA 90024, USA

In this chapter, jet impingement technique is discussed extensively under external flow boiling systems. Flow structure under impinging jet is first presented, and then characteristics in boiling heat transfer as well as critical heat flux are discussed. Correlations are presented for predicting critical heat flux in several configura tions. Another external flow boiling system investigated here is the falling film evaporation and boiling. Finally, techniques to improve heat transfer and CHF in external flow boiling systems are reviewed.

14.1 IMPINGING JETS AND OTHER GEOMETRIES Boiling provides an efficient heat transfer mechanism for applications such as the cooling of electric components (Incropera, 1 988), metal processing (Viskanta and Incropera, 1 992), and fusion components (Boyd, 1 983), where heat transfer coefficients commonly exceed 1 0,000 W/(m2 K) at relatively high heat fluxes. The critical heat flux (CHF), which defines the upper limit of the efficient range of boiling heat transfer, limits the maximum heat flux that may be dissipated from the components. For enhancement of the CHF, it is important to utilize the boiling 331

332 HANDBOOK OF PHASE CHANGE: BOLING AND CONDENSA nON

Nozzle

Ambient Gas

/k .. �W/J .... a. Free-surface

c.

Figure 1

Ambient Gas f??J&j(foVAWM --... Liquid b. Wall (free-surface)

Submerged

d. Confined

Schematic of various jet configurations.

heat transfer properly and to satisfy the demands of efficient cooling processes. Jet impingement systems offer a very attractive means to enhance the CHF.

14.1.1 Configurations of Jet Impingement Systems The jet system may be categorized into four different configurations: free-surface jets, submerged jets, confined jets, and wall jets, as schematically shown in Figure 1. From a viewpoint of enhancing heat transfer as well as CHF, one has to first focus on the free-surface jets, especially the circular one, because it provides a simple and basic geometry that can be extended to other configurations. Circular free-surface impinging jets have been extensively studied (Azuma and Hoshino, 1 984; Liu et aI., 1 99 1 ; Stevens and Webb, 1 993; Watson, 1 964). Figure 2 shows representative flow conditions of velocity profile, pressure change, and the liquid layer thickness. The flow structure in the radial layer is demarcated into impinging zone, laminar region, and turbulent region. Within the impinging zone (r/ (d/2) < 1 .57, r is radius and d is jet diameter; Liu et aI., 1 99 1 ), the liquid is continuously accelerated with decreasing static pressure due to the dynamic

AUGMENTATION TECHNIQUES AND EXTERNAL FLOW BOILING

I

333

: Impinging Region

II : Laminar flow Region III : Turbulent flow Region

Turbulent flow

--------��--;---+---+_- r

III

2

Figure 2

ro *

r/(d/2)

Distribution of pressure, velocity, and boundary layer thickness for a circular, free surface impinging jet along with the r-direction.

pressure of jet impingement, and then it approaches the jet velocity at the nozzle exit. For the laminar region ( 1 .57 < r/ (d/2) < r; (=600(uN d/ v) - 0.422 , U N is jet velocity; Liu et aI. , 1 99 1 ), the liquid spreads radially in laminar flow, and the streamwise velocity is essentially that of the jet. The hydrodynamic effects of impingement completely disappear within the flow in this region. The laminar boundary layer is developed along the r-direction and then reaches the free surface at r� (=O.282(uN d/ v) - 1 /3 (see Azuma and Hoshino, 1 984). Beyond this point, the thickness of the boundary layer starts decreasing, while the streamwise velocity in the layer is almost the same as the jet velocity. For the turbulent region (r; < r/(d/2», the liquid flow becomes turbulent where the streamwise velocity starts decreasing while the liquid film thickness increases. For the planar free surface jet, the flow characteristics are similar to that of a circular jet, except for a change in the liquid film thickness. For the wall free surface jet (Figure 1 b), the flow conditions are similar to that of a planar jet, except for the existence of an impingement zone. For submerged and confined arrangements (Figures 1 c and d), the flow con ditions may be largely influenced by geometrical conditions, such as the distance

334

HANDBOOK OF PHASE CHANGE: BOLING AND CONDENSATION

between nozzle exit and wall and the confined space between nozzle plate and wall. Consequently, these flow conditions become more complicated. For this reason, much interest has been paid to the free surface jets. In addition, most studies have focused on heat transfer problems in the laminar and turbulent flow regions of the free-surface jets, shown in Figure 1 . Heat transfer problems related to the free-surface jet will be mainly discussed in the following sections.

14.1.2 Heat Transfer Characteristics Occurrence of nucleate boiling is generally suppressed or delayed due to the forced convection effects. Therefore, the wall superheat (D. Tsa t = Tw Twt ) needed for nucleate boiling to commence is increased, depending on the liquid velocity and subcooling of the jet. Nucleation is delayed in the film due to improved heat transfer from forced convective effects. After commencement of nucleate boiling, a further increase in heat flux activates more sites, eventually leading into fully developed nucleate boiling. -

Incipience of nucleate boiling. Few experiments have been conducted on the in cipient point of nucleate boiling with impinging jets, so a parametric understanding about it is rather limited. From an application point of view, however, the incipient point is of less interest than understanding the CHF phenomenon and extending the fully developed boiling region further. Fully developed nucleate boiling for free-surface jets. Free-surface jets have been extensively investigated in literature (Copeland, 1 970; Furuya et al., 1 995 ; Katto and Ishii, 1 978; Katto and Kunihiro, 1 973; Katto and Monde, 1 974; Monde and Katto, 1 978; Monde et al., 1 980; Nonn et al., 1 988; Ruch and Holman, 1 975; Vader et al., 1 992). For fully developed nucleate boiling, the heat flux was almost independent of impinging jet velocity, diameter, heater dimension, and surface orientation, and depended only on the wall superheat. Katto and Monde ( 1 974) pointed out that even for a jet velocity range as large as 5.3-60 m/s with saturated water, the fully developed nucleate boiling curve roughly lay on an monotonic extension of the data for pool boiling to larger heat fluxes and wall superheats. In addition, Monde et al. ( 1 980) reported that even for multiple impinging jets, flow conditions are more complicated compared with a single impinging jet, the number or placement of jets has little effect on the boiling curve except for the degree of scatter in the data, and these data eventually merged into the extension of pool boiling curve (see Figure 3). For a planar free-surface jet of saturated water, the characteristics of boiling heat transfer were reported to be consistent with this result (Furuya et al., 1 995; Katto and Ishii, 1 978).

AUGMENTATION lECHNIQUES AND EXTERNAL FLOW BOILING 335

3

Water at atmospheric pressure Katto and Monde (1974); Monde and Katto (1978); Monde (1 980) (Free)

�/

Katto and Ishii (1978 (Free)

Furuya et al.(1995) (Free)

I

Katto and Kunihiro (197 (Free & Submerged)

Copeland (1970 (Free)

: Circular - - - : Plane : Pool

--

10 Figure 3

_

.

/

/ ./Kutateladze � (1 952) // .

(Pool)

/� //

/ /

Nishikawa et al. (Pool)

-

51!-----'- ----'-----'----L.- � ---'-- '-::!----'- -----L- -----'- --'--'----'-- � 1 00 !:l. Tsat [K]

Comparison of nucleate boiling correlations for free-surface and submerged jets of water with circular. plane. and wall jet configurations. (The lines presented here are chosen to pass through the mean of their respective data. The data scatter around the corresponding line) .

Figure 3 depicts a comparison of nucleate boiling curves for free-surface jets for saturated and subcooled water at atmospheric pressure. Despite significant differences between jet configurations, there is generally good agreement between their respective curves and the pool boiling curve (Kutateladze, 1 952; Nishikawa and Yamagata, 1 960). It should be mentioned that the agreement for Rl 1 3 is rather poor as compared to that for water (Monde and Katto, 1 978; Monde et aI., 1 980; Ruch and Holman, 1 975).

336 HANDBOOK OF PHASE CHANGE: BOLING AND CONDENSATION Roughly speaking, the heat transfer characteristics for the fully developed nucleate boiling are not significantly different in the jet impingement and pool boiling configurations (Katto and Kunihiro, 1 973; Monde and Katto, 1 978). The difference between the two is mainly attributed to the different surface conditions, such as roughness and wettability. The correlations for pool boiling are recommended to predict the heat transfer coefficient for the fully developed nucleate boiling with free-surface jets.

14. 1 .3 Critical Heat Flux Impinging jet system is capable of cooling at extremely high heat fluxes, more than 1 0 MW1m2 • Effects of system parameters on CHF for some jet configurations are studied extensively, and specific correlations for CHF are available for a number of these configurations with high accuracy. Theoretical understanding of the CHF, however, still is not clear, in spite of some efforts (Kandula, 1 990; Lienhard and Eichhorn, 1 979; Monde, 1 985; Mudawar and Wadsworth, 1 99 1 ) to model the CHF mechanism for each jet configuration. Circular impinging jet. A number of studies are available for the circular im pinging geometry over a wide range of experimental conditions.

Single jet impingement. Several authors (Katto and Kunihiro, 1 973; Katto and Monde, 1 974; Katto and Shimizu, 1 979; Monde, 1 987; Monde and Katto, 1 978; Monde and Okuma, 1 985) obtained experimental data for this configuration and categorized the CHF characteristics into four regimes, referred to as the L-, V-, 1-, and HP-regimes. The CHF dependence on parameters such as jet velocity and heated diameter markedly changes in each regime (Katto and Shimizu, 1 979; Monde, 1 987; Monde and Okuma, 1 985). As for the general feature of each region, region L appears for low velocity and large disk, either region V or I appears at moderate pressure (whether the CHF depends on the jet velocity or not), and region HP appears at high pressure. Generalized correlations for the CHF in the respective regimes are proposed to predict them with a good accuracy, while the boundaries between them remain ambiguous; the boundary between 1- and HP regimes especially is not yet specified. This is attributed mainly to a shortage of the CHF data in HP-regime. The vast majority of the CHF data belongs to the V-regime, and their reproducibility is also very good. The following correlations and boundaries are proposed by Monde ( 1 985, 1 987), and Monde and Okuma ( 1 985): For the L-regime ( 1 4. 1 - 1 )

AUGMENTATION TECHNIQUES AND EXTERNAL FLOW BOILING 337 Table 1 Exponent and constant in Eq. (14.1-2) for each regime

V-regime I-regime HP-regime

m

n

k

C

0.645 0.466 1 .27

0.343 0.42 1 0.28

-0.364

0.22 1 0.69 1 0. 1 72

-0.303 -0. 1 0 1

where t = 0.0389(ptf pg )O. 674 (dl Ja I g ( PI For the V-I-and HP-regimes,

_ C (PI IPg )m ( 2 (D2a - ) 2 ( 1 d) Pg h I gU N Plu N

Eq. Eq. Eq.

-

( 14. 1 -2a) ( 1 4. 1 -2b) ( 1 4 1 2c )

.

- pg)) 1 . 24

.

qeo

+

Dld)k

(14. 1 -2)

where each component, m, n, and k, and the constant, C, are listed for the respec tive regimes in Table 1 . The correlation for the HP-regime should be considered tentative because it is derived from only 24 CHF data points. For the boundaries between L- and V-regimes, the CHF only in the L-regime appears for

D id :::: 1 8.4(pt fpg )- O. 1 94 (dI Jalg(PI - Pg » -O.76 (2alpl u� (D

_

d » - O. 209 ( 14. 1 -3)

otherwise, the CHF only in the V-regime appears under the following condition for the boundaries between V- and I-regimes:

PI/Pg 67. 1 or Pl /pg 67. 1 2al plu� (D - d) :::: 4.5 x 10-7 (pt! Pg ) 2 . 29 ( 1 >

( 14. 1 -4a)

DO

'roIl .-<M

-- Chun-Seban - - - - Dolder

1.0 0.8 0.6

...\.0 � \. _ -

Plots : Chun-Seban ( 1971 )

0.4

-

"

- - -

�

0. 2 0. 1 Figure 1 1

2

4

6 8 10'

2

2 4 Ref = 4r/T'l 1

Heat transfer coefficients of falling fi lms (Tj

=

0).

4

352


I II " J I II II I

1 1 1 1 1\\ \ \ I

I

I

"

I

I

I

, � �+ , ,

I

I

I

I

! � , "

"

"

I

111 1 I \\\ 1 \\\ \ 1111 \\\\ 11I1 III1 \\'1 1 111 1111 "

1

"

, , "

\

\

\

1

\ \ "

• I I ,

\ , I I

I I t .

I 1 1 I

' HI

I I I I

1 1 11

• I I I

Figure 12

Force balance at the top edge boundary of a dry patch.

However, when the film thickness becomes very thin as the film falls down, a dry patch is formed or the film disrupts into rivulets. The minimum flow rate at which the film can flow as a continuous film is called the minimum wetting rate (MWR). At the front edge of the permanent dry patch, the dynamic pressure of the film flow is balanced by the upward component of the surface tension (see Figure 1 2; Hartley and Murgatroyd, 1 964). If the film is laminar with a smooth surface and uniform thickness Yi , l

-

1 Plu2dy Y;

2 0

= a ( 1 - cos e )

1 Plu2dy = __1_3 2-2 y;

( 1 4.2-42)

From Eqs. ( 1 4.2-22) and ( 14.2-23), l

� Pf = -2 r=

respectively. Then,

Y;

lY; Pludy = - Pt

rc =

0

1

o MWR r c

1 p g 15

g

-

3 17 1

3

Yi

171

and

cos e ) ] 3/5 ( 17;1 )1 /5

( 1 4.2-43) ( 1 4.2-44)

is expressed from these as

1 .69[a ( 1 -

(1 4.2-45)

AUGMENTATION lECHNIQUES AND EXTERNAL FLOW BOILING 353 I

01

I

I

Eq . ( 14 . 2 . 3 - 4 8 ) - 400

Acrylic resin pipe o D = 6mm

-

6. D = 10mm o D = 25mm

�

�

- 200

8 � o

Figure 13

I

0. 5

I

Distance

1 z

I

1.5

m

o

Minimum wetting rates of isothernlal falling films.

However, it has been known that Eq. ( 1 4.2-45) gives too high values compared with the measured MWRs (Norman and McIntyre, 1 960). Similar results have been obtained by Hewitt and Lacey ( 1 965) for a climbing film. Koizumi et al. ( 1 996) obtained MWRs by blowing an air jet to a water film to artificially form a dry patch and measuring the lowest film flow rate at which the dry patch was rewetted by the film flow (Figure 1 3). The MWRs measured the decrease with distance z from the top of the film, which corresponds to the growth of height of waves on the film. Since the film is associated with many waves and the dynamic pressure fluctuates largely, the appropriate value to apply to the left hand term of Eq. ( 14.2-42) seems to be the maximum of the fluctuating dynamic pressure. It has been observed that the contact angle () increased with increasing the film flow rate, and the dry patch was rewetted when () reached the advancing contact angle ()A (()A = 1 20 degree for water). Therefore, the force balance at the MWR condition should be

[ � Jjilmr

2 PI U dY

]

= a ( 1 - COS ()A ) ( 1 4.2-46) max Koizumi et al. suggested that the above equation is approximately reduced to !:1

( )5 =

Ymax Pf ----y;-

a ( 1 - cos ()A )

where Yma x is the maximum film thickness, including the wave height.

( 1 4.2-47)

354


The other analytical approach to the MWR is to look into energy of the film flow (Bankoff, 1 97 1 ; Hartley and Murgatroyd, 1 964). Considering that the critical thickness of the film below which the film disrupts is given by the condition that the sum of the kinetic and the surface energy becomes minimum, Hartley and Murgatroyd derived the following equation

rc = 0. 8030' 3/5

( 7J;' )

1 /5 ( 1 4.2-48)

This correlation overpredicts MWRs like Eq. ( 14.2-45), since the wavy motion of the film is not reflected.

14.2.4 Falling Film Boiling When a falling film on a heated surface is at saturated condition, temperature over the film surface is almost uniform and close to the saturation temperature. Relations between the heat flux qw and the wall superheat Ts = Tw - Ts mea sured for boiling-falling water films (Fujita and Veda, 1 978b) are presented in Figure 1 4. Since the film thickness is thin, bubbles generated form hemispheri cal domes on the film and the domes grow to burst while flowing down with the film. In a range of qw < 0.80 X 1 05 W/m2 , the number of nucleation sites is small and qw ex T s. The heat transfer coefficient

�

�

( 1 4.2-49) is dependent on the film flow rate, and the convective heat transfer is dominant. The heat transfer coefficients show values about 1 0% higher than those of Chun and Seban ( 1 97 1 ) correlations for the falling film evaporation, Eqs. ( 1 4.2-40) and ( 14.2-4 1 ) . I n the range o f q w > 1 .2 x 1 05 W/m2 , the number o f the nucleation sites increases greatly. The relation between qw and Ts shows no dependency on the film flow rate and is expressed by a unique line of saturated pool nucleate boiling. In this nucleate-boiling-dominant condition, the amount of the droplet entrainment due to the bubble burst cannot be neglected. Although the droplet entrainment by the bubble burst has not been well under stood, the surface tension or the kind of liquid has a large effect on the droplet entrainment rate. Petrovichev et al. ( 1 97 1 ) measured droplet entrainment rates m EB from a boiling water film flowing down on the outer surface of 50 mm in heated length and proposed the following correlation:

�

m EB = 2.09 x

10 3 Yi

(

qei qw 1 06 1 06

)207 2

[kg/m s]

( 14.2-50)

AUGMENTATION lECHNIQUES AND EXTERNAL FLOW BOILING

4

355

fin kg/ms

A 0. 141

3

(Water)

o 0. 1 70

o 0.242 0 0.349 'V 0.508 X 0. 723

2

N

EI :a

...... ;.. C7' >
-o

10

----- Lockhart-Matinclli ( 1 9 4 5)

.. _

(1966) (1 98 1)

'-'-��-'6 d'o-c1

5 Figure

5 0

X

0

10

Comparison of experimental two-phase pressure drop data with correlations.

In summary, the two-phase friction multiplier in subcooled flow boiling is close to that in saturated flow boiling and can be estimated reasonably if the true mass fraction is adequately evaluated.

15.3 SATURATED FLOW BOILING 15.3.1 Overview The process of converting liquid into vapor is basic to many thermodynamic cycles. It is encountered in applications ranging from steam generation in oil or coal-fired boilers, nuclear reactors, refrigeration, and air-conditioning applications, to pro cesses in chemical and petrochemical industries. Saturated flow boiling refers to the condition when both the liquid and vapor phases are at the saturation condition and heat supplied at the wall provides the latent heat for the evaporation process. The focus here is on the boiling inside smooth circular tube. The concepts devel oped here are later extended to augmented tubes, compact heat exchangers, and external flow boiling. Consider the flow of a liquid through a circular tube. Saturated boiling begins from the point where the thermodynamic quality reaches zero. Heat transfer prior to this location is covered earlier under subcooled boiling in Section 1 5 .2. In the saturated region, the liquid and vapor are close to saturation conditions, and the non-equilibrium effects become less pronounced. These conditions become

FLOW BOILING IN CIRCULAR TUBES

387

important in the region close to x = 0 and after dryout condition is reached at higher qualities.

15.3.2 Flow Patterns in Flow Boiling The saturated flow boiling starts when the thermal equilibrium quality or vapor mass fraction x becomes positive. The flow pattern changes with the increase of the quality. In the low quality region, just at the start of the saturated flow boiling, the "bubbly flow" appears as the continuation from the subcooled flow boiling. As the quality increases, the bubbles coalesce and form large slugs. The flow pattern is named the "slug flow" when the shape of the slug is not disturbed so much or "chum flow" when it is disturbed by the fluid motion. Details of each flow pat tern are described in Chapter 9. As the quality increases, the slugs in churn flow coalesce in the flow direction and form continuous vapor flow in the core region and the liquid film flow on the heated wall. The flow pattern is named the "annu lar flow." With the increase of the vapor and liquid mass velocities, liquid droplets generated from the wavy liquid film on the wall are dispersed in the vapor in the core region. The flow pattern is named the "annular flow with entrainment." With further increase in quality, the liquid film on the heated wall becomes thin and finally disappears or dryout occurs. Beyond this point, the heated wall temperature increases sharply above the saturation temperature, and the liquid and vapor are in a non-equilibrium state. The flow pattern is named as the "mist flow," which continues beyond x = 1 due to non-equilibrium effects. Actual single phase vapor flow starts at the point where all of the mist evaporates and disappears.

15.3.3 Heat Transfer in Saturated Flow Boiling-Wetted Wall Region Heat transfer in the saturated flow boiling is a combination of the convective heat transfer from the wall to the liquid (and subsequent film evaporation at the liquid vapor interface) and nucleate boiling at the wall. The modeling of the convective heat transfer is complicated by the two-phase flow in the bulk. The effect of the two-phase flow on the nucleation, growth, and departure characteristics of bubbles is also very complex and does not easily lend itself to exact analytical treatment. The presence of nucleation in the flow has been a controversial issue in the research conducted in the past decade. The bubble activity tends to be very rapid under saturated boiling conditions, and the presence of the two phases further complicates visual examination of the heater surface. Some studies are reported in which glass tubes are heated with thin transparent gold coatings. It seems rea sonable to assume that bubble nucleation will occur in the flow so long that the nucleation criterion shown in Figure 2 is satisfied and that cavities of the required

388


sizes are available on the heater surface. In the absence of the detailed local descrip tion of the temperature field, the actual two-phase heat transfer coefficient could be employed in this criterion in place of the single-phase heat transfer coefficient. Estimation of the resulting nucleate boiling heat transfer under such conditions is an area where further research is warranted. There are two approaches possible for estimating heat transfer in two-phase flow. In the flow pattern based approach, the existing flow pattern is first predicted and the specific model for that flow pattern is employed. In the second approach, the heat transfer is directly predicted from the flow parameters and properties; since the parameters governing the flow pattern are also responsible for heat trans fer, the uncertainty associated with flow pattern prediction is not carried forward. Many researchers are conducting numerical study of the two-phase flow and heat transfer phenomena, and some specialized codes have been developed for specific applications. Overview of available correlations. There are a large number of correlations presented in literature on flow boiling of saturated liquids. A number of good surveys reviewing these correlations are available, e.g., Melin ( 1 996), Wattelet ( 1 994), and Darabi et al. ( 1 995) . Table 2 lists three of the more commonly used correlations. The Chen ( 1 966) correlation has been employed most commonly for water. It is based on the additive model for the nucleate boiling and convective boiling contributions. The convective contribution is enhanced due to the two-phase ef fects, while the nucleate boiling is suppressed due to the flow effects. Chen used 600 experimental data points for water, cyclohexane, and pentane in vertical flow in arriving at his correlation. The Chen correlation works well with low-pressure water data, but large deviations are observed with refrigerants. Schrock and Grossman ( 1 962) also proposed an additive form of correlation using Bo and Xtt as parameters. They also proposed a chart, which was later refined by Shah ( 1 972) by removing the viscosity ratio term from Xtt . Shah ( 1 982) proposed equations to fit the chart, but it lacked accuracy, and the trends in heat transfer coefficients were not accurately represented. The Gungor and Winterton ( 1 987) correlation combines the Chen model and the Shah ( 1 982) model. Other recent correlations, such as Liu and Winterton ( 1 99 1 ) and Steiner and Taborek ( 1 992), employ an asymptotic model i n which a �p = a�on + a �uc with n = 2 and 3, respectively. The aeon and anue are obtained from a combination of earlier models. These models work well in some cases; however, they do not correctly represent the trends of a vs. x. Also, it becomes difficult to identify the role of different parameters in studying the parametric effects. Kandlikar ' s ( 1 990a) correlation is developed using over 1 0,000 data points for refrigerants, water, and cryogens. The data covered wide ranges of density ratio between the two phases, heat flux and mass flux. The effect of density ratio and

FLOW BOILING IN CIRCULAR TUBES 389 Table 2 Summary of some important correlations in saturated flow boiling InvestigatorlDetails/ Correlation/Comments

l

[

]

Chen ( 1 966), Bennett and Chen ( 1 980) : Vertical flow, water, cyclohexane, pentane; 600 data points prl + 1 0.444 _ - 0.5 . . . ., amac _ . _ aTP _ - amac + amIC· - al E ., amlc - ap h 5 , E - ( 1 + Xu ) 1 78 -2I 2 5 = 0.9622 - tan- ( ReI E . 5 /6. 1 8 . 1 04 ) ; al = 0 .023 Re · 8 Pr . 4 A t / d ap b

. 45 0 .49 0.00 1 22 A [0. 79 c0 p.1 pI a

O. s

0. 29 , 0 . 2 4 0 24 Z ig Pg '

. Tit

?

?

b. TO . 24 b. 0.75

sat

Psat

Works well with water a t relatively low pressure, but large deviations observed with refrigerents; above equations are modified form of the original correlation proposed by later investigators using curve fits. Gungor and Winterton ( 1 987): water, R- 1 1 , R- 1 2, R-22, R- 1 1 3 , R- 1 1 4, ethylene glycol, 3,600 data points aTP / a l

lo

5 [ I + 3000 BoO . 86 j E2 + 1 . I 2 [x / ( 1 - x ) jO. 7 [pt/ pg j 0 4 1 52

2F For horizontal tubes with Frio < 0.05 . E2 Fr · 1 - r ) and S 2 = Frp�5 ; otherwise, E2 I and 1 . The agreement with experimental data is good; however, the trends in heat transfer coeffi cient with quality are not correctly represented. Kandlikar ( 1 990a, 1 99 1 a): refrigerants, water, cryogens, over 1 0,000 data points =

=

S2

=

aTP / a lo aTP/alo INBD

=

aTP / a[o l cBD

=

a lo

=

alo

=

=

o

0.6683 (pt/ pg ) O. 1 xO. 1 6 ( 1 - x ) 0.64 h CFrto) + 1 058 BoO. 7 ( 1 - x ) 0.8 Fi/

o

1 . 1 36(pt/ Pg )0.45 XO.72 ( 1 - x ) 0 . 08 h(Frlo ) + 667 . 2 Bo . 7 ( l - x ) 0. 8 Fit 2/ 1

1 .07 + 1 2.7 (Prl . - 1 ) ( //2)0.5 (Relo - 1 000) pr/ ( //2) (At /d) 2/3

1 + 1 2 . 7 (Prl =

=

higher of the NBD and CBD correlations given below

RelvPr/ ( //2) (At /d)

h (Frlo )

:�

- 1 ) ( //2) 0.5

(25Frlo )0.3 for Frio

0.4. For FrIo ::s 0.4, f2 (FrIo ) is given by the following equation.

=

=

f2 (Frlo)

=

{(

25 Fno)O. 3 ,

for Fno < 0.04 in horizontal tubes

I , for vertical tubes and for Frio :::: 0.04 in horizontal tubes ( 1 5 .3-4)

The single phase all liquid heat transfer coefficient a l o is obtained from the Gnielinski ( 1 976) correlation in the range 0.5 ::s Prl ::s 2000 and 2300 ::s Relo < 1 04 . lo - 1000)PrL (f/2) ( At/d) alo = (Re I + 1 2.7 (Pr;/ 3 - 1 ) ( f12)°· 5

( 1 5.3-5)

The Petukhov and Popov ( 1 963) correlation is employed for 0 5 ::s Prl ::s 2000 and 104 ::s Re lo ::s 5 x 106. .

alo =

-�;;----1 )(fI2) 0 . 5

Relo Prl (f/ 2) ( A l l d)

----

1 .07 +

1 2.7 (Pr;/ 3

--

( 1 5.3-6)

The friction factor f is given by the following equation.

f=

[ 1 .58 In(Relo) - 3 .28r 2

( 1 5 .3-7)

FFJ is a fluid-surface parameter that depends on the fluid and the heater surface characteristics. Table 3 lists the FFI values for several fluids in copper tubes. These values should be considered as representative for commercial tubing. For stain less steel tubes, FFJ should be taken as 1 .0. The value for the azeotropic mixture R-32/R- 1 25 is reported by Wattelet ( 1 994). His experimental data for refrigerants R- 1 2, R- 1 34a, R-22, and R-32/R- 1 25 could be predicted by an earlier version of the above correlation (with the Dittus-Boelter correlation for aLO) to within 1 2.6- 1 4 . 8 percent mean deviation. Introduction of the improved single-phase cor relation is expected to improve the agreement further. Introduction of FFJ in the nucleate boiling term is an important aspect of the Kandlikar correlation. It has been recognized that nucleation characteristics and associated heat transfer depend on contact angle, surface tension, available cavity

FLOW BOILING IN CIRCULAR TUBES 391 Table 3 FFI recommended by Kandlikar (1990a, 1991a) Fluid

FFI

1 .00 1 .30 1 .50 1 .3 1 2.20 1 .30 1 .24 \ .63 1.10 3 . 30

Water R- l l R- 1 2 R- 1 3B l R-22 R- l 1 3 R- 1 1 4 R- 1 34a R- 1 52a R-32/R- 1 32 60%-40% wt. Kerosene

fluids.

For stainless steel tubes, use FFI

0.488 =

1 .0 for all

FFI Kerosene from Chen and Cheng ( 1 998).

FFI depends on the surface-fluid combination.

The values listed are for commercial tubing.

sizes, and their distribution on the heater surface. It is therefore reasonable to expect a surface-fluid effect on the nucleate boiling contribution. Kandlikar and Spiesman ( 1 997) studied the effect of surface structure on the heat transfer in flow boiling with water in a rectangular test section. The same aluminum heater surface was tested with different surface roughness in the range of 0.3 to 1 4 micrometer. Although the surface roughness varied considerably, a microscopic study of the surface revealed that the cavity size distribution was essentially similar. The dif ferences between the surfaces were mainly seen in the larger cavity sizes that are not activated. The heat transfer performance for all surfaces was within 1 0 per cent of each other. Further work is warranted to include the effect of cavity size distribution on flow boiling heat transfer. Melin ( 1 996) conducted experiments with a fluid heated copper test section for a number of fluids. From his data, the values of FFl were obtained for many refrigerants. For the tubes tested by Melin, in general, the FFl values are higher than the values obtained by Kandlikar ( 1 990a) using extensive data sets reported in literature for R- 1 2, R-22, and R- 1 34a. The FFJ values recommended by Melin for additional fluids are: 1 .88 for R-32, 2.27 for R- 1 25 , 1 .74 for R- 1 42b, 2.39 for R- 143a, 2. 1 5 for propane, and 1 .50 for butane. An independent check is advised for specific tubes before using these values. The parametric dependence of the heat transfer coefficient on quality is an important consideration in studying the relative contributions of the two mecha nisms in flow boiling. The nucleate boiling contribution is expected to decrease with increasing quality, while the convective contribution is expected to increase.

392 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION 1 50 �--�--�--. •

_

•

_

....

