Coupled Transport in Multiphase Systems: A Theory

ADVANCES IN HEAT TRANSFER, VOLUME 31

Coupled Transport in Multiphase Systems: A Theory of Drying

STEPHEN WHITAKER Department of Chemical Engineering and Material Science University of California at Davis Davis, California

1. Introduction

,.\

Multiphase transport phenomena playa dominant role in nearly every branch of engineering, and in many situations the multiphase transport equations for heat, mass, and momentum transfer are inferred from the single-phase analogs. This leads to the use of equations containing effective transport coefficients that one hopes can be measured experimentally, and in some cases this approach is successful. Darcy's law is a classic example in which a spatially smoothed momentum equation can be deduced by the solution of Stokes' equations for laminar flow in a tube, and reliable experiments can be carried out for porous media to determine the permeability tensor. However, single-phase flow in a rigid porous medium is a rather benign process in which the interface is fixed in space and subject only to the no-slip condition. If we move on to consider two-phase flow, we find an entirely different situation involving severe theoretical and experimental challenges. One can contrast the simplicity of the measurement of the permeability for single-phase flow with the very complex experiments required for the determination of the coupling permeability tensors for two-phase flow [lla]. Coupling in two-phase flow occurs at two levels: at the macroscopic or Darcy scale, where the two momentum equations are joined by the capillary pressure-saturation relation, and at the microscopic or closure scale, where the coupling permeability tensors originate [12]. Understanding the coupling at both levels requires a theoretical framework and thoughtful experiments. Drying represents a process in which heat, mass, and momentum transport occur simultaneously and may be coupled in complex ways [4]. A direct experimental attack is usually successful only when experiments can be carried out in the parameter space associated with the design application; however, experiments can be designed that circumvent some of the typical problems

ISBN 0·12·020031·7

ADVANCES IN HEAT TRANSFER, VOL. 31 Copyright © 1998 by Academic Press. All rights of reproduction in any form reserved. 0065·2717/98 $25.00

2

STEPHEN WHITAKER

associated with coupled transport processes. The methodeflash [10] is an example of an experiment that can be used to measure the effective thermal conductivity in a partially saturated porous medium. If one subjects a uniform, partially saturated porous medium to a steady-state heat conduction process, evaporation, condensation, and capillary flow occur and the liquid distribution within the system becomes nonuniform. The interpretation of experimental results for such a system is difficult. With the methode flash, a uniform, partially saturated porous medium is subjected to a short thermal disturbance that allows one to probe the system without significantly disturbing the liquid distribution. In this manner, one can obtain values of the effective thermal conductivity as a function of the saturation [1] for well-defined conditions. An experiment analogous to the methode flash that could be used to measure the effective diffusivity in a partially saturated porous medium does not appear to exist. When the method of volume averaging was first used to analyze the drying process, a rigorous method of closure was unknown. Since that time, we have gained a great deal of experience with closure problems and solutions of those problems. It is the purpose of this article to apply that experience to a multiphase transport process that illustrates significant coupling. In the drying process, important coupling occurs at the macroscopic level, or the Darcy scale, and important coupling occurs at the microscopic level, or the level of closure. The theoretical framework for the coupling that occurs at the closure level was first presented by Moyne et al. [13], and the experiments that were performed in conjunction with that work are described by Azizi et at. [I]. In this study, we consider a partially saturated porous medium composed of rigid, impermeable particles. Adsorption at the gas-solid surface is neglected, and the gas phase is treated as a pseudo-binary system. The system under consideration is illustrated in Fig. 1, where we have suggested a typical drying process in which warm, dry air is passed over a wet porous medium. We have identified the rigid, impermeable solid phase as the O"-phase, the pure liquid phase as the jJ-phase, and the air-water vapor phase as the y-phase. For the process of drying a wet porous medium, illustrated in Fig. I, or the process of drying wet porous particles, illustrated in Fig. 2, one can develop reasonably sound arguments in favor of a diffusion model of drying. This model is developed in Sec. XIII and it provides the following moisture transport equation:

as

-

= V . (K s ' VS)

at'

capillary flow

,

+V

. (Kg' pfJg)

