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A partially nonlinear finite element analysis ofheat and mass transfer in a capillary-porous bodyunder the influence of a pressure gradientR. W. Lewis and W. J. FergusonDepartment of Civil Engineering, University College of Swansea, Singleton Park,Swansea, U.K.

This paper present s a i nit e element analy sis for a coupled heat and mass t ransfer pr oblem under t hei nfl uence of a pressure gradi ent . The numeri cal modelpr esent ed is part ial ly nonli near, w here some of t hemat eri al pr opert ies are held constant duri ng t he soluti on process, and is compared w it h a ull y nonli nearmodel, w here all material properti es are permi t ted t o vary as t he soluti on progresses.Keywords: capillary-porous, pressure gradient

IntroductionThe process of the transfer of heat and mass of asubstance are amongst the most important sections ofmodern science and have great practical importance inmany technological areas such as chemical engineering,the construction industry, and soil science. The occur-rence of heat and mass transfer problems within in-dustry are wide and varied and the examples mentionedare just a few. A characteristic feature of the phenom-ena of heat and mass transfer in the areas mentioned istheir interdependence, when heat and mass transferbecome one combined process.It appears that the first engineering analysis of thedrying of solids was carried out by Lewis, who pro-posed that the drying of a solid material represents abalance between the processes of diffusion of moistureand of the evaporation of moisture from the materialsurface. The idea of moisture transfer by diffusion wasfurther developed by Sherwood2*3, who discussed thepossibility of moisture vaporizing within the porousbody and moisture transfer occurring by diffusion ofvapor to the surface.More sophisticated models were proposed, whichwere derived from either a classical approach,Krischer4, and Philip and De Vries, or in aphenomeno-logical manner, Luikov6, Huang et al., and Har-

Address reprint requests to Prof. Lewis at the Department of CivilEngineering, University College of Swansea, Singleton Park,Swansea SA2, 8PP, England.Received 18 December 1991; revised 21 April 1992; accepted 5 May1992

marthy. Only Luikov discusses the possibility of incor-porating the gaseous pressure within the porous mediainto his model. His system of coupled partial differentialequations was the more general and could be applied toany body, which can be idealized as capillary-porous inwhich heat and mass transfer takes place.Whitaker developed a system of equations from amechanical approach that demonstrated the necessityof taking into account the gas phase momentum equa-tion, even below the boiling point of water. His ap-proach formed the basis of models by Perre et al. andQuintard and Puiggali. Moyne and Degoviannii* andDegovianni and Moyne13 showed that above the boilingpoint of water the pressure gradient becomes a signifi-cant driving force.Utilizing the thermodynamics of irreversible pro-cesses, 0nsager4,5, Luikovlj, defined a coupled sys-tem of partial differential equations that described thevariation of temperature, moisture content, and pres-sure within a capillary-porous body. Analytical solu-tions to Luikovs system of equations exist in only onedimension8; hence they can be used for only the sim-plest of problems. To solve any realistic engineeringproblems having inherent intricate geometrical contig-urations and complex boundary conditions, resort mustbe made to a numerical technique.Comini and Lewis I9were the first to employ a numer-ical technique; the finite element method, to solve theLuikov equations in real engineering situations. Theytook as an example problems involving a foundationbasement and brick drying. The results obtained wererestricted to the case where the material propertieswere assumed to be constant. This work was developedby Thomas et al.* and Thomas* such that the materialproperties were fully nonlinear and could vary with

