Advances in Heat Transfer, Volume 13

ADVANCES IN HEAT TRANSFER Volume 13

Contributors to Volume 13 HOLGER MARTIN D. BRIAN SPALDING STEPHEN WHITAKER HORST H. WINTER

Advances in

HEAT TRANSFER Edited by

James P. Hartnett

Thomas F. Irvine, Jr.

Energy Resources Center University of Illinois at Chicago Circle Chicago, lllinois

Department of Mechanics State University of New York at Stony Brook Stony Brook, New York

Volume 13

@ 1977 ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 8 1977, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN A N Y FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWl

LIBRARY OF CONGRESS

CATALOG CARD

NUMBER:63-22329

ISBN 0-12-020013-9 PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

List of Contributors . . . . . . . . . . . . . . . . . . . . .

vii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Contents of Previous Volumes . . . . . . . . . . . . . . . . .

xi

Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces HOLGER MARTIN Introduction . . . . . . . . . . . . . . . . . . . . . . I. Hydrodynamics of Impinging Flow . . . . . . . . . . . . I1. Heat and Mass Transfer: Variables and Boundary Conditions I11. Local Variation of the Transfer Coefficients . . . . . . . . IV . Integral Mean Transfer Coefficients . . . . . . . . . . . . V . Influence of Outlet Flow Conditions on Transfer Coefficients for Arrays of Nozzles . . . . . . . . . . . . . . . . . . VI . Other Parameters Influencing Heat and Mass Transfer . . . . VII . Optimal Spatial Arrangements of Nozzles . . . . . . . . . VIII . Design of High-Performance Arrays of Nozzles . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

1

2 6 8 13

27 41

45 52 58 59

Heat and Mass Transfer in Rivers. Bays. Lakes. and Estuaries

D . BRIANSPALDING I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Two-Dimensional Parabolic Phenomena . . . . . . . . . . 111. Two-Dimensional Steady Jets and Plumes . . . . . . . . . IV . Two-Dimensional Steady Boundary Layers Adjacent to Phase Interfaces . . . . . . . . . . . . . . . . . . . . . . . V

62 70 73 79

vi

CONTENTS

V . One-Dimensional Unsteady Vertical-Distribution Models . . 89 VI . Two-Dimensional Floating Layers . . . . . . . . . . . . 98 VII . Concluding Remarks . . . . . . . . . . . . . . . . . . 113 Nomenclature . . . . . . . . . . . . . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . 114 Simultaneous Heat. Mass. and Momentum Transfer in Porous Media: A Theory of Drying

STEPHEN WHITAKER I . Introduction . . . . . . . . . . . . . . . . . . . . . . II . The Basic Equations of Mass and Energy Transport . . . . . 111. Energy Transport in a Drying Process . . . . . . . . . . . IV . Mass Transport in the Gas Phase . . . . . . . . . . . . . V . Convective Transport in the Liquid Phase . . . . . . . . . VI . Solution of the Drying Problem . . . . . . . . . . . . . VII . The Diffusion Theory of Drying . . . . . . . . . . . . . VIII . Conclusions . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

119 126 153 165 175 192 194 198 199 200

Viscous Dissipation in Shear Flows of Molten Polymers

HORSTH . WINTER I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Shear Flow (Viscometric Flow) . . . . . . . . . . . . . . 111. Elongational Flow; Shear Flow and Elongational Flow Superimposed (Nonviscometric Flow) . . . . . . . . . . . . . IV . Summary . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

205 212 260 262 263 264

........................

269

Subject Index . . . . . . . . . . . . . . . . . . . . . . . .

274

Author Index

LIST OF CONTRIBUTORS HOLGER MARTIN, Institut.Fr Thermische Verfahrenstechnik, der Universitat Karlsruhe (TH),D 75 Karlsruhe, Kaiserstraje 12, Germany D. BRIAN SPALDING, Mechanical Engineering Department, Imperial College of Science and Technology, Exhibition Road, London S W7 ZBX, England STEPHEN WHITAKER, Department of Chemical Engineering, University of California at Davis, Davis, California 95616 HORST H. WINTER, Institut f i r Kunststofftechnologie, der Universitat Stuttgart, 7 Stuttgart I , Boblinger Str. 70, West Germany

vii

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PREFACE The serial publication Advances in Heat Transfer is designed to fill the information gap between the regularly scheduled journals and university level textbooks. The general purpose of this series is to present review articles or monographs on special topics of current interest. Each article starts from widely understood principles and in a logical fashion brings the reader up to the forefront of the topic. The favorable response to the volumes published to date by the international scientific and engineering community is an indication of how successful our authors have been in fulfilling this purpose. The Editors are pleased to announce the publication of Volume 13 and wish to express their appreciation to the current authors who have so effectively maintained the spirit of the series.

ix

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CONTENTS OF PREVIOUS VOLUMES

Radiation Heat Transfer between Surfaces E. M. SPARROW

Volume 1

The Interaction of Thermal Radiation with Conduction and Convection Heat Transfer R. D. CESS Application of Integral Methods to Transient Nonlinear Heat Transfer THEODORE R. GWDMAN Heat and Mass Transfer in CapillaryPorous Bodies A. V. LLJIKOV Boiling G. LEPPERT and C. C. PITTS The Influence of Electric and Magnetic Fields on Heat Transfer to Electrically Conducting Fluids MARYF. ROMIG Fluid Mechanics and Heat Transfer of Two-Phase Annular-Dispersed Flow MARIOSILVESTRI AUTHOR INDEX-SUBJECT

AUTHOR INDEX-SUBJECT

INDEX

Volume 3 The Effect of Free-Stream Turbulence on Heat Transfer Rates J. KESTIN Heat and Mass Transfer in Turbulent Boundary Layers A. I. LEONT’EV Liquid Metal Heat Transfer RALPHP. STEIN Radiation Transfer and Interaction of Convection with Radiation Heat Transfer R. VISKANTA A Critical Survey of the Major Methods for Measuring and Calculating Dilute Gas Transport Properties A. A. WESTENBERG

INDEX

AUTHOR INDEX-SUBJECT

Volume 2

Turbulent Boundary-Layer Heat Transfer from Rapidly Accelerating Flow of Rocket Combustion Gases and of Heated Air D. R. BARTZ Chemically Reacting Nonequilibrium Boundary Layers PAULM. CHUNC Low Density Heat Transfer F. M .DEVIENNE Heat Transfer in Non-Newtonian Fluids A. B. METZNER

INDEX

Volume 4

Advances in Free Convection A. J. EDE Heat Transfer in Biotechnology ALICEM. STOLL Effects of Reduced Gravity on Heat Transfer ROBERTSIEGEL Advances in Plasma Heat Transfer E. R. G. ECKERTand E. PFENDER xi

xii

CONTENT^ OF PREVIOUS VOLUMES

Exact Similar Solution of the Laminar Boundary-Layer Equations DEWEY, JR. and C. FORBES JOSEPHF. GROSS AUTHOR INDEX-SUBJECT

INDEX

Heat Transfer in Rarefied Gases GEORGE S. SPRINGER The Heat Pipe and W. 0. BARSCH E. R. F. WINTER Film Cooling RICHARD J. GOLDSTEIN AUTHOR INDEX-SUBJECT

INDEX

Volume 5 Application of Monte Carlo to Heat Transfer Problems JOHN R. HOWELL Film and Transition Boiling DUANE P. JORDAN Convection Heat Transfer in Rotating Systems FRANKKREITH Thermal Radiation Properties of Gases C. L. TIEN Cryogenic Heat Transfer JOHNA. CLARK AUTHOR INDEX-SUBJECT

INDEX

Volume 8 Recent Mathematical Methods in Heat Transfer I. J. KUMAR Heat Transfer from Tubes in Crossflow A. ~ K A U S K A S Natural Convection in Enclosures SIMON OSTRACH Infrared Radiative Energy Transfer in Gases R. D. CESSand S. N. TIWARI Wall Turbulence Studies 2.Z A R I ~ AUTHOR INDEX-SUBJECT

INDEX

Volume 9 Volume 6 Supersonic Flows with Imbedded Separated Regions A. F. CHARWAT Optical Methods in Heat Transfer W.HAUFand U. GRIGULL Unsteady Convective Heat Transfer and Hydrodynamics in Channels E.K. KALININ and G. A. DREITSER Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties B. S. PETUKHOV AUTHOR INDEX-SUBJECT

INDEX

Volume 7 Heat Transfer near the Critical Point W. B. HALL The Electrochemical Method in Transport Phenomena T. MIZUSHINA

Advances in Thermosyphon Technology D. JAPIKSE Heat Transfer to Flowing Gas-Solid Mixtures CREIGHTON A. DEPEWand TEDJ . KRAMER Condensation Heat Transfer HERMAN MERTE,JR. Natural Convection Flows and Stability B. GEBHART Cryogenic Insulation Heat Transfer C. L. TIENand G. R. CUNNINGTON AUTHOR INDEX-SUBJECT

INDEX

Volume 10 Thermophysical Properties of Lunar Materials: Part I Thermal Radiation Properties of Lunar Materials from the Apollo Missions RICHARD C. BIRKEBAK

...

XI11

CONTENTS OF PREVIOUS VOLUMES Thermophysical Properties of Lunar Media: Part I1 Heat Transfer within the Lunar Surface Layer CLIFFORD J. CREMERS Boiling Nucleation ROBERTCOLE Heat Transfer in Fluidized Beds CHAIMGUTFINGER and NESIMABUAF Heat and Mass Transfer in Fire Research S. L. LEEand J. M. HELLMAN AUTHOR INDEX-SUBJECT

INDEX

Volume 11

Boiling Liquid Superheat N. H. AFGAN Film-Boiling Heat Transfer E. K. KALININ,I. I. BERLIN,and V. V. KOSTYUK The Overall Convective Heat Transfer from Smooth Circular Cylinders VINCENT T. MORGAN

A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection Problems G. D. RAITHBY and K. G. T. HOLLANDS Heat Transfer in Semitransparent Solids R. VISKANTA and E. E. ANDERSON AUTHOR INDEX-SUBJECT

INDEX

Volume 12

Dry Cooling Towers F. K. MOORE Heat Transfer in Flows with Drag Reduction YONADIMANT and MICHAEL POREH Molecular Gas Band Radiation D. K. EDWARDS A Perspective on Electrochemical Transport Phenomena AHARON S. ROY AUTHOR INDEX-SUBJECT

INDEX

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Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces HOLGER MARTIN Institut f u r Thermische Verfahrenstechnik der Universitat Karlsruhe ( TH 1, D 75 Karlsruhe, KaiserstraJe 12, Germany Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Hydrodynamics of Impinging Flow . . . . . . . . . . . . . . . . . . . 11. Heat and Mass Transfer: Variables and Boundary Conditions. . . . . . . . 111. Local Variation of the Transfer Coefficients. . . . . . . . . . . . . . . . A. Single Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . B. Arrays of Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . IV. Integral Mean Transfer Coefficients . . . . . . . . . . . . . . . . . . . A. Equations for Single Nozzles . . . . . . . . . . . . . . . . . . . . B. Equations for Arrays of Nozzles . . . . . . . . . . . . . . . . . . . V. Influence of Outlet Flow Conditions on Transfer Coefficients for Arrays of Nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . V1. Other Parameters Influencing Heat and Mass Transfer . . . . . . . . . . . A. Turbulence Promoters. Swirling Jets . . . . . . . . . . . . . . . . . B. Wire-Mesh Grids on the Surface of the Material . . . . . . . . . . . . C. Impinging Flow on Concave Surfaces. . . . . . . . . . . . . . . . . D. AngleofImpact. . . . . . . . . . . . . . . . . . . . . . . . . . VII. Optimal Spatial Arrangements of Nozzles . . . . . . . . . . . . . . . . VIII. Design of High-Performance Arrays of Nozzles . . . . . . . . . . . . . . Nomenclature.. . . . . . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 6 8 8

12 13 15 18 21 41 41 41

43 44 45 52 58 59

Introduction Heating or cooling of large surface area products is often carried out in devices consisting of arrays of round or slot nozzles, through which air (or another gaseous medium) impinges vertically upon the product surface. 1

2

HOLGFBMARTIN

Such impinging flow devices allow for short flow paths on the surface and therefore relatively high heat transfer rates. The annealing of metal and plastic sheets, the tempering of glass, and the drying of textiles, veneer, paper, and film material are some of their more important industrial applications. In order to achieve a suitable plant design, both from an economic and a technical viewpoint, knowledge of the dependence of the heat and mass transfer rates on the external variables is required. The gas flow rate, the diameter (or slot width) of the nozzles, their spacing, and their distance to the product surface are the main variables, which can be chosen to solve a given heat or mass transfer problem. During the past twenty years, research on momentum, heat, and mass transfer in impinging flow led to a multitude of experimental data obtained by different methods involving different hydrodynamic, thermal, or material boundary conditions. This contribution is intended to be a comprehensive survey emphasizing the engineering applications rather than a basic theoretical approach. Therefore empirical equations are presented for the prediction of heat and mass transfer coefficients within a large and technologically important range of variables. These equations are based on experimental data for single round nozzles (SRN) [11, arrays of round nozzles (ARN) [2-41, single slot nozzles (SSN) [5], and arrays of slot nozzles (ASN) [6-91. How to apply these equations in heat exchanger and dryer design as well as in optimization is also shown. I. Hydrodynamics of Impinging Flow

As shown by Schrader [lo], Glaser [ll, 121,and other authors [13-161, the flow pattern of impingingjets from single round and slot nozzles can be subdivided into three characteristic regions: the freejet region, the stagnation flow region, and the region of lateral (or radial) flow outside the stagnation zone, also called the wall jet region after the basic theoretical work of Glauert [17]. The velocity field of an impinging jet is shown schematically in Fig. 1. Under technically realistic conditions the free jet, developing from the exit of the nozzle (a) with diameter D or slot width B in general will be turbulent. By an intensive exchange of momentum with the surrounding gas over the free boundaries (6) the jet broadens linearly with its length z’ up to a limiting distance zg from the solid surface (c). The velocity profile (d), being nearly rectangular at the nozzle exit, spreads toward the free boundaries and for sufficient length of the free jet approaches a bell-shape that can be

IMPINGING JET FLOWHEATAND MASSTRANSFER

L-xg;rg

3

4

FIG.I . Flow field of impinging flow (schematically).

described approximately by a Gaussian distribution : w ( x , z‘)/w(o,z’) = exp{ -(X/CZ’)~}

(1.1)

The constant C in Eq. (1.1)has a value of about 0.1, slightly depending on the shape of the nozzle and the exit Reynolds number. The range of validity of Eq. (1.1)is limited on one hand by the so-called core length zk’ up to which the velocity on the axis (e) remains nearly constant (w(0, 2’) a wD for z’ < zk’)and on the other hand by the distance zg from the surface that marks the boundary to the stagnation region. The velocity on the jet axis can be calculated following SchlUnder [lS] for plane and axisymmetric free jets: Plane jet (SSN)

w(0, 4 = {erf($)}”’ WD

Axisymmetricjet (SRN)

Equations (1.2) and (1.3) are valid in the whole free jet region [0 5 z’ I ( H - ZJ. If one defines the core length zk’as that distance from the nozzle exit where the pressure head on the axis has fallen to 95% of its maximum vaIue ~ ’ ( 0zk’)/wD2 , = 0.95) a simple relation between the constant C and core

HOLGERMARTIN

4

length zk‘can be found: C

=

0.127(4B/zkf)

C

=

0.102(40/~~’) (SRN)

(SSN)

(1-4)

(1.5)

Core lengths of about four slot widths (or four nozzle diameters) are to be expected with normal well-rounded nozzles. With increasing jet length ( 2 >> zk’)Eqs. (1.2) and (1.3) asymptotically approach the well-known hyperbolic laws for the dependence between axial velocity and distance from the exit:

w(0, z’) wD

1-

1 +--

D Jzc z’

--

(1.3a)

(SKNJ

Stagnation flow just begins relatively close to the surface (according to Schrader [lo], the limiting distance zg is about 1.2 times the nozzle diameter for SRNs). Here the vertical velocity component is decelerated and transformed into an accelerated horizontal one. Exact analytic solutions of the Navier-Stokes equations of motion are known for the idealized limiting case of the infinitely extended plane and axisymmetric laminar stagnation flows. (see Schlichting [19, pp. 76-81]). For example, those flow patterns are also to be met in the immediate vicinity of the stagnation point of cylinders and spheres in cross flow. They are typical boundary layer flows, the influence of viscosity being restricted to a thin layer near the solid surface. The velocity components of stagnation flow outside this boundary layer are given by w, = -aEz

w,

( 1*6)

w, = -2a,z

= aEx w, = aRr (plane) (axisymmetric) (stagnation flow)

(1.7)

Here aEand a, are constant values; i.e., the velocity components are linearly proportional to the distance from the stagnation point. The boundary layer thickness do, defined as the distance from the surface where the lateral component reaches 99% of the value in Eq. (1.7), is found from the tabulated dimensionless velocity profiles in Schlichting [ 191: =

=

2.38(v/uE)’I2

(plane)

1.95(v/aR)”2

(axisymmetric)

(stagnation flow)

( 1-81

(1.9)

IMPINGING JET FLOWHEATAND MASSTRANSFER

5

Provided that ideal stagnation flow conditions were met, the boundary-layer thickness remained constant, i.e., did not depend on the lateral coordinate x (or r) as it would for parallel flow. For real stagnation flows due to impinging jets of finite breadth (or diameter), the constants uE and uR were found experimentally by Schrader [lo] and Dosdogru [20] [l s ( H / B ; H / D ) I lo]: UE = (w,/B)(1.02 - 0.024H/B) (1.10) UR =

(WD/D)(1.04- 0.034H/D)

(1.11)

The relative boundary-layer thickness ( divided by the nozzle diameter = 2B)are therefore inversely proportional to the exit Reynolds numbers:

D or hydraulic diameter S

d0.E - - (1.68/Re'12)(l.02 - O.O48H/S)- ' I 2 S

(1.12)

60,R - - (1.95/Re'12)(1.04 - 0.034H/D)-112

(1.13)

D

Since the Reynolds numbers in practical applications are mostly of an order of magnitude of lo4 or higher, the boundary layer thicknesses do in the stagnation zone will reach about one-hundredth of the nozzle diameter. Due to the flnite breadth of the jet and the exchange of momentum with its quiescent surroundings the accelerated stagnation flow finally must transform to a decelerated wall jet flow. So, the wall parallel velocity component w, (w,) initially increasing linearly from zero must reach a maximum value at a certain distance xg( r g )from the stagnation point and finally tend to zero with x - " ( r - " ) in the fully developed wall jet. The exponent n is about 0.5 for the plane [17, 21, 221 and about 1 for the axisymmetric [17,22,231 turbulent wall jet. Whereas the stabilizing effect of acceleration keeps the boundary layer laminar in the stagnation zone, transition to turbulence generally will occur immediately after xg (or r g ) in the decelerating flow region. Wall boundary layer and free jet boundary grow together, forming the typical wall jet profile where the boundary layer 6 is defined as the locus of the maxima of the velocity (z(wxmaX))(see (f)in Fig. 1). The impinging flow from arrays of nozzles generally shows the same three flow regions-free jet, stagnation zone, wall jet-but in addition there are secondary stagnation zones where the wall jets of adjacent nozzles impinge upon each other. Like the wake zones in the rear of cylinders or spheres in cross flow, these secondary stagnation zones are characterized by boundary layer separation and eddying of the flow. Especially for smaller spacings between the nozzles of an array, this can lead to a considerable reaction on the other flow regions.

