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.
111 Fifth Avcnut, New Ymk. New York 10003
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
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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
HID
>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 << 1).
IMPINGING JET FLOWHEATAND MASSTRANSFER
29
The factor TD assumes values from zero for good nozzles up to 1.69 for sharp-edged orifices [ 5 3 ] . Differentiating Eq. (5.5a) with respect to y yields -dPldY
=
(1
+ 5dPW d W Y
(5.5b)
Now we introduce some dimensionless parameters and variables: the relative outlet area f A :
FdBl
fA
a total friction factor for the outlet flow CA: [A
= jlRl/dh
(5.7)
and a (preliminary) dimensionless outlet flow-path coordinate :
From these, Eq. (5.la) becomes w( Y) = fA du/dy*
(5.lb)
and eliminating the pressure gradient from (5.3) and (5.5b) results in (5.9)
Replacing w in Eq. (5.9) by Eq. (5.lb), and dw/dy* by its derivative, yields a differential equation for the outlet flow velocity u( y) (5.10)
This equation becomes much simpler when friction is neglected against acceleration of the outlet flow (CA + 0). Combining the constant factor of the second derivative with y* to a new flow-path coordinate
or
further simplifies Eq. (5.10) (without friction) to u't -
=
0
(CA
= 0)
(5.12)
and generally to
(d' - u)u' = ([A/4P) u2
(5.13)
30
HOLGERMARTIN
where a prime denotes a derivative with respect to q. From the solution of that nonlinear, ordinary differential equation for the outlet flow velocity u(q) also the variations of the nozzle exit velocity w(q) and of the pressure p ( q ) underneath the array may be calculated by Eqs. (5.la) and (5.5a). The general solution of Eq. (5.12) for the frictionless outlet flow is given by a sum of hyperbolic functions: U(V) = CI
sinh q
+ ~2 cash q
(5.14)
The constants c1 and c2 are to be evaluated from the boundary conditions u(0) = 0
=> c2 =
(5.15)
0
The second boundary condition is given by Eq. (5.2): u(p) = u, = m/fA
c1 =
w/(fA sinh p)
(5.16)
So, the outlet flow velocity u ( y ) becomes
(5.17) From Eq. (5.la) the corresponding variation of the nozzle exit velocity w(y) (5.18)
and from Eq. (5.5a) the variation of pressure p(y) under the array are obtained immediately : (5.19)
The same functional variations of u, w, and p were found by Reichardt and Tollmien [54] for the manifold collection problem, i.e., the flow of a fluid through many parallel ducts into a main collecting duct perpendicular to these. Putting y = 1 in Eq. (5.19) and considering that p(1) (p/2)ue2equals the surrounding pressure pm, after some reorganization yields (with Eq. (5.16))
+
(5.20)
Equation (5.20) describes the gauge pressure in the plenum chamber required to yield a mean nozzle exit velocity W and to overcome the acceleration pressure loss of the outlet flowin addition. For large relative outlet area f A
IMPINGING
JET FLOW HEATAND MASSTRANSFER
31
the right-hand side of Eq. (5.20) tends to unity ( p -,0, see Eq. (5.11a)) (5.20a) and for small relative outlet area to infinity (5.20b) With Eq. (5.16) this limit may be written (5.20~) The flow resistance is then completely determined by the outlet flow. Subdividing the total pressure difference in Eq. (5.20) between plenum chamber and surroundings into the pressure difference over the nozzle exit (after Eq. (5.20a)) and the acceleration pressure loss of the outlet flow, this additional fraction may be formulated (5.21) Figure 18 shows the additional pressure loss due to the acceleration of the outlet flow versus the relative outlet area f A computed from (5.21) with = 0 (well-shaped nozzles).
cD
0
1
-5
2
3
4
FIG.18. Acceleration pressure drop of the outlet flow relative to the nozzle exit pressure drop vs. relative outlet area.
HOLGERMARTIN
32
An analytic solution for the complete differential equation (5.13)(with the friction of the outlet flow not neglected) was found by a power series approach [5,8]. The form of the solutions (Eqs. (5.17)-(5.19)) remains completely the same, only the hyperbolic functions (sinh q, cosh q) have to be replaced by new functions S(q,b) and C(q,b), where b is an abbreviation for the parameter in Eq. (5.13) containing the outlet flow friction factor iA: b = - =CA- f 4p
(+ ) 1
c.4
iD
112
4A--
(5.22)
It turns out that the power series.S(q,b) and C(q,b) contain the sinh or cosh series, respectively [9] : S(q,b) = q
+ 3!1
-q3
+ -7!1q 7 + -9!1q 9 + ' . .
1 5!
f -q5
(5.23)
+ (2b14 + . . a
(5.24)
C(% b) = - S(V, 6)
av
For .fA 2)-/, series are given by
(q I1) and CA 5 4 good approximations to these
S(q, b) 2 sinh q
+
(5.23a)
C(q,b) 2 cosh q
+
(5.24a)
Clearly, for negative values of q also p (or b) will be taken to be negative. Otherwise the symmetry relations S( -q) = - S ( q ) , C( - q ) = C(q) are lost. The experimental data show a surprisingly good agreement with the results of these calculations (see Figs. 19a-d). The pressure loss in the y direction due to acceleration and friction of the outlet flow lowers the difference between the pressure p o in the plenum chamber (being independent of y) and the local pressure p ( y ) in the center
IMPINGING
JET FLOWHEATAND MAss TRANSFER
H / S =25
H/S
-600 -400-2w 0 XO COO 600 y /mm
20
= 1.25
I
20
O
33
II
, , , -600 -4OO-Xx)
I
! , , 0 Mo 400 y/mm
H/S = 4.2 = 0.96 &-0)
I = 0035 9= 12.2m/s
O I , ,
,
-600 - 4 0 0 3 0 0
sbo
I
I
0 y/mm
,
,
,
400 600
FIG.19. Measured variation of the nozzle exit velocity in arrays of slot nozzles. Curves calculated after Eq. (5.18).
of the array ( y = 0), whereas it enlarges this difference at the edges ( y = k 1). Depending on that pressure difference, the nozzle exit velocity; and on their part, depending on the latter, the heat and mass transfer coefficients have to follow this variation. Qualitatively, the local variation of the transfer coefficients for impinging flow from arrays of slot nozzles is to be expected to look like that shown in Fig. 20, when the gas is forced to stream out parallel to the slots on both sides over the width of the material. Indeed, mass transfer experiments, carried out to investigate the influence of the outlet flow, turned out to prove this expected behavior [9]. Some of the results from this investigation are shown in Figs. 21 and 22 (32 additional figures of similar results may be found in Martin [S]).
34
HOLGERMARTIN
m
FIG.20. Variation of local transfer coefficients for arrays of slot nozzles (qualitatively) [9].
90
70 Shl
50
Ylm
FIG.21. Variation of Sherwood numbers across the width of the plate for arrays of slot nozzles [9]. ASN No. 4 ; f = 0.14; H / S = 1.25;fA = 1.29; Re varied.
36
28 Shl
20 12
FIG.22. Variation of Sherwood numbers across the width of the plate for arrays of slot nozzles [9]. ASN No. 5; f = 0.07; H/S = 1.25; Re = 1300; height of outlet channels varied (140 mm 2 A, 2 0).
IMPINGING JETFLOWHEATAND MASSTRANSFER
35
The direct interrelation between the variations of the exit velocities and the transfer coefficients may be seen clearly by comparing Figs. 19 with 21 and 22. The larger the relative outlet area f A , the smaller are the variations of w( y) and p( y). As a measure of the degree of nonuniformity, the ratio w( y = O)/w(y = 1) (or fi( y = O)/fi( y = Ip) respectively)is plotted against f A in Figs. 23 and 24. The curves in Fig. 23 were calculated from an equation derived from Eq. (5.18), with the hyperbolic functions replaced by the more general S and C functions (Eqs. (5.23) and (5.24)): W ( Y = O)/W(Y
= 0 = 1 / C ( ab)
(5.25)
-
For the mass transfer, the corresponding curves in Fig. 24 were obtained from these with /? w2I3 (see Eq. (4.22)) B(Y = O)/B(Y = 1,) = 1/c2’3(P(~p/~)9 b)
(5.26)
Even though there are considerable differences between measurement and calculation especially for small f A in the case of mass transfer, the general agreement is quite sufficient regarding the simplicity of the flow calculations
0
1
2
‘A
3
FIG.23. Nozzle exit velocity ratio vs. relative outlet area: 0 Reichardt and Tollmien [54];
Frenken and Gref [ 5 5 ] ; 0 own measurements [9] (see Fig. 19). Curves computed from Eq. (5.25).
HOLGER MARTIN
36 10
OR 0.6
41 0 0
1
2
?4
3
FIG.24. Mass transfer coefficient ratio vs. relative outlet area. Symbols as in Fig. 16. Curves computed from Eq. (5.26).
leading to these curves. From Fig. 24 or Eq. (5.26) one may see that the outlet cross section F A required to make the transfer coefficients sufficiently uniform across the width of the material ought to be at least thrice the nozzle exit cross section Bl. Relative outlet areas f A falling short of unity are to be avoided in any case. Of course, the above calculations concerning the influence of outlet flow on the variation of the nozzle exit velocity w ( y ) are valid analogously for arrays of round nozzles with similar outlet flow conditions. Nevertheless, the results obtained for arrays of slot nozzles are not directly applicable to round nozzles (or holes). This was shown by a new investigation on mass transfer with different arrays of holes (see Table 111) carried out by Kraitschev [4] with the experimental setup already used in Martin [9]. The typical variation of the ASN transfer coefficients across the width of the material, with a minimum in the center of the array, and increasing values in outlet flow direction (see Figs. 21 and 22) were not found here but for very small relative nozzle-to-plate distances (HID < 3). Becoming nearly uniform for H / D z 3, the variation is completely turned over for intermediate HID, now showing a maximum in the center with the transfer coefficients decreasing in the y direction. With further increasing relative nozzle-to-plate distances H I D the maximum decreases until uniformity is reached again (see Fig. 25). This was observed qualitatively with all arrays of holes investigated. One may explain this behavior (at least qualitatively) with the deflection of the round jets by the outlet stream. A jet will be the
TABLE 1x1
ARRAYS OF ROUNDORIFICES INVESTIGATED BY KRAITSHEV [4]
(k:
ARO-l\(aSymbol 1 V 5 2 V 5 3 0 25
4 5 6 7
8 A
A 0
8 9 10 11 12
AK BK CK /mml lmml /mml
Ef
feff
30 a015 0 90 I7O 0.458 30 0.015 75 90 170 30 0.035 0 L8 128 o.609 75 30 0.0351LO 48 128 10 30 0.061 0 90 170 o,L58 10 30 0.061 75 90 170 10 30 0.061 0 L0 128 o.609 10 30 0.061 140 48 128 2.5 15 0.015 5 30 I = 750mm 10 60 '* ~=0.61 20 120
+ 0
0 0
ARO No.7
Re ' = 7600 f, = 0.LL . . .2.18
I1
50
Sh,
1
LO
30
20 10
I
I
I
I
-0.7 -0.5 -a3 -0.1
l
l
I
I
I
0.1
0.3
QS
0.7
FIG.25. Variation of Sherwood numbers across the width of the plate for an array of round orifices [4].
HOLGER MARTIN
38
more deflected in outlet flow direction, the longer the undeflected jet axis (i.e., the larger H / D ) and the higher the ratio of the momenta of outlet flow and jet (uz/wz) are. The momentum ratio may be estimated from Eqs. (5.17) and (5.18) with f A HID H/D (5.27) f*= -
-
KFA
as a function of the relative nozzle-to-plate distance
(5.28)
[,,
where was put equal to 1 for the sake of simplicity. So, with increasing H / D the length of the undeflected jet axis (HID) on one hand and the momentum ratio (u/w)~on the other will act upon the jet deflection in an opposite way. This has to lead to a maximum jet deflection at an intermediate value of HID, probably corresponding to the maximum variation of the transfer coefficients across the width of the material. As a measure for the nonuniformity of mass transfer, the ratio b( y = O)/ /3(y = Ip) is plotted against H / D in Figs. 26 and 27. Unity, an upper limit to this ratio in the case of slot nozzles (see Fig. 24), is by far exceeded here. As one would expect from the above jet-deflection considerations, maxima of nonuniformity were found at certain intermediate values of HID. Position and extent of these maxima depend on KFA (defined by Eq. (5.27)), being 1.8
P(y=Ol
7.6
p fy= 1,)
1.4
1.2 1.0 0.8
1.4 1.2 1.0 0.8
n
1
I
10 __F
15
HID
FIG.26. Mass transfer coefficient ratio vs. relative nozzle-to-plate distance for arrays of round orifices with outlet channels. Symbols,see Table I11 [4].
IMPINGING JETFLOWHEATAND
MASS TRANSFER
39
3
Ply=Ol
P iy =Ip)
1. 1
0
0
3
20
10
30
0
HID
FIG.27. Mass transfer coefficient ratio vs. relative nozzle-to-plate distance for regularly spaced arrays of round orifices without outlet channels. Symbols, see Table I11 [4].
an individual constant for each array without outlet channels (AK = 0) and a weak function of H / D for arrays containing those channels ( A K # 0). From simple geometric considerations K,, may be expressed in terms of data given in Table 111: n KFA
=
D
r,
1
""HID
HID
+ AKBJCKD)
(5.29)
The nozzle spacing & was the same for arrays 1-8, so KF, is proportional to the hole diameter D, and to the effective fraction of the area f e f f . This effective fraction is the ratio of the sum of all hexagonal areas (see Fig. 7) to the total area of the array: (5.30)
Here n is the number of parallel lines of holes within one spacing C K and l/LD the number of holes in one line. The arrays investigated (1-8) had always three lines of holes per spacing CK (n = 3). In fact the maxima in Fig. 26 are the higher the greater the hole diameter D and the smaller the spacing C K . For arrays with outlet channels (full symbols) K, diminishes much more at small HID values than at large ones. (See Eq. (5.29).)Correspondingly, the equalizing effect of the outlet channels is restricted to smaller relative distances HID. For larger HID values, it makes no significant differenq whether the outlet channels were open or blocked.
40
HOLGER MARTIN
For simple perforated plates (arrays 9-12) with regularly spaced holes, the diameter-spacing ratio D/L, was kept constant, and the effective fraction f,,,= 1. Here K , , is proportional to the number of holes in the outlet flow direction (=I/L,) (see Fig. 27). From this it follows that the number of jets to be passed by the outlet flow is to be kept as small as possible to reach a uniform distribution of heat and mass transfer across the width of the material. It is important to remove the exhaust on the shortest possible path not only to provide for uniformity but also for the magnitude of the heat and mass transfer coefficients. Figure 28 shows mean mass transfer coefficients divided by those calculated from Eq. (4.21a) (here denoted by a subscript m i.e., for f A + m ) against the relative outlet area f A . With increasing number of nozzles (or holes) to be passed by the outlet flow, the transfer coefficients decrease considerably. The simple perforated plate has severe disadvantages against somewhat more expensive layouts with outlet channels or other means to lead off the exhaust on shorter paths. Since the channels, however, need some fraction ( z1 - jiff) of the total area, a certain reduction of performance against regularly spaced arrays has to be accepted. The mean transfer coefficients calculated from Eq. (4.21a) with a mean relative nozzle area 6 7 = cfleff are always somewhat higher than the measured ones. Using, however, only the local relative nozzle area c f in Eq. (4.21a) and multiplying the result by f,,,(as proposed by Lee [56]) yields values that are nearly always somewhat lower than the measured ones. So, these two different kinds of calculation for arrays of round nozzles
0
1
3
2
L
5
-4 FIG.28. Reduction of mean transfer coefficients by unfavorable outlet flow conditions. l/Lo = number of holes (jets) in outlet flow direction [4].
IMPINGING JETFLOWHEATAND MASSTRANSFER
41
(or holes) with irregular spacing due to outlet channels may be used to find an upper and a lower limit for the transfer coefficients. The true value will be the closer to the lower limit, the smaller the relative nozzle area J and vice versa.
VI. Other Parameters Influencing Heat and Mass Transfer A. TURBULENCE PROMOTERS. SWIRLING JETS The influence of the turbulence level at the nozzle exit has already been mentioned. This influence, which is never taken into account in our design equations, may cause much of the scatter of data from different investigators. It would make little sense, however, to insert this parameter into an engineering correlation for heat and mass transfer since it is not a design variable like geometric data and the gas flow rate, for example. Investigations with turbulence promoters (mesh screens closely upstream from the nozzle exit) in single slot nozzles [20,26] showed that heat transfer within the stagnation region may be considerably enhanced by these means. In certain cases the secondary peaks may then vanish, and the integral mean values over a larger area were found to be even lower than without turbulence promoters [20]. According to a personal communication of Professor Jeschar on the results of a new investigation, carried out by his assistant Mr. Potke [57], the insertion of swirl baffles into a single round nozzle had no significant influence on heat and mass transfer.
B. WIRE-MESH GRIDSON THE SURFACE OF THE MATERIAL In some types of drying plants the sheets of material are transported between continuous wire-mesh belts. This is necessary for example in veneer drying to keep the sheets even. Already in the investigation of Schllinder and Gnielinski [l] it was shown that rings of wire (so-called “trip wires”) on the surface may considerably enhance the mass transfer in certain regions. The rings used in Schliinder and Gnielinski [I] were made of wire 0.5 mm in diameter, placed concentrically to the stagnation point of a single round impinging jet. Figure 29 shows some results of these “trip-wire” experiments. Korger [SS] made some experiments with different types of sieves, grating, and mesh grids on a naphthalene surface exposed to the impinging flow from an array of three parallel slot nozzles. In each case he found a slight enhancement of the mean mass transfer coefficients,reaching about 14% for a certain light wire-mesh grid.
42
HOLGERMARTIN
I 50
I r
I
mm 150
FIG.29. Local variation of mass transfer coefficients under the influence of “trip wires” [l]. Single round nozzle: D = 40 mm; H I D = 3.75;Re = 78000; Sc = 0.59. - without wires; A-*-trip wires at positions 2 and 3; 0-- trip wires at positions 2-7.
Lhm) cp
FIG.30. Local variation of mass transfer coefficients for impinging flow from an array of slot nozzles with a wire-mesh grid on the sublimating surface. Sc = 2.5 [58].
IMPINGING JETFLOWHEAT AND MASS TRANSFER
43
-
FIG.30a. The wire-mesh grid used in the experiment whose results are shown in Fig. 30. Diameter of wire coils 13 mm; mass per unit area -4 kg/m2.
Figure 30 (from the unpublished manuscript of a lecture [St?] kindly given to me by Mr. Korger) presents the local variations of mass transfer obtained with (and without) a mesh grid on the surface, The thick broken line (curve 2) gives the variation with the grid on the surface, and the thin full line (curve 1) the corresponding one without the grid. The local minima of curve 2 have about the same spacing as the connecting wires of the mesh grid. As may be seen from the hatched areas, the positive deviations from the reference curve considerably predominate over the negative ones.
FLOWON CONCAVE SURFACES C. IMPINGING For cooling of the leading edge of gas turbine vanes, its concave inner side may be exposed to impinging flow. With regard to this very special application some experimental work was done on impinging flow heat transfer from concave surfaces. So, the impingement cooling of concave surfaces with a line of circular air jets was studied by a transient method by Metzger, Yamashita, and Jenkins [59]. The arrangement is shown schematically in Fig. 31. A similar arrangement with a slot nozzle instead of the line of holes was investigated by Dyban and Mazur [60]. Lohe [13] studied heat and mass transfer between a single round jet and a liquid surface being deformed by the impact of the jet. Because of the very special applications involved, discussing the results of these investigations [13,59,60] would be beyond the scope of this report. For details, we refer to the original papers.
FIG.31. Cooling of a concave surface by impinging flow from a line of holes.
44
HOLGERMARTIN
D. ANGLE OF IMPACT
The influence of the angle between jet axis and surface on heat transfer coefficients was already studied in one of the first papers [61] on impinging flow heat transfer. A hot round jet of air was made to impinge on a watercooled flat plate (see Fig. 32). The heat transfer coefficients were determined by means of a calorimeter flush with the surface of the plate. This calorimeter, whose diameter was less than or equal to the nozzle diameters used in [61] (r/D IO S ) , was always centered at the point of intersection between jet axis and surface. With the distance H between nozzle exit and this point of intersection kept constant, the heat transfer coefficientsdetermined in this way strongly decreased with decreasing impact angle cp. At an angle of = 15" the heat transfer coefficient was found to be about 43% lower than the corresponding one for normal impingement (cp = 900). This experimental finding, however, being restricted to a certain spot on the surface, says nothing about the dependence of the mean transfer coefficients on the angle of impact. Korger and KFiZek, in a new paper [62] on mass transfer to inclined plane impinging jets, pointed out that the stagnation point (or the point of maximum heat and mass transfer) does not coincide with the point of intersection between jet axis and surface. The stagnation point shifts by a length AX toward that part of the jet being deflected in the acute angle (see Fig. 32). This length may be determined [62] approximately by
AX
=
A cot q
(6.1)
where A was given by the authors for a slot nozzle 10 mm in slot width as a linear function of H by the equation A = 14 mm + 0.197. Generalizing this result for arbitrary slot widths B, one may easily recognize that A (and AX) is proportional to the breadth of the free jet: A = 1.4(B
+ O.llH)
FIG.32. Inclined impinging jet (schematic)
(6.2)
IMPINGING JETFLOWHEATAND MASSTRANSFER
45
With their naphthalene sublimation method Korger and KfiZek found that the integral mean mass transfer coefficients were practically independent of the impact angle cp for constant distance H (within the range 30” I rp I 9 0 O ) . Only the position of the maximum is shifted by AX and the local values to the left of the maximum (for negative values of X - AX;see Fig. 32) are higher than those to the right of it. Presuming these findings to be valid approximately also for round nozzles (because of the shift of the maximum this would not be inconsistent with Perry’s [61] results) the equations given in Section IV for normally impinging jets may also be used for impact angles smaller than 90”. Only the nozzle-toplate distance must not be measured normal to the plate but normal to the nozzle exit cross-section: H = Z/sin cp (6.3)
VII. Optimal Spatial Arrangements of Nozzles Here and in the following the notion “optimal spatial arrangement” means a combination of the geometric variables that yields the highest average transfer coefficient for a given blower rating per unit area of transfer surface. For uniformly spaced arrays of nozzles with good outlet flow conditions, there are always three independent geometric variables: (1) nozzle diameter D (or slot width B), (2) nozzle-to-nozzle spacing & (for ARNs L, may be L, or L,, see Fig. 71, (3) nozzle-to-plate distance H. The mean nozzle exit velocity i~ is to be expressed as a function of the blower rating per unit area of transfer surface P:
P
= ApV/A,
(5) 113
Ap
= 5(p/2)iij2,
V = WfA,
W
=
The pressure loss coefficient 5 may be considered the sum of all flow resistances between blower and nozzle exit (7.2)
where f A is the total nozzle exit cross-sectional area, and the ui are crosssectional areas of the ducts connecting the blower with the array of nozzles. Most of the flow resistances will be independent of the flow velocity, so their sum may be considered a constant. Furthermore, one of the three lengths involved in the problem is to be preset. Otherwise one will find that scaling down all three lengths simultaneously results in monotonically increasing transfer coefficients. To determine the optimal spatial arrangement, in the equations for the mean heat and
46
HOLCERMARTIN
mass transfer coefficients to impingingflow from arrays of nozzles (Eqs. (4.21) and (4.22)),the nozzle exit velocity (in the Reynolds number) is replaced by the right-hand side of Eq. (7.1). Thereby the mean transfer coefficients are described in terms of the relative blower rating P and the three lengths. According to the choice of the preset length, the optimization problem is solved under three different secondary conditions (D(or B) = const; L, = const; H = const) consequently yielding different optimal spatial arrangements [8, 91. The most important of these three possible secondary conditions is to preset the nozzle-to-plate distance H. A certain minimum distance H will often be imposed for constructional reasons, for example to avoid any contact with the nozzles when the sag of the material varies. Moreover, the arrangement optimized under this secondary condition (H = const) turns out to have the greatest relative outlet area f and is therefore better than the two other possible ones regarding the influence of the outlet flow. For the mentioned reasons, only this secondary condition H = const will be treated in the following. Equations (4.21)and (4.22)with E replaced by Eq. (7.1) will be reorganized such that the left-hand side, besides the mean transfer coefficient E (or B), contains only those parameters that may be considered constant (P, H, t, and physical properties): (7.3)
On the right-hand side there will remain a function of the geometric variables alone. This function GH of course can be represented in terms of other sets of two geometric ratios, e.g., GH(”)H’
ARN
,
GH(”)H ’ H
ASN
For the ARN one obtains from Eq. (4.21) (H/D)”3
G H ( f r H/D)ARN
=f2/9
K(f’
1 - 2.2J-s 1 + 0.2(H/D - 6 ) 8
and for the ASN from Eq. (4.22)
fO(H/S)
=
[60
+ 4(H/S - 2)2]-1’2
IMPINGING
JETFLOW HEAT AND MASS TRANSFER
47
The functions in Eqs. (7.4) and (7.5) each possess an absolute maximum, the position of which may be determined by putting their first derivatives equal zero. From aG,/a(H/S) = 0 and aG,/af = 0 (necessary conditions for the existance of an extremum) one obtains two equations of the form
the solution of which yields the set (f; H I S ) , , . The calculation is carried out first for the ARN in the following. With some abbreviations h Y
I
[l
HID;
cp
=fl
G2 = cp5/’;
G 1 -= h1I3. G4
E
+ 0.2(h - 6 ) q J - l ;
G3 = 1 G5
[l
-
2.2~
+ (h*q/0.6)6]-0.05
Eq. (7.4) may be written G, = G1G2G3G4G5:
ah
3h
8%-
51
0.2cp - 0.05 1 + 0.2(h - 6 ) q
acp - GH {G (p -
2.2 1 - 2.2q
11F,(cp, h) = 3h 5
-
0.2(h - 6 ) - 0.05 6(h/0.6)(rph/0.6)’] 1 + 0.2(h - 6)q 1 (~h/0.6)~
+
((Ph/o.6)5
cp
+ (h - 6 ) q - 0 . 5 ~1 + ( ( ~ h / 0 . 6=) ~
5 1 22 F,(cp, h) = - - 9 ~p 10 - 2 2 ~ 5
h-6 ((Ph/o.6)5 = + (h - 6 ) -~0.5h 1 + (40h/0.6)~ (7.9)
These are two equations for the two unknowns (H/D),, and fop,. The last terms, involving fifth and sixth powers, which originate from the array correction function, may be eliminated by multiplying Eq. (7.8) by - h / q and adding the result to Eq. (7.9): F,(q,
h
4 = -- F , ( q , h) + F,(q, h) cp
21 22 F,(q, h) = -- 10 - 2243 5 9q Equation (7.10) may be solved explicitly for h. +
6 + (h - 6)q =
o
(7.10)
HOLGERMARTIN
48
After some reorganization one gets h = ‘(12 + 25 22 - 169q (7.10a) 11 (11q)Z - 10q The result may be introduced in either of the two Eqs. (7.8) or (7.9) to find qoptby iteration (and simultaneously hoptfrom (7.10a)). From this the optimal spatial arrangement for an array of round nozzles becomes fop, = 0.0152
(H/D),,pt = 5.43 (ARN = 0.385 max GH = GH((H/D)opt,fop,) Krotzsch [3], using the different powers of Re according to Eq. (4.11) and Eq. (4.16) for the array correction function, from a similar calculation obtained the following values: 2000 < Re < 30,000, fopt = 0.0128, ( H / D ) , , = 5.6 30,000 < Re < 120,000, fop, = 0.015, (H/D)opt= 5 So, the modifications that were introduced here against the original representation after Krotzsch did not significantly change the position of the maximum. Correspondingly one may rewrite Eq. (7.5) for the ASN with the following abbreviations : h
HIS;
3
G,
GI
= f419;
G
h1I3;
G,
G 2 E fA7/”.
= (f2 + f02)-213,
2
GH = - 22‘3G,G,G,G, 3
;{ ; YfO2}
dG,= G , _ _ _ af
ah
i
-f 2
- GH -1 -1 + 17 1 df, 3h
12fo dh
-d f o - 4fO3(h- 2) dh
2 2f0 3f2 fo2dh
+
(see Eq. (7.5))
From this one obtains the two equations: F l u , h) = fo2 - 2f2 = 0 F 2 ( f , h) = 1
- h(h - 2)fO2{17 - 16
jo2 f 2
+fo2
}
(7.1 1) =
0
(7.12)
IMPINGING JET FLOWHEATAND MASSTRANSFER
49
f’ = fo2/2 is introduced in (7.12):
To eliminate f from (7.11),
1
19 -h(h 3
_ _ fo’
With l/fO2= 60 for h
-
2)
=
0
(7.13)
+ 4(h - 2)’ from Eq. (7.13) a quadratic equation results 7h’ + 10h - 288 = 0 (7.14)
This quadratic equation has the positive solution: h
=
(8/S)opt=
(J6484- 10)/14 = 5.037
From Eq. (7.11) the corresponding relative nozzle area results in fop, = 0.0718. Thus, the optimal spatial arrangement for an array of slot nozzles is given by
f o p , = 0.0718 (H/S)opt= 5.037
max GH
=
(ASN)
G H ( (8/S)opt,fop,)
=
0.355
The transfer coefficients that can be reached by the optimal ARN are by about 8% higher than those for the optimal ASN, provided the blower ratings, the pressure loss coefficients, and the nozzle-to-plate distances are equal in both cases. Thereby the nozzle exit velocities are in the ratio -’ARNopt ~ A S N opt
-
-
(FASN)’;’
(7.15)
1
opt ARN
whereas the required volumetric flowrates in contrast are related to each other by (7.16)
So, slot nozzles should be used preferentially if the material does not allow for high impact velocities, as for example in film drying. Since the optimal spatial arrangements were derived for preset nozzleto-plate distance H , it would be much more convenient to give the optimal dimensions in multiples of 8.Rewriting in this way yields for the ARN optimal nozzle diameter optimal spacing (hexagonal array) (orthogonal array)
Dopt
=
0.1848
LDopt= 1.4238 L,
opt
= 1.3248
HOLGERMARTIN
50
and for the ASN optimal (hydraulic)nozzle diameter ( = 2Bop,1 optimal spacing
Sopt = 0.199H
bOpt = 1.3838
Expressed in these terms, the optimal arrangements are described by nearly the same ratios for both round and slot nozzles. With a reasonable degree of approximation one may use the following rule of thumb for an optimal arrangement, being valid for both types of nozzles 1
‘V
-H
LToptz
I H 5
Sopt
(7.17)
- 5
7
(7.18)
(with S = D for round and S = 2B for slot nozzles). Keeping these optimal relations, the transfer coefficientsmay be calculated from the following simple equations ARN max
ARN max
[D = H / 5 , L, = 7H/5] (7.19)
ASN max
P I H sc-0.42
(
O.20(YYi3
)
ASN max
[B = H/10, LT = 78/53 (7.20)
Choosing diameters and spacings different from the optimal diminishes the transfer coefficients by the ratio: (7.21)
The functions in Eq. (7.21) are plotted in the form of “contour lines” in Figs. 33 and 34 to give a “map” showing the position, the extent, and the slopes of the “transfer-coefficient hill.” For convenience, the coordinates f and HIS (or H/D) are also shown in these figures.
IMPINGINGJET FLOWHEATAND MASSTRANSFER
51
FIG.33. Positions of the optimal spatial arrangements for arrays of round nozzles (ARN) with given blower rating per unit area and preset nozzle-to-plate distance. Here L , means the nozzle-to-nozzle spacing in a hexagonal array: L , = L, = La.)
FIG.34. Positions of the optimal spatial arrangements for arrays of slot nozzles (ASN) with given blower rating per unit area and preset nozzle-to-plate distance.
One may also consider these “maps” showing a “blower-rating trough” for given transfer coefficients. The point Z/Zmax= 1 then becomes P/Pmin= 1, and the lines ?$tmax = const have to be marked in values of PIPminwhich = ( E / Z , , , ~ ~ ) - ~(see ” Eq. (7.3)): may be calculated from (P/Pmin) -
ct/Zmax
1 0.99
0.95 0.90 0.85
PIPmin
1 1.05
1.26
1.61
2.08
52
HOLGER MARTIN
Thus, with an array of nozzles, the geometric data of which are outside the contour line ?X/Brnax= 0.85, more than twice the blower rating would be required to reach the same transfer coefficient attainable with the optimal arrangement.
VIII. Design of High-Performance Arrays of Nozzles High-performance arrays of nozzles will inevitably have narrow spacings L,. According to our rule of thumb, Eq. (7.18) for example, one obtains a best nozzle-to-nozzle spacing of only 70 mm when a nozzle-to-sheet distance of 50 mm can be realized. Thus, it becomes very difficult to provide sufficiently large outlet channels. In such a case a uniform distribution of transfer coefficients across the width of the sheet may be attained despite the obvious lack of outlet area by compensating for the nonuniformity by an appropriate local distribution of the nozzle exit cross section. For arrays of slot nozzles, a compensating distribution of slot width B( y) yielding uniformly distributed transfer coefficients was calculated [7, 9) (see Fig. 35) from the same one-dimensional flow model that was used in Section V to find the variation of the nozzle exit velocity w( y). Starting points of this calculation are again the elementary balances over the volume element of the flow tube (see Fig. 14) yielding three equations for the three unknown functions of y: w ( y ) , u( y), and p ( y ) (Eqs. (5.la), (5.3), and (5.5b)).Since the slot width B is now also considered a variable, the system has to be completed by a fourth equation. This additional equation has to express the required uniformity condition : a ( y ) = const
(or P ( y ) = const)
(8.1)
To formulate this condition, the empirically known relation between the transfer coefficients (a, p), the nozzle exit velocity w, and the slot width B according to Eq. (4.22) (or any other correlation) is solved for w : w( y ) = w(a(y), B( y), L,, H , physical properties)
(8.2)
FIG.35. Slot form B ( y ) to attain uniformly distributed transfer coefficients across the width of the material (qualitatively).
IMPINGINGJET FLOWHEATAND MASSTRANSFER
53
Now, dividing w( y) by w( y = 0) and taking into account the condition after Eq. (8.1) yield the needed fourth equation in the form:
-dP*/dq
=
2U d U / d q
-dP*/dq
=
(1
w
+ [D) W d W / d q
*(x)
(11)
(8.5)
(111)
(8.6)
(8.7) In the momentum balance for the outlet flow (11) friction was neglected against acceleration (cA = 0). From (11) and (111) the pressure gradient may be immediately eliminated: =
(IV)
This, combined with (I) gives 1dW
=
& ( J2
(8.9)
id?
Now Eq. (8.9) is differentiated with respect to q and introduced in (I) to eliminate U : (8.10)
From (IV), applying the chain rule,
dW dq
- - - _ a
-
d$dx dx dq'
d 2W dq2
d2$ dx
-=(&)
d$d2x
+dxp
(8.11)
also W may be eliminated, and a differential equation for the slot width B( y ) or x(q) is obtained: -0
(8.12)
HOLGER MARTIN
54
A solution of this nonlinear, second-order ordinary differential equation can be given without specifying explicitly the function $(x), which contains the empirical relation between a, B, and w, and the condition a # a(y). The special case *(x) = l/x (i.e., wB = const) gives a slot form suitable to blow out (or to suck off) with the flow rate constant over its length. This special slot form has been calculated earlier (and proven experimentally as a suction slot) by Gref [63]. The general solution to Eq. (8.12)is obtained in form of the inverse function q(x) [9] : (8.13) In this solution, one boundary condition [(dB/dy),,o = 01 has already been introduced yielding the - 1 under the square root in the denominator of the integrand. This may lead to numerical difficulties since the integrand has a pole at q = 0: q = o
(y=O)*x=l
* = 1 (w=w(O))
(B=Bo);
These difficulties can be avoided by putting ($2
drl/ - 1)1'2
=
d(arcosh *)
(8.14)
and integrating by parts (fu du = uu - fu du with u = l/x; du = d(arcosh
$1): ?(X)
=
arcosh
+
+
J
a r y 2 h$
dx
+C
(8.13a)
The constant C is found from the second boundary condition: x = 1;
q(1) = 0, =
V(X)
=
-S'
arcosh +(x)
arcosh 1 = 0
$(1) = 1 ; arcosh XZ $
-L 1
dx
arcosh 52
(8.15)
+(r)d5
(8.16)
The remaining integral may be evaluated without difficulty. To get numerical values for the slot form, the function $(x) has to be introduced in Eq. (8.16). This is found from Eq. (4.22) with B = xBo after some reorganization:
IMPINGING JET FLOWHEATAND MASSTRANSFER
55
with
and
y
:(
1/fo2(x) = 60 + --- 4
The ratio cp(x,. . .)/q(l,. . .) in Eq. (8.17) in general is not far from unity. Therefore $(x) may be approximated by $(x) z x-I’Z
(8.17a)
In this case the integral in Eq. (8.13) may be solved analytically [7]: 1 1 + (1 +-In 2 1 - (1
?l,Z(X) = -
- x)l/2 - x)”2
Figures 36 and 37 show slot forms computed from Eqs. (8.16) with (8.17) for H/Bo = 5 and 9 and different values of BOIL,. For comparison, the form after Eq. (8.18) is given as a dashed line. It must be noted that the two parameters (H/Boand BOIL,) cannot be chosen arbitrarily. To any value of H/Bo there is an upper limit max(Bo/LT).Exceeding this limit will cause imaginary values of u. Mathematically this follows from the condition that $(x) must not become less than unity within the range 0 < x 5 1 (see Eq. (8.13)).In the limit the minimum of $(x) may just be at x = 1. So, the condition (8.19)
1
9
2
3
FIG.36. Slot forms BIB&) computed from Eqs. (8.16) and (8.17). Dashed curve computed from Eq. (8.18) ($ = x-”’).
HOLGER MARTIN
56
6 80
0
1
9
2
3
FIG. 37. Slot forms BIB&) computed from Eqs. (8.16) and (8.17). Dashed curve computed from Eq. (8.18)(I/ = x- ' I 2 ) .
applied to Eq. (8.17) determines the maximum possible values of (BOIL,) for each H/B,. The slot form in this limit becomes acute at y = 0. Physically this means that the transfer coefficients for this combination of geometric parameters do not increase further with increasing slot width (aol/LB = 0). So, even by further widening of the slots in the center of the array, the effects of the lower nozzle exit velocity at this point are not compensated for. An array was built for an experimental test where the slots could be adjusted according to a form B ( y ) calculated from (8.16) and (8.17) for L, = 85 mm, H = 60 mm (see Fig. 38). The integral mean slot width was B = 5 mm. With B / H = 0.083 and L,/H = 1.42 this array was very close to the optimal arrangement (see Fig. 34). The slot width varied from z 8.1 mm in the center to 3 mm at the edges. For the same array with parallel slots of 5-mm slot width, the transfer coefficients were to be expected to vary across the width of the plate by about 25% of the center value ( f A = 1.49). The results of the mass transfer measurements with that array are shown in Figs. 39a-c. The measurements were carried out at three different mass
FIG.38. Slot form B ( y ) of array 18 (see Table 11). Computed from Eqs. (8.16) and (8.17) for H = 60 mm,L, = 85 mm. = 5 mm, = 0. Scale tenfold enlarged in B direction.
IMPINGINGJET FLOWHEATAND MASSTRANSFER
57
FIG.39. Variation of Sherwood numbers across the width of the plate for an array of slot nozzles with the slots widened toward the center according to the form shown in Fig. 38.
flow rates of air (Reynolds numbers) and three different nozzle-to-plate distances H . H,, denotes the distance of 60 mm for which the array had been designed. At the closer distance H < H , the widening of the slots to the center is not sufficient to attain a uniform distribution of the transfer coefficients. At H = H,, one may see that the predicted uniformity was reached to a very good degree of approximation. Finally, at H > H,,the variation of the transfer coefficients was reversed. For that larger distance, and consequently larger relative outlet area, the slots are too wide in the center. The integral mean transfer coefficients from these experiments were in good agreement with those computed from Eq. (4.22)(see Fig. 40). For arrays of round nozzles, in principle one may use similar measures to attain uniform transfer coefficients across the width of the material. It is not yet possible, however, to precalculate a distribution of nozzle diameters D(y) that yields the required uniformity. As the transfer coefficients here decrease ASN No. 18
0.059
o
L
0
5
10
-
HIS
FIG.40. Mean integral Sherwood numbers for array 18; curves from Eq. (4.22).
