Simultaneous Heat, ...Theory of Drying 1977 (1).pdf

7/28/2019 Simultaneous Heat, ...Theory of Drying 1977 (1).pdf

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Reprinted from:ADVANCES IN HEAT TRANSFER, VOL. 13@1977

ACADEMIC PRESS, INC.NEW YORK SAN FRANCISCO LONDON

Simultaneous Heat, Mass, andMomentum Transfer in Porous Media:A Theory of Drying

STEPHEN WHITAKERDepartment o/Chemical Engineering, University o/California at Davis,

Davis, California 95616

I. Introduction. . . . . . . . . . . . . . . . . .II. The Basic Equations of Mass and Energy Transport.A. Governing Point Equations .B. Boundary Conditions . . . . .C. Volume Averaged Equations ..III. Energy Transport in a Drying ProcessA. Total Thermal Energy Equation .B. The Effective Thermal ConductivityC. 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 PhaseA. Darcy's Law for a Discontinuous Phase.. B. A C o n ~ t i t u t i v e Equation for the Forces Acting on the Liquid Phase.VI. Solution of the Drying Problem.VII. The Diffusion Theory of Drying.VIII. Conclusions .NomenclatureReferences . . .

I. Introduction

119126128133137153154158164165166169175175184192194198199200

Man has been keenly aware of the importance of drying in a porous mediasince the first clay bowl was shaped by hand and set in the sun to dry. Underthe proper conditions [1] a rock-hard product was obtained and the art of119


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120 STEPHEN WHITAKERpottery was advanced. The dehydration of foodstuffs has also been practicedfor centuries, and the proper management of our agricultural products andour energy resources now requires an improved understanding of the dryingprocess in biological materials. The importance of the process of dryingsolids has been dramatized by Lebedev and Ginzburg [2] who stated in theirreview 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." I f this is indeed thecase, then surely our knowledge of the drying process deserves improvement.Fulford [3] has characterized our situation with the statement that "anyonefaced with engineering a new drying system rapidly realizes that solids dryingremains largely an art," and in the 1970 I & EC biannual review on dryingMcCormick [4] assumed a similar, pessimistic position with the statementthat there has been a "decline in quantity and quality of unit operationsresearch in the United States." Any decline that has occurred has only beenrelative to the enormous progress made in the analysis of the well-posedproblems of simultaneous heat and mass transfer. While computer solutionshave cut a swath through the maze of boundary value problems associatedwith engineering science research, the complexities of the porous structureencountered 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 thatof Lewis [5] who postulated that drying consisted of two processes: (1) thediffusion 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 termmoisture apparently referred to the liquid state, and the term diffusion clearlyreferred to a mechanism comparable to molecular diffusion in a multicomponent, single phase system. The suggestion by Lewis that drying was adiffusional process was picked up by Sherwood and a series of papers [6-9]on the drying of solids resulted. These were all based on a "diffusion" equation of the form

(1-1)where C represents the vaguely defined "moisture content" and D representsa parameter which is determined by experiment. Sherwood also discussedthe possibility of diffusion in the vapor phase and considered the heat transferphenomena associated with drying. However, neither of these phenomenawas considered to be particularly important, and it was the diffusion ofliquid through the porous solid that occupied the center of attention. Thesolutions given by Sherwood were extended by Newman [10, 11] to othergeometries using the appropriate modification of Eq. (1-1). Further work byGilliland and Sherwood [12] again made use of the diffusion equation toestimate the duration of the so-called constant rate period, and it appeared

J

,


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A THEORY OF DRYING IN POROUS MEDIA 121that a complete theory of drying was within reach using only the diffusionequation.*While the chemical engineering community was busy comparing dryingdata with solutions to the diffusion equation, other scientists were examiningthe motion of liquids through unsaturated porous media from an entirelydifferent point of view. Soil scientists, colloid chemists, and ceramicists wereexplaining the movement of moisture in porous media in terms of surfacetension forces or by capillary action. The work of Gardner and Widtsoe [13],Richards [14], Rideal [15], and Westman [16] clearly indicated that surfacetension effects could not be ignored in the study of liquid motion in anunsaturated porous media. Comings and Sherwood were quick to recognizethis, and a short note was published [17] providing a qualitative discriptionof the motion of liquid owing to capillary action and the motion of vaporowing to molecular diffusion. A brief set of experiments on the drying of claywas performed by Comings and Sherwood, and the results indicated thatcapillary action could indeed be an important mechanism in the movementof liquid during the drying of a porous solid. Other researchers were thenwarned that "the word diffusion must be used with care in referring to themovement of water through a soil."The warning of Comings and Sherwood [17] and the work of the soilscientists was not lost on other engineers interested in drying, and in 1937Ceaglske and Hougen [IS] began their paper on the drying of granular solidswith the statement: "The drying rate of a granular substance is determinednot by diffusion but by capillary action." Armed with the extensive work ofHaines [19,20] on the relation between capillary pressure and liquid content,Ceaglske and Hougen were able to calculate the saturation distributionduring the drying of a layer of sand 5.OS-cm thick. The results are shown inFigs. I-1a and I-1b and clearly demonstrate, for the particular case underconsideration, that capillary action can playa dominant role in the movement of moisture during the drying of a porous solid. Later we shall returnto a more detailed discussion of these results; however, for the present weneed only point out that the theoretical results were obtained using only thelaws of hydrostatics along with an experimental determination of the capillary pressure as a function of saturation. The work of Ceaglske and Hougenwas followed by an extensive survey by Hougen et al. [21] appropriatelyentitled "Limitations of Diffusion Equations in Drying." Comparisons weremade between solutions of the diffusion equation, calculations based oncapillary pressure-saturation curves, and experimental data. The comparisonfor sand is shown in Figs. 1-2, and the results strongly favor the capillarytheory. 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 MassTransfer, Sydney, Feb. 1977), the diffusion theory of drying is explored in considerable detail.


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122 STEPHEN WHITAKER100

XIX8027.50//z0i= X ,~ IX> 60t-V I

w

/ /13

1(,!)t- 40wU~w / vy. 20 / /:.-- 0,.0/o -O-o - - - - ;_ _ x 01_X -X - ~ O -- - 0

0 2DISTANCE (CM)

FIG.I-Ia. Experimental water distribution in a sand layer during drying. Adapted fromCeaglske and Hougen [18].

80z0i=~=> 60t-VIw(,!)t- 40wu~wa.

20

OL-____-L______ ____-L______ ____

DISTANCE (CM)FIG. I-lb. Theoretical water distribution in a sand layer during drying. Adapted from

Ceaglske and Hougen [18].


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A THEORY OF DRYING IN POROUS MEDIA 12340 ~ - - - - ~ - - - - - - , - - - ,

- - -Expe r imen t a l Values- - - Calculated from

Diffusion Equation35

30~V>co>-"" 25I -ZwI -Z0 20w""::>I -~0::e 15I -zWU""0..

10

DISTANCE FROM SURFACE OF DRYING -CM.

FIG. I-2a. Moisture distribution during the drying of sand--comparison with diffusiontheory. Adapted from Hougen et al. [21].

and lead shot; and in each case the diffusion equation could not be made tofit the experimental data using a constant diffusivity. For wood the situationis somewhat different, and reasonably good agreement between experimentaldata 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] arecompared with calculated values from the diffusion equation and rather goodagreement is obtained. Hougen et al. [21] point out, however, that the drying

"i:; of extremely wet wood cannot be modeled by the diffusion equation, and thateven below the fiber saturation point one should allow for variation of thediffusivity with moisture content, temperature, pressure, and density.


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124 STEPHEN WHITAKER40 - - Experimental Values

X X X X Calculated fromCapi Ilory Pressure

35 Saturation Curve

V> 30V>ell>-'">t- 25zUJt-z0uUJ 20'"::>t-V>0:et- 15UJU'"J"-

10

DISTANCE FROM SURFACE OF DRYING -CM.FIG. I-2b. Moisture distribution during the drying of sand-comparison with capillary

theory. Adapted from Hougen et at. [21].

While questions about capillary action and the diffusion of moisture werebeing raised by chemical engineers in the U.S.A., Krischer [23-25] wasexploring the very same ground in a series of papers on drying. While heattransfer had played a minor role in previous papers on drying, it appears thatKrischer was the first to consider seriously the intimate role that the transportof energy may play in a drying process.Although the diffusion theory of drying seemed to be in a state of disreputeafter the papers by Ceaglske and Hougen [18] and by Hougen et al. [21],

...

,w


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..

A THEORY OF DRYING IN POROUS MEDIA

f -ZWf -Zouwa:::::>f -V'>(5::;;f -ZwUa::wa.