0 1993 Butterworth-Heinemann Appl. Math. Modelling, 1993, Vol. 17, January 15

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Heat and mass transfer in capi l lary-porous body: R. W. Lewis and W. J. Fergusoneither temperature or moisture content as the numericalsolution progressed.Lewis and Ferguson2 developed this work further toinclude the pressure term in the coupled heat and masstransfer equations. Previously, the pressure term hadbeen assumed constant throughout the domain of inter-est, and its effect on the numerical solution negligible.However, during an intense period of drying, a totalpressure gradient arises within the material. As a resultof this pressure gradient an additional transfer of mois-ture and heat takes place because of the filtrationmotion of the liquid and vapor contained within thecapillary porous body. The total pressure gradient ap-pears within the material as a result of evaporation andthe resistance of the porous skeleton vapor motion. Asystem of partial differential equations resulted, whichdescribes the interaction between heat, moisture, andpressure in a capillary-porous body.For a nonhomogeneous problem, where the specificmass capacity varies from material to material, a dis-continuity in the moisture content will arise across thematerial boundaries. By introducing the concept ofmoisture potential, as demonstrated in the partiallynonlinear formulation, the discontinuity can be over-come. The relationship between moisture content andmoisture potential is analogous to that of enthalpy andtemperature. Both the moisture content and enthalpyare discontinuous across internal material boundaries,whereas moisture potential and temperature are C(0)continuous.In this paper we will demonstrate how the finiteelement method can be employed to solve the threedegree of freedom Luikov equations. A comparison ismade between two formulations of the Luikov equa-tions, first, a fully nonlinear formulation, which wasdeveloped by Ferguson and Lewis23, where all materialproperties are allowed to vary with the relevant workingvariable, and second, a partially nonlinear formulationwhere some material properties are held constant.

Conservation equationsIn the derivation of the partial differential equations, thefollowing assumptions were made in order to define thefully nonlinear model of temperature, moisture content,and pressure:l The mass is present only as liquid and vapor.l The movement of moisture in the capillary-porousbody is sufficiently slow so that in practice thetemperature of the liquid, the vapor, and the body areequal at coincident points.l Chemical reactions associated with water loss arenot taken into account.l Dimensional changes that occur within the material,due to a temperature or moisture content change, arecomparatively small and can be ignored.The following subscripts are used to describe the vari-ous components of the material: 0 = porous body skele-ton; 1 = vapor; 2 = liquid; 3 = solid; and 4 = inert gas.

16 Appl. Math. Modelling, 1993, Vol. 17, January

M ass tr ansfer equati onThe mass balance for one of the bound materials,vapor or liquid, follows from the law of mass conserva-tionG%P,)-= -divj,,,+Z;a7 ( 1)

The mass flux in a capillary-porous body is the sum ofthe mass flux due to diffusion and the mass flux due tofiltration. The filtration mass flux occurs because of thepresence of a total pressure gradient within the body,which brings about a filtration transfer of the liquid andvapor mixture. The diffusion mass flux is related notonly to the moisture content gradient but also to thetemperature gradient; this effect is more commonlyknown as the Soret effect. The mass flux can be de-scribed as

j, = j d , + j f i lwhere

( 2 )

jd, = - a,p,(Vm + 8VT)jfi, = jifi, + jzf;/ = - k,VP

Equation (1) represents the mass balance for the i bound material. However, by summing for all the mate-rial components, i.e., i = 0, 1, 2, 3, and 4, the massbalance for the material as a whole can be obtained. Thesummation of the source and the sink terms equals zero.Hence, after substituting equation (2) into equation (1)we obtaina(mp,)- = div (a,p,Vm + a,p,SVT + k,VP)a7 (3)

Luiko@ introduced the concept of moisture potential,which is related to moisture content by the followingrelationship

m=c,U (4)In addition, the coefficient of moisture diffusivity isrelated to the coefficient of moisture conductivity by

ka, =--!L (5)PO mOn introducing equations (4) and (5) into equation (3)and rearranging becomes

pOc,,,g = k,,,SV2T + k,,,VU + k,V2P (6)Heat transfer equation

The balance of the thermal energy within the capil-lary-porous body is given by Luiko@cppOE = -divj, - i h iZ i i c,j,VT (7)i=o i=o

where the isobaric specific heat is denoted by ci(8)