HOLGER MARTIN
58
toward the edges of the array-at least for intermediate values of HID-in contrast to the slot nozzle the nozzle exit cross section had to be enlarged in the outlet flow direction. Meier and Kunze [64] investigated one such arrangement with the hole diameters increasing in the outlet flow direction from 2.4 mm in the center up to 4.6 mm at the edge. They used a small-sized test setup and the naphthalene sublimation method. Their results showed that the somewhat arbitrarily chosen distribution of hole diameters overcompensated for the decrease of transfer coefficients in the outlet flow direction. Finding an exactly suitable distribution of nozzle (or hole) diameters D(y ) would require a series of tests in any case. NOMENCLATURE m2/s
thermal diffusivity height of the outlet channels (see Fig. 14) m2 area of the heat or mass exchanging surface width of slot nozzles m m slot width in center of array ( Y = 0) J/(kmol K) specific molar heat capac-
fij
m
A
ity
D
f
m
F G
1 1 1 1
h
1
fA
Jlkmol
H
m
K
1
1
m
‘P
m
LT m n
m kg/(rn’s) kmol/m3
diameter of round nozzles relative noule area relative outlet area function (e.g. F(Re)) function of the geometric variables abbreviation for HID and HIS (only used in Section VII) molar enthalpy of vaporization distance between nozzle exit and solid surface array correction function outlet length (see Fig. 14) length from center ofarray to center of last measuring plate in Fig. 14 ( = half breadth of material) nozzle spacing . mass flow rate molar density of the mixture
P Pi
P
4 r S u w 2, X
x
Y
z
z‘
kmol/(m2s) molar flux of ith component N/m2 total pressure N/m2 partial pressure of ith component W/mZ blower rating per unit area of material W/m2 heat flux m radial coordinate m hydraulic diameter of slot nozzle(S = 2B) m/s velocity of outlet flow m/s n o d e exit velocity 1 mole fraction of ith component I dimensionless slot width x = B/B, (Section VII) m coordinate from stagnation point lateral to jet axis (Fig. 1) m coordinate from center of array in outlet flow direction (Fig. 14) m coordinate from stagnation point in jet axis direction m :’ = H - :(see Fig. 1 )
Definition of dimensionless numbers used throughout the text: Nu
= US/,?.
Sh
Pr
= v/u
= ,B S / 6 , , . Re Sc = v/6,2 I
I _
E
WSIV
with S = D for round nozzles and S = 2 8 for
IMPINGINGJET FLOWHEATAND MASSTRANSFER slot nozzles; physical properties to be taken at 9, = (9, + 9,)/2. a
P 6
W/(m2K) m/s m
612
m2/s
c
1 1 1
i
v 9 i,
‘C W/(m K)
P
1
V
m2/s
4
1
heat transfer coefficient mass transfer coefficient boundary layer thickness (only in Section 1) diffusion coefficient contraction coefficient friction factor dimensionless outlet flow coordinate temperature thermal conductivity dimensionless length in outlet flow direction kinematic viscosity pressure loss coefficient
p kg/m3 c p 1 cp
$
rad 1
59
density abbreviation for f i (only used in Section VII) angle between jet axis and surface function
Indices evaporating component (e.g., water), with transfer coefficients: unidirectional mass transfer 2 receiving component (e.g., air) A outlet area D at the nozzle exit L in the surrounding air m arithmetic mean between nozzle exit and surface states 0 at the solid surface 1
REFERENCES 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
E. U. Schliinder and V. Gnielinski, Chem.-1ng.-Tech.39, 578 (1967). P. Krotzsch, Chem.-1ng.-Tech. 40, 339 (1968). P. Krotzsch, Verfahrensrechnik (Muinz) 3, 291 (1970). S. G. Kraitshev and E. U. Schliinder, Paper presented at the Fachausschuss Trocknungstechnik der Verfahrenstechnischen Gesellschaft im Verein Deutscher Ingenieure (VDI), Colmar, France, 1973; synopsis in Chem.-1ng.-Tech.45, 1324 (1973). E. U. Schlunder, P. Krotzsch, and F. W. Hennecke, Chem.-lng.-Tech.42, 333 (1970). H. Martin and E.U. Schliinder, Chem.-1ng.-Terh.42, 927 (1970). H. Martin, Chem.-lng.-Tech.43, 516 (1971). H. Martin and E. U. Schliinder, Chem.-1ng.-Tech. 45,290 (1973). H. Martin, Dissertation, University of Karlsruhe, 1973. H. Schrader, VDI-Forschungsh. 484 (1961). H. Glaser, Chem.-1ng.-Tech.34, 200 (1962). H. Glaser, Melliund Texfilher.44, 292 and 400 (1963). H. Lohe, Fortschr. Ber. VDI-Z., Reihe 3 No. 15 (1967). K. Petzold, Dissertation, TU Dresden, 1968; synopsis in Luft- Kalterech. 5. 175 and 256
(1969). 15. M. Kumada and I. Mabuchi, Bull. JSME 13, No. 55,77 (1970). 16. P. N. Romanenko and M. I. Davidzon, Inzh.-Fiz. Zh. I f , 791 (1969); In!. Chem. Eng. 10, 223 (1970). 17. M. B. Glauert, J . Fluid Mech. 1, 625 (1956). 18. E. U. Schliinder, Z.Nugwiss. 19, 108 (1971). 19. H. Schlichting, “Grenzschichttheorie,” 3rd ed. Verlag G. Braun, Karlsruhe, 1958. 20. G. A. Dosdogru, Dissertation, TH Darmstadt, 1974; synopsis in Chem.-lng.-Tech. 44, 1340 (1972). 21. R. A. Seban and L. H. Back, Inr. J. Heat Muss Transfer 3,255 (1961). 22. W . H. Schwartz and W. P. Cosart, J. Fluid Mech. 10, 481 (1961). 23. P. Bakke, J. Fluid Mech. 2, 467 (1957). 24. M. T. Scholz and 0. Trass, AIChE J. 15, 82 (1970). 25. E. M. Sparrow and T. C. Wong, Int. J . Heat Muss Transfer 18, 597 (1975).
60
HOLGER MARTIN
26. R. Gardon and J. C. Akfirat, h i . J. Heui Mass Transfer 8, 1261 (1965). 27. R. B. Bird, W. E. Stewart, and E.N. Lightfoot, “Transport Phenomena.” Wiley, New York, 1965. 28. E. U. Schliinder, in “VDI-Warmeatlas,” 2nd ed., p. A-28. Ver. Deut. Ing., Diisseldorf, 1974. 29. V. V. Rao and 0. Trass, Can. J. Chem. Eng. 42,95 (1964). 30. R. Gardon and J. Cobonpue, in “International Developments in Heat Transfer,” p. 454. Am. SOC.Mech. Eng., New York, 1962. 31. R. Jeschar and W. Potke, “Verbrennung und Feuerungen,” No. 146. Ver. Deut. Ing., Diisseldorf, 1970. 32. K. Petzold, Wiss. Z. Tech. Univ., Dresden 13, 1157 (1964). 33. M. Korger and F. Kiiiek, Verfahrenstechnik (Mainz) 7, 376 (1973). 34. M. Korger and F. Kfiiek, Int. J. Heat Mass Trander 9, 337 (1966). 35. R. Gardon and J. C. Akfirat, J . Hear Transfer 88, 101 (1966). 36. G. E. Myers, J. J. Schauer, and R. H. Eustis, J . Heat Transfer 85, 209 (1963). 37. S. P. Kezios, Ph.D. Thesis, Illinois Institute of Technology, Chicago, Illinois, 1956. 38. T. S. Kim, Ph.D. Thesis, Illinois Institute of Technology, Chicago, Illinois, 1967. 39. T. Nakatogawa, H. Nishiwaki, M.Hirata, and K.Torii, Proc. Int. Heat Transfer Con$, 4th, 1970 Preprint FC 5.2 (1970). 40. J. J. Schauer and R. H. Eustis, Tech. Rep. No. 3. Dep. Mech. Eng., Stanford University, Stanford, California, 1963. 41. A. A. Andreev, V. N. Dakhno, V. K. Savin, and B. N. Yudaev, Inzh.-Fiz. Zh. 18,631 (1970). 42. M. Wolfshtein, Ph.D. Thesis, Imperial College, London, 1968. 43. 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. 44. P. M. Brdlick and V. K. Savin, Inzk-Fiz. Zh. 8, 146 (1965). 45. V. A. Smirnov, G. E. Verevochkin, and P. M. Brdlick, Ini. J. Heat Mass Transfer 2, 1 (1961). 46. H. Brauer and D. Mewes, “Stoffaustausch einschliesslich chemischer Reaktionen,” p. 316. Verlag Sauerlander, Aarau-Frankfurt am Main, 1971. 47. H. Brauer and D. Mewes, Chem.-Ing.-Tech.44,741 (1972). 48. H. H. Ott, Schweiz. Bauztg. 79, 834 (1961). 49. E. Hilgeroth, Chem.-Ing.-Tech.37, 1264 (1965). 50. S. J. Freidman and A. C. Mueller, Proc. Gen. Discuss. Hear Transfer, 195 p. 138 (1951). 51. G. Hoppner, Dissertation, TU Dresden, 1969; synopsis in Lufi- Kulierech. 6 , 283 (1970). 52. E. Hilgeroth, Chem.-Ing.-Tech.41,731 (1969). 53. K. Huesmann. Chern.-Ing.-Tech. 38,877 (1966). 54. H. Reichardt and W. Tollmien. Miti. Max-Planck-Inst. Siriimungsforsch. No. 7 (1952). 55. H. Frenken and H. Gref, unpublished data on the variation of nozzle exit velocity in an industrial air jet dryer (personal communication). 56. C. S. Lee, Verfahrensierhnik (Mainz) 8, 164 (1974). 57. W. Potke, Dissertation, TU Clausthal, 1974. 58. M. Korger, Paper presented at the Fachausschuss Trocknungstechnik der Gesellschaft fur Verfahrenstechnik und Chemieingenieurwesen (GVC) im VDI, Darmstadt, 1974 (unpublished). 59. D. E. Metzger, T. Yamashita, and C. W. Jenkins, J . Eng. Power 91, 149 (1969). 60. E. P. Dyban and A. I. Mazur, Inzh.-Fiz. Zh. 17, 785 (1969). 61. K. P. Perry, Proc. Insi. Mech. Eng., London 168, 775 (1954). 62. M. Korger and F. Kiiiek, Verfahrenstechnik (Mainz) 6 , 223 (1972). 63. H. Gref, Schiyitechnik 3, 248 (1955-1956). 64. R. Meier and W. Kunze, Luft- Kaelreiech. 8, 323 (1972).
Heat and Mass Transfer in Rivers. Bays. Lakes. and Estuaries D . BRIAN SPALDING Imperial College of Science and Technoloyy. Mechanic.d Engineering Department. Exhibition Road. London S W7 ZBX. England
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A . The Problem Considered . . . . . . . . . . . . . . . . . . . . . . B . The Scientific Components of THIRBLE . . . . . . . . . . . . . . . C. Related Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . D . The Interface-Transfer Problem . . . . . . . . . . . . . . . . . . . E. Classification of THIRBLE Processes . . . . . . . . . . . . . . . . . F . Outline of the Present Contribution . . . . . . . . . . . . . . . . . . I1. Two-Dimensional Parabolic Phenomena . . . . . . . . . . . . . . . . A . Mathematical Characteristics . . . . . . . . . . . . . . . . . . . . B. Methods of Prediction of Two-Dimensional Parabolic Phenomena . . . . C . The GENMIX Computer Program . . . . . . . . . . . . . . . . . . 111. Two-Dimensional Steady Jets and Plumes . . . . . . . . . . . . . . . . . A . The Round Jet in a Surrounding Stream . . . . . . . . . . . . . . . . B. The Vertically Rising Warm-Water Plume in Stratified Surroundings . . . . IV . Two-Dimensional Steady Boundary Layers Adjacent to Phase Interfaces . . . A . The Air Boundary Layer above a Lake . . . . . . . . . . . . . . . . B. The Boundary Layer in the Water at the Lake Surface . . . . . . . . . . C . The Combined Air-Water Layer . . . . . . . . . . . . . . . . . . . V . One-Dimensional Unsteady Vertical-Distribution Models . . . . . . . . . . A . The Type of Problem Considered . . . . . . . . . . . . . . . . . . . B . The Mathematical Formulation . . . . . . . . . . . . . . . . . . . C . The Development of the Ekman Layer . . . . . . . . . . . . . . . . D . The Cooling of a Warm-Water Column . . . . . . . . . . . . . . . . VI . Two-Dimensional Floating Layers . . . . . . . . . . . . . . . . . . . . A . The Steady Two-Dimensional-Layer Model . . . . . . . . . . . . . . B. The “Hyperbolic” Layer . . . . . . . . . . . . . . . . . . . . . . C . The “Partially Parabolic” Layer . . . . . . . . . . . . . . . . . . . D . Some Useful Further Developments . . . . . . . . . . . . . . . . . . VII . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
61
62 62 62 63 64 65 69 10
70 72 72 13 73 16
79 19
82 85 89 89 89 93 95 98 98 106 109 111 113
113 114
62
D. BRIANSPALDING I. Introduction
A. THEPROBLEM CONSIDERED Power stations and industrial plants discharge thermally or chemically polluted water into rivers, bays, lakes, and estuaries all over the world. This discharge is widely recognized as damaging to man’s environment, which is also that of the animal and vegetable kingdoms. In recent years, therefore, much scientific attention has been devoted to study of the quantitative laws governing the behavior of the pollutant, once it has been discharged. The purpose of the study is the usual one of applied science: to understand, with a view to beneficial intervention. There are many ways of situating the discharge points, of promoting or hindering dispersal, and of varying the distribution of discharge with respect to space and time; the designer of the plant can choose from among these possibilities that combination which, while still attaining the technical and economic objectives, does the least damage to the environment. Alternatively, if no way is found of reducing this damage to a level that is acceptable to the authority representing the interests of the local community, permission to build the plant may be withheld. From their different viewpoints, therefore, both plant designers and local authorities are vitally interested in the processes of heat and mass transfer in natural waters; and their interests converge on the problem of quantitative prediction. If the environmental impact of a plant is found to be unacceptable after it has been built, the only remedy may be to close it down. The economic loss of doing so is often enormous. Therefore, it is highly desirable for all concerned to establish beforehand, and to agree upon, what the environmental impact will probably be. This is the task of prediction, and it can, in principle at least, be performed through the proper employment of scientificknowledge of heat and mass transfer. The present paper is a discussion of the extent to which this knowledge truly suffices for the task. B. THESCIENTIFIC COMPONENTS OF THIRBLE To avoid having to repeat the phrase: transfer of heat in rivers, bays, lakes, and estuaries, the acronym THIRBLE will be employed. Mass transfer in natural waters will be regarded as part of the same set of phenomena. The relevant scientific laws are: the laws of conservation of matter, energy, and momentum;
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
63
the laws of transport of these entities by diffusion, conduction and radiation, and viscous action; and source-and-sink laws concerned with the generation and disappearance of chemical species (i.e,, chemical kinetics), and the absorption and emission of radiation. For the most part, the fluids appearing in THIRBLE are in motion; and often the flow is turbulent. It is therefore necessary to include, among the scientific components of the subject, the body of knowledge concerned with “turbulence modeling.” This systematizes the quantitative regularities that have been observed to hold, approximately, between the various statistical properties of a turbulent fluid; and it is expressed in sets of differential and algebraic relations, which must be solved simultaneously with those representing the better known laws just mentioned. Quantitative prediction of THIRBLE is largely a matter of solving these general equations, together with the initial and boundary conditions which specify the configurations of land boundary, water discharge, and atmospheric state that are particularly in question. Therefore the techniques of numerical analysis, and their embodiment in computer programs, are of great importance to the THIRBLE specialist. C . RELATED SUBJECTS
The laws and models that have just been mentioned are not, of course, exclusive to THIRBLE: they are the same ones that influence the movement of liquid sodium in a fast-breeder nuclear reactor, or the flow of air between the blades of a jet-engine compressor, or the dispersion of smoke from a chimney. Therefore, much knowledge and skill, accumulated by mechanical and chemical engineers, as well as by meteorologists and oceanographers, is available for adaptation and use. In some respects, the THIRBLE phenomena are simpler than those that are dealt with in, say, aeronautical or combustion engineering; chemical reaction is either absent or unobtrusive; density changes are very small (albeit significant in respect of gravitational forces); two-phase effects are absent (unless sedimentation is to be considered); and the penetration distance of radiation is small compared with the dimensions of the system. In other respects, THIRBLE phenomena are complex. For example, no matter how precisely one calculates the heat-dispersion processes within the body of water, the heat that has been transferred to it must nearly always be dissipated to the atmosphere, in the last resort, through the water-air interface. This dissipation, however, is among the most complex of heat and mass transfer processes. The reasons will be explained in the next section.
64
D. BRIANSPALDING
D. THEINTERFACE-TRANSFER PROBLEM If every river, lake, etc. had a glass-smooth surface, it would be possible to calculate the local heat-transfer coefficients on the liquid and air sides of the interface, and hence the overall coefficient. It would not be uery easy because of two mass-transfer complications. First, water will normally vaporize at the interface, both affecting the heattransfer coefficient on the air side and introducing an enthalpy change into the heat-balance equation. Secondly, the salt which is (often) present cannot vaporize; so it accumulates just below the interface, causing the vapor pressure of water there to be smaller than it otherwise would be. The relevant equations are known and soluble; but the interactions between the processes render the problem more difficult than many heat-transfer practitioners find customary. The above remarks apply to the glass-smooth surface; however, in practice there are two additional complications :The peculiarities of turbulence decay near a two-fluid interface and the effect of wind stresses in first rippling and then disrupting the interface. These will now be discussed in turn. There is much knowledge of the way in which turbulence is diminished by the presence of a solid boundary to a fluid. For example, the well-known uniform-shear-stress layer is well documented; and it is known how a thin fully laminar region adjacent to the wall merges gradually into a fully turbulent one exhibiting a logarithmic velocity profile. Can one expect that the same behavior will be present in the vicinity of a two-fluid interface? There is reason to answer no; for, although gravitational forces may prevent vertical turbulent motions from significantly raising or denting the water-air interface, there is no reason for supposing that fluctuations of the horizontal components of velocity will be so strongly damped out. Turbulence decay near a liquid -air interface is therefore likely to be very different from that near a liquid-solid interface. The differences have not however been subjected to quantitative study, at least so far as the author is aware. As to the second difference, namely the ruffling, rippling, and disruption of the interface by wind stress, the phenomenon causes large, but almost unquantifiable, deviations from the smooth-surface transfer behavior. It is true that the latter may be the extreme case on which critical predictions can be conservatively based; but it is tiresome not to know how large is the “safety factor” resulting from this practice. Such disturbed interfaces are not exclusively features of THIRBLE; they occur also in chemical-engineeringflows, for example in wetted-wall absorption columns. However, chemical engineers are not sufficientlywell advanced in their studies of the phenomenon for more than qualitative guidance to be deduced regarding the environmental transfer process.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
65
E. CLASSIFICATION OF THIRBLE PROCESSES
To say that THIRBLE processes involve the same physical laws as those in other branches of engineering, although it gives some comfort to the would-be predictor, is very far from telling him how difficult his problem is. It is therefore useful to run through a series of kinds of problem, encountered in engineering, and to comment on the presence or absence of THIRBLE problems of the same kind. 1. Zero-Dimensional (Stirred-Tank) Problems
Chemical engineers, who often conduct reaction and transfer processes within vessels containing mechanical stirring devices, are accustomed to considering idealized situations in which the stirrer is so effective that the fluid temperature and composition are uniform throughout the vessel. It is convenient to do so; for the heat- and material-balance equations are then algebraic ones if the process is steady; and, if it is transient, the differential equations are of first order, with single-point (initial-time) boundary conditions. Such mathematical problems are easy to solve : the computer programs are small and so are the requirements of storage and time. This zero-dimensional idealization is not without its utility in THIRBLE problems also. It is sometimes useful to consider warm water, discharged into the environment, as flowing out in coherent “packets,” which move away from the discharge point, losing heat to the atmosphere, but not mingling significantly with surrounding bodies of water. This is the case, for example, when “layering” occurs, as it often does: the warm water floats as a fairly thin (e.g., 1-m thick) layer, over a deeper body of cooler water; and mixing is inhibited by the density stratification. Sometimes it is useful to consider a whole pond or lake as a “stirred-tank,’’ and to calculate what temperature such a tank would then attain. In reality, of course, the stirring brought about by jet injection, or by the wind, is rarely sufficient to eliminate temperature variations; all the same, the stirredtank temperature is often worth calculating as a limit, giving the order of magnitude, and revealing the degree of sensitivity to such factors as humidity of air and lake-surface area. Zero-dimensional models of THIRBLE processes are discussed at some length in Spalding [l], a reference which reveals much of the thinking on which the present survey is based. 2. Steady Axisymmetrical Jets The jet of one fluid, injected at a uniform rate into a surrounding fluid otherwise at rest, is a familiar object of study in fluid-mechanics textbooks and in engineeringresearch laboratories; and, perhaps with the complication
66
D. BRIANSPALDING
that the surrounding fluid may be in parallel motion, such jets are found also in equipment, for example, jet-engine exhausts, ejectors, mixing devices, gas flames, etc. Later in this article (Section 111), the detailed numerical modeling of axisymmetrical jets will be discussed. At the present point however, it is useful to remember that jet-mixing phenomena can be very simply described, and quantitatively predicted, by way of well-established algebraic formulas. For example, Abramovich [23 gives formulas for the angle of spread; and Ricou and Spalding [3] have provided an expression for the entrainment rate. With the aid of one or another of these formulas, many jet-mixing phenomena of engineering practice can be quantitatively predicted. This is done by chemical engineers, by heating-and-ventilating specialists,and many others. Axisymmetricaljet-mixing phenomena are also to be found in THIRBLE processes. For example, if the density-differenceeffect is small compared with the effect ofthe injected momentum, the rate ofmixing between a horizontally injected, warm-water stream and a coflowing river may be predicted by an algebraic formula of the above kind. Further, if the fluid is injected vertically upward and the surrounding water is substantially at rest, the development of the jet may be calculated from an entrainment formula; Morton et al. [4] have done this for flows in the atmosphere. Axisymmetrical jet phenomena are easy to predict; and, if all THIRBLE phenomena were of this kind, the present article would scarcely be necessary. 3. Two-Dimensional Steady Boundary Layers Aeronautical engineers have devoted much attention to the behavior of the thin layer of retarded fluid immediately adjacent to an airfoil wing. Heatand-mass-transfer specialists have been equally interested because it is in this layer that the major resistances to transfer are found. Finally, academic scientists and mathematicians have clustered around this group of phenomena because the mathematical problems are just too difficult to be called trivial, while still being soluble with modest labor. THIRBLE problems of this type can be discerned, but not easily. If a wind blows over the surface of a lake, the distributions of velocity, temperature, and humidity in the air are akin to those in a two-dimensional boundary layer of the kind mentioned above; and the surface distribution of heat-transfer coefficient can be deduced from the appropriate theory (provided surface rippling is not too great). There is a somewhat similar boundary layer beneath the water surface as well. However, although such two-dimensional boundary layers are present in THIRBLE, they are not common. Most of the practical problems are more troublesome.
HEATTRANSFER IN RIVERS, BAYS,LAKES, AND ESTUARIES
67
4. Three-Dimensional Steady Jets
The mixing of a jet with a steady stream, inclined to it at an angle, has been studied by combustion engineers, among others; such phenomena appear in the dilution region of gas-turbine combustors and in related devices. Some authors have extended the entrainment concept to enable them to predict such flows [S, 61 ;and their example has been followed by THIRBLE specialists [7, 81. The latter are concerned, for example, with the vertically upward injection of warm water into a river; or with the horizontal injection into a river, parallel to the stream, but in such circumstances that the buoyancy-induced bending of the jet causes three dimensionality. Mixing processes of this kind, which are common in environmental heat and mass transfer, are by no means easy to predict accurately, especially when “secondary flows” are present in the river (caused perhaps by a river bend) or when density stratification interacts with (and tends to damp out) the turbulence. Recently, the present writer and his co-workers have developed a numerical procedure for solving the relevant equations [9-113; and successful applications to environmental problems have been made [12,13].
5. Other Steady Three-Dimensional Processes Of course, many other steady three-dimensional fluid-flow processes are of practical engineering importance; and numerous methods have been devised for predicting them quantitatively. Until comparatively recently, the only method of prediction was to use a geometrically and dynamically similar model; for example, a small-scale model of an airplane was mounted in a compressed-air wind tunnel, on a six-component force balance. Such model experiments are extremely difficult to devise for THIRBLE flows because similarity laws for the thermal processes (including radiation), and their interactions with the fluid dynamics, are capable of being completely satisfied only on the full scale! Wholly theoretical means of prediction are therefore to be preferred. What means are available? The designers of compressors and turbines, largely influenced by Wu [141, have for many years been developing and using pseudo-three-dimensional calculation procedures. These ignork many effects, especially heat transfer and shear stresses. Such methods do not appear to be of any applicability to THIRBLE problems. More recently, however, specialists in numerical and fluid mechanics have extended their activities to three-dimensional steady flows of a general character. Publications with which the present writer has been associated include [15-181; the last-named concerns the mixing and combustion process in a gas-turbine combustor.
68
D. BRIANSPALDING
There are THIRBLE processes that can be solved by such means. One example is the mixing of warm water from an array of jets, discharging into a moving body of water, with such a geometry that recirculation occurs in the vicinity of the discharge points; such recirculation is very likely to occur in practice. Another example is the steady circulatory motion that is set up in a lake, over the surface of which is exerted a constant wind stress: the Coriolis forces associated with the earth’s rotation, and also the buoyancy forces associated with density stratification, complicate the process. Many THIRBLE processes, if realistically described, fit into this category. They can be solved; but doing so is expensive. This point will be emphasized below; and it provides the motive for the major contributions of the present article. 6. The Generul Case
If the last simplification, steadiness, is removed, even the most complex of the THIRBLE processes can be admitted. A good example of complexity is the mixing of warm water from a power station with the waters of a tidal estuary, under conditions of varying wind and solar radiation, Transient three-dimensional phenomena are rarely predicted theoretically in other branches of applied science (except meteorology); so THIRBLE processes can play a prominent part. They have also the distinction of having the maximum geometrical complexity; for estuaries never have rectilinear banks or smoothly sloping bottoms; so the numerical analyst must either make his grid nonuniform, or enure himself to boundaries that cut his grid lines in arbitrary ways. The methods of numerical analysis mentioned in the last subsection can be (and have been [I91 applied to solve three-dimensional transient THIRBLE problems also; but little work of this kind has been done, mainly for reasons of economics. I . Conclusion
It has been seen that, when the whole range of THIRBLE processes is considered, the resources of science and numerical analysis become seriously stretched. Mosr THIRBLE processes are three-dimensional and unsteady; and the geometries of practical rivers, bays, lakes, and estuaries are highly nonuniform. The consequence is that for an accurate description of the geometries both ofthe water-mass boundaries and ofthe injection devices, a very large number of grid points is needed, larger than is commonly available on existing computers. Further, if the computation is to simulate several tidal cycles, very
HEATTRANSFER IN RIVERS, BAYS,LAKES, AND ESTUARIES
69
large computer times are necessary. Numerical simulation of the thermalpollution process becomes an expensive matter. Of course, the numerical simulation may still be economically justified; but the number of computations that are practicable will be limited; and indeed the research that is necessary to validate the models before they can be relied upon for design is itself so expensive that very little has been carried out so far. It is for this reason that, in the major part of the present article, the question asked is: How can one extend the number ofpractically interesting THIRBLE processes that can be represented by simple numerical models, specifically those that involve only one-dimensional storage in the computer, and marching integration?
F. OUTLINE OF THE PRESENT CONTRIBUTION The remainder of the present article considers a range of THIRBLE problems that can be solved by means of computational procedures involving only one-dimensional storage for each variable. The reason is that such procedures are (or may be) sufficiently economical to be employed extensively, while still permitting adequate modeling of many physical processes. Attention will be concentrated, for concreteness, upon a particular computational procedure, that of Patankar and Spalding [20] and a particular computer program, GENMIX [21]. However, there is no inherent connection between the formulation of the problems and the particular method of solution; readers preferring their own methods, or computer programs, may still find the discussion to be of interest. In problems of the present kind, the “one dimension” referred to above is a space dimension: the values of the variables stored represent the distributions of the relevant fluid property (e.g., temperature) with respect to position, either at a definite time or at a definite cross section through the flow. The implication of the last remark is that there is a second dimension involved; and this may be either time (for a transient problem) or a second distance dimension (e.g., the longitudinal position of the cross section in question). The present problems are therefore two dimensional; but they are of that special kind known as parabolic: in the second dimension, influences travel only one way. The nature of two-dimensional parabolic problems, and methods of solving them, are described in Section I1 below. There follow sections in which particular applications of these methods to THIRBLE problems are discussed. Sections 111 and IV are concerned with rather conventional applications, namely to jets and boundary layers; Section V concerns the transient
D. BRIANSPALDINC
70
behavior of water layers of great width; and Section VI is concerned with steady floating layers of finite width. The treatment is far from exhaustive, either in the range of examples or the detail with which they are treated. However, it is hoped that sufficient information and interest will be conveyed to facilitate the reader’s making further explorations for himself. 11. Two-Dimensional Parabolic Phenomena
A. MATHEMATICAL CHARACTERISTICS 1. Difeerential Equations
The problems to be considered can all be expressed mathematically as obeying differential equations of the following type [20] : (2.1-1)
Here 4 is the dependent variable; it may stand for one or more velocity components, for temperature, for concentration, or for a turbulence characteristic; x is the “longitudinal dimension,” either distance (for steady-state problems) or time (for transient ones); o is the “cross-stream dimension” (i.e., the one for which one-dimensional storage of variables is needed); a, b are functions of x, to be described below; c is a quantity representing the effectiveness of transport processes occurring in the o direction (e.g., heat conduction); d is a term representing the source of the entity 4 in unit volume (e.g., the source of heat resulting from absorption of radiation). The parabolic character is represented by the presence of only first-order differentials with respect to x. The equation is often nonlinear because the quantity c is frequently a function of one or more of the 4’s. In most cases, several equations such as (2.1-1) have to be solved simultaneously. Linkages between them are effected through c, as just indicated, but also through d. For example, d for the energy equation (with 4 standing for temperature, or stagnation enthalpy) contains uelocity-gradient terms, representing kinetic heating. 2. Boundary Conditions
Problems of the present kind are well posed when values of the 4’s are given at the lowest (“starting”) value of x (say x = 0), and when further values of 4 (or of gradients of 4) are given at the upper and lower limits of o.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
71
It is a feature of the Patankar-Spalding procedure [20] that w is a “stretching” coordinate, so chosen that the upper and lower limits of w can be unity and zero respectively. Boundary conditions are not needed at the largest value of x ; for influences travel only in the direction of increasing .Y. This is a consequence of the presence of only the first-order derivative with respect to that variable.
3. Auxiliary Relations The cross-stream variable w is connected with the distance y in that direction by the relation w
=
rpu dy/J:E rpu dy
(2.1-2)
where r is the distance from the symmetry axis (if present; otherwise, put r = 1); p is the fluid density; u is the velocity in the x direction (in steady problems; otherwise, put u = 1, and x = time); yE is the y value of the “external” (o= 1) edge of the integration domain. The radius r and the distance y are connected by r
=
r,
+ y cos
c(
(2.1-3)
where r, is the r of the “internal” (w = 0) boundary of the integration domain; a is the angle between the constant-x line and the normal to the symmetry axis. The quantities a and b are defined by (2.1-4)
(2.1-5) here $ represents the “stream function”, i.e., 1/2a times the rate of mass flow. Thus (2.1-6) - $1 = rpu dY
+
J;
Here also u is set equal to unity for a one-dimensional transient problem. The quantity c is defined by
c
= r2pU reff/($E
-
(2.1-7)
where reflis the effective transport property of the entity for which q5 stands; is the effective (laminar + turbulent) for example, if 4 stands for u, reff for other variables is often connected with peffby way of an viscosity. refr
72
D. BRIANSPALDING
“effective Prandtl-Schmidt number oeff.Thus reff
=
Pefdceff
(2.1-8)
-
Other auxiliary relations, too numerous to list, are also employed to express fluid-property behavior (e.g., viscosity temperature linkages). OF PREDICTION OF TWO-DIMENSIONAL B. METHODS PARABOLIC PHENOMENA Predictions are obtained by solving the differential equations, together with the associated boundary conditions definingthe particular phenomenon, and the auxiliary relation representing fluid properties, etc. Solution is normally effected by expressing the equations in finite-difference form, and then solving the resulting algebraic equations in a “marching” manner, along successive lines of constant x,swept downstream from small x to large. The interested reader is referred to Spalding [21] for a discussion of possible and actual methods of solution. The most important point to understand is that methods exist which can solve problems of this type easily and economically.There is no need for anyone to invent new ones, or to fear that any problem of this kind is insoluble.
C. THEGENMIX COMPUTER PROGRAM [21] Spalding [21] describes and contains a computer program that has been designed for solving problems of the present type. Here only its main features are mentioned : the program handles chemically reacting flows rather prominently; however, it is easily adapted to THIRBLE phenomena by omission, or by skipping, of the chemical-reaction sequences; the program structure is modular, permitting fairly easy adaptation to many different problems; the language is Fortran IV; automatic grid-expansion techniques are built in so as to ensure that the w = O(1) and w = 1(E)boundaries just enclose the domain of interest; line-printer-plot output is provided. It is difficult to make informative general statements about the cost of GENMIX calculations. An example must suffice: a solution of the equation for four &s, with 40 x values and 20 o values, and with copious output, has required 17 sec execution time on a CDC 6400 machine. The mixing-length model of turbulence is built into the GENMIX code of Spalding [21]. However, versions containing more advanced turbulence models also exist. The same is true of versions embodying hyperbolic and semielliptic features (see Section VI below).
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
73
111. Two-Dimensional Steady Jets and Plumes
A. THEROUNDJET IN A SURROUNDING STREAM
A preliminary mention of two-dimensional jet phenomena, occurring in THIRBLE, was made in Section I.D.2. Now is the time to consider how such phenomena can be predicted by way of GENMIX and what success can be achieved. The discussion will follow the format of Chapters 9 and 10 of Spalding [21] ;the subheadings will be : geometry, physics, boundary conditions, importance, method adaptation, expected (or actual) results, remarks. It is impracticable to provide a comprehensive account of any of the problems; it is hoped that the information supplied will suffice for immediate understanding. The references must be turned to for the complete picture. 1. Geometry Let a jet of slightly polluted water emerge steadily from a thin-lipped circular-sectioned orifice into a steady uniform surrounding stream of water, flowing parallel to the direction of injection, but at a different velocity. The general flow direction may be horizontal; but, in that case, the density differences must be small enough for gravitational effects to be negligible. This implies that 2 uo I P O - PSI << I (3.1-1) gD0 Pm where uo is the velocity of injection; g is the gravitational acceleration : Do is the diameter of the orifice; po is the density of the injected water; pm is the density of the surrounding water. The distance x will be measured along the axis, from the orifice; y and r will be measured from this axis (a equals unity). The w = 0 boundary will coincide with the axis; and the w = 1 boundary will start at the orifice lip and extend therefrom into the surrounding stream, at such a radius (increasing with x) as just to enclose the jet of significantly disturbed fluid. Thus, a will equal zero; and b will be negative. 2. Physics In all practically interesting THIRBLE problems, the flow is turbulent. The turbulence has two sources: that already existing in the injected fluid and in the surrounding stream and that which is created by the shear stresses associated with the differing velocities or by the interactions of density gradients with the gravitational field. The latter interactions are absent from the present problem. In order to be able to predict the effects of both kinds of turbulence, it is necessary to solve an equation for the turbulence energy; and a second
D. BRIANSPALDING
74
equation from which the length scale can be deduced is also desirable. Turbulence models of the required kind have been developed by several authors; Launder and Spalding [22] contains a review. The energy equation, in the present case, will take the form ak
ao
pi[ (:;y
+ ( a + bo)= a. (c a. + -
peff
-
-
CDp?]
(3.1-2)
where k is the turbulence energy per unit mass; 1 is the length scale of turbulence, to be obtained, for each location, from another differential equation; C Dis a constant; and peffis to be obtained from the auxiliary relation peff = C,pk"'l
(3.1-3)
and C , is another constant. 3. Boundary Conditions The influences of initial and free-stream turbulence are represented, in this model, by the values ascribed to k and to 1 at the x = 0 and o = 1boundaries. Convection and diffusion spread the influences of these into the interior of the jet. Other boundary conditions of interest, to be ascribed values at the same boundaries, are of course the velocity and temperature of the fluid. If the free-stream velocity varies with longitudinal distance, an associated pressure gradient appears as a momentum-source term in the differential equation for u. The appropriate expression is (3.1-4) At the w = 0 boundary, the boundary conditions are
a4/ay
=
o
(3.1-5)
for all variables. 4. Importance
This problem has practical importance, as explained in Section I.D.2. It is also of great theoretical importance; for, if it should prove that this comparatively simple turbulent-mixing process could not be predicted correctly, what hope would there be of correct prediction of more complex (e.g., threedimensional) processes? As is therefore to be expected, considerable attention has been devoted to the problem; but this has not, it must be admitted, resulted in the settlement of all the associated questions.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
75
5. Method Adaptation There is little to be done by way of adaptation of the computer code of Spalding [21] to the present problem, apart from the insertion of the data and conditions mentioned above. The code is already constructed so as to make this insertion easy. The only point worthy of further mention is the determination of the b x function, which controls the rate of spread of the integration domain into the surrounding stream. It should be clearly understood that the location of the E boundary is to be settled only by reference to computational convenience and economy; the E boundary has no physical significance.GENMIX contains a built-in device for ensuring that the E boundary is located, through an appropriate computation of b, so that it lies where the fluid velocity is nearly, but not quite, that of the surrounding stream at infinity.
-
6. Expected Results Provided that the best recommendations are adopted for the constants and functions of the turbulence model, the predicted distributions of velocity, temperature, and turbulence energy, downstream of the injection plane, are likely to agree well with accurate experimental data. Although concerned mainly with gaseous flows, not THIRBLE situations, the study reported in Launder et al. [23] provides confirmation. This paper, and the associated ones of the NASA Langley Conference on Turbulent Shear Flows, showed that it is possible to predict a large number of experital data successfully if a two-equation turbulence model is employed. Oneequation models, or zero-equation ones like the Prandtl mixing-length hypothesis, required ad hoc adjustments to procure agreement with experiment. 1. Remarks
THIRBLE specialists wishing to predict two-dimensional steady jet phenomena may conclude that the theory of turbulent heat and mass transfer is now well able to meet their needs. Moreover, the theory has been economically embodied in a finite-differencecomputer program; there is therefore no need to turn to “integral,” or other more approximate, prediction procedures. These, though they may be even cheaper to execute than GENMIX, are not capable of predicting all effects, for example that of free-stream turbulence. It is true that insufficient systematic testing of free-stream-turbulence effects has been made in round-jet circumstances; further research would be useful. However, the work of Saiy [24] has shown that these effects can be well predicted for plane mixing layers; it therefore seems not unlikely that round-jet predictions will be reliable also.
D. BRIANSPALDING
76
B. THEVERTICALLY RISINGWARM-WATER PLUME IN STRATIFIED SURROUNDINGS The situation now to be considered is this: a warm-water jet is injected oertically upward into a reservoir; the water in this reservoir is at rest, but it is not of uniform temperature; instead, the reservoir temperature increases linearly with height, being lower than that of the injected fluid at the plane of injection, but then rising steadily. What is to be predicted is the rate of spread of the resulting turbulent plume, and also the height to which it will rise. 1. Geometry
This is an axisymmetrical situation. The distance x is measured vertically upward; y and rare measured horizontally; CI is zero. An important dimension is ro, the radius of the jet at the orifice.
2. Physics In this problem, the density differences interact with the gravitational field to produce a momentum source. The d term of the u equation is thus
d
=
(3.2-1)
(PU)-'g(P, - P)
wherein pE is the density of the reservoir fluid, and g is the gravitational acceleration. Unless the temperature of the water is in the vicinity of 4" C, the density and the temperature can be regarded as linearly related, e.g., by
P
=
(3.2-2)
Poll - P(T - To))
here p o and T o are the density and temperature of the water flowing through the orifice, and p is the thermal-expansion coefficient. Of course, there is no especial difficulty about incorporating a nonlinear p T relation into the computer code. In addition to affecting the jet momentum, the density gradients may influence the turbulence energy directly: when the less dense fluid lies above the more dense fluid (as it commonly does), turbulence tends to be damped out; the reverse relation promotes generation of additional turbulence. A discussion of the mechanism may be found in Launder and Spalding [22]. This phenomenon must be reflected at least by the inclusion of a term involving gP d v d x in the source term for k ; details, and a particular example, can be found in Chen and Rodi [25].