HOLGERMARTIN

6

11. Heat and Mass Transfer: Variables and Boundary Conditions

External variables influencing heat and mass transfer in impinging flow are on one hand mass flow rate, kind and state of the gas (for mass transfer problems also the kind of component transported) and on the other hand shape, size, and position of the nozzles relative to each other and to the solid surface. Additionally, the hydrodynamic, thermal, and material boundary conditions that may be kept constant for each particular case must be considered. The hydrodynamic boundary conditions are given by the distribution of velocities at the nozzle exit and at the surface. In the following it will be presumed that all velocity components vanish at the surface (surface at rest and impermeable)and that the gas velocity at the nozzle exit is equally distributed over the cross section. The first of these assumptions-surface at rest-nearly never will be met in practical applications. However, this assumption does not impose severe restrictions at all since the moving velocity of the material does not exceed a small fraction of the gas impact velocity in most cases. The second assumption-an equally distributed nozzle exit velocity-is fulfilled to a high degree of approximation by most nozzles used technically. The case of initially laminar jets with fully developed parabolic velocity profile at the exit-when long tubes or rectangular ducts were used instead of nozzles-is treated by Scholz and Trass [24] for the axisymmetric and by Sparrow and Wong [25] for the plane jet. A complete description of the hydrodynamic boundary conditions should contain the turbulence level at the nozzle exit, which can influence heat and mass transfer considerably in the stagnation zone [20,26]. In the following we shall knowingly renounce such completeness since the turbulence level cannot be chosen determinately by the designing engineer. As thermal (and/or material) boundary conditions we shall presume the temperatures (and/or partial pressures) to be constant over the surface and over the nozzle exit cross section. For single nozzles, the state of the surrounding gas can be chosen arbitrarily and therefore must be considered as a variable or a boundary condition [l]. For arrays of nozzles, it will be determined by the other boundary conditions and variables. are I defined here as flux The heat and mass transfer coefficients a and # per nozzle-exit-to-surface driving force difference:

= 4 / ( 8 D - 80) B = f i i , / m , o - %,d a

(2.1)

(2.2) These definitionsare valid for pure heat transfer and equimolal mass transfer, respectively. In order to maintain the analogy (between heat and mass

IMPINGINGJET FLOWHEAT AND MASSTRANSFER

7

transfer) for the case of coupled heat and unidirectional mass transfer too, a logarithmic driving force difference has to be introduced (see, e.g., Bird et al. [27] or Schlunder [28] :

p1 = f i , / [ n ln( 1 + x"1.0 - X"1.D 1 - x"l.0

>I

In many practical cases there will be no significant difference between these unidirectional transfer coefficients and the ones defined by Eqs. (2.1) and (2.2) since the dimensionless driving force parameters

and

often will be very small relative to unity. This can be seen more clearly by writing Eqs. (2.3) and (2.4) in the following way: a1

4 K9 = aDSo ln(1 + K , )

(2.3a) (2.4a)

For K < 0.1 the term K/ln(l + K) practically remains equal to unity (relative error less than 5%). With the boundary conditions fixed as above the heat and mass transfer coefficients generally can be described in dimensionless form: Nu (or Nu,) = F(Re, Pr, geometric relations)

(2.7)

Sh (or Sh,) = F(Re, Sc, geometric relations)

(2.8)

The characteristic length in Nu, Shyand Re is chosen here to be the hydraulic diameter of the nozzles ( S ) : S =D

(SRN,ARN)

S = 2B

(SSN,ASN)

(2.9) (2.10)

The Reynolds number is formed with the mean velocity at the nozzle exit calculated from the total mass flow rate. Since besides the nozzle diameter S,

HOLGERMARTIN

8

the vertical distance H between nozzle exit and surface and the lateral (or radial) distance x (or r) from the stagnation point are involved in the problem, Eqs. (2.7) and (2.8) must contain at least two geometric relations, e.g., HIS and x/S (or r/D respectively). 111. Local Variation of the Transfer Coefficients

A. SINGLENOZZLES To determine local mass transfer coefficients Schlunder and Gnielinski [l] as well as Schliinder et al. [5] used an experimental setup as shown schematically in Fig. 2. Air jets from single round [l] or slot [ 5 ] nozzles (a in Fig. 2) impinged upon a flat plate consisting of concentric rings-r parallel strips-(d) of porous stoneware. The elements (d) were separated from each other, each having its own water chamber (e) connected with a supplying flask (b) and a burette (c). During the experimental runs the surface could be kept uniformly moist by putting each flask at a certain level above the plate according to the local pressure head. Mass transfer rates were determined by shutting off the hose to the supplying flask and observing the then moving water column in the burette with a stopwatch. Thermocouples (f) allowed for measuring local surface temperatures.

IT

FIG.2. Experimental setup in Schlllnder and Gnielinski [ 11 and SchlUnder et al. [ S ] . a, nozzle; 6, supplying flask; c, burette; d, plate of porous stoneware; e, water chambers;f, thermocouples, P,, pressure head on the plate.

IMPINGINGJET FLOWHEATAND MASSTRANSFER

9

The air was preheated to a temperature of about 40°C to yield an adiabatic saturation temperature on the moist porous plate very close to the temperature of the surrounding air. So any heat transfer with the surroundings could be avoided nearly completely. The partial pressure of water at the nozzle exit was the same as in the surrounding air. Mass transfer coefficients determined under these boundary conditions are comparable to heat transfer coefficients measured with the nozzle-exit and the surrounding air at the same temperature. Most of the local measurements of impinging flow from single nozzles known to date have been carried out under this boundary condition which is referred to as case I1 in Fig. 3. The local variation of mass transfer coefficients-as determined by the method described above-is shown in Figs. 4a-d and 5a-f. Sherwood numbers Shl = plS/S,, were plotted versus the relative radial (or lateral) distance r / D (or x/S) from the stagnation point with Re = WS/v as parameter. The relative nozzle-to-plate distance HID (or HIS) was kept constant in each figure. For both round and slot nozzles, the local variation is qualitatively the same : monotonically decreasing bell-shaped curves for large relative nozzleto-plate distances HID ( H I S ) and curves with a more or less distinct hump or second maximum for small HID ( H I S ) .

FIG.3. Thermal (material)boundary conditions for impinging flow from single nozzles [I]: CaseI: Case 11:

9L

- 90 -

___ = I ; 9L

9, - 9,

~

P1.L

Pl,O

- 0;

- P1.D =1 - Pl.0

P1,L - P l . 0

P l . 0 - P1.D

=

HOLGER MARTIN

10

0

2

1

6

(a)

0

2

4

r/D

6

8

(C)

FIG.4. Local Sherwoodnumbers for impingingtlow from single round nozzles [11;Sc = 0.59.

The sharp increase of the transfer coefficients begins immediately after the end of the accelerated flow region where the disappearance of the stabilizing streamwise pressure gradient leads to a sudden steep rise in turbulence level [I, 261. Results similar to those shown in Figs. 4 and 5 can be found for single round nozzles in [14, 26, 29-33] and for single slot nozzles in [15, 20, 26, 34, 351 including heat transfer to air jets (Pr = 0.7) [14, 20, 26, 30, 32, 351, mass transfer naphthalene-air (Sc = 2.5) [lS, 31,33,34] and trans-cinnamic acid-water (Sc = 900) [29].

IMPINGING JETFLOWHEATAND MASSTRANSFER

I1

250 9 1

200 150

m 50 n "0

5

1y)

(a)

15

0

5

l0

x/s

(b)

15

0

5

1 0 1 5 3 0 (C)

FIG.5. Local Sherwood numbers for impinging flow from single slot nozzles [ 5 ] ; Sc = 0.59.

For comparison, some curves from Gardon and Akfirat [26] and Petzold [32] are given together with those from [l] in Fig. 4c. Generally the agreement is quite satisfactory in the wall jet region, whereas larger discrepancies occur in the stagnation zone probably due to the different turbulence levels at the nozzle exit. These discrepancies are of minor importance for the integral mean transfer coefficients, which are of predominant technical interest. Several theoretical or semiempirical approaches combining previously existing stagnation flow [19] and wall jet [17, 21, 22, 23, 361 analyses or directly applying boundary layer theory were carried out for single axisymmetric [37-391 and plane [15,40,41] impinging jets. They did not succeed however in predicting the observed nonmonotonic variation of the transfer coefficients for smaller nozzle-to-plate distances.

12

HOLGERMARTIN

Wolfshtein [42] applied a numerical method [43] to solve systems of elliptic partial differential equations on the problem of impinging flow from a single slot nozzle. The system of equations must be closed by more or less arbitrary hypotheses on the interrelation between the time --averaged products of turbulent fluctuating components (such as v ' d , v'p', v'T', for example) and the mean values of velocities, pressures, temperatures, and so on. The method yields plenty of detailed information on the whole flow field: stream lines, lines of equal vorticity, isotherms, and lines of equal turbulence energy. Unfortunately, the computations were only carried out for one fixed relative nozzle-to-plate distance HIS = 4.Nusselt numbers are in reasonable agreement with those measured by Gardon and Akfirat [35]. Their lateral variation, however, deviates significantly from the measured curves, especially for the lower Reynolds numbers. B. ARRAYS OF NOZZLES Local measurements of heat and mass transfer coefficients for impinging flow from arrays of nozzles [30, 34,351 show qualitatively similar results as from single nozzles. Additionally, there may occur further peaks in the lateral variation of the transfer coefficients where the wall jets of adjacent nozzles impinge upon each other forming secondary stagnation zones (for reference, see Fig. 8 in Gardon and Cobonpue [30], Figs. 9 and 10 in Korger and KfiZek [34], and Figs. 6 and 8 in Gardon and Akfirat [35]). For one array of parallel slot nozzles ( B = 10 mm, H = 60 mm, and nozzle spacing L, = 150 mm), these secondary peaks in the middle of the nozzle spacing were found to reach the same height as the primary peaks directly under the nozzles (Fig. 10 in [34]). The state of the gas surrounding the jets cannot be chosen arbitrarily as it could be for a single nozzle (see Section 11) since it is here determined by the mass and energy balances: (3.la) (3.lb) The right-hand sides of Eqs. (3.1) are the numbers of transfer units (NTU). In most practical applications of impinging jets NTU values are very small (high mass flow rates of gas), so that the thermal boundary conditions closely approach case I1 in Fig. 3. In many cases the local variation of the transfer coefficients will depend not only on the lateral coordinate x (or the radial r) but also on the transverse coordinate y (see Fig. 6), namely when the gas

IMPINGING

JET FLOWHEATAND MASSTRANSFER

13

n

FIG.6. Array of slot nozzles with the outlet flow laterally over the edges of the sheet [9].

cannot be led off directly upward between the nozzles but flows symmetrically to both sides (parallel to the slots in the + y directions) over the width of the material. Clearly this outlet stream will influence the whole flow field. The smaller the ratio of the outlet flow area F A (hatched area in Fig. 6 ) to the nozzle exit cross section BI (for slot nozzles) the higher will be the outlet flow velocity u and the less uniformly distributed will be the transfer coefficients over the width of the material. This influence of the outlet flow conditions will be treated in detail in Section V. IV. Integral Mean Transfer Coefficients

For practical engineering calculations integral mean heat and mass transfer coefficients are needed: (4.1) Equation (4.1)can be specified for the SRN a = - J2; r2

a(r')r' dr'

and for the SSN

For single round and slot nozzles, the mean heat transfer coefficients after Eqs. (4.2) and (4.3) and the corresponding mean mass transfer coefficients may be described in the following dimensionless form :

Nu = F(Re, Pr, r/D, H I D ) Sh

=

F(Re, Sc, r/D, H I D )

(for slot nozzles with XIS,HIS instead of r/D, HID).

14.4) (4.5)

HOLGER MARTIN

14 SRN

ARN

Jk

ASN

SSN I

I1

FIG.7. The spatial arrangement of nozzles in regularly spaced arrays.

For arrays of nozzles, the averaging in Eq. (4.1) must be carried out over those parts of the surface area attributed to one nozzle. For regularly spaced ARNs, these parts are squares with edge lengths L, or regular hexagons with edge lengths LD/2(see Fig. 7). The relative nozzle area f is given by the ratio of the nozzle exit cross section to the area of the square or the hexagon attached to it: n

-

4

f = 'quare

D2 (4.6) (hexagon)

The radius r in Eqs. (4.4) and (4.5) may be replaced by the radius RT of a circle enclosing the same area as the square or the hexagon: RT = D/@. For ASNs, the integration in Eq. (4.3) has to be taken from x = 0 to x = X, with XT = LT/2. This can also be expressed by the relative nozzle area, being f = Bl/L,l = s/2LT (4.7) in this case XT = S/4f. So, for arrays of nozzles, the relative distance from the stagnation point may be replaced by a function of the relative nozzle area:

In Table I the various expressions for the relative nozzle area f are listed for single nozzles and regularly spaced arrays of nozzles.

IMPINGING JET FLOWHEAT AND

MASS TRANSFER

15

TABLE I SRN

ARN,

ARN,

l(D>'

n (D)' 2J3 L,

"D)' -

4 1

SSN

A. EQUATIONS FOR

4 L,

ASN

SINGLE NOZZLFS

1. Single Round Nozzle ( S R N )

The local mass transfer measurements of Schlunder and Gnielinski [11, partly shown in Fig. 4, were averaged using Eq. (4.2). Together with the heat transfer measurements of Gardon and Cobonpue [30], Petzold [32], Brdlick and Savin [MI, and Smirnov et al. [45] these integral mean values could be correlated by the following empirical equation [13:

Sh

(&),kN

Nu

=( F ) s R N

D

=

1 - l.lD/r

71 + O.l(H/D - 6)D/r F(Re)

(4.10)

This multiplicative representation G(r/D, H/D)F(Re) was found to be applicable only for radial distances from the stagnation point of at least 2.5 nozzle diameters. The function F(Re),shown in Fig. 8, may be approximated by the following power functions [2,5] : 2000 < Re c 30,000,

F(Re) = 1 . 3 6 R e O ~ ~ ~ ~

30,000 < Re < 120,000,

F(Re) = 0.54Re0.667

120,000 < Re < 400,000,

(4.11)

F(Re) = 0.151Re0.775

To avoid discontinuities in the functional variation at the limits of these three Reynolds number ranges, the function may be better represented by the following smooth curve expression: F(Re) = 2Re'''

(

1f R;J0.5

(4.11a)

HOLGERMARTIN

16

Re FIG.8. Heat and mass transfer between a circular plate and an impinging jet from a single round nozzle [l] 2.5 < r/D < 7.5; 2 < H/D < 12. 0 Schliinder and Gnielinski; A Petzold; Gardon and Cobonpue; V Brdlick and Savin; 0 Smirnov et d,

Range of validity (Eqs. (4.10) with (4.11) or (4.11a)):

2000 IRe I 400,000

2.5 I rjD 5 7.5

(0.04 2 f 2 0.004)

2 I HID 5 12

Equation (4.11a) fits the data in Fig. 8 somewhat better than the function F(Re) = 1 S F e + 0.089Reo.*proposed by Brauer and Mewes, who recommended Eq. (4.10) in their book on mass transfer [46] and in a review article for engineering practice [47]. Especially in the midrange of Reynolds numbers, lo4 < Re < lo5, Brauer's function deviates systematically by up to 10% from the mean curve through the data points in Fig. 8. For distances from the stagnation point less than 2.5 nozzle diameters (or relative nozzle areas greater than 4%) one may use the graphic representation of the integral mean transfer coefficients given in Figs. 9 and 10, containing the whole range of variables investigated by Schliinder and Gnielinski [11. This graphic correlation is of the form

(K)mN = (E)= F,(Re, r/D)k(HID, rlD) ~ ~ 0 . 4 2

~ ~ 0 . 4 2 SRN

(4.12)

where k(HID, rlD) has the character of a correction function, describing the relatively weak influence of the nozzle-to-plate distance HID that vanishes completely for larger radial distances. For HID = 7.5, the correction function

IMPINGING JET

FLOWHEATAND MASSTRANSFER

17

Re FIG.9. Heat and mass transfer for impinging flow from single round nozzles at H I D = 7.5 Gardon and Cobonpue; V V Brdlick and Savin (r/D = 2.34, 3.75,6.25); +V Smirnov rt ul. ( r / D = 2.24, 3.75,9.6).

[l]. 00 Schlllnder and Gnielinski: A A Petzold;

1.2 1.1

k

1.0

0.9 08 D.7

0

6

2

75 8

10

12

ti/a

FIG.10. Correction function K ( H / D , r / D )to Fig. 9 for relative nozzle-to-plate distances H I D 1, - --2 , -3 . -.-.5, .- -.7. other than 7.5 [l], r / D : . . - - . . O .

takes the value of unity by definition: 47.5, r / D ) 3 1

(4.13)

The power function of Pr (or Sc) with exponent 0.42 was found by comparison of the mass transfer measurements [l] (water-air Sc = 0.59) with the data on heat transfer to air [30, 32, 441 (Pr = 0.7) and to water [45] (Pr c 7). According to Jeschar and PBtke [31] even the local mass transfer coefficients determined by Rao and Trass [29] with trans-cinnamic acidwater (Sc = 900) agree quite well in the wall jet region with the air-water

18

HOLGER MARTIN

or naphthalene-air (Sc = 2.5) data when reduced with SC'.~'. This agreement must be considered to be somewhat fortuitous since the trans-cinnamic acid surface probably was eroded by the water jet at least in the stagnation zone, resulting in an unusual high dependence on the Reynolds number. (Sh (stagnation zone) Re'*06).Furthermore, the material boundary conditions in Rao and Trass [29] contrary to the other investigations were of a type close to case I in Fig. 3.

-

2. Single Slot Nozzle (SSN) The local mass transfer data of Schliinder et al. [S], partly shown in Fig. 5, were averaged using Eq. (4.3) and correlated by the following empirical equation [ 5 ] : 1.53

~ ~ 0 . 4 2

SSN

m = 0.695

XIS

+ H / S + 1.39

Rem(X/S.H I S )

- (x/S + (H/S)1.33+ 3.06)-'

(4.14)

Range of validity : 3000 IRe I 90,000 2 Ix/S I 25

(0.125 2 f 2 0.01)

2 I HIS 5 10 The exponent of Re, being dependent here on the geometric variables-as it really should be for the round nozzle too (see Fig. 9)-varies from 0.56 to 0.68 in the given range of validity. Equation (4.14) fits nearly all data of Schliinder et al. [ 5 ] within & 15% as shown in Fig. 11. This equation, contrary to that for the SRN (4.10), may even be used in the stagnation zone (x/S < 2). For small nozzle-to-plate distances and high Reynolds numbers, heat and mass transfer coefficients calculated from Eq. (4.14) with x/S = 0 (stagnation point) may differ from measured ones by about 3540%.

B. EQUATIONS FOR ARRAYS OF NOZZLES 1. Arrays of Round Nozzles ( A R N ) Kratzsch [2] compared his own experimental results, obtained in an industrial impinging jet dryer, and the data of Glaser [12], Gardon and Cobonpue [30], Ott [48], Hilgeroth [49], and Freidman and Mueller [50] with the SRN equation (Eq. (4.10) with (4.11)) replacing r / D by l/Tf (see Eq. (4.8)).

IMPINGING JET FLOW HEATAND MASSTRANSFER

19

Shl,

100

measured

150

2oy)

FIG.1 1 . Mass transfer for impinging flow from single slot nozzles. Comparison between measured Sherwood numbers and those calculated from Eq. (4.14)[S]. a B = 20 mm; 0 B = 10 mm; 0 B = 5 mm; V B = 2 mm; Gardon and Akfirat V B = 3.2 mm; Korger and Kiiiek 0 B = 5mm.

Starting his comparison with Gardon’s [30] correlation for arrays of round nozzles, he found good agreement between the transfer coefficients for arrays and single nozzles as long as the relative nozzle-to-plate distance HID remains below a certain limiting value (H/D)lim.For larger distances H / D > (H/Dkmthe SRN transfer coefficients decrease more rapidly with HID than the corresponding single nozzle values. The limiting distance (H/&, is a function of the relative nozzle area f given by Krotzsch [2] as (H/D)Iim

=

0.61J7

(4.15)

To account for these differences between arrays and single round nozzles he developed a simple array correction function K, depending on nothing

HOLGER MARTIN

20

but the ratio of HID to its limiting value (H/D),i,,,.