-

-

...

--L .,., Schrock and Grossman's data, Water •

R1ID 293,P/Pg = 110, Bo = I.Ix10·3, D = 6 mm

•

R1Dl lS6,p/pg = 210, Bo = 3.2x10·4, D = 3 mm

o �--�--�----� 0.5 0.4 0.2 0.3 0. 1 0.0

Quality, x

Figure 13

Dependence of ex on x ; comparison of Schrock and Grossman ( 1 959) data with Kandlikar ( 1 990a) correction.

The overall trend is then dependent on the relative contributions from these two mechanisms. Figure 1 3 by Kandlikar ( 1 990b) compares the flow boiling data from the Schrock and Grossman ( 1 959) and Kandlikar ( 1 990a) correlations. It is seen that increases with x for one set, while it decreases with x in other. The density ra tio and boiling number are identified as the main parameters responsible for this trend. Low-pressure water with a high value of density ratio coupled with a low Bo yields an increasing trend, while the high-pressure data yields a decreasing trend. Similar results are obtained with the R- 1 1 4 data by Jallouk ( 1 974), the R- 1 1 3 data by Khanpara ( 1 986), and the R- 1 1 3 data by Jensen and Bensler ( 1 986). The exper imental data of Kenning and Cooper ( 1 988) near atmospheric pressure shows an increasing trend in vs x and is very well represented by the Kandlikar correlation. Further discussion on the parametric trends is presented in Section 1 5.3.4. Considering the accuracy of predicting the heat transfer coefficient and the parametric trends, the Kandlikar correlation is recommended in the wetted wall region below a quality of about 0.8. At higher qualities, local dryout may occur with significant reduction in the heat transfer coefficient.

a

a

15.3.4 Flow Boiling Map-Subcooled and Saturated Regions The parametric relationship among heat transfer coefficient, heat flux, mass flux, and quality are often displayed on a map. The basis of an earlier map presented

FLOW BOILING IN CIRCULAR TUBES 393 by Collier ( 1 98 1 ) was that the heat transfer at low qualities was essentially by nucleate boiling without any effect of mass flux or quality, and that in the high quality region, heat flux had a negligible effect. Kandlikar ( 1 988) reported that there are many data sets in literature for which heat transfer coefficient in fact decreases with quality, and the Collier ( 1 98 1 ) map does not represent this trend. It was postulated that the nucleate boiling and convective mechanisms are both present, but their contribution depends on the flow rate and heat flux conditions. Two regions are identified as nucleate boiling dominant and convective boiling dominant, indicating the relative strength of the respective mechanisms. In the nucleate boiling dominant region, increasing quality causes a reduction in the nucleate boiling component, and since it is the dominant contributor, the overall heat transfer coefficient also decreases. On the other hand, in the convective boiling dominant region, the contribution due to nucleate boiling is small, and the increase in the convective contribution with increasing quality is reflected in the overall heat transfer coefficient. The Kandlikar ( 1 990a) correlation captures this trend, and it is used as the basis in developing a flow boiling map shown in Figure 14 by Kandlikar ( 1 99 1 a). The thermodynamic equilibrium quality is plotted on the x-axis, and the non dimensionalized heat transfer coefficient, aTP / alo , is plotted on the y-axis. Negative values of quality indicate the subcooled region. The combined effect of heat flux o and mass flow rate is represented by a modified boiling number, Bo* = Bo . F�� / . 7) , and the density ratio of the two-phases, PI / Pg , represents the two-phase effects. In the subcooled region prior to the onset of nucleation, heat transfer is by single phase mode, aTP / alo = 1 . The heat transfer coefficient then gradually increases at a given heat flux as the quality increases (subcooling decreases in this region). The curves in this region can be obtained from the equations presented for the partial boiling, fully developed boiling, and significant void flow regions in Section 1 5 .2. As the heat flux increases, the onset occurs earlier at lower qualities (higher values of subcooling). The thermodynamic non-equilibrium effects are not shown on the map but are important in the significant void flow region. In the saturated region, the heat transfer is divided into nucleate boiling dom inant (NBO) and convective boiling dominant (CBO) regions. In the NBO region, for high values of Bo* and low values of density ratio (high pressure data with high heat flux), the nucleate boiling component is significantly higher than the convective boiling component. The convective component in this case does not in crease as rapidly (due to small difference in the phase densities), resulting in a decrease in aTP /al o with increasing quality. In the CBO region, at low Bo and high values of density ratio (low pressure data with low heat flux), the nucleate boiling contribution is small, and an increase in quality results in a significant increase in the flow velocity, resulting in an increasing aTP /al o trend with quality. For intermediate values of Bo* and pI ! Pg , the plots would lie between the two

f...,j Ie ""

8o:z

803

=

=

3

and 4,

=

=

3

1 0-'4

10-

=

1

Figure 14

•

-

I t

. . .....

t

-

0.0

-, '4\.

Quality,

x

0.2

� � I/_� -=_ =!._ ..-__ '\... 1 r-

##' 3

-/ ./' 2

I tt

/1

FDB

/

1/.

k

NVG'] NVG,2

� ..... "'"

--

1

•-••

_

- 0.2

--

"oNB.2

PB

x=O

x' = -80/Stlo 4X' = 4Tsa'.ONBCp,tlilg XDNB = x· -4'"

SATURATED REGION

0.4

Flow boiling map for the subcooled and saturated flow boiling in smooth tubes (Kandlikar, 1 99 1 a).

- 0.4

h..,Jh1o

SPL

FDB reac hed before

For

Stlo•3 < Stlo...

803

Slto.1 < Stlo.:z

801

SUBCOOLED REGION

0.6

0.8

o

5

0.

3'

o

t""

� =: n m :J �

n o CD

m

:J fI)

Q. :J" CD

CD

:J

!!!. ;if

CD :J fI) 0-

z o :J

!!t R-

R "c:1.oj lO ....

15

20

FLOW BOILING IN CIRCULAR TUBES 395 extremes. Kandlikar ( 1 99 1 a) derived these plots from his ( 1 990a) correlation and verified them with the experimental data for water, refrigerants, and other chem ical compounds. The map can also be used directly to estimate the heat transfer coefficient.

15.3.5 Effect of Flow Geometry Flow boiling in rectangular geometry. The effect of channel dimension on flow boiling heat transfer has been studied by a number of investigators. Tran et al. ( 1 996) provide a good summary of these studies. Their experimental results for R- 1 2 in a circular brass tube 2.46 mm diameter and a rectangular channel (4.06 x 1 .70 mm) with a hydraulic diameter of 2.4 mm were similar, indicating that the hydraulic diameter concept could be employed for these cases. They also found that the nucleate boiling contribution was significant in the rectangular channels. Other investigators, e.g., Mandrusiak and Carey ( 1 989), found very little nu cleation in compact heat exchanger passages. Another aspect noted by Tran et al. ( 1 996) was that the heat transfer was higher than that predicted from the flow boiling correlations. These correlations employ turbulent flow correlations for single-phase flow, while the flow in narrow passages is generally laminar. To elim inate some of the uncertainties, it is suggested that all single phase liquid flow experiments should be carried out to obtain the single phase coefficients. Also, to account for the surface effects, the fluid-surface parameter FH for each geometry should be established before employing the flow boiling correlation. The pres sure drop in the small diameter channels also changes considerably along the flow length, and its effect needs to be included in determining the local conditions and properties.

15.3.6 Heat Transfer Mechanism in Flow Boiling Heat transfer in flow boiling has long been recognized due to the nucleate boiling and convective mechanisms. Dengler and Addoms ( 1 956) presented a systematic study measuring temperature difference profile along the length of a steam evap orator tube. They observed that the nucleate boiling was present at low qualities and low mass flow rates. At higher qualities, the heat transfer coefficient was insensitive to temperature difference. Complete suppression of nucleation was as sumed at this location. The heat transfer in the entire region was correlated with Martinelli parameter, Xu, with a nucleate boiling correction term based on the ra tio of actual temperature difference to the temperature difference at the location where nucleation was assumed to be totally suppressed. Mesler ( 1 977) observed

396


nucleate boiling in thin film flows and argued that nucleation is still present in the flow past the point of suppression as identified by Dengler and Addoms. Schrock and Grossman ( 1 962) conducted a dimensional analysis and proposed an additive mechanism for heat transfer using the boiling number, Bo, and Xtt as parameters. Shah ( 1 982) extended the original charts proposed by Schrock and Grossman, and simplified the convective component by removing the weak viscosity ratio tenn. The Kandlikar correlation ( 1 990a) employs the same basic fonn as the Schrock and Grossman correlation, with the nucleate boiling being represented by Bo and density ratio identified as an important parameter. The presence of nucleation in flow has been a major issue in arriving at the current understanding of flow boiling. The nucleation was assumed to be suppressed by many investigators who em ployed high speed photography to detect the presence of bubbles. Since the heat transfer coefficients are generally very high, the nucleation criterion presented in Section 1 5 .2 indicates that cavities as small as 1 - 1 0 micron will be activated. Kandlikar and Stumm ( 1 995) emphasized the need to combine high speed pho tography with high resolution microscopes to detect the nucleating bubbles. They observed the bubbles in the flow, which seemed to disappear as the heat flux was increased. However, the bubbles were visible again as the camera speed was increased. The bubbles will be generated on the heater surface so long as the nucleation criterion for the given flow conditions is satisfied and there are cavities in the active range present on the heater surface. Some of the areas where further research is still needed are heat transfer under two-phase conditions, effect of flow on nucleation, bubble growth, and related heat transfer at the microscopic level, and the cavity size distribution on flow boiling heat transfer.

15.3.7 Heat Transfer in Post-Dryout Region This topic is covered in Section 1 7.3 .2.

15.3.8 Pressure Drop and Void Fraction The relation between void fraction ( 15 .3-8) using the drift flux model.

8

and actual quality x

--�------�-------

x

is expressed in Eq.

( 1 5 .3-8) pg + Pg :gj m PI PI where Co is the distribution parameter and Vgj is the vapor phase drift velocity. Co is usually taken between 1 . 1 3 and 1 .2. Vgj depends on the existing flow pattern. 8

=

Co

[ x /j.p

+

]

FLOW BOILING IN CIRCULAR ruBES 397 Zuber and Findlay

( 1 965) present the following expressions for Vgj : Bubble flow:

Slug flow :

Vgj

1 .53 7

[a g !:l.P l I /4 g !:l.P-D- ] 1 /2 0.35 [ PI

Vgj

=

=

( 1 5.3-9) ( 1 5.3- 1 0)

The relation between the void fraction and the actual quality is expressed by using slip ratio S.

x P x + ....! S ( 1 - x) PI ( ) 00 1 1 + 0.07 ( ;� )00561 0.93 ;�

( 1 5.3- 1 1 )

€ =

where the slip ratio

S by Thorn is expressed as follows : S=

( 1 5.3- 1 2)

Two-phase friction pressure drop is evaluated by using the two-phase friction mul tiplier

¢10 .

!:l.PI ¢120 -D..plO

( 1 5.3- 1 3)

=

Several models have been proposed for the multiplier. Homogeneous flow model. Two-phase flow is assumed as a homogeneous mix ture flow of vapor and liquid. In estimating the friction coefficient for the flow, modeling of the viscosity in the Reynolds number is the key issue. The following relation is obtained by assuming the same viscosity as the liquid phase:

¢120

=

1+

( PI - 1 ) x Pg

( 1 5 .3- 14)

Lockhart- Martinelli model. As already explained, Lockhart and Martinelli ( 1 945) proposed the relation between two-phase friction multiplier and Lockhart Martinelli parameter X . They derived the relation assuming two-phase separate flow under adiabatic condition. Vapor and liquid flows are either in laminar or tur bulent conditions. Then they obtained the relation for four combinations of flow and X . In usual two-phase conditions. Figure 1 5 shows the relation between

¢10

¢Io


1000 oS -a-

100 10

Figure 15

1000

� 100 g

�

¢ versus X tt .

x

��ri-----�--�--�

0.1 Figure 16

10 20

Pressure, MPa

Pressure drop versus pressure for steam at different exit qualities (inlet quality is zero) .

FLOW BOILING IN CIRCULAR TUBES 399 flow, both vapor and liquid flows are under turbulent condition. For the turbulent turbulent flow, the Lockhart-Martinelli parameter X is expressed as follows:

x"

s+ :::: 0 Tw - T = 5Q { Pr + In [ 1 + Pr (S; - I ) ] } 30 > s+ 5 Tw - T = 5Q [Pr + In(l + 5Pr) + 0.5 1n G�)] s+ 30 Q = qlPICplUT Intionstheatabove stemperature distribution equations, Cpl is calculated at saturated condi + is the non-dimensional distance from the wall,Twandis obtained. U T is the friction velocity. From the above calculation, the wall temperature Using m, P out.

.

a.

Pou t.

�

�

P out.

CHF AND POST-CHF (POST-DRYOUT) HEAT TRANSFER 48S

the value aboveoftemperature equations, its,isatpossible to calculate s*, that is,is the the distancedistribution from the heated wall, which the fluid temperature equal to the saturation value at Pout. Calculation of DB : _ 32 af(fJ)Pl DB - f rh2 where f(fJ) 0.03 and the friction factor f comes from 1 (0.72aP2 l 9.35 ) 2. 0 log fDrh Jf ReJf Calculation of 8: 8 = s* DB Calculation of CD : CD = 3"2 DB ) .5 g(P1 � pg» 0 ( Calculation of UB and LB (linked to each other) through an iterative procedure: (2LB g(P - pg ) ) 0.5 0.125 (8 DB ) frh2 s+ 5 UB 2 PI"7l PlCD =

=

1. 14

-

+

-

=

l

+

+

o t:.

c

C F C 1 1 (PrL=4 .2> C F C l 1 3 ( PrL=7 .5) H C F C 1 23 ( PrL=5 . 1 >

+ x

•

C F C 1 1 (PrL= 3 . 8) C F C 1 1 3 (Prr= 8 . 1 > Ethan o l CPrL= 1 2)

l/5Rem*

Pr

Figure 4

H C F C 2 1 ( PrL=3.5) Wate r CPrL= 1 .75)

Comparison of Eq.( l 9.2-43) with experimental data.

---

-20%

E q . ( 1 9 . 2-43)

FILM CONDENSATION 543

( 1974), Kutateladze and Gogonin ( 1 979), Uehara et al. ( 1 983), and Uehara and Kinoshita (1997). 19.2.2 Theoretical Study on Transition Point and Heat Transfer of Wave Flow and Thrbulent Flow

The condensate filmflow, first flows l a minarl y at the top portion of the surface, then changes to sine wave harmonic wave flow, and finally to turbulent flow, if the length of cooling surface is longTheenough. hydrodynamic stability analysis on bycondensate film onLeea vertical plate for body forced convection has been conducted Marshall and ( 1 973), and they have shownthethatoverall a lamieffect nar ofcondensation film ismass unstabl e andtends wavesto will appear. However, the condensation transfer stabi lize the film. 19.2.2.1 Stability analysis of condensate film flow.

Kapitzafilm(1948) firston conducted simple anal y tical treatment of heat transfer i n a wavy based the assumption ofAccording regular sinusoidal wave and considering onlyheat heattransfer conduction across the film. to Kapitza' s analysis, the enhanced rate for wavy film i s 2 1 % above that for the smooth film. Soflata ( 1980, 1 98 1 ) proposed the correlations of the mean heat transfercoeffi ciandentmoved for thewith sine awave condensation constant velocity. film, assuming that the waves were sinusoidal Ueharafilmet and al. ( 1two982)phase analyboundary zed the wavylayerfilmandcondensation Soflat'correla s model fortions:wavy proposed theusing following 19.2.2.2 Theoretical analysis of wavy film condensation.

( 1 9.2-44)

for body forced convection,

/ 3 1 /2 }1 1 / Pr L 2 2 / 1 Nux 0.45 ( 1 - ,B ) { 1 .2 + RJa(1 - ,B2) - 2 Re t for forced convection, and / 1/ 3 { ( 1 - ,B 2 ) - 2 PrL } 4 /2 Nux Xf3 1 4 1aFrx Relx 4 Xf3J _ -

.

=

+

(1 9.2-45)

( 1 9.2-46)

S44 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSAnON

for combined convection, where U2

(19.2-47) { PrL (1Ria(2 ) 1 /2 } 1 /3 (19.2-48) = 0. 45(1 - f3 2 ) -1/2 1. 2 ( ) 1 /2 (19.2-49) R = PLJJ.L PvJJ.v f3 is a dimensionless amplitude of waves and can be expressed in the following equation from the result of an analysis by Penev et al. (1973). (19.2-50) f3 = 0)/0 O 2 1 f3 = 0.47We- . 138 (19.2-51) where We is Weber number: (19.2-52) mean (over the cross section of the film and the wavelength) film velocity f3 : mean (over the wavelength) film thickness f3max : film thickness in cross section corresponding to the crests of wave Hydrodynamic and heat transfer analysis of the1982b). wavyThe film flow condensation were conducted by Hi r shburg and Florshuetz (1982a, dynamics andby heat transfer of the wave film condensation on a verti c al plate were analyzed solving the time-dependent Navier-Stokes and energy equations as well as the Poisson equation for the pressure with finite schemes and non-periodic boundary conditions by Stuhltrager et al. (1993,1995). Various turbulent modelstheforvelocity the eddyprofile, diffusivity of momentum and heator mean have been proposed for solving film thickness, and local condensation heat transfer coefficient forfilmthecondensation body forced convection. The main models in the turbulent without vapor flow are shown as follows: 1. The et al.von(1956)Karman-Nikurade model, as used by Seban (1954) and Rohsenow CM/V == 0 t /5 - 1 0 ::::: yytt :::::::::: 530 cM/V (19.2-53) cM/V = yy t /2.5 5y t::::: 30 PrT = 1 Frx =

�

gx

-

+

Xf3

( omax

-

-

uO :

= I

ax

x =

+

=

=

=v=

0

=

cH )

=

- = - =

=

=

+

0 =

=

=

w


where

Kl

( 19.2-65)

= 0.33Pr�, 08 .

Using Nu� , Re� , and PrL, we can obtain the following expressions: 2 ) /5 0.25 ) p Pr� p -3) /5 ( 1 0.905 PrL l l ) - ( l -p Re? Nu; = 0.221 l - p ( 1-p for 0.01 PrL 1 1 ( 1 9.2-66) +

x

:::;

::s

( 19.2-67)

where

p=1

-

PrL o , 08 /0.99.

( 19.2-68)

The mean Nusselt number is expressed by the following equations: ( 19.2-69) 1 /5

2/ 5

1 /5 ( 1 9.2-70) for 1 1 PrL 1 00 . Using Nu� , Re� , we can obtain the following equations. 2 ) /5 0.25 ) P Pr� p- 3)/5 ( 1 0.905PrZ l l / 8 ) - ( 1 -P Re:i Nu� = ( 1 - P )0.22 1 l - P ( 1-p for 0.01 PrL 1 1 ( 19.2-7 1 )

nUL 1\.1,

_

-

0 027PrL Grl

:::;

r Ja

x

::s

+

::s

:::;

( 19.2-72)

In Figure 5,( 1the988,experimental data ofthe local heat transfer obtained by Uehara and Kinoshita 1 994) are compared with the predictions of Dukler ( 1960), correlations ( 1 9.2-22) to ( 1 9.2-25), and correlation ( 1 9.2-64) for PrL = 1 , 5, and 10. The Uehara-Kinoshita correlations agrees very well with the experimental data, the Dukler are larger than experimental data obtained by Hebbardbut and Badger predictions ( 1 934), Meisenburg et al. ( 1 935), Gregorig et al. (1974), Kutateladze and Gogonin ( 1 979), Uehara et al. ( 1 983), Uehara et al. ( 1 988), and Uehara and Kinoshita ( 1 997). In Figure 6, the experimental data for the mean heat transfer coefficient are compared wi t h correlation ( 1 9.2-39) to ( 1 9.2-42) and ( 1 9.2-7 1). The correlations ( 1 9.2- 39) to ( 1 9.2-42) and correlation ( 19.2-7 1 ) agree very well with experimental data.


3

E q . < 1 9 . 2-22)

U e h a r a-K i n o s h i t a

E q . < 1 9 . 2-23) E q . ( 1 9 . 2-24)

'* :..: :J �

E q . ( 1 9 . 2-25)

1 O.B 0 .6

c.

C F C 1 1 3 (PrL=7 .5)

Uehara et a 1 . < 1 9BB) • &

HC FC 1 23 (PrL=5. 1 >

'

m m

_

m Ol m

�______�___�___�� 2 3 4 5 6

_______________

Re� x 10- 3 (b)

Resistance coefficient through tube banks with condensation (a) inline and (b) staggered_

FILM CONDENSAnON 563

Zukaruskuson (1972) showed thebanksgraphis calculated of CD forbynon-condensing gas. CDvaluefor condensation staggered tube multiplying the CD obtai n ed from Zukaruskus' graph with 0. 7 5, and CD for condensation on in-line tube banks by multiplying the CD value obtained from Zukaruskus' graph with 0.is 4calculated . 19.4 CONDENSATION OF MIXTURES

The condensation mUlti-component vapory applied mixturesin iair-conditioning/refri s widely encounteredgerin chemical and foodofindustries and is recentl ation low-temperature-Ievel In this section, describande theoretical treatment onenergy steady utilizati laminaronfilmsystems. condensation of a multiwe component vapor inmixture onanda flatmassplatetransfer in orderof tothedeepen our understanding of basicWhen phenomena the heat vapor mixture. vapor mixture contactsof athecooled temperaturea lmUlti-component ower than the dew-point temperature vapor,solidthe wall liquidwithfilma appears the cooledThewall, and outsidethetheboundary film, a boundary layer ofthethethermo vapor midynamic xture onisstate formed. film thickness, l a yer thickness, at theby thevapor-liquid interface, the condensation massvaporflux,veletc.ocit, yareis strongl y affected velocity of mai n vapor flow. When the very w, thety condensation characteristics arethecontrolled by a buoyancy force due toandthethelodensi di ff erence. The former is called forced convection condensation, lattercondensation is called thereginatural convection condensation. Condensation between these two o ns is called combinedforced and natural convection condensation. Toi-component qualitativelvapor y explmixture ain basionc acharacteristics of the filambicondensation ofana mul t flat pl a te, we consider n ary system as example. Figurenormal 17a shows thecooled distributions oftemperature andtemperature vapor massandfraction infraction the direction to the wall. These variations of massthe on a diagram of phase equilibrium are also shown in Figure 17b. In case of the condensation of a binary vapor mixture, the less volatile component (k = 2)equilibrium condenses rulmoree. Consequentl than the volati le volcomponent (k = 1),isaccordi ng to theat phase y , the a tile component concentrated the vapor-liquid interface in the vapor phase. The balance between the diffusion and convecti vesame masstitransfer makes a steadyof mass fraction distridecreases bution in thefromvapor phase. At the m e, the temperature the vapor mixture the bulk to the vapor-liquid interface because the phase equilibrium is valid at the vapor-liquid interface.areIn theformed. liquidThefilm,temperature generally, both fraction distributions dropthein thetemperature condensateandfilmmassis based on nearlas yinheatthe conduction due to extremely liquid film. This mechanism is the same case of condensation of a purethinvapor.


T

A

�

._._._._._._

. _. _ . _ . _ . _ . _

*� .

T

_

Tb V

Tb V

A: Liquid film B: V apor boundary layer

I;

J--\----f't..I...

C: Bulk vapor flow

y

Figure 17

(a)

0

WI L W1 Vb

WI Vi WI

(b)

Condensation of a binary vapor mixture on a vertical flat plate (a) Temperature and con centration distributions around the cooled wall (b) Phase equilibrium diagram.

Thea multi-component most effective method tomixture quantitatively analyze theboundary condensation prob lem of vapor is the two-phase layer theory, which containing was established by Sparrow gases. and Eckert (1961) fortheory, the filmnumerous condensation of steam noncondensable Based on this theoret ical studies on the steady laminar film condensation of a mUlti-component vapor mixture on a were flat plate havetobeen carrisimilarity ed out, assolutions summarized in Table 2. Mostmanyof these studies devoted solving and demonstrated numerical resultsandinlackthe form graphs.FujiiTheietr al.conclusions, are qualitative generaliofttables y. Roseand(1980), (1977, 1978,however, 1987, 1991a), and Koyama et al. (1986) proposed general correlations that can be ap plied to algebrai c ally predict the condensation characteristics of bi n ary vapor mix tures without rigorously applying thewassimilarity solutions. The algebraic condensa prediction method for binary vapor mixtures extended to a mUlti-component tion problemparts,by theFujiialgebraic et al. (1984, 1989) method and Koyama et al.by(1987a, 1987b). In the following prediction proposed Fujii and coworkers are introduced briefly. The numerical results of the similarity solutions and the


Table 2 Theoretical studies of steady laminar film condensation of a multicomponent vapor mixture on a flate plate Authors

Year

Mode

Method

S ubstances

Sparrow-Eckert Koh Sparrow-Lin Minkowycz-Sparrow Sparrow et al. Minkowycz-Sparrow Sparrow-Marashall Rose Denny et al. Denny-Jusionis

1 96 1 1 962 1 964 1 966 1 967 1 969 1 969 1 969 1 97 1 1 972a

Natural Forced Natural Natural Forced Forced Natural Natural Combined Combined

Similarity solution Similarity solution Similarity solution Local similarity sol. Similarity solution Similarity solution Similarity solution Integral method Finite difference Finite difference

Denny-J usionis

1 972b

Combined

Finite difference

Mori-Hijikata Hijikata-Mori

1 972 1 972

Natural Forced

Integral method Integral method

Tamir Taitel-Tamir

1 973 1 974

Natural Natural

Integral method Similarity solution

Lucas

1 976

Combined

Sage-Estrin

1 976

Natural

Finite difference, Integral method Similarity solution

Air + H2 O Parametric Air + H2 O Air + H 2 O Air + H 2 O Air + H2O CH 3 0H + H 2 O Any gas-vapor Air + H2 O Air + (H2 0, NH3 , C2 Hj OH, etc.) CH 3 0H + H2 0, C3 H7 0H + H2 0, etc. Air + H2 O Air + H 2 O Air + KCI CH3 0H + H 2 O Air + CH 3 0H + H2 O, CH3 CO + CH3 0H + H 2 O CH3 0H + H 2 O

Fujii et al .

1 977

Forced

Fujii et al.

1 978

Natural

Stephan-Laesecke

1 980

Rose Fuji-Koyama

1 980 1 984

Forced, Combined, Natural Forced Forced

Kotake Fujii et al.

1 985 1 985

Natural Natural

Koyama et al.

1 986

Natural

Fujii et al .

1 987

Forced

Koyama et al. Koyama et al. Fuj ii et al. Fujii et al .

1 987a 1 987b 1 989 1 99 1 a

Forced Natural Natural Forced

Similarity solution, Correlation Similarity solution, Correlation Stagnant film theory

Correlation Similarity solution, Correlation Perturbation method Similarity Solution, Correlation Similarity Solution, Correlation Similarity Solution, Correlation Correlation Correlation Similarity solution Correlation

Ne + N2 + H2 0, N2 + CCIzF2 + H2 O, N2 + C� + H2 0 Air + H2O Air + H2 O CH 3 0H + H2 0, Air + H2 O

Air + H 2 O Air + CH 3 0H + H2 O Multi-component Air + R 12 + R 1 1 4 C2 Hj OH + H2 0, Air + H 2 O C2 HjOH + H2 O, Air + H 2 0, etc. Multi-component Multi -component CH 3 0H + C 2 HjOH + H 2 O Binary vapor


algebrai( 1 99c prediction method by Fujii and his coworkers are descried in detail by Fujii 1 ). 19.4.1 Laminar Film Condensation of a Binary Vapor Mixture

We theplate,steady laminarthefilvapor m condensation ofel and a binary vapor mixture on1 8 ashows verticonsider ctheal flatphysical formodelwhich flows paral l downwards. Fi g ure vapor-liquid two-phase boundary layer and the coordiEmploying nate system.some relevantof theassumptions and theof total boundary concept,energy, we can derive the governing equations for conservation mass,layer momentum, and mass of component 1 for a binary vapor mixture; these equations are called the two-phase boundary layer equations. By using the similarity transfonnation, the governing equations canof thebe reduced ton ofa settheoftwo-phase simultaneous ordinlaryayerdiequations fferential equations. The details deri v ati o boundary are described by Koh ( 1 962), Minkowycz and Sparrow ( 1966), and Fujii ( 1 99 1 ). 19.4.1.1 Prediction method for heat and mass transfer characteristics in forced convection condensation. ( 1 977, 1 987, 1 99 1 1 99 1

Fujiifor ettheal.laminar forced convection a, b)filmsolved numerically the similarity solutions con densation of saturated and superheated binary vapor mixtures on a flat plate and proposed the following algebraic method to predict the heat and mass transfer characteristics. ��-------. y

[ r� v L

Vv

\,

\

�

\

x

Figure 18

Interface

WIV () Iy,

Vapor boundary layer

Ambient vapor

U�

(TVb , W1Vb )

p=const.

uVb

Physical model and coordinate system for condensation of a binary vapor mixture.


(

Heat balance at the vapor-liquid interface

0.433 1 .367

_

PrL M FL Ph .

0.432

V2RMFL

+

1

V2RMFL I

)

1 2

( 1 + 0.320 PhO . 87 ) - 1

( ) TVb - Ii Ti - Tw { 1 2.6Pr�66 [RMFL ( 1 - � C; V (WI Vi - WILl ) t05 } ( 1 9.4- 1 ) where thenumber M FL , the phase p - J-L ratio R, the dimensionless condensation mass change Ph, the liquid Prandtl number PrL , the vapor Prandtl number Prv , the functiheat on ofdifference the vaporcPrandtl number CF (Prv ), and the dimensionless specific isobaric ; v are defined as ! ( 19.4-2) R = ( PL J-L L ) =

x

+

A v \.IL CF (PrV ) - AL \.I v

2

+

flux

Pv J-L v . mx x M FL = J-L L Re Lx cpL (Ii - Tw ) -- --Ph = ---=t:. h L v L PrL = J-L CpL AL v Cpv J-L Prv = Av Prv ! CF (Prv ) = (27.8 + 75.9Prv O. 306 + 657Prv ) 6 Cp l V - Cp2 V Cp v = c pv ( 19.4-3), Re Lx U VbX ReLx = - \.I L

( 19.4-3)

I

2

( 19.4-4) ( 1 9.4-5) ( 19.4-6) ( 1 9.4-7)

I

( 19.4-8)

*

In Eq.

the two-phase Reynolds number

is defined as

( 19.4-9)

Mass balance of more volatile component at the vapor-liquid interface

2.5 . C (Sc v ) M= RScv 1 .5 + WR

(

) m ( WR - 1 )

( 1 9.4- 10)


where theRvapor Schmidtas number Scv , the exponent m , and the ratio of concen tration are defined W Vv D12v = 0.5 + 0.05Scv - 0.2/ R "2 Scv

m

( 1 9.4- 1 1 )

= -

I

( 1 9.4- 1 2)

WIVi -- WIWILL ---WIVb and the function of the vapor Schmidt number

( 1 9.4- 1 3)

WR =

form as Eq. ( 19.4-7). Miscible condition

CF (Scv )

is the same functional

x WI L = W I Li = m-mx. -ix = mIxmi+m2x .