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Heat and mass transfer in capillary-porous body: R. W. Lewis and W. J. FergusonHowever, in capillary-porous bodies, provided that theequivalent Reynolds number, Reey, is ~20, the con-vective heat transfer component is small in comparisonwith the conductive component. In the majority ofcases of heat and mass transfer in capillary-porousbodies the equivalent Reynolds numbers are considera-bly less than unity. Therefore the convective heat trans-fer term is assumed to be negligible

i c,j,VT-0 (9)i=oTherefore the simplified form of the thermal energybalance within a capillary-porous body is representedby

Pl,c, g = - divj, - 2 h,Z, (10)i=oThe heat flux is usually related to the temperaturegradient, but in the case of coupled heat and masstransfer, the heat flux is also related, albeit weakly, tothe moisture gradient. This effect is more commonlyknown as the Dufour effect; however, this is usuallyconsidered to be insignificant for capillary-porous bod-ies. We can therefore write the heat flux as

j, = - k,VT (11)The source or sink term is due to the phase change of theliquid or vapor contained within the body structure.Hence

f: h,Z;i=o= -hdiv amzpo Vm + NT + &VP( [

k I) (12)amp0Substituting equations (4), (5), (1 I), and (12) into equa-tion (10) and rearranging we obtain

= (k, + eAk,S)V*T + lk,V2U + lk,V2P (13)Pressure equation

For a closed system, an equation representing thechange in pressure is required. Diffusion of vapor andair in the capillaries is small in comparison with filtra-tion transfer.j, + j, = -k,VP

C, = POCSK, = (k, + lk,,$)K2, = k,,,SK3, = -&k,

Cm = PoCmK,, = Ehk,Kx = kmK32 = - l,,,

(14)

c& = PoCpK,, = EhkpKz = kgKx3 = k,(l

Summing the mass balance equation (1) with respect toi = 1 and 4, we obtain

d ( p , (ml + mJ) = -div(j, + j,) + I, + Z4 (15)&-The specific mass content of the vapor-gas mixture isdetermined by the pressure and temperature

(16)On differentiating equation (16) and assuming T* >> c,and T >> db and letting

Mllbc zz-PR TP,, (17)

we obtaind(m,+mA)=c dp

dT p ar (18)On substituting equations (14) and (18) into equation(15) we obtain

p,.cD$ = k,,V=P - P,.E~ (19)Introducing equation (4) and rearranging, we obtain thegoverning equation for the pressure variation within thecapillary-porous body.

p$$ = -ek,SV=T - l,VU + k,,(l - lV*P(20)

Governing system of equationsEquations (6), (13), and (20) form the governing systemof equations that describe the variation of temperature,moisture potential, and pressure within a capillary-po-rous body. This system of equations was used by Fer-guson and Lewis22,23 to solve practical engineeringproblems.

Cq; = K,,V2T + K,,V=U + K13V2P

Cm: = K,,V*T + K2,V2U + K2,V2P

C$ = K,,VT + K,,V2U + K3,V2P(21)

where

Appl. Math. Modelling, 1993, Vol. 17, January 17

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Heat and mass transfer in capillary-porous body: R. W. Lewis and W. J. FergusonBoundary conditionsThe boundary conditions associated with the partially nonlinear system of equations are

T= T,

kqg +& + a,(T - T,) + (1 - E)(Y,A(U U,) = 0u= u,

k,,,g + j, + k,,$$ + cxu(U - U,) = 0

P = P,

on I, (22)on I2 (23)on I3 (24)on I4 (25)

on rs (26)

The first term in equation (23) is the amount of heatpassing into the body; the second and third terms are theheat supplied at the surface; and the last term is theamount of heat expended in the phase change of theliquid.For equation (25), the first term describes the supplyof moisture to the surface under the influence of atemperature gradient, whereas the final two terms de-scribe the amount of moisture drawn off from the sur-face.Equations (22), (24), and (26) describe the Dirichletboundary condition for the moisture, temperature, andpressure, respectively.Although the surface boundary conditions, equa-tions (22) to (26), are expressed in terms of moisturepotential, U, which is unique to Luikovs system ofcoupled partial differential equations, a more widelyaccepted method of defining the boundary conditions isto express them in terms of absolute humidity, as de-tailed by Keey 24 However, the major drawback is thatdiscontinuities would arise across the internal materialboundaries. By introducing the concept of moisturepotential as the driving force, discontinuities do notoccur.Equations (23) and (25) may be written in a general-ized form as follows