-0.3

,

o m

1

0.6/$

(4.16) -

The geometric function G(r/D, HID) from Eq. (4.10) with 1/J4f instead of r/D becomes =

1

+

1 - 2.2J7 0.2(H/D - 6 ) f l

(4.17)

So, the mean integral heat and mass transfer coefficients for arrays of round nozzles may be computed using the following equation [2]:

(K)

~ ~ 0 . 4 2 = ARN

(=) ~ ~ 0 . 4 2 ARN

= KA(H/D,f)G(H/D, f)F(Re)

(4.18)

A comparison of heat and mass transfer data of several investigators with Eq. (4.18)as given in Fig. 12a after Krotzsch [2] shows an agreement that is sufficient for most technical applications. In the lower range of Reynolds numbers, however, the retention of the Reynolds function from the single round nozzle (Eq. (4.11) ) leads to systematic deviations from the experimental data. A better representation of experimental findings will be obtained when Krotzsch's equation is slightly modified by replacing the SRN Reynolds

, FIG.12a. Heat and mass transfer for impinging flow from arrays of round nozzles (and orifices) [2].

IMPINGING JETFLOWHEATAND MASSTRANSFER

,.

2 0

21

/

011

FIG.12b. Same representation of data as in Fig. 12a with the modified Re function after Eq. (4.19).

function (after Eq. (4.11)) in the whole range by one simple power relation F(Re),,,

= 0.5Re2’3,

2000

-= Re < 100,OOO

(4.19)

The straight line in Fig. 12b represents Eq. (4.19), the data points being the same as in Fig. 12a. The discontinuous description of the array correction function K, (Eq. (4.16)) as given by Krotzsch may be replaced by the superposition formula (see Fig. 13): (4.20)

0

, 0

FIG.13. Representation of KriStzsch’s [2] array correction function by a single expression, Eq. (4.20).

HOLGER MARTIN

22

This modified correction function has the advantage of avoiding the discontinuities in the functional variation of the transfer coefficients. In a computer program, for example, this will save relational test and branching operations. With these two modifications to the representation in Kratzsch [2] the mean heat and mass transfer coefficients for impinging flow from regular (square or hexagonal) arrays of round nozzles may be calculated from ~ ~ 0 . 4 2

ARN

with K(H/D, f)from Eq. (4.20). Range of validity:

2000

< Re I100,000

0.004 If I0.04 2 IHID I12

For arrays of round orifices (ARO) consisting of more or less sharp-edged holes instead of nozzles, the jet contracts immediately after the orifice exit. Equation (4.21) may then be used here too if the contracted cross-sectional area is used instead of the geometric one. So, the exit velocity mythe orifice diameter D,and the relative orifice area f must be replaced by the corresponding values W‘ = f i l e ;

D’ = DJ;;

f ‘ = fi

Here E means the contraction coefficient, i.e., the ratio of the narrowest cross section of the jet to the geometric orifice exit cross section.

(4.21a) 2. Arrays of Slot Nozzles ( A S N )

Mainly to investigate the influence of the outlet flow conditions (see Section

V), new mass transfer experiments with arrays of slot nozzles were carried

out [9]. The experimental setup is shown schematically in Fig. 14. Table I1 contains the geometric data of the arrays used in this investigation. All

IMPINGING JET FLOWHEAT AND

MASSTRANSFER

23

FIG.14. Experimental setup used in [9]. TABLE I1

ARRAYS OF SLOT NOZZLES INVESTIGATED LN [9]" (mm) 3

LT (mm)

f

(%)

A , (mm)

ASN number

85

3.5

210

1.5 1.5 1.5

170 125 85

0.9 1.2 1.8

140

3 3 3

170 125 85

1.8 2.4 3.5

10 14 6

6 6 6

170 125 85

3.5 4.8 7.1

9 13 5

12 12 12

170 125 85

7.1 9.6 14.1

8 12 4

15 18

85 85

17.6 21.2

16 17

8>B>3 B( v )

85

5.9

5

3b

11 15 7

18'

' All arrays, I

= 750 mm. Used in preliminary test and flow measurements only. ' See Section VIII and Fig. 38.

arrays had a base area of 1.02 x 1.5 m2and consisted of six, eight, or twelve parallel slots with a slot length 21 = 1.5 m. Arrays of this size may be used as modules in industrial dryers. To simulate conditions encountered when processing continuous sheets of material

HOLGER MARTIN

24

passing underneath the nozzles the outlet flow was restricted to two directions (ky in Fig. 14) by boxing in two of the sides of the rectangular arrays. The slots were slightly converging with well-rounded intakes (not shown in Fig. 14). The height of the outlet channels AK was 140 mm (except for arrays 3 and 18), providing relatively good outlet flow conditions even for small nozzle-to-plate distances H. Some of the experimental runs were carried out with the outlet channels partly blocked by boards (140 mm 2 A , 2 0). This allowed varying of the cross-sectional area for the outlet flow F A independently from the nozzle-to-plate distance H . The plate, 1.02 x 1.4 mz in size, was made of a 5-mm-thick capillary porous ceramic material cemented on an insulating, impermeable base of polystyrene hard foam, and divided in the y direction into 14 strips each 100 mm wide (see Fig. 14). The strips were wetted with distilled water and then positioned under the array of nozzles from which jets of air impinged upon them. The amount of water evaporating per unit time and area was measured by weighing the strips twice in a certain time interval during the period of constant drying rate. Thermocouples fitted closely under the ceramic surface were used to measure surface temperatures So. From these, the partial pressures P , , ~or the mole fractions Zl,o = pl,o/p were calculated after the saturation vapor pressure relation by putting = pi”(9,). Also, the temperature, moisture, and flowrate of the air were measured. From the measured quantities mass transfer coefficients p1were calculated using Eq. (2.4a). These are integral mean PI values over x (one strip covers at least six nozzle-to-nozzle spacings LT) and local ones in the outlet flow direction y. In practical applications the continuous movement of the material will totally equalize the local variation of the transfer coefficients over the x direction. Therefore knowledge of integral mean values over x is sufficient. In contrast, any variations in the y direction, if existing, are not equalized by the movement of the material and may lead to the product being overheated at certain points and a corresponding worsening of the product quality. The results concerning the local variation in the y direction will be treated in Section V. It could be shown [9] that the integral mean values, averaged over the whole area, were not affected by the outlet flow when the relative outlet area fA, i.e., the ratio of the outlet cross section FA to the nozzle exit cross section B1 is greater than about one. This relation ( f A = FA/(BI) > 1) was met in nearly all runs except those with the outlet channels blocked. The nozzle-to-plate distance H was varied in four or five steps from 15 to 210 mm. The mass flowrates of air ritZ(relative to the base area of 1.02 x 1.5 mZ)had the following values: m2 [kg/(m2 s)]

0.176 0.333 0.667

1.000

1.333 2.0

IMPINGINGJET FLOWHEATAND MASSTRANSFER

25

From these the nozzle exit velocity is calculated by m = k J p 2 f.Theoretically, it would have been possible to reach nozzle exit velocities up to about 200 m/s (f = 0.009, p2 z 1.1 kg/m3). This would have exceeded, however, the power of the blower motor ( - mZ3/f2). So the highest mass flow rates could not be reached for arrays with small relative nozzle area, and the nozzle exit velocities were in the range 100 m/s 2 w 2 1.5 m/s. As in the earlier experiments [l, 51 the air was preheated to about 40°C yielding surface temperatures on the moist porous plate very close to the temperature of the surroundings to avoid any heat losses or gains. For more details on the experimental setup and the measuring technique, see Martin [9]. An attempt was made to correlate the integral mean mass transfer coefficients for the arrays of slot nozzles by the SSN equation (4.14)accounting for eventual differences between array and single nozzle by a correction function. From a comparison of the data with Eq.(4.14)with x/S replaced by 1/4f it was found that such a correction function would depend on all three variables HIS, f,and Re. A representation in this form would have become unnecessarily complicated, all the more since it turned out that the dependence on Reynolds number, contrary to the case of a single nozzle, was better fitted by a constant exponent. Within the range of Reynolds numbers investigated, rn was found to be two-thirds, independent o f f and HIS. Plotting %,/Re2'3 vs. the relative nozzle area f with the relative nozzle-to-plate distance H / S as parameter, showed maxima at certain values f = fo, depending on HIS (see Fig. 15).

0 10 Re2'3

0 05

002

0 01

002

005

01

0.2

f

FIG.15. Mass transfer for impinging flow from arrays of slot nozzles. Influence of the geometric variables [ 9 ] .

26

HOLGER MARTIN

The correlation was given in terms of these values f :

with fo(H/S) = [60 + 4(H/S - 2)2]-1/2.Range of validity: 1500 IRe 5 40,000

0.008 If

S

2Sfo(H/S)

1 5 HIS I40

The curves in Fig. 15 are calculated from Eq. (4.22). As shown by Fig. 16 this equation represents the own data within about f15%. Most of the data available from other investigators [34, 35, 50-521 are also in reasonable agreement with this correlation (see Fig. 17). For more or less sharpedged slot orifices instead of nozzles, the corresponding values for myS,and f are w’ = W / E , S’ = SC,and f’= f. (see Section IV.B.l for ARO). Contrary to the round orifice, here the Reynolds number is unaffected by the jet contraction (w’S’ = mS).

FIG.16. Mass transfer data from [9] compared with Eq. (4.22).

IMPINGING

JET FLOWHEAT AND MASSTRANSFER

5

n

20

50

NTi ~5, c apro.(3 ~,

calculated

27

roo

FIG.17. Heat and mass transfer data from various authors compared with Eq. (4.22).

V. Influence of Outlet Flow Conditions on Transfer Coefficientsfor Arrays of Nozzles

The correlations for mean heat and mass transfer coefficients to impinging flow from arrays of nozzles (Eqs. (4.21) and (4.22)) are valid for good outlet flow conditions. Clearly a free outlet directly upward between the nozzles would be most favorable. When the air is forced, however, to flow laterally over the width of the material (as shown in Figs. 6 and 14), the outlet stream may significantly influence the whole flow field and consequently also the temperature and concentration fields. The effect of this outlet flow will be treated first for arrays of slot nozzles since the flow pattern is somewhat simpler to understand in this case. Inside a “flow tube” of cross section FA and length 21 (see Fig. 14) the variables of the flow field (w, u, p) are considered to be functions of y only [6, 91. This simplifying treatment will be sufficient to clear up the leading features of the influence of outlet flow on heat and mass transfer. A mass balance over a volume element FAd y of the flow tube (with constant density p ) gives uFA

+ w B d y = (u + du)FA

(5.1)

HOLGER MARTIN

28

or

F A du w(y) =-(5.la) B dY The continuous stream of air into the flow tube over its whole length (w(y)) yields an acceleration of the outlet flow in the y direction (duldy). So, the velocity u of the outlet flow will increase monotonically from u ( y = 0) = 0 to a final value u ( y = 1) = u, which may be obtained by integrating the mass balance Eq. (5.1) : u

=-s’ B FA

W(Y) 0

BZ dy = FA

The pressure gradient in the y direction, resulting from the acceleration and the friction of the outlet flow, may be determined by applying a momentum or force balance on the volume element:

The fraction of this pressure gradient caused by the friction force K , is assumed to be proportional to the square of the local outlet flow velocity. Analogously to tube flow we write

where l Ris a friction factor and dh is the hydraulic diameter of the flow tube. The missing third equation to determine the three unknown functions of y (w(y), u( y), p( y ) ) may be obtained by writing an energy balance over the nozzle volume between a cross section L, dy upstream the nozzle in the plenum chamber and the nozzle exit cross section B d y : PO

+

(P/2)w02

=P

+ (p/2)w2 f

CD(p/2)w2

(5.5)

Herein p o is the constant pressure in the plenum chamber upstream of the nozzles and wo is the very small velocity there (wo = fw by continuity). The factor CD accounts for pressure losses in the nozzle. PO - P = (l - f

2

+ CD)(P/2)w2

(5.5a)

The square of the relative nozzle area f 2 = (B/LT)’ is negligible against 1 in practical cases ( f 2 = (T,>fl(V,> + (T,T,) The term is sometimes referred to as a dispersion vector, and is discussed elsewhere in great detail [43, 491. Substitution of Eq. (1I.C-49)

(CV,)

* Note that d(C,)/at

= C,(af,/at) and V(C,) = Coveowhen C , is a constant.

' See Eqs. (10)-(16) of Gray [47].

A THEORY OF DRYING IN POROUS MEDIA

145

into Eq. (1I.C-45)and rearranging leads to

a -

at

(Pshj?)

+ v - (PphpVp)

Substitutionof this equation into Eq. (1I.C-43)allows us to write the thermal energy equation for the liquid phase as*

* Here we have treated [ l i p . over the area A B y .

+ ( C , ) ~ ( ( T ~-) T,,')] P as a constant with respect to integration

146

STEPHEN WHITAKER

Here we have again used Eq. (1I.C-44)to represent h, in terms of h," and (cP),(TB- Tp").The fourth and fifth terms on the left-hand side of Eq. (1I.C-52)will cancel when the temperature in the liquid phase is uniform, i.e., T , = (T,)O = constant

but for the general case we must again use Eq. (1I.C-47a) to obtain the simplified form of (II.C.-52):

-

Here we have expressed V ( q p ) in terms of T , by referring to Eq. (1I.C-20) and interchanging the subscripts 0 and /?.We should note that the governing differential equation for the liquid temperature (T,)p contains several terms for which constitutive equations are required. The dispersion term pP(cp), V (F,VB) is usually modeled as a diffusion mechanism, although more complicated models have been suggested [43, 491. The area integrals of T , on the right-hand side of Eq. (1I.C-53)are generally taken to be proportional and incorporated into an effective thermal conductivity. The to V(C,( interphase flux terms can be modeled as indicated by Eq. (II.C-21), thus requiring the expetimental determination of film heat transfer coefficients. For the present we shall ignore these difficulties and simply note that Eq. (I1.C-53)is the proper form of the liquid phase thermal energy equation in a porous media. We now turn our attention to the y phase, and being with the continuity equation as given by Eq. (1I.A-12):

3at3 + v

*

(puvy) = 0

(1I.C-54)

We need only repeat the analysis given by Eq. (II.C-22)-(11.C-26)to obtain at

+V

(pyvy>

+

1 JAY@

py(vy- w) * nys d A = 0

(1I.C-55)

A THEORY OF DRYING IN POROUS MEDIA

147

Further simplification of this result is not possible since py depends on both the temperature and composition of the gas phase. Representation in terms of the intrinsic phase average is accomplished by expressing the point functions as p, = (p,)' + p, in they phase (1I.C-56a) p, = ij, = 0

v, = (v,)

v,

=

I

v,

=

in the aandpphases

(1I.C-56b)

in the y phase

(1I.C-57a)

+ 7, 0

in the a and

p phases

(1I.C-57b)

In addition we can use the definition (P,) = .,(P,>'

(1I.C-58)

along with Eqs. (1I.C-56) and (1I.C-57)to express Eq. (1I.C-55) as

(1I.C-59) There is no advantage in representing the area integral in terms of the intrinsic phase average (p,)' and the phase average (v,), so we have left that term unchanged. In general we expect p, and V, to be much smaller than (p,)' and (v,) in the y phase, and we state this as a generally plausible assumption: A.1.

$m

"

0,

p, and Y

and qm
C, a(T>/at +

Aktp(k)

=

v * {[EUkU+ k , ( K ,

-

K,)1 V ( T ) ) (1II.B-22)

This result is nearly identical to the thermal energy equation of Berger and Pei*; however, they assumed that &, + k,(K, - KY)was constant. On the basis of the functional dependence of 7, one can construct arguments to justify their assumption provided the functional dependence on a( T)/at is taken to be negligible. In general we would expect that conduction through both the solid and liquid phases would be important, and that Keff as given by Eq. (1II.B-15) would be the appropriate representation for the effective thermal conductivity tensor. Further simplification of our total thermal energy equation can be obtained if we make the reasonable assumption that diffusive transport of thermal energy in the gas phase is negligible: Diffusive transport of thermal energy in the gas phase is negligible. This allows us to write

A.7

(Pivi)

= (Pivy)

(1II.B-23)

and the representations given by Eqs. (1I.C-57)and (1I.C-68)lead to (Pivi)

= (Pi>’(Vy>

* See Eq. (7) of Berger and Pei [31].

+

(1II.B-24)

A THEORY OF DRYING IN POROUS MEDIA

163

Invoking assumption A.2 allows us to further simplify this expression to (Pivi)

= '(vy>

(1II.B-25)

where the dispersion term, if considered important, could be lumped in with the representation given by Eq. (1II.B-16). Using Eq. (1II.B-25) we now express the total thermal energy equation as

+

(1II.B-26) V . (KTff V( T ) ) (a) The traditional definition of the heat capacity of a gas mixture is given by =

i=N ~ r ( ~ p = ) y

1

Pi(cp)i

(I1I.B-27)

i= 1

and we shall again make use of assumption A.2 to derive immediately (1II.B-28) Eq. (1II.B-26)can now be expressed as

= V * (KTff * V(

T))

+ (a)

(1II.B-29)

This certainly appears to be an attractive result, for our total thermal energy equation now has essentially the same form as Eq. (11-5) when that equation is expressed in terms of the temperature. Certainly a great deal of expertise has been developed in the methods of solving equations such as Eq. (1II.B-29); however, we have buried most of our problems in the total effective thermal conductivity tensor K&, and these problems are not at all trivial. The experimental determination of the nine components of KTff represents an overwhelming and somewhat unreasonable task upon which one might embark. It seems plausible that most anisotropic materials encountered in drying could at least be modeled as transversely isotropic (wood is a classic example), thus there would be a principle axis coordinate system in which only the diagonal elements of Kzff would be nonzero and only two of the three components would be distinct. Even with that simplification (or with the simplification of isotropy) the experimental determination of the total thermal conductivity tensor remains a very difficult task. It should be kept in mind that K;f,, consists of a conductive part and a dispersive part as indicated in Eq. (1II.B-18).A review of the subject of thermal

164

STEPHEN WHITAKER

conductivity of heterogeneous materials has been given by Gorring and Churchill [SO] and some recent experiments on the effective thermal conductivity of saturated metal wicks are described by Singh et al. [51]. The problem of dispersion in two phase systems has received a great deal of attention*; however, dispersion in three-phase systems comparable to that encountered in drying processes does not appear to have been studied either experimentally or theoretically.? C. THERMODYNAMIC RELATIONS In order to connect the total thermal energy equation with the gas phase diffusion equation given by Eq. (1I.C-72) we need to state some thermodynamic relations. We have already inferred that the gas phase is to be treated as ideal (see Eq. (1I.A-3) and restriction R.12 following Eq. (1I.C-75) so that species density can be determined by pi = piRiT, i = 1,2,. . . (1II.C-1) Here pi is the partial pressure of the ith species and R, is the gas constant for the ith species. We can form the intrinsic phase average (see Eq. (1I.C-5)) of Eq. (111.C-l),impose assumptions A.2 and A.3 to obtain (Pi)' = (Pi)YRi(T), i = 1,2,. . . (I1I.C-2) The only other thermodynamic relation required is the vapor pressuretemperature relation for the vaporizing species. When the gas-liquid interface is flat, the vapor pressure can be adequately represented by the ClausiusClapeyron equation$: P1 =

PI0 exp[

;( k)]

-R, Ah",,

-

(II1.C-3)

Here plo is the vapor pressure at the reference temperature To.When surface tension effects are important, one must take into account the effect of curvature and surface tension on the vapor pressure-temperature relation. Defay et a/. [40, p. 2371 suggest that in the capillary condensation region the vapor pressure isotherms can be represented by the Kelvin equation in the form (II I.C-4) P1 = PI0 exP(-2opylrPpw-) Here r is a characteristic length which must be determined experimentally as a function of E ~ and , opyis the interfacial tension of the gas-liquid interface. * See, for example, Whitaker [43,49], Greenkorn and Kessler [52], Miyauchi and Kikuchi [53], and Subramanian e t a / . [54].