-
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
77
3. Boundary Conditions The values of u and k will be zero at the E boundary (i,e., the outer “edge” of the plume, where w = 1); but T , will vary with x in accordance with the prescribed stratification law. The inlet (x = 0) and symmetry-axis (w = 1) conditions will be the same as for Section 1II.A. A linear pE x variation was mentioned above; but, of course, any variation can be incorporated. The time is long past at which restriction to linearity was the price that had to be paid for obtaining a solution at all.
-
4. Importance
Vertical discharge of warm water into large water masses does occur in practice, although seldom in conditions of such perfect axial symmetry as are here postulated. This problem therefore has some practical importance. Its theoretical importance is also great, partly because of the interactions between gravity and turbulence mentioned in Section 3 above, but also because of a feature which can be expected to be predicted only approximately by a noniterative “parabolic” procedure. This feature is the jinite height to which the jet can actually rise because of the density stratification of its surroundings. The injected water, together with all that it has entrained, must therefore spread out horizontally when it has reached its ceiling. It is interesting to find out how well this effect can be predicted by a solution procedure that marches only in the vertically upward direction.
5. Method Adaptation Apart from the inclusion of the terms for the sources of momentum and turbulence energy and of the appropriate procedure for calculating density from temperature, there is no adaptation needed. Naturally, the appropriate boundary-condition information must be supplied. It is useful to make the integration process approach the upper part of the jet gradually; therefore, the increments of x should be made rather small when the velocities in the jet become close to zero.
6. Expected Results When the density stratification in the surrounding water is slight (pE = constant), the jet behavior will be found to depend upon the value of the dimensionless quantity uO2fl(T,- TE)/(grO). If this quantity is large, the value of u on the axis, u,,varies as x- over an x range many times as great as r , ; the temperature excess on the axis, T, - T E , follows a similar law.
78
D. BRIANSPALDING
When uo2/?(To- TE)/(gro)has a moderate value, this x-l dependence is short-lived: at large x/ro, uo varies as x-1/3 and T, - TE as x - 5 1 3 . This is the consequence of buoyancy, which increases the velocity of the water and which also, because of the increased dilution by entrained liquid from the surroundings, lowers its temperature. If uo2/3(T0- TE)/(gro)is small, the x - l dependence range is not observable at all. Buoyancy effects dominate; and the momentum with which the warm water is injected loses all significance. This (and the earlier) predictions agree well with experimental data. When the density stratification of the surroundings plays a significant part, further interesting features appear in the predictions. Thus, radial temperature profiles may be found exhibiting a peak on the axis, a minimum at an intermediate radius, and then a rise to a secondary maximum at the outer edge of the layer. These come about as follows: at small x, cooler water is entrained into the jet; the temperature profiles takes up the familiar near-sinusoidal form. At larger x, however, the temperature of the surroundings TEis greater than at lower depths; and cooler water from below has been carried up by the jet momentum. This cooler water is found at moderate radii. At smaller radii, warm water remains, still relatively undiluted; and outside lies the warm water of the surroundings. Of course, the momentum source at the intermediate radii is now negative (see Eq. (3.2-20)).Consequently, u tends to diminish; and eventually a value of the height x is reached at which it is zero at one radial location. At this point the marching integration must be terminated for two reasons. First, it has ceased to correspond with physical reality; for, from now on, the downward-flowing (u < 0) water conveys influences from above, about which the marching integration knows nothing. Secondly, as soon as u becomes zero, the radial distances between grid points become infinite; for these are computed from Eq. (2.1-2) in its inverted form: (3.2-3)
A similar phenomenon, ie., the cessation of the plume’s upward penetration, actually occurs in practice. However, the details are bound to be somewhat different from those of the prediction; indeed there is a tendency for some streamlines to “overshoot,” i.e., to rise at first to a height exceeding that which they finally adopt. This cannot be predicted by GENMIX, which has no provision for negative M’S. 7. Remarks
Computations of the present kind have been performed by Rodi and coworkers [25]. However, the whole range of possible phenomena was not studied by them. It appears that further research is desirable.
HEATTRANSFER IN RIVERS, BAYS,LAKES, AND ESTUARIES
79
This remark must be made about most of the phenomena that have recently become amenable to detailed mathematical modeling. The experiments were performed, for the most part, before this possibility existed; and they were therefore not performed quite as the would-be model-validator desires. Since the mathematical models have become available, neither time nor resources have sufficed for mounting a new experimental investigation. It has to be admitted, also, that experimenters and mathematical modelers have only recently begun to communicate cooperatively. 8. Closure
Many books have been written and some theses have been written (e.g., Rodi [ 2 6 ] )about the numerical modeling of turbulent shear flow, its achievements so far, and its possibilities for the future. The foregoing pages merely give a brief indication of the present status of the subject.
IV. Two-Dimensional Steady Boundary Layers Adjacent to Phase Interfaces A. THEAIR BOUNDARY LAYER ABOVE A LAKE The next topic to be considered is the numerical modeling of twodimensional boundary layers adjacent to interfaces between phases. These have been extensively studied in mechanical and aeronautical engineering [27-301; the task is now to consider some relevant THIRBLE problems. The examples to be considered have already been mentioned in Section I.E.3. They concern the boundary layers in the air above, and in the water below, the surface of a wide lake. The shore line is to be thought of as straight, and aligned horizontally to the direction of the wind. Conditions are supposed to be steady. As in Section 111, concreteness will be given to the discussion by the supposition that the GENMIX computer code will be used for the solution of the relevant equations; however, there is no reason why other codes should not be employed. The boundary layer in the air will be dealt with first. 1. Geometry Let x be the distance from the shore line, in the direction of the wind; and let the distance above the lake surface be y. These two coordinates suffice as dependent variables; the flow is planar. The Z(o = 0) boundary of the integration domain coincides with the lake surface (presumed rectilinear). The I ( o = 1) boundary, on the other hand, lies at a height above the lake which increases with x; for it must enclose
D. BRIANSPALDING
80
the whole of the stream of air, of increasing thickness, which is influenced by contact with the water. Marching integration will take place, when GENMIX is employed, from the shore line in a downwind direction. 2. Physics The dependent variables needed to characterize the air conditions at every location will be: u, the horizontal velocity component; T, the temperature; m, the mass fraction of water vapor; k, the turbulence energy; and 1, the turbulence length scale (or more probably E , the dissipation rate, from which, when k is known, 1 can be computed [31,32]). Possibly, however, it may be permissible to dispense with the- latter variable, for oneequation turbulence models [33-351 can predict the behavior of boundary layers near phase interfaces fairly well. It will almost certainly be necessary to include the effect of the density gradient in the air on turbulence generation (or decay); and the influence of humidity (m)on p will be appreciable. The magnitude of the effect will be measured by the Richardson number Ri defined for present purposes by Ri
.l/!’E({’E
- pI)/(/’E‘E2)
(4.1-1)
Since the boundary-layer thickness yE will increase with x, the absolute value of the Richardson number will increase, the greater is the extent of the lake in the direction of the wind. Strictly speaking, the Coriolis forces associated with the earth’s rotation require to be included; then, indeed, the second horizontal velocity component must be computed. The GENMIX code is capable of solving the associated second momentum equation simultaneously with the other equations; however, this refinement will be omitted from the present discussion. It is mentioned in a somewhat different context below (Section V).
3. Boundary Conditions The conditions at the x = 0 station are the velocity, temperature, humidity, and turbulence-property profiles in the wind. If these are not specified, they must be guessed. Uniform values are convenient; but the numerical procedure can accept any distributions. At the E boundary, i.e., the upper edge of the integration domain, the values of u, T, m, etc. will again be those specified for the undisturbed wind, at higher and higher altitudes. At the I boundary, i.e., the water surface, u and T can be taken as those of the water. For the time being they may be supposed known; but, when Section 1V.B has been read, it will later be recognized that their values must follow from a separate computation. The humidity of the air can be taken
HEATTRANSFER IN RIVERS, BAYS,LAKES, AND ESTUARIES
81
as that which is in thermodynamic equilibrium with the water (which may of course contain dissolved salts affecting its vapor pressure). The turbulence energy, and the length scale, are both zero at the interface. However, there is much more to be said about this than can be fitted into the space available here. The following remarks must suffice:
A complete analysis of the region very close to the interface, in which the turbulence dies away under the influence of viscous effects, would be possible if the interface were entirely smooth. The methods of analysis described in the following papers are relevant, and available [34, 36-40]. However, even for smooth surfaces there is much merit in employing the “wall-function” approach described in Spalding et ul. [20-22, 32) and elsewhere; and, when the interface is rough or rippled, it is essential to do so. This wall-function approach can best be understood, in a qualitative way, as involving the fitting of the whole low-Reynolds-number region close to the interface into a single grid interval, to which special Reynolds-numberdependent transport properties are ascribed. Once incorporated into GENMIX, the wall functions give rise to no difficulty, computational or conceptual. There is much successful experience of using them. 4. Importance The very least that an analysis of the present kind will lead to is a prediction of the rate of heat transfer between the air and the water. If the problem is to determine whether the lake can dissipate the heat supplied to it by a power station, without excessive temperature rise, this is of great importance. However, numerical modeling provides much detailed insight into the processes in addition. This insight may stimulate ideas, both scientific and practical ; their importance cannot be known in advance.
5. Method Adaptation GENMIX can handle both plane and axisymmetrical phenomena; it is necessary simply to change the value of a single index. Incorporation of the property and boundary-condition data, mentioned above, presents no difficulty; and the rate of spread of the integration domain into the air is automatically handled by the program. The present problem is thus a simple one for this computer code. 6. Expected Results
Although many computations have been performed on heat transfer from surfaces beneath turbulent boundary layers with the aid of GENMIX [e.g., 41 -441, no study has been made, to the writer’s knowledge, of precisely
82
D. BRIANSPALDING
the problem presented presented here. Nevertheless, it is easy to foresee qualitatively what the result will be; and then the task will be to make comparisons with experimental data, for example, those of Liepmann and Laufer [45]. The most unusual feature of the solutions, from the point of view of the mechanical and aeronautical engineer, will be the changes in the turbulence structure and the velocity and temperature profiles, brought about by the Richardson-number effect. If the water is warmer than the air, the friction and heat-transfer coefficients will be enhanced; but, if it is colder, so that the density gradient tends to damp out turbulence, these coefficients will be reduced.
7. Remarks THIRBLE researchers, meteorologists, and others interested in the fluid mechanics of heat and mass transfer have here an opportunity to make quick advances in a little-explored but important area. Some branches of applied science have become so overcrowded that it seems appropriate to draw attention to such opportunities whenever possible. LAYER IN B. THEBOUNDARY
THE
WATER
AT THE
LAKESURFACE
The shear stress on the interface, exerted by the wind, sets the surface water of the lake in motion; this motion generates shear stresses within the water; the consequence is that momentum is transferred to deeper and deeper layers, the greater is the distance from the shore. Once again, the steady state will be considered; and effects of the earth’s rotation (Coriolis effects) will be neglected. The restriction to the steady state is necessary if variations in two space dimensions are to be considered; but the neglect of the rotational effects is no more than a minor, and needless, economy. The discussion will proceed in the same manner as before. 1. Geometry
Let x again be the horizontal distance from the shore; and let y be the vertical distance measured from the lower edge (I boundary) of the integration domain. This boundary will coincide with the airwater interface at the shore line; but it will fall increasingly below it as x increases and the fluid is set in motion at greater and greater depths. It will be supposed that the depth of the lake is many times as great as the depth of the I boundary. The E boundary of the integration domain will be the interface itself.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
83
2. Physics The processes occurring in the water are similar in principle to those already described as occurring in the air; but there are some detailed differences. Thus, the counterpart to the diffusion of water vapor through the air is the diffusion of dissolved salt through the water; and the conduction of heat through the water is significantly augmented by short-distance emission and reabsorption of thermal radiation, in a manner which is insignificant in the air boundary layer. An explanation of the latter effect will be found in Spalding [l]. It may be necessary to explain more fully the salt-diffusion phenomenon. If there were no vaporization of water from the surface, the diffusion of salt would not require consideration. However, when water does vaporize, the salt, which is involatile, has to diffuse downward against the (small) upward flow of water; it can do so only if its concentration in the upper part of the liquid layer becomes larger than that in the lower part. This concentration rise will somewhat lower the vapor pressure of water at the surface; it therefore tends to reduce vaporization rate. If the diffusion coefficient of the salt were very low, a nearly impervious “skin” would form on the surface, inhibiting further transfer. The dependent variables of the differential equations to be solved are the mass fractherefore: u, the horizontal velocity; T, the temperature, e, tion of salt; k, the turbulence energy; and perhaps also a second turbulence quantity, e.g. E, from which the length scale 1 can be computed. Probably, however, it will suffice to insert 1 as an algebraic function of distance from the interface, in the manner of Prandtl [33], Glushko [34], and others. The physics of the semilaminar region immediately below the airwater interface must be regarded as uncertain. This point has already been explained, in Section 1.D above. 3. Boundary Conditions At the lower (I) boundary of the layer, the velocity u is zero, as is the turbulence energy ; the salt concentration and the water temperature are known. Of course, the depth of this boundary is adjusted so to ensure that the enforcement of these conditions produce no discontinuities of gradient. There will be an upflow (entrainment) of water across the I boundary, in order to satisfy the requirements of continuity; for the quantity of water in horizontal motion will increase, the larger is the distance from the shore. However, this upflow rate does not have to be specified beforehand; it will be calculated in the course of the solution of the differential equations. At the upper (E) boundary, an important input to the momentum equation is the wind stress; this may be taken as known, as a consequence of the
D. BRIANSPALDING
84
analysis of the atmospheric boundary layer, whether this has been made as in Section 1V.A or by more approximate methods (e.g., the presumption that the “friction factor” has a particular value). Also to be prescribed are (preferably) the surface temperature and so, from the analysis of the air boundary layer, the heat flux and the vaporization rate. The boundary condition for salt transfer is y = y,:
msm”= gSpamJay
(4.2-1)
where m“ is the vaporization rate at the interface; this implies that the upward rate of convection of salt is exactly equal to its downward diffusion rate, and it therefore expresses the fact that salt does not vaporize. There is a degree of “implicitness” about these interface boundary conditions that renders it rather unsatisfactory to be solving the equations of the air and water boundary layers separately. An escape from the difficulty will be discussed in Section 1V.C.
4. Importance One of the most influential components of the process, namely the rippling of the interface and its effect on the transport of momentum, heat, and mass in its vicinity, is little understood; therefore, the sophistication of analysis that is being envisaged for other components is somewhat excessive, when judged purely from a practical point of view. If therefore the present topic is judged important at all, it is for its theoretical significance: the relative magnitudes of the influences of various defining conditions can be quantitatively explored; and the detailed calculation and display of the resulting profiles can deepen understanding.
5. Method Adaptation There is little to say about how the Patankar-Spalding procedure [20] and the GENMIX computer code [21] are to be adapted for solving the present problem. The only nonobvious feature concerns the introduction of the wind-stress condition; and even this becomes obvious when it is recognized that the wind stress is simply a source of momentum, which is present in the computational cell adjacent to the surface. GENMIX possesses a “source-term’’ array; so it is merely necessary to insert the proper value in the proper location. 6. Expected Results
Although computations of the present kind have not been carried out, so far as the author is aware, their outcomes are easily imagined in a semiquantitative fashion. A few of their features will now be mentioned.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
85
The velocity profile will have a nearly uniform shape, steep near the surface and flattening as the I boundary is approached. The actual magnitudes of the velocities will vary little (along lines of constant o)as x increases; but the vertical extent of the profile will increase. yE can be expected to increase in proportion to x”, where n is of the order of 0.8, until, if pE < p,, turbulence is damped out by the increase in Richardson number. That Ri is likely to increase is indicated by its definition, Eq. (4.1-1):for yEcan be expected to increase, while the other quantities change rather little. Of course, if Ri grew so large that yE ceased to increase, entrainment would also have ceased; this means that the value of #E, the surface velocity, would increase because of the momentum imparted by the wind stress. Obviously, this could lead to a decrease of Ri, which would lead to a renewed entrainment. It is easy to see that, at large x, #E and yE will grow together, so as to maintain a constant value of the Richardson number. It is probable that, in problems of interest to THIRBLE specialists, the direction of heat transfer will be from water to atmosphere. In such a case, pEwill exceed p , ;and there will be no limit on depth imposed by Richardsonnumber considerations. Vaporization of water from the interface will have the same tendency; for the increased salinity at the interface will make pE greater than it otherwise would have been.
7 . Remarks For the sake of brevity, many features which could be introduced into the mathematical model have been disregarded. However, the reader can probably recognize for himself that the wind stress could be allowed to vary with x, that solar radiation (also varying with x) could be allowed for, and that empirically based “wall functions” allowing for rippling and wave formation, could be introduced. The basic computer code is a capacious one, hard to overburden.
C. THECOMBINED AIR-WATERLAYER The air boundary layer influences, and is influenced by, the water layer. It is a nuisance to have to guess the water-surface velocity before computing the wind stress; and then to find, when the water-surface velocity is computed from the wind stress, that the original guess was slightly wrong. Yet this happens inevitably; and similar difficulties beset the guessing and computation of interface temperature, surface salinity, etc. Of course, the computations of Sections 1V.A and 1V.B can be performed successively, and repeatedly, the results of the last becoming the starting point of the next. Iteration, if skillfully carried out, will surely bring convergence. Yet there is a better solution.
86
D. BRIANSPALDING
To hit upon this better solution, it is necessary only to recognize that the combined air and water boundary layers may be considered as comprising a single mixing layer, in a medium which happens to exhibit a strong discontinuity of density. Mixing layers without such discontinuities have been studied by many workers [46,47]; and GENMIX [24] (and no doubt other codes) have been used to predict their development. There is no difficulty about allowing for strong continuous variations of density in GENMIX; so why should discontinuous ones not be accounted for also? This question will now be answered. Specifically, the problem will be addressed of solving the problem of heat, mass, and momentum interchange between the air and the water in a single marching integration. The water-air interface will lie within the integration domain; and the grid will extend both above it and below it.
1. Geometry Let it be decided that the interface shall lie upon a line of constant w. This is permissible because the locations of the I and E boundaries of the computation are both arbitrary; all that is necessary is to determine what is a suitable value for the interface. Let this be called wi; then, as a consequence of the definition, the ratio of the mass flowing in the air layer to that in the water layer equals (1 - wi)/wi. Since the momentum lost by the air is equal to that gained by the water, it can be concluded that (4.3-1)
where u, is the average velocity increase of the water and (- 6ua)is the average velocity decrease of the air. The ratio of the quantities on the right-hand side is usually of the order of 100, being proportional to the square root of the density ratio for water and air. It follows that a value of around 0.01 is suitable for wi. Whatever value is chosen for mi, the choice limits the freedom with which the I and E boundaries can be located: the entrainment functions a and b, of Eqs. (2.1-4)and (2.1-5),must be chosen so that mri u + bwi = ___ (4.3-2) $B
-
$1
where iWi is the vaporization rate. The quantities a (which is always positive) and b (which is always negative) must be chosen so that Eq. (4.3-2)is obeyed and so that the grid stretches far enough into both the air and the water for the gradient at its edges to be small.
HEATTRANSFER IN RIVERS,BAYS,LAKES, AND ESTUARIES
87
2. Physics There is nothing essentially new to say about the physics of the process. However, in the computation the transport processes at the interface play roles which appear to be different. Thus, the “wall-functions” appear as special effective-transport-propertyformulas for the grid link joining the two grid points lying on either side of the interface; and, in the turbulenceenergy equation, the interface; and, in the turbulence-energy equation, the interface acts as a source or sink of energy, making the value of the energy rather strictly proportional to the turbulent shear stress. 3. Boundary Conditions The conditions at the free boundaries in the air (E) and water (I) are those that have already been described in Section 1V.A and 1V.B.As just indicated, the interface conditions have become internal ones. 4. Importance This particular computational device (the study of the combined layer) will reduce the computational cost (as compared with iterative computations of the individual layers), for a given accuracy. It also possesses a conceptual value; for consideration of the conjoined boundary layers as a single mixing layer draws attention to the features that they share with familiar free turbulent flows.
5. Method Adaptation In the analysis of Section IV.A, the concentration variable was the mass fraction of water vapor in air; in that of Section IV.B, it was the mass fraction of salt in water. Unless two concentration equations are to be solved in the combined-layer problem, this anomaly is best eliminated; specifically, the mass fraction of HzO, mHzO,in the mixture is best used as the concentration variable in both phases. Of course, mHZ0is very close to unity in the liquid phase, differing from it only by the amount of the mass fraction of salt. At the interface, mHzOexhibits a discontinuity by reason of the thermodynamics of interphase equilibrium. This fact must be incorporated into the program. One possible method of doing so will now be described. Subscript H 2 0 is omitted for ease of writing. Let the subscripts 1, 2, 3, and 4 denote respectively the grid point next below the interface; the immediate vicinity of the interface on the water side; the immediate vicinity of the interface on the air side; the grid point next above the interface. Then the upward flux of H 2 0 across the interface can be represented, on the water side, as mi‘= m“ m2 - D l z ( m l - m 2 ) (4.3-3)
88
D. BRIANSPALDING
where D12 is the relevant product of diffusion coefficient, density, and reciprocal distance; and, on the air side, it can be represented by m" = &I'm3 - D,,(m3
- m4)
(4.3-4)
Elimination of m" leads to the following relation between the m's: m3 - m4 D,,(ml - mz) = D34 1 - m2 1 - m3
(4.3-5)
Now m2 and m, are also related by the interphase equilibrium relation; this also involves the interface temperature. Thus m3 = m3fT23, m2)
(4.3-6)
With the aid of these equations, it is possible to represent the diffusion flux of H 2 0 between grid points 1 and 4 as
flux
=
D14(m1
- m4) + El4
(4.3-7)
wherein the derivation of expressions for 0 1 4 and El4 is left to the interested reader. The consequence of these manipulations is that the transfer of water vapor between grid points 1 and 4 is to be represented by a term of the usual diffusion type, viz. Dl,(ml - m4), plus an additional term E14; this will appear as a sink in cell 1 and as a source in cell 4. Similar devices are needed to account for the enthalpy change associated with vaporization. But no other method adaptation features are sufficiently recondite to require mention. 6. Expected Results
The predicted profiles of velocity, temperature, etc. in the air and water boundary layers will, of course, be the same as indicated in Sections 1V.A and 1V.B. In the present section, it is only the method of obtaining the solution that is different. 7. Remarks
This completes the discussion of two-dimensional steady boundary layers in the THIRBLE context. Although such phenomena are not very common, an extended discussion was presented because they serve as a bridge between the well-understoodjets and plumes and the less frequently studied floating layers, which occupy attention in the remainder of this article. The mathematical forms, the solution procedure, and the details of the recommended computer program are similar in all cases.
HEATTRANSFER IN RIVERS,BAYS,LAKES,AND ESTUARIES
89
V. One-Dimensional Unsteady Vertical-DistributionModels A. THETYPEOF PROBLEM CONSIDERED Attention will now be turned to another group of two-dimensional parabolic THIRBLE problems, those in which one dimension is the vertical distance and the other dimension is time: transient phenomena in one space dimension are in question. Specifically, means will be presented of calculating the distribution with depth of the following properties within a layer of water, differing in some of these properties from the larger body of water upon which it floats: temperature, two horizontal velocity components; salinity; turbulence quantities; radiation flux. Problems of this type are of interest from many points of view: oceanographers need to understand quantitatively the structure of the upper layers of the oceans; ecologists are concerned with the factors governing the thermocline in a lake, during the yearly cycle of seasons; and THIRBLE specialists need to calculate the rate of cooling of supposedly coherent “packets” of warm water which float away from the discharge point over the surface of an estuary. The last concept is particularly apposite to this article: if the warm-water “packets” can be regarded as coherent, albeit squashing or stretching in the vertical dimension, a separation is effected between two questions, a thermal and a hydrodynamic one. The first is, Where is the packet (i,e., what are its horizontal coordinates) at a given time after discharge? The second is, Whut are its average and surface temperatures? Ability to pose and answer these questions one at a time eases the analyst’s task. Problems of this type are mathematically similar to those of Section IV; but time has replaced the horizontal dimension x. It will be the tasks of the remainder of Section V to demonstrate this similarity, to explain how the problems can be solved, and to discuss some particular solutions.
FORMULATION B. THE MATHEMATICAL 1. T h e Generul DifSerential Equation
Equation (2.1-1) describes the phenomena in question. However, it can be written somewhat more transparently as follows:
D. BRIANSPALDING
90
Here, the following simplifications have been made, not out of necessity but to aid present understanding: (i) x has been replaced by t, and u has been put equal to unity because the problem is a transient one. (These substitutions were foreshadowed in Section 1I.A.) (ii) The mass transfer through the interface (vaporization) has been regarded as negligible (in respect of mass balance, not in respect of its thermal effect); hence, with the E (o= 1) boundary defined to lie on the interface, b must be equal to - a (see Eqs. (2.1-4)and (2.1-5)). (iii) The density has been regarded as uniform (which does not preclude allowance for influence of its nonuniformity on the generation of turbulence energy); hence, and from the definitions of w, 4, c, and a and from the knowledge that the geometry is planar so that r can be omitted: (5.2-2) (5.2-3)
a
=
(l/YE)
=
rP/YE2
dYE/dt
(5.2-4) (5.2-5)
(iv) The symbol S has been introduced for the volumetric source of the entity 4.
The general variable 4 stands for any of the two horizontal velocity components U and V ; the temperature T; the turbulence energy k ; and perhaps for other variables (e.g., turbulence dissipation rate, radiation flux). The symbols U and V have been adopted to make plain the distinctions from u and u. It is not the case, for example, that U is the same as the u appearing in Eq. (2.1-2). 2. Particular Differential Equations
The differential equations of horizontal momentum are
au
-
at
1 dyE + --(1 Y E dt
dv 1 dyE -+--(I at Y E dt
- w) =
(5.2-6)
- 0)= --
(5.2-7)
Here thefV and -f U terms are those resulting from the earth's rotation. The quantity f is the so-called Coriolis parameter, defined by
f = 2QsinII/
(5.2-8)
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
91
where R is the rotational speed of the earth (=2n/(24 x 60 x 60) rad/sec), and II/ is the latitude. It is supposed that U is the west-to-east velocity component, and V is the south-to-north component. No pressure-gradient terms are included; it is supposed that these, if present, are exactly balanced by the gravitational body forces associated with an appropriate “tilting” of the surface of the ocean or lake. The temperature equation can be written
Here Aeff is the effective thermal conductivity, Q is the heat source per unit volume, and c is the specific heat of the water. It is shown in Spalding [I] that, if an appropriate radiative component is contributed to the effective conductivity, the radiative sources and sinks can be omitted, except as an energy-flux boundary condition at the air-water interface. This practice is recommended here. The turbulence-energy equation can be written
Here nT and ok are respectively the Prandtl-Schmidt numbers, for the diffusion of temperature and turbulence energy; I is the length scale of turbulence, which (if the Prandtl [33] one-equation turbulence model is used) is proportional to the layer thickness; and C, is a function of the local Reynolds number of turbulence which tends to a constant value when this number is large. ( N o t e : It is impossible, in the present article, to explain all these matters fully; readers desiring more complete explanations are referred to Launder and Spalding [22].) In the interests of brevity, the salinity equation will be omitted. 3. Auxiliary Relations
The effective viscosity perfis to be calculated from the Prandtl-Kolmogorov relation perf= C,pk‘’21
(5.2-11)
wherein C, is a constant, or function of the local Reynolds number of turbulence.
D. BRIANSPALDING
92
The length scale 1 will be deduced, within the framework of the PrandtlGlushko [33, 341 model, from an algebraic relation of the form 1/6 =
Lf ( Y E - Y ) / @
(5.2-1 2)
where 6 is the thickness of the turbulent layer, and the Lf function is defined in advance to ensure that 1 falls to zero in the vicinity of the upper and lower boundaries of the layer and is of the order of 0.16 in the middle. Often of course, the grid-expansion process of GENMIX will be arranged so that 6 and yE approximately coincide. The effective conductivity will be computed from (5.2-13) where Arad is the radiative component to conduction, calculated from Lrad
= 8 K T 3 / ( af
S)
(5.2- 14)
Here k is the Stefan-Boltzmann Constant, and a and s are respectively the absorptivity and the scattering coefficient per unit depth. 4. Boundary Conditions At the airwater interface (E; o = l), the boundary conditions are: wind-stress components in the U and I/ directions; the appropriate turbulence energy (see Launder and Spalding [32]); a heat flux expressed as a linear (or linearized nonlinear [l]) function of the surface temperature. This function can be arranged so as to allow for effects of vaporization and radiation.
All these conditions can be allowed to vary with time. At the lower (I; o = 0) boundary, the boundary conditions are (probably):
u=v=o; T = T o ,a fixed quantity or a function of depth below the interface; k = 0.
The depth of the lower boundary will increase with time, in accordance with the grid-expansion practices of the Patankar-Spalding [20] procedure. This arranges that the I boundary lies at a depth at which influences emanating from the upper layer have only just become significant. 5. Solution Procedure
The equations and boundary conditions pose mathematical problems of a kind that can be easily solved by way of the GENMIX computer code [21]. Other methods and codes can of course be used.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
93
Whatever the particular method that he prefers, almost every analyst will transform the equations first into finite-difference form, and then proceed to solve them in a time-marching manner. The details need not further concern us. In the following sections, in which particular examples are considered, the discussion will be based on the use of the GENMIX code, for the sake of concreteness.
C. THEDEVELOPMENT OF THE EKMAN LAYER In a famous paper [48], Ekman demonstrated that a steady wind stress over the surface of the ocean leads to the buildup of distributions of U and I/ which vary with depth in a remarkable way. Specifically, if the wind blows from south to north, the steady-state surface current is directed (in the northern hemisphere) from south-west to north-east. But, at lower depths, the U’s and V’s vary in such a way that every other possible direction of flow is represented. Specifically, if the effective viscosity of the ocean is taken as independent of depth, Ekman’s analysis yields U = ( U z + T/z):i2e-“ I/ =
cos(n/4 - a)
(5.3-1)
( U 2 + I/Z);/2e-a sin(n/4 - a)
(5.3-2)
where : a
= (YE
-
Y)E(fP/2Peff)1’2
(5.3-3)
Further, the surface “drift” velocity is given by (U2+
v2)A”= (zE/P)(P/Xpeff)”2
(5.3-4)
where z is the wind stress. In reality, of course, the wind stress is never constant for long periods; and peff is by no means uniform with depth. It is therefore interesting to make a transient analysis of the buildup of the Ekman layer and to investigate the influences of varying perfThe interactions with the temperatureproduced density gradients are also of interest. All this can be accomplished by the method now to be described. The same format will be adopted for the description as was used in Section IV. 1. Geometry
A quick resume will be made: the E boundary is on the surface, the I boundary below it at a depth y, increasing with time; y is measured upward from the I boundary. U and V are the horizontal components of velocity.
94
D. BRIANSPALDING
2. Physics
Only the equations for U , V , and k need be solved for the straightforward Ekman problem. If density-gradient effects are to be accounted for, the T and m, (salinity) equations are also needed. The physical processes involved have all been discussed, and formulated mathematically in Section V.B. 3. Boundary Conditions In the classical Ekman problem, at the start of the process, the fluid is at rest ( U = V = k = 0). Then, at t = 0, a finite wind stress zE,is applied at the surface (E boundary), and maintained constant. At the I boundary, U and V remain at zero. Temperature and salinity effects are not discussed. Other sets of boundary conditions are also if interest.
4. Importance A calculation of the classical Ekman (perf = uniform) problem can usefully be performed so that the computed steady-state ( t + m) solution can be compared with the analytical one; this will engender confidence. Then the more realistic turbulence-model solution can be computed and compared with the constant-perfone. Thereafter the transient solution might be examined, so that the rate of buildup of the Ekman layer could be computed; for departures of experimental observation from the steady-state solutions must in part be the result of the fact that real winds seldom stay constant for long enough. Subsequently, arbitrary wind-stress time variations can be supplied, and thermal and salinity effects included. It should be clear that the method presented can provide a very complete understanding of oceanic boundary layers of this transient one-dimensional type. The importance of providing this need not be stressed further.
-
5. Method Adaptation
For all the examples of Section V, the main modification required to GENMIX is the incorporation of the simplifications: u = 1 and r = 1. The first replaces x-direction convection by time dependence; the second (employed also in the problems of Section IV) corresponds to the planar geometry. Otherwise, the only adaptive steps needed are the obvious inclusions of initial and boundary conditions. 6. Results
-
Svensson [49] has used the GENMIX program for the classical Ekman problem, both with Ekman’s uniform-perfassumption and with the k Q turbulence model [31] as an alternative. For the former case, his results
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
95
agree well with those of the analytical solution, as of course they are bound to do if the numerical work is correctly performed. For the latter case, Svensson showed that the steady-state deviation of the surface drift velocity from the wind direction is only about one-half of the 45” predicted from the constant-pe,, assumption. This prediction is in agreement with some experimental data; but a comprehensive study remains to be carried out. Other outputs of Svensson’s study include: the variation of the effective viscosity with vertical height and with time; the variation of the magnitude and direction of surface velocity with time; the profiles of U and V with depth and their variations with time; etc. To mention only one point, the surface velocity at first has the direction of the wind stress. However, a finite I/ (say) puts a momentum source CfV) into the U equation; so U becomes finite; and as a consequence, the V equation receives a negative momentum source (-fU).The result is that the vector of the surface velocity (or of the velocity at a lower depth, for that matter) traces out a converging spiral path.
7. Remarks The suitability of the GENMIX method for analyzing the generalized Ekman problem has been demonstrated by Svensson’s work. Further investigations may be left to oceanographers.
D. THECOOLING OF A WARM-WATER COLUMN Thermal effects can, as has been indicated, easily be included in the analysis of the Ekman problem; then perhaps the emphasis of the analyst may shift toward the study of the thermocline in wide expanses of water. In the present subsection, a still further shift will be illustrated. Specifically, attention will be concentrated on a “coherent column” of warm water, floating as part of a layer over a supporting cold-water mass. For this to be an acceptable model, variations of U and I/ through the layer must necessarily be small; for nonuniformity destroys the coherence. However, although gradients of T , k, etc. in the horizontal directions must now be admitted (for the column is to be regarded as “drifting,” and cooling as it does so), they must be assumed to be so small that neither in respect of conduction nor convection do they have any appreciable effect. This neglect of actually existing temperature variations by no means invalidates the model; for warm-water layers in THIRBLE problems are immensely wider than they are deep: the temperature gradients in the horizontal directions are often only one-thousandth of those in the vertical. The discussion follows the same pattern as before.
D. BRIANSPALDING
96
1. Geometry The same one-dimensional layer is considered as in Section V.C. Its thickness will probably grow with time; although, if the critical Richardson number is exceeded at the lower edge, as is probable in practice, the rate of entrainment of cold water into the warm layer will be very small. The experiments of Ellison and Turner [50] have shown this; and the diminution of the entrainment rate is an easily understood consequence of all turbulence models which make use of the energy equation [e.g., 31-43], when the heat-flux buoyancy interactions are included.
-
2. Physics
The basic physical processes have already been described above. Here however, in partial preparation for the further development of layer models in Section VI, it may be mentioned that even small variations of layer thickness with horizontal position have the effect, when density variations are present, of providing significant horizontal momentum sources. Such momentum sources can maintain finite values of U or K even though wind stress is absent; and indeed they are actually necessary, to ensure that discharged warm water does actually move away from the point of discharge. 3. Boundary Conditions Many sets of conditions are of interest. The simplest of relevance to the cooling of a warm-water column involves: U and V initially uniform in the layer; T uniform in the layer and larger than both the temperature of the supporting water mass and the equilibrium temperature of water in contact with the prevailing atmosphere. The latter temperature can be computed from the air temperature, the air humidity, and other factors, as explained in spalding [l]. The heat flux from the water surface to the atmosphere will not ordinarily be given directly; but a surface heat-transfer coefficient will be prescribed. This depends largely on wind speed; but other factors (salinity, temperature level) are significant. 4. Importance
This model is of great importance to THIRBLE specialists. For all its comparative simplicity, it does pay attention to the essential processes of heat transfer from the warm water to the surroundings; and it is sufficiently easy to analyze exactly for numerical computations to be economically practicable.
HEATTRANSFER IN RIVERS,BAYS,LAKES, AND ESTUARIES
97
In order that this point can be clearly perceived, let the spread of a warmwater plume into an estuary be envisaged. A full time-dependent threedimensional analysis is enormously expensive; and, indeed, because of the strong influence of “false diffusion” [l], the results obtained with even the finest of practicable grids may be seriously inaccurate; for a finite-difference calculation of conventional kind can ascribe only one value to the temperature in each cell, so that precisely those differencesthat it is desired to compute are suppressed when the grid size is excessive. However, it is possible to proceed by asking three separate questions, namely : (i) How does the bulk of the water move? (ii) How does the warm-water layer move over its surface? (iii) How do conditions within columnar elements of the warm-water layer vary with time? In the present section, it is question (iii) that is under consideration. Section VI gives partial answers to question (ii).Answers to question (i) can be fairly approximate in character; and they may be obtained by several methods.
5. Method Adaptation Apart from what has been stated already, the GENMIX code requires only one additional feature to be supplied, namely the horizontal pressure gradient, if present. There are obvious source-term locations in the code permitting this.
6. Results Some calculations of this kind have been performed by Spalding and Svensson [51]; and the results of others can easily be obtained, or (in qualitative features) imagined. Only one point will here be mentioned, namely that the heat-loss situation proves to be one in which the temperature within the layer is nearly uniform (but diminishing slightly as the surface is approached); at the lower boundary of the layer, however, the temperature falls sharply to that of the cold-water mass. The cause, of course, is the gravitational source term in the turbulenceenergy equation (5.2-10):whenever aT/aw is negative, as it always is where heat is being lost to the surface, the turbulence level tends to increase; and this tends to diminish the temperature gradient. One consequence is that it is the air-side heat-transfer coefficient that mainly controls the rate of cooling of the layer, not the transfer of heat within the layer. This conclusion is arrived at without numerical analysis in Spalding [11 also.