( 1 9.4- 14)

.

where mkx (k = 1 , 2) is the local condensation mass flux of component k.

W I LVi : WWLv((IiIi,, p)p) } ( 1 9.4- 1 5) WI i When vaporparameters, condition ofsimultaneous (p, TVb , WI Vb) and the wall temperature Tw areEqs.gi(v1theen9.4-asbulk known algebraic equations composed of 1 ), ( 1 9.4- 1 0), ( 1 9.4- 14), and ( 19.4- 1 5) can be solved by the following phase equilibrium procedure: calculaterepresentative L using theproperties W I Vi and WIphysical relation Eq. ( 1 9.4-guess1 5)Tiand; (2)calculate and dimen

Phase equilibrium relation at the vapor-liquid interface

(1)

sionless numbers c; v ' Ph, Prv , R, Scv and WR ; (3) obtain it FL from Eq. ( 1 9.4- 10); until the above satisfy Eq. ( 1 9.4- 1 ), modify the value Ii and repeat cal culation, from stepsarevalues (2) to (4); and finally (5) all physical quantities such as Ii , Furthermore, the localcomponent condensationi mass flux wall , itFL obtained. mx , WIVilocalWILcondensation theheat mass fl u x of more volatile the local , m x flux qwx and the local convective heat flux qvx are calculated as (4)

mx. = -J.1.L M· FLReLx 2 mIx = mxWI L 1 ) "2 AL . + . qwx = 0.433 ( 1 .367 - V0.432 2R M FL 2R M FL 1

( 1 9.4- 1 6)

x

( 1 9.4- 17)

I

- ( T; - T )Re X

w Lx

1

2

( 1 9.4- 1 8)

FILM CONDENSATION

569

( 19.4- 1 9)

where the vapor Reynolds number Re vx is defined as Revx

U VbX Vv

( 1 9.4-20)

= -

19.4. 1.2 Prediction method for heat and mass transfer characteristics in natural convection condensation. ( 1978) ( 1986)

Fujii etforal.the laminarandnatural Koyamaconvecti et al. on film numerically solved the similarity solutions condensation of saturated and superheated binary vapor mixtures on a vertical flat plate. Then, they proposed algebraic method for predicting the heat and mass transfer in natural convectionthecondensation. Heat balance at the vapor-liquid interface

TVb - Ti ( ) ( .* Prv ) � T Tw AL Vv

if -NLI" = PrL MNL + hcN (Pr v ) AV

{

.

Ph

X

0 66

1 + 1 .25Prv

( . )1 .17 R MNL n i�

1-'L

I

"2

j

X,

_

[ 1 - 0.85cp* v ( W I Vi - wi d ]

}

( 19.4-21 )

where thenumber dimensionless condensation mass flux if NL , the function of the vapor Prandtl CN (Pr v ), the parameter of buoyancy force xt (introduced by Tanaka, 1 985) and the parameter of buoyancy force are defined as .

M

_

NL

-

(

( )

ni

I

mx x Ga LX - 4 -- --4 i-t L

( 1 9.4-22)

� Prv ( 19.4-23) � 4 2.4 + 4.9JPrv + 5Pr v ( 1 9.4-24) x t = n j + Ww ( WI Vi - W l vb ) [(Prv /Sc v ) � - 1 ] ni = wT ( Tvb 1j ) + [ I - W r CTvb - 1j ) ] Ww ( W I Vi - W I Vb ) ( 19.4-25)

CN (PrV )

=

)

-

In the due aboveto equations, numberonGadistributions Lx and the parameters of buoyancy force temperaturetheandGalileo concentrati in the vapor boundary


layer WT and Ww are defined as

( 19.4-26)

1 WT = TVb MI - M2 WW = ( Ml - M I - M2 ) W I Vb

( 19.4-27)

----

( 19.4-28)

Mass balance of more volatile component at the vapor-liquid interface

. M

·is the same functional fonn

1

(WR - I ) 2CN (Sc V ) ( Xi S C V ) 2 = R 1 ) 0.5 W� 2 Scv R (W CN (Scv) ( 19.4-23) Xi NL

+

( 1 9.4-29)

where functioand n ofthethe parameter Schmidt number as(1985) Eq. theis defined of buoyancy force introduced by Tanaka as Xi = Qi wT (Tvb - Ii ) [(Scv Prv ) 1 - I ] ( 1 9.4-2 1 ) ( 1 9.4-29) ( 1 9.4- 14) ( 1 9.4- 1 5) (p, TVb , W I Vb )

/

+

( 19.4-30)

By solving Eqs.and the phase equilibrium and relation togetherEq. with the miscible condition Eq. under a given bulk vapor condition of and wall temperature Tw, the local heat and mass transfer characteristics of binary mixture condensing on a vertical flat plate can algebraically. Then, all physical quantities, such as Ti , WI Vi , W I L , and beifandpredicted NL are obtained. The local condensation mass flux mx, local wall heat flux , local convective heat flux qvx are also calculated from the following qwxequations.

--- ( ) •

L M NL Ga LX mx 4 x .

qvx

=

A v (Tvb - Ii )

x

{

x

_

J.1.

1

4

( 1 9.4-3 1) ( 19.4-32)

( XtQi ) 4 CN (Prv) - (GrVx PrV ) 4

1 + 1 .25Prv0 66

1

( ) 1.17 . R M NL Qi�

1

[ 1 - 0.85 cp ( W I Vi - W I d] *

}

( 1 9.4-33)


where the Grashof number Grvx is defined as Grvx =

gQiVvx3

(1904-34)

---

2

19.4.2 Laminar Film Condensation of a Multi-Component Vapor Mixture

We steadyThelaminar film condensation ofinanSubsection n-component vapor mixture onwhenaconsider vertical flattheplate. same physical model as 190 4 .1 is employed , and WIVb in1),Figure 18 are replaced by WkL , WkLi , , , , L L WIV WI WI WIVi i , and (k = 1, 2, . , n respectively. , WkViToorWkV(1964)WkVband Stewart and Prober (1964) derived the boundary layer equa tions for the n-component mass transfer problem by linearizing thetherelevant diffu sion coefficient and then showed that this problem can be treated as combination of related binary problem by the theorthogonal transformati olayer n (e.gequations ., Wylie, for1966).the Fujii and Koyama (1984) treated two-phase boundary laminusing ar forced convection film condensation of orthogonal a ternary vapor mixture onThen, a flat plate the similari t y transformati o n and the transformation. they proposed an algebraic method to predict the heat and mass transfer character istics by applying prediction methoda, 1987b) for a binary vapor extended mixture descri bed in Subsection 1904.1. theKoyama et al. (1987 successfully this method to the forced and natural convection condensation problem of a mUlti-component vapor mixture.vapor In theirmixture method,arethetreated governing equations forsimilari the condensation of the n-component as (n 1) sets of t y solutions for the equivalent bi n ary systems, in whi c h the Schmidt numbers for the binary systems, SCk v (k = 1, 2, . . . ,n -1), are determined by the orthogonal transformation of the multi-component diffusion matrix The matrix is defined as (1904-35) where (W/Vi--WkVb) WIVb) (1904-36) Qkl DVkVIV(WkVi (1904-37) DZIV DknV - DklV (1904-38) where is the diffusivity of the paiby rreferring of component n-component, - inoftheHirshfelder the valueDt{ofv which can be calculated to text kbooks et al. . .

-

A.

=

A

---.:.:c.:..._ .:..____

=

I


(1954) matrix andisBird givenetbyal. (1960). The diagonal matrix which is obtained from the (19.4-39) (Okt!SC)� :::: � = (19.4-40) = p- 1 (qkl) �:::: � (19.4-41) = (pkl)�:::: � whereIn Othekl isfollowing the Kronecker delta and 1/Sqmethod v is the eigenvalue of the matrix parts, the algebraic proposed by Koyama et al. are described. 8,

A,

B

QAP

==

Q

=

P

A.

19.4.2.1 Prediction method for heat and mass transfer characteristics in forced convection condensation.

Koyama etandal. (1987a) proposed the laminar follow ing algebrai c method for predicting the heat mass transfer for the forced plate. convection film condensation of the n-component vapor mixture on a flat

Heat balance at the vapor-liquid interface

(

432 0.433 1. 367 V0.2RMFL _

1

+

1. ) (1 + 0. 3 20Pho.87 )- 1 2R M FL 2

Dkl V (WIVi - WIVb) * {� k=l 1= 1

(19.4-42)

In Eq. (19.4-41), 4>i is the function for the diffusion term, expressed as �.

0/ 1

� � cpkn =� X

Vv

wmWm VbL ) ] } (19.4-43) wWmmViVi --_ [I:j=l Plj SCjV (mI:=1 qjm _

-

where the dimensionless specific isobaric heat difference c;kn v is defined as C kVCpVcpnv (19.4-44) cpkn V = p and the other dimensionless parameters are the same as in Section 19.4.1.1. *


= CRF(SCSCkkVv ) ( 1.5 2.5Wk R ) m (WkR - 1) (k 1, 2, . . . , n - 1)

Mass balance of component k at the vapor-liquid interface .

M

FL

where

=

+

(19.4-45)

1 ___ n-I Wm Vi __ Wk R = 1 �qk - Wm L m m =1 Wm Vi - Wm Vb

(19.4-46)

WkL = WkLi = mmx. kx (k = 1, 2, . . . , n - 1)

(19.4-47)

=_1 1_

_

__

_

_

�

Miscible condition

--

Phase equilibrium relation at the vapor-liquid interface

WkV� : WkV (Ti, WIVi, W2 Vi, . . . , wn-2V.i ) } (19.4-48) WkLI - WkL WI v , , W2 V" . . . , Wn-2 VI ) When the bulk vapor condition of parameters, b, WI Vb, W2 V b, . . . , Wn - I Vb ) and the wall TV temperature are given as known all physical quantities, such as Tw and are obtained by simultaneously solving Eqs. (19. 4 -42), ,(19. Ii WkVi 4-45),, WkL, (19.4-47),M and (19.4-48). Then, in the same manner as in Section .1.1, the localmkx, condensation flux mfluxx , the localandcondensation massvflux of the19.4k-component the local mass wall heat the local convecti e heat qwx, flux qv x are calculated as . J.1-L M· ReLx (19.4-49) mx = x (19.4-50) mkx = mx WkL (k = 1, 2, . . . , n - 1) ( 0.432 1 ) -AL (Ii - Tw )Relx qwx = 0. 4 33 1.367 x 2 R M J2RM (19.4-51 ) . [ ( 1 - 3if>2 i )] I 05} A V (TVb - Ii ) CF(PrV )Revx qvx = 1 2. 6Prv066 R M' (19.4-52) p,

(T" p ,

(p ,

FL ,

FL

�

I

+

.

{

FL

+

FL

.

1

'2

FL

X

�


19.4.2.2 Prediction method for heat and mass transfer characteristics in natural convection condensation.

Using the1 987same method asa forprediction the forcedmethod con vection film condensation, Koyama et al. proposed b ) ( forcondensing the heat naturally and massontransfer a verticalcharacteristics flat plate. of an n-component vapor mixture

CN (Prv ) AA L ( VVv ) TT;vb -Tw (X,.* Prv ) . ) 1.17 } R �NL 1 .25 Pr�66 ( 19.4-53) (

Heat balance at the vapor-liquid interface •

NL = PrLPh

M NL

. - .!. 3

M

X

where xt =

{

I +

n - I n- I n - I

+ v'2

I

�

---.£

7.

_

T I

I

4

rli Lk=1 L1=1 m=1L wWl(WlVi - WlVb)Plk qkm[(Prv/ScKv) 1 - 1] ( 1 9.4-54) 1 1 Qi = wT(Tvb - Ti ) + [ 1 - wT( Tvb - Ii )] Lk=1 L1= 1 m=1L Wwl (WI Vi - WlVb) Plk qkm +

n-I n-I

( 1 9 .4-55) 1 9.4. 1 .2.

and the( 1 9.4-54) other dimensionless parameters are the same as in Section In Eqs. and ( 1 9.4-55), the parameter of buoyancy force due to the concen tration distribution Wwk is defined as

Wwk

( 19.4-56)

=

1) SCkV)4 (WkR ki X MNL 2CN(SCkV)( (k 1,2, ... , n - 1 ) SCkV R (WkR ( 1 9.4-57) where Xki = Qi wT(Tv1 b- 1- Ii ) [(SqV /Prv) 1 - 1 ] + L L L Wwm(Wmvi - Wmvb)Pml qlp [ (SqV/SCIV) 1 - 1] ( 19.4-58) 1= 1 m=1 p=1

Mass balance of the component k at the vapor-liquid interface I

=

+

n-I n-I

+

1 ) 0 . 5 w�i

=


9.4-53), ( 1 9.4-57), ( 1 9.4-47), and ( 1 9.4-48) for aBygisimultaneously ven mixture at solving a givenEqs. wall( 1temperature and a given bulk vapor condition, wecalculate can obtain all physical quantities, such as Ti , Wk V i , WkL , M NL . Then, we can the values of mx , qwx , and qvx as i-LLM NL ( GaLx) ( 1 9.4-59) mx .

.

I

_

4

x

4

( 19.4-60)

x

{

1

+

1 .25Pr� 66

( RMtL Q

I

)1 . l7 }

( 19.4-61 )

19.4.3 Future Directions

Since Nusselt ( 1 9 1 6) proposed the film theory for the natural convection conden sation of water steam on a experimentally vertical flat plate,andmany researchersIn thehavepresent investistage, gated thewe can condensati o n problem theoretically. understand only the basic characteristics of the condensation of a multi component below: mixture on a flat plate. The remaining problems to be solved are listed 1 . The thermophysical properties of a multi-component mixture. 2. More rigorously theoretical treatment of buoyancy force due to temperature and concentration difference in a multi-component system. 3. Effect of the multi-component diffusion in the condensate film. 4. Condensation of a multi-component vapor mixture on a tube or in a tube-bank. 5. Augmentation technology of the multi-component condensation. 6. Acquisition of experimental data on the multi-component condensation. REFERENCES Bergelin, O. P., Brown, G. A., and Doberstein, S. C. 1 952. Heat Transfer and Fluid Friction During Flow Across Banks of Tubes-Iy' Trans. ASME 74:953-960. Bird, R. B . , Stewart, W. E., and Lightfoot, E. N. 1 960. Transport Phenomena. New York: John Wiley.


Butterworth, D. 1 98 1 . Inundation Without Vapor Shear. Power Condenser Heat Transfer Technology. Eds. P. J. Marto and R. H. Nunn. New York: Hemisphere Publishing Corporation. Cess, R. D. 1 960. Laminar Film Condensation on a Flat Plate in the Absence of Body Force. Z Angew Math. Phys. 1 1 :426-433. Chen, M. M. 1 96 1 a. An Analytical Study of Laminar Film Condensation: Part I -Flat Plates. Trans ASME J. Heat Transfer 83 :48-54. Chen, M. M. 1 96 1 b. An Analytical Study of Laminar Film Condensation: Part 2-Single and Multiple Horizontal Tubes. Trans ASME J. Heat Transfer 83:55-60. Colburn, A. P. 1 934. Note on the Calculation of Condensation when a Portion of the Condensate Layer is in Turbulent Motion. Trans. Am. Inst. Chern. Engrs. 30: 1 87- 1 93 . Denny, V . E. and Jusionis, V . J. 1 972a. Effects o f Noncondensable Gas and Forced Flow o n Laminar Film Condensation. Int. J. Heat Mass Transfer 1 5(2):3 1 5-326. Denny, V. E., and Jusionis, V. 1. 1 972b. Effects of Forced Flow and Variable Properties of Binary Film Condensation. Int. J. Heat Mass Transfer 1 5 ( 1 1 ) :2 1 43-2 1 53. Denny, V. E., Mills, A. F., and Jusionis, V. J. 1 97 1 . Laminar Film Condensation From a Steam-Air Mixture Undergoing Forced Flow Down a Vertical Surface. Trans. ASME C. J. Heat Transfer 93(3) :297-304. Dukler, A. E. 1 960. Fluid Mechanics and Heat Transfer in Vertical Falling-Film System. Chern. Engng. Prop. Symp. Ser. 56: 1 - 1 0. Fishenden, H. and Saunders, O. A. 1 950. An Introduction to Heat Transfer. London: Clarendon Press, p. 1 32. Fujii, T. 1 98 1 . Vapor Shear and Condensate Inundation: An Overview. Power Condenser Heat Transfer Technology, Eds. P. J. Marto and R. H. Nunn, pp. 1 93-223. New York: Hemisphere Publishing Corporation. Fujii, T. 1 99 1 . Theory of Laminar Film Condensation. New York: Springer-Verlag. Fujii, T. and Koyama, S. 1 984. Laminar Forced Convection Condensation of a Ternary Vapor Mixture on a Flat Plate. Fundamentals of Phase Change: Boiling and Condensation-ASME HTD 38:8 1 87. Fuj ii , T. and Uehara, H. 1 972. Laminar Film Condensation on a Vertical Surface. Int. J. Heat Mass Transfer 1 5 : 2 1 7-23 3 . Fujii, T. and Uehara, H. 1 973. Condensation Heat Transfer. Advance of Heat Transfer 1 : 1 . Yokendo, Tokyo. (in Japanese). Fujii, T., Honda, H . , and Oda, K. 1 979. Condensation of Steam on a Horizontal Tube-The Influence of Oncoming Velocity and Thermal Condition at the Tube Wall. Condensation Heat Transfer. Proc. 18th Nat. Heat Transfer Conf ASME, 35-43. Fujii, T., Koyama, S., and Goto, M. 1 985. Effects of Air upon Gravity Controlled Condensation of Nonazeotropic B inary Refrigerant Vapor on a Horizontal Tube. Trans. Jpn. Soc. Mech. Eng. 5 1 (46B):2442-2450 (in Japanese). Fujii, T., Koyama, S., and Watabe, M. 1 987. Laminar Forced-Convection Condensation of B inary Mixtures on a Flat Plate. Trans. Jpn. Soc. Mech. Eng. 53(486B ) :54 1 -548 (in Japanese). Fujii, T., Koyama, S., and Watabe, M. 1 989. Laminar Film Condensation of Gravity-Controlled Convec tion for Ternary Vapor Mixtures on a Flat Plate. Trans. Jpn. Soc. Mech. Eng. 55(5 1 OB) :434-44 1 (in Japanese). Fujii, T., Shinzato, K., and Lee, J. B. 1 99 1 a. A Proposal of a New Equation for the Mass Concentration at the Vapor-Liquid Interface in the Case of Forced-Convection Laminar Film Condensation of a Binary Vapor Mixture on a Flat Plate. Engineering Sciences Reports, Kyushu University 1 3(3):293-296. Fujii, T., Shinzato, K., and Lee, J. B. 1 99 1 b. A Condensation of the Similarity Solution for Laminar Forced-Convection Condensation of Saturated Vapors. Engineering Sciences Reports, Kyushu University 1 3(3): 305-3 1 6.


Fujii, T., Uehara, H., Hirata, K., and Oda, K. 1 972. Heat Transfer and Flow Resistance in Condensation of Low Pressure Steam Flowing Through Tube Banks. Int. 1 Heat Mass Transfer 1 5(2) :247-260. Fujii, T., Uehara, H., Mihara, K., and Kato, T. 1 977. Forced Convection Condensation in the Presence of Noncondensables-A Theoretical Treatment for Two-Phase Laminar Boundary Layer. Reports of Research Institute of Industrial Science, Kyushu University 66:53-80 (in Japanese). Fujii, T., Uehara, H., Mihara, K., and Takashima, H. 1 978. Body Force Convection Condensation in the Presence of Noncondensables. Reports of Research Institute of Industrial Science, Kyushu University 67:23-4 1 (in Japanese). Fujii, T., Uehara, H., and Oda, K. 1 972. Film Condensation on a Surface with Uniform Heat Flux and Body Force Convection. Heat Transfer lap. Res. 4:76-83. Gaddis, E. S . 1 979. Solution of the Two-Phase Boundary-Layer Equations for Laminar Film Con densation of Vapor Flowing Perpendicular to a Horizontal Cylinder. Int. l. Heat Mass Transfer 22:37 1-382. Gogonin, I. I., Kabov, D. A., and Sosunov, V. I . 1 983. Heat Transfer in Condensation of R- 1 2 Vapor on Bundles of Finned Tubes. Kholodil'naya Teknika 1 :26-29. Gregorig, R., Kern, J., and Turek, K. 1 974. Improved Correlation of Film Condensation Data Based on a More Rigorous Application of Similarity Parameters. Wiirme unt Stof{iibertragung 7( 1 ): 1 . Griggul, U . 1 942. Wannelibertragung beider Kondensation mitt Turbulenten Waserhaut, Forsch. Ing. Weser 1 3 :49-57. Gunningham, J. and Ben Boudinar, M. 1 990. The Effect of Condensate Inundation on the Perfonnance of Roped Tubes. Condenser and Condensation: 4 1 7-429. Hebbard, G. H. and Badger, W. L. 1 934. Steam Film Heat Transfer Coefficient for Vertical Tubes. Trans. A1ChE. 30: 1 94-2 1 6. Hijikata, K. and Mori, Y. 1 972. Forced Convective Heat Transfer of a Gas Containing a Condensing Vapor on a Flat Plate. Trans. lpn. Soc. Mech. Eng. 38(3 1 4) : 2630-2640 (in Japanese). Hirshfelder, J. 0., Curtiss, C. F., and Bird, R. B. 1 954. Molecular Theory of Gases and Liquids. New York: John Wiley & Sons. Hirshburg, R. I. and Florshuetz, L. W. 1 982a. Laminar Wavy Film Flow. Part I : Hydrodynamic Analysis. Trans. ASME 1 04:452-45 8. Hirshburg, R. I . and Florshuetz, L. W. 1 982b. Laminar Wavy Film Flow. Part II: Hydrodynamic Analysis. Trans. ASME 1 04:459-464. Honda, H. 1 997. Tube Banks, Condensation Heat Transfer. In: International Encyclopedia of Heat Mass Transfer. Eds. G. F. Hewitt, G. L. Shires, and Y. V. Pohezhaev, pp. 1 1 77- 1 1 8 1 . New York: CRC Press. Honda, H. and Fuj ii, T. 1 984. Condensation of Flowing Vapor on a Horizontal Tube-Numerical Analysis as a Conjugate Heat Transfer Problem. Trans. ASME, l. Heat Transfer 1 06:84 1 -848. Honda, H., Nozu, S., and Takeda, Y. 1 989a. A Theoretical Model of Film Condensation in a Bundle of Horizontal Low Finned Tubes. Trans. ASME 1 Heat Transfer 1 1 1 : 525-532. Honda, H., Takamatsu, H., and Kim, K. 1 994. Condensation of CFC- l l and HCFC- 1 23 in In-Line Bundles of Horizontal Finned Tubes: Effect of Fin Geometry. 1 Enhanced Heat Transfer 1 : 1 97. Honda, H., Takamatu, H., Takada, N . , and Makishi, D. 1 996. Condensation of HCFC 1 23 in Bundles of Finned Tubes: Effect of Fin Geometry and Tube Arrangement. Int. l. Refrig. 1 9( 1 ) : 1-9. Honda, H., Uchida, B., Nozu, S., Nakata, H., and Fujii, T. 1 988. Condensation of Downward Flowing R- 1 1 3 Vapor on Bundles of Horizontal Smooth Tubes. Trans. lSME 54: 1 453-1 460; Heat Transfer-lap. Res. 1 8(6): 3 1 -52. Honda, H., Uchida, B., Nozu, S., Nakata, H., and Fujii, T. 1 989b. Condensation of Downward Flowing R- 1 1 3 Vapors on Tube Bundles of Horizontal Smooth Tubes. Heat Transfer-lap. Res. 1 8 : 1 3 1 1 52. Honda, H., Uchida, 8., Nozu, S., Nakata, H., and Torigoe, E. 1 99 1 . Film Condensation of R- l 1 3 on In-Line Bundles of Horizontal Finned Tubes. Trans ASME 1 Heat Transfer 1 1 3 :479-486.


Honda, H . , Uchida, B . , Nozu, S., Torigoe, E., and Imai, S. 1 992. Film Condensation of R- I 1 3 on Staggered Bundles of Horizontal Finned Tubes. Trans ASME I Heat Transfer 1 14:442-449. Kapitza, P. L. 1 948. Wavy Flow of Thin Films of Viscous Liquids. Zh. Eksoerim. Teor. Fiz. 1 8 : 3- 1 8 . Kays, W. M., London, A. L., and Lo, R. K. 1 954. Heat Transfer and Friction Characteristics for Gas Flow Normal to Tube Banks. Trans. ASME 76:3 87-396. Kern, D. Q. 1 958. Mathematical Development of Loading in Horizontal Condensers. AichE. 1 4: 1 571 60. Kinoshita, E. and Uehara, H. 1 995. Turbulent Film Condensation of B inary Mixture on a Vertical Plate. Proc. of 4th ASMEllSME Thermal Engineering 2: 367-373. Kirkbride, C. G. 1 93 3 . Heat Transfer by Condensing Vapor on Vertical Tube. Trans. Am. Inst. Chern. Engrs. 30: 1 70- 1 86. Koh, J. C. Y. 1 962a. Film Condensation in a Forced-Convection Boundary-Layer Flow. Int. 1. Heat Mass Transfer 5 : 94 1 -954. Koh, J. C. Y. 1 962b. Laminar Film Condensation of Condensible Gases and Gaseous Mixtures on a Flat Plate. Proc. 4th u.s. National Congress ofApplied Mechanics 2: 1 327- 1 3 36. Kotake, S . 1 985. Effect of a Small Amount of Noncondensable Gas on Film Condensation. Int. 1. Heat Mass Transfer 28(2):407-4 1 4. Koyama, S., Goto, M., and Fujii, T. 1 987a. Laminar Film Condensation of Multicomponent Mixtures on a Flat Plate-First Report, Forced Convection Condensation. Reports of Institute of Advanced Material Study, Kyushu University 1 ( 1 ) :77-83 (in Japanese). Koyama, S . , Goto, M., Watabe, M., and Fujii, T. 1 987b. Laminar Film Condensation of Multicomponent Mixtures on a Flat Plate-Second Report, Gravity Controlled Condensation. Reports of Institute ofAdvanced Material Study, Kyushu University 1 ( 1 ) :85-89 (in Japanese). Koyama, S . , Watabe, M., and Fujii, T. 1 986. The Gravity Controlled Film Condensation of Satu rated and Superheated Binary Vapor Mixtures on a Vertical Plate. Trans. lpn. Soc. Mech. Eng. 52(474B):827-834 (in Japanese). Kutateladze, S . S . and Gogonin, 1. 1 . 1 979. Heat Transfer in Film Condensation of Slowly Moving Vapor. Int. 1. Heat Mass Transfer 22( 1 2): 1 593-1 597. Lee, W. C. and Rose, J. W. 1 982. Film Condensation on a Horizontal Tube-Effect of Vapor Velocity. Proc. 7th Int. Heat Transfer Conf , Munich, Vol . 5, 1 0 1 - 1 06. Lucas, K. 1 976. Combined Body Force and Forced Convection in Laminar Film Condensation of Mixed Vapors-Integral and Finite Difference Treatment. Int. 1. Heat Mass Transfer 1 9( 1 1 ) : 1 273-1 280. Marshall, E. and Lee, C. Y. 1 973. Stability of Condensate Flow down a Vertical Wall. Int. 1. Heat Transfer 1 6 :4 1-48. Marto, P. J. 1 984. Heat Transfer and Two-Phase Flow During Shell-Side Condensation. Heat Transfer Engineering 5 : 3 1 -6 1 . Marto, P. J . 1 988. An Evaluation of Film Condensation on Horizontal Integral Fin Tubes. Trans ASME I Heat Transfer 1 1 0: 1 287- 1 305 . Mayhew, Y. R., Griffiths, P. J., and Phillips, J. W. 1 965. Effect of Vapor Drag on Laminar Film Condensation on a Vertical Surface. Proc. Instn. Mech. Engnrs 1 80(Pt.3J): 280-287. Meisenburg, S. J., Boarts, R. M., and Badger, W. L. 1 935. The Influence of Small Concentration of Air i n Steam on the Steam Film Coefficient of Heat Transfer. AIChE. 1. 31 :622-63 8. Memory, S . B . and Rose, J. W. 1 99 1 . Free Convection Laminar Film Condensation on a Horizontal Tube with Variable Wall Temperature. Int. 1. Heat Mass Transfer 34:2775-2778. Minkowycz, W. J. and Sparrow, E. M. 1 966. Condensation Heat Transfer in the Presence of Noncon densables, Interfacial Resistance, Superheating, Variable Properties, and Diffusion. Int. 1. Heat Mass Transfer 9( 1 0): 1 1 25- 1 1 44. Minkowycz, W. J. and Sparrow, E. M. 1 969. The Effect of Superheating on Condensation Heat Transfer in a Forced Convection Boundary Layer Flow. Int. 1. Heat Mass Transfer 1 2(2): 1 47-1 54.