K,,+;+J,:= 0

whereJ; = A, (T - T,) + A,(U - U,) + J,JZ,=A,(T-T,)+A,(U-U,)+J,

K IF,A, = -k,

(27)

A, = F(l - lKI ,4Kz+qSA,= -~

kqC$(l _ l

4 1J , = f $ j q

4 slj,,,, = Kz2 f - kr 14Finite element formulationThe variation of the temperature, pressure, and mois-ture potential throughout the domain of interest, Cl, isapproximated in terms of the nodal values, T,y, Us, andP, as

T- i: Ns k y ) T , W (29)s=l

u = Ii N, (x,YW, (0 (30)s=l(31)

If the approximation given by equations (29), (30), and(31) are substituted into equations (21) a residual isobtained, which is then minimized using the Galerkinmethod. This requires that the integral of the weightederrors over the domain, R, must be zero, with the shapefunctions, N,, being used as the weighting functions, asdescribed by Zienkiewicz25.


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Heat and mass transfer in capi l lary-porous body: R. W. Lewis and W. J. Ferguson

+ K,ZV2U + K,,V2P - Cqz 1a = 0f K2,V2U + K,,V2P - C,,,: 1a = 0 (32)+ K,,V2U + K3,V2P - Cp;

Idf l = 0

The application of Greens theorem (integration byparts) and the introduction of the generalized boundaryconditions to equations (32) produces a system of differ-ential equations that may be written in matrix form asK(O)@ + C(@)$ + J(Q) = 0 (33)

where K(Q) and C(Q) are solution-dependent matrices.

K= (3; ;i E;.)

Typical matrix elements are

Timestepping algorithmsThe numerical solution of equation (33) is achieved byusing the Lees three-level time stepping scheme26,which employs the finite difference technique in time.

K ( + pP+* - W-J)3 2Ar+ J = 0 (34)

The superscript n refers to the time level and A T refersto the timestep. The Lees three-level time steppingscheme has the advantage of solving for the time level n+ 1, by evaluating the coefficient matrices at time leveln, which avoids the need for an iterative solver. Wood2and Wood and Lewisz8 showed that this scheme wasstable even though oscillations appeared in the solutionwhen a convective boundary condition was used. Thenoise can be dampened to an acceptable level by intro-ducing a maximum permissible timestep.

ApplicationThe problem to be investigated consists of a cross-section through a container wall, which is designed toprevent the temperature and moisture levels from ex-ceeding preset levels above which damage to the con-tainers contents would occur. The cross-section of thecontainer, shown in Figure I, is one-dimensional. Theexample consists of an epoxy resin container wall withair inside. Inserted into the epoxy resin container wall isa layer that acts as a moisture barrier. The moisturebarrier is positioned within the epoxy resin wall toprevent excessive moisture migration in the x-directionfrom the epoxy resin exterior to the air entrapped withinthe container. To numerically achieve the effect of amoisture barrier tilm, a small value, compared to thevalues used for the epoxy resin and air, was used for thecoefficient of moisture conductivity, k,. Although themoisture barrier is designed to prevent an excessivemigration of moisture, it is not intended to act as athermal insulator nor to provide an air-tight seal; there-fore the gradients of temperature and pressure are notaffected by the presence of the barrier film.Luikovs three degree of freedom system of partialdifferential equations describes the behavior of temper-ature, moisture content, and pressure within a capil-lary-porous body. Clearly air is a gas and not a capillary-porous body, but, treating it as a material similar to solidmaterials and idealizing it as a capillary-porous body, itis intended to obtain the moisture variation within thecentral section of the container to a first order of ap-proximation. An obvious limitation in treating the en-trapped air in this manner is that we cannot predict theair movement or circulation by convection within the


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Heat and mass transfer in capi l lary-porous body: R. W. Lew i s and W . J. FergusonI8 Insulated. Epoxy resin.