D. BRIANSPALDING
98 7. Remarks
The last conclusion is of very great significance indeed; for it justifies further simplifications in the analysis of THIRBLE processes. If the temperature differences in the layer are small, why bother about them? Attention and computing power may be more profitably applied elsewhere. This possibility is exploited in the “layer models” of Section VI, which now follows. VI. Two-DimensionalFloating Layers
A. THESTEADY TWO-DI~NSIONAL-LAYER MODEL 1. Description
Let it be supposed that warm water is discharged steadily into a steadily flowing river; the injection velocity may have a component transverse to the river-flow direction, but its downriver component must be positive. Either the warm water is injected at or near to the river surface; or, if it is injected appreciably below this surface, analysis starts at the downstream station at which buoyancy has lifted the warm water (by now slightly diluted) and spread it over the surface sufficiently for the vertical temperature gradients to be much greater than the horizontal ones. It will now be assumed, on the basis of the findings of Section V.D (or of experimental evidence) that the vertical profiles consist of a nearly uniform higher temperature region near the surface, a lower region in which the temperature is that of the undisturbed river, and a very thin intermediate region of steep temperature gradient. The first region will be called “the layer”; and it will also be supposed that the horizontal velocity components of the water have vertical profiles of corresponding shape. It is thus meaningful to regard the layer conditions as being characterized by the following two-dimensional variables: Tfx, Y ) ;
ufx, Y ) ;
ufx, Y ) ;
hfx, Y )
Here x and y &rerespectively the horizontal distances in the directions along and normal to the flow of the river; and h is the thickness of the layer. When these four functions have been computed, the thermal-pollution problem can be regarded as solved. In the “coherent-column” analysis of Section V.D, it was postulated that interactions between neighboring columns were negligible: all the heat flowed out through the upper surface; and all entrainment occurred through the base of the column. This presumption will be expressed by the absence of second-order differential coefficients from the equations. The setting up of the equations is the next task.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
99
2. Mathematical Formulation The layer, seen from above, will be regarded as having definite boundaries, labeled I and E as before. At the I boundary, y will be set equal to zero; so the width of the layer will be y,. The distance y will be measured along lines of constant x ; but the x y grid will not be quite orthogonal if the I boundary is curved. The name “stream function,” and the symbol $, will again be reserved for the mass flow rate. The definition will be
-
*
= Jo’wh dY
(6.1-1)
which implies that $ is defined to be zero at the I boundary. The latter feature is a departure from the GENMIX practice, permitted because there is no question of horizontal entrainment: both the I and E boundaries are streamlines. However there is question of entrainment from below, and it must be allowed for. It follows that the distinction must be made between streamlines on the one hand, and lines of constant stream function t,b on the other; the former are lines (or rather surfaces formed by vertical generators through lines drawn on the water-air interface) across which there is no mass transfer within the layer. Let the subscript s denote a stream line. Then the inclination of a stream line to a normal through a constant-x line is tan-’(u/u); and its gradient in the x y plane is given by
-
(6.1-2)
This result will be used below; it shows that the streamline inclination can be connected, through u and u, with the two momentum equations. The valuation of t,b along such a line is given by (6.1-3)
where m“ is the rate of entrainment of river waterfrom below. This quantity will be regarded as a function of the local Richardson number; thus (6.1-4)
Here the subscript r refers to the river; and the function ffRi) will be deduced from the experiments of Ellison and Turner [50] to be approximately given
D. BRIANSPALDINC
100
by
Ri < 0.8:
f = 0.075(1 - Ri/0.8)2
(6.1-5)
Ri 2 0.8:
f =0
(6.1-6)
The basic differential equation for the conserved property 4 can now be given. It is (6.1-7) wherein the quantity S represents the source of q5 per unit volume. If IC# stands for temperature, the quantity S is given by
4
S = @(T,,- T)/(ch)
T:
(6.1-8)
where a is the surface heat-transfer coefficient, T,, is the temperature which the water would take up if it were in equilibrium with the atmosphere above it, and c is the specific heat of the watel. If 4 stands for a velocity component, the S terms represent the sum of wind-stress and hydrostatic momentum sources, thus f#)
= u:
4
V:
7, s=- gpP(T h
z
dh Tr)-
ax
(6.1-9)
ah
gp/l(T - T,)(6.1- 10) h dY Here z, and z y represent the wind stresses on the E surface in the two directions. S
=2-
-
3. Discussion The task is to solve Eq. (6.1-7), for (b = T , u, and v, numerically; and this involves also solving for the layer depth h, from (6.1-1)by way of (6.1-3) and (6.1-4). The nature of the task can be perceived by consideration of three cases; they may be labeled, without serious impropriety : “parabolic,” “hyperbolic,” and “elliptic.” Only the second and third are of practical significance; but the first forms a useful starting point for the discussion. 4. Solution Procedure: The Parabolic Case
Let it be supposed that the lateral velocity v is known at every point (never mind how); and let the ahlax term in S for u be negligible. The calculation procedure can then be conducted as follows: (1) The space between the I and E boundaries is divided by N-1 stream lines, so that the layer consists of N stream tubes, in each of which 4 is
HEATTRANSFER IN RIVERS,BAYS,LAKES,AND ESTUARIES 101 regarded as depending upon x alone. The stream tubes can be numbered 2,3,. . . , i, . . . , N-1 in the manner of GENMIX. (2) Since u is supposed known, there is need to solve only for T , u, and h. The first two can be computed from a finite-difference version of (6.1-71, which runs
Here subscripts U and D denote “upstream” and “downstream” values, respectively. Obviously the equations can be solved explicitly; there is not even any need to employ the tridiagonal matrix algorithm because of the absence of the lateral-convection and turbulent-exchange terms. (3) Then the finite-differenceform of Eq. (6.1-3)is solved, for each stream tube, from (6.1-12) wherein ‘Pi and & are the differences of $ and y respectively across the stream tube. (4) Equation (6.1-2)is now involved for the calculation of the downstream values of the & by way of
4
where the subscripts i + and i - 3 denote the locations of the stream lines dividing the stream tubes. ( N o t e that 2 - f denotes the I boundary, and N - 1 + denotes the E boundary.) ( 5 ) Finally the new values of h are deduced from the finite-difference form of the equation (6.1-1), namely
4
I7i,D = @ i , D / ( p u y ) i , D
(6.1- 14)
This calculation, being explicit, presents absolutely no difficulties. Although it may be convenient to fit the actual computations into the framework of the GENMIX code, the full capabilities of GENMIX are not called upon. It should be noted that it was essential to neglect the dh/dx term in the u-momentum equation; for, otherwise, hi,o would be needed before it had been computed.
5. Solution Procedure: The “Hyperbolic” Case The presumption of the v field is never practicable; for, in practice, the layer will spread laterally at a rate that is influenced by the lateral gradient of h. In other words, it is unlikely that the u’s which were prescribed in the
D.BRIANSPALDING
102
“parabolic” case would, if associated with the computed h’s, actually fit the v-momentum equation. In the “hyperbolic” case, the v’s are not prescribed; but, as a consequence, the problem becomes appreciably more complex. It is soluble however, without abandonment of the marching-integration procedure, by a technique introduced by the present author for supersonic gas flows [52]. This technique employs a special version of the SIMPLE algorithm of Patankar and Spalding [15]. The essential feature is that the downstream values of the u’s are computed from the relevant momentum equations, into which are supplied guessed values of the downstream h’s. Then, when it is discovered that these lead by way of the continuity equation to diJ6erent values of the downstream h‘s, a correction procedure is introduced. The relevant steps are as follows: (l),(2), (3) As for the “parabolic” case. (4) The u’s are calculated from guessed (“starred”) downstream h’s from the relevant finite-differenceversions of equation (6.1-7), namely
(5) Equation (6.1-13) is used, as for the “parabolic” case. (6) Equation (6.1-14) is used, with htDinserted, to produce Y?,’S that are not “starred.” These differ from the X,,’S of Eq. (6.1-15);and their differences, the Ai, are computed from
Ai
E
K,D -
Y*i,D
(6.1-16)
(7) Corrections (primed symbols) are now calculated for the h’s, u’s, and Y’s from the following set of equations, the origin of which is explained below,
Dihi’ = Aihft
x’=
+ &hi-, + Ci
(2) hi’
(6.1-17) (6.1-18) (6.1- 19)
Solution of the equation set (6.1-17) proceeds by way of the tridiagonal matrix algorithm; then the Y’s and u’s are computed directly.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES 103 (8) The corrections, i.e., the primed quantities, are added to the starred quantities to produce the values that are accepted as being the solution to the equation set (6.1-20) hi,D = h z D hi'
+
+ x'
X,D
=
Y?D
Vi+1/2,D
=
u?+l/2.D
(6.1-21)
+ vi'
(6.1-22)
(9) The marching integration then proceeds to the next forward step. The origin of the h-correction equation (6.1-17)is as follows: (i) Differentiation of Eq. (6.1-13)leads to (6.1-23) (6.1-24)
(ii) Differentiation of (6.1-14) leads to (6.1-25)
(iii) Differentiation of (6.1-15) leads to
av?+112.D-ahtD
gPfl(Ti+112.D- Tr) (Pu)i+ I/Z(Yi + 112 -Yi - 1/2)U{l/(xD -
+(rh"/puh)i+ 1/2,U 1
(6.1-26) (6.1-27)
(iv) The influence of hZD on Yi,D through Eqs. (6.1-13) and (6.1-15) can be written
This may be written more compactly as (6.1-29) 6 Y,,D = Ei 6hzD Ai Sh?+ 1,D Bi 6hf- 1 , ~ where Ai, Bi, and Ei are defined as the comparison of terms dictates.
+
+
104
D. BRIANSPALDING
(v) There should be no discrepancy Ai between Y z Dand &,D; and there would be none if the htD had been correctly guessed. Therefore the heights should be changed so that the discrepancy is eliminated. The requirement can be written (6.1- 30) 6Y,,D-6Y?D= - A i where 6 Y t D and 6YzD are the changes that result from a change in hzD, namely 6hzD.(Later, the symbol h i is used for the latter quantity.) (vi) Now 6 Y t D varies in accordance with (6.1-31) wherein the differential coefficient is given by Eq. (6.1-25). (vii) Combination of Eqs. (6.1-29)-(6.1-31) now yields (6.1-32) i.e., (6.1-33) (viii) This equation may now be identified with equation (6.1-17) if the simpler symbol hi’ is used for dh?,, if (-Ai) is equated to Ci,and if the bracket on the left-hand side is equated to D,. (ix) The differential coefficientsin Eqs. (6.1-18)and (6.1-19)are of course deducible from Eqs. (6.1-25) and (6.1-26) and (6.1-27). (x) The set of equations is therefore now complete. 6. Solution Procedure: The “Elliptic” Case (a) Equations. The effect of the ahlay terms was accounted for in the “hyperbolic” case, but not that of the ay/dx terms. It is true that, in cases in which the river velocity is so large that the layer is strongly elongated, the ahlax terms will be much smaller than the ahlay terms; however, the case in which they are not so must also be considered. The neglect of the ahlax terms occurred when Eq. (6.1-31) was derived. A more complete differentiation would have been
(6.1-34) i.e.,
(6.1-35)
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES 105 Here, utD is the value of u ~obtained , ~ from (6.1-11)with u in place of d, and with hzD appearing in the finite-differenceform of (6.1-9).The relevant equations are
+-(Puh)i,u + g P p ( T i , D - Tr)(hi,U zx
(Pu)i.U(xD
- hZD)
(6.1-36)
XU)
and that which is obtained from differentiation, when only the main influences are considered, namely, (6.1-37) The best approach to Eq. (6.1-35) is by a return to Eq. (6.1-14), which, on differentiation, yields (6.1-38) i.e., (6.1-39) where f is defined by gp(Ti,D
JGlu:D{ui,U 3
- Tr)hzD
+ m!dxI3
1 - Ri*, say.
- xU)/(phi,U)> (6.1-40)
Here the symbol Ri* has been used for the expression in f because, as comparison with Eq. (4.1-1) makes clear, it has that nature. Evidently the "hyperbolic" model discussed in Section 5 above is valid, strictly speaking, only for Ri* = 0. (b) D i s c u s s i o n . Now it might be thought that all that is needed, to make the analysis of Section 5 more general, is to introduce the Ri* term. This is the case, provided that Ri* is less than unity; otherwise, however, f becomes negative; and, unless Ei is sufficiently large, the coefficient Di in Eq. (6.1-17) becomes negative also. This radically changes the mathematical nature of the problem. The consequences and the necessary modifications to the solution procedure will be explained only in general terms, as follows. Further explanation can be found in Spalding [l].
D. BRIANSPALDING
106
A negative Di coefficient makes the problem “elliptic”; and this permits effects from downstream to travel upstream. However, these influences are conveyed only by the pressure gradient associated with gradients of h; convective influences continue to flow only from small x to large; and horizontal diffusive influences are totally absent. This feature renders the problem, “semielliptic” or “partially parabolic,” these terms being synonyms [53, 541. A consequence is that the equations cannot be solved in a single marchingintegration sweep; for the corrections to h have to be made upstream of the x station at which and are being brought into conformity; and this upstream correction disturbs the balances in the equations that were achieved at an earlier stage in the calculation. Repeated marches, and successively smaller adjustments to the h‘s, must be made before the finite-difference equations are in balance over the whole domain of integration. This implies that the height distribution must be stored two dimensionally in the computer: a value of h must be in store for each pair of x and y values. However, the other variables (u, u, T) require only one-dimensional storage; and it is this that justifies inclusion of the partially parabolic problem in the present article. This calculation scheme can be, and has been, accommodated within the framework of the GENMIX computer code.
x
x*
B. THE“HYPERBOLIC” LAYER
Consideration will now be given to the spread of a warm-water layer over a river surface under the conditions of large longitudinal velocity: lateral spread, under hydrostatic influences, will be taken into account; but such influences will be regarded as negligible in the longitudinal direction. It will be supposed that the phenomenon is to be predicted by means of an adaptation of the GENMIX computer code. The discussion will be presented in accordance with the pattern employed elsewhere in this article. 1. Geometry
A resume will be given of what has already appeared in Section 1V.A: the river flows in the x direction, carrying the layer with it; this layer spreads laterally, i.e., in the y direction; the total width yE increases with x. At the start of the calculation, where x equals zero (say), yE has the value yE,D. The layer depth ho, can be supposed independent of y there (but an arbitrary h,fy) variation could be handled). All properties are regarded as independent of depth, within the layer, at the bottom of which however they jump discontinuously to the values characteristic of the river.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES 107 2. Physics Once again, a resume suffices: the physical processes are entrainment from below, heat loss and wind stress at the surface, and lateral movement under the joint influence of inertia and hydrostatic forces. Horizontal exchanges of heat and momentum are regarded as negligible. 3. Boundary Conditions The initial conditions uo, uo, and To must be given; they may depend upon y . The conditions of the river (ur, ur, Tr)and of the atmosphere (windstress components, surface heat-transfer coefficient, equilibrium temperature) must also be specified; they may vary with x and y , but probably will not do so in practice. Special boundary conditions, not so far mentioned, apply to the lateral velocity u. At the boundaries, discontinuities in h are to be expected; so the spread of the layer counts as the movement of a large-amplitude wave. Experiment shows that such waves move at a speed, relative to the supporting fluid, given Iue = 1.1{ghe(-6pe/p)}”2 (6.2-1) where subscript e denotes the edge (either I or E) and 6p, is the density difference there, given by (6.2-2) - ~ P , / P = P ( T e - Tr) Equation (6.2-1) needs to be incorporated into the solution procedure, in the form of a linearized relation between ue and he, for example,
This replaces Eq. (6.1-15) for the outer edge v’s. 4. Importance Problems of this kind have high practical importance; for warm water from a power station is often injected near the bank, with a finite velocity. It is carried downstream, spreading laterally as it does so. The river may be curved, so that “secondary flows” are present. Calculating the resulting surface temperature distributions becomes a matter of appreciable concern. Of course, a full three-dimensional analysis, of the kind reported by McGuirk and the author [12,13] can be made if this is desired. However, such computations are expensive. If quantitatively realistic predictions can be made with the help of a simple program such as GENMIX, they will certainly be preferred.
108
D. BRIANSPALDINC
In this regard, it should be mentioned that input data, for example the wind and river conditions, are rarely of high precision in THIRBLE problems. It is therefore foolish to incur the expenses entailed by using computational methods of a higher order of precision.
5 . Method Aduptation The GENMIX computer code requires modifications of two kinds if it is to be used for this problem; but its general structure can be preserved and utilized. First, there are the modifications associated with the “parabolic” calculation, particularly the modified distance calculation (h is not present in the standard GENMIX) and the inclusion of entrainment (which, though present in the standard GENMIX, has a different significance there). Secondly, there are the modifications needed to make the computation “hyperbolic,” i.e., the computations of the discrepancies between the two kinds of Y’s, the setting up of the coefficients in the h-correction equation, the solution of this equation by the tridiagonal-matrix algorithm, etc. These adaptive modifications, which can be carried out in various ways, present no essential difficulty. Description of further details can be spared. 6. Expected Results
It is useful to consider what rate of spread is probable. As has been seen, the edge is likely to move at a speed of around (g/.?(T - Tr)h)’’’.Insertion of the values: y = 9.81 m/sec2, j3 = 2 x lop4(“C)-’, T - T , = l O T , h = 0.1 m, gives a velocity of spread of 0.044 m/sec. Since a fast-moving river will have a velocity of the order of ten times this value, it is indeed probable that the layer, seen from above, will have an elongated shape. Other results can be readily imagined: the layer will fairly quickly take up the longitudinal velocity of the river water which supports it because of entrainment from below; there will be little lateral variation of temperature at any particular x; but the tendency will be for the lower temperature to appear near the edges because h will be lower there, and the surfactvolume ratio therefore higher. There are many interesting effects to be explored, resulting from the many possible combinations of wind stress, initial conditions, supporting-stream behavior, etc. These must be deferred for later study, with the aid of the computer program, and especially of its graphical-output features. 7 . Remarks
The present approach, it should be mentioned, is not the only one that it is possible to make. A method of a different kind, specifically one in which the profile shapes of u, T, and h are fixed, has been reported by Harleman and Stolzenbach [ 5 5 ] ; it has significant successes on its record.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES 109 The present method is more flexible than that of Harleman and Stolzenbach [ S S ] ; for it can handle arbitrary profile shapes, oblique wind directions, and other practically occurring complexities. However, it awaits practical exploitation. C. THE“PARTIALLY PARABOLIC” LAYER
If neither the injection velocity nor the river velocity are large compared with the quantity {ga(T - T,)h}’’’, the neglect of the ahlax terms on the longitudinal velocity cannot be justified. It is therefore necessary to consider this case also, lest erroneous predictions should be used as the basis of design or environmental-protection decisions. This is the task of the present section. 1. Geometry
The situation considered is the same as in Section V1.B; however, conditions will be such that the layer, seen from above, is not a highly elongated region: the “angle of spread may be large.
2. Physics No new physical processes are in question; but the formerly neglected longitudinal pressure gradient is now included in the model. 3. Boundary Conditions
Because effects from downstream can propagate upstream, there is now a requirement for conditions at large x to be specified, at least in terms of h. This may be recognized if it is imagined that an “inverted weir” could be placed at the downstream boundary, sufficiently deep to prevent the warm water from moving further until h had increased to the depth of the lower edge of the weir below the free surface; such a weir would increase the lateral spread of warm water. Alternatively, a warm-water-suction device might be imagined, which would remove warm water from the river surface as soon as it reached the given x station. This would cause h to equal zero there; and this would certainly accelerate the flow of the warm water layer in the x direction. Whereas the first condition would make ahlax greater than zero at the downstream boundary, the second condition would make it less than zero. It is therefore reasonable to take, as an intermediate boundary condition to be used whenever more detailed information is absent, the downstream boundary condition: x = x,,,:
-ah= o
ax Otherwise, the boundary conditions are as in Section V1.B.
(6.3-1)
110
D. BRIANSPALDING
4. Importance
Problems of this kind are of great theoretical importance, both in THIRBLE contexts and elsewhere in engineering (for example, turbomachinery [56] or shipbuilding [57]). This importance derives from their peculiarity of requiring maximum computer storage for only one (pressure, or layer depth) of the many dependent variables. This feature permits much greater accuracy to be attained, when computer storage is the limitation, than if all variables had to be stored for all grid points. It should be stressed that this “partially parabolic” (“semielliptic”) class of phenomena always involve steady processes. Now, many authors have regarded it as convenient to compute steady processes by way of timemarching methods: the steady state is reached at the end of a sufficiently prolonged computation of the flow, with arbitrary starting conditions but fixed boundary conditions. This use of time-marching methods has two points in its favor: it is simple to understand; and convergent and stable numerical procedures are known to many practitioners. These advantages lose significance, however, in comparison with the great disadvantage of time-marching methods: all dependent variables must be stored for all grid points. Therefore, whenever economy and accuracy are strongly desired, steady-state processes having a predominant direction of flow should be treated by steady-state partially parabolic procedures. 5. Method Adaptation When GENMIX has once been adapted for the “hyperbolic” problem of Section VI.B, it is not hard to adapt it further for the present problem. The following features must be incorporated: (i) The finite-difference equations must be modified in the manner indicated, but not completely described, in Section VI.A.6. This will require clarification of the grid notation, and specifically a reversion to the complete “staggered grid explained in Spalding [l] and elsewhere. The result will be a set of difference equations having the implication that an increase in the depth h tends to increase the net flow out of the computational cell, both in the y and the x directions. In this way the negative-coefficient problem of Section VI.A.6.b is resolved. (ii) Two-dimensional storage must be provided for h. This is easy to do. (iii) Provision must be made for the marching-integration sweeps, from small x to large, to be repeated until the imbalances remaining in the finitedifference equations are small enough to be ignored. Otherwise, the GENMIX program can remain as it is. Input and output subroutines will not be affected in their structure.
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
11 1
6. Expected Results
Qualitatively, the results will be similar to those of Section VI.B, for most sets of boundary conditions. However, the forward velocity u will be somewhat greater; h will be somewhat smaller; and the lateral spread will be slightly reduced. All these effects are consequences of now admitting the influences of ahlax, which will be negative, in giving the warm water a (probably slight) forward push). The computations will be somewhat more expensive, in respect of both storage and time; but only by factors of two or three.
I. Remarks No computation of this kind has yet been performed; the author’s current research plans do however include work of this kind.
D. SOMEUSEFULFURTHER DEVELOPMENTS 1. The Radial Pool
If the injection velocity and the “river” velocity are both zero, and if there is no wind, there is no preferred direction for the warm water to take: it will therefore spread radially outward. This is also a problem that can be handled by the GENMIX program, and preferably (in this case) by a time-marching technique; for there is little to be gained by reducing one-dimensional storage to zero dimensional; and, in any case, it is not always true that a steady-state solution actually exists. Only the briefest of descriptions will be provided here of how the radiallyspreading pool is to be analyzed. The main points are The domain of integration will be a narrow-angled wedge, lying on the water surface, with its apex at the point of discharge. This apex will represent the I boundary; and the E boundary will move radially outward in the course of time. Time t will take the place of longitudinal distance x. The “grid lines” will be placed at such radii, increasing with t, that no mass crosses them. However, the amounts of mass in the layer between pairs of these lines will increase with time, as a rule, as a consequence of entrainment from below. The problem will be of the “hyperbolic” type; for there is no question of events at large time iduencing those which took place earlier. Analyses of this kind may be usefully employed to throw light on much more complex phenomena. For example, when a large quantity of light pollutant is dumped into the ocean, it will drift with the ocean current, and perhaps under the influence of wind. However, if the coordinate system
112
D. BRIANSPALDING
moves with the pool, treatment as a radially spreading layer may still be realistic, at least until a shore line is approached. How long a time must elapse, it may be asked, until the maximum surface temperature has fallen to a definite level? The procedure described can answer this question without difficulty; and, even if actually existing departures from circularity reduce the quantitative accuracy of the answer, its order of magnitude will be correct; and great insight will have been provided. 2. Steady Nearly Radial Outflow The desirable feature of the partially-parabolic method of Section V1.C is that it avoids the necessity of storing any dependent variable but h two dimensionally; and this results from the fact that the velocity of flow is always positive in the x direction, so that convective effects flow always from small x to large. The marching integration can proceed in the same direction. The same feature is possessed by a layer emanating from a source in an ocean exhibiting a slow but steady drift, even when both Cartesian direction velocities exhibit positive and negative values. All that is required is that the radial component of velocity of the layer fluid is everywhere positive; i.e., the drift velocity of the ocean is not large enough to reverse the tendency of the layer fluid to spread. A GENMIX-based model of this process would employ radius in place of x; and the distance y would be measured around the circumference of concentric circles of increasing radius. If the drift velocity and the wind stresses allowed symmetry to exist, the I and E boundaries would be the two branches of the symmetry axis, joined at the center of the circles. If they did not allow symmetry, the I and E boundaries would actually be coincident, the integration domain being like the surface area of a completely open fan. In this case, a “cyclic tridiagonal matrix algorithm” [58] would be needed for solving the difference equations. Marching integration would proceed in the radially outward direction. This model is useful for the analysis of many thermal pollution problems; but of course the need for the flow to be steady does limit its applicability. 3. Unsteady Layers
It may be said as well that the extension of the layer analysis to transient phenomena is not very difficult. Two-dimensional storage is of course required for all dependent variables; and the repeated marching integration is necessitated by stepping forward in time as well as by the procuring of convergence. Computer codes which are like GENMIX in general structure, but which are equipped for transient flow in two space dimensions, are now
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES 113 available [59]. They can handle also recirculating flow, i.e., that in which both velocity components exhibit both positive and negative values. At this point, however, the boundaries set for this article have been reached, and slightly transgressed. This is a good point to stop. VII. Concluding Remarks The number of practically interesting THIRBLE processes, which are two-dimensional parabolic, hyperbolic, or partially parabolic, is very large; and there has been little exploration of them. This is why so much of the present article has had to be concerned with possibilities and expectations rather than demonstrated achievements. When this large number is contemplated, and the still larger number is imagined of problems of a three-dimensional character, it is evident that exploration of THIRBLE processes by heat- and mass-transfer experts has scarcely begun. This is no proof in itself that large-scale exploration should begin; for the costs and likely benefits need to be estimated carefully in advance before every enterprise, no matter how intellectually attractive it may seem. The author’s view, after some experience of applying three-dimensional elliptic computational procedures to THIRBLE problems, is that it is easy to make demonstrations of ability to solve such problems in principle, but at present very hard to make the procedures simultaneously accurate and cheap enough for practical use. The immediate future may therefore lie with models of the type discussed in the present article. At any rate, it seems wise to investigate thoroughly their capabilities and limitations before, in advance of proved need, devoting large resources to more elaborate models. Fashions come and go in applied science; and their careers are often determined more by the existence of solution procedures than by the urgency of the problems. In this article also, the existence of a certain class of solution procedure has been put in the forefront of attention; but the importance of the practical problems which the procedure can solve, it is truly believed, can serve as a justification. NOMENCLATURE Latin symbols: equation or section of first mention
f 9
a b c
J
2.1-1, 5.3-1, 6.1-8 2.1-1 2.1-1, 5.2-9 2.1-1
h k
I m
5.2-6 3.1-1 6.1-1 3.1-2 3.1-2 Section IV.A.2
D.BRIANSPALDING
114
S
Section IV.B.6
n 4 r S
t U V
X
Y A B C D E F G H
K L
v p 0
‘ I
* r A
Y
n
V
6.1-12
U
Greek symbols: equation or section of first mention a
j
Y
2.1-13 3.2-2
energy dissipation (with C) downstream external boundary any boundary effective equilibrium internal boundary i grid point 0 orifice, origin r river rad radiation s salt T temperature U umtream x x direction y y direction a~ at infinity * guessed value (except for Ri*, Eq. (6.140) ) correction D D E e eff eq I
5.2-9
X Y
T
6.1-16 6.1-12 5.2-8
Subscripts and superscripts
5.2-14 5.2-12
4.1-1 5.2-1 3.2-2 5.2-6 5.2-6
S
2.1-2
p
N
Q
0
I
2.1-2 5.2-14 2.1-2 2.1-2 6.1-2 2.1-1 2.1-2 6.1-17 6.1-17 3.1-2, 6.1-17 3.1-1, 6.1-17,4.2-1 4.3-7, 6.1-29
M R Ri
f#J
5.2-12 Section IV.A.l 5.2-9 3.1-2 2.1-2 2.1-8 5.3-4 5.3-4 2.1-1
REFERENCES
1. D. B. Spalding, “Transfer of Heat in Rivers, Bays, Lakes and Estuaries-THIRBLE,”
Rep. HTS/75/4. Mech. Eng. Dep., Imperial College, London, 1975. 2. G. N. Abramovich, “The Theory of Turbulent Jets.” MIT Press, Cambridge, Massachusetts, 1963. 3. F. P. Ricou and D. B. Spalding, Measurements of entrainment by axisymmetrical turbulent jets. J. FluidMech. 2, Part 1,21-32 (1961). 4. B. R. Morton, G. I. Taylor, and J. S. Turner, Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. SOC.London, Ser. A 234, 171 and 200 (1956). 5. J. F. Campbell and J. A. Schetz, Analysis of the injection of a heated turbulent jet into a cross dow. NASA Tech. Rep. R-413 (1973).
HEATTRANSFER IN RIVERS,BAYS,LAKES,AND ESTUARIJS
115
6. R. L. Stoy and Y. Ben Haim, Turbulent jets in a confined cross flow. J. Fluids Eng. 95, NO. 4, 551-556 (1973). 7. G. B. McBride, Numerical solutions of the equations governing submarine discharge of liquid waste. In “Numerical Methods in Fluid Dynamics” (C. A. Brebbia and J. J. Connor, eds.), pp. 494-511. Pentech Press, London, 1974. 8. R. C. Y. Koh and L. N. Fan, “Mathematical Models for the Prediction of Temperature Distributions Resulting from the Discharge of Heated Water into Large Bodies of Water,” Rep. 16130 DWO 10/70, p. 219. US.. Environ. Prot. Agency, Research Triangle Park, North Carolina, 1970. 9. A. D. Gosman and D. B. Spalding, The prediction of confined three-dimensional boundary layers. Symp., pp. B90-B97. Inst. Mech. Eng., London, 1970. 10. L. S. Caretto, R. M. Curr, and D. B. Spalding, Two numerical methods for threedimensional boundary-layers. Comput. Methods Appl. Mech. & Eng. 1,39-57 (1972). 11. S.V.Patankar and D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787-1806 (1972). 12. J. J. McGuirk and D. B. Spalding, “Mathematical Modelling of Thermal Pollution in Rivers,” Rep. HTS/75/18. Mech. Eng. Dep., Imperial College, London, 1975. 13. J. J. McGuirk, Prediction of turbulent buoyant jets in co-flowing streams. Ph.D. Thesis, University of London, 1975. 14. C.-H. Wu, A general through-flow theory of fluid flow with subsonic or supersonic velocity in turbomachines of arbitrary hub and casing shapes. NASA Tech. Nore TN-2302(1951). 15. L. S. Caretto, A. D. Gosman, S. V. Patankar, and D. B. Spalding, in “Two Calculation Procedures for Steady, Three-dimensional Flows with Recirculation” (J. Ehlers et al., eds.), p. 60. Springer-Verlag, Berlin and New York, 1973. 16. S. V. Patankar and D. B. Spalding, “A Computer Model for Three-dimensional Flow in Furnaces,” pp. 605-614. Combustion Inst., New York, 1973. 17. S. V. Patankar and D. B. Spalding, Simultaneous predictions of flow pattern and radiation for three-dimensional flames. In “Heat Transfer in Flames” (N. Afgan and J. Beer, eds.), p. 73. Scripta, Washington, D.C., 1974. 18. S. V. Patankar and D. B. Spalding, A finite-difference procedure for solving the equations of the two-dimensional boundary layer. Inr. J. Heat Muss Transfer 10, 1389-1411 (1967). 19. D. B. Spalding, “Environmental Impact of Energy Production: Heat and Mass Transfer Problems,” Rep. HTS/75/22. Mech. Eng. Dep., Imperial College, London, 1975. 20. S. V. Patankar and D. B. Spalding, A calculation procedure for the transient and steady state behaviour of shell-and-tube heat exchangers. In “Heat Exchangers: Design and Theory Sourcebook” (N. Afgan and E. U. Schliinder, eds.), pp. 155-176. Scripta, Washington, D.C., 1975. 21. D. B. Spalding, “GENMIX: A General Computer Program for Two-dimensional Parabolic Phenomena,” Rep. HTS/75/17. Mech. Eng. Dep., Imperial College, London, 1975. 22. B. E. Launder and D. B. Spalding, “Mathematical Models of Turbulence.” Academic Press, New York, 1972. 23. B. E. Launder, A. P. Morse, W. Rodi, and D. B. Spalding, The prediction of free shear flows-a comparison of six turbulence models. NASA Tech. Rep. SP-311(1972). 24. M.Saiy, Turbulent mixing of gas streams: An experimental and computational investigation of turbulence in plane two-stream mixing layers with various levels of free stream turbulence. Ph.D. Thesis, University of London, 1974. 25. C.-J. Chen and W. Rodi, A mathematical model for stratified turbulent flows and its application to buoyant jets. Proc. Int. Assoc. Hydraul. Res., 16th, 19 p, 31 (1975). 26. W. Rodi, The prediction of free turbulent boundary layers by use of a two-equation model of turbulence. Ph.D. Thesis, University of London, 1972.
116
D. BRIANSPALDING
27. H. Schlichting, “Boundary-layer Theory,” 4th ed. McGraw-Hill, New York, 1960. 28. A. Walz, “Boundary Layers of Flow and Temperature.” MIT Press, Cambridge, Massachusetts, 1969. 29. T. Cebeci and A. M. 0.Smith, “Analysis of Turbulent Boundary Layers.” Academic Press, New York, 1974. 30. S. J. Kline, M. V. Morkovin, G. Sovran, and D. J. Cockrell, “Computation of Turbulent Boundary Layers-1968,’’ AFOSR-IFP-Stanford Conf. Thermosci. Div., Stanford University, Stanford, California. 1969. 31. F. H. Harlow and P. I. Nakayama, “Transport of Turbulence Energy Decay Rate,” LA-3854. Los Alamos Sci. Lab., University of California, 1968. 32. B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows. Compur. Methods Appl. Mech. & Eng. 3,269-289 (1974). 33. L. Prandtl, Uber ein neues Formelsystem fur die ausgebildete Turbulenz. Nachr. Akad. Wiss.Gijerfingen, Math.-Phys. KI. pp. 6-19 (1945). 34. G. S. Glushko, Turbulent boundary layer on a plane plate in an incompressible fluid. Izv. Akad. Nauk SSSR, Ser. Mech. No. 4, pp. 13-23 (1965). 35. P. Bradshaw, D. H. Ferris, and N. P. Atwell, Calculation of boundary-layer development using the turbulent energy equation. J . Fluid Mech. 28,Part 3 , 593-616 (1967). 36. D. B. Spalding, Heat transfer from turbulent separated flows. J. Fluid Mech. 27, Part 1 , 97-109 (1967). 37. M. Wolfshtein, Convection processes in turbulent impinging jet. Ph.D. Thesis, University of London, 1967. 38. A. K. Runchal, Transfer processes in steady, two-dimensional separated flows. Ph.D. Thesis, University of London, 1969. 39. W. P. Jones and B. E. Launder, The prediction of laminarisation with a two-equation model of turbulence. Inr. J. Heat Mass Transfer 15, 301 (1972). 40. S. Hassid and M. Poreh, “A Turbulent Energy Model for Flows with Drag Reduction,” ASME Paper, 75-FE-H. Am. SOC.Mech. Eng., New York, 1975. 41. K. H. Ng and D. B. Spalding, Turbulence model for boundary layers near walls. Phys. Nuids 15, No. 1, 20-30 (1972). 42. A. K. Singhal and D. B. Spalding, “Prediction of Two-dimensional Boundary Layers with the Aid of K--E Model of Turbulence,” Rep. HTS/75/15. Mech. Eng. Dep., Imperial College, London, 1975. 43. M. M. Gibson and D. B. Spalding, “A Two-equation Model of Turbulence Applied to the Prediction of Heat and Mass Transfer in Wall Boundary Layers,” AIChE-ASME Heat Transfer Conf., ASME Publ. 72-HT-15. Am. SOC.Mech. Eng., New York, 1972. 44. R. J. Moffat and W. M. Kays, The turbulent boundary layer on a porous plate: Experimental heat transfer with uniform blowing and suction. Int. J. Heat MQSSTransfer 11, 1547-1566 (1968). 45. H. W. Liepmann and J. Laufer, Investigation of free turbulent mixing. NASA Tech. Note TN-1257(1947). 46. M. Coantic and A. Favre, Activities in, and preliminary results of, air-sea interactions research at IMST. Ado. Ceophys. 18A,391 -405 (1974). 47. I. Wygnanski and H. E. Fiedler, The two-dimensional mixing region. J. Fluid Mech. 41, Part 2,327-362 (1970). 48. V. W. Ekman, On the influence of the earth’s rotation on ocean currents. Ark. Mat., Astron. Fys. 2, 53 (1905). 49. U. Svensson, private communication (1976). 50. T. H. Ellison and J . S. Turner, Turbulent entrainment in stratified flows. J . Fluid Mech. 6,423-448 (1959).
HEATTRANSFER IN RIVERS, BAYS,LAKES,AND ESTUARIES
117
51. D. B. Spalding and U. Svensson, “The development and Erosion of the Thermocline,” Rep. HTS/76/7. Mech. Eng. Dep., Imperial College, London, 1976. 52. D. B. Spalding, Mech. Eng. Dep., Imperial College, London, 1973 (unpublished work), 53. D. B. Spalding, Numerical computations of steady boundary layers-a survey. In “Computational Methods and Problems in Aeronautical Fluid Dynamics” (8.L. Hewitt et al., eds.). Academic Press, New York, 1975 (to be published). 54. V. S. F’ratap and D. B. Spalding, Numerical computations of flow in curved ducts. Aero Q.26, 219-228 (1975). 55. K . D. Stolzenbach and D. R. F. Harleman, Three-dimensional heated surface jets. Water Resour. Res. 1, 129- I37 ( I 973). 56. A. K. Majumdar and D. B. Spalding, “A Numerical Investigation of Flow in Rotating Radial Diffusers,” Rep. HTS/76/4. Mech. Eng. Dep., Imperial College, London, 1976. 57. D. B. Spalding, “Calculation Procedures for Three-dimensional Parabolic and Partiallyparabolic Flows,” Rep. HTS/75/5. Mech. Eng. Dep., Imperial College, London, 1975. 58. C. Temperton, Algorithms for the solution of cyclic tri-diagonal systems. J . Comput. Phys. 19, 317-323 (1975). 59. W. M. Pun and D. B. Spalding, “A General Computer Program for Two-dimensional Elliptic Flows,” Rep. HTS/76/2. Mech. Eng. Dep., Imperial College, London, 1976.
This Page Intentionally Left Blank
Simultaneous Heat. Mass. and Momentum Transfer in Porous Media: A Theory of Drying STEPHEN WHITAKER Department of Chemical Engineering. University of Calgornia at Davis. Davis. California 95616
. .
I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 The Basic Equations of Mass and Energy Transport . . . . . . . . . . . A . Governing Point Equations . . . . . . . . . . . . . . . . . . . . . B. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . C. Volume Averaged Equations . . . . . . . . . . . . . . . . . . . . I11 Energy Transport in a Drying Process . . . . . . . . . . . . . . . . . A . Total Thermal Energy Equation . . . . . . . . . . . . . . . . . . B. The Effective Thermal Conductivity . . . . . . . . . . . . . . . . C. Thermodynamic Relations . . . . . . . . . . . . . . . . . . . . . IV Mass Transport in the Gas Phase . . . . . . . . . . . . . . . . . . . . A . The Gas Phase Diffusion Equation . . . . . . . . . . . . . . . . . B. Convective Transport in the Gas Phase . . . . . . . . . . . . . . . V . Convective Transport in the Liquid Phase . . . . . . . . . . . . . . . A . Darcy's Law for a Discontinuous Phase. . . . . . . . . . . . . . . B . A Constitutive Equation for the Forces Acting on the Liquid Phase . . . VI Solution of the Drying Problem . . . . . . . . . . . . . . . . . . . . . VII . The Diffusion Theory of Drying . . . . . . . . . . . . . . . . . . . . . VIII Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
. .
.
. . . . . . . .
119 126 128 133 137 153 154 158 164 165 166 169 175 175 184 192 194 198 199 200
.