40 ~ - - - - ~ - - - - ~ - - - - - - ~ - - - - - ,

30

20

10

Experimental Values_____ Calculated fromDiffusion Equation

1/4 1/2 3/4

DISTANCE FROM SURFACE OF DRYING-IN

125

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 tomodel the latter stages of drying. Presumably the liquid at this stage is inwhat Haines [19, 20] referred to as the pendular state and motion owing tocapillary forces is greatly reduced if not completely halted. Under thesecircumstances the convection and diffusion in the vapor phase are the onlymechanisms by which the moisture content can be reduced. This situationseems to be especially important in the drying of biological materials to verylow moisture contents. In 1947 Van Arsdel [26] pursued this line of attackand investigated the effect of a variable diffusivity on drying rates predictedby the diffusion theory. Equation (1-1) was essentially modified to the form

(1-2)and solved numerically to yield results similar to those found in the labora-tory. Phillip and DeVries [27] and DeVries [28] extended previous treat-ments of drying to include effects of capillary flow and vapor transport, andincorporated the thermal energy equation into the governing set ofequationsthat describe the drying process. The transport owing to capillary forces wasrepresented in terms of gradients of the moisture content and temperature;thus a diffusion like equation resulted. Similar equations for heat and masstransfer in porous media were also published by Luikov [29]. Churchill andGupta [30] continued the use of diffusion equations in their study of the


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126 STEPHEN WHITAKER

freezing of wet soils, but were quick to note that "the formulation of the ratesof moisture transfer in both the vapor and liquid phases in terms of diffusivities and concentration gradients is highly arbitrary and cannot be rationalizedon theoretical grounds." Berger and Pei [31 J continued the practice ofrepresenting liquid and vapor fluxes in terms of diffusivities and concentration gradients, and more recently Husain et al. [32J have applied the diffusionequations for heat and mass to the drying of biological materials. Goodagreement between theory and experiment is obtained provided one allowsthe "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 inferredin a purely intuitive manner from the well-known point equations of continuum physics.* Often this intuition was enhanced by the use of shellbalances [33J to construct the differential equations, but the final result wasnevertheless an intuitive product. In the following sections a rigorous, butlimited theory of drying will be presented based on the well-known transportequations for a continuous media. These equations will be volume averagedto provide a rational route to a set of equations describing the transport ofheat and mass in a porous media. The theory will be limited in two ways: byrestrictions and by assumptions. The former represent clear cut limitationsin the form of the governing point equations and boundary conditions andcan be removed by further analysis. The latter are concerned mainly with thetopology of the three-phase system and the order of magnitude and functional dependence of various terms that arise in the theoretical development.I t seems likely that the validity of the assumptions can be tested only bycomparison with experiment, although in some cases they may be acceptedor rejected on the basis of further theoretical analysis. In order to clearlyidentify 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.

II. The Basic Equations of Mass and Energy TransportIn our analysis we consider the motion of a liquid and its vapor through arigid porous media such as that shown in Fig. II-I. There the (j phase represents a rigid solid matrix, the f3 phase is the liquid, and the y phase representsthe gas phase which consists of vapor and some inert component (usually air)

that is insoluble in either the (j or the f3 phase. There is no liquid containedin the (j phase, thus our development excludes what is sometimes referred toas "bound" moisture [29, p. 234]' Although our attention is directed toward* The one exception to this statement is the use of modified forms of Darcy's law.

i

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A THEORY OF DRYING IN POROUS MEDIA 127

Averaging volume. VFIG. II-I. Drying process in porous media.

the gas-liquid-solid system, one may interpret the f3 phase as a second solidphase undergoing sublimation so that the analysis also includes the case offreeze drying. The situation shown in Fig. 11-1 would represent the pendularstage as opposed to the funicular stage since the liquid phase is discontinuous.The analysis is not restricted by the particular liquid distribution shown inFig. 11-1, and in subsequent developments both the pendular and the funicular stages will be discussed.

In the analysis ofdrying phenomena we are generally interested in knowingthe moisture content and temperature as a function of space and time. Thesequantities will be determined by application of the appropriate laws ofphysics which we list as follows:

continuity equation: op- + V ' (pv) = 0otith species continuity equation:

OPi- + V (p.v.) = rot ' , ,

(II-I)

(11-2)


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128 STEPHEN WHITAKERlinear momentum principle:

DvP- = pg + V 'TDt

angular momentum principle:T = Ttthermal energy equation:

Dh Dpp Dt = - V q + Dt + Vv : t +

(11-3)

(11-4)

(11-5)

In Eq. (11-2) the term ri represents the mass rate of production of the ithspecies owing to chemical reaction, and Vi represents the ith species velocity[33, p. 497]. In Eq. (11-3) the total stress tensor [34, p. 113] is represented byT and P represents the transpose oft [35, p. 103]. In Eq. (11-5) we have used tto represent the viscous stress tensor, p is the pressure, h is the enthalpy perunit 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. 5.4]. Weshould note that Eq. (II-I) can be derived from Eq. (11-2) by summing overall N species in the system and imposing the definitions for the total densityp and mass average velocity V,

i=NP = L Pi

i= 1i=N

pv = L PiVii= 1

(11-6)

(11-7)in addition to imposing the restriction on the chemical rates of reaction that

i=NL ri = 0 (11-8)i= 1

In Section II of this article we shall focus our attention on Eqs. (II-I), (11-2),and (11-5) in developing the relevant volume averaged transport equationswhich describe the drying process. In Section IV we shall be concerned withthe transport of momentum in the gas phase, and there we shall begin ourinvestigation of the laws of mechanics as they apply to the drying process.A. GOVERNING POINT EQUATIONS

In this section we shall examine Eqs. (II-I), (11-2), and (11-5) and list theforms that they take in the three separate phases. We shall use a, {3, and}' assubscripts to denote the phase in question, thus Ta will represent the point

"-..

.


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temperature in the solid phase, and vp will represent the point mass averagevelocity in the liquid phase.1. (J 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 (J phase is zero:R.l (II.A-l)thus Eqs. (11-1)-(11-4) are of no consequence. This is perhaps one of the mostimportant limitations of the entire analysis, for most biological materialsand many nonbiological materials undergo a change in volume upon drying.It is possible that the change in volume upon drying can be accounted for byallowing various derived parameters to be a function of the moisture content;however, a rigorous attack on the problem would require that Eq. (II-I) forthe solid phase be incorporated into the analysis.

On the basis of restriction R.l we can immediately simplify Eq. (11-5)for the (J phase to (ah,,)p" at = - V . q" + " (II.A-2)Here we have dropped the reversible and irreversible work terms in Eq. (11-5),but retained the source term owing to electromagnetic radiation so that thetheory includes drying processes of the type described by Lyons et ai. [37]and by Raiff and Wayner [38]. Throughout our analysis of all three phaseswe shall assume that the enthalpy is independent of pressure:R.2 h = h(T) in the (J , /3, and y phases (II.A-3)and that all heat capacities are constant. This means that h can be replacedby cpT to within an arbitrary constant*:R.3 h = cpT + constant in the (J , /3, and y phases (II.A-4)On the basis of R.2 and R.3 we can express Eq. (II.A-2) as

(aT,,)p"(C p ), , T t = - V q" + " (II.A-5)and apply Fourier's law to obtain the final form of the thermal energy equa-tion for the solid phase

(aT,,) 2p"(C p ), , T t = k" V T" + " (II.A-6)* In the gas phase where there is more than one component present we interpret R.2 and

R.3 as h, = h,(T) and h, = (cp),T + constant. Here h, represents the partial mass enthalpy[39, p. 279].


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130 STEPHEN WHITAKERHere we have assumed that the thermal conductivity is constant. Thisassumption will also be imposed on the liquid and gas phases, and we denoteit as our fourth limiting restriction:R.4 The thermal conductivities are constant in the (J , {3, and y phases.2. {3 Phase (Liquid)

We assume the presence of only one component in the liquid phase:R.5 The {3 phase contains only a single component.so that Eqs. (II-I) and (11-2) are identical and written for the liquid phase as

(II.A-7)For the time being we shall avoid any discussion of Eqs. (11-3) and (11-4) asthey apply to the liquid phase. In considering the thermal energy equationwe assume negligible compressional work and viscous dissipation:R.6 (II.A-8)so that Eq. (11-5) takes the form

Pp Dhp/Dt = - V qp + p (ILA-9)While neglecting reversible work and viscous dissipation is often satisfactoryin heat transfer calculations [36, Section 5.5], Defay et al. [40, p. 236] suggestthat the reversible work Dpp/Dt may become important for systems with avery small pore structure such as activated carbon or silica gel. In suchsystems surface tension can cause large pressure changes in the liquid phaseas drying takes place and the accompanying reversible work may represent asignificant term in the thermal energy equation. The material derivative inEq. (II.A-9) can be expanded, leading topp(ohp/ot + vp Vhp) = - V qp + clip (1I.A-1O)Representing the enthalpy hp as indicated in Eq. (II.A-4), and making useof Fourier's law along with restriction R.4 leads to

(ILA-ll)3. y !Phase (Gas)

We consider the gas phase to be made up of the vapor and an inert component that is insoluble in either the (J phase or the {3 phase. The continuityequation is expressed as(II.A-12)

".