'------v------'

L'

gravitational flow

a'lfo [eyO~ff +V { '-

a(T) pfJ(1 - eO')

+

eyO:ff pp(1 - err)

-

eyO~ pfJ(1 - err)

J·VT (>}

~~~

passive diffusion

enhanced diffusion

multiphase thennal diffusion

(I)

COUPLED TRANSPORT IN MULTIPHASE SYSTEMS

wann,drY,lJ[

FIG.

I. Drying porous media.

,.

FIG.

2. Drying porous particles.

3

4

STEPHEN WHITAKER

Here S is the saturation and the first term on the right-hand side of Eq. (1) represents the liquid-phase transport owing to capillary forces. The second term represents the flow caused by the gravitational force, and this term is often neglected when it should not be [22]. The second set of terms on the right-hand side of Eq. (1) describes the gas-phase transport, which has been represented in terms of three separate diffusivities that originate from related physical processes. In the derivation of Eq. (1), we have made use of the intrinsic average vaporphase density of water and its gradient in the form (2) Here (T) is the spatial average temperature, and the function ?:fo is given explicitly by

{l1hva II)}

p~y

p (

?:fo(T») = RA(T) exp -~ (T) - TO

(3)

Equations (2) and (3) are based on the conditions of local mass equilibrium (Sec. VIII) and local thermal equilibrium (Sec. IX). The former allows us to make use of an interfacial equilibrium relation to determine the volume-averaged gasphase density of water, and the latter allows us to represent the temperature of all three phases in terms of the single-volume average temperature (T). While the Kelvin effect has been incorporated in the general analysis, the contribution ofthis effect has been discarded in Eqs. (2) and (3). The three diffusivities that appear in Eq. (1) have been estimated at 40°C as Gy O~ff --=-Cy CfD AB 2

(4)

The first of these representations is quite reliable, since it is confirmed by both theory and experiment. The second and third estimates are less reliable, since they are based on an interpretation of the complex coupled closure problems described in Sec. XII. Both O~ff and 01 are weak functions of temperature, whereas O~ff depends directly on the parameter (l1hvapCfDABlky)(a?:fola(T») and thus increases with increasing temperature. When convective heat transport in both the liquid and gas phases can be neglected, the energy transport equation takes the form

(P)Cp

a(T) ° = V . (Keff ' V(T») at '-----,,0--0"

-

+ V . [(l1hvapCyOeff)(a?:fola(T») . V(T)]

passive conduction

+ V . (KtJ . V(T) coupling at the closure level

'I

,

coupling at the macroscopic level

(5)

5


in which the contributions to the coupling at the closure level are given by (6) For a typical air-water-solid system at 40°C, the first two terms representation for K:ff have been estimated in Sec. XIII as

K:ff = O[Kia2Fla(T»)] = 0 {(

I - e )epe (k Ik) } a l~ " y K~ff er + ep

III

this

(7)

and this suggests, but does not prove, that these two terms may make an important contribution to the overall effective thermal conductivity. The estimate for the last term in Eq. (6) is given by ~hvaperD

*T(a'!:Yla en;; ( ) _ T )-

0

{(1 - ea)epe K;;ffo} y

ey + IOep

(8)

and this is smaller by a factor of k/k" than the first two terms. All of these estimates are for a temperature of 40°C, and since the terms that make up K:J depend on a2Fla(T), they will increase with increasing temperature. The estimate of the evaporation-condensation term in Eq. (5) is given in terms of the passive effective diffusivity tensor according to

o

en;;

~hvapeyDerr(a'!:Yla(T»)

_

- 0

{(1 - e )ey K;;ffo} a

(9) er + IOep This result suggests that evaporation-condensation is not very important at 40°C; however, the effective diffusivity that controls the energy transport by evaporation and condensation at the macroscopic level is not D~ff but the total effective diffusivity Deff . This coefficient is larger than D~ff and increases with increasing temperature; thus Eq. (9) underestimates the importance of evaporation and condensation. The experimental work of Azizi et al. [1] clearly indicates that the two coupling terms in Eq. (5) make a major contribution to the overall effective thermal conductivity for temperatures above 40°C, and this is consistent with the theoretical