I Introduction Man has been keenly aware of the importance of drying in a porous media since the first clay bowl was shaped by hand and set in the sun to dry. Under the proper conditions [l] a rock-hard product was obtained and the art of 119
120
STEPHEN WHITAKER
pottery was advanced. The dehydration of foodstuffs has also been practiced for centuries, and the proper management of our agricultural products and our energy resources now requires an improved understanding of the drying process in biological materials. The importance of the process of drying solids has been dramatized by Lebedev and Ginzburg [2] who stated in their review article that “according to some data, in the fuel balance of the U.S.S.R., the fuel consumption for drying is 10 percent of the total.” If this is indeed the case, then surely our knowledge of the drying process deserves improvement. Fulford [3] has characterized our situation with the statement that “anyone faced with engineeringa new drying system rapidly realizes that solids drying remains largely an art,” and in the 1970 I & EC biannual review on drying McCormick [4] assumed a similar, pessimistic position with the statement that there has been a “decline in quantity and quality of unit operations research in the United States.” Any decline that has occurred has only been relative to the enormous progress made in the analysis of the well-posed problems of simultaneous heat and mass transfer. While computer solutions have cut a swath through the maze of boundary value problems associated with engineering science research, the complexities of the porous structure encountered in drying processes have proved to represent a relatively impenetrable barrier to this type of analysis. The first engineering analysis of the drying of solids was apparently that of Lewis [S] who postulated that drying consisted of two processes: (1) the diffusion of moisture from the interior of the solid out to the surface, and (2) the evaporation of the moisture from the surface of the solid. The term moisture apparently referred to the liquid state, and the term diffusion clearly referred to a mechanism comparable to molecular diffusion in a multicomponent, single phase system. The suggestion by Lewis that drying was a diffusional process was picked up by Sherwood and a series of papers C6-91 on the drying of solids resulted. These were all based on a “diffusion” equation of the form aqat
=
D(a2qaxz)
(1-1)
where C represents the vaguely defined “moisture content” and D represents a parameter which is determined by experiment. Sherwood also discussed the possibility of diffusion in the vapor phase and considered the heat transfer phenomena associated with drying. However, neither of these phenomena was considered to be particularly important, and it was the diffusion of liquid through the porous solid that occupied the center of attention. The solutions given by Sherwood were extended by Newman [lo, 111 to other geometries using the appropriate modification of Eq. (I- 1). Further work by Gilliland and Sherwood [12] again made use of the diffusion equation to estimate the duration of the so-called constant rate period, and it appeared
A THEORY OF DRYING IN
POROUS
MEDIA
121
that a complete theory of drying was within reach using only the diffusion equation.* While the chemical engineering community was busy comparing drying data with solutions to the diffusion equation, other scientists were examining the motion of liquids through unsaturated porous media from an entirely different point of view. Soil scientists, colloid chemists, and ceramicists were explaining the movement of moisture in porous media in terms of surface tension forces or by capillary action. The work of Gardner and Widtsoe [13], Richards [14], Rideal[15], and Westman [16] clearly indicated that surface tension effects could not be ignored in the study of liquid motion in an unsaturated porous media. Comings and Sherwood were quick to recognize this, and a short note was published [17] providing a qualitative discription of the motion of liquid owing to capillary action and the motion of vapor owing to molecular diffusion. A brief set of experiments on the drying of clay was performed by Comings and Sherwood, and the results indicated that capillary action could indeed be an important mechanism in the movement of liquid during the drying of a porous solid. Other researchers were then warned that “the word diffusion must be used with care in referring to the movement of water through a soil.” The warning of Comings and Sherwood [17] and the work of the soil scientists was not lost on other engineers interested in drying, and in 1937 Ceaglske and Hougen [181began their paper on the drying of granular solids with the statement: “The drying rate of a granular substance is determined not by diffusion but by capillary action.” Armed with the extensive work of Haines [19,20] on the relation between capillary pressure and liquid content, Ceaglske and Hougen were able to calculate the saturation distribution during the drying of a layer of sand 5.08-cm thick. The results are shown in Figs. I-la and I-lb and clearly demonstrate, for the particular case under consideration, that capillary action can play a dominant role in the movement of moisture during the drying of a porous solid. Later we shall return to a more detailed discussion of these results; however, for the present we need only point out that the theoretical results were obtained using only the laws of hydrostatics along with an experimental determination of the capillary pressure as a function of saturation. The work of Ceaglske and Hougen was followed by an extensive survey by Hougen et ul. [21] appropriately entitled “Limitations of Diffusion Equations in Drying.” Comparisons were made between solutions of the diffusion equation, calculations based on capillary pressure-saturation curves, and experimental data. The comparison for sand is shown in Figs. 1-2, and the results strongly favor the capillary theory. Results were presented for the drying of clay, soap, paper, paper pulp, * In a later paper (“Drying in Porous Media,” 2nd Australasian Conf. on Heat and Mass Transfer, Sydney, Feb. 1977), the diffusion theory of drying is explored in considerable detail.
STEPHEN WHITAKER
122
z P I-
a
P 3 I-
a
v)
w
c)
4
ku
k
w
a
0
1
2
3
4
5
DISTANCE (CM)
FIG.1-la. Experimental water distribution in a sand layer during drying. Adapted from Ceaglske and Hougen [18]. 100
80
z 0 t a
2 K
60
3 w
2 t
z W
40
u K w
n
20
0 0
1
2
3
4
5
DISTANCE ( C M )
FIG.I-lb. Theoretical water distribution in a sand layer during drying. Adapted from Ceaglske and Hougen [18].
A THEORY OF DRYING IN POROUS MEDIA 40
---
-Experiment01
123
'I
Volues
Colculoted
from Diffusion Equation
35
Y
3 > K
O
30
25
I-
z W
I-
z
0
U
20
W
K
3
I-
s
=
0
15
I-
z w u K
w n
10
5
;'-
4
L--
0 0
1
2
DISTANCE FROM SURFACE OF DRYING - 0 .
FIG.I-2a. Moisture distribution during the drying of sand-comparison with diffusion theory. Adapted from Hougen et af. [21].
and lead shot; and in each case the diffusionequation could not be made to fit the experimental data using a constant diffusivity.For wood the situation is somewhat different, and reasonably good agreement between experimental data and the diffusion equation are obtained provided the moisture content is below the fiber saturation point. In Fig. 1-3 the data of Tuttle [22] are compared with calculated values from the diffusion equation and rather good agreement is obtained. Hougen et al. [21] point out, however, that the drying of extremely wet wood cannot be modeled by the diffusion equation, and that even below the fiber saturation point one should allow for variation of the diffusivity with moisture content, temperature, pressure, and density.
STEPHEN WHITAKER
124 40
35
%? v)
3c
m < >
K
n i-
z
25
w
I-
z
0
U Y
3 K
2t
I-
L"
0 i
I; w
11
V
0: W
a 1L
I
0
1
2
DISTANCE FROM SURFACE OF DRYING-CM
FIG.I-2b. Moisture distribution during the drying of sand--cornparison with capillary theory. Adapted from Hougen et al. [21].
While questions about capillary action and the diffusion of moisture were being raised by chemical engineers in the U.S.A., Krischer [23-251 was exploring the very same ground in a series of papers on drying. While heat transfer had played a minor role in previous papers on drying, it appears that Krischer was the first to consider seriously the intimate role that the transport of energy may play in a drying process. Although the diffusion theory of drying seemed to be in a state of disrepute after the papers by Ceaglske and Hougen [18] and by Hougen et al. [21],
A THEORY OF DRYING IN POROUS MEDIA
125
40 I-
z W
I-
z
8
30
W
K
3
!i
20
8 I-
z
x w w
10
Q
0 0
1/4
1/ 2
3/4
1
DISTANCE FROM SURFACE OF DRYING-IN.
FIG.1-3. Moisture distribution during the drying of wood. Adapted from Hougen et al. [21].
there was some definite evidence that the diffusion equation could be used to model the latter stages of drying. Presumably the liquid at this stage is in what Haines [19,20] referred to as the pendular state and motion owing to capillary forces is greatly reduced if not completely halted. Under these circumstances the convection and diffusion in the vapor phase are the only mechanisms by which the moisture content can be reduced. This situation seems to be especially important in the drying of biological materials to very low moisture contents. In 1947 Van Arsdel [26] pursued this line of attack and investigated the effect of a variable diffusivity on drying rates predicted by the diffusion theory. Equation (1-1)was essentially modified to the form
and solved numerically to yield results similar to those found in the laboratory. Phillip and DeVries [27] and DeVries [28] extended previous treatments of drying to include effects of capillary flow and vapor transport, and incorporated the thermal energy'equation into the governing set of equations that describe the drying process. The transport owing to capillary forces was represented in terms of gradients of the moisture content and temperature; thus a diffusionlike equation resulted. Similar equations for heat and mass transfer in porous media were also published by Luikov [29]. Churchill and Gupta [30] continued the use of diffusion equations in their study of the
126
STEPHENWHITAKER
freezing of wet soils, but were quick to note that “the formulation of the rates of moisture transfer in both the vapor and liquid phases in terms of diffusivities and concentration gradientsis highly arbitrary and cannot be rationalized on theoretical grounds.” Berger and Pei [31] continued the practice of representing liquid and vapor fluxes in terms of diffusivities and concentration gradients,and more recently Husain et al. [32] have applied the diffusion equations for heat and mass to the drying of biological materials. Good agreement between theory and experiment is obtained provided one allows the “diffusivity” to be a complex function of the moisture content. In all of the previous theoretical studies of drying the governing differential equations which it was hoped applied to the porous media were inferred in a purely intuitive manner from the well-known point equations of continuum physics.* Often this intuition was enhanced by the use of shell balances [33] to construct the differential equations, but the final result was nevertheless an intuitive product. In the following sections a rigorous, but limited theory of drying will be presented based on the well-known transport equations for a continuous media. These equations will be volume averaged to provide a rational route to a set of equations describing the transport of heat and mass in a porous media. The theory will be limited in two ways: by restrictions and by assumptions. The former represent clear cut limitations in the form of the governing point equations and boundary conditions and can be removed by further analysis. The latter are concerned mainly with the topology of the three-phase system and the order of magnitude and functional dependence of various terms that arise in the theoretical development. It seems likely that the validity of the assumptions can be tested only by comparison with experiment, although in some cases they may be accepted or rejected on the basis of further theoretical analysis. In order to clearly identify the limitations of the theory the restrictions will be denoted as R.l, R.2, etc., while the assumptions will be designated by A.l, A.2, A.3, etc. 11. The Basic Equations of Mass and Energy Transport
In our analysis we consider the motion of a liquid and its vapor through a rigid porous media such as that shown in Fig. 11-1. There the c phase reprephase is the liquid, and the y phase represents sents a rigid solid matrix, the /I the gas phase which consists of vapor and some inert component (usually air) that is insoluble in either the u or the /3 phase. There is no liquid contained in the u phase, thus our development excludes what is sometimes referred to as “bound” moisture [29, p. 2341. Although our attention is directed toward
* The one exception to this statement is the use of modified forms of Darcy’s law.
A THEORY OF DRYING IN POROUS MEDIA
127
FIG.11-1. Drying process in porous media.
the gas-liquid-solid system, one may interpret the fl phase as a second solid phase undergoing sublimation so that the analysis also includes the case of freeze drying. The situation shown in Fig. 11-1 would represent the pendular stage as opposed to the funicular stage since the liquid phase is discontinuous. The analysis is not restricted by the particular liquid distribution shown in Fig. 11-1, and in subsequent developments both the pendular and the funicular stages will be discussed. In the analysis of drying phenomena we are generally interested in knowing the moisture content and temperature as a function of space and time. These quantities will be determined by application of the appropriate laws of physics which we list as follows: continuity equation: aP at
-
+v
*
(pv) = 0
(11-1)
(pivi) = ri
(11-2)
ith species continuity equation: api
-
at
+V
*
STEPHEN WHITAKER
128
linear momentum principle: (11-3) angular momentum principle : (11-4)
T = Tt
thermal energy equation:
Dh
p-=
Dt
'
- V ' q + - +DP VV:T+@ Dt
(11-5)
In Eq. (11-2) the term ri represents the mass rate of production of the ith species owing to chemical reaction, and vi represents the ith species velocity [33, p. 4971. In Eq. (11-3) the total stress tensor [34, p. 1131 is represented by T and T+ represents the transpose ofT [35, p. 1031.In Eq. (11-5)we have used z to represent the viscous stress tensor, p is the pressure, h is the enthalpy per unit mass, and @ represents the source or sink of electromagnetic radiation. The thermal energy equation is derived directly from the first law of thermodynamics, and a detailed discussion is given elsewhere [36, Sec. 5.41. We should note that Eq. (11-1) can be derived from Eq. (11-2) by summing over all N species in the system and imposing the definitions for the total density p and mass average velocity v, i=N
P =
c
Pi
(11-6)
Pivi
(11-7)
i= 1 i=N
PV
=
C
i= 1
in addition to imposing the restriction on the chemical rates of reaction that i=N
Cri=O
(11-8)
i= 1
In Section I1 of this article we shall focus our attention on Eqs. (11-l),(11-2), and (11-5) in developing the relevant volume averaged transport equations which describe the drying process. In Section IV we shall be concerned with the transport of momentum in the gas phase, and there we shall begin our investigation of the laws of mechanics as they apply to the drying process. A. GOVERNING POINTEQUATIONS
In this section we shall examine Eqs. (11-l), (11-2), and (11-5) and list the forms that they take in the three separate phases. We shall use 0, p, and y as subscripts to denote the phase in question, thus Towill represent the point
A THEORY OF DRYING IN
POROUS
MEDIA
129
temperature in the solid phase, and vg will represent the point mass average velocity in the liquid phase. 1. c Phase (Solid) We consider the solid phase to be a rigid matrix fixed in an inertial frame. Relative to this frame the velocity in the c phase is zero:
R.l
V, =
0
(1I.A-1)
thus Eqs. (11-1)-(11-4) are of no consequence. This is perhaps one of the most important limitations of the entire analysis, for most biological materials and many nonbiological materials undergo a change in volume upon drying. It is possible that the change in volume upon drying can be accounted for by allowing various derived parameters to be a function of the moisture content; however, a rigorous attack on the problem would require that Eq. (11-1) for the solid phase be incorporated into the analysis. On the basis of restriction R.l we can immediately simplify Eq. (11-5) for the c phase to (1I.A-2) Here we have dropped the reversible and irreversiblework terms in Eq. (11-5), but retained the source term owing to electromagnetic radiation so that the theory includes drying processes of the type described by Lyons et al. [37] and by Raiff and Wayner [38]. Throughout our analysis of all three phases we shall assume that the enthalpy is independent of pressure: R.2
h = h ( T )in the c,b, and y phases
(1I.A-3)
and that all heat capacities are constant. This means that h can be replaced by cpT to within an arbitrary constant*:
R.3
h = c,T
+ constant in the 6,p, and y phases
(1I.A-4)
On the basis of R.2 and R.3 we can express Eq. (1I.A-2) as (1I.A-5) and apply Fourier's law to obtain the final form of the thermal energy equation for the solid phase (1I.A-6) * In the gas phase where there is more than one component present we interpret R.2 and R.3 as f;, = & i ( T )and f;, = (T,),T + constant. Here f;, represents the partial mass enthalpy [39, p. 2791.
STEPHEN WIUTAKER
130
Here we have assumed that the thermal conductivity is constant. This assumption will also be imposed on the liquid and gas phases, and we denote it as our fourth limiting restriction: R.4 2.
The thermal conductivities are constant in the 0, p, and y phases.
p Phase (Liquid) We assume the presence of only one component in the liquid phase:
R.5
The @ phase contains only a single component.
so that Eqs. (11-1) and (11-2) are identical and written for the liquid phase as
+
(1I.A-7) ap,/at v (ppvp) = 0 For the time being we shall avoid any discussion of Eqs. (11-3) and (11-4) as they apply to the liquid phase. In considering the thermal energy equation we assume negligible compressional work and viscous dissipation : R.6
DpplDt = V V :~~p = 0
(1I.A-8)
so that Eq. (11-5) takes the form ps Dh,/Dt = - V
qs
+ @p
(11.A-9)
While neglecting reversible work and viscous dissipation is often satisfactory in heat transfer calculations [36, Section 5.51, Defay et al. [40, p. 2361 suggest that the reversible work Dp,/Dt may become important for systems with a very small pore structure such as activated carbon or silica gel. In such systems surface tension can cause large pressure changes in the liquid phase as drying takes place and the accompanying reversible work may represent a significant term in the thermal energy equation. The material derivative in Eq. (11.A-9) can be expanded, leading to pa(dh,/dt
+ v p . Vh,) = - V
'qa
+
(1I.A-10)
Representing the enthalpy h, as indicated in Eq. (1I.A-4), and making use of Fourier's law along with restriction R.4 leads to
+
p a ( ~ p ) a ( d T s / 8 t v P VT,) = k, V 2 T ,
+
(1I.A-11)
3. y Phase (Gus)
We consider the gas phase to be made up of the vapor and an inert component that is insoluble in either the G phase or the phase. The continuity equation is expressed as ap,p
+ v ' (pvvv) = 0
(1I.A-12)
A THEORY OF DRYING IN POROUS MEDIA
131
and we write the species continuity equation as dpi/dr
+V
*
i = 1,2, . . .
(pivi) = 0,
(1I.A-13)
thus suggesting that no chemical reaction takes place in the gas phase:
R.7
There is no chemical reaction in the y phase.
Here p, and v, are given by Eqs. (11-6) and (11-7) as
+ P2 = PlVl + PZV2
(LA-14)
Py = P I
(1I.A-15) PyVy Writing the species velocity vi in terms of the mass average velocity v, and the diffusion velocity ui, (1I.A-16) vi = v, + ui allows us to write Eq. (I1.A-13) as dpip
+ v .(piv,) = - V
1 = 1,2, . . .
(p,~,),
(1I.A-17)
The diffusive flux piui can be expressed as [33, p. 5021 PiUi
=
-P
Y
V(Pi/Py) ~
(1I.A-18)
.
(1I.A-19)
so that our final form of Eq. (1I.A-13) is* api/at
+ V * (Piv,)
=V
V(pi/~,)l
In attacking the thermal energy equation for the gas phase we must refer to the work of Bird er ul. [33] or the work of Slattery [39] where it is shown that the appropriate form of Eq. (11-5) for a multicomponent system is?
+ 0,+
c piui
i=N
' fi
i= 1
- p
D i = NI p , -c--u, Dt,=l 2py
(1I.A-20)
Here we use fii to represent the partial muss enthalpy which is simply the partial molar enthalpy divided by the molecular weight of the ith species.
* Caution is recommended here for the case of freeze drying for that process is usually accompanied by Knudsen diffusion and Eq.(1I.A-IS) is not applicable. Note that q; will be expressed as - k;. VT7 in keeping with Eq. H in Table 18.3-1 of Bird et u / . [33], and Eq. J in Table 8.3.5-1 along with Eqs. (4-1)and (4-7) in Sec. 8.4.4 of Slattery [391.
'
132
STEPHEN WHITAKER
It is quite reasonable to neglect compressional work and viscous dissipation in the gas phase :
R.8
Dpy ~
Dt
=
VV, :z,
=
(1I.A-21)
0
so that Eq. (1I.A-20) simplifies to
+ ic piu, = 1 i=N
*
fi - p
c
1) i = N 1 p . -2ui2
Dti,l 2py
(1I.A-22) The last two terms in Eq. (1I.A-22) represent the diffusive body force rate of work and the time rate of change of the diffusive kinetic energy. Both these terms can be safely neglected:
R.9
1p
i=N
i= 1
I
~* f.,1 = p
1
D
lpi 1 ui2 = 0 D t i = l 2p, i=N
-
(1I.A-23)
--
so that Eq. (1I.A-22)takes the form
Making use of Eqs. (1I.A-16) and (II.A-l2), along with the definition i=N ~ $
7=
C pi&
(1I.A-25)
i= 1
allows us (after some algebraic manipulation) to express Eq. (1I.A-24) as =
-V.q,
-
V.
($
piui?'ii)
+ Oy
(1I.A-26)
Application of Fourier's law and Eq. (1I.A-4) now leads to the familiar form
Here the gas phase heat capacity ( c J Yis given by
and we have again imposed restriction R.4.
A THEORY OF DRYING IN POROUSMEDIA
133
Up to this point we have listed the appropriate forms of the heat and mass transport equations for each of the three phases subject to the restrictions R.1 -R.9. Our next task is to state the boundary conditions that apply at the various phase interfaces.
B. BOUNDARYCONDITIONS Referring to Fig. 11-1 we note that there are three interfacial areas contained within the averaging volume, Y".We denote these as,
A,,, solid-liquid interfacial area; A,?, solid-vapor interfacial area; A,,, liquid-vapor interfacial area; and note that A,,
A,,,
=
A,, = A,,,
A,, = A,,
(1I.B-1)
The boundary conditions for the solid-liquid interface are quite simple and may be represented as
B.C.l
vB = 0 on A,,
B.C.2
q,
B.C.3
T , = T , on A,,
*
nus
+ q,
( I I. B-2) *
n,,
(11.B-3)
= 0 on A,,
(11.B-4)
Here nu, represents the outwardly directed unit normal for that portion of the cr phase in contact with p phase. Thus nu, points out from the cr phase phase and is related to n, by into the /I nap = -n,,
on A,,
(II.B- 5)
We note at this point that there is no source or sink of energy on the A,, surface, thus the interfacial energy of A,, is taken to be negligible. We shall impose similar restrictions on the boundary conditions at the o-y and b-y interfaces and we list this as our tenth limiting restriction:
R.10
Interfacial energies for the o-B, negligible in the thermal sense.
P-r,
and y-o interfaces are
This restriction is easily removed by application of the work of Slattery [41] and may be an important consideration in the drying of highly porous materials such as activated carbon and silica gel. DeVries [28] has incorporated a heat of wetting in his analysis of drying, but the fundamental approach of Slattery [41] has not yet been applied to this problem.
STEPHEN WHITAKER
134
At the solid-vapor interface we can write a set of boundary conditions similar to those given by Eqs. (II.B-2)-(II.B-4):
B.C.4 B.C.5 B.C.6
v,
=
Oon A,,
+
q, * nor q, * n,, T , = T , on A,,
=
(1I.B-6) (1I.B-7) (1I.B-8)
0 on A,,
The boundary conditions for the liquid-vapor interface will require some discussion, for A,, represents a moving, singular surface. The jump conditions that apply at singular surfaces have been discussed by Truesdell and Toupin [42, p. 5171 and more recently by Slattery [39]; however, the jump condition for the multicomponent form of the thermal energy equation has not been listed explicitly and will be presented here for completeness. Our development will follow the approach given by Slattery [39, p. 211 in which we focus our attention on the material volume illustrated in Fig. 11-2. This volume Ilr,(t) contains both /? and y phases, which are separated by the singular surface, A,, = A,,. The velocity of this surface is denoted by w. Returning to Eq. (1I.A-lo), we add h,[dp,/at + V (p,~,)] = 0 to the left-hand side in order to represent the liquid phase thermal energy equation as
.
a
t(Pph,)
+ v * (p,h,v,)
=
-v
*
q,
+ @,
(1I.B-9)
A similar operation on Eq. (1I.A-26) allows us to express the gas phase thermal energy equation as
"0
y - ptiase ;.i.\ (vapor plus inert)
"Y
FIG.11-2. Material volume containing a singular surface.
A THEORY OF DRYING IN POROUS MEDIA
135
At this point we note that our special forms of the thermal energy equation given by Eqs. (ILB-9) and (ILB-10) are consistent with the integral representat ion
which applies to any material volume regardless of whether it contains a singular surface at which (ph) and q suffer jump discontinuities. It should be clear that in the B phase we have Ph
=ap
(1I.B-12a) (1I.B-12b) (1I.B-12~)
A
(1I.B-13a)
=
Pphp
Q =qp Q
while in the y phase Ph = P
i=N
=qy Q,
+
=ay
1 piuixi
(1I.B-13b)
i= 1
(1I.B-13~)
Returning to Eq. (1I.B-9), we integrate over the volume V,(t) shown in Fig. 11-2 to obtain
(1I.B-14) We can use the general transport theorem [34, p. 88; 42, p. 3471 to express the first term in Eq. (1I.B-14) as
Here A, represents the material surface of V,(t) and A,, represents the singular surface moving with a velocity w which may be different from the mass average velocity vB. Substitution of Eq. (1I.B-15) into Eq. (1I.B-14) and applying the divergence theorem to the second and third terms leads to
(1I.B-16)
STEPHEN WHITAKER
136
We can repeat these steps with Eq. (1I.B-10) to obtain the comparable expression for the y phase:
We now wish to add Eqs. (ILB-16) and (1I.B-17) and note that [39, p. 22)
where Eqs. (1I.B-12)and (1I.B-13)apply to the left-hand side. Since d,,,(t) = A , + A?, we also have
-
q n dA
= AP
-
qa n, d A
+
JAY
[qy
+
c piuifii] -
i=N
nr d A
(1I.B-19)
i= 1
and the addition of Eqs. (1I.B-16)and (1I.B-17)leads to*
(1I.B-20) Comparing this result with Eq. (1I.B-11) leads to the jump condition at the P-y interface
B.C.7
- w) nflu + prh,(v, - w) * nus
p,&(vp
(1I.B-21) The jump condition for the mass average velocities is easily shown to be 36, Sec. 10.2; 39, p. 241
B.C.8
p&p
-
w) * npy + py(vy- w) nup = o
* Here we use the fact that A,,
= A,?,
(I1 .B-22)
A THEORY OF DRYING IN POROUS MEDIA
137
while the tangential component is assumed to be continuous
B.C.9
VO
*
LO,,= v,, * A,,
(1I.B-23)
Here a,, is any tangent vector to the surface denoted by A,?. The derivation of the species jump condition follows that given for Eq. (1I.B-22)and we list the result as
+ pp(vp - w) * n p y= 0,
B.C.10
pi(vi - w) - n y p
B.C.11
pi(vi - w ) - n r p = 0,
i
=
2 , 3, . . .
i = 1
(1I.B-24) (1I.B-25)
Here we have referred to restriction R.5 and have designated the vapor by i = 1 and the components of the inert gas as i = 2,3 . . . . The boundary conditions for the three phase interfaces are now complete, and we can turn our attention to the problem of deriving the volume averaged form of the transport equations. This will give us a set of equations that apply at every point in space, not just in the three separate phases. This type of approach to the analysis of transport phenomena in multiphase systems has been discussed at length by Slattery [39] and special cases have been treated by Whitaker [43-451, Slattery [46], Gray [47], and Bachmat [48].
EQUATIONS C. VOLUMEAVERAGED With every point in space we associate an averaging vohme V , such as that shown in Fig. 11-1. We have chosen a sphere as the averaging volume, but any shape will suffice provided the dimensions and the orientation are invariant [44]. There are three types of averages that are useful in the analysis of transport phenomena in porous media. The first of these is the spatial average of some function $ defined everywhere in space. This average is represented.by ($) and is defined by (1I.C-1) More often we are interested in the average of some quantity associated solely with a single phase. For example, we may be concerned with the average temperature of the solid phase, and we define the phase average of Tu as 1 (1I.C-2) (To> = T o dV
F sy
Since T,, is defined in the normal way in the c phase and is zero in all other phases Eq. (11.C-2)reduces to (1I.C-3)
STEPHEN WHITAKER
138
The phase average has the drawback that if T , is constant, the phase average is not equal to this constant value. A quantity that is more representative of the temperature of the solid is the intrinsic phase average which is given by (1I.C-4) Once again since T, is zero in phases other than the reduces to
D
phase, Eq. (1I.C-4) (1I.C-5)
We define the volume fractions for the three phases as E,
=
E/V",
€ D ( t ) = I$(t)/.tr,
ey(t)
(1I.C-6)
= V,(t)/Y-
Clearly the sum of these fractions is one, 8,
+
+ .$)
€&)
=
(1I.C-7)
1
and the phase average and intrinsic phase average are related by
4TU)"
=
(T,)
(1I.C-8)
The prime tool in the formulation of the volume averaged equations is the so-called averaging theorem [39, p. 194;431, which can be written as
We begin our analysis with the solid phase and note that p, is constant so so that Eq. (1I.A-2)takes the form
a
z(Prrh,) = - v * q ,
+ 00
(1I.C-10)
Integrating over V, and dividing by Y- gives the initial form of the volume averaged equation :
We can interchange differentiation and integration in the first term and refer to the definition of the phase average given by Eq. (1I.C-3) to write
a
t(P,h,>
=
-(V*q,)
+ (0,)
(1I.C-12)
A THEORY OF DRYING IN POROUS MEDIA
139
Use of the averaging theorem, Eq. (1I.C-9),allows us to express the average of V * q, as
(1I.C-13) so that Eq. (1I.C-12)takes the form
The area integrals on the right-hand side of Eq. (1I.C-14)represent the rate at which heat is transferred from the /3 and y phases to the 0 phase over the interfacial areas A,, and A a y . In order to illustrate the physical significance of the term V (q,) we need only apply the divergence theorem to express (V q,) as
-
-
(1I.C-15) Here A,, represents the area of entrances and exits for the cr phase at the bounding surface of the averaging volume. The outwardly directed unit normal nu and a portion of the surface A,, are illustrated in Fig. 11-1. Comparing Eqs. (1I.C-13)and (1I.C-15)provides the physical significance of the term V (q,): 1 (1I.C-16) v (4,) = 7J*ucq' * nu dA
-
Returning to the analysis of Eq. (II.C-14),we invoke Fourier's law q, = - k , VT,
(1I.C-17)
and use the averaging theorem to write*
(1I.C-18)
* Here we have imposed restriction R.4 and treated k, as a constant.
STEPHEN WHITAKER
140
If we now substitute Eq. (1I.C-18)into Eq. (1I.C-14)and represent h, in terms of the heat capacity and temperature as indicated by Eq. (II.A-4),we obtain the following form of the thermal energy equation for the cr phase
As mentioned earlier it is the intrinsic phase average temperature that is most closely related to the actual temperature of the cr phase, thus we make use of Eq. (1I.C-8) to obtain the final form of the thermal energy equation:
(1I.C-20) Here we see that our differential equation for (7,)"contains several terms for which we require constitutive equations. For example, one might represent one of the interphase flux terms as* 1
7
JAOP
q,
"a@
dA
=
h,,a,(
"
- (T@>9
(1I.C-21)
however, it is difficult to represent interphase heat fluxes in terms of film heat transfer coefficients such as h,, when convective transport is small compared to conduction. The area integrals involving T , in Eq. (1I.C-20) are usually incorporated into an efSectiue thermal conductivity, and there is some justification [43] for that type of representation. For the present we shall leave Eq. (1I.C-20)as is, and note only that determination of the (T,)" field is a very difficult task. In attacking the p phase equations we must remember that the P-y interface may be moving at a velocity different from either v, or v,, and that we have designated that velocity as w. We begin with the continuity equation aP,/at
* Here amprepresents A,,,/Y,
+v
(Ppva) = o
or the 0-8 surface area per unit volume.
(I I.c-22)
A THEORY OF DRYING IN POROUS MEDIA
141
and form the integral over %(t),and divide by V to obtain
Application of the general transport theorem allows us to write
(lI.C-24) while the averaging theorem provides
(11.C-25)
Remembering that v, = w = 0 over A,,, we can substitute Eqs. (I1.C-24) and (1I.C-25)into Eq. (1I.C-23)to obtain
Here we have replaced the total time derivative in Eq. (1I.C-24)with the partial derivative because the time derivative of the average is associated with a fixed point in space. This allowed us to write (lI.C-27) At this point we wish to impose the entirely reasonable restriction that the density in the liquid phase is a constant: R.ll
(I1.C-28)
pa = constant
Referring to Eqs. (1I.C-2)and (1I.C-3)we note that restriction R . l l allows us to write (PpV,) = P&*) (1I.C-29) (Pa> = E B P , (1I.C-30) Here (v,) represents the superficial velocity vector that is to be used to calculate volumetric flow rates. For example, the volumetric flow rate of the p phase past the surface area A is given by
Qa
=
sA
(vg)
*
n dA
(1I.C-31)
STEPHEN WHITAKER
142
Substitution of Eqs. (1I.C-29)and (1I.C-30) into Eq. (1J.C-26) and division by ps leads to the appropriate form of the continuity equation for the liquid
phase* (1I.C-32) It is important to note that this result is one of the key transport equations in the analysis of drying processes where one of the prime objectives is the determination of ep as a function of time and position. Up to this point we have made repeated and vague use of the term moisture content; however, it will now be appropriate to put forth some precise definitions of quantities that are often used as a measure of the moisture content. If the amount of moisture in the gas phase is much less than that in the liquid phase ~ , , ( p ~ )
completely dry product
(1I.C-34)
This definition is of limited value since a completely dry product is rarely available; however, Luikov [29, p. 2351 uses this definition extensively. Another measure of the moisture content is simply the phase density of the liquid phase which is given by (pp)O = ~~p~ = mass of liquid per unit volume
(p s ) =
(1I.C-35)
This measure of the moisture content is also referred to by Luikov [29, p. 2341 and designated by w.By far the most common measure of the moisture content would appear to be either the fractional liquid saturation ss = E&, + E,,) = fractional liquid saturation (1I.C-36) which varies between 0 and 1.0, or the percentage liquid saturation sfl x 100 =
~
(9
:
x 100 = percentage liquid saturation
(1I.C-37)
6,)
which varies from 0 to 100. When the amount of moisture in the gas phase is not negligible compared to that in the liquid phase, one can modify all
* Note that for the case of freeze drying Eq. (11.C-32)simplifies to
A THEORY OF DRYING IN POROUS MEDIA
143
these definitions except that given by Eq. (1I.C-33)to include the effect of the gas phase. These new definitions are given as follows: mass of moisture per mass of dry solid (1I.C-38) mass of moisture per unit volume (1I.C-39) fractional moisture saturation
(1I.C-40) percentage moisture saturation
(1I.C-41) For the case where ~ ~ ( <
a
+v
$Psh,)
(Pj4pVp) =
-v
’ 4s
+ @p
(I I .C-42)
To obtain the volume averaged form of this transport equation we integrate over i$(t), divide by “Y, and make use of the general transport theorem for the first term and the averaging theorem for‘the second and third terms to obtain*
=
-v.(qa)
-s
- 1
“Y
q B - n a y d A-
q,.n,,dA “Y
+ (@)I,
*I%
(1I.C-43) Our analysis of the right-hand side of Eq. (1I.C-43)would be identical to that given previously by Eqs. (II.C-14)-(II.C-20).Thus one need only exchange G and /3 on the right-hand side of Eq. (1I.C-20)to get the right-hand side of Eq. (1I.C-43) in terms of T , and the two interphase heat flux terms. The
* Note that d(p,h,)/dt
= a(pah,)/dt since ( p s h , ) is associated with a fixed point in space.
STEPHENWHITAKER
144
terms on the left-hand side require that we express the enthalpy as indicated in Eq. (1I.A-4).This leads to (I1.C-44) h, = h," + (Cp)p(T,- T,") Here h," is the enthalpy at the reference temperature T,". Directing our attention to the first two terms in Eq. (1I.C-43),we substitute Eq. (1I.C-44) and remember that p,, (cp),, and [h," - ( C , ) ~ T are ~ ~ constant ] in order to obtain *
a
-(P,h,) at
+ V ' (P,h,V,)
=
a
P,(Cp)p
;ii('p(Tp>9
+ P,(Cp), v * (Tpv,)
+ P&?"
-
(CP),T,"lv * (Vp) (I 1.C-45)
Here we have used Eq. (1I.C-8) for the fl phase to express the phase average temperature as ( T , ) = q?(T,)a (1I.C-46) Our objective at this point is to represent the term V * ( T,v,) in terms of (T,), and (v,). In order to accomplish this we need to represent the point functions T , and vp in terms of the average values and deviations from these average values. This approach was first discussed by the author [43] in an analysis of diffusion and dispersion in porous media; however, the definitive treatment is available in the work of Gray [47] who detected a serious flaw in the representation suggested by Whitaker [43]. Using Gray's suggested representations we express the point functions as
T , = ( T o ) @+ T p= T , = 0
-
in the /?phase in the u and z phases
+
v, = ( v , ) Vfl in the B phase v, = V, = 0 in the u and y phases
(1I.C-47a) (I1.C-47b) (1I.C-48a) (1I.C-48b)
Application of Eqs. (1I.C-47)and (I1.C-48) allows us to write? (1I.C-49) (T,V,> = (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.
$m
<< (@,>"
0,
p, and Y
and qm<<
($m)
in the w phase where w refers, to (1I.C-60)
This allows us to express the gas phase continuity equation as
w) * nyBd A
=0
(1I.C-61)
since Eq. (1I.C-60) will allow us to write
v - ((P,>')
>'
v - (P,Y"J
(1I.C-62)
Here we have made a definite assumption about the order of magnitude of functions and it is best to list carefully this assumption as: A.2.
In general, the product of deviations (i.e., terms marked by a tilde) will be considered negligible in comparison to the product of averages.
We must use this assumption with care for there are situations in which we may not wish to drop the product of deviations relative to the product of
STEPHENWHITAKER
148
averages. Application of assumption A.2 to Eq. (1I.C-59)seems quite appropriate; however, we should note that the term V (FpS,) was not discarded relative to (v,) * V( Tp)p in Eq. (1I.C-53),for in that case the dispersion of thermal energy, while small compared to convection, may be large compared to the conduction of thermal energy and we may wish to take this into account. In going from Eq. (1I.C-55)to Eq. (1I.C-59)we have again made use of the work of Gray [47] and the interested reader is urged to study Eqs. (10W16) of that reference. We now turn our attention to the species continuity equation for the gas phase which was given by Eq. (LA-13) as
-
aPi -
at
+V
(pivi) = 0,
i = 1,2,. . .
(1I.C-63)
and a result analogous to Eq. (1I.C-55)is readily obtained:
(1I.C-64) Following Eqs. (1I.A-16)and (II.A-18), we represent vi in terms of the mass average velocity and the diffusion velocity and make use of Fick’s law to put Eq. (1I.C-64)in the form
*+ v
(pivy) t
at
sAVl
1 7
pi(vi -
w)
- nypd~ (1I.C-65)
It seems quite reasonable to neglect variations of 9 within the averaging volume while retaining the possibility that this quantity may very significantly over distance that are large compared to the characteristic length of the averaging volume. Under these circumstanceswe can express Eq. (1I.C-65) as 1 a(pi) V (pivy) at vp Avo pi(vi - w) nus d A
+
a
+
-s
-
(1I.C-66) At this point we need to develop an equation for the intrinsic phase average =
Cy(Pi)’
(1I.C-67)
A THEORY OF DRYING IN POROUS MEDIA
149
however, the remaining terms require that we again resort to Gray’s [47] representation scheme and we begin using Eqs. (1I.C-56)in order to write Here we have used assumptions A.l and A.2 in order to write
Using Eq. (1I.C-68) and the result ((P,)’
$7)
=
(P,>’<$y>
allows us to expand the diffusion term on the right-hand side of Eq. (1I.C-66) in order to obtain
- ( p , > ’ ( v [ ”(PJ’ ( q ] ) (PrY
+ (P’
“&)) (1I.C-69)
where assumption A.2 has again been imposed. Use of the averaging theorem on the first term on the right-hand side of Eq. (1I.C-69)gives* ( P , V ( P i / P y ) ) = (Py>y[V((pi>/(Py>y)
+ ail
(1I.C-70)
where
Here we have indicated that the area integrals resulting from application of the averaging theorem are on the order of deviations; later we will show why this notation is chosen. Substitution of Eqs. (1I.C-67) and (1I.C-70) into Eq. (1I.C-66) leads to our final form of the gas phase species continuity equation: a 1 z(ey’) + v * ((Pi>’) + 7JAid Pi(vi - W) n y p d~ =
v
{(P,>y~[V((Pi>/(P,>y) +
ail - (PiVy))
(1I.C-72)
Here the convective transport term has been handled in a manner analogous to that given by Eqs. (II.C-55)-(II.C-59). * Here we have used ( p i / ( p J Y )
= (pi)/(p,)’.
STEPHEN WHITAKER
150
We begin our analysis of the vapor phase thermal energy equation with Eq. (1I.A-24) which took the form
This equation is identical in form to Eq. (1I.C-42)for the liquid phase, and the result analogous to Eq. (1J.C-43) can be written immediately:
(1I.C-74) Following our analysis of the P-phase thermal energy equation, we express the partial mass enthalpy as
xi =
+
(1I.C-75) - TyO) In expressing the partial mass enthalpy in terms of the pure component heat capacity (Qi instead of the partial mass heat capacity (Zp)i, we are invoking the restriction of a thermodynamically ideal gas phase which we need to list as:
R.12
hi”
(C&Ty
The gas phase is ideal in the thermodynamic sense.