Mori, Y. and Hijikata, K. 1 972. Theoretical Study on Free Convective Condensation of a Steam Contain ing Noncondensable Gas on a Vertical Flat Plate. Trans. Jpn. Soc. Mech. Eng. 38(306):4 1 8-426 (in Japanese). Murata, K., Abe, N., and Hashizume, K. 1 996. Condensation Heat Transfer in a Bundle of Horizontal Integral-Fin Tubes. Proc 9 th In Heat Transfer Conf 4:259-264. Niknejad, J. and Rose, J. W. ( 1 98 1 ) . Interphase Matter Transfer-An Experimental Study of Conden sation of Mercury. Proc. R. Soc. Lond. A 378: 305-327. Nusselt, W. 1 9 1 6. Die Oberftachenkondensation des Wasserdampfes. Z. Vereines Deutsch. Ing. 60(27):54 1 -546 ; (28):569-575 . Penev, V, Krglov, V S . , Boyadjiev, C. H., and Vorotilin, V. P. 1 972. Wavy Flow of Thin Liquids Film. Int. J. Heat Mass Transfer 1 5 : 1 395- 1 406. Pierson, O. 1. 1 959. Experimental Investigation of the Influence of Tube Arrangement on Convection Heat Transfer and Flow Resistance in Cross Flow of Gasses over Tube Banks. Trans. ASME, PRO-59-6:563-572. Rohsenow, W. M., Webber, J. H., and Ling, A. T. 1 956. Effect of Vapor Velocity on a Laminar and Turbulent-Film Condensation. Trans. ASME 78: 1 637- 1 643. Rose, J. W. 1 969. Condensation of a Vapor in the Presence of a Non-Condensing Gas. Int. J. Heat Mass Transfer 1 2(2) :233-237. Rose, J. W. 1 979. Boundary-Layer Flow with Transpiration on an Isothermal Flat Plate. Int. J. Heat Mass Transfer 22: 1 243- 1 244. Rose, J. W. 1 980. Approximate Equations for Forced-Convection Condensation in the Presence of a Non-Condensing Gas on a Flat Plate and Horizontal Tube. Int. J. Heat Mass Transfer 23:539-546. Rose, J. W. 1 984. Effect of Pressure Gradient in Forced Convection Film Condensation on a Horizontal Tube. Int. J. Heat Mass Transfer 27 : 39-47. Rose, J. W. 1 988. Fundamentals of Condensation Heat Transfer: Laminar Film Condensation. JSME Int. Journal, Series 11 3 1 (3) : 357-375. Rose, J . W. 1 998. Condensation Heat Transfer Fundamentals. Trans. IChemE 76: 1 43- 1 52. Sage, E. F. and Estrin, J. 1 976. Film Condensation from a Ternary Mixture of Vapors upon a Vertical Sutface. Int. J. Heat and Mass Transfer 19(3): 323-333. Seban, R. A. 1 954. Remarks on Film Condensation with Turbulent Flow. Trans. ASME 76:299-303. Shekriladze, 1 . G. and Gomelauri, V 1 . 1 966. Theoretical Study of Laminar Film Condensation of Flowing Vapor. Int. J. Heat Mass Transfer 9:5 8 1 -59 1 . Shekriladze, 1. G . 1 977. Analysis of Laminar Film Condensation from a Moving Vapor. Inzhenerno Fizicheskii ZhurnaI 32:22 1 -225. Shklover, G. G. 1 990. Generalized Data of Steam Condensation Computation in Horizontal Tube Bundles. Condenser and Condensation, 203-2 1 2. Soflata, H. 1 980. Theoretical Study of Film Wise Condensation Considering Wave Initiation. Wiirme und StoJfubertragung 1 4:20 1 -2 1 0. Soflata, H . 1 98 1 . Improved Theoretical Model for Wavy Film Wise Condensation. Wiirme- und Stojfubertragung 1 5 : 1 1 7- 1 24. Sparrow, E. M. and Eckert, E. R. G. 1 96 1 . Effects of Superheated Vapor and Noncondensable Gases on Laminar Film Condensation. AIChE J. 7(3):473-477. Sparrow, E. M. and Lin, S. H. 1 964. Condensation Heat Transfer in the Presence of a Noncondensable Gas. Trans. ASME. J. Heat Transfer C 86(3):430-436. Sparrow, E. M. and Marshall, E. 1 969. Binary, Gravity-Flow Film Condensation. J. Heat Transfer 9 1 (2):205-2 1 1 . Sparrow, E. M., Minkowycz, J . W., and Saddy, M . 1 967 . Forced Convection Condensation in the Pre sence of Noncondensables and Interfacial Resistance. Int. J. Heat Mass Transfer 1 0( 1 2) : 1 8291 845 .


Stephan, K. and Laesecke, A. 1 980. The Influence of Suction on Heat and Mass Transfer in Conden sation of Mixed Vapors. Wiirme-und StofJiibertragung 1 3 ( 1 /2): 1 1 5- 1 23. Stewart, W. E. and Prober, R. 1 964. Matrix Calculation of Multicomponent Mass Transfer in Isothermal Systems. I & EC Fundament. 3(3):224-235. Stuhltrager, E., Miyara, A., and Uehara, H. 1 995. Flow Dynamics and Heat Transfer of a Condensate Film on a Vertical Wall-II . Flow Dynamics and Heat Transfer. Int. J. Heat Mass Transfer 38( 1 5):27 1 5-2772. Stuhltrager, E., Naridomi, Y., Miyara, A., and Uehara, H . 1 993. Flow Dynamics and Heat Transfer of a Condensate Film on a Vertical Wall-I. Numerical Analysis and Flow Dynamics. Int. J. Heat Mass Transfer 36(6): 1 677-1 686. Sukhatme, S. P. 1 990. Condensation on Enhanced Surface Horizontal Tube. Proc 9th In Heat Transfer Conf 305-328. Taitel, Y. and Tamir, A. 1 974. Film Condensation of Multicomponent M ixtures. Int. 1. Multiphase Flow 1 (5):697-7 1 4. Tamir, A. 1 973. Condensation of Binary Mixtures of Miscible Vapors. Int. J. Heat Mass Transfer 1 6(3) :683-685. Tanaka, H. 1 985. On Expression for Local Nusselt Number and Local Sherwood Number Concerning Simultaneous Heat and Mass Transfer in Free Convection from a Vertical Plate. Reports of Research Institute of Industrial Science, Kyushu University 7 8 :47-52 (in Japanese). Toor, H. L. 1 964. Solution of the Linearized Equations of Multicomponent Mass Transfer: II. Matrix Methods. A1ChE J. 1 0(4):460-465. Uehara, H. and Kinoshita, E. 1 994. Wave and Turbulent Film Condensation on a Vertical Surface (Correlation for Local Heat-Transfer Coefficient). Trans. ofJSM£. 60(577), 3 1 09-3 1 1 6 . Uehara, H . and Kinoshita, E. 1 997. Wave and Turbulent Film Condensation o n a Vertical Surface (Correlation for Average Heat-Transfer Coefficient). Trans. JSM£. , pp. 40 1 3-4020. Uehara, H., Egashira, S., and Taguchi, Y. 1 989. Film Condensation on a Vertical Smooth Surface in Flowing Vapor ( 1 st Report, Flow Patterns of Condensate Film and Local Heat Transfer Coeffi cients). Trans. JSME: 450-456. Uehara, H., Kinoshita, E., Ido, H., Sugimoto, K., and Hasegawa, H. 1 995 . Experimental Study on Forced Convection Turbulent Film Condensation of Binary Mixture. Proc. Japan National Syp. Heat Transfer: 7 1 7-7 1 8 . Uehara, H . , Kinoshita, E . , and Matuda, S . 1 998. Theoretical Analysis o f Turbulent Film Condensation on a Vertical Surface. Trans. JSME 64:41 37-4 1 44. Uehara, H., Kusuda, H., Nakaoka, T., and Yamada, M . 1 983. Experimental Study on Film Condensation on a Vertical Surface. Trans. JSME 49(439B) :666-675. Uehara, H., Nakaoka, T., Kusuda, H., and Nakashima, S. 1 982. Wave Film Condensation on a Vertical S urface (Combined Convection). Trans. JSME 48(433): 1 75 1 - 1 760. Uehara, H., Nakaoka, T., Murata, K., and Egashira, S. 1 988. Body Forced Convection Condensation on a Vertical Smooth Surface ( 1 st Report, Flow Pattern of Condensate Film and Local Heat Transfer Coefficient). Trans. JSME 54(505B):2537-2544. Webb, R. I. 1 988. Enhancement of Film Condensation. Int Comm Heat Mass Transfer 1 5 :475-507. Webb, R. I . and Murawski, C. G. 1 990. Row Effect for R- l 1 Condensation on Enhanced Tubes. Trans ASME J Heat Transfer 1 1 2 :768-776. Wylie, C. R., Jr. 1 966. Advanced Engineering Mathematics. 3rd editon, pp. 429-492. New York: McGraw-Hill. Zukauskas, A. 1 972. Advances in Heat Transfer, 8. New York: Academic Press, pp. 93- 1 6 1 .

CHAPTER

TWENTY DROPWISE CONDENSATION

John Rose University of London, London E1 4NS, United Kingdom

Yoshio Utaka Yokohama National University, Yokohama 240-8501, Japan

Ichiro Tanasawa Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan

20.1 INTRODUCTION

Dropwise condensation occurs when a vapordoescondenses on a butsurface thatdiscrete is not wetted by the condensate. The condensate not spread fonns drops, assurface shownpromoted in Figureby1 trilauryl , for dropwise condensation of ethylene glycol on a copper trithiophosphite. has longhigher beenheat-transfer recognized that for non-metal vapors, dropwise condensation givesItmuch coefficients thandropwise found condensation with film condensation. For instance, the tiheat-transfer coefficient for of steam i s more than twenty mes that for film Dropwise condensationcondensation at atmospheric pressure. involves a combination of several processes. Bareby surface is continually exposed to vapor by coalescences between drops and sweepity norg acti on shear of thestress. departing"Pridrops asdrops they areare fonned removedatfrom the surface bytheon thegravi vapor m ary" nucleation sites exposed surface (typical nucleation site densities are in the range 107 to 1 09 sites/mm2 ). The primary drops grow by condensation until coalescences occur betweenat sites neighbors. Thethrough coalesced drops continue to grow,continues, and new ones form and grow exposed coalescences. As the process coalescences occur between drops of varioussizesizes,(r while the largest drops continue to grow until they reach their maximum 1 mm), when they are removed from the surface. The diameter between1 06the. largest and smallest drops during dropwise condensation of steamratiis oaround ;::::::

581


0.5

rom

Figure 1

Dropwise condensation (ethylene glycol at 1 1 .5 kPa, photographed at angle 45 degrees to the copper condensing surface promoted by trilauryl trithiophosphite, Utaka et aI., 1 988).

Reviews( 1 969), of dropwise heat transfer have been given by Le Fevre and Rose Tanasawacondensation ( 1 991), Marto ( 1 994), and Rose ( 1 994). 20.2 PROMOTION OF DROPWISE CONDENSATION

Clean metalthat surfaces are wetted byin practice. non-metallic liquids, and filmknown condensation ismoters, the mode normally occurs Non-wetting agents, as pro are needed to promote dropwise condensation. Unfortunately, sufficiently reliable promoting techniques have not yet been developed, exceptononantheindustrial labora tory scale, and dropwise condensation has not so far been used scaleMethods to any significant extent. that have used for promoting dropwise condensation may be classified into the followibeen ng categories: 1 . Application of an appropriate non-wetting agent, i. e . , organic promoter, to the condenser surface before operation. 2. Injection of the non-wetting agent intermittently (or continuously) into the vapor. 3. Coating the condensing surface with a thin layer of an inorganic compound, such as metal sulfide. 4. Plating the condensing surface with chromium or a thin layer of a noble metal, such as gold. 5. Coating the condensing surface with a thin layer of an organic polymer, such as polytetrafluoroethylene (PTFE, Teflon).

DROPWISE CONDENSATION 583

Various promoters haveThebeenearliest identified andusedusedweresuccessfully inwhilaboratory tests, mainly wi t h steam. to be fatty acids, ch mustfor belonginjected i n termi t tently into the vapor to maintain dropwise condensation intervals. More promising promoters are longsurface chaintohydrocarbon moleculesc with an active group that bonds with the metal present a hydrophobi surface to the vapor.layerExcess promoterthermal is washedresistance. away byAthetypical condensate mono-molecular of negligible exampleto leave of thisa type of promoter is dioctadecyl disulphide, which has given satisfactory dropwise condensation etimes ainl application laboratory investigations. Successfullifindustri ofdropwise condensation has been prevented promoternon-wetting breakdownsurface; often associated wihasth surface oxidation. PTFE provi des an byexcellent however, it not been found possible to produce sufficieantlythermal thin residurable surface layers. A layer of thicknessof dropwise 0.02 to 0.over03 film mm offers s tance suffici e nt to offset the advantage condensation forcondensation steam. has frequently been observed on metal surfaces such Dropwise as chromium andhowever, gold, which arenon-wetting very smooth, withoutwere use present. of a promoting agent. It seems probable, that impurities In experi m ents wherefound great(see careErb, has been taken to ensure clean conditions, filmaI., condensation has been 1973; Erb and Thelen, 1966; Wilkins et 1973a; Woodruff and Westwater, 1979). 20.3 DROPWISE CONDENSATION OF STEAM 20.3.1 Measurements at Atmospheric Pressure

The coefficientbyof dropwise condensation of steam at a these pressureshowneara 1considerable atmheat-transfer has beenscattermeasured many researchers. Taken together, due to the effect of noncondensing gas and the accuracy of measuring method. Heat-transfer coefficients obtainable with dropwise condensa tion are hightheyandarehence prone to error in tomeasurement ofthethepresence small temperature diporfference; also very susceptible reducti o n by in thesmall va of noncondensing gas. In the absence of significant vapor velocity, very gas concentrations lead to appreci ambleental lowering of thehave heat-transfer coefficient.andIn more recent years, i m proved experi techniques led to reproducible reliable experimental data; see,At present, for instance, LebeFevre andtheRoseheat-transfer (1964, 1965)coeffi and Citakoglu and Rose (1968). it can said that cient of dropwise condensation for steam at 1 atm, under the normal gravi2 K)tatiinothenal accelerati on andofon0.1-1a vertiMWcal1mcopper aboutis 230 2 , provisurface, heat flux range ded thatisthere no effect50 ofkW/(m noncondensing ±


Surface sub cooling Figure 2

�T

K

Dropwise condensation of steam on copper surfaces at atmospheric pressure.

gases andbetween the steamthevelmore ocityrecent is smalinvestigations ler than aboutare1 0duemls.to Reltheaeffects tively smal l dif ferences of steam velocity and2surface characteristics. Figure shows several representati ve measurements for steam at atmospheric pressure together with the theory of Le Fevre and Rose ( 1966) (see Section 5) and anis notempirical equation for steamthe(seeoriginEq. shows [20.3- 1 ] below). The fact that Figure 2 a strai g ht line through that subcooling. the heat-transfer coeffic�ientofrange,dropwise condensation depends on the surface In smaller T the tendency of decreasing heat-transfer coefficient wi t h decreasing surface subcooling is due to the decreasing number density of active nucleation sites on the condensing surface. nearl In they constant range of with moderate coolingsurface intensity, the heat-transfer coefficient becomes increasing subcooling. 20.3.2 Effect of Pressure

Several sets surfaces of resultsatarepressures availablbele forowdropwise condensation of steam on copper condensing 1 atm, as shown in Figure 3. All data show a tendency of decreasing heat-transfer coefficientresiwisthtance decreasing pressure. Thithes isliquid-vapor caused mainly by the i n crease of the interfacial to mass transfer at interface at low pressure. Differences between different data sets are in part attri b utabl e to the dependence on �The T, with higher heat-transfer coefficients being foundof heat-transfer at larger surfacecoefficient subcoolings. solid lines


� 100

� �

N

u _

50

�o

��--��--�

0 Tanner et al. (1968)

t:S

10 � rJJ U

AT K

0. 1 - 1.7

0 Graham (1969) 0.1 - 1.7 'il Tsuruta & Tanaka (1983) 2.0 - 2.4 l:::. Hatarniya & Tanaka(1986)1.1 - 1 .5 ... Wilrnshurst & Rose (1970)1.5 - 5.6 • Stylianou & Rose (1980) 0.5 - 8.0

- Rose (1996) {Eq.(l)}

M

= �

M ..... I .....

5

� (])

::t:

Figure 3

0.5

Pressure p

kPa

Dependence of heat-transfer coefficient on pressure for steam [Note that the "error bars" indicated for the data of Wilmshu rst & Rose ( 1 970) and Stylianou & Rose ( 1 980) are due to dependece of a on � T ] .

are calculated by the empirical equation for steam proposed by Rose (1996); al(kW1m2 K) = TeO.8 (5 0.3 b. TIK). (20. 3 -1) where Te is Celsius temperature of the saturated vapor. +

20.3.3 Effect of Departing Drop Size

The coefficient ofsurface, dropwisewhich condensation is dependent onmaximum the dis tridropbutiheat-transfer odiameter n of dropor thesizes on the in tum depends on the departi ng dropcoeffici diameter. Tanasawa etatmospheri al. (1976) cmeasured theon thedependence of the heat-transfer e nt for steam at departing drop diameter under almost constant heat flux of 600 pressure kW/m2 , using gravitational, centrifugal, and vapor shear forces to change the departi ng drop diameter. The results are shown in Fi g ure 4. It is seen that the heat-transfer coefficient is proportional to the departing drop diameter to the power of about rangecoeffici of b.Tentin ontheb.measurements was quite small, so the dependence of-0.heat3 . The -transfer T was not significant. It is very interesting to note that the theori of Le Fevrein andSection Rose5,(1966) Rose, the1976,same1988) (1975a, 1975b),es discussed lead to(seevirtually result.and Tanaka


0

8

�

/

a

=

O.25 D - O.32

Q)

� """

�

g

�

0 Vapor shearing force 0

Centrifugal force Inclined surface

� °d.�1--��--�-�5 t::.

Departing drop diameter D

mm

Figure 4 Dependence of heat-transfer coefficient for steam on departing drop diameter [Tanasawa et al. ( 1 976)].

20.3.4 Effect of Thermal Properties of Surface Material

In thedrops dropwise condensation process, those parts oflarge the surface under relatively large are nearly adiabatic, while the heat flux is on those regions having very small drops. Also, drops move on the surface due to the coalescences. Hence, theandtemperature andeffect the heatof thermal flux on theconductivity condensingis expected surface must bethenonuniform fluctuating. An due to forthewhichheat-transfer a model hascoeffi beencientproposed bylower Mikicon(1969). According tomatthe Mierial.kicAntheory, must be a poor conducti v e heat capacity (transient conduction) the surface materialto should alsoeffectariseoffrom the unsteady natureheatof process. Bothofwould be expected lead Figure to an effective additional resistance to heat transfer. 5 shows the experimental and theoretical results. There has been con siderable controversy on thisbuttopic,rathernotover overtheir the factmagnitudes. that suchThe resistances are ex pected on physical grounds, measurements ofTogashi Wilkins(1989) and Bromley (l973b),quite Hannemann and Mikic (1976a) and Tsurutacoeffi and have suggested a strong dependence of heat-transfer cient on(1980), thermalwhoconductivity ofthetheaccuracy surface materials, while data ofmeasurement, Stylianou and Rose questioned of some of the earlier and Aksanof heat-transfer and Rose (1973) have indicated insignificant dependence. Theabove depenis dencies coefficient on temperature di ff erence discussed also measurements show lowlowheat-transfer coefficients atiments lowrelevant, surfacesince thermalthoseconductivity have that necessarily � while those exper T, performed at higher � T have generally shown small or negligible effect. constric

tion effect,


�TK

p kPa o •

o o

.. \l f::.

Wilkins & Bromley (1973b) 101 1 .0 -2.0 Stylianou & Rose (1980) 101 Aksan & Rose (1973) 101 Hannemann & Mikic (1976a) 101 Hatamiya & Tanaka (1986) 1 .0-101 1 . 1 - 1 .5 2.5 101 Tanasawa et al. (1976) Tsuruta & Togashi (1989) 9.0 Hannemann & Mikic (23] Tsuruta & Tanaka (1988)

p

o --

. --

0 _ -: .... · ..",. "

-

101 kPa

I _ . _ ._ . . ·- � o. - t- - -;.--'f) .... . ....

.

_ ..

=

.

- - --

-

-

-

0

p = 1 .0 kPa

--_

.-.-.

;I

-._

._ -

_.

500

Figure 5

-

1 000

Surface thermal conductivity A W/mK

Dependence of heat-transfer coefficient on thermal conductivity of surface material for

steam.

A possiblerate, explanation ishigh, that and at a thehightemperature � T , the condensation rate, andcondensing hence co alescence are very distribution on the surfacewould is effectively smoothed soreverse that dependence on thermal properties of the sur face be small, whereas the is true at low � T, when the condensation rate and henceaccurate the ratedataof coalescence isresolve relativelythe small. Further are needed to question ofofthesurface extentmaterial to whichis the dependence of heat-transfer coefficient on conductivity affected surface by condensation rate. Forbethisof sufficient purpose, accuracy measurements withthediffsufficiently erent con densing materials should and cover wide range of heat intensive flux, whichcooling shouldwhen be theusingsamelowforconductivity all surfaces.testSuchsurfaces. experi mentsTheoretical may require studies of this assumingandsteady have been made by Hannemann and problem Mikic (1976a) Tsurutaheatandconduction Tanaka (1988),


which showmaterial. dependentBothheat-transfer coefficient on thermalwithconductivity of�the condensing model s showed good agreement data for low T. No dependence on heat capacity (transient heat conduction) has been establ i shed either experimentally or theoretically. 20.3.5 Condensation Curve

With increasing surface subcooling, it has been shown by Takeyama and Shimizu ( 1974) for steam and Wilmshurst and Rose (1974) for organic vapors that the heat flux in dropwise condensation has a peak value, and that the transition to filfluxm plotted condensation occurs at high � T. Figure 6 shows the measurements of heat against bysurface subcoolandingUtaka ("condensation curves") for steam at at mospheric pressure Tanasawa ( 1 983) and Takeyama and Shimizu using special condensing blocks to achiofevemeasurement. large heat flux. It is thought that the(1974)latter may be subject to some i n accuracy It may upbe toseena veryfromhighFigure maintained heat fl6uthat x. very high heat-transfer coefficients are Max. drop Departing V apor diameter

0.025 0.055 o 0.11 ... 0. 1 9 .0 0.54 l:l. •

N

� �

8 10

mm

droo diam. velocity

mJs 76

mm

0.05 0.11 0.23 0.40 1 .5

51 32 20 7

-

'/

""""" - ", -

\

� >< ;:::) � ..... ro !l)

1

::r:

/

/

/

/

/

/

/

\ y\Y 7V

/

Takeyama & Shimizu

1

10

Surface subcooling Figure 6

(1974)

1 00 !1T

K

Conduction curves for steam at atmospheric pressure (Tanasawa & Utaka, 1 983).


20.4 DROPWISE CONDENSATION OF ORGANIC VAPORS

Tocondensation date, mostcanmeasurements have beenanddone with steam, forlaboratory which dropwise be readily promoted maintained in the owing to theof other high surface tension of water. To promote and maintain dropwise condensation fluids is more difficult. Experiments have beenRose performed with severalandvapors. Figure 7onshows the results of Wilmshurst and (1974) for aniline ni t robenzene a PTFE coated surface, Stylianou andglycol, Rose (1983) forglycol, ethyleneandglycol, and onUtakaa copper et al. (1987, 1994) for propylene ethylene glycerol, surface using a mono-layer type promoter at reduced pressures. � S -

50

N

Ethylene glycol

0

�

Propylene glycol Glycerol Aniline

�

tS � � 0 Co)

25

pkPa 1 1 .5 • 6.9 [] 4.8 • 4.8 6 1 .4 .. 1 .4 v 0.6 o 1 1 .0 1 .7 o

z mm 25 Stylianou & Rose (1983) 25

18 Utaka et al. (1987, 1994) 66 18 66 4 19 Wilmshurst & Rose(1974) 19

....

� tI:l c:: CI:I

1:: I

� � ::r::

0 c. 0 0

N

1

� S

COb C))

:;E

� � �

t:;:::

o.

� � ::r::

Surface sub cooling /). T Figure 7

Heat transfer characteristics for organic vapors.


As incoefficient the case forof dropwise steam, in condensati the moderateon iscooling intensitylargerrange,thantheforheat transfer significantly film condensation and decreases with decreasing pressure. Quanti t ati v ely, the heat transfer for these organiMainly c vaporsduedifftoerlower fromliquid those thermal for steamconduc due to thetivity wideofcharacteristics range of physical properties. the are organic fluids, relatively low heat-transfer coefficients in comparison with steam seen. It can also be seen that the surface subcooling ranges of ihigher deal dropwise condensation differ widely, depending on the fluid. For a fluid of tension such ascompared glycerol,withdropwise condensation is maintained upas topropylene largersurface surface subcooling, a lower surface tension liquid such glycol.heat fluxes are factors of 10 to 100 smaller than for steam at at The peak mospheric pressure dueof these to highfluids.viscosity, low surface tension, andcurves low latent heat of condensation As a result, the condensation (peak heat-transfer coeffici e nt) are readil y obtainable without special methods to obtain intensive cooling. Thefrompeaktheheattopfluxof thedecreases with decreasing pressuretransition and with increasing distance condensing surface. Typically, from dropwise to film condensation occurs as follows. As the cooling intensity is increased, following theandrangesubsequently of "ideal" dropwise condensation, thinlower rivuletsdownof condensate are formed, film condensation appears the surface. The condensate film extends upwards with an increase in cooling in tensity until the surface is wholly covered with condensate film. Other transition modes haveis reduced. been observed, in some cases showing is"hysteresis" as and the cooling intensity The phenomenon of transition complicated, mechanisms have been suggested (see Stylianou and Rose, 1983 ; Takeyamavarious and Shimizu, 1 974; Utaka et aI. , 1 988, 1 990). 20.5 HEAT TRANSFER THEORY 20.5.1 Le Fevre and Rose Theory

This flux theoryfor (Le Fevrecondensation and Rose, 1966), first proposed in 1 966,forderithevheat es thetransfer mean heat dropwise by combining expressions through a sintransfer gle dropresiof sgiven the distribution of dropare sizes. In derifor:v ing the heat tancesiofzeaandsinglefor drop, the following accounted ( 1 ) the influence of surface curvature on the phase equilibrium temperature, (2) the mass transferthrough resistance at theTheliquid-vapor interface,theandheat(3)fluxthethrough heat conduction resistance the drop. relation between the base ofdisatridrop and the surface subcooling /),. T is given by Eq. (20.5- 1 ) in Table 1 . The bution of drop sizes is given by Eq. (20.5-2), which expresses the fraction

�

(It \C

R�ax � T

n

R�, = KJ

p, ) g

J (PI ':

+

�T T

-

- - --

T K2 Vg T Y + 1 � -21f h}g Y 1 qB

2a v/ r h Jg r n - ' dr K2 Vg Y + 1 j?[[K]r + 21f AT h}g Y 1

A

Kl r

Rmax iRmin

r

2a vg T Rmin = h Jg � T

ex

=

-

f = 1

( r / Rmax )n

2a v/ T + r h Jg

�T =

Le Fevre-Rose ( 1 966) , Rose ( 1 976, 1 988 )

-- ( - -- - - - )

Table 1 Dropwise condensation theories

(20.5-5)

-

( 20.5 4)

(20.5 -3)

(20.5 -2)

( 20.5 - 1 )

=

a (Nra_ ) __ at

1

-

1fp 2 N(p,t)dp

1

21fp N (p , t )dp

----

)(

1

-

ex Rmax/A = 5 . 3 ( Rmax / D ) O. 7

-

) r rE (r, t)

re)

(20.5. 1 1 )

(20.5- 1 0)

(20.5-9)

(20.5-8 )

(20.5-7 )

(20.5 -6)

21fp{ra (r) + ra (p) } IJI (r ; N) N(r)N (p)dp

"p 'N (p . t)dp

( ir RmaX l_ i Rm� 2-"-Rm---'n a-x i

rE C Rmin , t) = D

Rmin

ir Rmax

2 3 1fp3 1f r [ra (r) + ra (p») IJI (p ; N ) N(p)dp = 21f r 2 (ra

rE (r , t) =

J.r

_

+ 1f R�ax N (Rmax )ra ( Rmax ) N (r )

-

lJI (r , t; N) =

-

aN at

Tanaka ( 1 975a, 1 975b)


of surface area covered by1 drops having base radius 0).greater thantheser. (Note that, From equations, for any n, for r 0, and for r thesmallest surfaceviableheatdroptransfer andcoefficient is givendropby Eq. were (20.5taken -3). Theas Eqs.radii(20.of5the-4) the maximum and (20. 5-5), respectively. The dimensionless constants KJ , K2 are shape factors, and K3 arises from dimensional analysis. For dropwise condensation of steam KJ = 2/3, K2 = 1/2, K3 = 0. 4 , and n = 1/3. The integral in Eq. (20. 5 -3) can be given in closed form. (See also Rose, 1988). f

�

f�

�

Rmax , f � Rmax

Rmin

20.5.2 Tanaka's Theory

A theory proposed by Tanaka (n197to account 5a, 1975bthe) consi ders theoftransient change of local drop size distribution, taki n g i processes growth and coalescence of drops. Drops formed on the bare surface grow by direct condensation with a rate ofthereapparent (r) , but because coalescence between drops takes place at the same time, rateng radi of growth is asr anandaverage, r (r , The asnumber of drops per uniis thet areatimehavielapsed iafterbetween r dr i s denoted N (r , )dr. Here, thea situation, area of thethecondensing surface was5-6)-(20. swept 5off-8) byis aderideparti n g drop. In such relation of Eqs. (20. v(byed forcoalescence drops withandradius r by considering the balance between growth and loss sweeping). The vari a ble defined by Eq. (20. 5 -8) i s the "equivalent diameter for coalescence" and is the measure of how closely packed theradiusdropsr when are. Equation (20. 5 -9) is deri v ed for the volume increase of a drop wi t h thendrop growsthebyincorrect coalescence. It may be noted thatKatzthe(1949) rate ofasdropfor growth was obtai ed from theory of Fatika and re, but the quantitative effect on the model of Tanaka is not clear. It was assumed that rE is equalby Eq. to the(20.average sites D between as expressed 5-10), anddistance Eqs. (20. 5-6) andneighboring (20.5-9) solvednucleation numerically. The relation for the heat-transfer coefficient was derived as Eq. (20.5-11). a

t) .

t

+

t

rE

REFERENCES Aksan, S. N. and Rose, J. W. 1 973. Dropwise Condensation-The Effect of Thermal Properties of the Condenser Material. Int. 1. Heat Mass Transfer 1 6:46 1 -467. Citakoglu, E. and Rose, J. W. 1 968. Dropwise Condensation-Some Factors Influencing the Validity of Heat-Transfer Measurements. Int. 1. Heat Mass Transfer 1 1 :523-537. Erb, R. A . 1 973. Dropwise Condensation on Gold. Gold Bull. 6:2. Erb, R. A. and Thelen, E . 1 966. Dropwise Condensation Characteristics of Permanent Hydrophobic Systems. U.S. Off. Saline Water Res. Dev. Rep. No. 1 84. Fatika, N. and Katz, D. L. 1 949. Dropwise Condensation. Chern. Eng. Prog. 45 :66 1 -674.