Air.

4B \

///////////////// PI 1,,,,,,,,,,1 I I I I I /v//I/. %,I,Boundaryedge.

DI Insulated. Moisture barrier.* 45.0mm 4* 47.5mm e0. 50.0mm -

Figure 1. Cross-section of the epoxy resin container

Table 1. Material properties for the fully nonlinear formulationMaterial Moistureproperty S.I. units Epoxy resin barrier Air

PO kg/m3 1170.0 55.0 1.19 at 20C1.13 at 35C1.08 at 50C

k,J/kg 2.30*106 2.30*1 O6 2.30*106J/hr.m.K 512.0 at 20C 154.8 92.04 at 20C794.0 at 50C 99.45 at 50C

kg kg/hr.m.N.m-* 4.50*10m5 4.20*10-* 6.00*10-a, m/h? 3.52*10m6 at 50% mc 1.69*10m6 1.86*10~* at 40% mc1.86*10m5 at 100% mc 9.33*10m2 at 80% mcc, J1kg.K 1400.0 2343.0 1012.0cP kg/kg.N.m- 5.00*10~2 5.00*10-2 5.00*10~2E 0.3 0.3 0.3 at 20C

1.0 at 50C6 kg/kg.K 6.00*10m3 6.00*10-3 6.00*10~3

container, which are brought about by the temperaturedifference and variation in density of the air within thecentral enclosed section of the container. An alternativeto modelling the air as a capillary-porous body would beto apply a boundary condition to the air/resin interface.However, Figure 1 shows only a cross-section throughthe container wall; the internal temperature, moisturepotential, and pressure equilibrium values are notknown. Hence, a boundary condition applied to the air/resin interface would not be practical.Air has highly nonlinear material properties, andthese are shown alongside the material properties forthe epoxy resin and the moisture barrier in Tabl e 1. Thematerial properties presented in Table I are those usedin the numerical simulation employing the fully nonlin-ear model. The material properties used in the partiallynonlinear solution were those given in Table 1 exceptthat pO , E, 8, and c, were held constant and were takenas the mean of those values used for the fully nonlinear

model. As previously stated, this epoxy resin containerexample is a one-dimensional problem. Flow occursonly in the x-direction and not in the y-direction; hence,faces AC and BD are assumed to be insulated becausethere is no flow either into or out of the body throughthese faces.The initial conditions throughout the domain of inter-est were temperature 2oC, moisture potential SOM,and pressure 1 atmosphere. Along face CD a flux typeboundary condition, with a ramp loading, was appliedto the temperature and moisture potential, whilst for thepressure term there was a fixed-point boundary condi-tion, also with a ramp loading. The boundary conditionsincreased linearly from the initial conditions to the ep-oxy containers steady state conditions of temperature5oC, moisture potential 100M and pressure 2 atmo-spheres. The ramp loadings of the tixed point pressureboundary condition and of the ambient temperature andmoisture potential are shown in Figure 2. Along sides


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Heat and mass transfer in capi l lary-porous body: R. W. Lewis and W. J. FergusonAC and BD there was assumed to be a nonconductingboundary condition. The finite element mesh used forthe numerical solutions of both the fully and the par-tially nonlinear formulations is shown in Figure 3.

Figures 4 to 6 show the variation of temperatureagainst time; Figures 7 to 9 show the variation ofmoisture potential against time; and Figures 10 to 12show the variation of pressure against time for nodes 3,75, and 123, respectively. The nodes for which resultsare presented lie along the x-axis, i.e., the center line ofthe section of the epoxy resin container wall underanalysis. The problem is one-dimensional, in the X-direction; therefore, no flow occurs in the y-direction.