We should remember that a previous restriction, R.3, required that all heat capacities be constant. We now focus our attention on the first two terms of Eq. (1I.C-74) and substitute Eq. (1I.C-75) to obtain
3
i=N
i=N
Our task here is a bit more difficult than it was in the comparable analysis for the phase since the pi cannot be treated as constants. In order to proceed, we again resort to representing the point functions in terms of average values and deviations from the average as we did earlier in Eqs. (1I.C-47)
A THEORY OF DRYING IN
POROUS
MEDIA
151
and (1I.C-48). In this case the representations take the form
T,
=
(T , ),
+ F,
I
T,
=
T,
=
pivi = (pivi)
in the y phase
in the 0 and /?phases
0
+ p?,
N
p.v. I 1 = p.v. 1 1 = 0
in the y phase in the 0 and /l phases
(1I.C-77a) (1I.C-77b) (1I.C-77~) (1I.C-77d)
along with Eqs. (1I.C-68).Application of these representatives yields
We can now substitute Eqs. (I1.C-78)into Eq. (1I.C-76),evaluate the derivatives of the various products, and rearrange to obtain
"F
at
+v
i=l
($
pivii;i)
i=N
The last two terms on the right-hand side represent a source term which can be neglected on the basis of assumption A.2 and a dispersion term which generally cannot be neglected relative to conductive transport. If we now multiply Eq. (1I.C-64) by [hio ( c J i ( ( T y ) ,- T,")] and sum over all N components, we use the result to express Eq. (1I.C-79)as
+
a i=N
i=N
(1I.C-80)
152
STEPHEN WHITAKER
Substitution of Eq. (1I.C-80) into Eq. (1I.C-74) and following the analysis given by Eqs. (1I.C-52)and (1I.C-53)leads to
(1I.C-81) Here we have expressed the right-hand side of Eq. (1I.C-74)by interchanging the subscripts p and y on the right-hand side of Eq. (1I.C-53). As was the case with the liquid phase thermal energy equation, there are a number of terms in Eq. (1I.C-81)that must be determined experimentally before it is of any use to us. Nevertheless, it is the correct form of the gas phase thermal energy equation for simultaneous heat and mass transfer in porous media and will prove useful in the construction of less complicated theoretical results. At this point in our theoretical development we have used the well-known point equations for heat and mass transfer (Eqs. (11-l),(11-2), and (11-5) to derive the appropriate volume averaged form of these transport equations for the solid, liquid, and gas phases in a rigid porous media. These are given by Eqs. (1I.C-20), (1I.C-32), (II.C-53), (II.C-61), (1I.C-72), and (1I.C-81). Nothing has been said about the determination of (va) and (v,) or the thermodynamic relations that must be considered, and it is clear that the theory in its present form is a very complicated one. Aside from the difficulty of solving several partial differential equations simultaneously, we are also confronted with several unknown terms in the governing equations. The interphase heat flux terms that appear in each of the three thermal energy equations must be accounted for in terms of heat transfer coefficients, and it is not at all clear how one would perform experiments in a three-phase system which could be used to determine the three interphase heat transfer coefficients. The conductive heat flux terms and the diffusive mass flux terms also contain unknown area integrals that must be accounted for by appro-
A THEORY OF DRYING IN POROUS MEDIA
153
priate models and experimental determination of the model parameters. While the theoretical formulation up to this point appears to be intolerably difficult, it does represent a solid foundation upon which more attractive theories can be constructed. In Section 111 of this article we shall simplify our thermal energy equations by forming the totul thermal energy equation, and in Sections IV and V we shall attack the problem of determining the mass average velocities in the gas and liquid phases. 111. Energy Transport in a Drying Process
While there are many processes in which the gas temperature ( T , ) ) ' is different from either the solid ( T , ) , or liquid temperature (Ta)P most drying processes are characterized by relatively low convective transport rates, and under these circumstances one is encouraged to assume that conductive transport is sufficient to eliminate significant temperature differences between the separate phases. Thus the solid liquid-gas system is considered to be in "local equilibrium." This assumption is sometimes stated explicitly [30; 31; 39, p. 4041 in studies of the drying process, but as often as not it is an unstated restriction of the analysis. The circumstances under which this assumption is valid certainly need to be explored; however, at this point we shall progress along the traditional lines and list our third limiting assumption as:
A.3
The solid-liquid-gas system is assumed to be in local equilibrium.
A logical consequence of this assumption is that the intrinsic phase average temperatures are equal: (T,)"
=
=
(TI,)'
(111-1)
Since the spatial average temperature is defined* as
(T)
<,(T,)"
+ ~p(Tp)' + E ~ ( T , : ) '
(111-2)
Eq. (111-1) requires thatt
(T,)"
=
= (T')? = ( T )
(111-3)
* See Eqs. (1I.C-1)-(II.C-8). A somewhat more attractive approach might be to define deviations from the spatial average temperature by equations of the type (T,)" = (T) + Tc,and then replacing assumption A.3 by the statement that all terms involving the deviations are negligible. Thus one would state that p O ( c p ) J ? ( T)/?r) >> p s ( c p )JJTJ?I), etc.
STEPHEN WHITAKER
154
In the following section we shall impose assumption A.3 on the three thermal energy equations in order to develop a total thermal energy equation.
A. TOTALTHERMAL ENERGYEQUATION We begin our analysis of the transport of thermal energy in a porous medium during drying by adding Eqs. (1I.C-20), (1I.C-53), and (1I.C-81) and imposing the assumption of "local equilibrium" as indicated by Eq. (111-3). This leads directly to the total thermal energy equation given by
V[(~,E,
+ k , ~ , + k,~,)(T)1 + (k, - k , )v-1s
A@
Tun,, d A
In this result we have made liberal use of the fact that n,, = -nay
n,, = -nys,
nap = -np,,
(1II.A-2)
in addition to
T,
T,
over
A,,
=
A,,
(1II.A-3a)
T, = T,
over
A,,
=
A,,
(1II.A-3b)
T , = T,
over
A,, = A,,
=
(1II.A-3~)
A THEORY OF DRYING IN POROUS MEDIA
155
These conditions were given in Section I1 as Eqs. (1I.B-l), (II.B-4), (II.B-5), and (1I.B-8). The source term in Eq. (1II.A-1) is defined as
and the combined liquid and gas phase dispersion has been represented in terms of V * (t) where V
- (6)
-
= pp(c,), V ( F p T p )
+V-
i=N
(1II.A-5)
( ~ , ) i ( p ~ ~ ~ ~ ) i= I
We should also note that restriction R.4 has been used in rearranging the conductive transport term. While Eq. (1II.A-1) hardly appears to be an improvement over Eqs. (1I.C-20), (1I.C-53), and (II.C-81), it can be simplified to the point where it becomes quite tractable. In the following paragraphs we shall examine separate portions of the total thermal energy equation with the intent of developing a form that is useful in analysis of drying problems. The spatial average density is a measurable and well-defined quantity given by* i=N
(P> =
~
u
(
~
+ u )~ ~f i
+
( ~ p Y cy
C
i=l
(Pi>’
(1II.A-6)
and it is appropriate to use ( p ) to define a mass fraction weighted average heat capacity byt
The first term on the left-hand side of Eq. (1II.A-1) can now be expressed as
(1II.A-8) Turning our attention to the interphase flux terms on the right-hand side of Eq. (1II.A-l),we note that two of them are identically zero by Eqs. (1I.B-3) and (1I.B-7) of Section 11. In order to attack the third term, involving qP - qy, we need to rewrite Eq. (1I.B-21) in the form$ i=N
~ p h p ( ~-p W) *
+ i1 pik(vi =
- W) n y p = -(qp
- 97)
*
(1II.A-9)
1
* Note that ( p ) is the total mass per unit volume. Remember that (pa)“ = pa and (pp)8 = pp since the densities of the solid and liquid phase are taken to be constant. See Eqs. (II.A-24)-(II.A-26) and (ILB-10). +
STEPHEN WHITAKER
156
Using this result and Eqs. (1I.B-3) and (1I.B-7) we have
1
i=N
(1II.A-10) Substitution of Eqs. (1II.A-10) and (1II.A-8) into (1II.A-1) and rearranging leads to a slightly more attractive form of the total thermal energy equation:
1
-
Y fi,, {Pe[hr - ( c p > p ~ p ' a l ( v p- w) i=N
+ i =C1 p i [ h =
v-
-
(cp)iFy](vi
{
-
I
npy
W) * n y ~ d A
+ kpep + k p , ~ ~ )+] (k,
w,€,
+ (k,
-
-
1 k y ) Y J A O Tpnpy y dA
- v (5)
1
- kp)TJs8 TunoPd~ 1
+ ( k y - k " ' ~ J A y oTynrad A ]
+ (a)
(1II.A-11)
Returning to Eqs. (1I.C-44)and (I1.C-75) of Section 11, we express the enthalpies as h, = h," + (Cp)&? - TpO) (1II.A- 12a)
?ii= h i "
+ (cp)2(T), - TY0)
(1II.A- 12b)
so that the integrand on the left-hand side of Eq. (1II.A-11) becomes i=N
A THEORY OF DRYING IN POROUS MEDIA
157
From Eqs. (1I.B-24) and (1I.B-25) of Section I1 we have
pi(vi - w) - nvp = 0,
- w) * nyb + pfi(vfi- w) nSr = 0
(1II.A-14a) i = 2, 3,
.. .
(1II.A-14b)
so that the right-hand side of Eq. (1II.A-13) can be simplified to* i=N
~ f i [ h f i- (cp)fiTfiI(vfi - W) ' nfiy = [hpO - h,"
+
C
.
pipi - ( c p ) i T y I ( v i - W) nyfi
i= 1
+ (cp)fi((T>- Tp")- ( c p ) A < n- T,")IPfi(Vfi - w) '
"By
(1II.A-15) We now identify the enthalpy of vaporization per unit mass at the temperature ( T ) by (1II.A-16) Ah,,, = [h," - hp" + ( c p ) 1 ( ( T )- T") - ( c ~ ) ~ ( (-TT) o ) ] Here we have at last specified the reference temperatures Tyoand Tpoas the temperature at the normal boiling point To. Substitution of Eqs. (1II.A-15) and (1II.A-16) leads to further simplification of the total thermal energy equation :
In writing Eq. (1II.A-17) we have identified the mass rate of vaporization per unit volume as
and (riz) will be a positive quantity for a drying process. We can now turn our attention to the conductive transport term in Eq. (1II.A-17)and attempt to arrange it in a somewhat simpler form. To do so we must make use of an important theorem developed by Gray [47] which
* Here we have used ( T 8 ) @= ( T,),
=
(T).
STEPHENWHITAKER
158
gives (for example)*
Note that Eq. (1II.A-19)is valid only when the point functions are represented as suggested by Gray. We can use this result to eliminate the temperatures T,, T,, and T, in Eq. (1II.A-17)and replace them with T,, Tfl,and Fy.The algebraic effort required to rearrange the conductive transport terms is considerable, and we shall note only that Eq. (1II.A-19) is used for the 0, p, and y phases to eventually reduce Eq. (IILA-17)to the form
-
v - (0 + (@>
(1II.A-20)
At this point we have made considerable progress for Eq. (1II.A-20)is beginning to look like something that we could use to predict the temperature field during a drying process. Only the conductive transport terms appear in a form that does not permit a comparison between theory and experiment. In the next section we shall try to remedy this difficulty.
B. THEEFFECTIVE THERMAL CONDUCTIVITY In the previous section we found that the total thermal energy equation contained terms of the type
where 7, is defined as
Ffl = To - (T S ) , in the jphase Fp= 0 in the ts and y phases
(1II.B-1a)
(1II.B-1b)
In order to construct a complete theory for the transport of thermal energy in porous media we need a representation of the terms involving fm,Tfl, and 7ythat is amenable to experimental interpretation. Slattery [39, p. 4051
* See Eq. (26) of Gray [47]. The original result was given for a two-phase system, but it is easily extended to a three-phase system.
A THEORY OF DRYING IN POROUS MEDIA
159
has suggested that empirical correlations for these terms should satisfy the principle of material frame indifference [42, p. 7001 and indicated that V( T,), is a likely correlating variable for terms involving T,. In the following paragraphs we wish to put forth some ideas about the functional dependence of ?, which strengthen earlier suggestions* that ?, is strongly dependent on V( T,Y. To obtain the governing differential equation for T, we substitute T , = (T,), + ?into , Eq. (1I.A-11) in order to obtain
(1II.B-2) Since ( T p ) p= (T) we can think of Eq. (1II.A-20)as the governing differential equation for ( T,),, thus leaving Eq. (1II.B-2)as the governing differential equation for T,. The functional dependence of T , can now be deduced from Eq. (III.B-2), i.e., T , depends on: (1) the independent variables: x, y, z, and t ; (2) the parameters appearing in the governing differential equation : P,, (c), v,, k,, a,, a(T,)P/at, v(T,),, and v [ ~ < ~ , > ~ 1 ; (3) any parameters that appear in the boundary conditions.
-
It is not at all clear what types of boundary conditions one would impose on T,; however, it is clear that F p depends on the usual dimensionlessvariables (the Reynolds and Prandtl numbers) in addition to being a function of ( T p ) @This . means that 7, in turn depends on all the parameters in Eq. (1II.A-20). On the basis of the form of Eq. (1II.B-2) we put forth the conjecture that the functional dependence of can be expressed as? ~ . 4
T , = %(a(
T,)P/at, V( T,),)
(1II.B-3)
where the dependence on the other variables, p,, (c), etc, is understood and the dependence on V [V( T,),] can be neglected. From the definition of ( T,), we know that ( T,),
=
T,
when T, is independent of the spatial variables (1II.B-4)
* See Appendix A of Whitaker [43]. + One could be more general and state only that pendence on the other variables understood.
4;8 is a functional of
with the de-
160
STEPHENWHITAKER
and it follows that
-
when (To), is independent of the spatial variables
T, = 0
(1II.B-5)
In addition to our first conjecture about the functional dependence of'?, given by Eq. (1II.B-3)we now make a second conjecture on the basis of Eq. (1II.B-5) which is stated as
7, = 0 when V(T,)S
AS
(1II.B-6)
=0
While we know that Eq. (1II.B-5) is true by definition, and it follows that ( T,)P is independent of the spatial variables when V( T,), = 0 everywhere, the restriction given by Eq. (II1.B-6) cannot be derived from Eq. (1II.B-5). In fact, if we impose V( T P ) , = 0 on Eq. (III.B-2), the equation for FPdiffers only to the extent that v, * V( To)@is eliminated. Certainly 7, = 0 is a possible solution to that equation, but the mathematical evidence does not strongly suggest that this is the case. Nevertheless we are highly motivated by intuition to extend the global constraint of Eq. (1II.B-5) to the local constraint of Eq. (1IJ.B-6). If we now expand Eq. (1II.B-3) in a Taylor series in V( T,>, about the point V(Tp)@= 0, we have =
B,
+ C,
*
V(Tp),
+ (D,
*
V( To),) * V(T,)P
+
*
..
(I1I. B-7)
Here we must remember that the coefficients in this expansion are all functions of d(T,)a/at in addition to all the other parameters implied in Eq. (1II.B-3). From Eq. (1II.B-6) we require that B, = 0, and if we limit our development to the first term in the Taylor series expansion, we obtain
T, = C,
V( T,),
(1II.B-8)
and it follows that
(1II.B-9) In going from Eq. (1II.B-7) to Eq. (1II.B-8) we have imposed the important limitation that 7,is a linear function of V(T,)@ and it would be best to list this as our sixth assumption:
A.6
T,is assumed to be a linear function of V( T,)P.
It seems reasonable that variations in V( T S ) , will be small compared to the spatial variations of Po, and Eq. (1II.B-9) can be expressed as (1II.B-10)
A THEORY OF DRYING IN POROUS MEDIA
161
Here we identify the second order tensor K, as K,
=-s
Ap:
n C @d A
(1II.B-11)
Equation (1II.B-10) can be immediately extended to the comparable terms involving Tuand Tyand we write
(1II.B-12) (1II.B-13) Substitution of these results into Eq. (1II.A-20) and remembering that ( T o ) @= (T,)" = (T,,)Y = ( T ) gives us
Here Keff represents the effective thermal conductivity tensor and is given by Keff =
+ ~ p k ,+ e y k y ) U + (k, - k,WU + (k, - kYWp + (k, - kUWy
(E&,
(1II.B- 15)
-
If one interprets the dispersion, indicated by V (t), in terms of a diffusion model [43] one writes (1II.B-16) (6) = -K, * V ( T ) and Eq. (1II.B-14) is simplified to
=
-
V * (KZff V ( T ) )
+ (0)
(111.B- 17)
where KZff = Keff
+ KD
(I I I. B- 1 8)
This form of the total thermal energy equation is quite appealing for all of our difficulties have been incorporated into a single second order tensor KTff which must be determined experimentally or on the basis of further theoretical developments. The total thermal energy equation given by Eq. (111.B-17) is similar in form to that suggested by Berger and Pei [31]; however, they were concerned
STEPHENWHITAKER
162
with a case in which heat conduction took place only in the solid phase. In addition they neglected any convective transport of thermal energy. These conditions can be illustrated with Eq. (1II.B-17) by dropping the convective transport term on the left-hand side of Eq. (1II.B-17) and replacing Kzff with Keffto obtain
Here we have also dropped the energy source term (a) in keeping with the work of Berger and Pei [31]. The case for which heat is conducted only through the solid phase can be obtained by setting k , and k , equal to zero so that Eq. (1II.B-15) leads to Keff = E,k,U
+ k,(K,
k,, k y
- K,),
+
0
(1II.B-20)
If the system is taken to be isotropic we can write Keff =
[ d u
+ ku(Ku - K y ) l U
(1II.B-21)
and Eq. (1II.B-19) takes the form (P>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>’/
We express this assumption as
-
Ri = DO * V ( ( p i ) ’ / ( p , ) ” )
(1V.A-7)
and write Eq. (1V.A-5) as, w) * nypd A
(1V.A-8) The total effective diffusivity D2if is given by
D:if
=
E,
9U
+ D$’ + DV
(1V.A-9)
where we expect that D‘& and D(2 will be strong functions of ey. Our choice of notation here indicates that we expect the total effective diffusivity to be different for the two different species in the gas phase. The forms of Eqs. (IV-6) and (IV-7) which are comparable to Eq. (1V.A-8)are given by
(1V.A-10) and
A THEORYOF DRYINGIN POROUSMEDIA
169
Once again we have swept all of our difficulties into one parameter, the total effective diffusivity, and experimental determination of this parameter, even for an isotropic media, will be a very difficult matter. Assuming that it can be done we are left with only the problem of determining (v,,) in order to complete our analysis of transport in the gas phase.
B.
CONVECTIVE
TRANSPORT IN THE GASPHASE
There are two possible approaches to the problem of determining the gas phase velocity (v,). The first approach is to preempt the laws of mechanics and use the continuity equation or the diffusion equation to determine the velocity. This requires that some constraint be placed on either the pressure field or the velocity field. Laminar boundary layer theory is a classic example of this type of analysis. There one determines one component of the velocity from an equation of motion, and then constrains the pressure to be constant so that the continuity equation can be used to determine the other component of the velocity vector. The Stefan diffusion tube problem [33, p. 522; 551 is another example in which the laws of mechanics are ignored and a suitable constraint placed on the species velocity (the film is “stagnant” or v2 = 0) in order to obtain a determinate set of equations that can be used to predict the mass average velocity. It is important to note that this kind of approach can be used to determine only one component of the velocity since the continuity equation is a scalar equation. In drying processes the pressure in the gas phase is likely to be essentially constant; thus the use of the continuity equation plus a suitable constraint is an attractive option. The obvious drawback is that such an approach is restricted to one-dimensional processes; however, it is more important to note that a suitable constraint on the velocity field is lacking. In boundary layer analysis the component of the velocity to be determined by the continuity equation is specified along one coordinate axis, while in the Stefan diffusion tube process the entire velocity field of one of the species is specified.* Neither of these constraints carries over directly to the drying problem, and at the present time there appears to be no obvious way in which this kind of analysis can be used to advantage in the determinanation of gas phase convective transport in the drying process. Another approach that allows one to ignore the laws of mechanics is to simply assume that all the convective transport (not just the dispersive part) can be modeled by a diffusive mechanism. This is the approach used by Berger and Pei [31] and by Gupta and Churchill [30] among others. The possibility that the gas phase velocity can be obtained without recourse to * Recent theoretical analysis by Meyer and Kostin [56]indicates that this approach is incorrect but leads to satisfactory results for the total flux.
STEPHEN WHITAKER
170
the laws of mechanics is an attractive one; however, there appears to be no reasonable way to accomplish this and we are forced into an application of the momentum equation. This is unfortunate for it gives rise to a coupling between the equations of motion and the diffusion equation, thus further complicating an already complicated analysis. In attacking the problem of determining the gas phase velocity field we shall follow the analysis developed previously by the author [44] for the treatment of single phase flow in porous media.* In so doing we shall make one plausible but very crucial assumption regarding the nature of the two phase flow process. We state this simply as: A.9
The gas phase is continuous.
This seems like a reasonable assumption for the drying process, but one must keep in mind that neither the development presented in this section nor that presented in the subsequent section is valid if this constraint on the topology of the gas phase is not satisfied. We begin our analysis with the equations of motion for the gas phase: (1V.B-1) (1V.B-2)
T, = T,+
neglect inertial effects, and require that the fluid be Newtonian with constant coefficients of viscosity so that Eqs. (1V.B-1)and (1V.B-2) reduce to
-by+ Pyg + Py v2v, + (ICY
-
(1V.B-3) V(V v,) Here p, is the coefficient of shear viscosity and IC,is the coefficient of bulk viscosity. Even though V * vy is not zero the last term in Eq. (1V.B-3) can be shown to be negligible, and we obtain P,@V,/W
=
P,(av,/w
=
- +PJ
-by+ Pyg + Py v2v,
(1V.B-4)
The characteristic time for flow in porous media is on the order of d2/vy where d is a characteristic pore diameter and v y is the kinematic viscosity. For d = 0.1 cm (a very large pore diameter) and vy = lo-' cm2/sec (the kinematic viscosity of air) the characteristic time associated with Eq. (1V.B-4) is on the order of 1 sec. Characteristic times for drying processes are generally on the order of minutes or hours; thus it is reasonable to treat the flow process as quasi-steady, and Eq. (1V.B-4) reduces tot V P , = py v 2 v y
(1V.B-5)
* The development has its origin in the unpublished work of Brenner [57] concerning the theory of transport processes in spatially periodic porous media. + Here we treat the flow as incompressible even though variations in p y are expected.
A THEORY OF DRYING IN POROUS MEDIA
171
Here the new pressure P , is given by (1V.B-6) p , = Py - P o + P,4 where p o is any reference pressure and 4 is the gravitational potential function given by g = -v4 (IV.B-7) We now focus our attention on an arbitrary curve lying entirely within the gas phase such as the curve shown in Fig. IV-1. It is important to remember that the illustrated curve can pass through every point in the gas phase without ever crossing a phase boundary. This will make our analysis quite simple and much different from the analysis of the liquid phase given in the next section. The arc length along the curve is Y and the unit tangent vector is represented by S(s). Forming the scalar product of Lwith Eq. (1V.B-5) gives
L. VP,
=
pyL
(VZV,)
(1V.B-8)
dP,/ds
=
p,L (V2v,)
(1V.B-9)
which can be expressed as
-
Arbitrary curve in the gas phase
tion of Are
FIG.IV-1.Gas phase flow for a two fluid system in a porous medium.
172
STEPHEN WHITAKER
Arguments have been given elsewhere [44] that the volume averaged velocity (v,) can be mapped into the point velocity v, by the linear transformation
v,
=
M,
- 0,)
(1V.B-10)
where the spatial variations of {v,) are negligible compared to those of M,. Under these circumstances we can substitute Eq. (1V.B-10) into Eq. (1V.B-9) to obtain (1V.B-11) dP,/ds = pyh (V2M1,) (v,)
-
We now integrate this result along the arbitrary curve shown in Fig. IV-1 from some reference point s = 0 to an arbitrary point s(r). This yields P,(r) = P,(O)
+ p, {j:I:r) I (V2M,)
*
1
(v,) dr]
(1V.B-12)
We again invoke the argument that spatial variations in (v,) are very small relative to those in M, or V'M, so that Eq. (1V.B-12) can be expressed as
We now identify the term in braces as the vector -myand note that the pressure in the gas phase is determined to within an arbitrary constant by the expression* (1V.B-14) P , = - pYm Y * ( vY )
* In a private communication, Professor W.G. Gray of Princeton University has presented some arguments against the carrying of arbitrary constants with terms to be averaged such as P,. The arbitrariness can be eliminated at this point by recognizing that Eq. (1V.B-14) must hold for the hydrostatic case for which it reduces to p,
=
Py
-
Po
+ P,$
=0
The gravitational potential function can be expressed as
4
= -g.r
+C
where C is an arbitrary constant. If we set C equal to zero we have py - p o - g . r = 0
Under these circumstances the reference pressure, po. is identified as the gas phase pressure at r = 0, and Pi in Eq. (1V.B-13)is given absolutely as
P,
= p,
- po - g a r
An alternate approach is to work directly with Eq. (1V.B-13) to arrive at
(v,,>
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)
= -
where (r) is the position vector locating the centroid of the gas phase volume VJt). Provided that gradients of cY are not outrageously large, (r) will coincide with the centroid of the averaging volume 9‘“ thus we think of (r) as locating the point at which all our volume averaged functions are defined. Substitution of Eq. (1V.B-24) into Eq. (1V.B-21) and carrying out the gradient operation yields*
Y.
(v,) =
-- Ky
.,[V(P’
P’
+ [(P’
-
Po)’
-
-
Po)’
- P’
-
P , O > 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, *
(V. A-4)
176
STEPHEN WHITAKER
curve
in
region
FIG.V-l . Liquid phase flow for a two fluid system in porous media.
and write Eq. (V.A-3) in a form analogous to Eq. (1V.B-11) dPslds
=
psA*(V2Mp)*(vp)
(V.A-5)
We can use Eqs. (V.A-5) and (1V.B-11) to determine the liquid phase pressure Ps at some arbitrary point on the curve s(r), provided we take into account the jump discontinuity that exists at the phase interfaces. We neglect the effects of surface viscosity [SS] surface elasticity [59] and convective momentum transport to express the jump discontinuity [39, p. 56) in the pressure as
(V.A-6) Here as,,represents the interfacial tension of the P-y interface (the “surface tension”) and rl and r2 are the principle radii of curvature at the point where the curve passes through the interface. Referring to Eqs. (1V.B-6) and (V.A-2),we see that Eq.(V.A-6) requires the jump in P to be given by
(V.A-7)
A THEORYOF DRYING IN POROUS MEDIA
177
Here A p is given by py - pa as we proceed along the curve from the liquid phase into the gas phase, and pa - pe as we proceed from the gas phase into the liquid phase. Integration of Eqs. (1V.B-11) and (V.A-5) and application of Eq. (V.A-7) leads to
(V.A-8) which can be expressed as
(V.A-9) where sN = s(r) and s N (V.A-7) we have
=
s(rN-l). In terms of Eqs. (1V.B-1 l), (V.A-5), and
(V.A-10) We again invoke the argument that spatial variations in (vs) and (v,) are negligible compared to those of V2M, and V2M, so that Eq. (111.3-10) takes the form
(V.A-11)
178
STEPHEN WHITAKER
We now identify the term in braces involving M, as the vector our expression for Ps takes the form
- m,
so that
Pfl(r) = Pp(0) - Pflmfl-(Vg)
i=N-1
(V.A-12) i= 1
Referring to Eq. (1V.B-13) we see that to within an arbitrary constant we can express the gas phase pressure at a position rN- as
Here we should note that the position vector r N - locates the point at which the arbitrary curve crosses the /3-y interface as we proceed from the gas phase into the liquid phase. It is important to remember that as long as r locates a point within a continuous subregion of the liquid phase, rN- is a constant and so is the gas phase pressure given by Eq. (V.A-13). This does not mean that this term can be treated as an arbitrary constant since within the averaging volume V the gas phase pressure will be evaluated at many different points in space. This will become clear later in our development. Returning to Eq. (V.A-12) we represent the capillary pressure pc to within an arbitrary constant as,*
(V.A-14) and represent the gravitational potential energy term to within an arbitrary constant as i=N- 1
To within an arbitrary constant we now express the liquid phase pressure as
* Here we follow the convention of Bear [60]who definesthe capillary pressure as the pressure in the nonwetting phase minus the pressure in the wetting phase.
A THEORY OF DRYING IN POROUS MEDIA
179
It is worthwhile to keep in mind that if the liquid phase is continuous, the analysis presented in Section 1V.B also holds for the fl phase and the pressure P, can be specified to within an arbitrary constant by the expression Pp(r) = -porn,
*
for a continuous liquid phase
(v,)
(V.A-18)
From this result we can deduce a constraint on some of the terms in Eq. (V.A-17) which takes the form Py(rN-I ) - pc(rN-
+ A p N - 14N-= constant, for a continuous liquid phase (V.A- 19)
Here we should note that rN-l represents any position vector locating a point on the B-7 interface. We have previously designated the reference pressure p o as the intrinsic phase average pressure in the gas phase at (r) = 0, and that the gravitation potential function is also zero at the origin* so that we require P,,=+=O, r=O (V.A-20) This allows us to specify the constant in Eq. (V.A-19) as the negative of the capillary pressure at the origin, and we write this as P,
=
(P,)~,
at r
=
(V.A-21)
0
We now rewrite Eq. (V.A-19) for a continuous liquid phase in the form Py(rN-A- Pc(rN-J+ A P N - I ~ N -=I -(P,)o
(V.A-22)
Clearly the appropriate form of Eq. (V.A-18) for a continuous liquid phase is now given by (V.A-23) P&) + (Pdo = - p p / I * (v,> 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)
- Pp(f>
’ 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
vEp
- k
V(T)
(~.~-17) (V.B-18)
Here we should note that if the gas and liquid pressure distributions are hydrostatic, it seems quite likely that (p,) can be determined from a curve such as that shown in Fig. V-2. Under these circumstances for a quasi-static process the coefficient k, is given by
(V.B-19)
188
STEPHEN WHITAKER
Substitution of Eq. (V.B-18) into Eq. (V.B-13) leads to a useable expression for the liquid phase velocity: (V,)
= -- KO
PLlS
*
E , k 6 VEP
i
+
Epk(T)
V(T)
- EP(PP
1
- Py)g (V.B-20)
We can now substitute this result into Eq. (1I.C-32) and make use of Eq. (1II.A-18) in order to obtain
We must keep in mind that Eq. V.B-21 is valid only when the liquid phase is continuous, and our next objective is to consider the discontinuous or pendular state. Returning to Eq. (V.A-30), we substitute Eq. (V.A-46) to obtain 1 (VP)
= --
P,
K,
*
V{E,[(P,
- Po>,
+ P p ( ( 4 > 8 - $1 - (Ti,
-
+ El}
Po)
(V.B-22) Carrying out the gradient operation and rearranging the terms leads us to* (vp> = -- K,
PLlS
+ [
a
i
r,[V(Pp)P - Ppg]
- POY
+ Pp(4>p
-
- E/J
vm,
- Po)
+ Po$
-
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 <+l,
M-1
and
- P,kl
5-0,
M+oo
Substitution of Eq. (V.B-30)into Eq. (V.B-27)leads to a very attractive representation for the liquid phase velocity
(V.B-31)
190
STEPHEN WHITAKER
The introduction of the function 5 must stand out as a crucial assumption in our development and we appropriately list it as the twelfth assumption: A.12
The forces acting on the liquid phase can be uniquely represented in terms of ( p , ) + (pa - p , ) ( 4 ) , and the function 5.
At this point our development follows that given previously by Eqs. (V.B-13)(V.B-20) so that the liquid phase velocity is given for the general case by
(V.B-32) Substitution of this expression for the velocity into the volume averaged continuity equation given by Eq. (1I.C-32) leads us to
a 9at
(V.B-33) From a practical point of view, this is a very attractive result since the analysis of drying processes is centered around the problem of determining the volume fraction of the liquid phase E, or the saturation s, = ~,/(1- E.,). The result given by Eq. (V.B-33) provides a transport equation for either E, or s, and thus greatly simplifies the analysis of drying. There are several special cases of Eq. (V.B-33) which should be discussed. 1. Quasi-Steady Transport in the Liquid Phase
For the quasi-steady case we have (&,/at)
+
0
and
(m)
+0
and Eq. (V.B-33) reduces to (V.B-34) k, V', + k(*) V ( T ) - (P, - P& = 0 If the flow owing to the temperature gradient is negligible, this further simplifies to (V.B-35) k, Vq? - (Pa - P& = 0 This result, along with the first of Eqs. (V.B-17), can be used to calculate the moisture distribution shown in Figs. 1-2. One must make use of experimental values of the capillary pressure as a function of E,, and one cannot use Eq. (V.B-35) to predict the course of a drying process. Nevertheless, it is an interesting special case of Eq. (V.B-33) which has been verified experimentally by Ceaglske and Hougen [18] among others.
A THEORY OF DRYINGIN POROUS MEDIA
191
2. Negligible Gravitational EfSect on the Liquid Motion In the drying of porous media with very small pores, one can neglect the effect of gravity and Eq. (V.B-33) simplifies to at
*
+ k(T) V(T)] + ( m ) / p ,
[k, Vr,
=
0 (V.B-36)
While somewhat simpler than Eq. (V.B-33), this result still contains the hard to evaluate tensor K,. 3. Negligible Gravitational Efect and un Isotropic System For an isotropic system we require that K, = U K ,
(V.B-37)
where U is the unit tensor. Under these circumstances Eq. (V.B-36) can be expressed as
where K , and K ( T ) are defined by K, K(T)
qltK,k,/P,
(V.B-39)
= E pe KB k
(V.B-40)
=
A result similar to Eq. (V.B-38) has been suggested* by Luikov [29] and used for extensive calculations by Husain et al. [32]; however, the equation suggested by Luikov contains the unlikely restriction that K , and K ( T ) are constant and does not contain the term (k). Since the continuity equation for the liquid phase must contain (m), it is difficult to see how any transport equation for E , could be constructed omitting this term. 4. Negligible Flow Owing to Temperature Gradients If the effect of temperature on surface tension opy is unimportant one can drop the term K ( T ) V( T ) in Eq. (V.B-38) to obtain (3E
at
=
V * ( K , Vrp) - ( k ) / p p
(V.B-41)
This result is essentially identical to the transport equation proposed by Berger and Pei [31] and given by their Eq. (4). The correspondence can be
* See Eqs. (6.4) and (6.50).
STEPHEN WHITAKER
192
seen by requiring that K , be a constant denoted by K, = K L P L
and that E, = u in the nomenclature of Berger and Pei. Since E ~ (, , K,, and k, are all strong functions of E, it seems unlikely that K, will be a constant in any real drying process.
VI. Solution of the Drying Problem In this section we summarize the previously derived transport equations, and tabulate the restrictions and assumptions that were made in the course of the theoretical development. Total thermal energy equation (1II.B-29) (P)Cp =
XT) 7 + [P,(Cp),(V,)
V * (Kzff * V(T))
+ (P,)y(Cp)y(v,)I
*
+ (0)
V(T)
+ Ah"apW) (VI-1)
Liquid phase equation of motion (V.B-32)
Liquid phase continuity equation (1I.C-32)
a,at + v - (v,) + (m)/p,
=
-
0
(VI-3)
Gus phuse equation of motion (1V.B-29) (VI-4) Gas phase continuity equation (IV-5)
The volume construint (1I.C-7) €g
+ .,(t) + .,(t)
=
1
(VI-7)
A THEORYOF DRYINGIN POROUSMEDIA
193
Thermodynamic relations (Section 1II.C) (Pl)'
=
(VI-8)
(P1)YR,(n
(VI-9) (VI-10) (VI-11)
In the equations of motion given by Eqs. (VI-2) and (VI-4) we must remember that it was assumed that the flow was incompressible. This means that in these equations we can replace py with (p,)' without incurring any significant error. The restrictions and assumptions that were imposed on the theoretical development leading to Eqs. (VI-l)-(V1-8) are listed in Table I. TABLE I RESTRICTIONS AND ASSUMPTIONS R. 1 R.2 R.3 R.4 R.5 R.6 R.1 R.8 R.9 R.10 R.ll R.12
The solid phase is a rigid matrix tixed in an inertial frame. The enthalpy of the u, p, and y phases is independent of pressure. and y phases is a linear function of the temperature The enthalpy in the u, /I, The thermal conductivities of the CT, p, and y phases are constant. The liquid phase contains only a single component. Compressional work and viscous dissipation are negligible in the liquid phase. There is no chemical reaction in the y phase. Compressional work and viscous dissipation are negligible in the gas phase. Diffusional body force work and kinetic energy are negligible in the gas phase. Interfacial energy for the u-p, p-y, and y-u interfaces is negligible in the thermal sense. The density of the liquid phase is constant. The gas phase is ideal in the thermodynamic sense.
3,
<< ($,)" and << ($,) in the w phase where w refers to u, fl, and y. In general, the product of deviations (i.e., terms marked by a tilde) will be considered negligible in comparison to the product of averages. A.3 The three phase system is in local equilibrium. A.4 = wa(Tj>a/at, V(T,),) ,= 0 when V( T,,)@ = 0 A.5 A.6 T , is a linear function of V( To)o A.1 Diffusive transport of thermal energy in the gas phase is negligible. A.8 R, is a h e a r function of V ( ( p , ) y / ( p , ) ' ) A.9 The gas phase is continuous. A.10 Concerning the forces exerted on the liquid phase, we shall assume that both the gas and the liquid pressure distributions are hydrostatic. A . l l The porous medium is homogeneous. A.12 The forces acting on the liquid phase can be uniquely represented in terms of ( p , ) + ( p e - p,) ($)"and the function 5.
A.l A.2
$u
zjT j
194
STEPHENWHITAKER
For any given problem we think of (0)and E, as being specified parameters along with all the physical properties and transport coefficients for all three phases. The theory provides us with 12 equations (Eqs. (VI-1)(VI-12) and 12 unknowns: eP, ey, , ( v J , ( T ) , (h), (p,)’, < p J Y , (pl>’, ( p 2 ) ’ , (pI)’, and ( P ~ The ) ~ .problem is not quite as difficult as it appears for the two vector equations (Eqs. (VI-2) and (VI-4)) can be substituted into the thermal energy equation and the continuity equations to give us four scalar transport equations, one scalar constraint on the volume fractions, and five thermodynamic relations. These ten equations can be used to solve for the ten unknowns: €yr <.T),(&), ( P y ) ’ , (py>’7 (Pl)’, (p2>’7 (P1 >’I and (p2>’. The central computational difficulty is certainly the simultaneous solution of the four scalar transport equations; however, numerical methods for the solution of these equations are well known, and it would appear that a rigorous analysis of a drying process is complicated but not impossible. what does appear to be overwhelmingly difficult at this point is the comparison between theory and experiment in order to determine the parameters that appear in the transport equations. Clearly we need some theoretical basis for estimating the value of these parameters or a series of simplified special theories that adequately describe various drying processes.
VII. The Diffusion Theory of Drying In the previous section we listed the volume averaged equations that govern the drying process in a porous media. The effort required to solve the coupled transport equations is significant, but can be accomplished by standard numerical methods. What appears to be an extremely difficult problem at this time is the comparison of theory with experiment in order to determine the value of the various parameters that appear in the governing equations. Clearly the general theory needs to be extended in order to place some constraints on the value of the parameters, and special theories need to be constructed so that various drying processes can be studied without recourse to an enormous computational effort. In this section we shall consider one of these special theories. In our summary of the governing equations given in Section VI, we described the gas phase transport in terms of the continuity equation (Eq. (VI-5)) and the diffusion for the inert species (Eq. (VI-6)). We now wish to make use of the diffusion equation for the vaporizing species, which was given earlier as Eq. (1V.A-10) and listed here as
(VII-1)
A THEORY OF DRYING IN
POROUS
MEDIA
195
In addition we need to use Eq. (V.B-33) which can be multiplied by pa and listed here as
(VII-2) We now make the following assumptions regarding these two equations with the hope that some drying processes may satisfy these restrictions: (1) negligible gravitational effects; (2) negligible effect of the temperature gradient on the liquid flow; (3) isotropic system; (4) negligible convective transport in the gas phase.
Under these circumstances Eqs. (VII-1) and (VII-2) reduce to
(VII-4) The restriction of negligible gravitational effects is certainly not applicable to the drying of granular material such as the sand used by Ceaglske and Hougen [18]; however, there are many other drying processes that would satisfy this restriction. The effect of the temperature gradient on liquid motion in porous media does not appear to have been studied experimentally, but it does not seem unreasonable that situations would exist for which this effect is negligible. The question of isotropy in a drying system would seem to be open to criticism. Certainly the rigid solid matrix could be isotropic, but can a three phase, nonequilibrium system be isotropic? The very nature of the process seems to exclude isotropy; however, forms similar to Eqs. (VII-3) and (VII-4) could be obtained if the process were one dimensional. The assumption that convective transport in the gas phase is negligible is certainly difficult to accept. This will occur when the mole fraction of the vaporizing species is small compared to one, yet this will be the case only during the latter stages of drying. During the early stages the assumption cannot possibly be satisfied; however, at that time convection in the liquid phase probably predominates and the fact that the assumption is not valid is of no consequence.