Graham, C. 1 969. The Limiting Heat Transfer Mechanism of Dropwise Condensation. Ph. D. Thesis, Massachusetts Institute of Technology. Hannemann, R. 1. and Mikic, B. B. 1 976a. An Experimental Investigation into the Effect of SuIiace Thermal Conductivity on the Rate of Heat Transfer in Dropwise Condensation. Int. J. Heat Mass Transfer 1 9: 1 309- 1 3 17. Hannemann, R. 1. and Mikic, B . B . 1 976b. A n Analysis ofthe Effect o f Surface Thennal Conductivities on the Rate of Heat Transfer in Dropwise Condensation. Int. J. Heat Mass Transfer 19: 1 299-

1 307.

Hatamiya, S. and Tanaka, H. 1 986. A Study on the Mechanism of Dropwise Condensation. ( I st Report, Measurement of Heat-Transfer Coefficient of Steam at Low Pressures). Trans. JSME, Ser. B

52(476): 1 828- 1 833.

Le Fevre, E. 1. and Rose, 1. W. 1 964. Heat-Transfer Measurement During Dropwise Condensation of Steam. Int. J. Heat Mass Transfer 7:272-273. L e Fevre, E. 1. and Rose, 1. W . 1 965 . A n Experimental Study of Heat Transfer Study b y Dropwise Condensation. Int. J. Heat Mass Tramfer 8: 1 1 1 7- 1 1 33. L e Fevre, E. 1. and Rose, 1. W . 1 966. A Theory o f Heat-Transfer by Dropwise Condensation. Proc. 3rd Int. Heat Transfer Con! 2:362-375. Le Fevre, E. 1. and Rose, 1 . W. 1 969. Dropwise Condensation. Proc. Symp. Bicentenary of the James Watt Patent, Univ. Glasgow, 1 66-1 9 l . Marto, P. 1 . 1 994. Vapor Condenser. McGraw Hill Year Book of Science and Technology, 428-43 1 . New York: McGraw Hill. Mikic, B. B . 1 969. On Mechanism of Dropwise Condensation. Int. J. Heat Mass Transfer 1 2: 1 3 1 1 -

1 323.

Rose, 1. W. 1 976. Further Aspects of Dropwise Condensation Theory. Int. J. Heat Mass Transfer

1 9: 1 363-1 370.

Rose, 1. W. 1 988. Some Aspects of Dropwise Condensation Theory. Int. Communications in Heat and Mass Transfer 1 5 :449-473 . Rose, 1. W. 1 994. Dropwise Condensation. Heat Exchanger Design Update. New York: Begell House Inc., 1-3. Rose, 1 . W. 1 996. Private communication. Stylianou, S. A. and Rose, 1. W. 1 980. Dropwise Condensation on SuIiaces Having Different Thennal Conductivities. Trans. ASME, J. Heat Transfer 1 02:477-482. Stylianou, S. A. and Rose, 1. W. 1 983. Drop-to-Filmwise Condensation Transition: Heat Transfer Measurements for Ethandiol. lnt. J. Heat Mass Transfer 26(5 ) :747-760. Takeyama. T. and Shimizu, S. 1 974. On the Transition of Dropwise-Film Condensation. Proc. 5th Int. Heat Transfer Con! 3 :274-278. Tanaka, H. 1 975a. A Theoretical Study on Dropwise Condensation. J. Heat Transfer 97:72-78. Tanaka, H. 1 975b. Measurement of Drop-Size Distribution During Transient Dropwise Condensation. J. Heat Transfer 97: 34 1 -346. Tanasawa, I . , Ochiai, 1., Utaka. Y., and Enya. S. 1 976. Experimental Study on Dropwise Condensa tion (Effect of Departing Drop Size on Heat-Transfer Coefficient). Trans. JSME 42(361 ) :2846-

2853.

Tanasawa, I. and Utaka, Y. 1 983. Measurement of Condensation Curves for Dropwise Condensation of Steam at Atmospheric Pressure. J. Heat Transfer 1 05:633-638. Tanasawa. I. 1 99 1 . Advances in Condensation Heat Transfer. Advances in Heat Transfer 2 1 :55- 1 39. Tanner, D. w.. Pope, C. 1., Potter, C. 1., and West, D . 1 965. Heat Transfer in Dropwise Condensation Part I: The Effects of Heat Flux, Steam Velocity, and Non-Condensable Gas Concentration. Int. J. Heat Mass Transfer 8:427-436. Tanner, D. w., Pope, C. 1., Potter, C. 1., and West, D. 1 968. Heat Transfer in Dropwise Condensation


at Low Steam Pressures in the Absence of Non-Condensable Gas. Int. 1. Heat Mass Transfer 1 1 : 1 8 1- 1 90. Tsuruta, T. and Tanaka, H. 1 983. Microscopic Study of Dropwise Condensation. Trans. lSME, Ser. B 49(446) :2 1 8 1-2 1 89 . Tsuruta, T. and Tanaka, H . 1 988. A Theoretical Study on Constriction Resistance in Dropwise Con densation. Trans. lSME, Ser. B 54:28 1 1 -28 1 6. Tsuruta, T. and Togashi, S. 1 989. A Study on the Constriction Resistance in Dropwise Condensation, Surface Chemistry. Trans. lSME, Ser. B 55 :2852-2860. Utaka, Y, Kubo, R., and Ishii, K. 1 994. Heat Transfer Characteristics of Condensation of Vapor on a Lyophobic Surface. Proc. 10th Int. Heat Transfer Can! 3:40 1 -406. Utaka, Y., Saito, A., Ishikawa, H . , and Yanagida, H. 1 987. Transition from Dropwise Condensation to Film Condensation of Propylene Glycol, Ethylene Glycol, and Glycerol Vapors. Proc. 2nd ASME-1SME Thermal Eng. Can! 4:377-384. Utaka, Y. , Saito, A., and Yanagida, H. 1 988. On the Mechanism Determining the Transition Mode from Dropwise to Film Condensation. Int. 1. Heat Mass Transfer 3 1 (5): 1 1 1 3- 1 1 20. Utaka, Y., Saito, A., and Yanagida, H. 1 990. An Experimental Investigation of the Reversibility and Hysteresis of the Condensation Curves. Int. 1. Heat Mass Transfer 33(4):649--659. Wilkins, D. G. and Bromley, L. A. 1 973b. Dropwise Condensation Phenomena. AIChE 1. 1 9 : 839-845. Wilkins, D. G., Bromley, L. A., and Read, S . M. 1 973a. Dropwise and Filmwise Condensation of Water Vapor on Gold. AIChE 1. 1 9 : 1 1 9- 1 23 . Wilmshurst, R. and Rose, J. W . 1 970. Dropwise Condensation-Further Heat-Transfer Measurements. Proc. 4th Int. Heat Transfer Can! 6:Cs l-4. Wilmshurst, R. and Rose, J. W. 1 974. Dropwise and Filmwise Condensation of Aniline, Ethandiol, and Nitrobenzene. Proc. 5th Int. Heat Transfer Con! 3 : 269-273 . Woodruff, D. W. and Westwater, J. W . 1 979. Steam Condensation on Electroplated Gold: Effect of Plating Thickness. Int. 1. Heat Mass Transfer 22:629--632.

CHAPTER

TWENTY ONE

DIRECT CONTACT CONDENSATION Ichiro Tanasawa Tokyo University ofAgriculture and Technology, Tokyo 184-8588, Japan

Yasuhiko H. Mori Keio University, Yokohama 223-8522, Japan

Yoshio Utaka Yokohama National University, Yokohama 240-8501, Japan

21.1 INTRODUCTION

Direct contact condensation is usually considered toa low-temperature be a mode of condensation observed when a vapor makes direct contact with liquideven and condenses on its surface. However, a vapor condenses onto the liquid surface in thecategory cases ofoffilmdirectandcontact dropwisecondensation. condensation,Theneither onedifference of which falling underdi the essential between rectlow-temperature contact condensation ., filminandthedropwise) is thata the liquidanditselfindirect acts condensation as the only heat(i.esink former, while coolant,of condensation which flows along thelatter. otherThus, side ofin adicondensing surface, absorbsthethemaxi latent heat in the r ect contact condensation, mum amount of condensate is limited by the heat capacity of the low-temperature liquid. condensationliquid is roughly classified one is thethe case Direct when thecontact low-temperature is dispersed in theintobultwo k oftypes: vapor phase; is theexample case whenof thethe vapor, whiis observed ch is to bewhen condensed, forms a dispersed phase. Aorothertypical former a low-temperature liquid drop a liquid jet fal l s down through a space filled wi t h a vapor. A steam condenser invented nearlysteam 300 years ago by Thomas Newcomen and improved later by James Watt for their engines employed this type of condensation. An example of 595

596 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSA nON

thein pool latterboiling. is the collapse of steamof bubbl bubbleses isinaccompanied a pool of subcooled waternoiseoccurring The collapse with a big that is heardInwhen water is boiled in a kettl e . both cases, the vapor and low-temperature liquid can be the same species (e.vaporg., steam condensing onsurface a waterof jet) ordroplets). different species (e.on,g.,influorocarbon condensing on the water In addi t i thebecaseeitherof dimiscible fferent orspecies, the low-temperature liquid and the condensate can immiscible. The characteristics of directtakes contact condensation vary greatlOnly y, dependi n g upon which type of condensation place. some fundamental s of direct contact condensation are described in this chapter due to the space limitation. Arti c les by Sideman ( 1 966) and Sideman and Moalem-Maron 982) and a book edited by Kreith and Boehm ( 1 988) are highly recommended for( 1more detailed information. 21.2 CASE WHEN A LOW-TEMPERATURE LIQUID FORMS A DISPERSED PHASE 21.2.1 Condensation on a Liquid Jet

Hasson et ofal.a(vapor 1 964) derived asymptotic Nusselt numbers for direct contact con densation cal jet, (2) a sheet jet, and (3) a fan jet. The results are expressedonto by Eqs.( 1 ) a(2cylindri 1 .2- 1 )-(2 1 .2-3) and shown in Figure 1 . 1 000 c-------�

10

Figure 1

Graetz number Gz

Dependence of the local Nusselt number on the Graetz number in direct contact condensa tion on jets.

DIRECT CONTACT CONDENSATION 597

{

(1) wiCylindrical jet: The asymptotic Nusselt numbers for a laminar-flow liquid jet th a constant velocity and diameter are expressed as follows: 5 . 784 when Gz 0 (21. 2-la and b) Nux = ,JGz/rr FJz when Gz (8/,JrrGz) rr where Nux (at=theaxdistance de / A) is the local Nusselt number, ax is the local heat transfer coefficient x from the nozzle exit, A is the thermal conductivity ofto thethe condensate, and de is the equivalent diameter of the liquid jet and equal diametervelocity of jet.ofInthethisjet,case,isGzthe(=thennal Ude / (K x)) is the Graetz number, U diffusivity of the condensate, iands thex mean K is the distance from the nozzle exit. (2) Sheet jet: The asymptotic Nusselt numbers for a laminar, constant-velocity, and uniform-thickness sheet jet are expressed as follows: when Gz 0 rr2 (= 9 . 780) (21.2-2a and b) Nux = ,JGz/rr FJz when Gz 1 (8/ ,JrrGz) rr where the thickness of the liquid jet sheet is used as the equivalent diameter de . Fan jet: The asymptotic Nusselt numbers for the laminar, constant velocity (3) fan (of which liquid filmasisfollows: inversely proportional to the jetdistance from thethe thickness nozzle exit)ofaretheexpressed when Gz 0 rr 2 (= 9.780) (21.2-3a and b) Nux = J3Gz/rr J3GZ when Gz 1 (8/,JrrGz) rr where twice the liquid film thickness is used as the equivalent diameter de .

{

{

---+

------ �

I -

-

---+ 00

---+

------ �

-

-

---+ 00

---+

------ �

-

--

---+ 00

21.2.2 Condensation on a Falling Liquid Drop

The ofresistance heat fromforthecondensation vapor to a low-temperature liquiddropdropandithes governed byresistance thetransport thennal at the surface ofthe thennal inside the drop. In most cases, however, the fonner is negligibly small when compared with the latter. are: There are three models proposed for the heat transport inside the drop. They (1) Solid drop, sphere model, which does not take into consideration the flow inside the

Circulatory flow model, which takes into consideration the flow inside the drop, and (3) Complete mixing model, which assumes that the temperature inside the drop is uniform. However, this model is not adequate in many instances. A typical example of the solid sphere model is an analytical solution by Newman ( 1 93 1 ) for the heat conduction in a sphere. In this solution, the change of temperature distribution iatn thethe surface sphere is(which obtainedcorresponds under thetocondition of a uniform and constant temperature zero external thermal resistance). Newman's solution gives the dimensionless mixed-cup temperature Em [ = (Tm - �)/(Tv - �) ] inside the drop as 1 1 ( JT 2n 2K t) Em = 1 JT 2 2: 2 exp - 2 (2 1 .2-4) n r where Tm is the mixed-cup temperature inside the drop, � is the initial (uniform) temperature ofandther drop, Tv is the vapor temperature, K is the thermal diffusivity of theA liquid, is the radius of the drop. much simplified approximation for Eq. (21 .2-4) is proposed by Vermeulen 598 HANDBOOK OF PHASE CHANGE: BOILING AND CONDENSATION

(2)

-

00

n=l

( 1 953):

(2 1 .2-5)

In Figure 2, the solid line represents Eq. (2 1 .2-4), and the circular symbols represent � � e

�

Q..

e � (;

.�� c:: 0

e � c:: 0

Z

1.0 0.8

: /

0.6

! (� 0.4 I ....�'-"""fi--t--+

Eq.(2 1 . 2-5) --;-t--+--t--+----1 I-f-!+-'+-+--+ ilr+_...-+ -- - Eq. (2 1 .2-6) -t--+--+--+-t-i 0.2 ... [-+--i-+-� Eq .(2 1 . 2-7) -11--+---+--+-t-i I I 1 I J

-- Eq . (2 1 . 2-4) --;-t--+--t--+----1

0

....

0

Figure 2

1 .0

2.0

Nondimensional time

7t

2 K

t/ r

3.0 2

Rise of the nondimensional mixed-cup temperature of a drop calculated by different models.


Table 1 Values of the constants An and the eigenvalues .An in Eq. (21.2-6) n

An

An

2 3 4 5 6 7

1 .656 9.08 22.2 38.5 63.0 89.8 1 23 . 8

1 .29 0.596 0.386 0.35 0.28 0.22 0. 1 6

values obtainedtofrom Eq.2-5).(21.2-5), indicating that Eq. (21.2-4) gives a good approximation Eq. (21. A good example of the circulatory flow model is the solution obtained by Kronig and Brink (1950). Theythatemployed thelinesHadamard stream function and derived a solution, assuming the stream and the isotherms inside the drop coincide. The expression they derived is 3 � A 2n exp ( 16AnK t ) Em = 1 8' (21. 2-6) r2 n= l where An's areoftheEq.constants andis aAn'littles arecomplicated, the eigenvalues shown inapproximation Table 1. Since theis proposed calculation (21. 2 -6) the following by Calderbank and Korchinski (1956): (21.2-7) 1 exp ( _ _2_.2_5r_�_2_K_t ) The brokenthelinevalues in Fiobtained gure 2 indicates Eq.(21.(21.2-7).2-6), whereas the triangular symbols represent from Eq. Hijikcatatedetbyal.Eq.(1984) a rate when of condensation one predi (21.2found -6) isthatobtained the vapor ofmuch a refrihigghererantthanCFC-the 1 13 is condensed a drop ofbecause CFC-113. According much higher condensation rate isonachieved the stream linestoandthem,the theisotherms inside the drop do not coincide with each other, and the stream lines take di ff erent paths for every ci r culati o n due to peri o di c al deformati o n of the drop whi l e falling down. Nakajima and condensation Tanasawa (1990) obtainedvapor an evenon ahigher condensation rateTheyfor direct contact of CFC-113 falling water droplet. observed that but the condensate the water droplet accumulatesofonCFC-113 the top ofdoesthenotdrop,coverjustthelikewhole a cap.surface A veryofhigh -

�

-

-

600


condensation rate istoattained are always exposed the vapor.because most parts of the surface of the water drop 21.3 CASE WHEN A VAPOR FORMS A DISPERSED PHASE

When a vaporcondensation is injected asoccurs discretewhilebubbles into a pool of subcooled liquid, direct contact the bubbles are rising toward the free surface. Such fonn of direct contact condensation can be classified into three cases, depending on the combination of liquid and vapor and their miscibility. Case vapor and and thethe low-temperature low-temperature liquid liquidarearedifferent the samesubstances substance.and are Case 2:1: The The vapor miscible when thethevapor is condensed.liquid are different substances and Case 3: The vapor and low-temperature immiscible when the vapor is condensed. Further, the case C can be subdivided into two cases, according to Mori (1985): Case 3a: The low-temperature liquid isthewellvapor wettedbubble, with theformicondensate, and the condensate entirely envelops n g a vapor-liquid bubble.does not wet the low-temperature liquid, and a drop (or Case 3b: two-phase The condensate drops) of condensate detaches from the vapor bubble. The following description applies to cases 1 and 3a. 21.3. 1 Condensation of a Single, Pure Vapor Bubble

The vapor to be condensed is assumed to be pure,atcontaining no thenon-condensable gases. Assuming a quasi-steady state established each instant, instantaneous heat transfer from (and the rise motion of) a bubble are expressed in Nusselt number Nu (= ad/Ac) and the drag coefficient as terms of the (21.3-1) Nu = (21. 3 -2) = respectively, where d is related the equivalent sphericalbubble-surface diameter of theareabubble, a is the heat transfer coefficient to the nominal d 2 and to the temperature difference /:). T between the saturation temperature of the condens ing substance and the temperature in the bulk of the continuous phase (i.phase, e., the low temperature liquid), Ac is the thennal conductivity of the continuous a l P�2Rea3

Cd

Cd

b 1 Reb2

7r


Pre Kclve) is theKePrandtl number of the continuous phase,the Rekinematic = Ud/vc) is the Reynolds number, and are the thermal diffusivity and viscosity, respectively, of the continuous phase, U is the velocity of rise of the bubble, and and are dimensionless numerical factors, which are evaluated later.The time t* elapsed before the vapor bubble (or the vapor phase in the two phase bubble) completely condensed, byandEqs.the (21. distance h* for the bubble to rise during this timeis period is approximated 3-3)-(21. 3-7) (Sudhoff et aI. , 1982): Kct* Kt { I } (21.3-3) d? 3E h* Kh } = (21.3-4) d; 1 3E { I where Et = / (2 (21.3-5a) Eh = (1 - ) / (2 (21. 3 -5b) (:�� ) K, = 2(\ (21. 3-6a) (=

(

Vc

a J , az, a3 , bl ,

bz

_

- 2-

t

+

t

_

a3

Kh

=

{ { (2(\

_

_

X ( \ / 3 )+Eh

+ bz)

a3

_

X (Z/ 3)-E,

+ bz)

E'

X )a Ja �' I

_

X a I Ja } - I

)

Pr�

-a,

}

-I

(:�� )

E.

}

(21.3-6b)

0 for Case A (21. 3-7a) for Case C I (21. 3 -7b) = v/ where dj is the initial value of d, Ga = g d; Iv;) is the Galileo number, g is the acceleration due to gravity, Ja = Pecp,epe b. T; /(pvhzg )) is the Jakob numger, and p ,c are the mass density and the specific heat capacity, respectively, of the continuous b. T varies with the rise is the initialin thevaluehydrostatic of b. T (note of the bubblephase, due tob.aT;reduction headthatexerted on the bubble), hlg ismassthe latent heat of vaporization of the condensing substance, and andrespectively, are the densities in the states of saturated vapor and saturated liquid, of theare condensing substance. Valuesnumber of theandnumerical factorsdepending upon andthe dependent on the Reynolds may change, phasesof and the extent to whiandch they(recommended are contaminated (Clift etsubstances aI. , 1978).ofThebothvalues coefficients by Isenberg and Sideman, 1970) are shown in Table 2. However, as mentioned in the table, x

=

X

C

P

(

Pd

(

Pc

Pv

Pd

a I , a2, a3 , bI ,

b2

aI , a2 ,

a3


Table 2 Values of the constants in Eq. (21.3-1) recommended by Isenberg and Sideman (1970) 2

°1

Case A

2 /fi

Case C ,

l /fi

°

Theoretical bases for the values of constants

°3

1 /2

1 /2

Theoretical solution for heat transfer to a sphere in potential flow Semi-empirical solution for evaporation of a drop in air flow

1 /2

1 /3

these values represent only rough approximations. When the continuous phase is water and most of condensation occurs in the range 1 0 3 � Re � 1 04 , one may take b 1 = 2.6 and b2 = 0 (Clift et aI., 1 978; Sudhoff et aI., 1 982).

21.3.2 Effect of Non-Condensable Gases When non-condensable gases are contained in the vapor phase, the bubble (or the gas phase in the two-phase bubble) does not vanish but reaches the liquid surface retaining some amount of the vapor phase. If both the vapor and the non condensable gases are assumed to be ideal gases and no non-condensable gases are assumed to be dissolved in the continuous phase, the fraction of the mass m v of vapor reaching the liquid surface is given by �

mv =

1

Psi Poo Xg (Ps lpoo) 1 - Xg Ps

-

_

(2 1 .3-8)

where Xg is the molar fraction of the non-condensable gases at the initiation of condensation, is the system pressure, and is the saturation pressure of the condensable vapor corresponding to the bulk temperature of the continuous phase. The concentration of the non-condensable gases increases as condensation proceeds and the rate of condensation is reduced. Generally speaking, t* and h* increase remarkably when the molar concentration o f the non-condensable gases, xg, is 0.005 or higher (Jacobs and Major, 1 982).

Poo

21.3.3 Condensation on a Train of Vapor Bubbles When the frequency of vapor injection into the continuous phase is increased, a train of bubbles (or two-phase bubbles) is formed. In such cases, the values of t * and h * become larger than the values given by Eqs. (2 1 .3-3)-(2 1 .3-7). Geometry of the bubble train, the rising velocity, and the temperature distribution in the liquid are all interrelated, and the numerical calculation by iteration is needed to predict t* and h* (Moalem et aI., 1 972).


REFERENCES Calderbank, P. H. and Korchinski, 1. J . 0 . 1 956. Circulation in Liquid Drops (A Heat-Transfer Study). Chem. Eng. Sci. 6:65. Clift, R., Grace, J. R., and Weber, M. E. 1 978. Bubbles, Drops, and Particles. New York: Academic Press, 1 7 1 . Hasson, D., Luss, D., and Peck, R . 1 964. The Theoretical Analysis of Vapor Condensation on Laminar Liquid Jets. Int. 1. Heat Mass Transfer 7:969. Hijikata, K., Mori, Y. , and Kawaguchi, H. 1 984. Direct Contact Condensation of Vapor to Falling Cooled Droplets. Int. 1. Heat Mass Transfer 27: 1 63 1 . Isenberg, J . and Sideman, S . 1 970. Direct Contact Heat Transfer with Change of Phase: Bubble Condensation in Immiscible Liquids. Int. 1. Heat Mass Transfer 1 3:997. Jacobs, H. R. and Major, B. H. 1 982. The Effect of Non-Condensable Gases on Bubble Condensation in an Immiscible Liquid. 1. Heat Mass Transfer 1 04:487. Kreith, F. and Boehm, R. E, eds. 1 988. Direct Contact Heat Transfer. New York: Hemisphere Pub lishing Co. Kronig, R. and Brink, J. C. 1 950. On the Theory of Extraction from Falling Drops. Appl. Sci. Res.

A2: 142.

Moalem, D., Sideman, S., Orell, A., and Hetsroni, G. 1 972. Condensation of Bubble Trains: An Approximate Solution. Progress in Heat and Mass Transfer 6: 1 55. Mori, Y. H. 1 985. Classification o f Configurations o fTwo-Phase VaporlLiquid Bubbles i n an Immiscible Liquid in Relation to Direct-Contact Evaporation and Condensation Processes. Int. 1. Multiphase Flow 1 1 :57 1 . Nakajima, H . and Tanasawa, 1 . 1 990. Direct Contact Condensation of the Vapor of an Immiscible and Insoluble Substance on Falling Liquid Droplets. Heat Transfer Enhancement and Energy Conservation. Ed. Deng, S .-J., p. 335. New York: Hemisphere Publishing Co. Newman, A. B. 1 93 1 . The Drying of Porous Solids: Diffusion Calculations. Trans. AlChE 27:3 1 0. Sideman, S. 1966. Direct Contact Heat Transfer Between Immiscible Liquids. Advances in Chemical Engineering 6:207. New York: Academic Press. Sideman, S. and Moalem-Maron, D. 1 982. Advances in Heat Transfer 1 5 :227. New York: Academic Press. Sudholf, B., Plischke, M., and Weinspach, P.- M. 1 982. Direct Contact Heat Transfer with Change of Phase-Condensation or Evaporation of a Drobble. Ger. Chem . Eng. 5:24. Vermeulen, T. 1 953. Theory for Irreversible and Constant-Pattern Solid Diffusion. I&EC 45: 1 664.

CHAPTER

TWENTY TWO

AUGMENTATION TECHNIQUES IN EXTERNAL CONDENSATION Hiroshi Honda Kyushu University, Kasuga, Fukuoka, Japan

John Rose University ofLondon, London El 4NS, United Kingdom

Heat transfer in film condensation is controlled by the thermal resistance of the condensate film. Thus, augmentation of condensation heat transfer is achieved by either thinning the condensate film or by augmenting turbulent mixing of the condensate, thereby thinning the viscous sublayer. A number of augmentation techniques have been proposed for external condensation. Among them, those techniques that utilize the surface tension effects are commonly employed in many practical applications. Finned tubes, fluted tubes, corrugated tubes, wire wrapped tubes, and tubes sintered with metal particles are included in this category. Other augmentation techniques include attaching promoters such as non-wetting strips, drainage skirts and strips, and electric fields.

22.1 VERTICAL FLUTED TUBE Vertical fluted tubes (often called grooved or finned tubes) with various flute geo metries have been developed for vertical shellside condensers. In many cases, the surface area enhancement of a fluted tube as compared to a smooth tube is less than 2. Figure 1 shows the physical model of film condensation on a vertical fluted tube originally proposed by Gregorig (1954). The tube has round ridges and troughs with small dimensions. Thus, the condensate film has a convex interface at the ridge and a concave interface at the trough. Due to the surface tension effect, the liquid 605


bt2

Figure 1

e direction of gravity

Film condensation on a vertical fluted tube.

pressure at the ridge is higher than that at the trough. As a result, condensate flow directing from the ridge to the trough is induced. This results in a very thin conden sate film (i.e., a very high heat transfer coefficient) near the ridge and a thick con densate film (i.e., a low heat transfer coefficient) near the trough. The condensate gathered at the trough is drained by gravity. When the trough is flat and the conden sate flow rate is small, the thick film region is formed onl y at the corner of the trough. Here we will consider the case shown in Figure 1 . For the thin film region near the ridge, the pressure gradient along the surface of a fluted tube is given by p = a ( 1 / r) where P is the liquid pressure, is the coordinate measured along the surface from the top of the ridge, and r is the radius of curvature of the liquid-vapor interface. Assuming that the liquid film thickness is much smaller than the flute pitch b and the flute depth e and assuming the laminar flow, the basic equation for the condensate film thickness is obtained as follows:

a / as a / as,

s

8

8

(22. 1 - 1 ) where z is the distance measured vertically downward from the top of the tube, Ts is the saturation temperature, and Tw is the wall temperature. The first and second terms at the left hand side of Eq. (22. 1 - 1 ) denote the increments of condensate flow rate in the z- and s-directions, respectively, and the right hand side denotes the condensation mass flux. The boundary conditions at the tube top and at the ridge are

8 30 3 a8/as a 8/as =

=

at z = =

0

0

(22. 1 -2) at

s

=

0

(22. 1 -3)

AUGMENTATION TECHNIQUES IN EXTERNAL CONDENSATION 607

Two other boundary conditions are derived from the condition that the thin conden sate film is connected smoothly with the thick condensate film at s = Sb . In the thick film region, the radius of curvature of the interface rb is almost constant because the condensate pressure in the horizontal cross section is almost constant. Thus,

a8/as = tan {3,

r = rb

at S =

Sb

(22. 1 -4)

where {3 is the angle shown in Figure 1 . Assuming laminar flow, the basic equation for the condensate flow in the thick film region is written as

(22. 1 -5) where w denotes the condensate velocity in the z-direction. The boundary condi tions are

a w/an = 0 at the liquid-vapor interface a w / a S = 0 at S = S W = (p, g8i i J).,e ) [(y/8b) - (y/8b ) 2 / 2] at S = Sb r

(22. 1 -6) (22. 1 -7) (22. 1 -8)

where n denotes the coordinate normal to the interface and y the coordinate mea sured vertically outward from the surface. The variation of Sb along z is obtained from the relation between the heat transfer rate Q in the region of 0 z and the vertical condensate flow rate M at z : '"'"'

(22. 1 -9) Then, the average heat flux based on the projected surface area q and the average heat transfer coefficient ex. are obtained from q =

Q (l)/bl,

ex. = q /(Ts - Tw )

(22. 1 - 1 0)

where l is the tube length. The flutes are gradually flooded with condensate as l increases. This results in a decrease in ex. . Mori et al. ( 1 98 1 a) proposed to shorten the effective tube length by attaching condensate drainage skirts, as shown in Figure 2. Figure 3 shows the numerical solutions of Eqs. (22. 1 - 1 )-( 22. 1 -9) obtained by Mori and Hijikata ( 1984). In Figure 3, the ex. values for three kinds of fluted tubes (i.e., flat-bottomed flute, triangular flute, and wavy flute) with a very small radius of curvature ro at the ridge are plotted as a function of b. For all tubes, ex. increases as b decreases and takes the highest value at b = 0.5'"'"'0.6 mm. Then it decreases with further decreasing b. This is due to the combined effects of the surface area increase and the decrease in the cross-sectional area of the drainage channel. The highest value of ex. is obtained by the flat bottomed flute with a relatively large drainage


drainage skirt

0

Figure 2

Vertical fluted tube with drainage

skirts.

20

A.!

flat-bottomed flute

N

1 7.5 triangular flute

15

.§ � � �

N

tS

�

1 2.5

waVY flute

10

R- I 1 3

N

,

Ts-Tw = 1 0K

7.5

e = 0. 6 1 mm

.

�

1 = 50 mm 5

ro = O.OI mm Nusselt

2.5 0

/ 0

1 .0 b

Figure 3

1 .5 mm

Variation of average heat transfer coefficient with flute pitch.