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and we write the species continuity equation as1 = 1,2, . . . (II.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 Py and Vy are given by Eqs. (II-6) and (II-7) as

Py = Pl + P2PyVy = P1V l + P2 V2

(II.A-I4)(II.A-I5)

Writing the species velocity Vi in terms of the mass average velocity Vy andthe diffusion velocity U i ,allows us to write Eq. (II.A-13) as

api/at + v . (PiVy) = - v . (PiUJ, 1 = 1,2, . . .The diffusive flux PiUi can be expressed as [33, p. 502]

PiUi = - p/!) V(Pi/Py)so that our final form of Eq. (II.A-13) is*

api/at + v . (PiVy) = V [Pygc V(Pi/P y)]

(II.A-I6)

(II.A-I7)

(II.A-I8)

(II.A-I9)In attacking the thermal energy equation for the gas phase we must referto the work of Bird et al. [33] or the work of Slattery [39] where it is shownthat the appropriate form of Eq. (11-5) for a multicomponent system ist

a ( i = N ) (i=N) Dpy- L piTii + V L Pivlii = - v . qy + -Dt + VVy : Tyat i = l i = li=N+ y + L PiUi fii= 1

D i=N 1p.- Py - L - ~ U i 2Dt i=12p y (II.A-20)Here we use hi to represent the partial mass enthalpy which is simply thepartial 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 Eg. (II.A-18) is not applicable.

t Note that q" will be expressed as -k" VT" in keeping with Eg. H in Table 18.3-1 of Birdet al. [33], and Eg. J in Table 8.3.5-1 along with Egs. (4-1) and (4-7) in Sec. 8.4.4 of Slattery[39].


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132 STEPHEN WHITAKERI t is quite reasonable to neglect compressional work and viscous dissipationin the gas phase:R.8 Dp-y = Vv :. = 0Dt y y (II.A-21)so that Eq. (II.A-20) simplifies to

a (i=N) (i=N )- I pJii + v I PivJii = - V qy +


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A THEORY OF DRYING IN POROUS MEDIA 133Up to this point we have listed the appropriate forms of the heat and masstransport equations for each of the three phases subject to the restrictions

R.l-R.9. Our next task is to state the boundary conditions that apply at thevarious phase interfaces.B. BOUNDARY CONDITIONS

Referring to Fig. II -1 we note that there are three interfacial areas contained within the averaging volume, Y. We denote these as,A.,.p, solid-liquid interfacial area;A.,.y, solid-vapor interfacial area;Apy , liquid-vapor interfacial area;

and note that(lI.B-l)

The boundary conditions for the solid-liquid interface are quite simple andmay be represented asB.C.lB.C.2B.C.3

Vp = 0 on A.,.pq.,. ".,.p + qp "p.,. = 0 on A.,.pT.,. = Tp on A.,.p

(II.B-2)(II.B-3)(I1.B-4)

Here ".,.p represents the outwardly directed unit normal for that portion ofthe a phase in contact with /3 phase. Thus ".,.p points out from the a phaseinto the /3 phase and is related to "p.,. by

(II.B-5)We note at this point that there is no source or sink of energy on the A.,.psurface, thus the interfacial energy of A.,.p is taken to be negligible. We shallimpose similar restrictions on the boundary conditions at the a-y and /3-yinterfaces and we list this as our tenth limiting restriction:R.lO Interfacial energies for the a-/3, /3-y, and y-a interfaces arenegligible in the thermal sense.This restriction is easily removed by application of the work of Slattery[41Jand may be an important consideration in the drying of highly porousmaterials such as activated carbon and silica gel. DeVries [28J has incorporated a heat of wetting in his analysis of drying, but the fundamentalapproach of Slattery [41] has not yet been applied to this problem.


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134 STEPHEN WHITAKERAt the solid-vapor interface we can write a set of boundary conditionssimilar to those given by Eqs. (II.B-2)-(II.B-4):

B.C.4 Vy = 0 on Auy (II.B-6)B.C.5 qu nuy + qy nyu = 0 on Auy (II.B-7)B.C.6 Tu = Ty on Auy (II.B-8)

The boundary conditions for the liquid-vapor interface will require somediscussion, for Apy represents a moving, singular surface. The jump conditions that apply at singular surfaces have been discussed by Truesdell andToupin [42, p. 517J and more recently by Slattery [39J; however, the jumpcondition for the multicomponent form of the thermal energy equation hasnot been listed explicitly and will be presented here for completeness.Our development will follow the approach given by Slattery [39, p. 21 Jin which we focus our attention on the material volume illustrated in Fig.

11-2. This volume 1/m(t) contains bothf1 and y phases, which are separatedby the singular surface, Apy = A yp . The velocity of this surface is denoted byw. Returning to Eq. (1I.A-1O), we add hp[8pp/8t + V' (ppvp)J = 0 to theleft-hand side in order to represent the liquid phase thermal energy equationas(II.B-9)

A similar operation on Eq. (II.A-26) allows us to express the gas phasethermal energy equation as:t (pyhy) + V (pyhyv y) = - V [qy + :K PiU})i] + y (II.B-10)

FiG. II-2. Material volume containing a singular surface.

;.

"


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At this point we note that our special forms of the thermal energy equationgiven by Eqs. (II.B-9) and (II.B-lO) are consistent with the integralrepresentation

~ t J j ; , - = ( t ) (ph) dV = - f....=(t) q . D dA + ff=(t) dV (II.B-ll)which applies to any material volume regardless of whether it contains asingular surface at which (ph) and q suffer jump discontinuities. It shouldbe clear that in the f3 phase we have

while in the')' phase

ph = pphpq = qp = p

i=Nq = qy + I Piuliii= 1

(II.B-12a)(II.B-12b)(II.B-12c)(II.B-13a)(II.B-13b)(II.B-13c)

Returning to Eq. (II.B-9), we integrate over the volume vp(t) shown in Fig.11-2 to obtain{' ~ (pphp) dV + {' V (pphpvp) dV = - {' V qp dV + {' p dVJVp(t) ut JVp(t) JVp(t) JVp(t)

(II.B-14)We can use the general transport theorem [34, p. 88; 42, p. 347] to expressthe first term in Eq. (II.B-14) asi a di ih dV = - h dV - h v D dAVp(t) at (pp p) dt Vp(t) (pp p) Ap Pp p p p

(II.B-15)Here Ap represents the material surface of vp(t) and Apy represents thesingular surface moving with a velocity w which may be different from themass average velocity vp. Substitution of Eq. (II.B-15) into Eq. (II.B-14)and applying the divergence theorem to the second and third terms leads todd {' (pphfJ) dV + i pphp(vp - w) Dpy dAt JVp(t) Ap,

= -i qfJ Dp dA - i qp. Dpy dA- + {' p dVAp Ap, JVp(t) (II.B-16)


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136 STEPHEN WHITAKERWe can repeat these steps with Eg. (lLB-lO) to obtain the comparable expression for the y phase:

dd f (pyhy) dV + f pyhy(vy - w) Dyp dAt JVy(T) Ayp(H.B-17)

We now wish to add Egs. (H.B-16) and (H.B-17) and note that [39, p. 22]Df, dl dlh dV = - h dV + - h dVDt f =(t) (p ) dt Vp(t) (pp p) dt Vy(t) (Py y) (H.B-18)where Egs. (H.B-12) and (H.B-13) apply to the left-hand side. Since dm(t) =Ap + Ay. we also havef q . D dA = r qp. Dp dA + f [qy + if PiU}?i] Dy dA (ILB-19)Jd=( t ) JA p JAy i= 1

and the addition of Egs. (H.B-16) and (II.B-17) leads to*~ f (ph) dV + f [pphp(v p - w) Dpy + pyhy(v y - w) Dyp] dADt J f= ( t ) JApy

- SAPY {qp. Dpy + [qy + :K PiU)?i] Dyp } dA (H.B-20)Comparing this result with Eg. (H.B-H) leads to the jump condition at thef3-y interfaceB.C.7 pphp(vp - w) Dpy + pyhy(vy - w) Dyp

= -{ p Dpy + [qy + :K PiU)li} Dyp } (II.B-2I)The jump condition for the mass average velocities is easily shown to be I36, Sec. 10.2; 39, p. 24]B.C.8 (H.B-22)

* Here we use the fact that A yp = Apy

,.


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while the tangential component is assumed to be continuousB.C.9 v p Apy = Vy Apy (II.B-23)Here Apy is any tangent vector to the surface denoted by A py . The derivationof the species jump condition follows that given for Eq. (II.B-22) and we listthe result asB.C.lOB.C.ll i = 2,3, . . .

i = 1 (II.B-24)(II.B-25)

Here we have referred to restriction R.5 and have designated the vapor byi = 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 averagedform of the transport equations. This will give us a set of equations that applyat every point in space, not just in the three separate phases. This type ofapproach to the analysis of transport phenomena in multiphase systemshas been discussed at length by Slattery [39J and special cases have beentreated by Whitaker [43-45J, Slattery [46J, Gray [47J, and Bachmat [48].C. VOLUME AVERAGED EQUATIONS

With every point in space we associate an averaging volume Y, such asthat shown in Fig. II-I. We have chosen a sphere as the averaging volume,but any shape will suffice provided the dimensions and the orientation areinvariant [44]. There are three types of averages that are useful in the analysisof transport phenomena in porous media. The first of these is the spatialaverage of some function t/J defined everywhere in space. This average isrepresented by (t/J) and is defined by

(II.C-l)More often we are interested in the average of some quantity associatedsolely with a single phase. For example, we may be concerned with theaverage temperature of the solid phase, and we define the phase average ofT" as

(II.C-2)Since T" is defined in the normal way in the (J phase and is zero in all otherphases Eq. (II.C-2) reduces to

(T,,) = ~ r Ta dV (II.C-3)Y Jy.