STEPHEN WHITAKER
196
Returning to Eqs. (VII-3) and (VII-4) we add these two equations, thus eliminating the term (ri~), and divide by the constant pp(ep + f y ) to obtain
=
V * { K . V[ Ppkp
]]
+ By)
In Section 1I.C we defined the fractional moisture saturation as S = pB'B -t
(pl)y'y
Pbk@ +
=
fractional moisture saturation
(VII-6)
where S represents the total mass of moisture (in both the liquid and gas phases) per unit volume divided by the maximum total mass of moisture per unit volume. Thus S = 1.0 for a completely saturated porous media and S = 0.0 for a bone-dry porous media. The latter state can only be achieved by extended contact with zero humidity air. We can see that Eq. (VII-5) has the makings of a transport equation for the total fractional moisture saturation, and we express this result as
We can think of a drying process as one for which Ep
=
(1 -
Ey
=
0,
En)
at the start of a drying process
and for which €0 + 0
at the end of a drying process (1 - E,) From our definition of 0::) given by Eq. (1V.A-9) and our definition of K , given by Eq. (V.B-39) we expect €7
+
K,
large +0 -+
at the start of a drying process
and Dl.:;
+
large
at the end of a drying process
A THEORY OF DRYING IN POROUS MEDIA
197
Furthermore, it is clear that at the start of a drying process
+ €7)
Ppkp
thus encouraging us to write
K , V[
”” Ppkp
+ .y)
]
=
K , VS
(VII-8)
We know that Eq. (VII-8) is not valid as S --r 0; however, that is unimportant since the entire term tends toward zero under those circumstances. Substitution of Eq. (VII-8) into Eq. (VII-7) leads to
);!(
=
v
(K,VS)
Here we are approaching closer and closer to a diffusion equation for the total fractional moisture content; however, the transport in the gas phase still presents a problem. While it is quite reasonable to approximate pp~p/p,,(eB e y ) by S during the early stages of a drying process, it does not seem reasonable to approximate ( p l ) Y e y / p p ( ~ B e y ) by S during the late stages ofa drying process. Even when the dominant mode of moisture transfer y the is diffusion in the gas phase, it is still possible that ppcp >> ( p l ) y ~ since density of the liquid will be many orders of magnitude greater than the density of the vaporizing species. Although we appear, at this point, to be prohibited from obtaining a transport equation for S, there is one further simplification that can be made in Eq. (VII-9). During the latter stages of drying it seems quite likely that the spatial variations in ey are small and that the gas phase density is essentially constant. Making this simplification allows us to write Eq. (VII-9) in the form
+
+
(E)
=
V ( K , VS)
+V-
{r?) I} V[
(P1)yEy
Pp(Ep
(VII-10)
+ Ey)
If, and only if, the gas phase diffusive transport becomes significant when ( ~ ~ )>>~ p rp P, , we can make the approximation
P P(pi)yey ( 4 + €7)
during the latter stages of a drying process
+
and write Eq. (VII-10) as
(g)
=
v
{[Kc
+
(331vs]
(VII-11)
STEPHEN WHITAKER
198
This situation is not likely to occur; however, there is a way around this difficulty. Making use of Eqs. (VI-8) and (V1-12), we can express the species density as
-[2cr,,lrp,Rl
- -exp{ R,(T)
+ (VII-12)
Here it becomes clear that during the latter stages of drying the species density will be a strong function of the characteristic length r, which in turn depends on eP. The dependence of ( p l ) ’ upon eB can be expressed as a functional dependence upon S for the case where pBtB>> ( p l ) y e yand we write (VII-13) (P1)Yry/Pa(E/l+ 4 = %(S)
For the case where ( p l ) Y e y >> ppeB we simply have %(S)= S. Substitution of Eq. (VII-13) into Eq. (VII-10) leads to
(z) {kc((g)
(VII-14)
V * (DVS)
(VII-15)
.
V + This in turn can be written in the form =
(asjat)
=
D ) ] VS]
where D is the drying difusion coeficient given by
D
= K,
+
[(g)?]
(VII-16)
We expect D to be large when S is near unity and to decrease rapidly with decreasing values of S . The development given in this section has not been at all rigorous, and the strict theoretician would consider it more wishful thinking than analysis. However, the pragmatist will view Eq. (VII-15) as an attractive alternative to the theory that is summarized in Section VI. VIII. Conclusions
The well-known transport equations for continuous media have been used to construct a rational theory of simultaneous heat, mass and momentum transfer in porous media. Several important assumptions regarding the structure of the gas-liquid system in a drying process were made which require theoretical or experimental confirmation. The general theory presented in this article should provide a starting point for the construction of
A THEORY OF DRYING IN POROUS MEDIA
199
special theories so that various drying processes can be studied analytically without recourse to an enormous computational effort. ACKNOWLEDGEMENTS This work was supported by NSF Grant P4K1186-000. The author would like to thank Professor W. G. Gray of Princeton University and Professor Blagoje Andrejevski of the the University of Skopje for the helpful comments. NOMENCLATURE
ROMANLETTERS
area [m’] A s p / V , surface area of the .-/I interface per unit volume [m-’] material surface [m’] constant pressure heat capacity [kcal/ kg OK1 mass fraction weighted average constant pressure heat capacity [kcal/ kg OK1 gas phase dispersion tensor [m’/sec] gas phase diffusion tensor [m’jsec] gas phase molecular diffusivity [m’/ sec] gas phase total effectivediffusivity tensor for the ith component [m’/sec] the drying diffusion coefficient [m’/ sec] body force per unit mass acting on the ith species [m/sec’] gravity vector [m/sec’] enthalpy per unit mass [kcal/kg] reference enthalpy [kcal/kg] partial mass enthalpy for the ith species [kcal/kg] heat transfer coefficient for the .-/I interface [kcal/sec mz OK] enthalpy ofvaporization per unit mass [kcawd thermal conductivity [kcal/sec m OK] liquid phase permeability tensor [m’/ sec] single scalar component of K, for an isotropic system gas phase permeability tensor [m’/ sec] single scalar component of K, for an isotropic system solid phase conductivity tensor
K,
single scalar component of K, for an isotropic system liquid phase effective termal conductivity tensor [kcalisec m OK] liquid phase thermal dispersion tensor [kcal/sec m OK] liquid phase total thermal conductivity tensor [kcal/sec m OK] -XP,>/~C, [N/m’l -Xp,>/d(T) [N/m’ “K] ~ , t K , k s / ~[mz/secl p e s t K , & < ~ , / ~[m’/sec a OK] mass rate of evaporation per unit volume [kg/sec m3] number of liquid continuous subregions in the averaging volume outwardly directed unit normal pressure [N/m’] p, - p,, capillary pressure [N/m’] reference pressure [N/m*] reference vapor pressure for component 1 [N/m2] relative pressure, p - po + prp [N/m’] volumetric flow rate [m3/sec] heat flux vector [kcal/sec mZ] mass rate of production of the ith species owing to chemical reaction [kg/sec m3] position vector [m] characteristic length of a porous media [m] gas constant for the ith species [N m/kg OK1 e8/(e8 + z,), fractional liquid saturation (P,€a + ( P l ) y ~ , ) / C P p ( e , + 41, fractional moisture saturation temperature YK] reference temperature [OK] reference temperature [OK]
STEPHEN WHITAKER gravitational potential function [mz/ sec'] unit tangent vector a gas phase mass fraction gradient [m-ll
total stress tensor [N/m'] time [sec] diffusion velocity of the ith species [mwl unit tensor mass average velocity [m/sec] velocity of the ith species [m/sec] volume of the rigid solid phase contained within the averaging volume
SUBSCRIPTS designates the ith species in the gas phase designates a property of the solid phase designates a property of the liquid phase designates a property of the gas phase designates a property of the a-fl interface designates a property of the fl-y interface designates a property of the y-ointerface
WI
volume of the evaporating liquid phase contained within the averaging volume [m'] volume of the gas phase contained within the averaging volume [m'] averaging volume [m3] material volume [m3] velocity of the 8-7 interface [mjsec] GREEK LETTERS V,/"v, volume fraction of the rigid solid phase V,(t)/Y, volume fraction of the gas phase bulk coeljicient of viscosity [N sec/
mZ1 a function of the topology of the liquid phase thermal dispersion vector [kcal/sec m31 shear coefficient of viscosity [N sec/ m'l density [kg/m3] density of the ith species [kg/m3] viscous stress tensor [Nlm'] rate of heat generation [kcal/sec m3]
MATHEMATICAL SYMBOLS total time derivative material time derivative partial time derivative spatial average of a function I) which is defined everywhere in space phase average of a function $p which represents a property of the fl phase intrinsic phase average of a function $, which represents a property of the B phase Note that ($), everywhere in space.
and < $ J yare defined
REFERENCES
E.J. Forsdyke, B. Rackham, et al., Pottery and porcelain. Encycl. Br. 18, 338-373 and XLIII (1945). 2. P. D. Lebedev and A. S. Ginzburg, General problems of drying theory and technique. 1.
Prog. Heat Mass Transfer 4, 55-16 (1971). 3. G.D. Fulford, A survey of recent Soviet research on the drying of solids. Can. J. Chem. Eng. 47,378-391 (1969). 4. P. Y. McCormick, Unit operations-drying. Ind. Eng. Chem. 62, 84-86 (1970). 5. W. K. Lewis, The rate of drying of solid materials. Ind. Eng. Chem. 13, 427-432 (1921). 6. T. K. Sherwood, The drying of solids. I. Ind. Eng. Chem. 21, 12-16 (1929). 7. T. K. Sherwood, The drying of solids. 11. Ind. Eng. Chem. 21, 976-980 (1929).
A THEORY OF DRYING IN POROUS MEDIA
20 1
8. T. K. Sherwood, The drying of solids. 111. Mechanism of the drying of pulp and paper. Ind. Eng. Chem. 22, 132-136 (1930). 9. T. K. Sherwood, Application of the theoretical diffusion equations to the drying of solids. Trans. Am. Inst. Chem. Eng. 27. 190-202 (1931). 10. A. B. Newman, The drying of porous solids: Diffusion calculations, Trans. Am. Inst. Chem. Eng. 27,310-333 (1931). 1I . A. B. Newman, The drying of porous solids: Diffusion and surface emission equation. Trans. Am. Inst. Chem. Eng. 27, 203-220 (1931). 12. E. R. Gilliland and T. K. Sherwood, The drying of solids. VI. Diffusion equations for the period of constant drying rate. Ind. Eng. Chem. 25, 1134-1 136 (1933). 13. W-. Gardner and J. A. Widtsoe, The movement of soil moisture. Soil Sci. 11, 215-232 ( 1920). 14. L. A. Richards, Capillary conduction of liquids through porous mediums. J . AppI. Phys. 1, 318-333 (1931). 15. E. K. Rideal, Philos. Mug. [6] 44, 1152 (1922). 16. A. E. R. Westman, J. Am. Cerum. SOC.12, 585 (1929). 17. E. W. Comings and T. K. Sherwood, The drying of solids. VII. Moisture movement by capillarity in drying granular materials. Ind. Eng. Chem. 26, 1096-1098 (1934). 18. N. H. Ceaglske and 0. A. Hougen, Drying granular solids. Ind. Eng. Chem. 29, 805-813 ( I 937). 19. W. B. Haines, Studies in the physical properties of soils. IV. A further contribution to the theory of capillary phenomena in soil. J . Agrir. Sci. 17, 264-290 (1927). 20. W. B. Haines, Studies in the physical properties of soils. V. The hysteresis effect in capillary properties and the modes of moisture distribution associated therewith. J . Agric. Sri. 20, 97-1 16 (1930). 21. 0.A. Hougen, H. J. McCauley, and W. R. Marshall, Jr., Limitationsofdiffusionequations in drying. Trans. Am. Inst. Chem. Eng. 36. 183-210 (1940). 22. F. Tuttle, A mathematical theory of the drying of wood. J . Franklin Inst. 200, 609-614 (1 925). 23. 0. Krischer, Fundamental law of moisture movement in drying by capillary flow and vapor diffusion. VDI Z.82,373-378 (1938). 24. 0. Krischer, The heat, moisture, and vapor movement during drying porous materials. V D I Z . , Beih. 1, 17-24 (1940). 25. 0. Krischer, Heat and mass transfer in drying. VDI-Forschungsh. 415 (1940). 26. W. B. Van Arsdel, Approximate diffusion calculations for the falling rate phase of drying. Trans. Am. Inst. Chem. Eng. 43, 13-24 (1947). 27. J. R. Phillip and D. A. DeVries, Moisture movement in porous materials under temperature gradients. Trans. Am. Geophys. Union 38, 222-232 (1957). 28. D. A. DeVries, Simultaneous transfer of heat and moisture in porous media. Trans. Am. Geophys. Union 39,909-9 I6 (1 958). 29. A. V. Luikov, “Heat and Mass Transfer in Capillary-Porous Bodies.” Pergamon, Oxford, 1966. 30. J. P. Gupta and S. W. Churchill, A model for the migration of moisture during the freezing of wet sand. AIChESymp. Ser. 69, No. 131, 192-198 (1972). 31. D. Berger and D. C. T. Pei, Drying of hydroscopic capillary porous solids-a theoretical approach. Int. J. Heat and Transfer 16,293-302 (1973). 32. A. Husain, C. S . Chen, and J. T. Clayton, Simultaneous heat and mass diffusion in biological materials. J , Agric. Eng. Res. 18, 343-354 (1973). 33. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena.” Wiley, New York, 1960.
202
STEPHEN WHITAKER
34. S. Whitaker, “Introduction to Fluid Mechanics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1968. 35. R. Ark, Vectors, “Tensors, and the Basic Equations of Fluid Mechanics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 36. S. Whitaker, “Fundamental Principles of Heat Transfer.” Pergamon, Oxford, 1977. 37. D. W. Lyons, J. D. Hatcher, and J. E. Sunderland, Drying of a porous medium with internal heat generation. Int. J. Heat Mass Transfer 15,897-905 (1972). 38. R. J. Raiff and P. C. Wayner, Jr., Evaporation from a porous flow control element on a porous heat source. Inr. J. Heat Mass Transfer 16, 1919-1929 (1973). 39. J. C. Slattery, “Momentum, Energy, and Mass Transfer in Continua.” McGraw-Hill, New York, 1972. 40. R. Defay, I. Prigogine, and A. Bellemans, “Surface Tension and Adsorption.” Wiley, New York, 1966. 41. J. C. Slattery, General balance equation for a phase interface. Ind. Eng. Chem., Fundam. 6, 108-118 (1967). 42. C. Truesdell and R. Toupin, The classical field theories. In “Handbuch der Physik” (S.Fliigge, ed.), Vol. 111, Part I , p. 226. Springer-Verlag, Berlin and New York, 1960. 43. S. Whitaker, Diffusion and dispersion in porous media. AIChE J. 13,420-427 (1967). 44. S. Whitaker, Advances in the theory of fluid motion in porous media. Ind. Eng. Chem. 61, 14-28 (1969). 45. S. Whitaker, The transport equations for multi-phase systems. Chem. Eng. Sci. 28, 139-147 (1973). 46. J. C. Slattery, Two-phase flow through porous media. AIChE J. 16, 345-354 (1970). 47. W. G. Gray, A derivation of the equations for multi-phase transport. Chem. Eng. Sci. 30,229-233 (1975). 48. Y. Bachmat, Spatial macroscopization of processes in heterogeneous systems. Isr. J. Tech. 10, 391-403 (1972). 49. S.Whitaker, On the functional dependence of the dispersion vector for scalar transport in porous media. Chem. Eng. Sci. 26, 1893-1899 (1971). 50. R. L. Gorring and S. W. Churchill, Thermal conductivity of heterogeneous materials. Chem. Eng. Prog. 57,53-59 (1961). 51. B. S. Singh, A. Dybbs, and F. A. Lyman, Experimental study of the effective thermal conductivity of liquid saturated sintered fiber metal wicks. In?. J. Hear Mass Trumfer 16, 145-155 (1973). 52. R. A. Greenkorn and D. P. Kessler, Dispersion in heterogeneous, nonuniform, anisotropic porous media. Ind. Eng. Chem. 61, 14-32 (1969). 53. T. Miyauchi and T. Kikuchi, Axial dispersion in packed beds. Chem. Eng. Sci. 30,343-348 (1975). 54. R. S.Subramanian, W. N. Gill, and R. A. Marra, Dispersion models of unsteady tubular reactors. Can. J. Chem. Eng. 52, 563-568 (1974). 54a. Okazaki, Effective thermal conductivities of wet granular materials. AIChE Meet., 1975 (1975). 55. S . Whitaker, Velocity profile in the Stefan diffusion tube. Ind. Eng. Chem., Fundam. 6, 476 (1967). 56. J. Meyer and M. D. Kostin, Circulation patterns in the Stefan diffusion tube. Int. J. Heat Mars Transfer 18, 1293-1297 (1975). 57. H.Brenner, “Elements of Transport Processes in Porous Media,” Monograph. SpringerVerlag, Berlin and New York (to be published). 58. R. J. Mannheimer, Surface rheological properties of foam stabilizers in nonaqueous liquids. AIChE J. 15, 88-93 (1969).
A THEORY OF DRYING IN POROUS MEDIA
203
59. S. Whitaker, Effect of surface active agents on the stability of falling liquid films. Znd. Eng. Chem.,Fundam. 3,132-142 (1964). 60. J. Bear, “Dynamics of Fluids in Porous Media.” Am. Elsevier, New York, 1972. 61. A. E. Scheidegger, “The Physics of Flow Through Porous Media,” 3rd ed. Univ. of Toronto Press, Toronto, 1974. 62. F. A. L. Dullien, New network permeability model of porous media. AIChEJ. 21,299-307 (1975).
This Page Intentionally Left Blank
Viscous Dissipation in Shear Flows of Molten Polymers HORST HENNING WINTER Institut f u r Kunststoff(echno1ogie. Universitat Stuttgarr, Stuttgart, West Germany
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. System of Equations. . . . . . . . . . . . . . . . . . . . . . . . . B. Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . C. Rheological Constitutive Equation. . . . . . . . . . . . . . . . . . . 11. Shear Flow (Viscometric Flow). . . . . . . . . . . . . . . . . . . . . . A. Thermal Boundary Condition. . . . . . . . . . . . . . . . . . . . . B. Steady Shear Flow with Open Stream Lines. . . . . . . . . . . . . . . C. Shear Flow with Closed Stream Lines . . . . . . . . . . . . . . . . . 111. Elongational Flow; Shear Flow and Elongational Flow Superimposed (Nonviscometric Flow) . . . . . . . . . . . . . . . . . . . . . . . . . IV. Summary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205 207 209 211 212 222 221 250 260 262 263 264
I. Introduction Polymer processing and applied polymer rheology occur at relatively high temperatures and often at high temperature gradients. In molten polymers, large stresses are required to maintain the flow and, additionally to convective and conductive heat transfer, temperatures essentially depend on viscous dissipation, i.e., on conversion of mechanical energy into heat. Velocity and temperature fields influence each other: the temperatures influence the flow through the temperature-dependent rheological properties, and the velocities influence the temperatures through convection, through dissipation, and through anisotropical effects (which are investigated very little) on the thermal properties. 205
HORSTH. WINTER
206
Research in rheology and in thermodynamics related to heat transfer problems is mostly done separately, by rheologists on molten polymers or polymer solutions at constant temperature, and by thermodynamicists on polymers at rest. The difficult task of combining the two areas is left to the polymer engineers (see for instance [1-41). A number of assumptions have to be introduced into the heat transfer analysis before applied problems can be solved. The different flow problems involving heat transfer and viscous dissipation can be classified in groups as shown in Table I. Each of the groups is characterized by different rheological phenomena, and one has to choose very different rheological constitutive equations to describe them. The two main groups are channeljow (includingflow geometries with partly solid and partly free boundary) and free surface flow (with no solid boundary). In polymer processing, channel flow of molten polymers occurs in a large variety of flow geometries. The polymer is forced through a channel by a pressure gradient (flow in an extruder die, for instance),or it is dragged along by a moving wall (rotating screw in a stationary cylinder, for instance). Very often both types of flow are superimposed on each other. Free surface flow for example occurs in film blowing or fiber spinning.
TABLE I
CLASSIFICATION OF
HEATTRANSFER AND VISCOUS DISSIPATION IN MOLTENPOLYMERS heat transfer and viscous dissipation in molten polymers
h free surface flow
channel flow
shear flow shear flow with open streamlines
shear flow with closed streamlines
(shear free flow)
shear flow and elongational flow superimposed (non-viscometricflow)
For rheological reasons, channel flow problems are subdivided into shear (also called viscometric flow) and nonuiscometric flow. The separation of the shear flow problems into one group with open stream lines and one with closed stream lines has to be made since their thermal development is different. Throughout the first section, the heat transfer problem will be considered in general, i.e., the relevant equations are listed in a general form and the properties are described. The rheological properties of the molten polymers have to be formulated differently according to the various flow types since
jow
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
207
the length of the following sections is supposed to reflect the degree of understanding of the respective flow and heat transfer problems. In Section 11, heat transfer in shear flow will be analyzed. A large emphasis will be laid on replacing the commonly used idealized boundary conditions, i.e., constant wall temperature or constant wall heat flux (with the limiting case of the adiabatic wall), by more general conditions. In practical applications, the idealized conditions will rarely occur; actually it is difficult to achieve them even in especially designed model experiments. To make the analysis applicable, heat transfer in a flowing polymer should not be studied separately inside the fluid, but together with the surrounding wall. In this analysis the heat transfer at the wall is described by an outer temperature difference (temperature of the surroundings minus temperature at the boundary) and the Biot number, which otherwise has been used successfully for describing the boundary conditions for temperature calculations in solids. The Biot number is appropriate for describing boundary conditions between isothermal and adiabatical, as they occur in real processes. Additionally, the thermal capacity of the walls is included in the analysis by introducing the capacitance parameter C . Heat transfer in viscometric flow has been studied quite extensively in the literature, and at the present state it seems to be necessary to show the many common aspects of the different studies. Thus, as the main goal of this study a unifying concept will be developed. This concept makes it possible to comprise the most important shearjow cases into a single one, which can be solved with one numerical program. For Section I11 on nonviscometric flow in channels and flow with free boundaries, the description will not go much further than stating the problem, showing the present methods of solution, and listing references. Since nearly all of the results in this report are on shear flow, the title is taken to be “in shear flows” even if the problem is stated in a general form and Section I11 is on nonviscometric flows. Heat transfer in non-Newtonian fluids at negligible viscous dissipation is not included in this report (see instead [5-7]), although it can be treated as a limiting case of the corresponding flow with viscous dissipation. A. SYSTEM OF EQUATIONS
The problems are governed by the equations describing the conservation of mass appt v * (pv) = 0 (1.1)
+
and the conservation of energy p DelDt = V ( k V T ) 9
+ o :Vv,
( 1.2)
HORSTH . WINTER
208
by the stress equation of motion p DvlDt = V * u
+ pg,
(1.3)
and by the constitutive equation which will be described below, together with the appropriate flow geometries. The three equations above are derived and tabulated in textbooks (see for instance [S]) for different coordinate systems. a/a denotes the partial and DID the substantial derivative; V is the “nabla” operator. Density p and thermal conductivity k are properties of the fluid. Velocity v, internal energy e, temperature T , time t, and stress CT are the variables. The stress CT is defined in such a way that the force on the positive side of a surface element of unit area and normal vector n is n 0. The equation of energy says that the rate of gain of internal energy per unit volume ( p De/Dt) is equal to the rate of internal energy input by conduction per unit volume V (k V T ) plus the rate of work by the stress on the volume element u :Vv, which is being partly stored and partly dissipated during the flow. For heat transfer studies, the internal energy has to be defined in terms of the fluid temperature and the strain and stress variables. IncornpressibleJiuid: In rheology the fluid is usually supposed to be incompressible (even when properties such as the viscosity are allowed to depend on pressure). The flow geometry, the temperature, and the rheological properties of the fluid determine the stress completely, except for an arbitrary added isotropic pressure [9]. Therefore, the stress is commonly separated into an arbitrary pressure p and the extra stress 7 , which is defined in the rheological constitutive equation, viz. a = -pd+z. (1.4) 6 denotes the unit tensor. In some flow problems, it is convenient to define the isotropic pressure p to be equal to one of the normal stress components in a certain coordinate system ( p = -ell, p = -crz2, or p = - L T ~ ~ )while , in other flow problems it might be preferable to define p = -(trace a)/3. For an “incompressible”fluid, the change in internal energy and the work of the stress per unit time are determined by p DelDt = cp DTIDt,
u:Vv = - p
v
*
v
+ 7:vv,
(1.5)
and Eq. (1.2) becomes cp DTIDt = V * ( k V T )
+ z:Vv.
(14
The specific heat capacity c is defined as the thermal energy needed per unit mass and Kelvin degree for changing the temperature of a material. Since the density is taken to be constant, c has to be measured at constant
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
209
density. If the fluid were really incompressible, the specific heat should be the same for measurements at constant density (c,) or at constant pressure (c,). From thermodynamic data at rest (Eq. (l.ll)),however, one finds that c, and c, of polymer melts differ by about 10%. Compressible fluid: There are difficulties in relating strain and stress in deforming materials that are slightly compressible. One commonly assumes that the deformation can be separated into two parts: a deformation at constant density and the volume change [lo]. Neglecting the influence on each other, the deformation at constant density is described by the constitutive equation, and the density of the flowing polymer is determined from equilibrium data p(T, p ) measured on the fluid at rest (taking p = -(trace 4/31. There also seem to be difficulties in defining the internal energy e of a compressible flowing fluid: one assumes that e can be described in terms of p and p only, independently of the other stress and strain variables (see for instance [81): e = e(p, P). Applying this relation to the flowing polymer melt, the substantial derivative of the internal energy then becomes De
PK= - P V
*V
+ € TDP -+ Dt
DT pepDt
(1.7)
with E = -p-l(i?p/aT),, the coefficient of thermal expansion. E and c, are evaluated from rest data at temperature T and the “pressure” p = -(trace 4 3 . The equation of energy takes the form usually shown in the literature [ll]: DT DP PC - = V .(k V T ) + ET- + t:Vv. Dt Dt From this equation, calculated temperature fields in channel flow (see Fig. 13, p. 244) show large temperature decreases due to cooling by expansion. The assumptions made above (concerning the density changes and the internal energy) are rather severe, and further experimental studies are needed to investigate their validity.
B. THERMAL PROPERTIES The properties in the analysis are the density p, the specific heat c, and the thermal conductivity k : the thermal diffusivity is defined as a = k/pc. Rheological properties are defined separately in the constitutive equation. In a stationary fluid, the density p ( p , T ) is a function of pressure and temperature. It can be described by the equation of Spencer and Gilmore
HORSTH. WINTER
210
TABLE I1 CONSTANn OF EQ. (1.9) MEASURED BY MATERIAL SPENCER AND GILMORE [121
polymer Id.PE
PS
PMMA CAB
b*
m3/kg]
W
p* [N/m']
0.875 0.822 0.734 0.688
3.275 1.863 2.157 2.844
kg/g-mole]
10'
28
x 10' x 10' x 10'
104 100
x
54
[12]: ( l / P - b*)(P
+ P*)
= RT/W,
(1.9)
where b*, p*, and W are material constants. Their values are tabulated (Table 11) for some examples of the most widely used polymers; R = 8.314 [J/K g-mole] is the gas law constant. From Eq. (1.9) one can evaluate the term ET of the equation of energy ET = 1
since b* is always smaller than
p-l,
- pb*;
(1.10)
the dimensionless product CT adopts
positive values smaller than unity.
The density generally is measured on the polymer at rest and in thermodynamic equilibrium. Dynamic measurements by Matsuoka and Maxwell [131, however, show a very delayed response of polyolefines to sudden pressure changes. Thus, the use of equilibrium density data restricts the analysis to flows of slowly changing pressures. The reaction to temperature changes is similarly delayed [141. Additionally the flow might influence the density. The specific heat commonly is measured at constant pressure. Using the equation proposed by Spencer and Gilmore, Eq. (1.9),one can determine the c, from specific heat data at constant pressure specificheat at constant density.~ C, =
cP - R/W.
(1.11)
The thermal conductivity k and the specific heat capacity cp are slowly varying functions with temperature and they also depend on pressure. In flowing polymers the thermal conductivity possibly varies with direction. For most polymers the temperature dependence can be expressed in a linear form
k
-
=
E(1 + uJT - To)),
cP = Tp(l + a,( T - To)).
(1.12)
k and Fp are values at some reference condition (temperature T o ) ,while ak and a, are the temperature coefficients, which might be positive or negative
VISCOUS DISSIPATION IN FLOWING MOLTEN POLYMERS
21 1
TABLE 111
THERMAL CONDUCTIVITY AND SPECIFIC HEAT CAPACITY OF SOMEMOLTEN POLYMERS AT TEMPERATURES To (see [ 15 -231) (‘P
k
Polymer
rC]
[lo3 Nm/kg K]
[N/K s]
1.d.PE h.d.PE
150 150 180
2.51 2.65 2.80
0.241 0.255
100
1.53
150 150
2.04
0.166 0.167 0.195
To
PP PVC
PS PMMA
-
-
depending on the polymer in question and on the temperature range. Table I11 shows some values of k and c p .More detailed data on the properties can be found in references [15-231, for instance. C. RHEOLOGICAL CONSTITUTIVE EQUATION For a large number of fluids, which can be regarded as incompressible, the stress can be described by the Stokes equation u = -p6
+ qy,
(1.13)
the simplest tensor generalization of Newton’s law of viscosity. Here p is the isotropic pressure, and i, = (Vv) + (Vv)’ is the rate of strain tensor; the viscosity q depends on temperature and on pressure, but not on the time t or on any kinematic quantities such as 9. Fluids that show this behavior are called Newtonian. The Stokes equation has been generalized by taking the viscosity q($) to be a function of the second invariant of the rate of strain tensor [24] : u = -p 6
+ q($)i,;
i,
= (+i,:i,)l’Z.
(1.14)
This equation defines the “generalized Newtonian fluid.” It has been applied quite successfully to molten polymers in steady shearjlow for calculating the shear component of the stress tensor in an appropriate coordinate system; however, the normal stress components calculated from this equation are known to be unrealistic for molten polymers. In general, the equation might be misleading in its tensor form because it does not allow one to calculate meaningful stress components in arbitrary coordinate systems. The appropriate statement of the rheological equation for molten polymers in steady shear flow is given by Criminale et al. [25]; their equation will be applied in Section 11.
21 2
HORSTH.WINTER
The rheological properties of elastic fluids (such as molten polymers) at any given position depend on the strain and temperature history of the fluid elements when they arrive at that position, independently of the history of neighboring particles. Translational and rotational movements do not influence the stress [9]. Depending on the type of flow, the rheological behavior of molten polymers is more or less different from the behavior of Newtonian fluids. Up to now there exists no general constitutive equation to describe all the phenomena known for a given polymer melt. Additionally, the temperature effects on the rheological properties have not been studied at all or only at different levels of homogeneous temperatures. The constitutive equations used will be stated in the beginning of Sections I1 and 111. 11. Shear Flow (Viscometric Flow)
In shear flow at constant density, material surfaces move “rigidly” (i.e., without stretching) across each other. These surfaces are called shear suduces [26]. Pipe flow, which is an example of shear flow, sometimes is called telescopic flow since its rigidly moving surfaces are concentric cylinders. In Fig. 1 the deformation of particle Po at the origin is described by the relative motion ofneighboring planes. On the left (Fig. la) an orthonormal coordinate system is chosen, such that x1 = direction of shear, x2 = direction of velocity gradient,
xg = neutral direction.
shear direction
tan
D
E
x21
= dtan
,shear direction
E,J*+
[tan c3d2 ‘
FIG.1 . Unidirectional shear flow in Cartesian coordinates. Shear flow in the 1-3 plane as superposition of shear flow along x, and along xj.
VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS
213
The direction ofshear (or shear direction) and related quantities are defined following [26]: an orthonormal coordinate system is chosen to have its x1 axis and its x3 axis in a shear surface (see Fig. 1). The x2 axis is perpendicular to the shear surface; it projects point Po onto neighboring shear surfaces. During shear flow, shear surfaces move past each other, and the normal projection of Po draws a line on neighboring shear surfaces. This line is called the shear line. (Note: In the examples of Fig. 1, the shear surfaces are planes, and the shear lines are straight.) The tangent to the shear line at time t is defined to be the shear direction at time t, and the angle of the shear line with the x1 coordinate is the shear angle cp; if the direction of shear remains constant with time, the flow is called unidirectional. For most applications, the direction of shear is identical with the direction of flow; an exception is the othogonal rheometer [26, p. 761, for instance. The shear rate, the extra stress, and the temperature are supposed to be uniform in the direction of shear, but they may change perpendicularly to the direction of shear (even within shear surfaces). If one allows for shear in the 3 direction (Fig. lb) additionally to the shear in the 1 direction, the shear direction is in the 1-3 plane (Fig. lc); for some steady shear flow geometries (as in helical flow), it is convenient to choose a global coordinate system with the velocity vector in the 1-3 plane and the velocity gradient normal to the 1-3 plane. The matrix of the rate of strain tensor becomes
[tl
=
[
Pl2
2:
:23]?
(2.1)
Q23
and the second invariant of the rate of strain tensor defines the shear rate:
In unidirectional shear flow at constant temperature T o , constant pressure p,,, and constant volume, the stress is given by the shear rate Q(t) and the three shear-rate-dependent viscometric functions [25,26] : viscosity q ( j t T o ,po), first normal stress coefficient m , T o ,po), second normal stress coefficient $2(QCm, T o ,po). Here symbolizes the “history of shear rates” to which the medium was submitted up to the present time t. Up to now, there do not seem to exist heat transfer studies on shear flow in general; however for steady shear flow, the publications are numerous. In Section II.C.3 an example of heat transfer in unsteady unidirectional shear flow will be shown.
HORSTH. WINTER
214
Steady Shear Flow and Shear Viscosity
For unsteady shear flow the shear direction and/or the value of the shear rate change with time; the shear surfaces, however, are maintained. If the shear direction and the shear rate are kept constant over some time (cp = const; j~ = const), the shear stress approaches a constant value, i.e., the viscosity adopts a constant value:
?(t,To,P o ) = lim ?(?‘-m, t+m
To,P o ) ,
(2.3)
called the viscosity, the shear viscosity, or the apparent viscosity. The flow becomes steady (unidirectional)shearpow. The index on To and po indicates that the viscosity is dejned at constant temperature and pressure. The elastic properties of the fluid are represented in steady shear flow by the two viscometric functions t,bl, t,b2. These functions, however, do not have any influence on heat transfer and viscous dissipation; this will be shown in the following. The only two terms in the system ofequations (Eqs.(1.1)-(1.3)) that contain the stress are a:Vv and V u. The rate of work by the stress a:Vv, which in steady shear pow is dissipated completely, sometimes is called the dissipation function. Taking the coordinate system of the shear flow, one can evaluate the two terms [8, p. 7381; u:Vv is a scalar 0:Vv = 212?12 + 2 2 3 7 2 3 , and V u is a vector with the three components
(2.4)
-
[V.U]l
=
[V-a], =
[V
.I3
=
The stress u has been decomposed into p and z as shown in Eq. (1.4).If x1 is the shear direction (Fig. la), the 1component of V u is used for calculating the velocity and the pressure gradient; due to the symmetry of the flow (at/ax, = 0); and since 7 1 3 = 0, the 1 component reduces to
-
[v*u]l =
--a P ax,
+ -.ax,
a212
For a shear direction in the 1-3 plane (Fig. lc),the 1 and the 3 components are used for calculating the velocity and the pressure gradient; since az/ax, = 0
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
215
and &/ax3 = 0, they reduce to [V.UIl
=
--aaPx ,
+ -,a x ,
az12
[V*6I3 =
--
ax3
+ax,
(2.7)
For calculating the velocity, the pressure gradient, and the rate of work by the stress, one finds from Eqs. (2.4), (2.6), (2.7) that one needs only the shear components z12,zZ3of the stress matrix 212
= q(?,
912,
723
= q(?,
(2.8)
j23*
The shear component 213 = sin 2q1($~+ $,)p2/2 and the normal stress components do not contribute to the analysis. The viscosity q is the only rheological property needed for solving heat transfer problems in unidirectional steady shear flow. Thus the heat transfer analysis of Section I1 is not restricted to purely uiscousftuids, even if the normal stresses are not mentioned further. The pressure and the normal stresses can separately be determined from the pressure gradient, the 2 component of the stress equation of motion (which contains [V * el2),and the appropriate boundary conditions. Some typical curves of viscosities referring to different temperatures are shown in Fig. 2 [27]. At low shear rates (p < 10 s-’) one measures the viscosity in Couette or in cone and plate rheometers, and at high shear rates ( j ~> 1 s-’) one uses a capillary or slit viscometer; the temperature in the test section is kept as uniform as possible.
10’
shear
rate 7
Is-’]
10’
103
l@
FIG.2. Typical viscosity curve of molten polymer (low density polyethylene) measured by Meissner [27].
HORSTH. WINTER
216
The pressure and temperature dependence of the viscosity usually is described by the corresponding coefficients pressure coefficient temperature coefficient
=
(3) .
-1
v aT
(2.10)
b,P
(Note: Some authors define the temperature coefficient p at constant shear stress instead of at constant shear rate.) In the expression for the viscosity, the variables are separated. The pressure dependence is described by an exponential function. For incorporating the temperature dependence one may take just an exponential function
v(P, T, PI
= f ( P 9 P ) exp( - BT)
(2.1 1)
or use an Arrhenius type expression V(P, T, P) = f(P, P ) e x p ( E / W
(2.12)
whose “activation energy” E has been reported to be a material constant over wide temperature ranges [28, 291. The temperature coefficient of the viscosity at temperatures around To can then be determined as (if one expands (EIRT) around To) P(T0) = E/RTo2
(2.13)
i.e., for constant E, the temperature coefficient is proportional T-’. The temperature dependence of P normally is neglected in analytical studies, which is acceptable if the deviations AT from the temperature level T o are not too large. The relative difference of the two expressions, Eqs. (2.11) and (2.12),is
= 1
- exp[-fl(AT)2/(To
+ AT)];
(2.14)
as an example AT = 10 K, fl = K-’, T o = 400 K gives a relative difference of less than 0.25%. At very low shear rates (p < 10-1-10-3 depending on temperature, polymer, molecular weight distribution) the viscosity adopts a value inde-
VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS
217
pendent of shear rate, the zero viscosity qo(T,p ) (see Fig. 2). At medium and high shear rates (jJ > lo), as they occur in polymer processing, the viscosity curve q ( j ) is nearly a straight line in the log-log plot. Ranges of the curve can be approximated by a power law [30] which will be formulated as
(2.15) The power law exponent is different for different ranges of i), T , p ; for molten polymers, the value of m is between 2 and 5. In Eq. (2.15),q = q(T, p o , T o ) is a reference viscosity within the power law region, i.e., q at the reference shear rate 7, at the reference pressure p o , and at the reference temperature To.In the literature the power law model has been used very widely because its form allows direct integration of the equation of motion for several flow geometries to be carried out. There are very few data on CI and p available; however, a0 and Po values of the zero viscosity ro have been published for several molten polymers. Thus, a relation will be derived in the following between CI and a, and between p and Po. The viscosity curves measured at different levels of T and p can be condensed to a single one, the so-called muster curue [31,32] q(B3
7-9
p)/qo(T,P) = f (jJ
*
?O(T
P) 1.
(2.16)
In the transformation process, the viscosity curves are moved in the log-log plot in the direction of -45“; the shape of each curve remains the same. Therefore, the shape of the master curve is identical with the shape of the other viscosity curves. At larger values qoi) (> lo4 N m-’), ranges of the master curve can be fitted again by the power law shown above q ( k T, p)lqo(T, P) = w J q o ) ( l / m ) -l ,
(2.17)
and the viscosity becomes
~ ( 9 T, , p ) = f+h/mj(l/m)-l.
(2.18)
K is a “material constant” whose dimension depends on the value of the power law exponent m ; K will be replaced by introducing the reference viscosity Fj of Eq. (2.15). For certain ranges, the power law exponent m is independent of temperature and pressure (since the master curve is independent of temperature and pressure), but it slowly increases with higher values of jq0 ranges. The pressure and temperature dependence of the viscosity q is comprised by the pressure- and temperature-dependent zero viscosity qo only. The pressure coefficient a,the temperature coefficient B, and the activation energy E of the viscosity in the power law region are then related to
HORSTH. WINTER
21 8
TABLE IV E , AND PRESSURE COEFFICIENT a,, OF ACTIVATION ENERGY THE ZERO VISCOSITY qoo EO
Polymer
a0
[lo-’ m2 N-I]
[lo4 J/gm mole]
1.d.PE
5.44
h.d.PE PS PMMA
7.08-8.99 -
-
3.25-3.98 4.28-9.07 2.45-4.08
14.23-19.1 3
Literature 1281 C331 ~321 1331
The temperature coefficient jo of the zero viscosity around the temperature To is equal to E o / R T o 2 ,see Eq. (2.13). a and j of the power law region can be determined from a. and Po by dividing with m, see Eq. (2.19).
ao, Po, and Eo of the zero viscosity by a = ao/m,
P
=
Po/m,
E = Eo/m.