For a given x-coordinate value along the center line, thevariation of temperature, moisture potential and pres-sure versus time is identical for any position along they-axis with the same x-coordinate. Two curves are pre-sented in each figure, one represents the numericalsolution from the fully nonlinear model, where thethermophysical properties are permitted to vary, andthe other represents the numerical solution from thepartially nonlinear model, where, in this example, thethermophysical properties are held constant.The variation of moisture within the capillary-porousbody with time is expressed in terms of moisture poten-tial, Figures 7 to 9. The fully nonlinear heat and mass

C Temperature. Moisture Potential. Pressure.51 atms

0.01 Time. (HE)Figure 2. Ramp loading boundary conditions

1 Time. (Hrs) 1 Time. (Hrs)

3 75 123I

Figure 3. Finite element discretization of the epoxy resin container

E I / I I I I I 10.0 0.03 0: 3.5 02 015 02 05s 0.4 CA5 03Time. (Hrs)

8: I 1 1 I 10.0 0.05 O.? 0.8 01 025 03 035 0.4Time. (Hrs)

Figure 4. Temperature vs. time at node 3 Figure 5. Temperature vs. time at node 75


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Heat and mass transfer in capillary-porous body: R. W. Lewis and W. J. Ferguson

Sl I 1 I I I00 0.M c.t 0.8 02 OZS Oil 0s 0.4

Time. (Hrs)Figure 6. Temperature vs. time at node 123

Figure 7. Moisture potential vs. time at node 3

0 m 200 do do Ga &I 40 &I s& &ITime. (Hrs)

Figure 8. Moisture potential vs. time at node 75

I Jo-tidy Nm-inw.-.- ;lAy No0-bw.r I. I I I0 m :m x8 do & 6io &I &a 930 c&o

Time. (Hrs)Figure 9. Moisture potential vs. time at node 123

Ti me. (Hrs)Figure 10. Pressure vs. time at node 3

0 50 m 50 2s 250 mTime. (Hrs)

Figure 11. Pressure vs. time at node 75


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Heat and mass transfer in capillary-porous body: R. W. Lewis and W. J. Fergusoncontainer, node 3, Figure 4, away from the externalwall, where the boundary conditions are applied.The numerical solutions presented in the moisturepotential versus time graphs, Figures 7 to 9, demon-strate and highlight the effect of varying the materialproperties as the solution proceeds toward steady state.Once again, the fully nonlinear solution reaches steadystate equilibrium faster than does the partially nonlinearsolution. This effect is most prominent, although notsignificant, at node 3, Figure 7, where the fully nonlin-ear solution reaches a value of 90M in 480 hrs, whereasthe partially nonlinear solution reaches the same valuein 640 hrs, clearly indicating the consequences of non-linear thermophysical properties on the numerical solu-tion.

Figure 12. Pressure vs. time at node 123

transfer model calculates the moisture distributionwithin the body in terms of moisture content, while thepartially nonlinear model evaluates the moisture distri-bution in terms of moisture potential. Moisture contentis related to moisture potential according to equation(4), provided that the specific mass capacity, c,, isconstant. In a nonhomogeneous body, the moisturecontent would produce a discontinuity across the mate-rial boundaries of the body, which, unless there was afinely graded finite element mesh at these boundaries,would cause problems while solving the fully nonlinearmodel. Luikov introduced the concept of moisture po-tential to overcome this difficulty, because moisturepotential is a continuous function across the internalmaterial boundaries, i.e., it is C(0) continuous. Thediscontinuity in moisture content and the relationship tomoisture potential is analogous to heat content andtemperature. However, in this example, the specificmass capacity is constant throughout the whole of thedomain of interest-the specific mass capacity for theepoxy resin is equal to the specific mass capacities forboth the moisture barrier and the air. Hence no discon-tinuities occur across material boundaries, because themoisture content and the transient moisture contentsolutions obtained by the fully nonlinear model canbe converted to moisture potential according to equa-tion (4).