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138 STEPHEN WHITAKERThe phase average has the drawback that if T" is constant, the phase averageis not equal to this constant value. A quantity that is more representativeof 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 (J phase, Eq. (II.C-4)reduces to

(1I.C-5)We define the volume fractions for the three phases as

(1I.C-6)Clearly the sum of these fractions is one,

(1I.C-7)and the phase average and intrinsic phase average are related by

(II.C-8)The prime tool in the formulation of the volume averaged equations is theso-called averaging theorem [39, p. 194; 43], which can be written as

We begin our analysis with the solid phase and note that p" is constant soso that Eq. (II.A-2) takes the form(1I.C-10)

Integrating over v". and dividing by 1/ gives the initial form of the volumeaveraged equation:

We can interchange differentiation and integration in the first term andrefer to the definition of the phase average given by Eq. (1I.C-3) to write(II.C-12)

j

."


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Use of the averaging theorem, Eq. (ILC-9), allows us to express tHe aventge ofV q" as


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140 STEPHEN WHITAKERI f we now substitute Eq. (1I.C-18) into Eq. (1I.C-14) and represent h" in termsof the heat capacity and temperature as indicated by Eq. (II.A-4), we obtainthe following form of the thermal energy equation for the a phasep"(cp ), , a


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and form the integral over vp(t), and divide by 1/ to obtain~ f (a pp) dV + ~ f V . (p v ) dV = 01/ JVp(t) at 1/ JVp(t) p p

141

(II.C-23); Application of the general transport theorem allows us to write

- - dV - - - dV - - W D dA(a pp) d [1 ~ ] 1/ vpit) at - dt 1/ Vp(t) Pp 1/ Apy Pp py(II.C-24)

while the averaging theorem provides~ f V (ppvp) dV =


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142 STEPHEN WHITAKERSubstitution of Eqs. (II.C-29) and (II.C-30) into Eq. (II.C-26) and division byPp leads to the appropriate form of the continuity equation for the liquidphase* OEp 1 f+ V'


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'.

A THEORY OF DRYING IN POROUS MEDIA 143these definitions except that given by Eq. (II.C-33) to include the effect ofthe gas phase. These new definitions are given as follows:

S x 100

( PPEP + (PI)YEy) .= mass of mOIsture per mass ofP"E" dry solid(II.C-38)

(PI) = EpPp + ElpI)Y = mass of moistureper unit volume (II.C-39)

S PpEp + (PI)YEy f . I .= ( ) = ractIOna mOIsture saturatIOnPp Ep + Ey(II.C-40)

[ PPEp + (PI)YEY] x 100 = percentage moisture saturationpP(Ep + Ey)(II.C-41)

For the case where Ey(PI)y EpPfJ one can easily see that Eqs. (II.C-38)(II.C-41) reduce to Eqs. (II.C-34)-(II.C-37). For the present we shall leaveEq. (II.C-32) in terms of EfJ and note that the saturations sp and S will probablybe most useful in comparing theory with experiment.We now direct our attention to the thermal energy equation for the (Jphase, which was given earlier by Eq. (II.B-9) asa .a/pfJhfJ) + v . (pphpvfJ) = - v . qp + fJ (II.C-42)To obtain the volume averaged form of this transport equation we integrateover vp(t), divide by "1/, and make use of the general transport theorem forthe first term and the averaging theorem for \the second and third terms toobtain*

(II.C-43)Our analysis of the right-hand side of Eq. (II.C-43) would be identical to thatgiven previously by Eqs. (II.C-14)-(II.C-20). Thus one need only exchange(J and (J on the right-hand side of Eq. (II.C-20) to get the right-hand side ofEq. (II.C-43) in terms of Tp and the two interphase heat flux terms. The

*Note that d


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144 STEPHEN WHITAKERterms on the left-hand side require that we express the enthalpy as indicatedin Eq. (II.A-4). This leads to

(II.C-44)

Here h/ is the enthalpy at the reference t,emperature T/. Directing our attention to the first two terms in Eq. (II.C-43), we substitute Eq. (II.C-44)and remember that Pp, (cp)p, and [h / - (cp)pT /] are constant in order toobtain*a a pot


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into Eq. (II.C-45) and rearranging leads toaat


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146 STEPHEN WHITAKERH ~ r e we have again used Eq. (1I.C-44) to represent hp in terms of hp 0 and(cp)p(Tp - T/). The fourth and fifth terms on the left-hand side of Eq.(p.C-52) will cancel when the temperature in the liquid phase is uniform, i.e.,

Tp = ( Tp / = constantbut for the general case we must again use Eq. (1I.C-47a) to obtain thesimplified form of (II.C.-52):

a(Tp)p pEppp(Cp)p at + pp(cp)p(vp) V(Tp)

(II.C-53)

Here we have expressed V . (qp) in terms of Tp by referring to Eq. (1I.C-20)and interchanging the subscripts (J and [3. We should note that the governingdifferential equation for the liquid temperature (Tp / contains several termsfor which constitutive equations are required. The dispersion term pp(cp)pV (f/vp) is usually modeled as a diffusion mechanism, although morecomplicated models have been suggested [43, 49]. The area integrals of T pon the right-hand side ofEq. (1I.C-53) are generally taki:ll to be proportionalto V(Ep(Tp)P) and incorporated into an effective thermal conductivity. Theinterphase flux terms can be modeled as indicated by Eq. (1I.C-21), thusrequiring the experimental determination of film heat transfer coefficients.For the present we shall ignore these difficulties and simply note that Eq.(1I.C-53) is the proper form of the liquid phase thermal energy equation in aporous media.We now turn our attention to the y phase, and being with the continuityequation as given by Eq. (II.A-12):

(1I.C-54)We need only repeat the analysis given by Eq. (lI.C-22)-(II.C-26) to obtain

(II.C-55)

'.


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"

A THEORY OF DRYING IN POROUS MEDIA 147Further simplification of this result is not possible since Py depends on boththe temperature and composition of the gas phase. Representation in termsof the intrinsic phase average is accomplished by expressing the point functions as

Py = (py)Y + Py in they phasePy = Py = 0 in the (J and /3 phasesVy = (Vy) + Vy in the y phaseVy = Vy = 0 in the (J and /3 phases

In addition we can use the definition

(II.C-56a)(1I.C-56b)(1I.C-57a)(1I.C-57b)

(Py) = E/py)Y (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 intrinsicphase average (Py)y and the phase average (Vy), so we have left that termunchanged. In general we expect py and Vy to be much smaller than (Py)yand (Vy) in the y phase, and we state this as a generally plausible assumption:A.I. 0ro (t/Jro)ro and 'i'ro ('i'ro) in the w phase where w refers, to

(J , /3, and y (1I.C-60)This allows us to express the gas phase continuity equation as

(1I.C-61)since Eq. (1I.C-60) will allow us to write

V ' (Py)Y(Vy) V ' (pyVy) (1I.C-62)Here we have made a definite assumption about the order of magnitude offunctions 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 theproduct of averages.We must use this assumption with care for there are situations in which wemay not wish to drop the product of deviations relative to the product of


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148 STEPHEN WHITAKERaverages. Application of assumption A.2 to Eq. (1I.C-59) seems quite appropriate; however, we should note that the term V < pvp) was not discardedrelative to


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however, the remaining terms require that we again resort to Gray's [47Jrepresentation scheme and we begin using Eqs. (ILC-56) in order to write

(pJp y) = (p;/(py)Y)[1 - (py/


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150 STEPHEN WHITAKERWe begin our analysis of the vapor phase thermal energy equation withEq. (II.A-24) which took the form

(II.C-73)This equation is identical in form to Eq. (II.C-42) for the liquid phase, andthe result analogous to Eq. (II.C-43) can be written immediately:

(II.C-74)Following our analysis of the p-phase thermal energy equation, we expressthe partial mass enthalpy as

hi = hie + (cpMTy - T/) (II.C-75)In expressing the partial mass enthalpy in terms of the pure component heatcapacity (Cp)i instead of the partial mass heat capacity (Cp)i, we are invokingthe restriction of a thermodynamically ideal gas phase which we need tolist as:R.12 The gas phase is ideal in the thermodynamic sense.We should remember that a previous restriction, R.3, required that all heatcapacities be constant.