'

A recent presentation by Okazaki er ul. [53a] has greatly improved our ability to predict effective thermal conductivities for drying porous media. t Here we have used i = 1 to designate water in the gas phase.

A THEORY OF DRYING I N POROUS MEDIA

165

Hysteresis can be important so that r should always be determined for decreasing values of cp when Eq. (1II.C-4) is to be used in the analysis of a drying process. Combining the Kelvin equation and the Clausius-Clapeyron equation would suggest that vapor pressure data for porous media could be correlated by the expression

(+ k)]} -

(1II.C-5)

Forming the intrinsic phase average and making use of assumptions A.l and A.3 eventually leads to

(11I.C-6) Here we have modified assumption A.l to read that ‘7. is not only small compared to ( T ) but that it is negligible compared to ( T ) . This should be quite satisfactory in Eq. (1II.C-6) where we are dealing directly with absolute temperature and not time or spatial derivatives of the absolute temperature. At this point we are ready to proceed from our study of energy transport during drying to the study of mass transport in the gas phase. IV. Mass Transport in the Gas Phase In Section I1 we derived the volume average form of the gas phase continuity equation to obtain

(IV-1) and the species continuity equation was expressed as

(IV-2) where

166

STEPHEN WHITAKER

These three equations were given previously as Eqs. (II.C-61), (1I.C-72), and (II.C-71), respectively. If we make use of Eqs. (1I.B-22) and (1II.A-18) to write

v J4 8

(m) = --

Py(Vy

- w) * " y S d A

(IV-4)

we can simplify the notation of Eq. (IV-1) to obtain

a $'y(Py>y)

+ v * ((Py>'(Vy>)

= (fit>

(IV-5)

Use of the boundary conditions given by Eqs. (1I.B-24) and (1I.B-25) allows for similar simplifications in the two diffusion equations leading to

+

q

+ n,] The total continuity equation, as given by Eq. (IV-5), is ready for application to practical problems; however, the two diffusion equations are in need of further analysis which will be given in the following paragraphs. A. THEGASPHASEDIFFUSION EQUATION

Our objective here is to put the diffusion equations, Eqs. (IV-6) and (IV-7), in a form that is suitable for analysis or comparison with experiment. In and ( P i V Y ) . The species particular we are concerned with the terms dispersion term is handled in exactly the same manner as the thermal dispersion term, and following Eq. (1II.B-16) we write

ai

(Pivy)

= - ( ~ y ) ~ D g*) V((Pi>'/(Py>')

(1V.A-1)

The arguments in favor of this representation are essentially identical to those presented in Section 1II.B and will not be repeated here.

A THEORY OF DRYING IN

POROUS

MEDIA

167

In attacking the area integrals in the expression for fiigiven by Eq. (IV-3) we make use of Eq. (1II.A-19)in order to write

This allows us to express fiias

(1V.A-2)

(1V.A-3) Note that in going from Eq. (IV-3) to Eq. (1V.A-3) we have freely moved the term (p,)’ inside and outside the area integrals. If we express the phase density ( p i ) in Eq. (IV-2) in terms of the intrinsic phase density ( P i ) = Cy(Pi)’

(1V.A-4)

and substitute Eq. (1V.A-3), we obtain the following form for the species continuity equation:

(1V.A-5) Here we have also incorporated Eq. (1V.A-I) to represent the dispersion and fii now has a new definition given by

STEPHEN WHITAKER

168

where the term involving Vey now appears in the molecular diffusion term. At this point we again follow the arguments given in Section 1II.B and hope that fii can be adequately represented in terms of gradients of ( p i ) y / ( p y ) y . While this seems plausible for the terms involving pi, what about the latter two terms involving p,? The functional dependence of FY can be inferred from Eqs. (IV-1) and (II.A-12), and it is clear that tiywould depend on gradients of (p,)’, not ( p i ) ’ . The gradient of (p,)’ would be related to gradients of (pl)y and (p2)Y through Eq. (11-6). If we are concerned with species 1, we could relate gradients of ( p 2 ) ’ to gradients of ( T ) and ( p ) the total pressure, through Eqs. (1II.C-2) and (1II.C-6). Thus we can see vague arguments arising in favor of expressing fil (for example) in terms of gradients of ( p l ) ? , ( T ) , and (p); however, we shall assume that the dominant factor is the particular species in question and list our eighth assumption as: A.S

fii is a linear function of

v((pi>’/

1 = --

in place of Eq. (1V.B-21).

PY

K , *V[E~((P;, - p,/r=o>Y + ~ ~ - (d~lr=o)')I 4

A THEORY OF DRYING IN POROUS MEDIA

173

We would now like to make use of a special form of the averaging theorem given in Section 1I.C as Eq. (1I.C-16). For the y phase we would express this result as (1V.B-15) where A,, represents the y phase entrances and exits contained in the averaging volume V .If we substitute P, for t,b, in the left-hand side of Eq. (1V.B-15) and -p,m, (v,) for t,b, in the right-hand side, we obtain

-

v(py>=

1

--

A:,

p,m,

(vr)ny d A

(1V.B- 16)

Here we can argue that the spatial variations in (v,) are negligible over the surface A,, and take ,u),to be a constant in order to write Eq. (1V.B-16) as (1V.B-17) We now identify the term in braces as K:,

I

and express our result in the form

VV,> = - P g

(1V.B-18)

(VJ

Following arguments given elsewhere [44] we assume that K.;’ inverse designated by K, allowing us to write 1

(v,)= --K PY

y *

VV,>

has an

(1V.B- 19)

Expressing the phase average in terms of the intrinsic phase average leads to

(v,>

1

= -- K,

Pr

V[r,(Py>’]

(1V.B-20)

and use of Eq. (1V.B-6) yields 1

(V,>

= -- K,

Ps

‘ V [ E ~ ( ( P-, p0>’

+ p,(+)’)]

(1V.B-21)

To within an arbitrary constant, we can express the gravitational potential function as 4 = - r’g (IV. B-22) and the intrinsic phase average becomes (1V.B-23)

STEPHEN WHITAKER

174

Since g is a constant, this reduces to

(4)’

(1V.B-24)

= - 81 V.’]

V ’81

(1V.B-25)

The interpretation of the quantity Vr is given by Vr

=

U

(1V.B-26)

where U is the unit tensor. In dealing with volume averaged functions it seems consistent to continue this interpretation inasmuch as (r) is the position vector for the volume averaged functions. Thus we write V(r) = U (1V.B-27) and Eq. (1V.B-25) takes the form

At this point we must be careful to choose the reference pressure p o as the intrinsic phase average pressure at the point (r) = 0 so that the term (p y - po)’ - py(r) g is zero under hydrostatic conditions. From a practical point of view it is generally assumed that the first part of the right-hand side of Eq. (1V.B-28) dominates and the gas phase velocity is expressed as 1 (1V.B-29) (v,) = -- K y .{€’[V(P, - Po)’ - P$I} K thus neglecting the term involving gradients in the gas phase volume fraction. Whether this is a suitable approximation for drying processes remains to be seen.

-

* Note once again that we consider it to be sufficient to treat the gas flow as incompressible.

A THEORY OF DRYING IN POROUS MEDIA

175

At this point our analysis of the gas phase mass transport is complete but complex. In the next section we shall go on to the study of convective transport in the liquid phase and suggest some simplifications that one might make in order to provide a more tractable theory. V. Convective Transport in the Liquid Phase

During the initial stages of the drying of a saturated porous media, it seems clear that liquid motion by capillary action is the dominant mechanism of moisture movement. A description of the physical phenomena was given by Comings and Sherwood [17], and a theoretical analysis is required in order to complete our treatment of the drying process. In this section we shall derive Darcy’s law for the discontinuous fluid in a two-fluid system, and then go on to suggest a constitutive equation for the forces acting on the liquid phase. A. DARCY’S LAWFOR

A

DISCONTINUOUS PHASE

The development given here will parallel that of the previous section; however, there will be an added difficulty because the liquid phase is taken to be discontinuous and the complications of multiphase flow phenomena are encountered. The subject has been discussed before from a theoretical point of view by Slattery [46]; however, Slattery does not obtain the traditional form of Darcy’s law which was given in Section 1V.B and which will be incorporated into our analysis of the liquid phase motion. We begin by following Eqs. (IV.B-l)-(IV.B-5) to obtain for the liquid phase VP, = p, v 2 v s (V.A-1) where (V.A-2) p/J = Ps - P o + Ps4 Here the reference pressure po is the same reference pressure given in Eq. (1V.B-6). To obtain a general expression for Pa we refer to the arbitrary curve shown in Fig. V-1 and form the scalar product of Eq. (V.A-1) with the unit tangent vector I to obtain dPsfds = p,A-(V2vp)

(V.A-3)

Referring to Eq. (1V.B-10) we express the point velocity as vj3

=

M, * Here the pressure P,(r) is no longer given to within an arbitrary constant but is an absolute value. This can be seen by considering the hydrostatic case where requiring that (v,) = 0 leads to, for a continuous liquid phase in hydrostatic equilibrium, Pp -

Po

+ PS4 + (Pdo

=

0

(V.A-24)

Remembering that ( P , ) ~is given by (Pch

= Po

- P,lr=o

(V.A-25)

quickly leads us to the following result for a continuous liquid phase in hydrostatic equilibrium (V.A-26) Pa = Ps(,=o + Pa4 * See Eqs. (1V.B-22) and (1V.B-24).

STEPHEN WHITAKER

180

which is, of course, the hydrostatic pressure distribution in a continuous liquid phase. We can now rewrite Eq. (V.A-17) as

(V.A-27) P&) - P y ( r N - l ) + P o ( r N - 1 ) - A p N - 1 4 N - 1 = -&mp ’ (vo) where P&) is now given absolutely by Eq. (V.A-27). In order to obtain Darcy’s law for the discontinuous liquid phase we make use of an equation for the fl phase which is analogous to Eq. (1V.B-15) and is written as (V.A-28) Substitution of the left-hand side of Eq. (V.A-27) for t,bp on the left-hand side of Eq. (V.A-28) and substitution of -pflmfl (vp> for @fl on the right-hand side of Eq. (V.A-28) yields

(V.A-29) Repeating the development given by Eqs. (IV.B-16)-(IV.B-19) immediately leads to

(V.A-30) Before going on to a discussion of an appropriate constitutive equation for the terms on the right-hand side of Eq. (V.A-30) we need to examine carefully each of the pressure terms. The phase average can be expressed explicitly as

p,Ar)

-

P?(~N I ) -+ ~~(rrv1) - A

~ r v - 1 4 ~1 -

We should remember that when r locates any point in a continuous region of the fi phase contained in “Y, the vector r N - I is a constant vector and 4Nis a constant function. Let us now think of the volume G(t)as consisting of M distinct volumes within which the discontinuous p phase is contained. We express Vp(t)as qt)= p)+ 1 / ( 2 ) + v(3)+ . . . Jmf) (V.A-32)

A THEORY OF DRYING IN POROUSMEDIA

181

and write Eq (V.A-31) as

Here we should note that rN-l locates the position at which the arbitrary curve shown in Fig. V-1 enters t h e j r s t continuous subregion within the averaging volume, and that rN- locates the position where the curve enters the second continuous subregion, etc. The first term in Eq. (V.A-33) represents the standard /3-phase average which can be expressed as

If we now turn our attention to the second term in Eq. (V.A-33), we note that r N - is constant for integration of the region V1), rN- is constant for integra, This requires that we express this term as tion over the region V 2 )etc.

(V.A-35)

STEPHENWHITAKER

182

Clearly this represents a different kind of an average than the phase average or the intrinsic phase average. For the special case where P , is everywhere a constant designated by Po, Eq. (V.A-35)takes the form

=

Pow. 54)= Po&)

(V.A-36)

We should also note that

V,>' = Po

(V.A-37)

when P , is the constant Po. For the present we will define the average given by Eq. (V.A-35) as Hyes and write 3+!

=

;{J"[*)P7(rN-l)dv

+Jv[*)Py(rN-2)dv+-.

.+SylM,P y ( r N - M ) d v ] (V.A-38)

If there are many subregions of the /lphase within the averaging volume, i.e., M is large, then it would appear that the average given by Eq. (V.A-38) very closely approximates the intrinsic phase average times the liquid volume fraction, i.e., P7yz,,-,( P y ) y ~ B as M -+ 00 .(V.A-39) If there is but one continuous region of the p phase, then Eqs. (V.A-22) and (V.A-23) apply and our interpretation of the average given by Eq. (V.A-38) is irrelevant. But what if there are one, or two, or three subregions of the /3 phase in the averaging volume? Under these circumstances the average given by Eq. (V.A-38) represents a new variable for which we have no governing differential equation and a constitutive equation is in order. We now continue with our analysis of Eq. (V.A-33)and apply the definition given by Eq. (V.A-38) to write (P&)

-

P y h - 1 )

= -

@y

+ Pc(rN-1) - 4%--I&-I) + - (Ps - Py)d.s

where A p N - , has now been explicitly identified as ps where M becomes large we have the interpretations

Py

+

-

Pc

+

- pu.

(PJ'

For the case (V.A-41a)

(PJ'

- (Pp>fl

d (&Jb = +

(V.A-40)

@c

for M

-b

03

(V.A-41b) (V.A-41~)

A THEORY OF DRYING IN POROUS MEDIA

183

The expression for pc results from Eq. (V.A-6) which can be expressed as

The appropriate average of p , would be taken over the area A,, and expressed as

These are intrinsic averages in that they are equal to the function itself if the function is a constant. Because of this it seems reasonable to express the average capillary pressure as (V.A-44) and accept the reasonable interpretation of p, as for M

-+

co

(V.A-45)

If we now express (P,) in terms of the intrinsic phase average and use Eq. (V.A-2),we can express Eq. (V.A-40) as

€,[(P/J - Po)’ + Pp((4>’ - $1 - (By- P o ) + tr,] (V.A-46) On the basis of Eqs. V.A-22) and (V.A-41) we can express the left-hand side of Eq. (V.A-46) as =

- Py(‘N

-

Py(rN-l)

+ Pc(‘N-1)

- APRL4N-1)

for M

00

-+

(V.A-48)

These are important implications for Eq. (V.A-48) indicates (through Eq. (V.A-30)) that (v,) + 0 as M + 00. This indicates that as the /Iphase becomes segmented into smaller and smaller continuous subregions, the liquid flow decreases. This agrees with the observation that little, if any, liquid movement takes place in the pendular state. At this point we can consider Eq. (V.A-30) to be an appropriate form of Darcy’s law for the discontinuous p phase, with the obvious special case

184

STEPHEN WHITAKER

for a continuous liquid phase (V.A-49) resulting when the fl phase is continuous. Clearly a constitutive equation is required for the various pressure and body force terms in Eq. (V.A-30), and in the following paragraphs we shall take up this matter.

B. A CONSTITUTIVE EQUATION FOR ACTING ON THE LIQUID PHASE

THE

FORCES

One of the main difficulties that we encountered in the previous section was the occurrence of the average quantities F,, ii,, and for which there were no governing differential equations or workable definitions. Clearly a constitutive equation, or equations, is in order and it is best to begin our analysis with an examination of the case where the /? phase is continuous, i.e., M = 1. Under these circumstances we can use Eq. (V.A-22) to express the gas phase pressure for a continuous liquid phase as

6

P,(r) - P C W

+ (Pp - P , ) m

=

-(Pc)o

(V.B-1)

and the liquid phase velocity for a continuous liquid phase as given by Eq. (V.A-49) (V.B-2) Strictly speaking the position vector r in Eq. (V.B-1) locates any position on the P-7 interface. The special case where the system is in hydrostatic equilibrium is of considerable interest, for under those circumstances Eq. (1V.B-6) can be used to show that for hydrostatic equilibrium (V.B-3)

P,(r) = 0

and for continuous liquid phase and hydrostatic equilibrium Eq. (V.B-1) reduces to (V.B-4) -PcW + ( P p - P J c m = -(Pc)o This result is essentially identical to that given by Scheidegger [61, p. 671 or Bear [60, p. 4481. The gravitational potential function is usually expressed as* =

(V.B-5)

QZ

so that Eq. (V.B-4)can be written in the form

P&)

= (Pp

- PJSZ

* Here the gravity vector would be expressed as g =

- kg.

(V.B-6)

A THEORY OF DRYING IN POROUS MEDIA

185

where the origin is located at the point where the capillary pressure is zero. If the capillary pressure is measured as a function of saturation sp, then one can use Eq. (V.B-6) to calculate the saturation as a function of z. This is precisely what Ceaglske and Hougen [18] did in their study of the drying of granular solids. One must impose a constraint or a boundary condition on Eq. (V.B-6) and the experimental relation

(V.B-7)

Pc = P A S S )

in order to calculate sp as a function of z. Ceaglske and Hougen imposed the constraint that the nuerage moisture distribution was identical to that determined experimentally. Here we see that the results presented in Figs. 1-2 for the drying of sand indicated that both the liquid and gas phase are continuous and the liquid is in a state of hydrostatic equilibrium. Since drying processes are usually quite slow, it seems reasonable that the liquid phase is in hydrostatic equilibrium. The fact that the liquid phase is continuous is supported by the capillary pressure-saturation curve obtained by Ceaglske and Hougen. Representative curves for drying and imbibition for sand are shown in Fig. V-2. Of particular importance is the fact that the capillary pressure arises extremely rapidly for fractional saturations less than 0.1. The general interpretation [29, p. 2181 of the abrupt rise in p c is the breakdown of the funicular (continuous) state to the pendular (discontinuous) state. It is in this region that we expect M to be increasing from 1 to 2 to 3, etc.

Y G?

1.2

a K

n

tl

>.

3 02

1

Drainage

Imbtbttion

0

0

20

40

60

80

100%

PERCENTAGE LIQUID SATURATION

FIG.V-2. Capillary pressure as a function of saturation.