The difference between the two numerical solutionsfor pressure versus time, Figures 10 to 12, is not signifi-cant and is caused by the effect of the moisture gradient.For both numerical models the values of the moisturefiltration coefficient, k,, and the coefficient of humid aircapacity, c,, were held constant. Because the system ofpartial differential equations that describe heat andmass transfer are coupled, both the temperaturegradient and the moisture gradient affect the transientpressure solution. However, in this example the tem-perature has reached steady-state equilibrium at allnodes by 0.4 hr, and, thereafter the temperaturegradient will be zero. The temperature gradient there-fore has little or no effect on the transient pressuresolution, which is affected solely by the moisture andpressure gradients.Conclusions

The effect of permitting the thermophysical proper-ties to vary can be seen in the transient temperaturesolutions. Figures 4 to 6. which were obtained from thetwo numerical models. In this example, permitting thematerial properties to vary with either temperature,moisture content, or pressure causes the solution ob-tained from the fully nonlinear model to reach steadystate equilibrium conditions, from the assumed initialconditions, faster than does the partially nonlinear solu-tion, where all thermophysical properties were heldconstant. Although the difference in the transient tem-perature solutions is not significant, the effect becomesmore prominent as we proceed toward the interior of the

In the example presented in this paper, the numericalsolutions obtained from the fully nonlinear model fortemperature, moisture potential, and pressure reachedsteady state equilibrium more quickly than did the nu-merical solution from the partially nonlinear model.Although the difference between the two solutions wasinsignificant, this is not always the case. The fullynonlinear solution is dictated by the nonlinear behaviorof the materials under examination, and the degree ofvariation between the fully and partially nonlinear nu-merical solutions will be dependent on these materialproperties. For other engineering problems, where thespecific mass capacity is not constant between materi-als, the partially nonlinear model would be used to solvethe problem so as to avoid a discontinuity in moisturecontent across the material boundaries.AcknowledgmentsWJF is grateful for the support of the Science andEngineering Research Council for a research scholar-ship to carry out this work.Nomenclature1; volumetric capacity of the source of the mate-rial


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Heat and mass transfer in capi l lary-porous body: R. W. Lewis and W. J. FergusonMPRRe,,Tu

2

%SEPOnh

molecular mass of humid airtotal pressure of humid air inside the body,kN/muniversal gas constantequivalent Reynolds numbertemperature, Cmoisture potential, Mmoisture diffusivity, m/sdegree of saturation of the capillaries withinthe bodymoisture capacity, kg/kg*Mcoefficient of humid air capacity, kg*m*/kgkNheat capacity, J/kg.Cspecific enthalpydensity of mass transfer flowdensity of diffusion mass transfer flowdensity of filtration mass transfer flowheat fluxcoefftcient of moisture conductivity, kg/rn.s-Mmoisture filtration coefficient, kg*m/s+kNcoefficient of thermal conductivity, W/m.Cmoisture contentconvective mass transfer coefficient, kg/m-s.Mconvective heat transfer coefftcient, W/C~m*thermogradient coefficient, M/Kratio of vapor diffusion to total diffusiondry density, kg/m3bulk porosity of the bodylatent heat of vaporization of water, J/kg

ReferencesLewis, W. K. The rate of drying of solid materials. J. Ind. Eng.Chem. 1921, 13, 427-432Sherwood, T. K. Application of theoretical diffusion equationsto the drying of solids. Trans. Am. Ins?. Chem. Eng. 1931, 27,190-202Sherwood, T. K. The drying of solids III. J. Ind. Eng. Chem.1930, 22, 132-136Krischer, 0. Die wissenschafthchen Grundlagen derTrocknungstechnik. Springer, Berlin, 1932Philip, J. R. and De Vries, D. A. Moisture movement in porousmaterials under temperature gradients. Trans. Am. Geophys.Union 1957, 38, 222-232Luikov, A. V. Heat and mass transfer in capillary-porous bod-ies. Pergamon Press, Oxford, 1966Huang, C. L. D., Siang, H. H., and Best, C. H. Heat andmoisture transfer in concrete slabs. Int. J. Heat Mass Transfer1979, 22, 257-266

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