We now focus our attention on the first two terms of Eq. (II.C-74) andsubstitute Eq. (II.C-75) to obtain

a (i=N) (i=N )-;- .L Pihi + V ' .L PiVihiut ,=1 ,=1a i=N i=N= at


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and (II.C-48). In this case the representations take the formTy =


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152 STEPHEN WHITAKERSubstitution of Eq. (II.C-SO) into Eq. (II.C-74) and following the analysisgiven by Eqs. (II.C-52) and (II.C-53) leads to

i=N+ V I (cp);


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priate models and experimental determination of the model parameters.While the theoretical formulation up to this point appears to be intolerablydifficult, it does represent a solid foundation upon which more attractivetheories can be constructed. In Section III of this article we shall simplifyour thermal energy equations by forming the total thermal energy equation,and in Sections IV and V we shall attack the problem of determining themass average velocities in the gas and liquid phases.

III. Energy Transport in a Drying ProcessWhile there are many processes in which the gas temperature


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154 STEPHEN WHITAKERIn the following section we shall impose assumption A.3 on the three thermalenergy equations in order to develop a total thermal energy equation.A. TOTAL THERMAL ENERGY EQUATION

We begin our analysis of the transport of thermal energy in a porousmedium during drying by adding Eqs. (1I.C-20), (1I.C-53), and (1I.C-81) andimposing the assumption of "local equilibrium" as indicated by Eq. (11I-3).This leads directly to the total thermal energy equation given by

= V {V[(kaEa + kpEp + kyEy)


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These conditions were given in Section II as Eqs. (II.B-l), (II.B-4), (II.B-5),and (II.B-8). The source term in Eq. (IILA-l) is defined asa i=N _


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156 STEPHEN W HITAKERUsing this result and Eqs. (II.B-3) and (II.B-7) we have

- ~ Lap (qa - qp) Dap dA - ~ L py (qp - qy) Dpy dA- ~ fA ya (qy - qa) Dya dA

= ~ Ly [Pphp(vp - w) Dpy+ :K pldvi - w) Dyp ] dA (IILA-lO)

Substitution of Eqs. (IILA-lO) and (IILA-8) into (IILA-l) and rearrangingleads to a slightly more attractive form of the total thermal energy equation:

J


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From Eqs. (II.B-24) and (II.B-25) of Section II we havePl(V 1 - w) Dyp + pp(vp - w) Dpy = (IILA-14a)

Pi(Vi - w) Dyp = 0, i = 2,3, . . . (III.A-14b)so that the right-hand side of Eq. (IILA-13) can be simplified to*

i=Npp[hp - (cp)pTp](vp - w) Dpy + L pJni - (Cp)iTy](Vi - w) Dypi= 1

= [h / - h1 0 + (cp)pT) - T/) - (cpM


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158 STEPHEN WHITAKERgives (for example)*

Note that Eq, (IILA-19) is valid only when the point functions are representedas suggested by Gray. We can use this result to eliminate the temperaturesT a, T p, and Ty in Eq. (IILA-17) and replace them with Ta, Tp, and Ty- Thealgebraic effort required to rearrange the conductive transport terms isconsiderable, and we shall note only that Eq. (IILA-19) is used for the ( j ,{3, and y phases to eventually reduce Eq. (IILA-17) to the form

8


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has suggested that empirical correlations for these terms should satisfy theprinciple of material frame indifference [42, p. 700] and indicated thatV(T p)P is a likely correlating variable for terms involving T p. In the followingparagraphs we wish to put forth some ideas about the functional dependenceof Tp which strengthen earlier suggestions* that Tp is strongly dependent onV(Tp)P.To obtain the governing differential equation for Tp we substitute T p =(Tp)P + Tp into Eq. (II.A-U) in order to obtain

( aTp -)pp(cp)p T t+ v p VT p

2 - {(a(Tp)p p) [ P]}kp V Tp + C!>p - pp(Cp)p at + Vp V(Tp) - kp V V(Tp)(III.B-2)

Since (Tp)P = (T ) we can think ofEq. (1II.A-20) as the governing differential equation for (Tp)P, thus leaving Eq. (III.B-2) as the governing differentialequation for T . The functional dependence of T can now be deduced fromEq. (III.B-2), i.e., Tp depends on:(1) the independent variables: x, y, Z, and t;(2) the parameters appearing in the governing differential equation:

Pp, (cp)p, vp, kp, C!>p, a(Tp)p/at, V(Tp)P, and V [V(Tp)P];(3) any parameters that appear in the boundary conditions.It is not at all clear what types of boundary conditions one would impose onT ; however, it is clear that T depends on the usual dimensionless variables(the Reynolds and Prandtl numbers) in addition to being a function of(Tp)p. This means that Tp in turn depends on all the parameters in Eq.(1II.A-20).On the basis of the form of Eq. (III.B-2) we put forth the conjecture thatthe functional dependence of T can be expressed as tAA Tp = '(a(Tp)p/at, V(Tp)P) (III.B-3)where the dependence on the other variables, Pp, (cp)p, etc, is understood andthe dependence on V [V(Tp)P] can be neglected. From the definition of(Tp)P we know that

(Tp)P = Tp when Tp is independent of the spatial variables(1II.B-4)* See Appendix A of Whitaker (43).t One could be more general and state only that Tp is a functional of (Tp)P with the dependence on the other variables understood.


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160 STEPHEN WHITAKERand it follows that

when < p)l1 is independent of the spatial variables (III.B-S)In addition to our first conjecture about the functional dependence of Tgiven by Eq. (III.B-3) we now make a second conjecture on the basis of Eq.(lII.B-S) which is stated asA.S (III.B-6)While we know that Eq. (III.B-S) is true by definition, and it follows that


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Here we identify the second order tensor Kp as

Kp = ~ f "pyep dAj/ ApT (III.B-ll)Equation (I1I.B-lO) can be immediately extended to the comparable termsinvolving T and T and we write

(III.B-12)

(III.B-13)Substitution of these results into Eq. (III.A-20) and remembering that


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162 STEPHEN WHITAKERwith 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. (III.B-17) by dropping theconvective transport term on the left-hand side of Eq. (III.B-17) and replacing K ~ f f with Keff to obtain

(III.B-19)Here we have also dropped the energy source term


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Invoking assumption A.2 allows us to further simplify this expression to(III.B-25)

where the dispersion term, if considered important, could be lumped inwith the representation given by Eq. (III.B-16). Using Eq. (III.B-25) wenow express the total thermal energy equation as8


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164 STEPHEN WHITAKERconductivity of heterogeneous materials has been given by Gorring andChurchill [50] and some recent experiments on the effective thermal conductivity of saturated metal wicks are described by Singh et al. [51]. Theproblem of dispersion in two phase systems has received a great deal ofattention*; however, dispersion in three-phase systems comparable to thatencountered in drying processes does not appear to have been studiedeither experimentally or theoretically.tC. THERMODYNAMIC RELATIONS

In order to connect the total thermal energy equation with the gas phasediffusion equation given by Eq. (II.C-72) we need to state some thermodynamic relations. We have already inferred that the gas phase is to betreated as ideal (see Eq. (II.A-3) and restriction R.12 following Eq. (II.C-75)so that species density can be determined byPi = PiRiT, i = 1,2,. . . (III.C-l)

Here Pi is the partial pressure of the ith species and Ri is the gas constantfor the ith species. We can form the intrinsic phase average (see Eq. (II.C-5))of Eq. (III.C-l), impose assumptions A.2 and A.3 to obtain


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Hysteresis can be important so that r should always be determined for decreasing values of EfJ when Eq. (III.C-4) is to be used in the analysis of adrying process. Combining the Kelvin equation and the Clausius-Clapeyronequation would suggest that vapor pressure data for porous media could becorrelated by the expression

o { [ L!!.hvap ( I I ) ]}1 = P1 exp - (2(JfJy/rpfJR 1T) + R;- T - To (I1I.C-5)Forming the intrinsic phase average and making use of assumptions A.Iand A.3 eventually leads to

(P1)y = P1oexP{-[(2(JfJy/rPfJR1(T) + L ! ! . ~ v 1 a p ( ~ ) - ;J]}(III.C-6)Here we have modified assumption A.I to read that T is not only smallcompared to (T ) but that it is negligible compared to (T). This should bequite satisfactory in Eq. (III.C-6) where we are dealing directly with absolutetemperature and not time or spatial derivatives of the absolute temperature.

At this point we are ready to proceed from our study of energy transportduring drying to the study of mass transport in the gas phase.