AUGMENT ATJON TECHNIQUES IN EXTERNAL CONDENSATION 609

channel. This tube shows the a value about 1 0 times as large as the prediction of the Nusselt ( 1 9 1 6) equation for a vertical surface:

NUl = 0.943 G ? o25

(22. 1 - 1 1 )

where NUl = ai/A, and G, = Plghl� l3 /AI V, ( T.\O Tw) . On the basis of the approximate solution in which r = rw was assumed in Eq. (22. 1 - 1 ), where rw is the radius of curvature of the flute surface, Adamek ( 1 98 1 ) and Kedzierski and Webb ( 1 990) proposed fin profiles that gave a high heat transfer performance. Honda and Fujii ( 1 984) developed a semi-empirical expression for the average heat transfer coeffi c ient. Adamek and Webb ( 1 990b) developed a theoretical model of film condensation in the drainage channel. -

22.2 HORIZONTAL FINNED TUBE Horizontal finned tubes are commonly used to enhance shell-side condensation of refrigerants and other organic fluids with relatively low values of surface ten sion. The fin shapes of commercially available tubes are divided into two cate gories: two-dimensional fins and recently developed three-dimensional fins. In many cases, the surface area enhancement of a finned tube ranges from 2.5 to 5. A wide range of experimental data has been accumulated for both two-dimensional and three-dimensional fins. A number of theoretical models have been proposed for the two-dimensional fin tubes. A comprehensive review of relevant literature is given by Marto ( 1 988). Figure 4 shows examples of the fin geometry. Tubes A and B have two dimensional fins, while tubes C to F have three-dimensional fins. It is generally

tube A

tube B

tube C

tube D

tube E

tube F

Figure 4

Example of fin geometry.


Figure 5

Film condensation on a horizontal two-dimentional fin tube.

believed that the three-dimensional fin is superior to the two-dimensional fin because the former has more comers than the latter. However, experimental results (Honda et aI., 1 99 1 ; Webb and Murawski, 1 990) have shown that the three dimen sional fin tubes are subject to a much higher effect of condensate inundation, and the heat transfer coefficient decreases more rapidly than the two-dimensional fin tubes. This is due to the fact that the two-dimensional fins with relatively large values of fin height and fin spacing prevent the falling condensate from spreading axially, and the inter-fin space acts as a good drainage channel. Figure 5 shows the cross-section of a horizontal two-dimensional fin tube at the mid-point between adjacent fins on which film condensation occurs. The condensate profile in the fin cross-section is basically the same as that for a vertical fluted tube shown in Figure 4. The condensate generated on the fin surface is driven by the combined effects of gravity and surface tension toward the fin root. Then, the condensate is drained circumferentially by gravity. Thus, a thin condensate film is formed on the fin surface, and a relatively thick condensate film is formed on the fin root tube surface. At the lower part of the tube, the surface tension acts to retain condensate between fins. Thus the inter-fin space at the angular portion of ¢! :::: ¢ :::: JT is almost completely flooded with condensate. For a trapezoidal fin tube with fin height e, fin pitch b, fin spacing at fin tip c, fin half-tip angle e, and tube diameter at fin tip d, the flooding angle ¢! is given by Honda et aI., 1 983:

¢! = cos - l (X -

1)

for O :::: X :::: 2

(22.2- 1 a)

AUGMENTATION lECHNIQUES IN EXTERNAL CONDENSATION 611

and ¢! = 0 where X =

4a sin e

�cos e ) for c( l sm e p[ gdc

for 2

400 kg/(m2 s), the forced convective condensation dominates the heat transfer characteristics. The effect of gravity can be neglected. For the condensation in this region, the correlations proposed by Cavallini and Zecchin ( 1 974) and Shah ( 1 979) are commonly used in industry. Cavallini and Zecchin ( 1 974) presented a dimensionless correlation for forced convective condensation of pure saturated vapor inside tubes based on a simple hydrodynamic model of the annular flow pattern and on the momentum and heat transfer analogy. Their analysis was limited to the ranges of density ratio PL / Pv from 1 0 to 2000, viscosity ratio J1- v / J1-L from 0.0 1 to 1 , Reynolds number from 5,000 to 500,000, Prandtl number from 0.8 to 20, vapor quality from 0. 1 to 0.9, thermal group cpd Tv - Tw ) / h1g from 0.0 1 to 0.2, and Froude number from 1 5 to 4,000. From the regression analysis of the data, they found that the N usselt number


mainly depends on the Reynolds number, the Prandtl number, and quality group 1 + x ( ( PL I Pv ) 0 . 5 - 1 ) . Other parameters, such as density ratio, viscosity ratio, thermal group, and the Froude number, do not influence remarkably within the ranges of parameters investigated. For 460 data points of CFC 1 1 3, CFC 1 2, and HCFC22 from 8 different sources, the analytical and experimental results agree well with their following correlation: (23 .3 - 1 2) By modifying the Dittus-Boelter equation for condensation in smooth tubes, Shah ( 1 979) proposed a very simple correlation based on 474 experimental data from many sources, covering water, R 1 I , R 1 2, R I B, methanol, ethanol, benzene, toluene, and trichloroethylene in horizontal, vertical, and inclined tubes of dia meters ranging from 7 to 40 mm.

Nu = 0 . 023 ReLo. 8 ProLA

[

(1

3 . 8x O . 76 ( 1 - X ) 0 .04 - x ) 0 . 8 + -----0 38 (Psatl Pc ) .

]

(23 .3- 1 3)

B ased on his recommendation, the Shah correlation is applicable in the range of reduced pressure from 0.002 to 0.44, saturation temperature from 2 1 to 3 1 0°C, vapor velocity from 3 to 300 mis, vapor quality from 0 to 1, mass flux from 10.8 to 1 599 kg/(m2 s), all liquid Reynolds number over 3 5 0 for tube and over 3000 for annulus, liquid Prandtl number over 0.5, and without the limitation for heat flux. From his analysis, the mean deviation for the 474 data points was only 1 5 .4%. Recently, Moser et al . ( 1 998) proved the reliability of the Shah correlation using 1 1 97 data points for six kinds of refrigerants from 1 8 sources. They found that the deviation of the Shah correlation is only 1 4.37% for all data points. Besides, Akers et al. ( 1 959), Boyko and Kruzhilin ( 1 967), Azer et al. ( 1 972), and Moser et al. ( 1 998) also developed correlations for the condensation of pure refrigerants in horizontal smooth tube. These correlations are mainly available for the refrigerant and can predict the condensation heat transfer characteristics very well in many cases. In the mass flux range from 1 00 to 400 kg/(m2 s), the forced convective conden sation and the gravity controlled convective condensation are of same importance. In the case of low mass flux and low vapor quality, the gravity controlled convective condensation dominates all the characteristics. For the condensation in this region, Fujii et al. ( 1 980) proposed an empirical correlation concerning both effects. From the same consideration, Haraguchi et al. ( 1 994b) proposed a correlation for the refrigerants using the asymptotic model. This model offers a smooth transition between the two different limiting phenomena: forced convective condensation and gravity controlled convective condensation. The result based on this model is applicable in the entire range between these two limiting phenomena. Haraguchi et al. tested three kinds of refrigerants, HCFC22, HCFC I 23, and HFC 1 34a, in a

HEAT TRANSFER AND PRESSURE DROP IN INTERNAL FLOW CONDENSATION 627

ad ( = NUF AL

horizontal smooth tube with inner diameter of 8.4 mm and developed the following correlation using their own experimental data:

Nu == where

2 + NU2 ) 1 / 2 B

(

Ga pr NUB = 0.725 H (� ) --L PhL H (�)

= � + ( l 0 [ ( l - � ) O . I - 1]

+

1 .7

x

)

(23 .3- 1 4)

(23 .3- 1 5) t

10-4 Re} v1 ( l - v1)

(23.3- 1 6)

(23 .3- 1 7)

G ( l - x)d ReLo = --- ILL

(23 .3- 1 8)

Gd Re = ILL

(23 .3- 1 9)

(23 .3-20)

This correlation is applicable for the refrigerant condensation in a horizontal smooth tube in the range of mass flux from 90 to 400 kg/(m2 s), vapor quality from 0. 1 to 0.9, heat flux from 3 to 33, liquid Prandtl number from 2.5 to 4.5, Reynolds number for liquid-only flow R eLo from 200 to 20,000, liquid Reynolds number Re from 3000 to 30,000, and the parameter Ga PrL/ PhL from 4.8 x 109 to 9.5 x 1 0 1 0 .

23.4 CONDENSATION HEAT TRANSFER OF BINARY MIXTURES During the condensation of zeotropic binary vapor mixture in tubes, the compo sitions as well as the temperatures in the vapor and liquid phases change along the length of the tube. The mass fraction of more volatile component in the vapor core increases, and the temperature of the bulk vapor mixture decreases along the flow direction. Due to the heat and mass transfer resistance, there is a state of non-equilibrium between the phases. The phase equilibrium is only established at


P = Const

u

o

Figure 2

y

[kglkg]

Phase equilibrium diagram.

the vapor-liquid interface, as indicated in the diagram of Figure 2. For the addi tional mass transfer resistance in the vapor and liquid phases, the condensation heat transfer coefficient of vapor mixture would be lower than that of pure vapor. But the heat transfer characteristic in the liquid film can be considered the same as that for the condensation of pure vapor. Usually, the heat transfer resistance can be measured from the temperature distribution in a cross-section of tube, which would be the same as that in the pure vapor case. But the mass fraction distribution in a cross-section is hardly ever measured, especially for in-tube condensation. This resistance is difficult to estimate. One of the most important aspects in the predic tion of heat transfer in condensation of binary mixtures is therefore to clarify the mass fraction distribution and mass transfer characteristic during the condensation process. The pioneering work to predict simultaneous heat and mass transfer charac teristics of binary vapor mixture is well established as the film theory, which is summarized by Bird et al. ( 1 960). Combining this theory and the Nusselt the ory, von Es and Heertjes ( 1 956) analyzed their experimental data on the con densation of a binary vapor mixture (benzene/toluene) flowing downward in a vertical tube. They found that the local mass transfer coefficient was in propor tion to R e� 8 ScV 3 and the local vapor-liquid interface temperature lay just below the bubble-point line. von Es and Heertjes ( 1 962) also followed the upward con densation of a bina�' vapor mixture. After their studies, the main research target moved to the mUlti-component system, and the film theory has been applied to


the design of many types of condensers for binary systems for a long time in the chemical engineering field. In the past ten years, binary vapor mixtures have received attention again to find alternative refrigerants in heat pump/refrigeration systems, and many researchers have engaged in experimental and theoretical studies on the condensation of alter native refrigerant mixtures in tubes. Zhang et al. ( 1 995) proposed a non-equilibrium model of condensation in a horizontal smooth tube. They tested the condensation heat transfer characteristics of HFC32/ 1 34a and HFC32/ 1 25/ 1 34a mixtures and developed a predictive method using their model. Their predicted and experimental results on heat transfer co efficient agreed well for both mixtures. But the mass fraction distribution and the mass transfer characteristic were not clarified in their study. Koyama et al. ( 1 998) developed a non-equilibrium model for condensation of binary refrigerant mixtures inside horizontal smooth tubes. In their model, it is assumed that the phase equilibrium is only established at the vapor-liquid interface, while the bulk vapor and the bulk liquid are in non-equilibrium. They considered the effect of liquid subcooling on condensation heat transfer and pressure drop but neglected the effect of vapor superheat. They derived the governing equations for heat and mass transfer characteristics of refrigerant mixture in two-phase region as follows:

Momentum Balance of Refrigerant

dp = - ( 4Wvi2 n ) 2 d [ nd dz dz

x2 � 2 pv

(1 +

(1

x )2 � ) 2 PL

-

-

The void fraction � and the frictional pressure change Eqs. (23.3-20) and (23 .2-7), respectively.

1 dPf dz dpf/dZ +

(23.4- 1 )

are calculated using

Heat Balance of Refrigerant (23.4-2) The liquid film heat transfer coefficient C¥L is calculated using Eq. (23 .3- 1 4).

Mass Balance of More Volatile Component in Vapor Core

.

ml

=-

d nd dz

WV in

--

- (XYlb ) = -

WV in Yli dx - - {3V ( Yli - Ylb )

nd dz

(23.4-3)


The vapor mass transfer coefficient f3v is calculated using the correlation proposed by Koyama et al . ( 1 998) as follows:

vd Shv = f3 Pv D

=

O.023 v1�Re�8 ScU3

(23.4-4)

Mass Balance of More Volatile Component in Liquid Film

ml

WV i n d = -- - { ( 1 - X ) X l b }

rrd dz

From the assumption

(23.4-5):

=

f3L -+ 00 ,

-

WV in X l i

rrd

dx dz

+

f3dxli - Xlb )

(23.4-5)

the following relation is reduced from Eq.

Xlb = X li

(23.4-6)

Relation Between Vapor Quality and Mass Fraction

(23.4-7) where Y l b in is the bulk mass fraction of more volatile component at the refrigerant inlet. The local values of vapor quality and the thermodynamic states of refrigerant bulk vapor, vapor-liquid interface, bulk liquid, and wall heat flux were obtained by solving Eqs. (23.4- 1 ) to (23.4-7), starting with the conditions of refrigerant vapor at the inlet of the condenser and the wall temperature distribution along the tube. The detailed calculation procedure was described by Koyama et al. ( 1 998). sing their method, Koyama et al. obtained the local heat transfer and pressure drop characteristics, as well as the local mass fraction distribution and the mass transfer characteristics during the condensation process. Figures 3, 4 and 5 show some of their results. It is found in Figure 4 that the vapor diffusion mass flux of more volatile component at the vapor-liquid interface (m · Y li - m 1 ) decreases in the refrigerant flow direction, while the liquid mass flux of that (m I - m . X l i ) increases in the refrigerant flow direction. This reveals that the mass transfer rate is controlled by the diffusion in the vapor side at the beginning point of condensation, and the diffusion in the liquid side increases gradually as the condensation proceeds. Figure 5 shows that the values of Y l b , Yli, and x li ( = X l b) increase in the downstream section. It is also shown that � ( = m 1 / m) coincides with X I b at the beginning point of condensation and approaches to Y lb i n downstream region. This trend o f � corresponds to the mass transfer characteristics shown in Figure 4.

U

HEAT TRANSFER A N D PRESSURE DROP IN INTERNAL FLOW CONDENSAnON 631

1 00

�� (nfS

YvbO

80

=

0.348 kg/kg

Equilibrium Quality X Exp. value Cal. value

-

�

El" values Tr • Tw Cal. values TVb -

--- T

. . . . . TLb _.-

Twi

40 20 o

o

1

2

z

3

[m]

Figure 3 Distributions of tempratures,

4 heat

5 flux

and

vapor

quality

along

tube

axis

(HFC 1 34a!HCFC 1 23).

For the condensation of binary mixtures inside a microfin tube, Koyama et al. ( 1 999) extended the same model as for the smooth tube to calculate the local values of vapor quality, the thermodynamic states of refrigerant bulk vapor, vapor-liquid interface and bulk liquid, and wall heat flux. In this case, the correlations for heat transfer coefficient in liquid film and pressure drop are changed to those for microfin tubes proposed by Yu and Koyama ( 1 998) and Haraguchi et al. ( 1 993). Condensation of binary mixtures in enhanced tubes has been experimentally studied by Kedzierski and Kim ( 1 998) for twisted tape inserts for single-component refrigerants and their zeotropic and azeotropic mixtures. Kedzierski and Goncalves ( 1 999) present local convective condensation measurements for R41 OA (R321R 1 25 , 50/50% mass) and pure refrigerants i n a micro-fin tube. Both heat transfer and pressure drop measurements are provided. The heat transfer degradation associ ated with R4 1 0A was shown to be relatively small and believed to be mostly due to nonlinear property effects.


0.3

YVbO

Po Gr

=

=

=

0.348 kg/kg 0.499 MPa 200.4 kg/(m �s) �

--

m - - mYvi • . . . . rnA _ . - mYLi

.�

0. 1 o

Figure 4

......

-

-

-

-

-

-

. . . . . . .. . .. --- : �:.-.-.-.-.-.-.-.-

0.4 0.6 1 - X [-]

0.2

o

0. 8

1

Distribution of condensate rate along tube axis (HFC 1 34a1HCFC 1 23).

1

= 0.348 [kgtkg] Po 0.499 [MPa] Gr 200.4 [kg/em 2s)]

YvbO

0. 8

=

=

--

Yvb

- - - - Yvi · · · · · � (= mA;;n)

0.2 o

Figure 5

o

0.2

0.4

I -x

0. 6

[-]

0.8

1

Distribution of mass fraction along tube axis (HFC 1 34a1HCFC 1 23).


23.5 CONDENSATION HEAT TRANSFER OF MULTI-COMPONENT MIXTURES Multicomponent mixtures are encountered in many chemical and process indus tries. The search for alternative refrigerants has resulted in a ternary refrigerant mix ture R407C, composed of 23wt%HFC32, 25wt%HFC 1 25 and 52wt%HFC 1 34a as a possible candidate for replacing R22, which has been widely used in air conditioning and refrigeration systems until now. In the case of multi-component mixture condensation, the treatment of the mass transfer is very complicated due to the coupling effect between components. Schrodt ( 1 973) applied the film theory to a ternary system (methanol/water/air) condensing in a vertical tube. In his theory, the ternary system was treated as the combination of two pseudo-binary systems. Toor ( 1 964) developed the linearized method to analyze the mUlti-component mass transfer program. His method treats the multi-component mass transfer equations as the combination of equivalent bi nary mass transfer equations by using the orthogonal matrix transformation, based on the assumption that the multi-component diffusivity matrix and the other phys ical properties are constant. In the treatment of the multi-component diffusivity matrix, Krishna and Standart ( 1 976) developed a more rigorous method to solve the multi-component mass transfer problem. Based on the film theory, Webb and Sardesai ( 1 98 1 ) compared these three multi-component mass transfer models with the experimental data on the condensation of two different ternary vapor mixtures in a vertical smooth tube. Their comparisons showed that these models were appli cable to predict the total condensation rate, but they recommended Toor's linearized model and the Krishna-Standart model. Koyama and Lee ( 1 998) extended the non-equilibrium model for the conden sation of binary refrigerant mixture in a horizontal smooth tube (Koyama et aI., 1 998) to multi-component refrigerant mixtures. The assumptions used in this sec tion are the same as those in Section 23 .4. The governing equations for heat and mass transfer characteristics in the two-phase region are derived as:

Momentum Balance of Refrigerant

(23 .5- 1 ) The void fraction � and the frictional pressure change dpf / dZ are calculated using Eqs. (23 .3-20) and (23 .2-7), respectively.


Heat Balance of Refrigerant qw

in d = - WV - {xhvb + ( 1 Ted dz --

-

x)hLb } = adTi - Tw)

(23 .5-2)

The liquid film heat transfer coefficient aL is calculated using Eq. (23 .3- 14), which was developed from the condensation of pure refrigerants in horizontal smooth tube, as mentioned in Section 23 .3-2.

Mass Balance of Component k(k = 1 , 2, . . . , n - 1) in Vapor Core

. WV in d ( WV in Yki dx (23 .5-3) ) )=.B ( mk = - -Ted dz XYkb Ted dz - kV Yki - Ykb The vapor mass transfer coefficient .Bkv is calculated using the following equation: .B (23 .5-4) Shy = kV� = 0.023 ft � Re� 8 Sc�V PV Dkk where Rev

=

Gxd

-- ,

J-Lv

S CkV =

J-Lv * ' Pv Dkk

-

(23 .5-5,6,7)

(23 .5-8)

Mass Balance of Component k(k = 1 , 2, . . . , n - 1) in Liquid Film (23.5-9) From the assumption equation:

.BkL -7 00 , the following relation is reduced from the above (23.5- 1 0)

Relation Between Vapor Quality and Mass Fraction

x = ( Ykb in - Xkb )/(Ykb - Xkb ) where Ykbi n is the bulk mass fraction at the refrigerant inlet.

(23.5- 1 1 )


The details of the calculation method is given by Koyama and Lee ( 1 998). From their calculation of the condensation of ternary zeotropic refrigerant mixtures composed of HFC32IHFC I 25IHFC 1 34a, including R407C, the local values of vapor quality, thermodynamic states at bulk vapor, vapor-liquid interface and bulk liquid, mass flux, etc., were obtained for a constant wall temperature and constant wall heat flux conditions. The above prediction method can be easily extended to the condensation in microfin tube by using the correlations of heat transfer coefficient in liquid film and pressure drop for the corresponding tube.

23.6 FUTURE RESEARCH NEEDS During the past 30 years, the research on condensation heat transfer inside tubes has made considerable progress. Many investigators have carried out experimental and theoretical studies in this field. The condensation heat transfer and pressure drop characteristics in the smooth tube can be predicted reasonably well from equip ment designers' viewpoint. Some of the areas where further research is warranted are listed here: 1. 2. 3. 4.

The effect of superheated vapor on condensation heat transfer in tube. Heat transfer characteristics in low mass flux region. Mass transfer characteristics in the liquid phase. Augmentation technology for in-tube condensation and appropriate predictive methods. 5. Development of simple and reliable prediction correlation for vapor mixtures in general design practice.

REFERENCES Akers, W. w., Deans, H. A., and Crosser, O. K. 1 959. Condensation Heat Transfer Within Horizontal Tubes. Chern. Eng. Prog. Symp. Ser. 29: 1 7 1 - 1 76. Azer, N. Z., Abis, L. v. , and Soliman, H. M. 1 972. Local Heat Transfer Coefficients During Annular Flow Condensation. ASHRAE Trans. 78(2): 1 35- 143. Bird, R. B., Stewart, W. E., and Lightfoot, E. N. 1 960. Transport Phenomena. John Wiley & Sons, Inc. Borchman, J. 1 967. Heat Transfer of High Velocity Vapors Condensing in Annuli. ASHRAE Trans. 73. Boyko, L. D. and Kruzhilin, G. N. 1 967. Heat Transfer and Hydraulic Resistance During Condensation of Steam in a Horizontal Tube and in a Bundle of Tubes. Int. J. Heat Mass Transfer l O(3):36 1 373. Cavallini, A. and Zecchin, R. 1 974. A Dimensionless Correlation for Heat Transfer i n Forced Convection Condensation. Proc. 5th Int. Heat Transfer Conference 3 :309-3 1 3 .


Fauske, H. K. 1 96 1 . Critical Two-Phase Steam Water Flows. Proc. of Heat Transfer & Fluid Mech. Inst. , Stanford, CA: Stanford Univ. Press, pp. 79-89. Fujii, T, Honda, H., Nagata, S . , Fujii, M., and Nozu, S . 1 976. Condensation of R I I in Horizontal Tube ( 1 st Report: Flow Pattern and Pressure Drop). Trans. JSME (B) 42(363) :3541 -3550 (in Japanese). Fujii, T, Honda, H., and Nozu, S. 1 980. Condensation of Fluorocarbon Refrigerants Inside a Horizontal Tube (Proposals of Semi-Empirical Expressions for the Local Heat Transfer Coefficient and Interfacial Friction Factor). Refrigeration 55(627):3-20 (in Japanese). Goodykoontz, J. H. and Brown, W. F. 1 967 . Local Heat Transfer and Pressure Distributions for Freon- I 1 3 Condensing in Downflow in a Vertical Tube. NASA TN D-3952. Haraguchi, H., Koyama, S., Esaki, J., and Fujii, T 1 993. Condensation Heat Transfer of Refrigerants HFC I 34a, HCFC I 23 , and HCFC22 in a Horizontal Smooth Tube and a Horizontal Mi crotin Tube. Proc. 30th National Heat Transfer Symp. of Japan, Yokohama, pp. 343-345 (in Japanese). Haraguchi, H., Koyama, S . , and Fuj ii, T 1 994a. Condensation of Refrigerants HCFC22, HFC 1 34a, and HCFC 1 23 in a Horizontal Smooth Tube ( 1 st Report, Proposal of Empirical Expressions for the Local Frictional Pressure Drop). Trans. JSME (B) 60(574):239-244 (in Japanese). Haraguchi, H., Koyama, S., and Fujii, T 1 994b. Condensation of Refrigerants HCFC22, HFC I 34a, and HCFC 1 23 in a Horizontal Smooth Tube (2nd Report, Proposal of Empirical Expressions for the Local Heat Transfer Coefficient). Trans. JSME (B) 60(574):245-252 (in Japanese). Hayashi, T 1 998. Enhanced Condensation Heat Transfer of Refrigerant HFC 1 34a in Horizontal Tubes. Master Thesis, Kyushu University, 1 998 (in Japanese). Kedzierski, M. A. and Goncalves, J. M. 1 999. Horizontal Convective Condensation of Alternative Refrigerants Within a Micro-Fin Tube. Journal of Enhanced Heat Transfer 6( 1 ). Kedzierski, M. A. and Kim, M. S. 1 998. Convective Boiling and Condensation Heat Transfer with a Twisted-Tape Insert for R 1 2, R22, R I 52a, R 1 34a, R290, R321R 1 34a, R321R 1 5 2a, R2901R 1 34a, and R 1 34a1600a. Thermal Science and Engineering 6( 1 ): 1 1 3- 1 22. Koyama, S. and Lee, S . M. 1 998. A Prediction Model for Condensation of Ternary Refrigerant Mixtures Inside a Horizontal Smooth Tube. Proceedings of l Ist Int. Heat Transfer Conference 6:427-432, Aug. 23-28, Korea. Koyama, S . , Yu, J . , and Ishibashi, A. 1 998. Condensation of Binary Refrigerant Mixtures in a Horizontal Smooth Tube. Thermal Science & Engineering 6( 1 ): 1 23- 1 29 . Koyama, S . , Yu, J . , and Ishibashi, A. 1 999. Heat a n d Mass Transfer o f Binary Refrigerant Mix tures Condensation in a Horizontal Microtin Tube. Proceedings of 5th ASMEIJSME Thermal Engineering Joint Conference, March 1 5- 1 9, San Diego. Krishna, R. and Standart, G. L. 1 976. A Multicomponent Film Model Incorporating a General Matrix Method of Solution to the Maxwell-Stefan Equations. AlChE J. 2(2) :3 83-389. Moser, K. w., Webb, R. L., and Na, B . 1 998. A New Equivalent Reynolds Number Model for Condensation in Smooth Tubes. ASME Trans., J. Heat Transfer 1 20(2):4 1 0-4 1 7 . Schrodt, J . T 1 973. Simultaneous Heat and Mass Transfer from Multicomponent Condensing Vapor-Gas Systems. AlChE J. 1 9(4):753-759. Shah, M . M . 1 979. A General Correlation for Heat Transfer During Film Condensation Inside Pipes. Int. J. Heat and Mass Transfer 22:547-556. Smith, S. L. 1 97 1 . Void Fraction in Two-Phase Flow: A Correlation Based Upon an Equal Velocity Heated Model. Heat and Fluid Flow 1 ( 1 ) :22-39. Soliman, M . , Shuster, J. R., and Berenson, P. J. 1 968. A General Heat Transfer Correlation for Annular Flow Condensation. ASME Trans., J. Heat Transfer, 90(2):267-276. Toor, H . L. 1 964. Solution of the Linearized Equation of Multicomponent Mass Transfer: II. Matrix Methods. AlChE J. 1 0(4):460-465 .


Traviss, D. P., Rohsenow, W. M . , and Baron, A.B. 1 973. Forced Convection Condensation Inside Tubes: A Heat Transfer Equation for Condenser Design. ASHRAE Trans. 79( 1 ) : 1 57- 1 65 . von Es, J. P. and Heertjes, P. M . 1 956. On the Condensation o f a Vapor o f a Binary Mixture i n a Vertical Tube. Chemical Engineering Science 5 :2 1 7-225. von Es, J . P. and Heertjes, P. M. 1 962. The Condensation of a Vapor of a B inary Mixture. British Chemical Engineering 8:5 80-5 86. Webb, D. R. and Sardesai, R. G. 1 98 1 . Verification of Multicomponent Mass Transfer Models for Condensation Inside a Vertical Tube. Int. J. Multiphase Flow 7(3):507-520. Yu, J. and Koyama, S. 1 998. Condensation Heat Transfer of Pure Refrigerants in Microfin Tubes. Proceedings of the 1998 Int. Refrig. Con! at Purdue, pp. 325-330. Zhang, L., Hihara, E., Saito, T., Oh, J. T., and Ijima, H. 1 995. A Theoretical Model for Predicting the Boiling and Condensation Heat Transfer of Ternary Mixture Inside a Horizontal Smooth Tube. Proceeding of Thermal Engineering Conference, 95, JSME, pp. 94-96 (in Japanese). Zivi, S. M. 1 964. Estimation of Steady-State Steam Void-Fraction by Means of the Principle of Minimum Entropy Production. ASME Trans. , J. Heat Transfer 86:247-252.

CHAPTER

TWENTY FOUR AUGMENTATION TECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 24. 1

Moo Hwan Kim

Pohang University of Science and TechnoLogy, Pohang, Korea

24.2

790- 784

Vijayaraghavan Srinivasan

Praxair, Inc., Tonawanda,

24.2 DeLphi Harrison ThermaL Systems, GM, Lockport,

NY 14150-7891.

USA

Ramesh K. Shah NY 14094-1896.

USA

24.1 AUGMENTATION TECHNIQUES IN INTERNAL FLOW GEOMETRIES This section considers augmentation techniques for internal forced flows. Aug mentation techniques used in external condensation are generally applicable to condensation inside tubes, but stratification of the flow by gravity needs to be con sidered at low vapor velocities in horizontal tubes. During the last several decades, many augmentation techniques inside tubes have been tested and established. Re cently, the use of non-azeotropic refrigerant mixtures (NARMs) has resulted in lower heat transfer coefficients compared to when pure refrigerants are used. Spe cial attention has been paid to improving the condensation heat transfer coefficient of NARMs in the refrigeration industry. These techniques include the use of inserts and internal fins. A special type of tube called micro-fin tube was developed in the late 1 970's and is currently widely used in industry. The tubing cost and pres sure drop must be considered when selecting enhanced condensation heat transfer techniques.

24.1.1 Fundamentals Condensation heat transfer augmentation in enhanced geometries occurs due to three reasons. First, mixing becomes more efficient due to recirculation and vortex 639


shedding when inserts and fins are present. Second, the wetted perimeter increases greatly while the cross sectional area of the flow passage decreases. In addition to that, the curvature of the enhanced geometries causes the surface tension force to become comparable to the inertia force (the 'Gregorig effect' ). For micro-fins, the surface tension force can be dominant due to the small radius of curvature. Third, spirally grooved micro-fin tubes also make the annular film more uniform along the tube circumference. The basic concept behind enhanced condensation heat transfer is to reduce the condensate film thickness. When fins are attached to the inner walls of the tube or when inserts are installed, the flow path becomes longer than for a plain tube, resulting in an increase in heat transfer. The main difference between inserts and internal fins is that inserts can affect the entire flow field, while internal fins can only affect the flow near the wall. Inserts work better when the heat transfer resistance is more or less evenly distributed throughout the flow cross section. On the other hand, internal fins are more effective if the heat transfer resistance is concentrated near the walls. For condensate films, internal fins may be more effective if the film is thin and turbulent, while inserts may be more effective if the condensate film is thick and laminar. However, the pressure drop penalties associated with the enhancement in heat transfer must be considered.