(2.19)
Table IV [28,32,33] lists E o and a. values of some polymers; the data can be used to determine a and P of the power law which describes the viscosity in the jqo range of the application in question. Semjonow [29] collected Eo data from the literature which is quite extensive. Due to the small values of a, the pressure dependence of the viscosity usually can be neglected up to moderate pressure levels ( < 300 bar, depending on the value of a) as they occur in extrusion. In injection molding studies, however, the pressure may adopt values up to 1500 bar and the pressure dependence should be included. In the following study, the pressure dependence of viscosity is not taken into consideration, although the numerical procedure would not have to be much different: it just would require one or more iterations of the whole computation procedure, until the axial pressure profile along the channel is known. The concept of steady shear flow is a theoretical one and it can only be approximated. However, the rheological properties of molten polymers do not seem to be too sensitive to some deviation from steady shear flow; and for a large number of applications, the results from steady shear flow calculations agree with flow experiments reasonably well. The shear flow might be unsteady (a/& # 0 ) ; during startups a constant stress is achieved only after some time of development. But even if the flow is steady (a/& = 0), it still might deviate from shear. Deviations occur as slowly relaxing entrance effects; temperature changes along stream lines, which induce changes in shear rate, are in contradiction to “steady shear flow,” which is defined to be isothermal and at constant shear rate; in Poiseuille flow, pressure changes
VISCOUS
DISSIPATION IN FLOWING MOLTENPOLYMERS
219
actually influence the density, while the fluid supposedly is incompressible. Flows of this kind are called nearly steady shearjlows, where the adverb nearly may refer to the word steady or the word shear. These limitations of the applicability of the shear flow concept will be mentioned again, when all the assumptions are listed together with the system of equations, see Sections II.B.l and II.C.l. Shear Flow Geometries with Open or with Closed Stream Lines
The most important shear flow geometries are shown in Fig. 3; Table V [34-951 lists heat transfer studies on those flow geometries. The flow due to the relative motion of one of the surfaces is called Couettejlow, while Couette
flow
Poiseuille
Couette flow and
flow
‘fl
Poiseuille flow superimposed
@
_.. v g
,
0
\
0
- positive pressure gradient
____
negative pressure gradient
FIG.3. The main simple shear flow geometries [2]: (a) drag flow in the narrow slit between two parallel plates (plane Couette flow), no pressure gradient; (b) axial drag flow between two coaxial cylinders (annular Couette flow), no pressure gradient; (c) flow through a pipe with constant circular cross section (Poiseuille flow); (d) flow through a narrow slit (Poiseuille flow); (e) axial flow through an annulus (Poiseuille flow); (f) helical flow (flow through an annulus with rotating inner cylinder); (9) axial drag flow in an annulus with nonzero axial pressure gradient; (h) drag flow in the narrow slit between two parallel plates with nonzero pressure gradient; (i) angular drag flow in the annulus between two coaxial cylinders (circular Couette flow), no pressure gradient; (k) flow in a cone-and-plate or in a plate-and-plate viscometer. Geometry a will be referred to as a, if the stream lines are open, or a2 if the stream lines are closed (limiting case K + 1 of geometry i).
TABLE V : HEAT TRANSFER STUDIES IN STEADY SHEARFLOW
Temperature field Fully developed
Thermal boundary condition
a1
b
d
C
see a,
34-41
15. 37. 42
mixed
see a2
55E
42
constant wall temperature
60.61
62-64.65E. 66,67E, 68E, 69-71, 72E, 73, 74,75E, 76-79, 133
17,80-82
constant wall temperature
Flow geometries with closed stream lines
Flow geometries with open stream lines e
f
g
k
h
a2
43
31, 38,42, 44-48
38, 42, 45, 49-51E
52, 53E, 54
38, 42. 45, 46, 55E. 56,57E
38,4549, 51E,55E, 58E, 59E
54
i
constant heat flux at wall, adiabatic wall
Developing
constant heat flux at wall, adiabatic
63,67E, 68E, 70, 71, 72E, 76, 78, 79
mixed
90E, 93E. 94E, 95E
77, 83
82, 84, 85
89
83
17,90E,
77,90E,
86E, 87,88
84.85, 91E
89,92
Letters u, b, . . . ,k refer to Fig. 3, where flow geometry a occurs with open and with closed stream lines. The numbers refer to the list of literature. Experimental studies or studies which contain experiments are marked with an E.
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
22 1
the flow due to a pressure gradient is called Poiseuilleflflow.In the literature, most emphasis has been laid on the fully developed temperature field in pipe flow and in Couette flow, and on the developing temperature field in pipes with circular cross section. Experimental studies or studies that contain an experimental part are marked by an E. Poiseuille flow in a pipe with constant but irregular cross section or Poiseuille flow in curved channels with constant cross section induce some small secondary flow in the cross section [96]; this will be mentioned in Section I11 on nonviscometric flow. A detailed description of the historical development of shear flow analysis can be found in the introduction of original papers on the different problems (see for instance [55] for the fully developed temperature profiles and [77, 781 for developing temperature profiles in pipe flow). This study tries to describe the various aspects of shear flow analysis and give credit to the different authors in connection with the arguments in the analysis. Stream lines are lines whose tangents are everywhere parallel to the velocity vectors. In steady flow, the stream lines describe the paths of fluid elements. The heat transfer in the various shear flow geometries depends on whether the stream lines are open (type a,-h in Fig. 3) or closed (type a,, i, k in Fig. 3). In steady flows with open stream lines, the temperature is locally constant with time (dT/dt = 0); for displacements along the flow direction, however, it changes until a fully developed temperature field (DT/Dr = 0 for T , = const) is reached, where conduction and viscous dissipation balance. In processing equipment, the fully developed temperature field is achieved rarely since the flow channels are not long enough and the thermal boundary conditions usually change in the flow direction. Nevertheless, the calculated fully developed temperature field is very useful as a reference state. The degree of development can be estimated from the value of the Graetz number. The unsteady developing temperature ( d T / d r # 0, aT/dz # 0) in flows with open stream lines has been studied very little, possibly because the numerical or experimental techniques are very involved. In shear flows with closed stream lines, the temperature is assumed to be uniform along the stream lines, but locally changing with time (dT/a@= 0; dT/dt # 0) during the starting phase. The stream lines are supposed to be circles, and 0 is the coordinate in the flow direction. Convective heat transfer has no influence on the temperature field. After some developing time, a constant temperature field is reached where viscous dissipation and conduction balance. The degree of development can be estimated from the value of the Fourier number. Drag flow in a narrow slit (plane Couette flow) is introduced twice (al and a, in Table V). One might treat it as an entrance value problem (as in a,) and study the axial development of the temperature field beginning from
222
HOWTH. WINTER
some inlet temperature distribution. On the other hand, one might treat plane Couette flow as a limiting case of circular Couette flow (as in a2); i.e., the stream lines are thought to be closed, and there are no changes in flow direction (d/d@ = 0). The temperature develops with time, beginning with some initial temperature distribution. The fully developed temperature field is the same for both cases. A. THERMAL BOUNDARY CONDITION
The specific heat flux 4 at the boundary is given by the thermal conductivity kRuidof the fluid together with the temperature gradient (aT/dr), in the fluid layer next to the wall 4 = -ktluici(JT/ar)w*
(2.20)
The problems of this section are described in cylindrical coordinates z, r, 0, where I is the coordinate perpendicular to the wall. If the thermal boundaries are not taken to be isothermal ( T , # const), the thermal development in the fluid is connected with the thermal development in the wall. The heat flux at the boundary is determined not only by the conduction to the outside of the channel, but also by the thermal capacity of the wall. Both effects will be analyzed separately in the following. 1. Biot Number and Conduction to Surroundings
If the effect of energy storage in the wall is of no influence, the heat flux at the boundary generally depends on the difference of the temperature level of the experiment to some temperature of the surroundings. In the analysis, the temperature gradient in the fluid layer next to the wall is taken to be proportional to the outer temperature difference (T, - T,); T, is the temperature of the surroundings, and T , is the wall temperature, i.e., the temperature at the boundary between melt and containing wall. The coefficient of proportionality is the Biot number [69,71] of equation (dT/Jr), = Bi(T, - TJh.
(2.21)
Bi is already well known for describing the thermal boundary condition during the heating or cooling of solid bodies (see for instance [97,98]). h is a characteristic length of the flow channel, i.e., the gap width of a slit or the radius of a pipe. Equation (2.21) describes just the radial heat flux in the wall; the axial heat conduction in the wall is neglected in the Biot number. For shear flow applications with closed stream lines, the validity of this assumption has to be verified in each case. However, the assumption seems to be reasonable
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
223
for shear flow applications with open stream lines, where the axial temperature gradient is much smaller than the radial one; during thermal development, the heat flux into the wall changes in the flow direction, but beyond a certain distance from entrance into the channel, these changes are small due to a small axial gradient. The value of Bi for certain applications can be derived from a heat balance for the wall. I n some cases Bi is a function of geometry and of the thermal conductivities only. The heat flux through the wall of a pipe, for instance, can be determined from the inner radius rp of the pipe wall, the wall thickness s, the thermal conductivity kwall,and the inner and outer temperatures T , and T , [98, p. 711. (2.22) On the other hand, the heat flux into the wall is determined by the thermal boundary condition (Eqs. (2.20) and (2.21)) (2.23) Thus, for steady pipe flow with controlled temperature at the outer wall ( T I = T w ;T , = Ts), the Biot number can be calculated by equating Eqs. (2.22) and (2.23): (2.24) Applying this formula to capillary viscometry (rp= 0.5 cm, s = 4.5 cm), one finds values of Bi w 20. Examples for pipe flow with 1 I Bi I 100 are given in Fig. 8 of Section II.B.4. Similarly, the outer Biot number Bi, for annular flow (with ri and r, as the inner and outer radius) would be (2.25) and some examples for pipe extrusion give 1 < Bi, < 10. Figure 4 illustrates the geometrical meaning of Bi. The tangent to the temperature curve T(r/h)at the wall passes through a guide point outside the flow channel; the distance between the guide point and the wall is Bi-', and the ordinate is the surrounding temperature T,. For Bi = 10, for instance, the distance of the guide point from the boundary is 1/10 of the gap width h for annular flow or 1/10 of the radius r, for pipe flow. When the Biot number changes in flow direction, one can visualize this by the appropriate displacement of the guide point.
HORSTH.WINTER
224
FIG.4. Thermal boundary condition for channel flow described by the Biot number Bi and the surrounding temperature T,, i.e., by a guide point outside the channel.
The boundary condition for the temperature field is not known in general. If there are temperature data T,(z) available, they can be used in a numerical program. But often one has to guess these conditions to make an estimate of the temperature profiles possible. Most of the studies shown in Table V prescribe idealized conditions such as: constant wall temperature T,
=
const
Bi, -, - co, Bi, +
or
00;
(guide point at the wall),
constant heat flux at wall (dT/dr),
=
const
or
Bi(T, - T,) = const,
adiabatic wall (aT/dr), = 0
or
Bi = 0; (guide point at infinity).
The use of the Biot number allows one to adopt more realistic thermal boundary conditions, and one goal of further experimental heat transfer studies should be the measurement of Bi in various engineering applications. For the examples shown throughout this analysis, the boundary condition at the wall will be described b y Bi and T, independent of z (for steady flow with open stream lines) or independent oft (for flow with closed stream lines), respectively. If the value of Bi is finite, the wall temperature T,(z) or T,(t) changes according to the development of the temperature field, and it reaches in the fully developed temperature field. a constant value Tw,m
VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS
225
2. Thermal Capacity of the Wall The Biot number is appropriate for describing heat conduction to the surroundings. However, if the wall stores some energy during thermal development, a different boundary condition is needed to describe this effect. The thermal development should be calculated for the fluid and the wall together. This has been done by Powell and Middleman [92] for plane Couette flow with one wall having a finite mass, which absorbs part of the heat generated by viscous dissipation. The thermal development was found to be significantly retarded by the response of the boundary; the parameter characterizing the retardation is the ratio of mass times heat capacity of the solid wall and that of the fluid: (mcp)wa,,/(mcp)R,,d. In this study, however, detailed calculation of the temperature field in the wall will be-avoided by introducing a capacitance parameter C . For flow with closed stream lines, the wall temperature changes with time during the thermal development. The rate of thermal energy stored in the wall is assumed to be proportional to the time change DT,/Dt of the temperature at the boundary. The temperature gradient in the fluid layer near the wall becomes
(2.26) The capacitance parameter C is dimensionless, and the ratio hla,,, is used in Eq. (2.26) because below the whole boundary condition will be made dimensionless. C is determined by the geometry and by the capacitance of both the fluid and the wall. The heat flux to the surroundings is kept proportional to the outer temperature difference T, - T,. Assuming constant thermal properties of the wall material, the rate of energy storage in the wall is proportional to the time change of the average temperature of the whole wall. The time change of the average temperature of the wall might not be proportional to the time change DT,/Dt at the boundary. Thus, the capacitance parameter describes the effect of energy storage in the wall only approximately. In many polymer engineering applications, however, the thermal development in the wall is much faster than the thermal development in the fluid, and the temperature at the boundary is representative for the whole wall. An example where uniform temperature in the wall is assumed will be given in the following. Example for C: The temperature of the inner cylinder of a Couette system (geometry i in Fig. 3, ri = inner radius, r, = outer radius, h = r, - r i ) changes during the thermal development of a shear experiment. The temperature of the inner cylinder is assumed to be uniform and equal to the
HORSTH. WINTER
226
temperature at the boundary; this is justified, if the ratio of the Fourier numbers (see Section II.C.2) is small: (2.27)
Axial heat conduction is neglected and, of course, the Biot number is equal to zero. The heat flux into the inner cylinder is balanced by the temperature raise of the inner cylinder: 2nri(aT/ar)w
kRuid
=
nri2(PC)cylindcr
aTw/at.
(2.28)
By comparing Eq. (2.28) with Eq.(2.26)one finds the capacitance parameter for the Couette system: (2.29) The influence of C on the thermaI development will be shown in Figs. 23 and 24 of Section II.C.3. For most of the steady heat transfer problems with open streamlines, the walls are stationary or just rotating about the z axis. The capacitance of the wall has no influence on the temperature (DT,,,/Dt = 0). An exception would be the inner boundary of axial Couette flow (geometry b or g of Fig. 3) as occurs in the wire coating die. The corresponding thermal boundary condition is vwh aTw Ts - Tw + ci--. = Bi, (2.30) h athid aZ u, is the axial velocity of the wall (inner cylinder). For the example of a wire coating process, a heat balance for the inner cylinder (wire) leads to the same formula for the capacitance parameter as for the Couette system above, Eq. (2.29). The assumptions made were uniform temperature in cross section of wire and no axial conduction in the wire. The Biot number for the wire is zero. The geometrical meaning of the boundary condition with capacity and conduction to the surroundings is shown in Fig. 5. The guide point is not at a constant position (as for flow with DT,/Dt = 0; see Fig. 4),but moves during the thermal development along T, = const. For the fully developed temperature field, the wall temperature does not change any more; the capacitance of the wall is of no influence, and the guide point is, as in Fig. 4, at a distance Bi- from the boundary. An adiabatic or perfectly insulating wall would be a wall without thermal capacitance (or with an appropriate heat source of its own); the corresponding Biot number and the capacitance parameter are both equal to zero.
);r
W
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
227
displacement of guide point during thermal development
FIG.5. Thermal boundary condition for a wall with thermal capacitance.During the thermal development,the guide point of the tangent on the temperature field moves toward the position (Bi- T J .
’,
B. STEADYSHEARFLOWWITH OPENSTREAMLINES Steady shear flow with open stream lines could be analyzed now by going into each of the shear flow geometries al-h in Fig. 3. Instead, it will be demonstrated here that the helical flow geometry is representative since all the other geometries are limiting cases of helical flow. In helical flow, the fluid flows through an annulus between two concentric cylinders (Fig. 6). Axially, the fluid flows due to a pressure gradient and/or due to the axial movement of the inner (or outer) cylinder. In the circumferential direction, the fluid flows due to the rotation of the inner cylinder. Fluid elements move on helical paths; the angle of the helices flow due to pressure gradient /and due to axial movement
path of !bid particle /
/
/
1
//// / / /// / / / /’/ ’,’,.’//,’/,’////
/ / / / ’ i//
FIG.6. Helical flow geometry.
flow due to rotation of inner cylinder,
HORSTH.WINTER
228
depends on the ratio of the axial to the circumferential velocity component, which both depend on the radial position of the fluid element. The annular geometry is characterized by the ratio of the radii IC =
(2.31)
ri/ra,
The limiting cases are the pipe ( I C = 0) and the plane slit ( K tion in the annulus is given by dimensionless coordinates : r - ri ra - ri
radially:
Y
= ___ ,
axially:
2
=
Z ~
1 Gz’
-, 1). The posi-
O I Y I 1 ;
(2.32)
0 I Z I Gz-’.
(2.33)
The value of Z indicates, as will be shown in the following, to what degree the temperature field is developed along the channel. The Graetz number Gz will be defined in Eq. (2.56). 1. Assumptions and System of Equations
The equations of change, Eqs. (1.1)-(1.3), have to be simplified before they can be solved. First the assumptions will be listed, then they will be commented upon: incompressible fluid with constant thermal conductivity and diffusivity ; steady laminar flow (a/& = 0); rotational symmetry (a/a@ = 0 ) ; velocity gradients
no slip at walls; inertia negligible ; kinematically developed velocity at z = 0; gravity negligible; viscosity measured at constant temperatures and constant shear rates gives applicable local values of the viscosity during temperature changes and during small changes in shear rate; rheologically developed stress at z = 0; convective heat transfer much larger than conduction in flow direction; heat transport toward the walls by conduction only. Throughout this section the molten polymer is taken to have constant density ( p / p = 1; E = 0), constant thermal conductivity (k/E = l), and constant thermal diffusivity (a/a = 1). In the general system of equations for helical flow, however, these properties have been kept as variables, and one might
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
229
evaluate them in the numerical program using p ( p , T ) , Up, T ) ,and u(p, T ) data from measurements on the fluid at rest. The density is assumed to be constant since in actual experiments (with T # const, p # const) density changes are delayed, i.e., the changes are overestimated if one applies p data of equilibrium thermodynamics. For this reason, the effect of expansion cooling is not considered in all but one example. For the one exception, the supposed expansion cooling term (containing E T )of Eq. (1.8) is kept in brackets in the energy equation; the effect of expansion cooling is estimated in an example of pipe flow; see Fig. 13 with E # 0. During the axial development of the temperature field, the temperaturedependent viscosity is changing and causes the shape of the velocity profiles to change accordingly. For continuity reasons, the changes of the axial velocity require some radial flow. Using the equation of continuity, Eq. (l.l), the radial velocity components have been estimated from the change of the lo-’ ij). axial velocity component and have been found to be small (lo,( Throughout Section ILB, the influence of the radial velocity components on the radial heat transfer, on dissipation, and on the viscosity will therefore be neglected. Due to rotational symmetry and due to du,/dr and r a(u,/r)/dr being the largest gradients, the isotropic pressure is taken to be a function of z only:
-=
P
=
P(4.
The shear rate becomes
(2.34) which is the root of the second invariant of the rate of strain tensor. In most applications, molten polymers do not slip at the wall, and in all the published heat transfer studies this assumption has been made. Polymeric materials such as high density polyethylene, polyvinylchloride, or polybutadiene, however, seem to slip in certain ranges of the normal stress and shear stress at the wall [99,100]; the velocity field is then drastically changed and additional frictional heating occurs on the sliding surfaces. The velocity at the entrance ( z = 0) is assumed to be fully developed; i.e., inertial effects are neglected, and the stress is assumed to be governed by the three viscometric functions (of steady unidirectional shear flow) at the local shear rate and the local temperature. For low Reynolds number pipe flow of inelastic liquids, the kinematic development is practically completed after a length of 1 = O.lra Re [loll. Neglecting inertia might therefore be justified for entrance flow of molten polymers, which is low Reynolds number flow.
230
HORSTH. WINTER
The rheological properties of the polymer entering the annulus are determined by a flow and temperature history, which obviously is different from that for steady shear flow. Judging from the measured pressure profiles along a slit die, steady shear flow might be reached practically at l/h = 20-30 (depending on geometry, flow rate, and polymer melt). Therefore, the heat transfer study of this section may give unrealistic results for flow in short annuli and short circular holes, which rheologically should be treated as an entrance flow problem. The equations for conservation of mass, momentum, and energy in helical flow are
(2.35)
The average axial velocity is u, =
2 ra2 - riz
u,r dr.
(2.36)
The initial and the boundary conditions are T(r,0) = T,(r)
(2.37)
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
23 1
The meanings of Bi and T , have been described already in Section 1I.A on the thermal boundary condition (Bi, 0). The capacitance parameter C of Eq. (2.30)has been omitted here since the temperatures are assumed to be steady and since the walls are stationary for most shear flow applications with open stream lines. For the dimensionless presentation of the equations, a reference velocity B can be defined by vector addition of the mean axial velocity b, and the mean circumferential velocity ~ , , ~ / 2 : -
v =
[vz2 + (VeJ2) 2 ] 112 .
(2.38)
The reference length is taken to be the gap width
h = r, - r i . (2.39) For pipe flow the reference length becomes equal to the pipe radius (h = ra). Using the reference velocity 3 and the reference length h, one can define a reference shear rate p = b/h (2.40) and a reference viscosity (2.41) rl = v(74 To). T o is a characteristic temperature level of the experiment, for instance the average melt temperature at the inlet ( T o = Te).Flow problems with viscous dissipation do not have a characteristic temperature difference to which temperature changes can be related. Some authors relate the temperature to the temperature level T o ;this, however, seems to be rather arbitrary since T / T o might assume different values in otherwise similar processes (e.g., at different temperature levels T o and To').The value of T / T oadditionally depends on the choice of temperature scale. Therefore, the temperature coefficient of the most temperature-sensitive property, the viscosity, has been used to define the dimensionless temperature: (AT)ref= 0-l. The dimensionless variables are:
velocity
(vR>
v@,
=
( v ~ / u@/B, ~ , uz/D)?
(2.42)
pressure gradient
(2.43)
shear stress
(2.44) (2.45)
radial position (2.46)
232
HORSTH . WINTER
viscosity
temperature
The dimensionless form of the equations becomes
(2.49) (2.50) (2.51 )
The dimensionless average axial velocity is
VZ =
2
1
VzR dR,
(2.53)
and the initial and boundary conditions are
(2.54)
VISCOUS
DISSIPATION IN FLOWING MOLTENPOLYMERS
233
In the expansion cooling term of the energy equation, the absolute temperature T is not replaced by the dimensionless temperature 9 since the dimensionless product ET can be considered as constant within the accuracy of the calculations. The system of equations is formulated in cylindrical coordinates, and R = r/ra is kept as a dimensionless coordinate, even if h = r, - ri is the reference length and not rs. The results are presented using the dimensionless coordinate Y . If one wants to avoid this inconsistency, the substitution
R = Y(l -
K)
+ K,
dR = dY(1 -
K)
eliminates R from the equations; with this substitution the equations look unnecessarily complicated and therefore both coordinates are kept : R in the equations and Y in the graphical presentation of the results. For pipe flow, R and Y are identical.
2. Dimensionless Parameters The problem as stated in Eqs. (2.49)-(2.54) is completely determined by six dimensionlessparameters (Na, Gz, K , m, Vz,L ) together with the boundary conditions. A general description of the dimensionless parameters has been given by Pearson [3]. If the pressure dependence of the viscosity or nonconstant thermal properties would be included, the number of parameters would increase accordingly. The equation of motion and the equation of energy, Eqs. (2.35),are coupled by the temperature-dependent viscosity. The extent of the coupling increases with the value of the Nahme number [44] : Na
=
j02q/X
(2.55)
which compares the dissipation term with the conduction term in the equation of energy. For values of Na greater than 0.1-0.5 (depending on geometry and thermal boundary conditions), the viscous dissipation leads to significant viscosity changes, i.e., changes reflected in the T and u fields. For smaller values of Na, isothermal conditions can be achieved practically; in this case, the equation of motion can be integrated independently of the energy equation. In some studies the Brinkman number [62] Br = B2ij/kTo has been used instead of the Nahme number. However, Br contains the arbitrary temperature level To (since no characteristic temperature difference is available) and may, therefore, have very different values for similar processes. The value of Br does not give any information on the extent of the coupling between the equation of motion and the energy equation. (Note that the Nahme number sometimes is called the GrBith number after Griffith [102], who used the same dimensionless group in one of the later applications.)
HORSTH. WINTER
234
The energy equation contains a convection, a conduction, and a dissipation term. By comparing the convection and the conduction terms one arrives at the Graetz number [1031
GZ = Vzh2Jzil
(2.56)
which has been included in the dimensionless form of the z coordinate. The Graetz number can be understood to be the ratio of the time required for heat conduction from the center of the channel to the wall and the average residence time in the channel [75]. A large value of Gz means that heat convection in flow direction is more important than conduction toward the walls, Gz = 100, for instance, is a common value for extruder dies. (Note that some authors define the Graetz number Gz n.) The Gz number has been defined with the average axial velocity and the length of the annulus, instead of the reference velocity ij and some mean path length Tfor the fluid elements in the annular section. One might, however, define Tto be T = 1ijpzwhich would result in a Graetz number Gz = ijh2/uTequal to the one defined in Eq. (2.56). The value of the dimensionless average axial velocity 2
Y.=B=[I+(gJ]
-1/2
,
V
O
(2.57)
describes whether the flow tends to be closer to axial flow in an annulus
(Vz= 1) or closer to circular Couette flow (Vz= 0). The dimensionless length of the annulus is
L
=
l/h
=
l/(ra -
Ti).
(2.58)
Another dimensionless parameter originates from the shear dependence of the viscosity: the power law exponent m in Eq. (2.47).
3. Universal Numerical Shear Flow Program The system of equations is solved by an iterative implicit method (aJaZ described by a backward difference; alaR and a2faR2 described by center differences;gradients at a boundary are calculated from a parabola through three points), similar to the one used in an earlier study on helical flow [85]. A network is superimposed on the annulus. Difference equations are then derived for each node point, which fulfill the condition of conservations of mass, momentum, and energy. The method described in the following was found to converge rapidly; for example, a run with 60 radial and 250 axial steps requires a computation time of about 30 s. The solution procedure is an iterative one, in which the coupled equations are linearized and solved separately. The nonlinear terms and the coupling
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
235
conditions have to be satisfied by alternating improvements on the velocity and on the temperature field. The iteration is terminated when the relative change in successive steps becomes smaller than one thousandth ( A ~< ~ ,1 0 - 3 ) . The flow chart in Fig. 7 describes the structure of the program. The velocity field at the enrrunce, which is assumed to be fully developed kinematically, is calculated by taking, as a first guess, a Newtonian viscosity according to the entrance temperature field. The shear dependence of viscosity is included then by iteration, using the improved values of the velocity field. After about 6-20 iterations, the velocities reach values that are practically constant. geometry. material and process data, boundary conditions Bi, , Bi., G.,, c,o ve,, , v,,, ; entrance conditions T (r); z=o; t,=l, v e z v,= 0 .
.
dimensionless parameters No , Gz,Vz ,x , dR = const. logarithmic steps sizes AZ, , number of steps no , n -- 0 I
conditions
;
4,. 0
end
FIG.7. Flow chart of universal shear flow program.
236
HORSTH. WINTER
The axial velocity V, and the pressure gradient P' are calculated from the Z component of the equation of motion together with the integral of the flow rate (Eqs. (2.51) and (2.53)). The circular flow V, is evaluated from the 0 component of the equation of motion (Eq. (2.50)).The R component V, of the velocity supposedly does not influence the viscosity and the convection; thus it is calculated separately at the end of each step using the equation of continuity (a numerical method that allows for positive or negative radial flow contributions has been suggested by Gosman et al. [104]). The entrance conditions are then stored, and the fully developed temperature jield is calculated so as to be available as a reference state for the developing temperature field. In the fully developed temperature field, the convective term of the energy equation is zero. The iteration starts out with the viscosities and velocities at the entrance. They give a first approximation of the dissipation term and of the fully developed temperature field. Using this solution, one gets improved values of the viscosities and the velocities by iteration. These values of the viscosities and velocities lead to the second approximation of the fully developed temperature field, and so on. After the values of (9, - ~ ( I c00)) , and satisfying the condition of (A,,J < of (9, - 9(1, 00)) are stored as reference values for the developing temperature field. Then the program goes back to the entrance temperature field and starts calculating the developing temperature, velocity, shear stress, and pressure. If both walls are adiabatic (Bi, = Bi, = 0), there does not exist a fully developed temperature field; the program then starts calculating the developing temperatures immediately. For flow in a capillary (IC= 0), the velocity and the temperature have a zero gradient at R = 0. The power law model fails in describing the viscosity at low shear rates, and for computational purposes at least one has to set an upper limiting value of the viscosity (this has been done in the numerical program of this study). For more accurate calculations, one has to approximate ranges of the viscosity curve by several power law and temperature coefficients. Also a viscosity table could be used instead. The numerical program has been checked with analytical solutions of the fully developed temperature and velocity field in plane Couette flow and with isothermal flow in a pipe and in an annulus [lOS]. Besides helical flow with its steady, but developing temperature field, the system of equations, Eqs. (2.35)-(2.37), describes the flow geometries of all other steady shear flows with open stream lines (type a,-h in Table V). Therefore, it actually is possible to use one numerical program for all these flow cases. The appropriate values of the ratio of the radii IC,the axial velocity of the inner cylinder V,(IC,Z ) , and the average axial velocity Y' are listed in Table VI.
VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS
237
TABLE VI SHEAR
FLOW GEOMETRIES AS LIMITING CASES OF HELICAL FLOW"
Flow geometry as described in Fig. 3
u
V.(JG z)
VZ
a1 b
0.999 o
2 determined by iteration for P = 0
1
0 0.999 O < X < l O<S
-
1
0 0 0
1 1
finite finite
0.999 O<W
0
0
0
0
a2 1
1
O
Listing of the corresponding geometry (described with K ) and kinematics (described with the velocity of the inner cylinder I/Z(K.Z)and the average axial velocity Vz).Geometries a,-h are with open, and geometries a2 and i with closed stream lines.
Due to the parabolical character of the solution procedure for the equation of energy, the helical flow program can be applied only to flows with nonnegative velocity components. Thus, axial drag flow in an annulus with nonzero axial pressure gradient (type g) and drag flow in a narrow slit between two parallel plates with nonzero pressure gradient (type h) can be analyzed only up to moderate positive pressure gradients. A solution procedure that allows for back flow is described in the literature [104], but it does not seem to have been applied to these types of flow. 4. Calculated Results There is a large variety ofheat transfer problems solvable with the universal shear flow program. Some examples follow, mainly concerning the thermal boundary conditions (Biot number) and the kinematics for various shear flow geometries. Similar examples of helical flow or annular flow calculations have already been published, however, with idealized thermal boundary conditions [85]. In all the examples of this section, the entrance temperature (at 2 = 0) is taken to be 9,(R) = 0. The thermal boundary conditions influence the developing temperatures and velocities to a large extent. In analytical studies generally, idealized conditions are assumed, i.e., isothermal or adiabatical wall; in real flow
HORSTH. WINTER
238 I
2
No.5 Bi =lo0
rn = 2 . 5
rn =2.5
=m
31R.Z
1
0.5
A
.?=lo-’
z 40”
C
R
0,5-
R
a5
R
1
1
Ro.8. Influence of the thermal boundary condition, described by a guide point outside the channel given by Bi and 9, = 0, on the developing temperature field in pipe flow. Bi = 100 is close to the isothermal wall condition, while Bi = 1causeslarge changes of the wall temperature.
VISCOUS bISSIPATION IN
FLOWING MOLTENPOLYMERS
239
situations the thermal boundary condition is somewhere between the two. The strength of the Riot number in describing a more realistic kind of boundary condition will be demonstrated on pipe flow: the fluid supposedly enters the pipe at a constant temperature equal to the temperature of the surroundings (9, = 9, = 0). Due to viscous Pissipation, the temperatures increase in flow direction (Fig. 8). For Bi = 100, the wall temperature stays nearly constant, while for Bi = 10 and Bi = 1, the wall temperatures already increase in early development. The fully developed temperature at Bi = 10 has a value between that at Bi = 100 and Bi = 1. At large Bi the temperature in the layer near the wall is kept low; the viscosity and hence the viscous dissipation is large, and a large temperature gradient is needed to conduct away all newly dissipated energy. At small Bi the wall temperatures have to rise signifieantly before the heat flux at the wall can balance dissipation: the temperature gradient can still be relatively small since the viscosity and hence the viscous dissipation become small at high temperatures. At large Bi the temperathreh are high because viscous dissipation is most pronounced. At small Bi the temperatures are high because the conduction toward the surroundings requires large wall temperatures. For intermediate Bi, the fully developed temperature has a minimum. The corresponding pressure gradient P ( Z ) decreases due to the decrease of the temperature dependent viscosity (Fig. 9). The decrease of P'(Z1 below its value at the entrance P(0)is most pronounced at small values of the Biot number, where the wall temperatures increase the most. The temperature gradient at the wall is defined with the Biot number and an outer temperature difference 9, - 9, (see Eq. (2.54)),where $,(Z) itself depends, besides the other parameters, on Bi. The dimensionlesstemperature
FIG.9. Pressure gradient in pipe flow at different thermal boundary conditions; decrease due to the thermal development described in the previous figure. The pressure gradient for isothermal pipe flow can be calculated analytically: P(0) = - 2 ( m + 3)"".
240
HORSTH. WINTER
gradient [as(&2)/dRlw increases with Z till it reaches its fully developed value at about Z = 1 (see Fig. 8).
a. Nusselt Number. For engineering calculations, the specific heat flux q often is described by means of the Nusselt number [lo61 (2.59) and a characteristic temperature difference (AT)ref.The product Nu k/h sometimes is called the heat transfer coefficient. The value of the Nusselt number depends much on the choice of the temperature difference ( A T ) r e f . For flow without signijcant viscous dissipation, one takes it to be the local average temperature difference ( T , - T ( Z ) ) or the average temperature change ( T ( Z ) - T(0))in the flow direction. The average temperature of the melt is chosen to be the “cup mixing temperature”
T ( Z ) = ___ 1 - uz
T ( R , Z)Vz(R, Z ) R dR
(2.60)
which would be the temperature of the homogeneous fluid after mixing (constant specific heat per volume assumed). The Nusselt number in its usual definition is not adequute for describing the wall heat flux in flows with signijicant viscous dissipation. One disadvantage of the use of Nu is the fact that both Nu@) and T ( Z )have to be known to calculate the wall heat flux q ( Z ) ;the main disadvantage, however, is that in Nu an attempt is made to describe two fairly unrelated quantities as a function of each other, the temperature gradient at the wall and the average temperature difference (Tw(Z)- T ( Z ) ) .This will be explained in the following, using pipe flow as an example. Figure 10 shows developing temperature profiles in pipe flow, on the left-hand side with negligible viscous dissipation (Na = 0.001) and on the right-hand side with significant viscous dissipation (Na = 1). The wall temperature is above the entrance temperature (9, = 0; 9, = 0.1); developing temperatures at constant wall temperature (Bi = 00) are drawn as solid lines, while for a thermal boundary condition with Bi = 10 dashed lines are.used. For flow without signijcant dissipation (Na = 0.001), the temperature of the fluid gradually approaches the wall temperature; the temperature gradient at the wall (shown in Fig. 11) decreases monotonically till it becomes zero. However, if there is significant viscous dissipation (Na = l),the temperature in the layer next to the wall increases drastically even before the average temperature is changed much. The temperature gradient changes its sign; the fluid heats the wall, even if the average fluid temperature is below the
VISCOUS
DISSIPATION IN FLOWING MOLTENPOLYMERS
24 1
9IR.ZI
0.:
0.2
~
----- Bi =10 -Bi
=w
S(R, 3, = 0.1
01
O!
z=10-’
0.5
R
1
C
FIG.10. Calculated temperatures in pipe flow with wall temperatures above the entrance temperature. Comparison of the thermal development without (Na = 0.001) and with (Na = 1) significant viscous dissipation. The wall temperature is taken to be constant (Bi = a)or described by a guide point with Bi = 10.
wall temperature. This corresponds to a negative Nusselt number or a negative heat transfer coefficient, which is an unrealistic result. The dimensionless average temperature difference (Fig. 12)
S ( Z ) - 3,(2)= p ( T ( Z ) - T , ( Z ) ) = B A T
(2.61)
is negative at the entrance of the pipe (prescribed initial condition) and at least for Bi = co, it increases monotonically with Z . For Na = 0.001, the fluid approaches the wall temperature (9, = LJ,+,); for Na = 1, the average temperature difference goes through zero and approaches a constant value greater than zero. The Nusselt number Nu(Z), which conventionally is defined with this temperature difference, has a singularity when the average
242
HORSTH.WINTER
FIG.11. Wall temperature gradients for pipe flow with viscous dissipation (Na = 1) and without (Na = 0.001).The wall temperature is above the entrance temperature.
FIG.12. Development of the difference between the average temperature and the wall temperature in pipe flow with viscous dissipation (Na = 1) and without (Na = 0.001); the wall temperature is above the entrance temperature.
VISCOUSDISSIPATION IN FLOWINGMOLTEN POLYMERS
243
temperature difference AT is equal to zero [69,76,82]. The specific heat flux at the wall obviously is finite (see the wall temperature gradient in Fig. 11). This seeming singularity (at Z of AT = 0) suggests that the usual choice of the reference temperature difference A T cannot be applied meaningfully to flow problems with viscous dissipation [69]. The same argument is valid if, instead of a constant wall temperature, a guide point is chosen outside the channel, see Figs. 10-12 with Bi = 10. In the definition of the Nahme number (Eq. (2.55)), p-’ is taken to be a characteristic temperature difference, and for defining a Nusselt number for flow with viscous dissipation one similarly may take (ATLeF = B-’.
(2.62)
Introducing this reference temperature difference into Eq. (2.59),the Nusselt number becomes Nu = hg(dT/dr), = (ds/aY),, (2.63) and it then is identical with the temperature gradient at the wall. Equation (2.63) seems to be an adequate definition of Nu for flow with viscous dissipation. The relation between Nu and Bi is given together with Eq. (2.54).For fluids with practically temperature-independent properties, an adequate definition of the Nusselt number for dissipative flow can be made by means of the “recovery temperature” [98, p. 4171. b. Expansion Cooling. In steady flow of compressible fluids with nonzero pressure gradient, the density will change in the flow direction; the equation of energy, Eq. (1.7),contains a term that describes the cooling or heating due to those density changes cT DplDt, where r T can be determined from Eq. (1.10). In the example of Fig. 13 the influence of expansion cooling is shown on developing temperature profiles in pipe flow (with 9, = 9, = 0). Values of ET = 0, 0.1, 0.2, 0.3 have been used since values of this magnitude can be evaluated from equilibrium thermodynamic data of molten polymers at rest. The applicability of equilibrium data to regions of rapid pressure changes is still an open question. c. Thermal Development. For constant inlet temperature equal to the temperature of the surroundings, the average temperature increases during the development of the temperature field with increasing Z. The temperature is fully developed at 2 = 0.5-2. Although the absolute value of the average temperature depends on the power law exponent m, or Na, and on Bi, the relative development is nearly the same for the different examples of pipe
244
HORSTH. WINTER
i
0
0.5
R
FIG.13. Calculated temperature profiles in a pipe with constant wall temperature equal to the entrance temperature. The magnitude of ET determines the amount of cooling due to expansion with decreasing pressure p; T is the absolute temperature.
flow (Fig. 14).In annular flow, however, the relative thermal development depends on the thermal boundary conditions (Fig. 15); if the dissipated heat can be conducted to both walls (Bi, = -10'; Bi, = lo5), the developing length is much smaller than if one wall is nearly adiabatical (Bi, = -1; Bi, = lo5).For Newtonian fluids (m = l), the channel length required for thermal development is shorter than for fluids with shear dependent viscosity (m = 3; rn = 5, for instance).
d . Zero Pressure Gradient or Zero Wall Shear Stress. The versatility of the program will be demonstrated on some velocity profiles of isothermal flow in an annulus, including the limiting cases of a plane slit (K:+ 1) and pipe (K = 0). Annular flow with zero velocity gradient at the inner wall (which also means zero shear stress at the inner wall) can be achieved with the appropriate pressure gradient and the appropriate velocity V'(K) at the inner wall (Fig. 16). An application of this type of velocity field could be die flow in the wire coating process: operating at low shear stress at the surface of the wire prevents ruptures of the wire. If the pressure gradient in the annulus
VISCOUS DISSIPATION IN FLOWING MOLTEN POLYMERS
245
1 I
10-2
axial
1o-~
10-1 position
1o-z axial
7
10” position
7
1
10
1
10
FIG.14. Development of the average temperature in pipe flow at different Bi and m ;(a) at Na = 1 and (b) at Na = 5.
axial position Z
FIG.15. Development of the average temperature in annular Poiseuille flow at different Bi,, Na, and m. K = 0.5; Q = 0.
HOWTH. WINTER
246
9 I1
x
1.0
-
z
->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
HORSTH. WINTER
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
This Page Intentionally Left Blank
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