IV. Mass Transport in the Gas PhaseIn Section II we derived the volume average form of the gas phase continuity equation to obtain

8 1 i8 Ey(Py)Y) + V ' (Py)Y(Vy) + - py(vy - w) n yfJ dA = 0t 1/ Ayp(IV-I)and the species continuity equation was expressed as

(IV-2)where

(IV-3)


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166 STEPHEN WHITAKERThese three equations were given previously as Eqs. (ILC-6I), (ILC-72),and (ILC-71), respectively. If we make use of Eqs. (II.B-22) and (IILA-I8)to write

(IV-4)we can simplify the notation ofEq. (IV-I) to obtain

8 .8t(Ey


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In attacking the area integrals in the expression for fi i given by Eq. (IV-3)we make use ofEq. (III.A-19) in order to write

:/" r PiDyp dA + ~ f PiDyu dA = ~ f PiDyp dAr JAyp 1/' Aya 1/' Ayp(IV.A-2)

This allows us to express fii asfii = - G ; : ~ : ) VEy + ~ ty p :i)y)DYP dA

+ ~ f A , a :i)y)Dyu dA - \V[


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168 STEPHEN WHITAKERwhere the term involving V y now appears in the molecular diffusion term.At this point we again follow the arguments given in Section III.B and hopethat ni can be adequately represented in terms of gradients of


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I

A THEORY OF DRYING IN POROUS MEDIA 169Once again we have swept all of our difficulties into one parameter, the totaleffective diffusivity, and experimental determination of this parameter, evenfor an isotropic media, will be a very difficult matter. Assuming that it can bedone we are left with only the problem of determining


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170 STEPHEN WHITAKERthe laws of mechanics is an attractive one; however, there appears to be noreasonable way to accomplish this and we are forced into an applicationof the momentum equation. This is unfortunate for it gives rise to a couplingbetween the equations of motion and the diffusion equation, thus furthercomplicating an already complicated analysis.In attacking the problem of determining the gas phase velocity field weshall follow the analysis developed previously by the author [44] for thetreatment of single phase flow in porous media.* In so doing we shall makeone plausible but very crucial assumption regarding the nature of the twophase 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 onemust keep in mind that neither the development presented in this sectionnor that presented in the subsequent section is valid if this constraint on thetopology of the gas phase is not satisfied.We begin our analysis with the equations of motion for the gas phase:

Dvy _ "PyDr - Pyg + V Ty (IV.B-l)(IV.B-2)

neglect inertial effects, and require that the fluid be Newtonian with constantcoefficients of viscosity so that Eqs. (IV.B-l) and (IV. B-2) reduce to

Here /l y is the coefficient of shear viscosity and Ky is the coefficient of bulkviscosity. Even though V " Vy is not zero the last term in Eq. (IV.B-3) can beshown to be negligible, and we obtain(IY.B-4)

The characteristic time for flow in porous media is on the order of d2Ivywhere d is a characteristic pore diameter and Vy is the kinematic viscosity.For d = 0.1 cm (a very large pore diameter) and Vy = 10- 1 cm2/sec (thekinematic viscosity of air) the characteristic time associated with Eq. (IV.B-4)is on the order of 1sec. Characteristic times for drying processes are generallyon the order of minutes or hours; thus it is reasonable to treat the flow processas quasi-steady, and Eq. (IV.B-4) reduces tot

(IV.B-5)* The development has its origin in the unpublished work of Brenner [57) concerning thetheory of transport processes in spatially periodic porous media.t Here we treat the flow as incompressible even though variations in P, are expected.

",


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A THEORY OF DRYING IN POROUS MEDIA 171Here the new pressure Py is given by

Py = Py - Po + PlP (IV.B-6)where Po is any reference pressure and is the gravitational potential functiongiven by

g = -V (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-I. I t is important to remember that the illustrated curve can pass through every point in the gas phasewithout ever crossing a phase boundary. This will make our analysis quitesimple and much different from the analysis of the liquid phase given in thenext section. The arc length along the curve is s and the unit tangent vectoris represented by A(S). Forming the scalar product of Awith Eq. (IY.B-5) gives

A VPy = flyA. (V 2vy ) (IV.B-8)which can be expressed as

(IV.B-9)

FIG. IV-I. Gas phase flow for a two fluid system in a porous medium.


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172 STEPHEN WHITAKERArguments have been given elsewhere [44] that the volume averaged velocity


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A THEORY OF DRYING IN POROUS MEDIA 173We would now like to make use of a special form of the averaging theoremgiven in Section II.C as Eq. (II.C-16). For the y phase we would express this

result as (IV.B-15)where Aye represents the y phase entrances and exits contained in the averaging volume "Y. I f we substitute Py for I/Iy in the left-hand side ofEq. (IV.B-15)and - ,uymy 0


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174 STEPHEN WHITAKERSince g is a constant, this reduces to

()Y = - ( r ) . g (IV.B-24)where ( r ) is the position vector locating the centroid of the gas phasevolume Yy(t). Provided that gradients of lOy are not outrageously large, (r)will coincide with the centroid of the averaging volume "f/ thus we think of(r) as locating the point at which all our volume averaged functions aredefined. Substitution of Eq. (IV.B-24) into Eq. (IV.B-21) and carrying outthe gradient operation yields*

(Vy) = - ;y Ky {EY[V(PY - Po)Y - Py V(r) g]+ [(py - Po)y - p / r ) g] VE Y} (IV.B-25)

The interpretation of the quantity Vr is given byVr = U (IV.B-26)

where U is the unit tensor. In dealing with volume averaged functions itseems consistent to continue this interpretation inasmuch as (r) is the position vector for the volume averaged functions. Thus we writeV(r) = Uand Eq. (IV.B-25) takes the form

(Vy) = - ~ Ky {EY[V(PY - Po)Y - pyg]fly

(IV.B-27)

+ [(p y - Po)Y - py(r) g] VEY} (IV.B-28)At this point we must be careful to choose the reference pressure Po asthe intrinsic phase average pressure at the point (r) = 0 so that the term(py - Po)Y - Py(r) g is zero under hydrostatic conditions. From a practical point of view it is generally assumed that the first part ofthe right-handside of Eq. (IV.B-28) dominates and the gas phase velocity is expressed as

(Vy) = - ;y Ky {lOY [V(py - Po)y - pyg]} (IV.B-29)thus neglecting the term involving gradients in the gas phase volume fraction. Whether this is a suitable approximation for drying processes remainsto be seen.* Note once again that we consider it to be sufficient to treat the gas flow as incompressible.


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A THEORY OF DRYING IN POROUS MEDIA 175At this point our analysis of the gas phase mass transport is complete butcomplex. In the next section we shall go on to the study of convective trans

port in the liquid phase and suggest some simplifications that one mightmake in order to provide a more tractable theory.

V. Convective Transport in the Liquid PhaseDuring the initial stages of the drying of a saturated porous media, itseems 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 isrequired in order to complete our treatment of the drying process. In thissection we shall derive Darcy's law for the discontinuous fluid in a two-fluidsystem, and then go on to suggest a constitutive equation for the forcesacting on the liquid phase.A. DARCY'S LAW FOR 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 takento be discontinuous and the complications of multiphase flow phenomenaare encountered. The subject has been discussed before from a theoreticalpoint of view by Slattery [46]; however, Slattery does not obtain thetraditional form of Darcy's law which was given in Section IV.B and whichwill be incorporated into our analysis of the liquid phase motion.We begin by following Eqs. (IV.B-I)-(IV.B-5) to obtain for the liquidphase(V.A-I)where (V.A-2)

Here the reference pressure Po is the same reference pressure given inEq. (IV.B-6). To obtain a general expression for Pp we refer to the arbitrarycurve shown in Fig. V-I and form the scalar product of Eq. (V.A-I) withthe unit tangent vector A to obtain

Referring to Eq. (IV.B-lO) we express the point velocity asvp = Mp.


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Averaging volume, V

FIG. V-I. Liquid phase flow for a two fluid system in porous media.

and write Eq. (V.A-3) in a form analogous to Eq. (IV.B-ll)dPfJ/ds = J1fJ/l(V2 MfJ )


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A THEORY OF DRYING IN POROUS MEDIA 177Here dp is given by Py - Pp as we proceed along the curve from the liquidphase into the gas phase, and Pp - pyas we proceed from the gas phaseinto the liquid phase.Integration of Eqs. (IV.B-ll) and (V.A-5) and application of Eq. (Y.A-7)leads to

Ap f . ~ = s ( r ) (dPp) d+Ll 2+" '+ - I]~ = S N - l dl] (V.A-8)which can be expressed as

i=(N+1)/Z _ (dP) i=(N-1)Pp(r) = Pp(O) + L r ~ : S 2 i - l d p dl] + L dP ii=1 S2.-2 I] i=1

'\ ' S2. Y d=(N-1)/Z i = . (dP)+ L... _ -I]i = 1 - S2 i - I dl] (V.A-9)where SN = s(r) and SN-1 = s(rN - 1). In terms of Eqs. (IY.B-ll), (V.A-5), and(V.A-7) we have

i=(N+1)/ZPp(r) = Pp(O) + IIp l ~ = = s : ~ i _ - 2 ' ) " VZMp' (vp> l]

i =N - 1 [ (1 1) ]L (Jpy - + - + dPicPii= 1 r1 rz i (Y.A-lO)We again invoke the argument that spatial variations in (vp> and (vy> arenegligible compared to those of VZMp and VZMy so that Eq. (I1I.3-1O)takes the form

i = N - 1 [ (1 1) ]L (Jpy - + - + dpicPii= 1 r1 r2 i (Y.A-ll)


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We now identify the term in braces involving Mp as the vector -mp so thatour expression for Pp takes the form

(V.A-12)Referring to Eq. (IV.B-13) we see that to within an arbitrary constant we canexpress the gas phase pressure at a position rN-l as

Py(rN- d = Ily ~ ( : t l l ) / Z L q ~ ~ S : ~ ; - l AVZMy dry}.