24.1 .2 Inserts Royal and Bergles ( 1 976) tested the effects of tube inclination, twisted tape in serts, and internally finned tubes on condensation heat transfer with low-pressure steam. Their results showed that the internally finned tubes were most effective in augmenting heat transfer. Royal and Bergles ( 1 978) also investigated the aug mentation of horizontal in-tube condensation by means of twisted tape inserts and internally finned tubes. The twisted tape inserts were found to increase the average heat transfer coefficients by as much as 30% above smooth tube values. Internally finned tubes increased the average heat transfer coefficients by 1 50% when compared to the smooth tube. Luu and Bergles ( 1 979) performed an experiment on augmentation of in tube condensation of R 1 1 3 with twisted tape inserts and internally finned tubes. They found that twisted tapes increased the heat transfer coefficient and pressure drop by about 30% and 250% over smooth tubes, respectively, while internally finned tubes increased the heat transfer coefficient by 1 20% over smooth tubes with a modest increase in pressure drop. Azer and Said ( 1 974) investigated the heat transfer coefficient during condensation in a smooth tube, a tube with inner fins, and a tube with static mixer inserts. Their results showed that heat transfer coefficient increased above that for the smooth tube by 1 30 - 1 50% for the inner

AUGMENTA TION lECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 641

finned tube and about 1 20% for the tube with mixer inserts. Lin et al. ( 1 980) tested static in-line mixer (called Kenix mixer) inserts using R - 1 1 3. The maximum ratio of the heat transfer coefficients with mixers to those without mixers was 1 85%, but the pressure drop penalty of about 400% was felt to be too high. Generally speaking, the film thickness is small in tubes during condensation, so inserts may not be appropriate to maximize film condensation heat transfer. Inserts may be more economical than inner finned tubes, however, and may be used to an advantage when thick laminar films are present.

24.1.3 Internal Fins Several types of internal fins have been used to increase the surface area and pro mote heat transfer in fluid flows. Vrable et al. ( 1 974) studied in-tube condensation of R- 1 2 in horizontal internally finned tubes. The condensation heat transfer coef ficient based on the smooth bore area increased by about 200%. They proposed the following correlation for condensation heat transfer coefficients inside internally finned tubes: (24. 1 - 1 ) where (24. 1 -2) and Pr = reduced pressure, P / Pc r . They developed this correlation from the smooth tube correlation of Cavallini and Zecchin ( 1 974) by replacing the conventional hydraulic diameter Dh by twice its value. Reisbig ( 1 974) obtained similar re sults using R- 1 2 containing some oil. As mentioned in Section 24. 1 .2, Royal and Bergles ( 1 976, 1 978) and Luu and Bergles ( 1 979) investigated the condensation heat transfer and pressure drop in internally finned tubes as well as tubes with twisted tape inserts. Their results indicated that the internally finned tubes were better than twisted tape inserts for condensation heat transfer and pressure drop. A review of their results is given by Webb ( 1 994). Royal and Bergles ( 1 978) developed the following correlation by modify ing the smooth tube correlation of Akers et al. ( 1 959) with the correction factor (b2 / WDj ), where b is fin height and W is the channel width between internal fins. They developed a correlation for condensation heat transfer coefficients from their steam condensation data inside horizontal tubes with straight and spiral fins, as follows: (24. 1 -3)


where

�

Reeq

= riz

eq D h

--

1] /

and

m eq

= riz [ ( 1

-

x ) + x ( Pt! pg ) 0 . 5 ]

(24. 1 -4)

Luu and Bergles ( 1 979) suggested a correlation from their data of R- l 1 3 condensed inside internally finned tubes, as follows:

a h where

= 0.024Reo . 8 rO . 8 [ [ ( p / Pm ) ?�5 +

P

riz Dh

Re = -1] /

and

( P/Pm )�;;t ] / 2] (b 2 / WDi ) - 0.22

P / Pm =

1 + x ( pt ! Pg

-

1)

(24. 1 -5 )

(24. 1 -6)

This correlation was obtained by modifying the smooth tube correlation of Boyko and Kruzhilin ( 1 967) by the correction factor (b 2 / WDi ) -0. 22 . Said and Azer ( 1 983), Kaushik and Azer ( 1 988, 1 989, 1 990), and Sur and Azer ( 1 99 1 a) suggested correlations and analytical models for condensation heat transfer and pressure drop in internally finned tubes. Hitachi has developed a 'micro-fin tube' that is widely used in the air-conditio ning industry. This is discussed below.

24.1.4 Micro-Fin 'lUbes Since the original micro-fin tube (Hitachi Thennofin ™ Tube) was first developed by Fujii et al. ( 1 977), many improved versions have been developed. Table 1 shows the specifications of the micro-fin tubes developed in Hitachi Cable, Ltd. Figure 1 shows the cross sectional shapes of micro-fin tubes developed in Hitachi Cable, Ltd. Figure 2 summarizes the development of the micro-fin tubes in Hitachi Cable, Ltd. As shown in these figures, the micro-fin shows better per formance with increasing fin height and helix angle for both evaporation and condensation, with one exception-that Thennofin-HEX-C does not enhance the performance for condensation against Thennofin-EX. Webb ( 1 994) summarized the chronological improvements in the Hitachi Thermofin™ Tube, citing Yasuda et al. ( 1 990). Hitachi researchers varied the fin height, spiral angle (helix angle), and fin apex angle of the original Thennofin ™ Tube (enhancement in heat transfer coefficient of 1 80% over plain tubes), resulting in an enhancement in heat transfer coefficient of 3 1 0% for the HEX-C during condensation for a given mass velocity. Because there are so many parameters that affect the heat transfer and pressure drop performance (e.g., spiral angle [helix angle], fin height, apex angle, mass velocity, and mass quality), few experiments and analysis have been systematically performed.

AUGMENTATION 1ECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 643

Table 1 Specifications of the micro-fin tubes developed in Hitachi Cable, Ltd. (from Yasuda et aI., 1990) Thermo fin

Thermo fin-EX

9.52

9.52

9.52

9.52

7.94

9.52

7.94

0.30

0.37

0.36

0.36

0.36

0.39

0.36

Plain tube Outer diameter D o [mm] Mean wall thickness tavg [mm] Weight per length [gmlmm] Bottom wall thicknes tw [mm] Groove depth Hr [mm] Number of grooves [n] Helix angle f3 [deg.] Apex angle [deg.] Increasing ratio of inner surface area

78

95

93

Thermofin -HEX

93

ThermofinHEX-C

77

99

77

0.30

0.30

0.30

0.30

0.30

0.30

0. 1 5

0.20

0.20

0.20

0.25

0.20

65

60

60

50

60

50

25

18

18

18

30

30

90

53

40

40

40

40

1 .28

1 .5 1

1 .60

1 .6 1

1 .75

1 .6 1

One of the most interesting parameters is the fin helix angle. Shinohara and Tobe ( 1 985) showed that the condensation heat transfer coefficient for the Thennofin-EX gradually increases as the helix angle increases from 7 to 30. Khanpara ( 1 986) tested a smooth tube and nine micro-fin tubes, resulting in en hancements in condensation heat transfer coefficients of 1 50 -3 1 0% for R - 1 1 3 and 1 60 - 1 75 % for R-22 over the smooth tube at the mass velocities of 1 50-600 kg/(m2 s). Shinohara et al. ( 1 987) showed that the R-22 condensation coefficient in a 9.5 mm O.D. tube increases 20% as the helix angle is increased from 10 to 35. Schlager et al. ( 1 990a) tested three micro-fin tubes having helix angles of 1 5 , 1 8, and 25, but the tubes were not in same fin heights. Table 2 shows the tested tubes ( 1 2.7 mm tubes) of their study. Figure 3 shows the condensation heat transfer results for smooth and micro-fin tubes. By their results, the highest condensation coefficient was obtained by the tube having the highest fin height and 1 8° helix angle, i.e., Micro-fin 1 . In a recent study, Yang et al. ( 1 995) developed a new-type of three dimensional inner microfin tubes. For R - 1 1 at inlet pressures of Pin = 1 47-265 kPa and conden sate mass flux ranging from 33 to 1 53 kg/m2 s, the average heat transfer coefficient was seen to increase by up to 1 40% over that of two dimensional micro-fin tube. Figures 4 and 5 are the schematics of a 2-D tube and a 3-D tube.


THERMOFIN-EX

Conventional THERMO FIN

THERMOFIN - HEX

*

Lengthwise cross-section

THERMOFIN-HEX-C

*

End cross-section A - A

- A

p'

1 •• _ _

A

Figure 1

Cross sectional shapes of micro-fin tubes developed in Hitachi Cable, Ltd. (From Yasuda et aI ., 1 990; reproduced with permission).

Figure 6 shows one of their results. In spite of the somewhat questionable decreasing trends of the average heat transfer coefficients against the mass flux, it is worthwhile to note the enhancement of the 3-D tube from the 2-D tube is quite high. It is speculated that the three most important factors that affect the enhance ment of condensation in micro-fin tubes are the area increase, secondary flow effects (enhancing flow mixing) by fins, and surface tension effects from fin cur vatures. When the condensate film is thin or the mass quality is large, the surface tension forces will dominate, and the fin shape becomes very important. When the

AUGMENTATION TECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 645

Table 2 Dimension of the tubes tested in Schlager et al. (1990a)

Do (mm) Dj,max (mm) wall thickness (mm) fin height (mm) number of fins helix angle (deg)

T .g Q)

.....

Smooth

Micro-fin 1

Micro-fin 2

Micro-fin 3

1 2.7 1 0.9 0.90

1 2.7 1 1 .7 0.50

1 2.7 1 1 .7 0.50

1 2.7 1 1 .7 0.50

0.30

0.20

0. 1 5

60

70

60

18

15

25

�

3. 5

C

'C\j 0: '-"

"HEX' . • .. . • • . . . . . . . . .

"HEX-C"

3,0

"EX·

�

'THERMOFIN' . . . . . . . -

Condensation

'

Figure 2

70

'75

• • • • • • • • • •

Evaporation

'8

' 90

'80

Year

5

Development of the micro-fin tubes in Hitachi Cable, Ltd. (From Yasuda et al. , 1 990; reproduced with permission).


6 r-------�

.. - Mpa > "". .' , . .- . '. . . .. . , . . .. ' . . . ' . .. ' ' . ...'.,..''- . . ..,

Condensati on P=1.5

5

.

1 .6

.

4

E

�

..ci

.

.

, .. . . , . .. , ,. . . , . . . . , . " ,. ,

3

. . , , ,

. ," ,

"

.

..

..

,

.

"

" .

.

,..

.

.

.

' ,

,

.

,.

.

"

.

.

.

2

Sl'TDOth t !.be ------

_ _ _ • • • • • • • • • •.•.

o

Figure 3

100

• •

200

3 00

G, kg/m'

• •

Micro-fin tube 1 Mia-o -fin tube 2 Mia-o -fin tube 3

400

50 0

• s

Condensation heat transfer results for smooth and micro-fin tubes a s a function o f mass flux. (From Schlager et ai., 1 990a; reproduced with permission).

condensate film is thick enough to cover the micro-fins, surface tension effects dis appear and the secondary flow effects of the film disturbance by the fins dominate. Systematic parametric studies are needed to obtain a fundamental understanding of the mechanisms by which heat transfer is enhanced during condensation in micro-fin tubes.

24.1.5 Oil Effects on Convective Condensation Oil flows through the components of an air conditioning system, including the evaporator and condenser. Pate ( 1 988) reports that the oil fraction ranges from


d Figure 4 Schematic diagram of a 2-D tube used by Yang et al. ( 1 995).

k

0.5% 2.0% (1985) 5%. 10% (1979): aD·' = [0 023 o.8 Al [0.8 8

5%

about to when an oil separator is present, and up to or higher without an oil separator. Tichy et al . investigated the effect of oil on condensation heat transfer and reported that the heat transfer coeffi c ient was reduced relative to oil free systems by for oil concentrations of and by for oil concentrations of Pressure drop was observed to increase only slightly with oil concentration. They proposed the following empirical correlation based on the Shah's correlation

.

Re'

x

2%

PrI° . 4 ]

23%

[(1 - x)O.8 + 3 8xO.76 (1O -8 X)O.04 ] . -----::-:,...:-:: Pr .

+ (Re/,ref/Re, ) 1 .99 ] e -5.0w

3

(24.1-7)

where Rei = Reynolds number based on liquid, Rei. ref = reference Reynolds number = and w = mass fraction of oil. The correlation results are shown in Figure Schlager et al. surveyed oil effects on condensation heat transfer and found that oil always degrades the heat transfer. Schlager et al. investigated

3650,

7.

(1987)

Figure 5

(1988)

Schematic diagram of a 3-D tube used by Yang et al. ( 1 995).


a:> 2' E

�

..

I I

I

15

10

SMOOTH TUBE

5

o

50

Figure 6

70

90

1 10

1 30

1 50

Mass flux ( kg 1 m· 8 )

Comparison of the average heat transfer coefficients of the 3-D tube, 2-D tube, and the smooth tube used by Yang et al. ( 1 995).

the condensation of R-22 and a 1 50SUS naphthenic oil mixture and found a 1 3 % decrease in heat transfer with a 5% lubricant content. Schlager et al. ( l 990b) also developed design equations for the oil effects on condensation heat transfer in smooth, microfin, and low fin tubes with the concentration variation of up to 5 wt.% of 1 50-SUS and 300-SUS oil. The mass flux ranged from 1 25 to 400 kg/m2 s. The condensing temperature was 4 1 °C. The mass quality ranged from 1 5 to 85% . They presented the following correlation for the enhancement factor for a microfin tube over a smooth tube performance: (24. 1 -8)

1m

where the subscripts Ir means liquid phase of pure refrigerant, means liquid phase of the refrigerant/oil mixture, and (J) means the mass fraction of oil. EF is defined as the ratio of heat transfer when refrigerant-oil mixture was used to heat transfer when pure refrigerant was used.

AUGMENTATION lECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 649 1 000 �------.

12 Saturation Pressure

R

9 00

-

800

�

Z

6 8 1 KPa ( 98 .8psia)

Wo -----------•

::s

=

-.-. -.-.-.-.-..

700

o 01 .05

.

s...

OJ ..c S

600

::s

Z ;::! OJ CI) CI) ::s

Z

500

400

300

200

R e IR = 5 000

1 00

.1

.2

.3

. 4-

.5

Quality , X

.6

.7

.8

.9

Figure 7 Correlation results for various Reynolds numbers and oil fractions (From Tichy et aI ., 1 985; reproduced with permission).

Sur and Azer ( 1 99 1 b) showed that the presence of 1 .2%, 2.8%, and 4.0% 1 50-SUS naphthenic oil reduced heat transfer coefficients by 7%, 1 2%, and 1 6%, respectively, compared with pure R- 1 1 3 . Eckels and Pate ( 1 99 1 ) reported a rough 1 0% degradation in heat transfer for R- 1 2 and a 5% 1 50SUS naphthenic oil mixture, but no significant effect of 1 65SUS PAG oil on heat transfer coefficient of R- 1 34a. Shao and Granryd ( 1 995) report that the heat transfer coefficient in the case of oil contained refrigerant is strongly dependent on the definition of the saturation


.-....

�

�E

�

4000 a b, based on pure refrigerant saturation temperature

3500

3000

Pure HFC I 3 4 a:

-......,

c:: Cl)

:9

� �

Q) 0 U lr-o Q)

p=O. 7834 MPa

�5

2500

Cr = 1 83kg/

2000 a a.

1 5 00

� CI)

c:: ro

)... ......,

......,

ro Q)

0

based

: 1000

OD

on mi x t ure saturation t e m p .

HF C I 3 4 a + oil ( J . 6 '11 bulk ) : V • +

:

p = 0 . 7 7 8MPa 500

�

0

6

-

--

-

Cr= 1 8 7 kg/m

0 0

0.2

0.4

2

s

0.6

0.8

1 .0

Liquid fracti o n

Figure 8 Typical profiles of local heat transfer coefficient along the condenser for pure R- 1 33a and oillR- 1 34a mixture. (From Shao and Granryd, 1 995 .)

temperature. The typical profiles of local heat transfer coefficients along the con denser is shown in Figure 8. From the result, the condensation heat transfer coefficients based on the mix ture saturation temperature of oil-refrigerant mixture are reduced by an average 1 0- 20% depending on oil concentrations, compared with pure refrigerant. Unlike evaporation heat transfer, where the effects of oil are unclear, it is generally accepted that heat transfer degrades with increasing oil fraction during condensation. The available data for condenser design are still not sufficient, how ever. Also, the physical mechanisms by which heat transfer degradation occurs are still not understood. In the case of evaporation heat transfer, Thome ( 1 995) explains for the trends of oil effect on heat transfer during flow boiling according to five categories: ( 1 ) thennodynamic effects on heat transfer coefficient (2) local physical properties of refrigerant/oil mixture (3) influence of oil on the nucleate boiling process in flow boiling (4) influence of oil on convective contribution to flow boiling (5) mass transfer effects on the evaporation process.


It is suggested that for condensation, the following effects of oil should be checked: ( 1 ) influence of oil on convective contribution to condensation: the change in viscosity of refrigerant/oil mixtures by addition of oil results in a decrease in Reynolds number and an increase of Prandtl number. (2) influence of oil on Gregorig effects: when the surface tension of the oil is much different from that of the refrigerant, the surface tension effects on condensation should be included. (3) the definition of saturation temperature: pure refrigerant saturation tempera ture or refrigerant/oil mixture saturation temperature.

24.1 .6 Convective Condensation of Non-Azeotropic Refrigerant Mixtures As a non-azeotropic refrigerant mixture condenses, the less volatile (the compo nent with the higher condensing temperature) condenses first, then forms a con centration boundary layer both in the vapor and the liquid region adjacent to the condensing interface. This boundary layer can be regarded as thermal resistance to condensation heat transfer. The condensing temperature of the mixture becomes lower as condensation proceeds, and this change of temperature is referred to as temperature g liding . It can be advantageous for the performance of counterflow heat exchangers in refrigeration and heat pump cycle when the gliding temperature is well matched to the heat source or sink temperature difference between the inlet and the outlet. Chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) have been rapidly replaced by alternative refrigerants due to concerns regarding ozone depletion and global warming . Some refrigerant mixtures have been suggested as potential candidates for replacing R-22. Relatively little research regarding phase change heat transfer of refrigerant mixtures has been performed, but this is changing. Bokhanovskiy ( 1 980) performed a condensation experiment on a non azeotropic mixture of R-22 and R - 1 2 and obtained the average heat transfer coef ficient as a function of heat flux. Four regions of heat transfer characteristics were observed, and the mass transfer effects were especially prominent in the low heat flux region. Tandon et al. ( 1 985, 1 986) proposed the following condensation heat transfer correlation for an R-22/R- 1 2 mixture: O . 365 cxm L (24. 1 -9) 2. 82Pr 1 /3 � Reog . 1 46

Al

=

( cp D. T )

This correlation, however, does not take into account the composition of the binary mixture.


Mochizuki et al. ( 1 988) measured the temperatures of the local wall, cooling water, and vapor along the tube axis, and obtained the local condensation heat transfer coefficients of a R- 1 1 3/R- I l refrigerant mixture flowing in the inner tube and cooling water in the annulus space in a counterflow direction. The following empirical correlation was suggested: Ctm X

Al

= (�) D -0.4 0.38

-

H - O . 6 R O . 8 PrO . 8 Re o . 8

(24. 1 - 1 0)

I

where H is c p /j. T / h 1g and R is ( pt ! pg ) 1 /2 ( ry t ! ryg ) 1 /2 . Koyama et al. ( 1 988, 1 99 1 ) carried out the experiment for the condensation of the R-22/R- 1 1 4 mixture in both the compressor and the pump systems. The local heat transfer coefficients were obtained by using a condensing apparatus consisting of 1 2 test sections. Also, they modeled mass transfer on the vapor side and could accurately predict the mixture condensation heat transfer in smooth and micro-fin tubes. Torikoshi and Ebisu ( 1 993) confirmed that the heat transfer degrades when using a non-azeotropic mixture of R-32/R- 1 34a (30:70 wt.%), and they also re ported that pressure drop of the binary mixture could be calculated using the ideal mixing rule. Shizuya et al. ( 1 995) investigated the condensation heat transfer of four single components, R-22, R- 1 42b, R- 1 14, and R- 1 23, and their mixtures condensing in 7mm ID smooth and grooved tubes. Figure 9 presents some of their results. They

!. .. . .. .!. .-...... (. . . . . . . -z:.;;..\ . r�·(3

. -'

••

•••

3

.

, :i .. :i

1 Iij

1

- -···

·· -

.8 .8

1.0

(R22)

MoleB22

Ct\

Figure 9

0.8

J

.. �

.a 0.8

0.4

0.2

fraction of R22 (R1 42b)

8142b I 822" 42b

1.0

__

_

0.8

_

..

1

.. .. ..

0.8

t

�

_

8ljj

#

•••• ·

··· ·

•

0.4

(R22) Mole fraction of R22 (R1 42b) Ct\ 822, A1A2D " 8221142b

..· -· .. . - . - .

.. .. .. .. ..

.�

1.0

...

· ·· · · · · . . .

_

i:� � .. .. . /

._

A22 fraction 0.8

(R22) Mole

(I)

·

.

�

...

__

t

·

,.

�

0.8

_

.

.. #.

·

#

·

.#

0.4

of R22 (R1 42b)

Al4@ " 822N4@

Comparison of heat transfer performance in annular flow regime (From Shizuya et al., 1 995 ; reproduced with permission).


found that heat transfer is reduced when using refrigerant mixtures in cases where the boiling temperatures of their components differ greatly. Also, heat transfer appears to improve for refrigerant mixtures by grooving the tube inner surfaces, which also compensates to a considerable degree the performance reduction asso ciated with mixtures. It is difficult to use the limited information in the open literature to design condensers using non-azeotropic mixtures, especially since the mechanisms by which reductions in heat transfer occur are not well understood.

24.2 CONDENSATION IN COMPACT HEAT EXCHANGERS 24.2.1 Introduction In a condenser, the process stream (single component or multicomponent with or without noncondensable gases) is condensed to a liquid with or without desu perheating and/or subcooling. Compact condensers are a special class of heat exchangers that have low hydraulic diameters or a heat transfer surface area den sity greater than 400 m2 /m3 on the condensation or liquid side and/or greater than 700 m2 /m3 on the gas side. The coolant stream in a two-fluid condenser can be a gas, a single-phase liquid, or a vaporizing liquid. In this chapter, only filmwise condensation is considered, since it is difficult to maintain dropwise condensation in industrial applications. Depending on the condenser design, con densation may take place inside or outside a circular tube or inside a noncircular, plain, or enhanced passage. In some cases, matrix type (plate-fin) surface may be employed to enhance condensation. A general classification of condensers is pre sented in Figure 1 0. While tubular condensers have been historically used in power,

Pool

Spray

Packed

and

Column

Tray

Shell-and-Tube

� I

Power Industry

Surface

Feedwater

Condenser

Heater

Exlended Surface

I

I

I

Process Industry

I

I I I I I

E G H J X

I

Total

�

Condensation

Reflux

Figure 10

Classification of condensers.

Knockback

I

I

Tube-fin

Plate-fin

air-roaled

cryogenic

condenser

acndenser

Plate-type

I Plate

I

I Spiral

I

Printed Circuit


process, and refrigeration/air-conditioning industries since early 1 900s, compact extended surface condensers are more commonly used in automotive and aerospace air-conditioning, and cryogenic processing systems are used for air separation. Other applications of compact condensers are: condensation of refrigerant mix tures, condensation of binary and multicomponent mixtures, condensation in new energy conversion systems, micro-gravity condensation in space applications, heat rejection unit in space vehicles, and vent condenser for power plant condensers. Some common constructions of compact condensers are: ( 1 ) individually finned round tubes with smooth or micro-finned surface inside to enhance conden sation heat transfer, (2) continuous flat fins (plain or maybe with waves, louvers, or other enhancements) on round or flat tubes (with or without flow interruptions inside to enhance condensation), (3) plate-fin heat exchangers that use corrugated fins such as perforated, wavy, or serrated fins (offset strip fins) brazed to parti tion sheets, and (4) plate-and-frame heat exchangers with corrugated flow pas sages. Other compact condenser constructions are welded plate (stacked plate), spiral plate, printed circuit, and dimple plate (welded, pressure bonded) heat ex changers. The flow passage geometries in most compact condensers are usually noncircular in cross section or have small diameter circular tubes. The flow ar rangement can be single-pass, conventional multipass, or serpentine. The flow passage orientation for condensing stream can be horizontal, vertical, or inclined. The major advantages of compact condensers are small volume and packaging, low mass, lower total cost (including installation), modular design approach, and the flexibility of selecting flow passage geometry, ultra-high performance, and close temperature approaches. The single most important problem with compact condensers is potential fouling in small size passages for a fouling fluid. The use of compact condensers is increasing in power, process, and other industries due to the above advantages in niche applications where fouling can be managed. In addition to a heat transfer device, compact plate-fin condensers are being used as heat and mass transfer devices, such as in distillation columns (Panchal, 1 993). Generally, multicomponent condensers are mass transfer devices. In this chapter, the following topics are addressed: compact condenser types and their applications, distinctive features of condensation through enhanced non circular passages, modeling methods employed for analyzing condensation through noncircular passages, predictive methods for friction and heat transfer, fin effi ciency calculations, and design methodology.

24.2.2 Compact Condenser Types The configurations of compact condensers depend primarily on the coolants used and specific applications. Table 3 summarizes the coolant media, principal appli cations, and typical configurations of condensers used in such applications.

AUGMENT ATION lECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 655

Table 3 Compact condenser types, applications, and coolants Coolant

Applications

Typical configurations

Single-phase gas

a. Automotive, Aerospace b. Chemical Process Systems

Single-phase liquid

a. Condensation of ammonia vapor in Energy Conversion Systems. b. Chemical /process systems-vapor condensers. c. Industrial refrigeration. Cryogenic heat exchangers

Tube-fin (circular tubes) and plate-fin (flattened enhanced tubes) heat exchanger. Plate-and-frame heat exchanger Plate-fin heat exchanger Printed circuit heat exchanger

Boiling (or vaporizing) liquid

Plate-fin heat exchanger

24.2.2.1 Thbe-fin type. Tube (or plate)-fin type compact condensers that primar

ily use gas as a coolant can be found in automotive and aerospace applications. Sim ilarly, condensation of overhead vapors from distillation columns use gas streams in the process as a coolant. The use of air as a coolant is also found in condensers employed in refrigeration and air-conditioning applications, where compact heat exchangers are used. The low values of the heat transfer coefficient ex normally associated with gas fl o ws necessitates the use of fins on the gas side. Typical designs are shown in Figures 1 1 a and 1 1 b. A circular tube and flat fin construction is used in most non-automotive applications, i.e., copper tubes and aluminum wavy or louvered fins. The flat or continuous fins are also sometimes referred to as plate fins. The tube can be flattened to give a rectangular or oval flow cross section. The tubes can have micro-finned surface inside the tubes to enhance condensation heat transfer (see Figure 1 2). In most condensers, fans are used for flowing air over the fins and tubes. Centrifugal fans are used when the air flows over the condenser through a duct; propeller fans are used with condensers installed outdoors. In a mUltipass condenser, the air and refrigerant flow in such a way as to have overall cross counterflow arrangement for a higher exchanger effectiveness and performance. 24.2.2.2 Plate-and-frame and printed circuit exchangers. Plate-and-frame and the printed circuit compact heat exchangers are used on those applications having liquid as a coolant. However, high pressure drops encountered in two-phase flows in the narrow passages in such exchangers restrict their widespread use. Some of the favorable application areas include: ( 1 ) steam heating of liquids (with condensing steam) such as in the pasteurization of milk, (2) condensing of refrigerants using water as a coolant, and (3) in chemical process systems. A review of the applications of plate-and-frame exchangers for condensing duty is given by Kumar ( 1 983). A

c:I'I U'I c:I'I

( a)

'" Exit Manifold

Downflow Condenser

Round Tubes

Fins

Condensing FICIN L! ...-

I , I

Z-ro.

R£TI,RN TANIC

--q

(b)

.... REFRIGERANT OUT (LIDUID)

I/II.£T/OUTLET TAIII(

'... REFRIGERANT IN (GAS)

Figure 1 1 Compact condensers: (a) round tube and flat fin construction, (b) flat tube and corrugated fin construction (courtesy of Delphi Harrison Thermal Systems).

...-I!..

I

'"

Inlet Manifold

DO CAP

AUGMENTATION lECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 657

(a)

(b)

Figure 12

Enhanced tubes used in compact condensers: (a) plain tube, (b) micro-fin tube, (c) inner-grooved tubes.

(c)

typical compact printed circuit condenser is shown in Figure 1 3 . Printed circuit heat exchangers are used in off-shore platforms where compactness becomes an important consideration. 24.2.2.3 Plate-fin type. These heat exchangers are used primarily as evapora

tors/condensers. Boiling liquid oxygen condenses nitrogen vapor in a main con denser/reboiler of a cryogenic air separation plant. Such a heat exchanger oper ates at very low T (effective temperature difference). Plate-fin heat exchangers (Figure 14) with a variety of fin-types (see Figure 1 5) have been designed for these applications. Condensation takes place vertically downward along the plate-fin passages. The corrugated fin surface in a plate-fin heat exchanger is also referred to as a matrix-type surface.

�

24.2.2.4 Flow geometry and orientations. Compact condensers used in applica tions described above typically have the following flow geometry and orientation for the condensing side passages of the heat exchanger: •

•

Condensation inside a round tube with or without microfins, oriented vertically (downward condensation) or horizontally. Condensation inside a small hydraulic diameter flattened extruded tube with and without microfins (see Figure 1 2).


Figure 13

Printed circuit heat exchanger.

nitrogen vapor

E>it "�rugen liquid (condensate)

Figure 14 boiling.

Plate-fin condenser for a cryogenic application with nitrogen condensing and oxygen


(a)

(b)

(d)

�)

(e)

(f)

Figure 15

(g)

Examples of plate-fin surface. (a) Rectangular (plain); (b) rectangular (offset); (c) offset strip; (d) triangular (perrorated) ; (f) louvred; (g) wavy (Polley, 1 99 1 ) .