186

STEPHEN WHITAKER

While one can compare theory and experiment using Eqs. (V.B-6) and (V.B-7), one cannot directly compute the moisture distribution as a function of time. In order to do this one needs the appropriate transport equations and our first order of business will be to develop these equations for the special case of a continuous liquid phase and hydrostatic equilibrium. Returning to Eq. (V.B-2) we repeat the development, step by step, given by Eqs. (IV.B-19)-(IV.B-28) to obtain

(v,)

7

E,[V(P,

= --KO*

PLp

+ [(Pp

- Po>,

- P O Y - Ppgl

+ (PA0

- p/?(r>‘81 V € , l j

(V.B-8)

Here we assume that the capillary pressure (pJ0 can be expressed in terms of the intrinsic phase averages (PA0

= ((PJ’

-

(V.B-9)

(Pp)p)(r>=o

and remember that the reference pressure p o is the intrinsic phase average gas pressure at the origin, so that Eq. (V.B-8) takes the form

+ [(Pp>p

- KPp)”(,,=O

- Pp(0

1

81 VEp

(V.B-10)

We can now express the liquid pressure in terms of the average capillary pressure and the gas phase pressure (V.B-11) ( P p Y = (PJ’ - ( P C ) so that Eq. (V.B-10) can be written in the form

+ [(Ps>p

- KPp>p)

’ 81

1

VEp

(V.B- 1 2)

Throughout this development we should keep in mind that we are dealing with the special case where the liquid phase is continuous. In Section 1V.B we alluded to the possibility that the pressure in the gas phase might indeed be essentially constant, or generally hydrostatic, during a drying process; however, there seemed to be no rational way in which that information could be utilized in order to determine the gas phase velocity field in absence of the laws of mechanics. Our treatment of convective transport in the gas phase indeed requires (see Eq. (1V.B-29)) that (v,) = 0 when the gas

187

A THEORY OF DRYING IN POROUS MEDIA

phase pressure is hydrostatic. Nevertheless we shall make the following assumption about the pressure forces acting on the liquid phase:

A.10

Concerning the forces exerted on the liquid phase, we shall assume that both the gas and liquid pressure distributions are hydrostatic. This assumption allows us to simplify Eq. (V.B-12) to the form,

(V.B-13) indicating that the liquid flow depends entirely on gravity and surface tension forces. The average capillary pressure in Eq. (V.B-13) is given by

(V.B-14) Fbr any given system we expect a,? to be a function of the temperature while rl and r2 will depend on E,, the contact angle, and the structure of the rigid porous matrix. We express these ideas as (p,)

=

.F(( T ) , E,, other parameters)

(V.B-1 5)

The structure of the porous media is difficult to characterize; however, Dullien [62] has been able to correlate permeabilities for single phase flow using the void fraction E, + E? and two pore size distributions. It seems clear that pinning down the “other parameters” for a drying process will be a most difficult task; however, if we restrict our development to the case where the porous media is homogeneous A.11 the “other parameters” in Eq. (V.B-15) are independent of the spatial coordinates and the gradient of (p,) is given by

v(P,> = ( a w a E , ) vE, + (a(~,)/a(v) V(T)

(V.B-W

We designate the two scalars in Eq. (V.B-16) as

k , = -(xPc)/aE,), and express V(p,) as v(Pc)

=

k 8 - $1 - (Ti,

-

+ El}

Po)

(V.B-22) Carrying out the gradient operation and rearranging the terms leads us to* (vp> = -- K,

PLlS

+ [p

-

- E/J

vm,

- Po)

+ Po$

-

P p 6 - (Ti, - Tio) + Bc1 V E ,

B C I

I

(V.B-23)

If we add and subtract the liquid phase reference pressure ((ps)8)(r,= last term on the right-hand side of Eq. V.B-23 we obtain

* See Eqs. (IV.B-22)-(IV.B-28)

for an analysis of the gravitational terms.

to the

A THEORY OF DRYING IN

POROUS

MEDIA

189

Imposing assumption A. 10 simplifies this expression somewhat to yield

-

ccrj, - P o ) + P p 6

- Pc

+ (Pc)ol

1

(V.B-25)

VEP

Here it is important to note that Eq. (V.A-22) provides the relation

(By- Po) + pP$

-

p,

for M

= -(P,)~

1

=

(V.B-26)

so that Eq. (V.B-25) reduces to the case for a continuous liquid film given by Eq. (V.B-10) when M = 1 and the liquid phase pressure distribution is hydrostatic. At this point we can repeat the development given by Eqs. (V.B-10)-(V.B-l3) to obtain

+ (Pp - P , k l

Here we must remember once again that

[(F, - Po) + Pp$

- Pc

+ (PJOl

+

for A4

0

+

1

(V.B-28)

and from Egs. (V.A-48) and (V.A-30) we deduce that

v {€pccPy - P o ) + P p 6 - Bc + (Pc)ol

I

(V.B-29) .p[V(P,> + ( P p - P&l for a In view of these limiting cases it seems appropriate to define a new function 5: by the relation +

v {€p"P,

-

- Po)

-+

+ Pp$

- Bc

1

+ (PA01

=

(5

- l)Ep[V(P,)

+ (Pp

(V.B-30)

where 5 has the property that aB -._ 0

0

2 0.5 -

0

0

as

1

radial position Y

FIG.16. Calculated velocities and pressure gradients for isothermal axial flow in an annulus with zero shear stress at inner wall. Parameter is the ratio of radii K.

is prescribed to be zero (P' = 0), the axial velocity of the inner cylinder V'(K) has to be the larger, the smaller IC is (Fig. 17); for a plane slit (K + l), the velocity gradient is constant and the velocity of the moving wall is twice the average velocity, obviously. Taking different values V,(K) in an annulus of K = 0.4 (Fig. 18), the pressure gradient P' adopts positive or negative values. A zero shear stress at the inner wall or a zero pressure gradient at isothermal flow does not mean that this condition applies to the whole flow channel: due to the thermal development the velocity changes, and accordingly a nonzero shear stress at the inner wall or a nonzero pressure gradient arises. 5 . Experimental Studies

The main motivations for undertaking experimental studies on heat transfer in steady shear flow with open stream lines seem to be: investigatingthe validity of the assumptions made in the analytical studies; information on the thermal boundary conditions, i.e., values of Bi in different applications. The flow geometries chosen for experiments were pipe flow and helical flow (see Table V). The measurable quantities were the flow rate, the pressure

VISCOUSDISSIPATION IN FLOWING MOLTEN POLYMERS

radial position

247

'k

FIG.17. Calculated isothermal velocity profiles in an annulus at zero pressure gradient. Parameter is the ratio of radii K.

"0

0.5

radial position Y

1

FIG.18. Calculated isothermal velocity profiles in an annulus of K = 0.4. Depending on the values Vz,i,the pressure gradient P' adopts positive or negative values.

248

HORSTH. WINTER

profile, the radial temperature distribution at the inlet and at the exit, the thermal boundary conditions ; additionally, for helical flow one could measure the torque and the angular velocity of the cylinders. As input data for the numerical program one needs the flow rate (or a pressure gradient), the properties of the polymer (viscosity q(P, T ) ,thermal diffusivity a(T),thermal conductivity k(T),density p(p, T ) ) ,the melt temperature at the inlet Te(r), and two boundary conditions each for the temperature and the velocity fields. The other data can be used for a check on the validity of the numerical solution. Several of the published experimental studies do not specify the data needed for comparing with analytical solutions. While wall temperatures can be measured quite accurately, the temperature measurements in the flowing molten polymer always contain some systematic errors. A thermocouple mounted on the tip of a probe is placed into the melt stream. The probe is supposed to adopt the melt temperature as closely as possible (zero temperature gradient in the polymer layer next to the probe). Apart from distorting the velocity profile by introducing the probe into the flow, two effects are influencing the temperature measurement: heat conduction along the probe, which requires a heat flux and a temperature gradient in the polymer layer next to the wall of the probe, and viscous dissipation in the polymer around the probe. The error due to conduction along the probe can be excluded by setting the base temperature, where the probe is mounted to the wall of the channel, equal to the temperature at the tip of the probe [107,108]. The error due to dissipation cannot be avoided, but it can be kept small by measuring the melt temperature at positions of very low velocities, i.e., after slowing down the flow in a wide channel and then calculating back to the corresponding temperatures at the exit of the narrow channel by means of the stream function [91,109]. Van Leeuwen [110] studied the applicability ofdifferent probe geometries and found that a probe that is directed upstream parallel to the streamlines of the undisturbed flow gives the most accurate temperature data of the melt. Gerrard et al. [67] pumped a Newtonian fluid (oil) through a narrow capillary ( I , = 0.425 mm and 0.208 mm, 33 I l/r, I 459). They measured the pressure drop, the flow rate, the inlet temperature, the wall temperatures, and the radial temperature distribution at the exit. The calculated values of the pressure drop and the temperature at the exit reportedly agree with the measured values within 5%. The viscosity was taken to be a function of temperature; expansion cooling was neglected in the analysis. Mennig [72] extruded polymer melt (low density polyethylene) through a capillary ( I , = 3.5 mm, l/r, = 225.7) at adiabatic wall conditions. Measured quantities were the temperature in the center of the entering polymer stream, the wall temperature distribution, the radial temperature distribution at the

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

249

exit, and the total pressure drop. The calculated values of the wall temperatures and of the radial temperature distribution exceeded the measured values by about 5%. The viscosity has been taken to be a function of shear rate and of temperature; expansion cooling has been included in the analysis. For capillary flow, Daryanani et al. [75] measured the average heat flux through the wall using an electrical compensation method. From the total pressure drop and the heat flux through the wall, they calculated the average temperature increase between entrance and exit of the capillary. Winter [91] extruded a polymer melt (low density polyethylene) through an annulus ( K = 0.955 and 0.972) with rotating inner cylinder. The measured quantities were the mass flow rate, the pressure distribution, the rotational speed of the inner cylinder, the radial temperature distribution at the entrance and at the exit, four temperatures each at the inner and at the outer wall. As shown in Fig. 19 the developing temperatures have been calculated beginning with the measured temperature distribution at the inlet. For the exit temperature distribution, measured and calculated values agreed up to Y zz 0.75 within 5% of the temperature increase (at the outer wall, 0.75 IY 5 1 the temperature distribution has not been desribed sufficiently with only four temperature readings). The measured and calculated pressure gradients agree within 8%. Expansion cooling has been neglected in the analysis.

FIG.19. Comparison of measured and calculated temperature profiles in helical flow [91].

250

HORSTH. WINTER

C. SHEARFLOWWITH CLOSEDSTREAM LINES The shear flow geometries with closed stream lines studied most widely are circular Couette flow and its limiting case, i.e., plane Couette flow (IC+ 1). The fluid is sheared in the annular gap between two concentric cylinders in relative rotation to each other (Fig. 20). The axial velocity component u, is zero. At time t = 0, the Couette system is started from rest at isothermal conditions with a step in shear rate (Q(t I 0) = 0 and Q(0< t ) = fo = const); alternatively the system might be started with a step in shear stress. Three types of development are superimposed on each other, each of them on a different time scale: Kinematic development: The fluid has to be accelerated until it reaches a velocity and a shear rate independent of time. The kinetic development can be calculated for a Newtonian fluid; a practically constant velocity field is reached after [1111 t

=

ph2/16q

(2.64)

(h is the gap width, q the constant Newtonian viscosity, and p the density).

For viscoelastic liquids an estimate on the duration of kinematic develop-

FIG.20. Flow geometry of circular Couette flow.

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

25 1

ment can be made from the loss and the storage modu1esG”andG’measured in periodic shear experiments at frequency w = l / t [1121 t

>> ~ ( P / G ” ) ” ~ or

t >> h(~/G’)l’~

(2.65)

For startup experiments on polymer melts, the kinematic development generally is assumed to be completed before the rheological and the thermal development has actually started. Rheological development: The viscosity ~ ( 9T, , t) needs some time of deformation at constant shear rate, until it adopts a constant value.

Thermal development: Due to viscous dissipation beginning at time t = 0, the temperatures in the gap rise until the temperature gradients toward the walls are large enough to conduct away all the newly dissipated energy. Convection does not influencethe temperature field because the temperatures along stream lines are constant. 1. Assumptions and System of Equations

The assumptions corresponding tothe ones listed in Section II.B.l are: incompressiblefluid with constant thermal conductivity and diffusivity ; no change in z direction; rotational symmetry (a/a@ = 0); velocity # 0;U, = v, = 0; no slip at walls; inertia negligible; kinematically developed velocity at t = 0; gravity negligible; viscosity measured at constant temperatures and constant shear rate gives applicable instantaneous values of the viscosity during temperature changes and during small changes in shear rate; rheologically developed stress at t = 0.

The stress‘equation of motion and the energy equation become (2.66) (2.67)

The reference values are chosen to be the same as in the helical flow analysis: v = veVi/2; h = ra - ri; 9 = ij/h; i j = ~ 6T o,) .

252

HORSTH . WINTER

The dimensionless variables are velocity

v,

radial position

R = r/ra = (1 - K ) r / h ,

=

y=shear stress

PR8

=

v,P, K

IR I 1,

r - ri , 05Y11, r, - ri

h Gc3==3

uvl

9 = P(T - TO).

temperature

The dimensionless form of the system of equation reads (2.68)

as

-pc

pE aFo

= (1

-

K ) ~ [ & (asR +~ ~ ) k

Na;(Rsxr]. a

v,

(2.69)

The initial conditions are

w,0) = 0,

V,(K 0) = [email protected](R) (2.70) where V',JR) is the kinematically developed velocity at the initial temperature. The boundary conditions are d 9 ( ~ Fo) , ss,i- ~ ( I cFo) , = Bii dR 1--K

1

( ~Fo) , +-1 Ci- K d 9dFo

a q i , FO) 9,, - 9(1, Fo) -- C, d9(1, Fo) = Bi, aR 1-u 1 - K dFo

0 = Fo. (2.71)

V@(K, Fo) = 2 Ve(1, Fo) = 0

The thermal boundary condition is an energy balance of the inner and of the outer wall. The heat flux into the wall is equal to the heat flux out of the wall minus the change of energy stored in the wall. The boundary condition has already been described in Section 1I.A. It is repeated here to show the complete mathematical problem at once (Bi, < 0; Bi,, Ci, C, > 0). 2. Dimensionless Parameters For a description of most of the dimensionless parameters, the reader is referred to Section II.B.2. The Nahme number, Eq. (2.55), compares the dissipation term and the conduction term of the equation of energy. The

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

253

ratio of radii K shows the influence of curvature, and m describes the shear thinning effect of the viscosity. Instead of the Graetz number one introduces as a dimensionless variable the Fourier number

FO = ta/h2

(2.72)

which can be understood as the ratio of the current time of the experiment and the time needed for heat conduction from the center of the channel to the wall. At Fo = 1-4, depending on the thermal boundary conditions, the thermal development is completed. The Fourier number corresponds to Z in the heat transfer problem with open stream lines, where one might define an average residence time f = z/O,:

Z

Z

=

1

za ai = -= h2 = Fo.

h2a,

(2.73)

3. Solution Procedure and Calculated Results The 0 componentLof the equation of motion is the same for annular shear flow with open and with closed stream lines. The kinematically developed velocity V,( Y, 0) at isothermal conditions can be calculated with the existing numerical program of Section 1I.B without any changes. The same is true for the thermally developed case at large times at constant thermal boundary conditions since the conduction and the convection terms are identical in both types of flow. If one replaces Z by Fo and sets V,(Y) = pz = (which is an arbitrary small value to avoid singularities in the program), even the developing velocity V@(Y,Fo), temperature S(Y, Fo), and shear stress PRe( Y , Fo) can formally be taken from the existing program without further considerations; see Table VI. The capacitance parameter, however, has to be included in the thermal boundary condition. The solution procedure is basically the same for steady shear flow with open stream lines and for unsteady shear flow with closed stream lines (Couette system), and it would have been possible to treat it in one special section in the beginning. For two reasons, however, this has not been done in this study: (1) shear flow with open stream lines is much more important for polymer processing; (2) the frequent change from Z to Fo would make the explanations difficult to comprehend. The solution procedure in Section 1I.B is meant to be an example, and it will not be described repeatedly for the corresponding problem in this section. The geometry of a cone-and-plate or a plate-and-plate viscometer cannot be described by the existing shear flow program. Turian and Bird [52-541 estimated the temperature effects in cone-and-plate systems by applying the maximal gap width (at the outer radius) to a plane Couette system with

HORSTH. WINTER

254

isothermal walls. The radial heat conduction, which might diminish the effect of dissipation, is neglected. The development of the temperatures in circular Couette flow is a function of the dimensionless parameters Na,Fo, IC,m,and of the thermal boundary conditions. In Figs. 21 and 22, the influence of the geometry on the developplane slit

- 0.8 >. -

annulus with x = 0.5

\

I Y = 0.999)

9

e

>.

.

6

\< \ \

-

* .

\ \ \\

0 . A\\

\

\ \

FIG.21. Comparison of developing temperatures for plane and for circular Couette flow. The outer wall is taken to be isothermal; the inner wall is close to isothermal (Bii = - 100) and close to adiabatical (Bi, = - 1). Na = 1; rn = 2; C, = 0.

I4 > 9 l%-1

2 c

e

E

2

(u

ff

0.5

6 W z

c

0 -

P O 10-~

lo-*

lo-'

1

dimensionless time Fo

FIG.22. Development of the average temperature in plane and in circular Couette flow. The solid lines correspond to the development with both walls close to isothermal (Bi, = - 1001, and for the dashed lines the inner wall has been taken to be close to adiabatical (Bii = - 1). Na = 1 ; m = 2;C, = 0.

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS 255 ing temperature 9(Y, Fo) will be demonstrated for plane Couette flow 1) and for circular Couette flow with K = 0.5, both with constant temperature of the surroundings equal to the initial temperature (9(Y, 0) = gS,,= 9,,, = 0); the outer wall is taken to be isothermal (Bi, = co) and the inner wall is taken to be close to isothermal (Bi, = -100) and close to adiabatic (Bi, = -l), respectively. The thermal capacitance of the wall is neglected (Ci = 0). For the plane slit the shear rate and hence the viscous dissipation are nearly uniform. The temperatures rise uniformly until the conduction toward the walls takes more and more heat out of the channel. When the temperature gradients are large enough to conduct away all the newly dissipated energy, the fully developed temperature field is reach. If the inner wall is nearly adiabatical (Bi, = - l), the temperature gradient has to adopt larger values since nearly all the dissipated energy has to be conducted to the other wall on the outside. The corresponding temperatures for circular Couettepow ( K = 0.5) are asymmetrical through the geometry of the system, additionally to the asymmetry of the thermal boundary condition. The shear rate and the viscous dissipation is much larger at the inner wall than at the outer one. The comparison of the average temperature ~ ( F oin) Fig. 22 shows that the development is much faster if both walls are cooled instead of one wall being nearly adiabatical (Bi, = - 1). The thermal development depends on the capacitance of the walls. In an example (Fig. 23) the outer wall of a circular Couette system is taken to be isothermal (9(1, Fo) = 0); the boundary condition at the inner wall is described by Bi, = - 1, ss,,= 0, and different values of the capacitance parameter Ci. The thermal development is delayed more, the larger the capacitance of the wall is taken to be. (K %

2 c , =o >,=O

, Bia=o

C,=IO

--

lo-’ Fourier number Fo

1

10

FIG.23. Thermal development of circular Couette flow depending on the capacitance parameter Ci at the inner wall; the outer wall is taken to be at constant temperature. Na = 1 ; m = 2; &.i = 0; S,,a = 0; Bii = - 1.

256

HORSTH. WINTER

FIG.24. Developing temperature in circular Couette flow for Ci = 0 and Ci = 0.1. The temperature of the outer wall is taken to be constant and equal to the initial temperature. K = 0.S;Na = 1 ; m = 2;Bii = -1.

The development of the temperature near the wall is determined by the value of C. For the example in Fig. 24 with C i = 0.1, the guide point initially is very close to the boundary. As the inner wall heats up, the guide point moves away from the boundary until the temperature of both the fluid and the wall reaches its full development. Dissipation and conduction balance and the temperature gradient at the inner wall becomes independent of Ci. a. Unsteady Plane Shear Flow with Closed Stream Lines. Analytical studies that include the time dependence of the viscosity q(fo, T, t ) do not seem to be available. Several authors calculated the developing temperature field in plane Couette flow of fluids with a viscosity independent of time: Gruntfest [89] : Krekel[86]: Powell and Middleman [92] :

q ( T ) = q(To)e-B'T-To), ~ ( j T) , = _sinh-'

AY

v]

=

'

-

(C;TJ7

const.,

Winter [SS]: Practical applications of their studies are the Couette rheometer [88, 89,921 and a shearing device for breaking up particles suspended in a fluid [86].

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

257

For the following example the assumption about the fully developed stress at t = 0 will be lifted. A Couette system is kept at rest, and the stress in the system is zero. At time t = 0 a shear experiment with constant shear rate is started. The shear stress 7 , z ( t )is found experimentally (see for instance [113]) to be governed by a time-dependent viscosity that increases gradually, goes through a maximum, and approaches a constant value. If these viscosity data are available, they might be used in the numerical program. For demonstrational purposes, the viscosity curve is approximated by q($, T, t )

=

[ ~ ~ ~ / ~ ) ( l ' m ) ~ ' e - " ' T --T e-'l')(l o)](l L

,

Y

+ cZe-'l'),

(2.74)

Eq. (2.47)

which qualitatively fits the measured curve shapes. The maximum viscosity is chosen to be three times the viscosity of steady shear flow; the time of the maximum is chosen to be Fo = 0.1, i.e., at about one-tenth of the thermal development time. The time-dependent viscosity contains an elastic contribution, which, however, is not specified unless one uses a complete rheological constitutive equation. In the calculation of the dissipated energy, the elastic part of the work of the stress is taken to be negligible compared to the viscous part. The stress growth curve as chosen in Eq. (2.74) is reproduced by the numerical program with Na = 0.001 (dashed lines in Fig. 25). If viscous dissipation is important (Na = 1, for instance) the stress reaches an earlier maximum at a lower value; the general shape of the curve is not changed through the I

.