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A THEORY OF DRYING IN POROUS MEDIA 179I t is worthwhile to keep in mind that if the liquid phase is continuous, theanalysis presented in Section IV.B also holds for the f3 phase and the pressurePp can be specified to within an arbitrary constant by the expression

for a continuous liquid phase (V.A-18)From this result we can deduce a constraint on some of the terms in Eq.(Y.A-17) which takes the form

Py(rN- 1) - pc(rN - 1) + f1PN-1


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180 STEPHEN WHITAKERwhich is, of course, the hydrostatic pressure distribution in a continuousliquid phase. We can now rewrite Eq. (V.A-17) as

(V.A-27)where Pp(r) is now given absolutely by Eq. (V.A-27).In order to obtain Darcy's law for the discontinuous liquid phase we makeuse of an equation for the f3 phase which is analogous to Eq. (IV.B-15) andis written as

(V.A-28)Substitution of the left-hand side ofEq. (V.A-27) for t/lp on the left-hand sideof Eq. (V.A-28) and substitution of - flpmp


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A THEORY OF DRYING IN POROUS MEDIAand write Eq (V.A-31) as

rp(r) - PirN-d + pc(rN - 1) - i1PN- 1N-l)= ~ { f V ( , ) p p ( r ) d V + fV(2)pp(r)dV + ... + fV(M) Pp(r) dV}

- ~ {fV(I)py(rN-1) dV + fV(2) Py(rN- Z) dV + ...+ r py(rN-M)dV} + ~ { r pc(rN- 1)dVJV(M) "y JV(I)+ fV(2)PC(rN- Z)dV + ... + fV(M)PC(rN-M)dV}- ~ { f V ( I ) ( i 1 P N - l N - l ) d V + fV(2)(i1PN-ZN-Z)dV + ...

181

+ fV(M) i1PN-MN-M dV} (V.A-33)Here we should note that rN _ 1 locates the position at which the arbitrarycurve shown in Fig. V-I enters the first continuous subregion within theaveraging volume, and that rN - Z locates the position where the curve entersthe second continuous subregion, etc. The first term in Eq. (Y.A-33) represents the standard f3-phase average which can be expressed as

~ { f V ( , ) p p ( r ) d V + fV(2) Pp(r) dV + ... + fV(M)pp(r)dV}= ~ f Pp(r) dV =


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182 STEPHEN WHITAKERClearly this represents a different kind of an average than the phase averageor the intrinsic phase average. For the special case where Py is everywherea constant designated by Po, Eq. (V.A-35) takes the form

~ { f V ( 1 ) p y ( r N - l ) d V + fV(2)Py(rN- 2)dV + ... + fV(M) PirN-M)dV}Vp(t)= Po Y = POEp(t) (V.A-36)

We should also note that (V.A-37)when Py is the constant Po. For the present we will define the average givenby Eq. (V.A-35) as PyEp and writePyEp = ~ { f V ( I ) P k N - d d V + fV(2)PkN-2)dV+ . .. + fV(M) PkN-M)dV}

(V.A-38)I f there are many subregions of the f3 phase 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 volumefraction, i.e.,

PyEp -+


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A THEORY OF DRYING IN POROUS MEDIA 183The expression for Pc results from Eq. (V.A-6) which can be expressed as

Pc = Py - Pp = ( J p y ( ~ + )r1 r2 (V.A-42)The appropriate average of Pc would be taken over the area A yp and expressedas

_1_S P dA = _1_ S P dA - _1_ S Pp dAA yp Ayp c A yp Ayp Y A yp Ayp (V.A-43)These are intrinsic averages in that they are equal to the function itselfif the function is a constant. Because of this it seems reasonable to expressthe average capillary pressure as

(V.A-44)and accept the reasonable interpretation of Pc as

Pc -4 -AI SA Pc dA =


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184 STEPHEN WHITAKERfor a continuous liquid phase

(V.A-49)resulting when the fJ phase is continuous. Clearly a constitutive equationis required for the various pressure and body force terms in Eq. (Y.A-30),and in the following paragraphs we shall take up this matter.B. A CONSTITUTIVE EQUATION FOR THE FORCES

ACTING ON THE LIQUID PHASEOne of the main difficulties that we encountered in the previous sectionwas the occurrence of the average quantities Py , Pc, and (f) for which therewere no governing differential equations or workable definitions. Clearly aconstitutive equation, or equations, is in order and it is best to begin ouranalysis with an examination of the case where the fJ phase is continuous,i.e., M = 1. Under these circumstances we can use Eq. (V.A-22) to expressthe gas phase pressure for a continuous liquid phase as

(V.B-l)and the liquid phase velocity for a continuous liquid phase as given byEq. (V.A-49)

1


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A THEORY OF DRYING IN POROUS MEDIA 185where the origin is located at the point where the capillary pressure is zero.I f the capillary pressure is measured as a function of saturation sp, then onecan use Eq. (V.B-6) to calculate the saturation as a function of z. This isprecisely what Ceaglske and Hougen [18] did in their study of the dryingof granular solids. One must impose a constraint or a boundary conditionon Eq. (V.B-6) and the experimental relation

(V.B-7)in order to calculate sp as a function of z. Ceaglske and Hougen imposedthe constraint that the average moisture distribution was identical to thatdetermined 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 arecontinuous and the liquid is in a state of hydrostatic equilibrium. Sincedrying processes are usually quite slow, it seems reasonable that the liquidphase is in hydrostatic equilibrium. The fact that the liquid phase is continuous is supported by the capillary pressure-saturation curve obtained byCeaglske and Hougen. Representative curves for drying and imbibitionfor sand are shown in Fig. V-2. Of particular importance is the fact that thecapillary pressure arises extremely rapidly for fractional saturations lessthan 0.1. The general interpretation [29, p. 218] of the abrupt rise in Pc is thebreakdown of the funicular (continuous) state to the pendular (discontinuous)state. I t is in this region that we expect M to be increasing from 1 to 2 to

: 3, etc.1.6

1.41.2UJ

Cl::~ 1.0 IJCl:: \.>- O.S \Cl:: \J:...J...J 0.6c:


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186 STEPHEN WHITAKERWhile 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 equationsand our first order of business will be to develop these equations for thespecial case of a continuous liquid phase and hydrostatic equilibrium.Returning to Eq. (V.B-2) we repeat the development, step by step, givenby Eqs. (IV.B-19)-(IV.B-28) to obtain

(vp) = - ~ Kp' {EP[V(PP - Po)P - ppg]J.lp+ [(pp - Po)fl + (PJo - pp(r) g] VEP]} (V.B-8)

Here we assume that the capillary pressure (PJo can be expressed in termsof the intrinsic phase averages(V.B-9)

and remember that the reference pressure Po is the intrinsic phase ayeragegas pressure at the origin, so that Eq. (V.B-8) takes the form(Vp) = - ~ Kp' {EP[V(PP)P - ppg]J.lp

+ [(pp)fl- (Pp)fl)


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:

A THEORY OF DRYING IN POROUS MEDIA 187phase pressure is hydrostatic. Nevertheless we shall make the followingassumption about the pressure forces acting on the liquid phase:A.I0 Concerning the forces exerted on the liquid phase, we shallassume that both the gas and liquid pressure distributionsare hydrostatic.This assumption allows us to simplify Eq. (V.B-12) to the form,

(vp) = :p Kp {EP[V(Pc) + (pp - py)g]} (V.B-13)indicating that the liquid flow depends entirely on gravity and surfacetension forces. The average capillary pressure in Eq. (V.B-13) is given by

(Pc) = __ f ( J p y ( ~ + ~ ) dAApy Apy r 1 r 2 (V.B-14)For any given system we expect (Jpy to be a function of the temperaturewhile r1 and r2 will depend on Ep, the contact angle, and the structure ofthe rigid porous matrix. We express these ideas as

(Pc) = :Y'((T), Ep, other parameters) (V.B-1S)The structure of the porous media is difficult to characterize; however,Dullien [62] has been able to correlate permeabilities for single phase flowusing the void fraction Ep + Ey and two pore size distributions. It seems clearthat pinning down the "other parameters" for a drying process will be amost difficult task; however, if we restrict our development to the case whereA.ll the porous media is homogeneousthe "other parameters" in Eq. (V.B-1S) are independent of the spatial coordinates and the gradient of (Pc) is given by

V(pc) = (o(Pc)IOEp) VEp + (o(Pc)lo(T) V(T)We designate the two scalars in Eq. (V.B-16) as

and express V


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188 STEPHEN WHITAKERSubstitution of Eq. (V.B-18) into Eq. (V.B-13) leads to a useable expressionfor the liquid phase velocity:


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'.