•

•

Condensation inside a flow passage formed in-between corrugated plates, as shown in Figure 1 6. The flow is predominantly vertical. Condensation in a plate-fin passage. The fins can be plain, perforated, wavy, or serrated. The flow orientation is predominantly vertical downward. However, in reflux condensers, the vapor flows upward countercurrent to the condensate draining downward.

24.2.3 Important Flow Characteristics of Condensation in Compact Passages The small channel dimensions, low mass flux values, and noncircular geometries influence the flow characteristics in a compact condenser. Table 4 shows the ranges ofthe mass flux G, Reynolds number Re for liquid and vapor phases, Weber number We (a ratio of inertia force to surface tension force) for liquid and vapor phases, Bond number Bo (a ratio of gravity force to surface tension force), and hydraulic diameter of some compact condensers. Data for shell-and-tube heat exchangers

Q\ Q\ =

Cryogenic main condensers (nitrogen) Automotive AlC condensers (R- 1 34a refrigerant) Energy conversion devices! chemical process systems Chemical process systems Powerplant (steam-water)

Plate-fin

Printed circuit Shell-and-tube

Flat tube and corrugated fin Plate-and-frame

Application

Condenser Type

3 .0-1 2.0 3.0-1 2.0 3 .0-85.0

0-450 0-200 0-5000

0-0.60 0 -0.30 0-1 00

0-30,000 0- 1 5,000 0-50,000

0- 1 200 0-600 0-20,000

2-40 2-20 20-500

1 .2-1 .5 1 2.7-25.4

2.0-8.0

1 .0-3.0

0.6-20.0 0-50

0-1 .0

0- 1 2,000

0- 1 000

1 0- 1 20

1 .5-3.0

1 .0-1 0.0 0-25 0-0.75 0-1 2,000

0- 1 000

1 5-50

Dh mm

Bo

Wev Rev

Rei

Range of G kg!m2 s

Wei

Table 4 Flow parameters in compact condensers

AUGMENTA nON TECHNIQUES AND CONDENSATION INSIDE ADVANCED GEOMETRIES 661

Figure 16

Corrugated Surfaces (Plate-and-Frame Heat Exchanger).

are also provided for comparison purpose; the larger G values are for small power plant condensers that typically use shell-and-tube condensers. As seen from Table 3, the hydraulic diameter (or the length scale) of a compact heat exchanger flow passage is roughly an order of magnitude less than that for a circular tube in conventional shell-and-tube heat exchangers. The smaller length scale increases the L j Dh ratio and therefore could increase the resultant pressure drop. In order to meet the specified pressure drop constraint, the mass flux G is reduced in these passages. Thus, it is a common practice to design compact condensers with low mass flux values. In Table 3, the Weber numbers for the liquid and the vapor phase are shown for compact heat exchangers. The low liquid Weber numbers in the operating regime of the condenser suggest that the surface tension force dominates at low qualities. At high qualities and in the vapor phase, the inertia force tends to dominate over the surface tension force. The B ond number at certain operating regimes of the heat exchanger shows that both the gravity and surface tension force have equal influence. Flow patterns in small noncircular passages were studied by Wambsganss et al. ( 1 993), Tran et al. ( 1 993), and Robertson et al. ( 1 986). It is important to note that flow patterns are strongly influenced by the small channel dimensions that


/

Partition Sheet

Trough

Local Thinning of Film

Figure 17

Condensation in the trough region of the fi n.

precludes a straightforward extension of the large diameter circular tube studies. The studies also indicate that the flow conditions at which transition occurs for compact heat exchanger passages are significantly different from those for round tube passages. Surface tension plays an important role by enhancing the two-phase flow heat transfer characteristics in compact condensers having small hydraulic diameter flow passages. It also produces the Gregorig effect, where the condensate is drawn into comers of the flow passages, thereby bringing down the film thickness (just outside of the trough region, as shown in Figure 1 7) and increasing the heat transfer coefficient. Hirasawa et al. ( 1 980) have also shown that the surface tension force through the condensate in the trough region induces a strong suction flow near the side surface of a fin, tending to produce a region of very thin liquid film as shown in Figure 1 7, thus enhancing the condensation heat transfer coefficient. Hirasawa et al. ( 1 980) have shown that the Nusselt number was increased by a factor of 3 due to the surface tension effect for one finned surface that they had investigated ! The study was done on a single finned passage. Additional features, such as non uniformity in the core flow and liquid shedding in the core flow at fin interruptions, are important for condensation in interrupted passages, such as in offset strip fin (serrated fin) channels. Nonuniformity of the core flow is found to alter the entrainment and deposition, which in turn changes the distribution of the liquid inventory over the perimeter of the passage. Carey ( 1 993) has covered these aspects in more detail for adiabatic two-phase flow. In addition to the above features, the analysis of two-phase flow inside compact condensers using finned passages becomes more complex due to the primary and secondary surfaces being at different temperatures and having significant variations in heat transfer coefficients.

24.2.4 Modeling and Predictive Methods for Heat Transfer Methods used for predicting heat transfer rates are slightly different for the two common flow orientations (horizontal flow and cocurrent vertical downflow) in


compact condenser flow passages due to the dominance of one of the condensa tion mechanisms (i.e., gravity, shear, and surface tension). Most of the predictive methods for complex flow geometries are based on experimental correlations and are described below.

24.2.4.1 Condensation heat transfer in vertical downflow

Circularplain tubes Two principal approaches have been used to predict internal convective condensation. In the first approach, an analytical model for the con densate film is derived that is based on the idealization that the flow pattern is annular (shear dominated flow). In this method, the Nusselt number is correlated for convective transport across the liquid film in terms of the Martinelli parameter X, the Reynolds number Rei for the liquid flowing alone, and the liquid Prandtl number Prl . (24.2- 1 ) In this approach, the shear at the liquid-vapor interface is linked to the transport of heat across the liquid film. A number of investigators have followed this approach for predicting the condensation heat transfer coefficient for a circular tube. For example, the correlations proposed by Akers et al. ( 1 959), Soliman et al. ( 1 968), and Traviss et al. ( 1 973) adopt the above approach. In the absence of interfacial shear, the condensation heat transfer coefficient is predicted by considering falling film flow (gravity dominated flow). The method proposed by Chun and Seban ( 1 97 1 ) analyzes the falling film using the Nusselt theory. However, this method allows for the enhancement caused by waviness of the liquid film, and this is correlated as a function of the Reynolds number Rei . For vertical downflow in a circular tube, Chen et al. ( 1 987) have combined the results of previous works of other investigators, such as Chun and Seban ( 1 97 1 ) and Soliman et al. ( 1 968), and proposed a comprehensive correlation for predicting condensation heat transfer coefficient. Their correlation considers the presence as well as the absence of vapor shear (i.e., annular and falling film flow). Chen et al. assumed the following functional form for the Nusselt number: (24.2-2) with (24.2-3)

=== =


where

Nuzs zero-shear Nusselt number Nus Nusselt number predicted with a vapor shear model NUlw Nusselt number for wavy-laminar flow NUt Nusselt number for turbulent film flow

=

=

U sing the data of several previous workers, Chen et ai. found the values of 2 and nz 6 as the best fit for all the data compiled in their study. In another approach, simple empirical relations have been developed that more directly correlate condensation heat transfer experimental data. For example, the work done by Boyko and Kruzhilin ( 1 966) on condensation follows this approach. The condensation heat transfer coefficient is given by nl

a eon

where flow.

Q'\o

=

a 10

[I - (;: - I ) r x

(24.2-4)

is the single-phase heat transfer coefficient for the total flow as liquid

Circular tubes with microfins Condensation inside circular tubes is enhanced by a number of techniques that include twisted-tape inserts, longitudinal fins, wire-coil inserts, and microfins. However, in compact condensers, microfin tubes are predominantly used. For example: microfin tubes are used in automotive condensers and residential air-conditioners. Condensing coefficients in microfin tubes are two to three times higher than that of a plain tube at the same mass velocity (Yasuda et aI., 1 990). In a recent study, Chamra et al. ( 1 996) have compared condensation heat transfer for different micro-groove geometries. Experimental friction and heat transfer data for R-22 are presented. However, it is surprising to note that no theoretical models or correlations have been proposed for the microfin tube in spite of its widespread use. Rectangularlflat tubes (with or without microfins) Flat tubes of Figure 1 2a are exclusively used in automotive condensers. The tubes contain webs between the flat surfaces that provide structural rigidity and pressure containment. These tubes are usually made of aluminum and are extruded. It is possible to extrude a smooth or a micro-finned inside surface. Condensation performance of flat extruded aluminum tubes was studied by Yang and Webb ( 1 995a). The tubes used in their study contained three internal webs that separate the flow into four separate channels. Two internal geometries were tested: one had a plain inner surface and the other had microfins (0.2 mm high). The tubes outside dimensions used in their study were 1 6 mm (width) x


3 mm (high) x 0.5 mm (wall thickness). R- 1 2 fluid was used in their tests. The data of Yang and Webb show that the condensation heat transfer coefficient increases with the heat flux to the 0.20 power. Even though this is contrary to the results from the Nusselt theory, where the heat transfer coefficient a is proportional to q- l /3 , the authors show that this increase in a is caused by the combined effects of vapor shear and momentum. The authors also show that surface tension drainage forces provide an additional enhancement at x > 0.5, which is additive to the effect produced by vapor shear. Even though no correlation is proposed, the following observation will be of help to designers: for the plain surface, at low mass velocities (up to 400 kg/m2 s), the Akers ( 1 959) correlation for circular plain tubes, which takes into account the effect of vapor shear, predicts the condensation coefficient well. However, at high mass velocities, the experimental data are 1 0 - 20% lower than those predicted by the Akers correlation. In the Akers method, an equivalent all liquid mass velocity (Geq ) is introduced that permits data for condensation within a tube to be correlated by the usual single phase heat transfer correlation. The liquid mass velocity Geq is defined such that it has the same wall shear stress as the actual two-phase flow. By this method, the effective mass flux Geq is obtained as G 'q

=

G

{

(1

_ x ) + x (;:) } I /2

(24.2-5)

where x is the vapor fraction and G is the actual mass flux. The condensation heat transfer coefficient is predicted using the single-phase heat transfer correlation (such as Gneilinski or Dittus-Boelter) using Geq as the mass velocity.

Plate-and-frame heat exchanger One of the most important configurations in a compact condenser is the plate type heat exchanger. This configuration is useful in situations where the coolant is a liquid and the pressure drop on the condensing side is not a severe constraint. However, there are only a few studies in the literature for condensation in plate heat exchangers. Experimental data for condensation in plate heat exchangers were obtained by Tovazhnyanski and Kapustenko ( 1 984) for steam, Uehara and Nakaoka ( 1 988) for ammonia, and Kumar ( 1 983) and Chopard et al. ( 1 992) for refrigerants. The experimental data of these investigators covered a wide range of the Reynolds number and different corrugation angles ranging from 30 to 60° . The authors correlated the data using the same approach that is used for circular tubes. However, in plate heat exchangers, the transition from a gravity-controlled regime to a shear-dominated regime is assumed to occur at Rev between 350 and 800 (Thonon and Chopard, 1 996) . The actual value depends on the corrugation angle.


In the gravity-controlled regime, Thonon and Chopard ( 1 996) recommend the use of the Nusselt equation for a plain surface. Therefore, in order to account for the effect of plate corrugations, the Nusselt value is multiplied with an enhancement factor that is obtained from single-phase data (a ratio of heat transfer coefficients in the corrugated surface to a plain surface at the same Re). In the shear-dominated regime, the Akers ( 1 959) method is recommended by Kumar ( 1 983) for calculat ing the condensation heat transfer coefficient. The effective liquid mass flux, as described previously in Eq. (24.2-5) for microfin flat tubes, is determined first. The single-phase flow relationships (for the corrugation geometry) are then used to cal culate the heat transfer coefficient corresponding to the effective liquid mass flux.

Plate-fin heat exchanger Plate-fin heat exchangers for condensing duties are ex tensively used in the cryogenic air separation plants. Gopin et al. ( 1 976) performed tests on plain fins with very low vapor inlet velocities ("-'0.3 m/s). Their data were in agreement with Nusselt's prediction for vertical plates. At higher vapor velo cities, vapor shear does affect the magnitude of the heat transfer coefficient. In a recent study, Clarke ( 1 992), investigated the heat transfer characteristics of down flow condensation in a serrated fin (offset strip fin) under conditions similar to those used in industrial cryogenic process plants. Clarke found that an approach similar to that of Chen et al. ( 1 987) (see Eq. [24.2-2]) described above gave better results for serrated fin channels. He recommends the use of Boyko and Kruzhilin ( 1 967) correlation for calculating Nus (with vapor shear effects) and the Chun and Seban ( 1 97 1 ) correlation for determing Nuzs (zero-shear Nusselt number). 24.2.4.2 Condensation heat transfer in horizontal flow. The prediction of heat

transfer for internal convective condensation in horizontal tubes is different at low and high vapor velocities. Gravity force dominates at low vapor velocities. Therefore, the condensate that forms on the upper portion of the tube drains and collects at the bottom region of the tube. This stratified flow is generally observed at low heat flux and vapor velocities. In view of flow stratification, heat transfer at the bottom of the tube where the liquid drains and collects is much less when compared to the upper portion, where a thin film exists. Correlations in the literature for predicting heat transfer for the stratified flow conditions in a circular tube may be summarized in terms of the following equation:

a

a=

0.725 K l

[ ki (PI - Pg)ghlVK2 ] 1 /4 I-LI Dh (ts - tw)

(24.2-6)

where is the local heat transfer coefficient for the condensate film. The parameter K I is a correction factor that takes into account the collection of liquid in the bottom of the tube where no condensation occurs. Various authors provide the value of Kl as a function of e where e is the angle subtended by the liquid surface formed by condensate retention. Chato ( 1 962), for example, suggested a value of e = 1 200 and

=


=

K\ becomes 0.78. Chato found that his data for R- 1 1 3 agreed well with the above equation with K\ 0.78. The parameter K2 takes into account liquid subcooling. K2 is detennined using the method described by Chato. A typical value of K2 is 1 . 1 5 . However, note that by setting K, and K2 1 , Eq. (24.2-6) becomes Nusselt's original equation for condensation outside of a horizontal tube. At higher vapor velocities, the liquid film gets evenly distributed around the tube and the annular flow gets established. When the condensate film is evenly distributed around the tube, many of the heat transfer models for vertical tubes may be used for horizontal tubes, such as the correlation of Akers ( 1 959) or Traviss et al. ( 1 973). Equations discussed above are summarized in Table 5.

24.2.5 Two-Phase Pressure Gradient in Condensing Flow The need for accurate methods for predicting pressure drop in compact condensers cannot be overemphasized because it affects both the fluid pumping power and condensation heat transfer (since the saturation temperature changes with the sat uration pressure). In heat exchangers with high thermal effectivenesses, such as the cryogenic main condenser/reboiler, the pressure drop across the condensing channel of a brazed aluminum plate-fin heat exchanger has an important effect on the economics of the process. Typically, in a main condenser of an air separation plant, a one psi (7 kPa) increase in pressure drop can result in a 0.5 to 1 .0% increase in the compressor power, which places a significant cost penalty. For example, typical compressor power required is 1 04 kW for a 1 000 tons/day oxygen plant; hence, a 7 kPa ( 1 psi) increase in pressure drop can have a significant impact on annual electric cost. The pressure loss in a main condenser can range from 3.5 to 1 4.0 kPa (0.5 to 2 psi). Two-phase pressure drop in condensing flow is calculated by integrating the local pressure gradient over the length of the flow channel. The local pressure gradient in tum is computed in tenns of the frictional, momentum, and gravitational components. For flow in a horizontal passage, the gravitational component of pressure drop is absent. The frictional component is traditionally evaluated from empirical correlations. Many of these correlations, such as that of Chisholm ( 1 983), are derived from adiabatic two-phase flow or boiling data. Several investigators have also calculated pressure losses based on flow patterns. A circular tube. For two-phase flow through circular tubes, a one-dimensional momentum balance yields the following expression for the local two-phase pres sure gradient (see Soliman et al. [ 1 968]):

�� = (p, y

+ (1

- y ) p,)g +

:z [ { � ;��) ;,: } 1 + :: ';' 2

l

,(

+

(24.2-7)

Horizontal

HorizontaVvertical downftow

2. Circular plain tube inside

3. Rectangular flat tubes

All

Annular

Stratified

All

Vertical downflow

1 . Circular plain

tube inside

Flow regime

Flow orientation

Type of compact HX surface =

=

[

( :t2)

[ (NUIW )n2 + (NUt ) n2 J n2

=

O.822 «Rel ) - O. 22 )

=

g

=

3.8 x

g

3 2

1 /3

for ReI S 300

( �t )

1 /3

Determine the heat transfer coefficient using Eq. (24.2-4). Alternatively, determine ex from the Dittus-Boelter equation using the mass velocity given by Eq. (24.2-5). Determine ex from the Dittus-Boelter equation using the mass velocity given by Eq. (24.2-5).

For finding Nus , determine the heat transfer coefficient using Eq. (24.2-4). Alternatively, determine ex from the Dittus-Boelter equation using the mass velocity given by Eq. (24.2-5). Use Eq. (24.2-6)

NUl

3

o 1 O- 3 (Re l )0 .4 (Prl ) . 65

and for all other values of ReI

NUl w

=

where NUlw and NUt are given by Chun and Seban ( 1 97 1 ) and n l 2 ; n2 6

Nuzs

Nu

] l /n l (Nuzs ) n l + (Nus) n l

Recommended method

Table 5 Heat Transfer Correlation for Condensation in Compact HX Surfaces

NUt is the Nusselt number of turbulent film.

NUlw is the Nusselt number of laminar, wavy film.

due to vapor shear.

Nus is the Nusselt number

Nusselt number.

Nuzs is the zero-shear

Comments

=" =" �

Gravity controlled Rev < 800

Shear-dominated regime Rev > 800

All

Vertical

Verticallhorizontal

Vertical downflow

5. Corrugated plates

6. Corrugated plates

7. Plate-fin geometry

All

Horizontal/vertical down flow

4. Rectangular flat tubes with microfins

[ ]

Use manufacturer's data. For typical microfin geometries, the enhancement in heat transfer including the surface area enhancement ranges from a factor of 1 .6 to 2.2 over a plain surface. In the absence of manufacturer's data, and for obtaining a rough estimate, determine Ci as for plain rectangular tubes and multiply by a factor of 1 .5 Use Nusselt's equation for condensation on a flat vertical plate and multiply by an enhancement factor that is obtained from manufacturer's single-phase data (a ratio of heat transfer coefficient in the corrugated surface to a plain surface at the same Rev ) Use Eq. (24.2-5) to determine Geq (effective liquid mass flux). Use single-phase relationships (for the corrugation geometry) to calculate the heat transfer coefficient corresponding to the equivalent liquid flux. Nu = NU �IK + Nu�k nl = 6 NUB K = Nusselt number from Boyko-Kruzhilin. Eq. (24.2-4) Nuzs = NusseIt number for Chun-Seban (See #1 above)


The three terms on the right-hand side of the above equation are the gravi tational, momentum I , and frictional components of the local pressure gradient, respectively. The total pressure drop is obtained by integrating the local pressure gradient over the entire length of the tube. Computation of the gravitational and momentum components of Eq. (24.2-7) requires know ledge of the void fraction y . The correlations for y available i n the literature are specific to the flow geometry in question. For round tubes, the void fraction y has generally been correlated with either the Martinelli parameter X (defined as { (dp/dz)I / (dp/dz) v } 1 / 2 , where the subscripts I and v represent the liquid and vapor phases) or the two-phase frictional multiplier OOt;j)oP�o 0 0 ootf'o ex>cP � o�

..,..,

.

.

q

•

•

.. ..etf'

•••

.......--.. .. -.............

..�

X 0 x

%00 00% 0 00 % ,pcPoo o e q o •

0.50

RG 2

=

=

RG 2

=

o

8 � .. .-..00 ..

..

0 • ....

0.1 5

" . o ��·�--��--------��--------�----------�--------��----�

1 .4

1 .4mm

Figure 1 0 Line void fractions R C I i n steady-state steam-water flow at high pressure (Martin, 1 969,

l 1 972). Pressure: 80 bar; heat flux density, 100 W cm -2 ; mass velocity: 220 g cm -2 s - ; rium quality.

Xeg :

equilib

E .s (/) (/) Q)

.:.:.c:(J £

o

o

50

••••••••

-----

Figure 1 1

X-ray generator

1 00 X-ray absorption technique Conductivity probe

Instantaneous liquid height (Solesio et aI ., 1 978).

Time (ms)


probe, which measures the liquid conductance between two electrodes mounted flush with the wall.

25.3.7 Design Parameters of an X-ray Densitometer Due to the constant use of this measuring technique, Fournier and J eandey ( 1 984) developed a computer program simulating the behavior of an X-ray beam from its emission to its detection. This program is able to optimize the major design parameters of an experimental setup by taking into account all possible sources of uncertainty, such as void fluctuations, Compton scattering, beam hardening, system drifts, positioning, and statistical uncertainty. The input data of the pro gram are the thickness and the material of the walls traversed by the beam, the components of the two-phase flow, and the characteristics of the detector (either the absorbing substance with its thickness or a detector with its efficiency). In addition, tables of absorption coefficients have been stored in the program for a wide range of chemical elements likely to be found on the beam path. The program enables the user to choose the best value for the tube high-voltage and to select filters to minimize the uncertainty.

25.4 AREA VOID FRACTION MEASUREMENTS 25.4.1 Photon Attenuation Techniques 25.4.1.1 One-shot technique. In the one-shot technique (Charlt�ty,

1 97 1 ; Gardner et aI., 1 970; Nyer, 1 969), the radiation crossed the whole of the tube cross section containing the two-phase mixture. The beam height must be small compared with the tube diameter in order to restrict the measurement to a given cross section. Calibration with lucite mockups are generally necessary due to the lack of a specific absorption law. A positive appraisal of the one-shot technique was given by Besset (1979). He showed that the accuracy t1R G2 could be better than ±0.05 for area void fractions ranging from 0 to 0.80. 25.4.1 .2 Single beam densitometer. The time-averaged area void fraction R G 2 can be determined from a profile of time-averaged chordal void fraction R G 1 . In a pipe of circular cross section, we have:

RG 2

=

1 2 7T R

-

lY=+ R 2JR 2 y=- R

-

y 2 R G I ( y) dy

(25.4- 1 )

where R i s the pipe radius and y the distance from the axis of the pipe to the photon beam. Note that if the flow is axisymmetric, the local void fraction CX G is a solution

INSTRUMENTATION IN BOILING AND CONDENSATION STUDIES 693 25 C u r i e C s 1 3 7

sou rce

tubes

Figure 12 Three-beam gamma densitometer (BaneIjee e t aI ., 1 978).

-

= R G (y) V R2 -

of the following Abel integral equation:

lR Y

ra G (r) dr Jr 2 y 2

---;:=.===========

where r is the radial coordinate.

-

1

. /

Y

2

(25.4-2)

25.4.1.3 Multibeam densitometers. The multibeam gamma densitometer has been used for many years by laboratories involved in water reactor safety studies. Figure 12 shows a schematic of the three-beam densitometer used by Banerjee and Jolly ( 1 978). A 25 Ci cesium 1 37 source provides three y -ray beams that are directed through the pipe in the same cross section plane. The beams are attenu ated according to the line void fractions, and the measurement of the attenuation of each beam can be used to determine the area void fraction R G2 . This technique requires a model connecting the area void fraction, the flow pattern, and the three measured line void fractions. Several laboratories have used multibeam densitometers for measuring area void fractions in transient two-phase flow. However, the size of the standard scintil lators has limited the number of beams to around ten. The development of medical X-ray tomography equipment removed this constraint and void fractions measure ments were made with up to 3 1 beams (Jeandey, 1 982). The equipment used by Jeandey consisted of a 1 40 kV electrostatic generator supplying the high-voltage of an X-ray tube. A flat, fan-shaped beam of X-rays covered the whole cross sec tion of the test conduit and impinged onto a series of 3 1 anti -Compton collimators.


Each collimator was followed by a xenon ionization chamber. The test conduit was a vertical cylindrical tube, 20 mm in diameter, followed by a divergent with a diameter of 60 mm. The conduit walls traversed by the X-ray beam were stainless steel, 1 .5 mm thick. The pressure was 1 00 bar and the temperature 300°C. The spatial resolution of each beam within the test section was I x 1 mm2 .

25.4.2 Impedance Void Meters Impedance void meters have proved to be sufficiently adequate for measuring area void fractions in a wide range of circumstances, especially in the study of void fraction waves. The most effective meters use flush-mounted electrodes at the pipe wall; two versions of these meters have been used so far. In the first version, a fixed electrical field is created between two electrodes, whereas in the second version, a rotating electrical field is created between three pairs of electrodes that are energized sequentially (Merilo et aI., 1 977; Snell et aI., 1 978). The advantage of the six electrode sensor is that its signal is less affected by the spatial distribution of the void fraction than that delivered by a two electrode sensor. An improved development of the six electrode sensor was reported by Toumaire ( 1 985, 1 986, 1 987) and Delhaye et al. ( 1 987) in connection with an experimental investigation of void fraction waves. It appears that impedance void meters are reliable, fairly accurate, and not too expensive. However, they require walls that are not electrically conducting. An advanced theoretical modeling of the impedance void meter can be found in Lemonnier et aI. ( 1 99 1 ), whereas practical applications are presented in de Cachard ( 1 989) and de Cachard and Delhaye ( 1 996) for air-lift pumping systems in Hervieu ( 1 994), Veneau ( 1 995), and Hervieu and Veneau ( 1 996) for liquefied propane flashing flow, and in Mi et aI. ( 1 997), Seleghim ( 1 996), and Hervieu and Seleghim ( 1 997) for flow regime identification. Several attempts have been made to use impedance tomography in hetero geneous media (Williams and Beck, 1 995). However, it has been proved that impedance tomography could not provide the local value of the void fraction with sufficient accuracy (Lemonnier, 1 995; Lemonnier and Peytraud, 1 995 , 1 996, 1 997; Peytraud, 1 995).

25.4.3 Neutron Scattering and Radiography Techniques X-ray attenuation techniques are not suitable for void fraction measurements in metallic piping when the void fraction is larger than 0.8. This is due to the fact that a small change in the liquid fraction is concealed by the absorption of the X -ray beam by the metallic wall. On the contrary, neutron scattering is a very sensitive technique in this case. For a stainless-steel tube 1 2 mm in diameter, the best choice found by

INSTRUMENTATION IN BOILING AND CONDENSATION STUDIES 695

Freitas ( 1 98 1 ) was the backward scattering of thennal neutrons. A neutron beam produced by a nuclear reactor was collimated in a conduit filled with helium to avoid scattering or attenuation by humidity. A bismuth shield was installed to stop the y rays. A boron-lined proportional detector was used to count the scattered neutrons. A 2.2 contrast ratio was obtained for void fractions ranging from 0.8 to 1 .0. Recently, significant improvements were reported in the use of neutron radio graphy in two phase flow (Mishima and Hibiki, 1 997; Takenaka et aI., 1 997). Applications of neutron radiography have also been proposed for many non destructive examinations, since the attenuation characteristics of neutron rays in materials are much different from those of X-rays and gamma-rays, i.e., an image difficult to be visualized by X-rays or gamma-rays may possibly be visualized by neutron rays. Figure 1 3 shows the mass attenuation coefficients of X-rays by a solid line and those of neutron rays by symbols against atomic numbers. The mass attenuation coefficients of X-rays increase with increasing the atomic number. Therefore, X rays are suitable to see an object made of high atomic number elements through that of low atomic number elements. However, the mass attenuation coefficient of thennal neutron rays greatly depend on the elements. Since the mass attenuation coefficient of neutron rays of water is higher than those of metals like aluminum and the elements of stainless steel, neutron radiography is suitable for the visualization of water two-phase flow in a metallic vessel or tube. It is also low for low melting metals, like lead, bismuth, and tin; therefore, these liquid metals flow can be also visualized by neutron radiography. Neutrons produced in a nuclear reactor or generated by nuclear interaction in an accelerator are thermalized in a moderator. A parallel beam is obtained by a collimator and the beam is irradiated to the test object. Neutrons are not visible or sensible to films for optical rays and X-rays; therefore, the neutron image of the object is converted to the other radioactive rays image, which is sensible to the films in film methods. It is also converted to the optical image by a scintillation converter in real-time methods and recorded by a high-sensitivity camera, like a Silicon Intensified Target (SIT) tube camera to make movies. An image intensifier is often used for high-speed movies. A cooled Charged Couple Device (CCD) cam era is used for high resolution and high dynamic range still imaging. More details on neutron radiography and its applications can be found in Domanus ( 1 992). Figure 14 shows photographs of a gas lighter taken by the film method using optical rays, thermal neutron rays, and X-rays to show the difference of the at tenuation characteristics (Hiraoka, 1 995 ; Yoneda et aI., 1 990). The surface of the gas lighter was covered with a metallic plate, lighter contained parts made from metals, plastic resins, and the oil. The oil and the structure made of the plastic resins can be seen by neutron-rays. The oil inside the metallic tank was not visible, but metallic pipes and springs were visible by X-rays.

Q\ \C Q\

Cl

H A

�

.N •

Sc

Fe

CO •

absorption

•

C o l d n e u t r o n s 0 . 0 0 3 eV

+ A b s o r p t i o n on l y

S

S

( 0A / o < l 0

( 0A / o > l 0

... P redomi n a n t l y a bs o r p t i o n

- P redom i n a n t l y scatter

• S catter and

o Ne

Mass attenuation coefficients of neutron and X-rays.

He

CI A

}

+ Cd

. Cd Sm

Ir

+ Er A T m LU A+ A Hf Ho

.Dy

A A EU

A Gd

. Gd

A Hg

APU

n e u t rons

T he r ma l

---

A tom i c n u mber

X - ra y s ( 1 2 5 kV )

+ Pa Kr Cs Nd • N i Se · • Tb + • Yb • X e W . La + p Ti n•• . Cu ·BrR b • Os ·. F .� a . V. As • • •• ZrMo Pd · T a• • • Pt Tl R I C K Mg Ga . a Th Sb � • • . Pr • S i· s • Sr W • • . + • • U02 • C a Fe .. . Ba Zn· · Ge • • Ru A Ph · Ce AI . - • • · Re . A � Ar U Y. Nb Ag Sn T e -. Au • . AI. Bl I I •Sn I I 10 20 30 40 50 100 90 60 80 70

Be D2O - .N •0 •

1-

. H 20

B

B

•

A Li

.H

•

0.01 I 0

0. 1

1

10

100

Figure 13

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