calculations of Moyne et al. [13]. The derivation of Eqs. (1) and (5) begins in Secs. II and III, where the basic differential equations describing heat, mass, and momentum transport will be reviewed and the essential features of the method of volume averaging will be presented. In Sec. IV, we will identify the special forms of the governing equations and boundary conditions that are applicable to the process illustrated in Figs. I and 2, and in Secs. V, VI, and VII we will derive the volume-averaged forms of the governing differential equations. The thermodynamic relations will be given in Sec. VIII, and there we will discuss the key concept of local mass equilibrium. This will be followed by a short section on local thermal equilibrium, and we will then move on to the closure problems for mass and energy in Secs. X and XI. The details of the coupled closure problem will be presented in Sec.

6

STEPHEN WHITAKER

..

'

~-

.

-;'"

'~

'.",

.)

",

, " ". ';!i~,. ." ,.,;,." ,"', ~ ,

'"t~

FIG,

'.~,

",

~

, ..

3, Drying porous media.

XII, and the results from that section will be used to develop the closed form of the mass and energy transport equations in Sec. XIII.

II. Basic Equations The general system under consideration is illustrated in Fig, 3, where we have shown a macroscopic region and an averaging volume containing the solid, liquid, and vapor phases. The solid is identified as the a-phase, the liquid as the ,B-phase, and the vapor as the y-phase. The gas phase typically consists of air and water vapor while the liquid phase is usually water, and this is the system that we will consider in our study. In order to predict some volume-averaged measure of the moisture content and the temperature, we need to develop the volume-averaged forms of the mass, momentum, and energy equations. These basic equations are given in the next section, A

MASS

The continuity equation for a multicomponent system can be expressed as

apA

-a t+,V ' (PAVA), = ~

accumulation

convection

rA

~

A

=

1,2, , .. ,N

(lO)

homogeneous reaction

and the constraint on the mass rate of production of all speCIes owmg to homogeneous chemical reaction is given by (11)

•

7


The total continuity equation takes the form

ap

- + V·

at

(pv)

(12)

0

=

in which the mass average velocity and the total mass density are defined by A~N

V =

L WAVA

mass average velocity

(13)

A~l

A~N

P =

L PA

(14)

total mass density

A~l

We will consider PA to be the mass density of water, and prediction of the volume average of this quantity is the key objective of any theory of drying. The solutions to Eq. (10) are generally accomplished by decomposing the species velocity into the mass average velocity and the mass diffusion velocity according to (15) This leads to the following form:

apA

at

+ V . (PAV)

'--..-'

accumulation

=

-V· (PAllA)

+

A = 1,2, ... ,N

rA

'----".----'

'-v-----'

'--..-'

convective transport

diffusion transport

homogeneous reaction

(16)

which indicates the need to determine both the mass average velocity and the diffusion velocity. These velocities can be determined by application of the laws of mechanics. B.

MOMENTUM

The species linear momentum can be expressed as [40]

a

B~N

- (PAVA) + V· (PAVAVA) = PAbA + V· TA + at '-------v-----'--,-' '-v--'

~ oca acceleration

convective acceleration

body force

surface force

+ rAVA

L PAB

B~ 1

~ 1 SlVe force

A = 1,2,3, ... , N

(17)

'--..-'

source of momentum owing to reaction

while the angular momentum equation takes the simple form given by A = 1,2, ... ,N

(18)

The N species momentum equations given by Eqs. (17) are needed to determine the N species velocities that appear in Eq. (10). Rather than work with these N

8

STEPHEN WHITAKER

equations, we make use of N - 1 of the momentum equations in the form of the Stefan-Maxwell equations, given by A

=

1,2, ... , N - 1

(19)

in addition to the total momentum equation, which can be expressed as

a at

- (pv)

+ V . (pvv)

= pb

+ V .T

(20)