A THEORY OF DRYING IN POROUS MEDIA 189Imposing assumption A.1O simplifies this expression somewhat to yield


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190 STEPHEN WHITAKERThe introduction of the function must stand out as a crucial assumption inour development and we appropriately list it as the twelfth assumption:A.12 The forces acting on the liquid phase can be uniquelyrepresented in terms of


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A THEORY OF DRYING IN POROUS MEDIA 1912. Negligible Gravitational Effect on the Liquid MotionIn the drying of porous media with very small pores, one can neglect

the effect of gravity and Eq. (V.B-33) simplifies to~ ~ - V . { ( E P ~ : P ) - [k, VEp + k(T> V/flp

(Y.B-38)

(V.B-39)(V.B-40)

A result similar to Eq. (V.B-38) has been suggested* by Luikov [29] andused for extensive calculations by Husain et al. [32]; however, the equationsuggested by Luikov contains the unlikely restriction that K, and K(T)are constant and does not contain the term


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192 STEPHEN WHITAKERseen by requiring that K, be a constant denoted by

K, = KLPLand that Ep = u in the nomenclature of Berger and Pei. Since Ep, K p,and k, are all strong functions of Ep it seems unlikely that K, will be a constant in any real drying process.

VI. Solution of the Drying ProblemIn this section we summarize the previously derived transport equations,

and tabulate the restrictions and assumptions that were made in the courseof the theoretical development.Total thermal energy equation (IILB-29)


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j

.'

A THEORY OF DRYING IN POROUS MEDIA 193Thermodynamic relations (Section III.C)


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194 STEPHEN WHITAKERFor any given problem we think of


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A THEORY OF DRYING IN POROUS MEDIA 195In addition we need to use Eq. (V.B-33) which can be multiplied by PfJ andlisted here as

:t (PfJEfJ) +


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196 STEPHEN WHITAKERReturning to Eqs. (VII-3) and (VII-4) we add these two equations, thuseliminating the term


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A THEORY OF DRYING IN POROUS MEDIA 197Furthermore, it is clear that

PpEp s}pp(Ep + Ey ) ---+ at the start of a drying processthus encouraging us to writeK V[ PpEp ] = K VS, pph + Ey ) , (VII-8)

We know that Eq. (VII-8) is not valid as S ---+ 0; however, that is unimportantsince the entire term tends toward zero under those circumstances. Substitution of Eq. (VII-8) into Eq. (VII-7) leads toe ~ ) = V (K, VS)

+ V'{


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This situation is not likely to occur; however, there is a way around thisdifficulty. Making use of Eqs. (VI-8) and (VI-12), we can express the speciesdensity as

Here it becomes clear that during the latter stages of drying the speciesdensity will be a strong function of the characteristic length r, which in turndepends on p. The dependence of


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A THEORY OF DRYING IN POROUS MEDIA 199special theories so that various drying processes can be studied analyticallywithout recourse to an enormous computational effort.

ACKNOWLEDGEMENTSI This work was supported by NSF Grant P4K1186-000. The author would like to thankProfessor W. G. Gray of Princeton University and Professor Blagoje Andrejevski of thethe University of Skopje for the helpful comments.

NOMENCLATUREROMAN LETTERS K. single scalar component of K. for anisotropic systemA area [m 2] K,ff liquid phase effective termal conduc-Q(1iJ A.p/1/", surface area of the a-fJ inter- tivity tensor [kcaljsec m OK]face per unit volume [m - I] KD liquid phase thermal dispersion tensordm(t) material surface [m 2] [kcaljsec m K]cp constant pressure heat capacity [kcalj K ~ f f liquid phase total thermal conduc-kg OK] tivity tensor [kcaljsec m OK]Cp mass fraction weighted average con- k, -iJ


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200 STEPHEN WHITAKERTU;

uvViVa

total stress tensor [N/m2Jtime [secJdiffusion velocity of the ith species

[m/secJunit tensormass average velocity [m/secJvelocity of the ith species [m/secJvolume of the rigid solid phase con-

tained within the averaging volume[m 3J

1>Aiii

gravitational potential function [m 2/sec2Junit tangent vector

a gas phase mass fraction gradient[m-IJSUBSCRIPTS

designates the ith species in the gasphase

Vp(t) volume of the evaporating liquidphase contained within the averaging volume [m 3J

rr designates a property of the solidphasefJ designates a property of the liquid

Yy(t) volume of the gas phase contained phasewithin the averaging volume [m 3Javeraging volume [m 3Jmaterial volume [m 3Jvelocity of the fJ-y interface [m/secJ

GREEK LETTERSVa/"Y, volume fraction of the rigidsolid phase

y designates a property of the gas phaserrfJ designates a property of the rr-fJ inter-facefJy designates a property of the fJ-y inter-faceyrr designates a property of the y-rr inter-

face

.,(t) V,(t)/"Y, volume fraction of the gasphase

MATHEMATICAL SYMBOLS

K

PPi,

bulk coe1j1cient of viscosity [N sec/m 2Ja function of the topology of the liquidphase

thermal dispersion vector [kcal/secm3Jshear coefficient of viscosity [N sec/m 2Jdensity [kg/m3Jdensity of the ith species [kg/m3J

viscous stress tensor [N/m 2Jrate of heat generation [kcal/sec m3J

d/dtD/Dta/at


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A THEORY OF DRYING IN POROUS MEDIA 2018. T. K. Sherwood, The drying of solids. III. 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, nans. Am. Inst.

Chem. Eng. 27, 310-333 (1931).11. 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, 1134-1136 (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. Appl. Phys.

1,318-333 (1931).15. E. K. Rideal, Phi/os. Mag. [6]44, 1152 (1922).16. A. E. R. Westman, J. Am. Ceram. Soc. 12,585 (1929).17. E. W. Comings and T. K. Sherwood, The drying of solids. VII. Moisture movement bycapillarity in drying granular materials. Ind. Eng. Chem. 26,1096-1098 (1934).18. N. H. Ceaglske and O. A. Hougen, Drying granular solids. Ind. Eng. Chem. 29, 805-813(1937).19. W. B. Haines, Studies in the physical properties of soils. IV. A further contribution to the

theory of capillary phenomena in soil. J. Agric. 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. Sci. 20,97-116 (1930).

21. O. A. Hougen, H. J. McCauley, and W. R. Marshall, Jr., Limitations of diffusion equationsin 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(1925).23. O. Krischer, Fundamental law of moisture movement in drying by capillary flow and

vapor diffusion. VDI Z. 82, 373-378 (1938).24. O. Krischer, The heat, moisture, and vapor movement during drying porous materials.

VDI z., Beih. 1, 17-24 (1940).25. O. 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, 909-916 (1958).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. AIChE Symp. Ser. 69, No. 131, 192-198 (1972).31. D. Berger and D. C. T. Pei, Drying of hydro scopic capillary porous solids-a theoreticalapproach. 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 bio-logical 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.


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202 STEPHEN WHITAKER34. S. Whitaker, "Introduction to Fluid Mechanics." Prentice-Hall, Englewood Cliffs,

New Jersey, 1968.35. R. Aris, 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. Int. 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. Chern., Fundarn.6, 108-118 (1967).42. C. Truesdell and R. Toupin, The classical field theories. In "Handbuch der Physik"

(S. Fliigge, ed.), Vol. III, 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. Chern.

61, 14-28 (1969).45. S. Whitaker, The transport equations for multi-phase systems. Chern. 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. Chern. Eng. Sci.

30,229-233 (1975).48. Y. Bachmat, Spatial macroscopization ofprocesses 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. Chern. Eng. Sci. 26,1893-1899 (1971).50. R. L. Gorring and S. W. Churchill, Thermal conductivity of heterogeneous materials.

Chern. 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. Int. J. Heat Mass Transfer 16,145-155 (1973).52. R. A. Greenkorn and D. P. Kessler, Dispersion in heterogeneous, nonuniform, anisotropicporous media. Ind. Eng. Chern. 61, 14-32 (1969).53. T. Miyauchi and T. Kikuchi, Axial dispersion in packed beds. Chern. Eng. Sci. 30, 343-348(1975).54. R. S. Subramanian, W. N. Gill, and R. A. Marra, Dispersion models of unsteady tubularreactors. Can. J. Chern. 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. Chern., Fundarn. 6,476 (1967).56. J. Meyer and M. D. Kostin, Circulation patterns in the Stefan diffusion tube. Int. J. HeatMass Transfer 18,1293-1297 (1975).57. H. Brenner, "Elements of Transport Processes in Porous Media," Monograph. Springer- "

Verlag, 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).


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A THEORY OF DRYING IN POROUS MEDIA 20359. S. Whitaker, Effect of surface active agents on the stability of falling liquid films. Ind.

Eng. Chern., Fundarn. 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. ofToronto Press, Toronto, 1974.

~ 62. F. A. L. Dullien, New network permeability model ofporous media. AIChEJ. 21, 299-307(1975).

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