'\

.oN'

\NO =0.001 1

dimensionless time Fo

I Fourier number 1

FIG.25. Thermal influence calculated for the startup experiment of plane Couette flow with time dependent viscosity as described in Eq. (2.74). The walls are taken to be at constant temperature equal to the initial temperature; rn = 2.5.

HORSTH. WINTER

258

effect of dissipation, and rheological and thermal effects seem undistinguishable in stress growth experiments. For comparison, the developing shear stress curves for (rheologically) time-independent viscosity (as described in Eq. (2.47)) are calculated and drawn as solid lines in Fig. 25. b. Fully Developed Temperature Field. The fully developed case has drawn much attention (see Table V), which is due to a double-valued solution, found in 1940 by Nahme [44]for plane Couette flow of Newtonian fluids. The shear stress in fully developed circular Couette flow (including plane Couette flow as a limiting case) cannot exceed a certain value, even if the shear rate is very large; for shear stresses below the maximum possible value, there are always two feasible shear rates 9, a small one at high viscosity and low temperature and a large one at low viscosity and high temperatures. Changes from one shear rate to the corresponding one require large temperature changes, and due to the heat capacity of the system together with the small thermal conductivity of the polymer, oscillations between the two states do not seem possible. For demonstrating the double-valued solution, Nahme [44] used a dimensionless shear stress o* and a dimensionless shear rate $*, whose definition can be extended to power law fluids: t* =

Nal/(I+m)p RdR,

oo)/PR@(R,

j,* = Nam/(l+m)

O),

(2.75) (2.76)

PRe(R, 00) and PR,(R, 0) are the dimensionless shear stress (see Eq. (2.45))

of the fully developed temperature field and of the isothermal case, respectively; the ratio of the two is independent of R. The dimensionlessshear stress

dim1 schear stress T*

FIG.26. Shear rate f* as a function of shear stress 7* (both defined in Eqs. (2.76) and (2.75)) of the fully developed temperature field; the parameter is the geometry.

Viscous DISSIPATION IN FLOWING MOLTEN POLYMERS

259

x =0939 4..

x =05

go5 0)

c

0) rn

e

>

1 diml. shear stress

r*

FIG.27. Average temperature gm of the fully developed temperature field for different geometries of circular Couette flow. Both walls are at constant temperature (9, = 0); m = 2.

P,,(R, co) is a monotonically descreasing function of Na, and it cannot be used by itself to demonstrate the double-valued solution. As an example, in Figs. 26 and 27 the double-valued solution jJ*(z*) and the corresponding average temperature 9,(.r*) of the fully developed temperature field are shown for circular Couette flow at K = 0.5 and IC x 1. Each shear rate p* has only one corresponding temperature 9,.

4. Experimental Studies The gap width of Couette systems is fairly small; and it is very difficult, if not impossible, to measure the temperature distribution by conventional means. The wall temperatures, however, can be measured quite accurately; other quantities measured are the torque on the system, the rotational speed of the cylinders, and the geometry. The double-valuedness of the shear rate seems to have been verified by Sukanek and Laurence [55] only. For viscosity measurements, the shear rate is prescribed and the average velocity in plane Couette flow is taken to be ti = jh/2. The experiment should be performed at conditions close to isothermal, which means that the Nahme number should be as small as possible: (2.77) The Nahme number is proportional to the square of the gap width, i.e., the Couette system should have a very narrow gap. Manrique and Porter [57] built a Couette rheometer with a gap of 5 x mm; reportedly they could eliminate the influence of viscous dissipation up to shear rates of 3 x lo6 s- '.

HORSTH. WINTER 111. Elongational Flow; Shear Flow and Elongational Flow

Superimposed (Nonviscometric Flow) The deformation during flow can be understood as a superposition of shear, elongation, and compression. If elongational components and density changes are negligible, the flow is shear flow, and the corresponding heat transfer problems can be analyzed as shown before. However, there are many engineering applications with a flow geometry different from shear flow; how the corresponding heat transfer problems are usually treated will be mentioned briefly. For a more detailed description, the reader will be referred to several examples in the literature. Other than for shear flow, there is no accepted rheological constitutive equation available for studying heat transfer. The proposed integral and differential constitutive equations are mostly tested in shear experiments at constant temperature, which might not be significant for nonviscometric flow during temperature changes. The main reason for not applying constitutive equations of elastic liquids is the fact that they require a detailed knowledge of the kinematics before the stress can be determined. But for other than Couette flow experiments, the kinematics of nonviscometric flows is not known in advance; it has to be calculated simultaneously with the stress. Presently a large emphasis of rheology is on solving nonviscometric flow problems at constant temperature. Rheological analysis is not advanced enough to incorporate temperature changes, and the present method of solution for nonviscometric engineering problems is practically identical with the one for steady shear flow, without care of the rheological differences. Elongational Flow Up to now, analytical studies on nonisothermal extensional flow have been done by means of a temperature dependent Newtonian viscosity, Eq. (1.13), and constant density. The studies are on melt spinning of fibers (see, for instance, [114,115]) and on film blowing (see for instance [116,1171). The measured stress and velocity indicate that the work of the stress a:Vv is very small (at least for film blowing [117]), and the heat transfer seems to be determined by convection with the moving film or thread and by conduction to the cooling medium. Shear Flow and Elongational Flow Superimposed In many different channel flows, as they occur in polymer processing, the rate of strain contains elongational components. The fluid elements are

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

261

FIG.28. Examples of converging and diverging flow: (a) Couette flow into a converging slit, which induces a pressure gradient for continuity reasons; (b) Couette flow in a converging annulus; (c) Poiseuille flow into a converging pipe or a converging slit; (d) radial flow in the gap between two parallel plates.

stretched while they are accelerated or slowed down along their paths. Examples (Fig. 28) are Couette flow into a converging slit or annulus, flow in a tapered tube, and radial flow between parallel plates. For describing the stress, one commonly uses the Strokes equation, Eq. (1.13), together with some average viscosity, or one takes the equation of the generalized Newtonian liquid, Eq. (1.14).The results of this kind of calculation seems to give relatively good estimates on temperature changes and viscous dissipation. Examples are heat transfer in screw extruders (see for instance [3, 102, 118-1221), in calendering [123], during mold filling [124-1291, and in melt solidification during flow [127-1291. If the deviations from shear flow are small, the stress might still be defined by the viscometric functions. An example of nearly viscometric flow is Poiseuilleflow in a pipe with constant but irregular cross section or Poiseuille flow in curved channels with constant cross section; the induced secondary flow in the cross section supports heat transfer toward the walls. The secondary flow, however, is very small. Whereas the improvement on the heat transfer for polymer solutions might be up to 30% [1303, for molten polymers (low density polyethylene in curved pipe) the influence of the secondary flow on the heat transfer was too small to be detectable with temperature probes in the melt [131]. Another example of nearly shear flow occurs in channels near a wall, even if the bulk of the fluid is mainly subjected to deformations other than shear

262

HORSTH.WINTER

[132]. For steady flow, the stress at the wall is described by the three viscometric functions and the wall shear rate, which of course can be determined only from the whole flow analysis including the nonviscometric part.

IV. Summary Heat transfer in flowing molten polymers is largely influenced by rheology, ie., by the rheological properties of the polymer and by the flow geometry. The rheology of steady shear flow is well understood, and hence the corresponding heat transfer problems can be treated most completely. However, heat transfer studies in flow geometries other than shear are, due to the present lack of an appropriate constitutive equation, only possible in very simplified form. The most important shear flow geometries are shown to be limiting cases of helical flow, and the corresponding heat transfer problems can be solved with one numerical program. Two groups of heat transfer problems are analyzed in the study: heat transfer in steady shear flow with open stream lines (represented by helical flow with a/& = 0) and the corresponding unsteady heat transfer problem with closed stream lines (represented by helical flow with d/dz = 0). The problem is completely determined by six dimensionsless parameters-the Nahme number; the Graetz number (or the Fourier number, respectively);the ratio of the radii of the annulus; the relative average axial velocity; the power law exponent of the viscosity; and the ratio of length to gap width-together with the boundary conditions. The commonly used: idealized boundary conditions are replaced by the Biot number for describing the heat conduction to the surroundings and by the capacity parameter for describing the thermal capacity of the wall during temperature changes with time. The conventional definition of the Nusselt number is not applicable to heat transfer problems with significant viscous dissipation, and a new definition has to be introduced. The shear dependence of the viscosity is described by a power law and the temperature dependence by an exponential function. The temperature coefficient of the power law region is shown to be directly related to the activation energy of the zero viscosity.

ACKNOWLEDGMENT

The author thanks Prof. G. Schenkel for his critical advice and many helpful suggestions; he has supported not only this work but also several specific studies of the author which were incorporated here. The author thanks Profs. A. S. Lodge, E. R. 0.Eckert, and K. Stephan for

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS

263

many critical comments and the colleagues G. Ehrmann and M. H. Wagner for helpful discussions on details of the study. The Deutsche Forschungsgemeinschaft is also acknowledged for having enabled the author to spend the time from August 1973 to November 1974 in Madison at the Rheology Research Center which was a fruitful preparation for this work.

NOMENCLATURE

a

Bi Cpr C"

C e

E

Fo Gz

h k

1, L

l/h

m

M Na Nu

P

P pRZ

4 r, R = r/r,

S thermal diffusivity [m2/s] Biot number [-I, see Eqs. r (2.21) and (2.26) T specific heat capacity at con- T stant pressure or at constant density [Jkg K] capacitance parameter of wall [-I, see Eqs. (2.26) and (2.30) internal energy [J/kg] activation energy [J/g-mole] D Fourier number, at/h2 [-I Graetz number, &h2/al [-] r, - ri = gap width [m]; h = r, for circular across sec- Y tion thermal conductivity [J/m s Z, Z U KI length of the slot power law exponent, see Eq. B (2.47) torque [mN] Nahme number, V 2 f i / k [-1, see Eq. (255) Nusselt number [-I, see Eq. s (2.63) pressure [N/m'], see Eq. E (1.13) dimensionless pressure gradient, see Eq. (2.43) 9 dimensionless shear stress components, see Eq. (2.44) and (2.45) specific heat flux at boundary [J/m2 s] radial coordinate (note: in Eqs. (1.9) and (2.12), R is the gas law constant) outer and inner radius of annulus [m]

wall thickness [m] time [s] temperature [K] average temperature [K], see Eq. (2.60) velocity components [m/s] angular velocity at inner wall [m/sl average velocity in z direo tion [m/s] reference velocity [m/s], see Eq. (2.38) dimensionless velocity components vep, v,P, vzp coordinate in r direction, see Eq. (2.32) = I/(/ GI) axial coordinate pressure coefficient of viscosity [m2/N1, 1- '(drt/aP)r.9 temperature coefficient of viscosity [K-'1, q-'(tlq/ 8T)P.V

rate of strain tensor [s-!] shear rate in simple shear flow [s-'1 unit tensor coefficient of thermal expansion, - p - '(dp/dT),

W-'l

dimensionless temperature, B(T - T o ) azimuth coordinate ratio of radii, rJra density [kg/m3] stress tensor [N/m2] extra stress tensor [N/m2] shear angle (see Fig. 1) first and second normal stress function in shear flow

HORSTH. WINTER INDICES 0

02

e

initial state, reference state, or related to the zero-viscosity (in a,, Po, E , ) fully developed state entrance

i, a r, R, z , Z , 0 S

W

inner or outer boundary coordinates surroundings wall, boundary of channel

REFERENCES 1. G. Schenkel, “Thermodynamik, Warmeerzeugung und Warmeiibertragung in der Extrudertechnik,” VDI-Bildungswerk BW 2185. Ver. Deut. Ing., Diisseldorf, 1972. 2. G. Schenkel, “Kunststoff Technologie,” unpublished lecture notes, Universitat Stuttgart (1970). 3. J . R. A. Pearson, Prog. Heat Mass Transfer 5, 73 (1972). 4. G. Schenkel, Kunstst, Gummi 7,231 and 282 (1968). 5. A. B. Metzner, Adv. Heat Transfer 2, 357 (1965). 6. J . E. Porter, Trans. Inst. Chem. Eng. 49, 1 (1971). 7. P. B. Kwant, Doctoral Thesis, Technische Hogeschool, Delft, 1971. 8. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” 2nd ed. Wiley, New York, 1962. 9. J. G. Oldroyd, Proc. R . Soc. London, Ser. A 200,523 (1950). 10. J. G. Oldroyd, Proc. R . SOC.London, Ser. A 202, 345 (1950). 11. H. L. Toor, Ind. Eng. Chem. 48,922 (1956). 12. R. S . Spencer and R. D. Gilmore, J . Appl. Phys. 21, 523 (1950). 13. S . Matsuoka and B. Maxwell, J. Polym. Sci. 32, 131 (1958). 14. R. S. Spencer and R. F. Boyer, J . Appl. Phys. 17,398 (1946). 15. R. H. Shoulberg, J . Appl. Polym. Sci. 7, 1597 (1963). 16. D. Hansen and C . C. Ho, J . Polym. Sci., Part A 3,659 (1965). 17. K.Eiermann, Kunststoffe 55, 335 (1865). 18. P. Lohe, Kolloid Z . & Z . Polym. 203, 115 (1965); 204, 7 (1965);205, 1 (1965). 19. V. S . Bil and N. D. Avtokratowa, Sou. Plast. (Engl. Transl.) H10,43 (1966). Kunstst. 7, 728 (1970). 20. F. Fischer, Gummi, Asbest, 21. W. Knappe, Adv. Polym. Sci7,477 (1971). 22. H. Wilski, Kolloid Z . & Z . Polym. 248, 861 (1971). 23. J. C. Ramsey, A. L. Fricke, and J. A. Caskey, J. Appl. Polym. Sci. 17, 1597 (1973). 24. K. Hohenemser and W. Prager, Z . Angew. Math. Mech. 12,216 (1932). 25. W. 0. Criminale, J. L. Ericksen, and G. L. Filbey, Arch. Ration. Mech. Anal. 1,410(1958). 26. A. S.Lodge, “Body Tensor Fields in Continuum Mechanics, with Applications to Polymer Rheology,” Academic Press, New York, 1974. 27. J. Meissner, Kunststoffe 61, 516 (1971). 28. 3. Meissner, Proc. Int. Congr. Rheol., 4th, 1963 Vol. 3, p. 437 (1965). 29. V. Semjonow, Adv. Polym. Sci. 5, 387 (1968). 30. W. Ostwald, Kolloid-Z. 36, 99 (1925). 31. G. V. Vinogradov and A. Y.Malkin, J . Polym. Sci.,Part A 2,2357 (1964). 32. K. H. Hellwege, W. Knappe, F. Paul, and V. Semjonow, Rheol. Acta 6, 165 (1967). 33. L. Christmann and W. Knappe, Colloid Polym. Sci.252, 705 (1974). 34. M. D. Hersey, Physics ( N . Y . )7,403 (1936). 35. H. Hausenblas, 1ng.-Arch. 18, 151 (1950). Rheol. 6,253 (1962). 36. E. A. Kearsley, Trans. SOC. 37. D. D. Joseph, Phys. Fluids 7 , 1761 (1964). 38. B. Martin, Inr. J. Non-linear Mech. 2,285 (1967).

+

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

265

J. C. J. Nihoul, Ann. Soc. Sci. Bruxelles, Ser. 185, 18 (1971). P. C. Sukanek, Chem. Eng. Sci. 26, 1775 (1971). P. C. Sukanek and R. L. Laurence, Ann. Soc. Sci. Bru-xelles, Ser. T 86, 11, 201 (1972). H. Schlichting, Z. Angew. Math. Mech. 31, 78 (1951). R. E. Colwell, in “Computer Programs for Plastics Engineers”(1. Klein and D. I. Marshall, eds.), p. 183. van Nostrand-Reinhold, Princeton, New Jersey, 1968. 44. R. Nahme, 1ng.-Arch. 11, 191 (1940). 45. J. Gavis and R. L. Laurence, Ind. Eng. Chem., Fundam. 7, 232 and 525 (1968). 46. R. M. Turian, Chem. Eng. Sci. 24, 1581 (1969). 47. J. C. J. Nihoul, Phys. Fluids 13, 203 (1970). 48. P. C. Sukanek, C. A. Goldstein, and R. L. Laurence, J . Fluid Mech. 57, 651 (1973). 49. R. Kumar, J. Franklin Insr. 281, 136 (1966). 50. J. M . Wartique and J. C . J. Nihoul. Ann. Sue. Sci. Bruxelles, Ser. T, 83, 111. 361 (1969). 51. G. Palma, G . Pezzin, and L. Busulini, Rheol. Acta 6, 259 (1967). 52. R. B. Bird and R. M. Turian, Chem. Eng. Sci. 17, 331 (1955). 53. R. M. Turian and R. B. Bird, Chem. Eng. Sci. 18,689 (1963). 54. R. M. Turian, Chem. Eng. Sci. 20, 771 (1965). 55. P. C. Sukanek and R. L. Laurence, AIChE J . 20,474 (1974). 56. D. D. Joseph, Phys. Fluids 8, 2195 (1965). 57. L. Manrique and R. S . Porter, Polym. Prepr. Am. Chem. Soc.. Diu. Polym. Chem. 13 992 (1972). 58. H. Zeibig, Rheol. Acta 1, 296 (1958). 59. G . M. Bartnew and W. W. Kusnetschikowa, PIaste Kaursch. 17, 187 (1970). 60. B. Martin, “Heat Transfer Coupling Effects Between a Dissipative Fluid Flow and its Containing Metal Boundary Conditions,” Reprint of the European Working Party o n non-Newtonian Liquid Processing (1970). 61. H. D. Kurz, “Programm fur die Berechnung der Druck- und Schleppstromung im ebenen Spalt.” Studienarbeit Inst. fur Kunststofftechnik, Universitat Stuttgart, 1973. 62. H. C. Brinkman, Appl. Sci. Res., Sect. A 2, 120 (1951). 63. R. B. Bird, S P E J . 11, No 7, 35 (1955). 64. H. L. Toor, Trans. Soe. Rheol. 1, 177 (1957). 65. R. E. Gee and J. B. Lyon, Ind. Eng. Chem. 49,956 (1957). 66. J. Schenk and J. van Laar, Appl. Sci. Res. Sect. A 7 , 449 (1958). 67. J. E. Gerrard, F. E. Steidler, and J. K. Appeldorn, Ind. Eng. Chem., Fundam. 4, 332 (1965); 5, 260 (1966). 68. J. E. Gerrard and W. Philippoff, Proc. Int. Congr. Rheol., 4th, 1963 Vol. 2. p. 77 (1965). 69. K. Stephan, Chem.-1ng.-Tech. 39,243 (1967). 70. R. A. Morette and C . G . Gogos, Polym. Eng. Sci. 8,272 (1968). 71. H. Schluter, Doctoral Thesis, Technische Universitat, Berlin, 1969. 72. G . Mennig, Doctoral Thesis, Universitat Stuttgart, 1969; Kunststofftechnik 9, 49, 86, and I54 (1970). 73. N. Galili and R. Takserman-Krozer, Isr. J . Technol. 9,439 (1971). 74. G . B. Froishteter and E. L. Smorodinsky. Proc. Inr. Semin. Heat Mass Transfer Rheol. Complex Fluids, Int. Center Heat Mass Transfer, Herzeg Novi (1970). 75. R. H. Daryanani, H. Janeschitz-Kriegl, R. van Donselaar, and J. van Dam, Rheol. Acta 12, 19 (1973). 76. G. Forrest and W. L. Wilkinson, Trans. Inst. Chem. Eng. 51, 331 (1973); 52, 10 (1974). 77. H. H. Winter, Polym. Eng. Sci. 15, 84 (1975). 78. N. Galili, R. Takserman-Krozer, and Z . Rigbi, Rheol. Acra 14, 550 and 816 (1975). 79. G . Mennig, Kunststoffe 65, 693 (1975). 80. E. M . Sparrow, J. L. Novotny, and S . H. Lin, AlChE J . 9, 797 (1963). 81. A. Seifert, Doctoral Thesis, Technische Universitat, Berlin, 1969. 82. J. Vlachopulos and C. K. J. Keung, AIChE J . 18, 1272 (1972). 39. 40. 41. 42. 43.