For drying processes, one can normally neglect the inertial terms and use the model of a Newtonian fluid to express the total momentum equation as (21) This form assumes that the only body force is the gravitational force, and this is a reasonable simplification for drying processes. Variations in the viscosity will certainly occur because of variations in the temperature; however, for most drying processes one can argue that (22) and Eq. (21) becomes an acceptable simplification for a drying process. If the system under consideration (air and water vapor) can be treated as a pseudo-binary system, the Stefan-Maxwell equations reduce to Fick's law and the mass transfer process is greatly simplified. For a three-component system (A, C, and D), the Stefan-Maxwell equation for species A can be expressed as

0=

-VXA

+

XAXcCVC -

VA)

XAXD(VD -

VA)

+ -----

CZlJ AC

(23)

CZlJ AD

Here we have in mind that species C and D represent oxygen and nitrogen, and if we accept the simplification that the species velocities for oxygen and nitrogen are equal, we can write (24)

Vc = VD = VB

This allows us to express Eq. (23) in binary form, given by

0=

XAXB(VB -

-VXA

VA)

+ -----'-------'--

(25)

CZlJ AB

in which the pseudo-binary diffusion coefficient can be expressed as

CZlJ AB

------ + -----(1 + XD/Xc)CZlJ AC (1 + XclXD)CZlJ AD

(26)


9

If the ratio of mole fractions is constant, variations in q]jAB owing to concentration changes can be ignored. From Eq. (25) one can extract all the classic variations of Fick's law [2], and for our purposes the most useful form is given by PA UA =

Here

WA

-Pq]jABVWA

(27)

is the mass fraction of species A, which is defined as WA = PA/P

(28)

Care must be taken when using the Stefan-Maxwell equations, or Fick's law, to note that various restrictions have been imposed on the species momentum equations in order to derive Eq. (27). These restrictions are discussed by Whitaker [40], and for the case of drying one of the most important is given by [47,48] fly q]jAB --2-

py ly

,, (310)

54

STEPHEN WHITAKER

In addition to the three intrinsic average temperatures that appear in Eqs. (310) through (312), we wish to make use of the spatial average temperature

(T)=~f "If

TdV

(313)

+ ep(Tp't + ey(Ty)Y

(314)

'V

which can also be expressed as (T)

=

ey(Tar

Under certain circumstances, the three averaged temperatures, (Tar, (Tp't, and (TyY, will be close enough so that they can be set equal, i.e.,

(Ta)"

=

(Tp't

=

(Ty)Y

=

(T)

(315)

When this represents an acceptable approximation, the condition of local thermal equilibrium is valid. In the original development of the volumeaveraged thermal energy equation for the drying process [38], the concept of local thermal equilibrium was suggested as a footnote. Since that time, local thermal equilibrium has been studied with increasing intensity [27, 32b, 39, 43,49]. In the most recent study [32b], the constraints associated with local thermal equilibrium were successfully compared with numerical experiments for transient heat conduction in two-phase systems. That type of intense examination of the condition of local thermal equilibrium has yet to be carried out for the drying process; nevertheless, we can draw upon all the previous studies to conclude that local thermal equilibrium is generally satisfied for drying processes that are diffusion-controlled. This means that we can add Eqs. (310) through (312) to obtain a one-equation model [32b] for the energy transport, and this requires the application of the appropriate interfacial boundary conditions. We begin our studies of the boundary conditions with the liquid-solid interface and recall the conditions given in Sec. IV.

/l-(J

interface

qp . na

=

qa . upa

Tp

=

Ta

at slpa

at slpa

(316) (317)


55

In tenns of the spatial deviation temperatures (see Sec. III.D), the second boundary condition can be expressed as (318) and the imposition oflocal thennal equilibrium, as indicated by Eq. (315), leads to the following condition for the spatial deviation temperatures: (319) This procedure can be repeated for the solid-gas and gas-liquid to obtain a-y interface

(320) (321) y-fJ interface

(322) (323) With these boundary conditions, we can add Eqs. (310) through (312) to obtain

kp - ky + -V-

J

-

Dpy Tp dA

ky - ka + -V-

Apy(t)

J

- J

Dya Ty dA

Aya(t)

- pp(cp)pV . (vpTp) - V . [(cp)y(py)Y(vyT)] - (cp)y(/'iyv y) . V(T) + (