266

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83. A. Brinkmann, Doctoral Thesis, Technische Hochschule, Braunschweig, 1966. 84. H. Rehwinkel, ”Stromungswiderstand und Warmeubergang bei nicht-Newtonschen

Flussigkeiten in Ringkanalen mit rotierendem Innenzylinder,” DFG-Abschlussbericht No. 260/24, Deutsche Forschungsgemeinschaft , Bad Godesberg, 1970. 85. H. H. Winter, Rheol. Acta 12, I (1973); 14, 764(1975). 86. J. Krekel, Doctoral Thesis, Technische Hochschule, Karlsruhe, 1964. 87. H. H. Winter, In!. J. Heat Mass Transfer 14, 1203 (1971). 88. H. H. Winter, Rheol. Acta 11, 216 (1972). 89. I. J. Gruntfest, Trans. SOC.Rheol. 7 , 195 (1963). 90. H. W. Cox and C. W. Macosco, AIChE J. 20,785 (1974). 91. H. H. Winter, Doctoral Thesis, Universitat Stuttgart, 1973. 92. R. L. Powell and S. Middleman, In!. J. Eng. Sci. 6,49 (1968). 93. R. G. Griskey and I. A. Wiehe, AIChE J. 12,308 (1966). 94. T. H. Forsyth and N. F. Murphy, Polym. Eng. Sci. 9, 22 (1969). 95. R. G. Griskey, M. H. Choi, and N. Siskovic, Polym. Ettg; Sci. 287 (1973). 96. J . L. Ericksen, Q.J. Appl. Math. 14, 318 (1956). 97. “VDI-Warmeatlas,” 2nd ed., Ver. Deut. Ing., Dusseldorf, 1974. 98. E. R. G. Eckert and R. M. Drake, “Analysis of Heat and Mass Transfer.” McGraw-Hill, New York, 1972. 99. J . L. den Otter, Rheol. Acta 14, 329 (1975). 100. E. Uhland, Rheol. Acta 15.30 (1976). 101. L. Schiller, Z. Angew. Math. Mech. 2, 96 (1922). 102. R. M. Griffith, h d . Eng. Chem., Fundam. 1, 180 (1962). 103. L. Graetz, Ann. Phys. Chem. 18,79 (1889). 104. A. D. Gosman, W. M. Pun, A. K. Runchal, D. B. Spalding, and M. Wolfshtein, “Heat and Mass Transfer in Recirculating Flows,” Academic Press, New York, 1969. 105. A. G. Fredrickson and R. B. Bird, Ind. Eng. Chem. 50, 347 (1958). 106. W. Nusselt, Z. Ver. Dsch. Ing. 54, 1154 (1910). 107. W. Tychesen and W. Georgi, SPE J . 18, 1509 (1962). 108. H. Janeschitz-Kriegl, J. Schijf, and J . A. M. Telgenkamp, J . Sci. Instrum. 40,415 (1963). 109. G. Schenkel, DOS 1,554,931 (1966). 110. J. van Leeuwen, Polym. Eng. Sci. 7 , 98 (1967). 1 1 I . H. Schlichting, “Boundary Layer Theory,” p. 65. McGraw-Hill, New York, 1955. 112. J. D. Ferry, “Viscoelastic Properties of Polymers,” 2nd ed., p. 121. Wiley, New York, 1969. 113. 114. 115. 116. 117. I 18. 119. 120.

121. 122. 123.

J. Meissner, Rheol. Acta 14, 201 (1975). Y.T. Shah and J. R. A. Pearson, Ind. Eng. Chem., Fundam. 11, 145 (1972). S. Kase, J. Appl. Polym. Sci. 18, 3267 (1974). C. J. S. Petrie, AZChE J. 21, 275 (1975). M. H. Wagner, Rheol. Acta 15.40 (1976). B. Martin, J. R. A. Pearson, and B. Yates, Vniv. Cambridge, Polym. Process. Res. Cent. Rep. No. 5 (1969). R. T. Fenner, “Extruder Screw Design,” Iliffe, London, 1970. 2. Tadmor and 1. Klein, “Engineering Principles of Plasticating Extrusion,” Van Nostrand-Reinhold, Princeton, New Jersey, 1970. G. Schenkel, Kunsfstofftechnik 12, 171 and 203 (1973). R. V. Torner, “Grundprozesse der Verarbeitung von Polymeren,” VEB Dtsch. Verlag Grundstoffind. Leipzig, 1974. V. J. Petrusanskij and A. I. Sachaev, Uch. Zap. Yarosl. Tekhnol. Inst. 23 (1971); cited by Torner [ 122).

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS 124. 125. 126. 127. 128. 129. 130. 131. 132. 133.

J. L. Berger and C . G. Gogos, Polym. Eng. Sci. 13, 102 (1973). M. R. Karnal and S. Kenig, Antec 18, 619 (1972). H. H. Winter, Polym. Eng. Sci. 15,460 (1975). E. Broyer, C. Gutfinger, and Z. Tadmor, Trans. SOC. Rheol. 19,423 (1975). J. Rothe, Doctoral Thesis, Universitat Stuttgart, 1972 C. Gutfinger, E. Broyer, and Z. Tadmor, Polym. Eng. Sci.15, 515 (1975). D. R. Oliver, Trans. Inst. Chem. Eng. 47T, 8 (1969). H. H. Winter, unpublished experiments. B. Caswell, Arch. Ration. Mech. Anal. 26. 385 (1967). H. Rehwinkel, Doctoral Thesis, Technische Universitlt Berlin (1970).

267

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Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. A

Abramovich, G.N., 66,114 Akfirat, J. C., 6(26), 10(26,35), ll(26). 12,

Broyer, E., 261(127), 267 Busulini, L., 219(51),265

26(35), 41(26), 59.60

Andreev, A. A,, 11(41),60 Appeldorn, J. K., 219(67), 248(67), 265 Ark, R.,128(35), 202 Atwell, N. P.,80(35), 116 Avtokratowa, N.D., 21 1(19), 264 B Bachmat, Y.,137,202 Back, L.H., 5(21), 11(21), 59 Bakke, P.,5(23), 11(23), 59 Bartnew, G . M., 219(59), 265 Bear, J., 178,184,203 Bellemans, A., 130(40), 164(40), 202 Ben Haim, Y., 67(6), 115 Berger, D., 126,153(31), 161,162,169,191, 20 1 Berger, J. L., 261(124), 267 Bil, V.S., 211(19), 264 Bird, R.B., 7,60,126(33), 128(33), 131,

169(33), 201,208(8), 209(8), 214(8), 219(52, 53), 23.6(105), 253,264,265, 266 Boyer, R. F., 210(14), 264 Bradshaw, P.,80(35), 116 Brauer, H.,16(46, 47),60 Brdlick, P.M., 15, 17(44,45),60 Brenner, H., 170,202 Brinkman, H. C., 219(62), 233,265 Brinkmann, A., 219(83), 266

C

Campbell, J. F., 67(5), 114 Caretto, L.S . , 67(10, 15), 102(15), I15 Caskey, J. A., 211(23), 264 Caswell, B.,262( 132), 267 Ceaglske, N. H., 121,122, 124,185,190,201 Cebeci, T.,79(29), 116 Chen, C.-J., 76,78(25), 115 Chen, C. S., 126(32), 191(32), 201 Choi, M.H., 219(95), 266 Christmann, L.,218(33), 264 Churchill, S. W., 125,164,169,201,202 Clayton, J. T., 126(32), 191(32), 201 Coantic, M.,86(46),116 Cobonpue, J., 10(30), 12,15, 17(30), 18, 19,

60

Cockrell, D. J., 79(30), 116 Colwell, R. E., 219(43), 265 Comings, E. W., 121,175,201 Cosart, W.P.,5(22), 11(22), 59 Cox, H. W., 219(90), 266 Criminale, W.O., 213(25), 264 Curr, R.M., 67(10), 115

D Dakhno, V. N., 11(41), 60 Daryanani, R.H., 219(75), 234(75), 249,265 Davidzon, M.I., 2(16), 59

269

AUTHOR INDEX

270

Defay, R., 130, 164,202 den Otter, J. L., 229(99), 266 DeVries, D. A., 125,133,201 Dosdogru, G. A., 5,6(20), 10(20), 41(20), 59 Drake, R. M., 222(98), 243(98), 266 Dullien, F. A. L., 187,203 Dyban, E. P.,43,60 Dybbs, A., 164(51), 202

E Eckert, E. R. G., 222(98), 243(98), 266 Eiermann, K., 21 1(17), 264 Ekman, V. W., 93,116 Ellison, T. H., 96,99, I16 Ericksen, J. L., 213(25), 221(96), 264,266 Eustis, R. H., 1 l(36, 40),60

F Fan, L. N., 67(8), 115 Favre, A., 86(46), 1I6 Fenner, R. T., 261(119), 266 Ferriss, D. H., 80(35), 116 Ferry, J. D., 251(112), 266 Fiedler, H. E.,86(47), 116 Filbey, G. L., 213(25), 264 Fischer, F., 21 1(20), 264 Forrest, G., 219(76), 243(76), 265 Forsdyke, E. J., 119(1). 200 Forsyth, T. H., 219(94), 266 Fredrickson, A. G., 236(105), 266 Frenken, H., 35.60 Fricke, A. L., 21 1(23), 264 Friedman, S.J., 18,26(50), 60 Froishteter, G. B.,219(74), 265 Fulford, G. D., 120,200 G

Galili, N., 219(73,78), 221(78), 265 Gardner, W., 121,201 Gardon, R., 6(26), lO(26, 30, 35), 1 I , 12, 15, 17(30), 18, 19, 26(35), 41(26), 60 Gavis, J., 219(45), 265 Gee, R. E., 219(65), 265 Georgi, W., 248(107), 266 Gerrard, J. E., 219(67,68), 248,265 Gibson, M. M., 81(43), 116

Gill, W. N., 164(54), 202 Gilliland, E. R., 120, 201 Ginzburg, A. S., 120,200 Glaser, H.,2, 18,59 Glauert, M. B.,2, 5(17), 11(17), 59 Glushko, G. S.,80(34), 81(34), 83,92, I16 Gnielinski, V.,ql),6(1), 8,9(1), 10(1), I I(l), 15, 16, 17(1), 25(1), 41,42(1), 59 Gogos, C. G., 219(70), 261(124), 265,267 Goldstein, C. A., 219(48), 265 Gorring, R. L., 164,202 Gosman, A. D., 12(43), 60,67(9, 15), 102(15), 115, 236, 237(104), 266 Graetz, L., 234,266 Gray,W. G., 137,144,148,149,157,158, 202 Greenkorn, R. A,, 164,202 Gref, H., 35,54,60 Griffith, R.M.,233,266 Griskey, R. G., 219(93,95), 266 Grunfest, I. J., 256, 266 Gupta, J. P., 125,169,201 Gutfinger, C., 261(127), 267

H Hansen, D., 211(16), 264 Harleman, D. R. F., 108, 109, 117 Harlow. F. H., 80(31), 94(31), 116 Hassid, S., 81(40), 116 Hatcher, J. D., 129(37), 202 Hausenblas, H., 219(35), 264 Hellwege, K. H., 217(32), 218(32), 264 Hennecke, F. W., 2(5), 8(5), I1(5), 18(5), 19(5),25(5), 32(5), 59 Hersey, M. D., 219(34), 264 Hilgeroth, E.,18,26(52), 60 Hirata, M.,11(39), 60 Ho, C. C., 211(16), 264 Hijppner, G., 26(51), 60 Hohenemser, K., 21 1(24), 264 Hougen,O. A., 121, 122, 124, 185, 190,20f Huesmann, K., 29(53), 60 Husain, A., 126,191,201

J Janeschitz-Kriegl, H.,219(75), 234(75), 248(108), 249(75), 265,266 Jeschar, R., 10(31), 17,60

AUTHORINDEX Jones, W. P., 81(39), 116 Joseph, D. D., 219(37, 56), 264,265

K Kamal, M. R., 261(125), 267 Kase, S.,260(1 IS), 266 Kays, W. M., 81(44), 116 Kearsley, E. A., 219(36), 264 Kenig, S.,261(125), 267 Kessler, D. P., 164, 202 Keung, C. K. J., 219(82), 243(82), 265 Kezios, S. P., 11(37), 60 Kikuchi, T., 164,202 Kim, T. S.,11(38), 60 Klein, I., 261(120), 266 Kline, S. J., 79(30), 116 Knappe, W., 21 1(21), 217(32), 218(32, 33), 264 Koh, R. C. Y.,67(8), 115 Korger, M., lO(33, 34), 12, 26(34), 41,42(58), 43,44,60 Kostin, M. D., 169, 202 Kraitshev, S. G., 2(4), 36, 37(4), 38(4), 39(4), 40(4), 59 Krekel, J., 219(86), 256,266 Krischer, O., 124,201 Kiiiek, F., lO(33, 34), 12, 26(34), 44,60 Krotzsch, P., 2(2, 3, 5), 8(5), 11(5), 18, 19, 20, 21, 22, 25(5), 32(5), 48,59 Kumada, M., 2(15), 10(15), 11(15), 59 Kumar, R., 219(49), 265 Kunze, W., 58,60 Kurz, H. D., 219(61), 265 Kusnetschikowa, W. W., 219(59), 265 Kwant, P. B., 207(7), 264

L Laufer, J., 82,,116 Launder, B. E., 75, 76,80(32), 81(22,32,39), 84, 91, 92, 115 Laurence, R. L., 219(41, 4 5 4 8 , 55), 221(55), 259,264,265 Lebedev, P. D., 120,200 Lee, C. S.,40,60 Lewis, W. K., 120,200 Liepmann, H. W., 82, 116

27 1

Lightfoot, E. N., 7(27), 60, 126(33), 128(33), 131(33), 169(33), 201,208(8), 209(8), 214(8), 264 Lin, S. H., 219(80), 265 Lodge, A. S.,211,212(26), 213(26), 264 Lohe, H., 2(13), 43, 59 Lohe, P., 211(18), 264 Luikov, A. V., 125. 126(29), 133, 142, 185(29), 191,201 Lyman, F. A,, 164(51), 202 Lyon, J. B., 219(65), 265 Lyons, D. W., 129,202

M Mabuchi, I., 2(15), 10(15), 11(15), 59 McBride, G. B., 67(7), 115 McCormick, P. Y.,120,200 McGuirk, J. J., 67(12, 13), 107, 115 Macosco, C. W., 219(90), 266 Majumbar, A. K., 110(56), 117 Malkin, A. Y., 217(31), 264 Mannheimer, R. J., 179(58), 202 Manrique, L., 219(57), 259,265 Marra, R. A., 164(54), 202 Martin, B., 219(38,60), 261(118), 264,265, 266 Martin, H., 2(6, 7, 8,9), 13(9), 22(9), 23(9), 24(9), 25,26(9), 27(6,9), 32(8, 9), 33, 34(9), 35(9), 36, 37(9), 46(9), 52(7, 9), 59 Mazur, A. I., 43, 60 Meier, R.,58, 60 Meissner, J., 215, 216(28), 218(28), 257(113), 264,266 Mennig, G., 219(72,79), 248,265 Metzger, D. E., 43, 60 Metzner, A. B., 207(5), 264 Mewes, D., 16(46, 47). 60 Meyer, J., 169,202 Middleman, S., 219(92), 225, 256,266 Miyauchi, T., 164,202 Moffat, R. J., 81(44), 116 Morette, R. A,, 219(70), 265 Morkovin, M. V., 79(30), 116 Morse, A. P., 75(23), 115 Morton, B. R., 66,114 Mueller, A. C., 18, 26(50), 60 Murphy, N. F., 219(94), 266 Myers, G. E., 11(36), 60

AUTHOR INDEX

272 N

Nahme, R., 219(44), 233, 258,265 Nakatogawa, T., rl(39), 60 Nakayama, P. I., 80(31), 94(31), 116 Newman, A. B., 120,201 Ng, K.H., 81(41), 116 Nihoul, J. C. J., 219(39, 47, 50), 264, 265 Nishiwaki, H., I1(39), 60 Novotny, J. L., 219(80), 265 Nusselt, W., 240,266 0

Okazaki, 164,202 Oldroyd, J. G., 208(9), 209(10), 212(9), 264 Oliver, D. R.,261(130), 267 Ostwald, W., 217(30), 264 Ott, H. H., 18,60

P Palma, G., 219(51), 265 Patankar,S. V., 67(11, 15, 16, 17, 18),69, 70(20), 71, 81(20), 84.92, 102, 115 Paul, F., 217(32), 218(32), 264 Pearson, J. R. A., 206(3), 233, 260(114), 261(3, 118), 264,266 Pei, D. C. T., 126, 153(31), 161, 162, 169, 191,201 Perry, K. P., 44(61), 45,60 Petrie, C. J. S., 260(116), 266 Petrusanskij, V. J., 261(123), 266 Petzold, K., 2(14), lO(14, 32), 11, 15, 17(32), 59.60 Pezzin, G., 219(51), 265 Philippoff, W., 219(68), 265 Phillip, J. R., 125, 201 Potke, W., 10(31), 17,41,60 Poreh, M., 81(40), 116 Porter, J. E., 207(6), 264 Porter, R. S.,219(57), 259, 265 Powell, R. L., 219(92), 225,256,266 Prager, W., 21 1(24), 264 Prandtl, L., 80(33), 83,91,92, 116 Pratap, V. S., 106(54), 117 Prigogine, I., 130(40), 164(40), 202 Pun, W.M., 12(43), 60, 113(59), 117, 236(104), 237(104), 266

R Rackham, B., I19(1), 200 Raiff, R. J., 129,202 Rarnsey, J. C., 21 1(23), 264 Rao, V. V., 10(29), 17, 18,60 Rehwinkel. H.,219(84), 266,267 Reichardt, H., 30, 35,60 Richards, L. A., 121,201 Ricou, F. P., 66, 114 Rideal, E. K., 121,201 Rigbi, Z., 219(78), 221(78), 265 Rodi, W.,75(23), 76, 78, 79, 115 Romanenko, P. N., 2(16), 59 Runchal, A. K., 12(43), 60, 81(38), 116, 236(104), 237(104), 266

S Sachaev, A. I., 261(123), 266 Saiy, M., 75, 86(24), I15 Savin, V. K., 11(41), 15, 17(44), 60 Schauer, J. J., 11(36,40). 60 Scheidegger, A. E.,184,203 Schenk, J., 219(66), 265 Schenkel, G., 206(1, 2,4), 210(4), 219(2), 248(109), 261(121), 264,266 Schetz, J. A,, 67(5), 114 Schijf, J., 248(108), 266 Schiller, L., 229(101), 266 Schlichting, H., 4, 11(19), 59, 79(27), 116, 219(42), 250(11 I), 265,266 Schlunder, E. U., 2(1,4, 5 , 6, 8), 3,6(1), 7, 8, W ) , 10(1), 11(1,5), 15, 16, 17(1), 18, 19(5), W, 5),27(6), 32(5,8), 36(4), 37(4), 38(4), 39(4), 40(4), 41,42(1), 46(8), 59.60 Schliiter, H., 219(71), 265 Scholz, M.T., 6,59 Schrader, H., 2,4, 5 , 5 9 Schwartz, W. H., 5(22), 11(22), 59 Seban, R. A., 5(21), 11(21), 59 Seifert, A,, 219(81), 265 Semjonow, V., 216(29), 217(32), 218,264 Shah, Y.T., 260(114), 266 Sherwood,T. K.,120,121, 175,200,201 Shoulberg, R. H., 21 1(15), 264 Singh, B. S., 164,202 Singhal, A. K., 81(42), 116

AUTHOR INDEX Siskovic, N., 219(95), 266 Slattery, J. C., 129(39), 131, 133,134, 136(39), 137, 138(39), 153(39), 158, 175, 176(39), 202 Smirnov, V. A., 15, 17(45), 60 Smith, A. M. O., 79(29), 116 Smorodinsky, E. L., 219(74), 265 Sovran, G., 79(30), 116 Spalding, D. B., 12(43), 60,65,66,67(9, 10, 1 1 , 12, 15, 16, 17, 18), 68(19), 69. 70(20). 71, 72, 73, 75, 76,80(32), 81, 83,84,91, 92,96,97, 102, 105, 106(53, 54), 107, 110, 113(59), 114,115,116,117, 236(104), 237(104), 266 Sparrow, E. M., 6,59, 219(80), 265 Spencer, R. S.,209,210,264 Steidler, F. E., 219(67), 248(67), 265 Stephan, K., 219(69), 243(69). 265 Stewart. W. E., 7(27). 60. 126(33). 128(33). 131(33), 169(33). 201,208(8). 209(8), 214(8). 264 Stolzenbach, K. D., 108, 109, 117 Stoy, R. L., 67(6), 115 Subramanian, R. S., 164,202 Sukanek, P.C., 219(40,41,48,55), 221(55), 259,264,265 Sunderland, J. E., 129(37), 202 Svensson, U., 94,97, 116, 117

T Tadmor, Z., 261(120, 127), 266,267 Takserman-Krozer?.R., 219(73, 78), 221(78), 265 Taylor, G. I., 66(4), 114 Telgenkamp, J. A. H., 248(108), 266 Temperton, C., 112(58), 117 Tollmien, W., 30, 35.60 Toor, H. L., 209( I l), 219(64), 264,265 Torii, K., 11(39), 60 Torner, R. V., 261(122), 266 Toupin, R., 134, 159(42), 202 Trass, O., 6, 10(29), 17, 18,59,60 Truesdell, C., 134, I59(42), 202 Turian, R. M., 219(46, 52,53,54), 253,265 Turner, J. S., 66(4), 96, 99, 114, 116 Tuttle, F., 123,201 Tychesen. W.,248(107), 266

273 U

Uhland, E., 229(100), 261(100), 266 V

van Dam, J., 219(75), 234(75), 249(75), 265 van Donselaar, R., 219(75), 234(75), 249(75), 265 van Laar, J., 219(66), 265 van Leeuwen, J., 248,266 Verevochkin, G. E., 15(45), 17(45), 60 Vinogradov, G. V., 217(31), 264 Vlachopulos, J., 219(82), 243(82), 265

W Wagner, M. H., 260(117), 266 Walz, A., 79(28), 116 Wartique, J. M.,219(50), 265 Wayner, P, C., Jr., 129,202 Westman, A. E. R., 121,201 Whitaker, S., 128(34, 36), 130(36), 135(34), 137, 138(43), 140(43), 144, 146(43,49), 159, 161(43), 164, 170, 172(44), 173(44), 176(59), 202,203 Widtsoe, J. A., 121,201 Wiehe, I. A., 219(93), 266 Wilkinson, W. L., 219(76), 243(76), 265 Wilski, H., 21 1(22), 264 Winter, H. H., 219(77,85,87,88,91), 221(77), 234(85), 237(85), 248(91), 249, 256, 261(126, 131), 265,266,267 Wolfshtein, M., 12, 60, 81(37), 116, 236(104), 237(104), 266 Wong, T. C., 6,59 WU,C.-H., 67, 115 Wygnanski. I., 86(47). 116

Y Yates, B., 261(118), 266 Yudaev, B. N., 11(41), 60

Z Zeibig, H.,219(58), 265

Subject Index A Acceleration pressure drop, in impinging flow, 31 Activation energy, in shear flow, 216-218 Air boundary layer above lake, 79-82 boundary conditions for, 80-81 geometry and physics of, 79-80 Air-water layer geometry and physics of, 86-87 in THIRBLE classification, 85-88 ARN, see Array of round nozzles ARO, see Array of round orifices Array correction function, for impinging flow, 21-22 Array of round nozzles, 46-47 transfer coefficients vs. outlet flow conditions in, 27-41 integral mean transfer coefficient for, 18-22 optional spatial arrangements for, 51 Array of round orifices integral mean transfer coefficient for, 22 Sherwood numbers and, 37 Array of slot nozzles high-performance, 52 integral mean transfer coefficient for, 22-26 optional spatial arrangements in, 51 variation of transfer coefficients for, 34-36 ASN, see Array of slot nozzles ASN transfer coefficients, 36

Nusselt number and, 243 in shear flow, 222-225 wall boundary condition and, 224 Boundary conditions in energylmass transport equations, 133-137 in shear flow, 222-227 at wall, 224 Boundary layer, in water at lake surface, 82-85 Brinkman number, in steady shear flow, 234 C

Capacitance parameter C, in molten polymer flow problems, 207 Capillary action, in porous media, 121, 124 Channel flow, defined, 206 Combined air-water layer, in THIRBLE classification, 85-88 Compressible fluid, defined, 209 Concave surfaces, impinging flow in, 43 Conservation laws, 62-63 Continuum physics, equations in, 126 Convective transport and constitutive equation for forces acting on liquid phase, 184-192 Darcy 's law in, 175 - 184 in gas phase, 169-175 in liquid phase, 175-192 Couette flow, 219-222,250,257-258 converging and diverging, 261 plane and circular, 254-259 Couette rheometer, 259 Couette systems, 253-259

D

B Biot number, 207, 238-239,262 calculation of. 223

Darcy's law, in convective transport, 175-184 Diffusion, in porous media, 120 274

SUBJECT INDEX Diffusivity, effective, in gas phase diffusion equation, 168 Dimensionless parameters, in shear flow, 233-235,252~253,262 Drag flow, 221-222 Drying process, 119-200 beginning and end of, 196- 197

and convective transport in liquid phase, 175-192

diffusion theory of, 124-125, 194-198 effective thermal conductivity in, 140, 158-164

energy transport in, 153 - 165 enthalpy of vaporization in, 157 heat capacity in, 155 liquid phase in, 184-192 and mass transport in gas phase, 165-175 moisture distribution in, 125 problem and solution in, 192-194 temperature gradients in, 191 -192 thermodynamic relations in, 164-165

E Effective thermal conductivity, in drying process, 140, 158-164 Ekman layer, development of, 93-95 Elongation flow, 260-262 Energy, conservation of, 62 Energy equation, total thermal, 154-158 Energy/mass transport equations, 126-1 53 see also Masslenergy transport equations Energy transport see also Masslenergy transport equations in drying process, 153 - 165 effective thermal conductivity in, 158-164 thermodynamic relations in, 164-165 total thermal energy equation in, 154-158 Enthalpy, of vaporization per unit mass, 157 Exit velocity, transfer coefficients and, 35 Expansion cooling, in shear flow, 229

F Fourier number, in shear flow, 221,262 Fractional moisture saturation, 143, 196 Free surface flow, defined, 206 Fully developed temperature field, 237 in shear flow, 258-259

275 G

Gas jet heat and mass transfer related to, 1-58 impinging, see Impinging flow; see also Array of round nozzles; Array of slot nozzles Gas mixture, heat capacity of, 163 Gas phase arbitrary curve in, 171 convective transport in, 169-175 mass transport in, 165-175 moisture in, 142 Gas phase continuity equation, 147 Gas phase diffusion equation, 166-169 Gas phase species continuity equation, 149 Gas phase velocity field, 170 Generalized Newtonian fluid, 21 I GENMIX computer program, 72-78 for air boundary layer above lake, 81 for Ekman layer problem, 94-95 for hyperbolic and partially parabolic layers, 106-1 10 for lake surface boundary layer, 84 for radial pool, 1I 1 -1 12 in THIRBLE problems, 69,72 for unsteady layers, 112 Governing point equations, in rnasslenergy transport, 128-133 Graetz number, 221,228, 262 defined, 234 Gravitational effects, in liquid phase, 191 Griffith number, in steady shear flow, 234 H

Heat capacity, in drying process, 155 Heat transfer impinging gas jets in, 1- 58 in rivers, bays, lakes, and estuaries, see THIRBLE Heat transfer coefficient in impinging flow, 6-7 integral mean, 13-26 outlet flow conditions vs. array of nozzles for, 27-41 swirling jets and, 41 turbulence promoters and, 41 wire-mesh grids and, 41-43 Helical flow geometry, 228 Hyperbolic layer, in two-dimensional floating layers, 106-109

276

SUBJECT INDEX I

Impinging flow acceleration pressure drop in, 31 arrays of round nozzles in, 46-49, 51 arrays of slot nozzles in, 49-51 on concave surfaces, 43 contour lines in, 50-51 heat transfer coefficient in, 6-7 high-pressure arrays of nozzles in, 52-58 hydrodynamics of, 2-5 integral mean transfer coefficients in, 13-26 jet length in, 4 mass transfer coefficient in, 6-7 mean heat and mass transfer coefficient correlations for, 27-41 nozzle array in, 12- 13 nozzle-to-plate distance in, 9 numbers of transfer units in, 12 and optional spatial arrangement of nozzles, 45-52 outlet flow conditions vs. transfer coefficients for array of nozzles in, 27-41 Sherwood number in, 9-12 for single nozzles, 8- 18 stagnation flow in, 4-5 swirling jets in, 41 turbulence promoters in, 41 variations in coefficients for, 8-13 velocity field of, 2-3 wall jet flow and, 5 wire-mesh grids and, 41 Incompressible flow, defined, 208 Integral mean transfer coefficients for array of round nozzles, 18-22 for array of round orifices, 22 for array of slot nozzles, 22-26 equations for single nozzles and, 15-1 8 impact angle and, 45 in impinging jet flow, 13-26 single slot nozzle in, 18 Interface-transfer problem, 64

J Jet see also Impinging flow gas, see Gas jet

round, see Round jet steady axisymmetrical, 65-66 swirling, 41 Jet flow, heat and mass transfer in, 1-58 see also Impinging flow

L Lakes and estuaries, heat and mass transfer in, see THIRBLE Lake surface air boundary layer above, 79-82 boundary conditions for, 83-84 boundary layer in water at, 82-85 GENMIX computer and, 84 geometry and physics of, 82-83 constitutive equation for forces acting on, 184-192 continuous, 186 convective transport in, 175-192 gravitational effects in, 191 hydrostatic equilibrium and, 186 moisture in, 142 quasi-steady state transport in, 190

M Mass/energy transport equations, 126- 153 boundary conditions in, 133-1 37 governing point equations and, 128-1 33 volume-averaged, 137- 153 Mass fraction weighted average heat capacity, 155 Mass transfer nonuniformity of in impinging flow, 38 in rivers, bays, lakes, and estuaries, see THIRBLE Mass transfer coefficient and array of slot nozzles, 42 in impinging flow, 6-7 integral mean, see Integral mean transfer coefficients vs. nozzle-to-plate distance, 39 swirling jets and, 41 trip wires and, 42 turbulence promoters and, 41 Mass transport convective transport in, 169-175 in gas phase, 165-175

SUBJECT INDEX Mass transfer coefficient, arrays of nozzles vs. outlet flow conditions for, 27-41 Matter, conservation of, 62-63 Mean transfer coefficients integral, see Integral mean transfer coefficients reduction of by unfavorable outlet conditions, 40 Moisture in liquids and gas phases, 142 porous media and, 120 Moisture content defined, 142 as function of space and time, 127 Moisture distribution, in sand, 122-123 Moisture saturation defined, 143 fractional, 143, 196 Molten polymers see also Shear flow; Steady shear flow Biot number and, 207, 222-225,238-239. 243, 262 elongation flow in, 260-262 heat transfer in, 206 nearly shear flow in, 261 Poiseuille flow in, 218-219 rheological properties of, 206-207, 230-231 shear flow in, 212-259 steady shear flow in, 227-250 thermal properties of, 209-210 viscosity curve for, 215 viscous dissipation in shear flows of, 205 -264 Momentum, conservation of, 62-63

N Nahme number, in steady shear flow, 234, 242-243,262 Navier-Stokes equations, in impinging flow, 4 Newtonian fluid, generalized, 21 1 Nonviscometric flow, 260-262 Nozzle exit velocity for array of slot nozzles, 25 blower rating and, 45 Nozzles high-performance arrays of, 52-58

277

optimal spatial arrangements of, 45-52 slot, see Arrays of slot nozzles Nozzle-to-plate distance in impinging flow, 36, 45 mass transfer coefficient and, 39 Numbers of transfer units, in impinging flow, 12 Nusselt number Biot number and, 243 in impinging flow, 7 as shear flow parameter, 262 in steady shear flow, 239-241 temperature difference and, 242 viscous dissipation and, 241 -242

0

One-dimensional unsteady vertical distribution models auxiliary relations in, 91-92 boundary conditions in, 92 differential equations for, 89-91 Ekman layer in. 93-95 mathematical formulation in, 89-93 type of problem in, 89 warm-water column cooling and, 95-98 Optical spatial arrangement, of nozzles in impinging flow, 45-52 Outlet flow conditions, mean transfer coefficients and, 40

P Partially parabolic layer, in two-dimensional floating layers, 109-1 I1 Partial mass enthalpy, 150 Poiseuille flow, 218-222 converging or diverging, 261 defined, 221 Pollution quantitative prediction of, 62 thermal or chemical, 62 Polymers, molten, see Molten polymers Porous media see also Drying process capillary action in, 121 characteristic time for flow in, 170 drying process in, 119- 120

SUBJECT INDEX

278

quantitative effects in, 191 heat, mass, and momentum transfer in, 119-200 liquid phase flow for two-fluid system in, 176 mass and energy transport in, 126- I53 temperature gradients in, 191-192 Power plants, pollution from, 62 see also THIRBLE Prandtl number, in impinging flow, 7

R Reynolds number for array of round nozzles, 20 for array of slot nozzles, 25 high performance arrays of nozzles and, 57 in impinging flow, 3, 7 in shear flow, 230 for single nozzles, 15-1 6 for single slot nozzles, 18 in stagnation flow, 5 Rheological constitutive equation, 21 1-212 Rivers and bays, heat and mass transfer in, 61-114 see ulso THIRBLE Round jet boundary conditions for, 74 expected results for, 75 geometry and physics of, 73-74 importance of, 74 method adaptation for, 75 in surrounding stream, 73-75 Round nozzles, vs. slotted, 36

S Sand layer, water distribution in, 122-123 Shear direction, 213 Shear flow, 212-259 activation energy and, 216,218 with closed stream lines, 250-259 Couette flow in, 219-222 dimensionless parameters in, 252-253 dimensionless variables in, 231 -233 elongation Row and, 260-262 expansion cooling in, 229 fully developed temperature field in, 258-259

heat transfer studies in, 220 kinematically developed velocity in, 253 master curve in, 217 nearly steady, 219 open or closed stream lines in, 219-222 Poiseuille flow in, 219-222 stead, see Steady shear flow stream lines in, 219-222 thermal boundary condition in, 222-227 unsteady, 256-258 viscous dissipation in, 205-264 wall thermal capacity in, 225-227 Shear flow program, universal numerical, 235-238 Shear surfaces, defined, 212 Shear viscosity, 214-219 Sherwood number for array of round nozzles, 20,37 for array of slot nozzles, 25-26, 34 in impinging flow, 7-1 2 mean integral for high performance arrays, 57 for single nozzles, 15-16 variation in for arrays of slot nozzles, 34 SIMPLE algorithm, 102 Single nozzles, integral mean transfer coefficients for, 15-18 Single slot nozzle, integral mean transfer coefficient for, 18 Slot nozzles vs. round, 36 single, 15- 18 Source-and-sink laws, 63 SRN, see Single round nozzle SSN,see Single slot nozzle Stagnation flow jet length and, 4 for single nozzles, 11 Stagnation point, vs. jet axislsurface point,

44 Steady axisymmetrical jets, in THIRBLE classification, 65-66 Steady shear flow defined, 2 1 8 dimensionless parameters in, 233-235 experimental studies in, 248-250 fully developed temperature field in, 237 heat transfer studies in, 220 Nusselt number in, 239-241 with open stream lines, 227-250

SUBJECT INDEX shear viscosity and, 214-219 velocity field in, 235 Steady two-dimensional layer model elliptic case in, 104-106 mathematical formulation of, 99-100 parabolic and hyperbolic cases in, 100-106 for two-dimensional floating layers, 98- 106 Stefan diffusion tube problem, 169 Stream function, 99 Stream lines, in shear flow, 221 Swirling jets, in heat and mass transfer, 41

T Temperature, as function of space and time, 127 Temperature field, fully developed, 237, 258-259 Thermal boundary condition, in shear flow, 222-227 Thermal conductivity, effective, 140, 158-164 Thermal energy equation, total, 154-158 Thermodynamic relations, in drying process, 164-1 65 THIRBLE (Transfer of Heat in Rivers, Bays, Lakes and Estuaries), 61 -1 14 classification in, 65-69 fluid motion in, 63 general case of, 68 GENMIX and, 69,72 interface-transfer problem in, 64 jet mixing phenomena in, 65-66 laws and models for, 62-63 one-dimension unsteady verticaldistribution models in, 89-98 scientific components of, 62-63 steady axisymmetrical jets in, 65-66 subjects related to, 63 threedimensional steady jets in, 67 two-dimensional floating layers in, 98-1 13 two-dimensional parabolic phenomena in, 70-72 two-dimensional steady boundary layers in, 66,79-88 two-dimensional steady jets and plumes in, 73-79 warm-water layer problems in, 95-98 zero-dimensional (stirred-tank) problems in, 65

279

Three-dimensional processes, in THIRBLE classification, 67-68 Total thermal energy equation, 154-158 Transport laws of, 63 mass, see Mass transport Trip-wire experiments, 41 Turbulence-energy equation, 91 Turbulence promoters, in heat and mass transfer, 41 Turbulent motions, water-air interface and, 64 Two-dimensional floating layers hyperbolic layer in, 106-109 partially parabolic layer in, 109-1 11 radial pool and, 1 11 - 112 steady nearly radial flow and, 112 steady two-dimensional layer model in, 98-106 in THIRBLE classification, 98-1 13 unsteady layers in, I 12- 113 Two-dimensional parabolic phenomena auxiliary relations in, 71 -72 boundary conditions in, 70-71 differential equation in, 70 mathematical characteristics of, 70-72 predictions for, 72 in THIRBLE, 70-72 Two-dimensional steady boundary layers, in THIRBLE classification, 66 Two-dimensional steady boundary layers adjacent to phase interfaces air boundary layer above a lake, 79-82 boundary layer in water at lake surface, 82-85 combined air-water layer for, 85-88 Two-dimensional steady jet phenomena predictions for, 75 round jet in surrounding stream, 73-75 in THIRBLE, 73-79 and vertically rising warm-water plume in stratified surroundings, 76-79 Two-fluid interface, transfer problem in, 64

U Unidirectional flow, 213 Universal numerical shear flow program, 234-238

SUBJECT INDEX

280 V

Vaporization, enthalpy of, 157 Velocity field at entrance, in steady shear flow, 235 Vertical-distribution models, onsdimensionai unsteady, 89-98 Vertically rising warm-water plume, 76-79 boundary conditions for, 77 expected results for, 77-78 geometry and physics for, 76 importance of, 77 method adaptation for, 77 Viscometric flow, 212-259 see also Shear flow Viscosity Newton’s law of, 21 1 pressure dependence of, 218 Viscous dissipation, Nusselt number and, 240 -24 1

Volume-averaged equations in mass/energy transport, 137- 153 phase average in, 138

W Wall, thermal capacity of, in shear flow, 225-227

Wall boundary layer, in impinging flow, 5 Warm-water column, cooling of, 95-98 Warm-water plume, vertically rising, 76-79 Water-air interface, turbulent motions at, 64

Wire-mesh grids, in heat and mass transfer, 41-43

Wood, moisture distribution during drying of, 125

2 Zero-dimensional (stirred-tank) problems, in THIRBLE classification, 65 Zero pressure gradient, in steady shear Row, 244

Zero wall shear stress, 246

A 8 7

c a D 9 E O

F 1

6 2 H 3