Cabeq 2011-03 verzija 6 · To generalize Darcy’s law to multiphase sys-tems one must assume that...

Modeling Plain Vacuum Drying by Considering a Dynamic Capillary Pressure

S. Sandoval-Torres,* J. Rodríguez-Ramírez, and L. L. Méndez-Lagunas

Instituto Politécnico Nacional, CIIDIR-Oaxaca, Hornos No. 1003,Col. Noche Buena, Santa Cruz Xoxocotlan, Oaxaca, Mexico

A coupled drying model for wood is proposed by introducing a dynamic capillarypressure. The pressures of non-wetting phase, the wetting phase, and the capillary pres-sure at equilibrium has been considered as non-static; this approach includes a two-scalemodel. According to numerical results, liquid, water vapor and air dynamics in the cham-ber have strong interactions with re-homogenization in the surface, controlled by capil-lary forces. The results at 60–100 bar and 70 °C are discussed. The phenomenologicalone-dimensional drying model is solved by using the COMSOL’s coefficient form and aglobal equation format. A good description of drying kinetics, moisture redistribution,and mass fluxes is obtained. A comprehensible transition at the fiber saturation point iswell simulated.

Key words:

Wood, vacuum drying, dynamic capillary pressure, COMSOL®

Introduction

Drying is a ubiquitous natural process. Thedominant stress during evacuation of free water inwood is the capillary pressure from the tinymenisci.1 Capillary pressure is normally written as�p = 2�cos�/r. In porous media, every meniscusacts like a low-pressure pump trying to suck liquidfrom other places. Because of the heterogeneity ofthe pore sizes, the menisci in small pores can pro-duce lower pressure and draw liquid from menisciin large pores. This flow would move the air-liquidinterface rapidly through the large pores, creatingbursts (Fig. 1).

The bursting process will terminate eitherwhen all the pores are small enough to balance thecapillary pressure or when the displaced liquidflows to the nearby menisci and reduces the staticcapillary pressure.

According to Lehmann et al.,2 drying of wetporous media is an immiscible displacement pro-cess of liquid phase by invading gas; hence princi-ples of invasion percolation in gas phase.

The drying kinetics of a porous medium isclassically described in three main periods, whichdepend on the interplay between the external andinternal mass transfers during evaporation. The firstperiod is described as a heating up period depend-ing on the external mass transfer; the second periodis described as a constant rate period, whereas thethird period is identified as a falling rate perioddominated by the internal mass transfer.3

In most traditional treatments of capillary pres-sure, it is defined as the difference between pres-sures of phases, in this case, air/water vapor andliquid water, and it is assumed a function of satura-tion. On one hand, recent theories have indicatedthat capillary pressure should be given a more gen-eral thermodynamic definition, and its functionaldependence should be generalized to include dy-namic effects.4 On the other hand, Beserer andHilfer5 affirm that experimental features betweencapillary pressure and saturations cannot be pre-dicted, since the theoretical derivation of the equa-tions from the well-known laws of hydrodynamicshas not yet been accomplished.

Surface areas and surface tensions known tocontrol capillary action and wetting properties donot appear in the mathematical formulation of thetraditional theory:

vk

p gr��

( ) (1)

S. SANDOVAL-TORRES et al., Modeling Plain Vacuum Drying by Considering …, Chem. Biochem. Eng. Q. 25 (3) 327–334 (2011) 327

*Email: [email protected]

Original scientific paperReceived: August 17, 2010Accepted: August 15, 2011

F i g . 1 – Capillary pressure and flow displacement

To generalize Darcy’s law to multiphase sys-tems one must assume that the flow of either fluidis hydrodynamically independent of the other flu-ids. The so-called relative permeability kr� accountfor the fact that the flow medium and its permeabil-ity for each fluid phase � is altered by the presenceof all other phases. Based on this idea, Darcy’s lawis generalized to the velocity field v of each phase �giving:

vk k

p gr��

� � �

�( ) (2)

In most applications, the relative permeabilitiesare treated as functions of saturation k f S� �� ( ).6

But in practice, these relationships are more com-plicated because of the dependence on physicalquantities like surface tensions, contact angles, vis-cosities, mass densities, pore structure, and the flowconditions have to be considered. Relative perme-ability is known to be affected by the flow velocity(or pressure gradient) at which the measurementsare carried out. Mathematically, relative permeabil-ity is also affected by the boundary conditions.7

Capillary pressure–saturation and relative perme-ability–saturation relationships are highly nonlin-ear, and their determination is often a very difficulttask. In this work, we are interested in including adynamic capillary pressure in a two-scale vacuumdrying model. Our study reveals the flow pattern ofdrying which may ultimately afford a means to con-trol drying in porous media. A deep understandingof drying may provide more fundamentals to deter-mine moisture transport and distribution in porousmedia.

Mathematical model

In particular, the reader must be aware that allvariables are averaged over the REV (Represen-tative Elementary Volume),8 hence the expression“macroscopic”. This assumes the existence of sucha representative volume, large enough for the aver-aged quantities to be defined and small enough toavoid variations due to macroscopic gradients andnon-equilibrium configurations at the microscopiclevel. The approach proposed by Whitaker9 andPerré10 was followed in this work. The physicalmodel is based on heat, mass and momentum trans-port at Darcy’s scale as obtained by volume averag-ing the corresponding pore scale balance equations.The average value of variable is defined as:

� � � � 1

VV

REV REV

d (3)

In our equations, instead of measuring pc and Sat given equilibrium distributions of the phases,they are determined continuously over time as theflow occurs. This leads to non-uniqueness in the re-lation between capillary pressure and saturation. Onthe pore scale, the dynamic contact angle is oftengiven as one reason for the dynamic effect in capil-lary pressure. The contact angle decreases with in-creasing flow velocity for drainage and increaseswith increasing velocity.

Assumptions

We established these considerations:

– Gravitational effects are negligible

– Temperature and pressure in the dryer are ho-mogeneous

– The gas phase is water vapor, which behavesas an ideal gas.

– The bound water has physical properties sim-ilar to those of free water.

– Thermodynamic equilibrium, so averagetemperatures for each phase are the same, and par-tial vapor pressure is at equilibrium.

– Lack of heat and mass losses assuming idealisolation.

Then, we can write:

T T Tv l� � Thermodynamic equilibrium

p a pv w vsat� � Vapor pressure

Compressibility effects in the liquid phase wereneglected, meaning l

ll cste� � and the gas

phase was considered as an air/water vapor idealmixture.

Mass transport

i

g i i

gm p

RT� (4)

p p pgg

ag

vg� � (5)

gg

ag

vg� � (6)

Pour i = a (air) or v (vapor)

Free water transport is explained by Darcy’slaw. The velocities of the gaseous and liquid phasesare respectively, expressed using the generalizedDarcy’s law, but as mentioned, gravitational effectsare neglected:

Vk k

p gl

rl

lll

l��

� � �

( ) (7)

Vk k

p gg

rg

ggg

g��

� � �

( ) (8)

328 S. SANDOVAL-TORRES et al., Modeling Plain Vacuum Drying by Considering …, Chem. Biochem. Eng. Q. 25 (3) 327–334 (2011)

Capillary pressure

Traditionally, one assumes that this relation-ship is determined under quasi-static or steady stateconditions but can be applied to any transient flowprocesses. The relationship between capillary pres-sure and saturation might not be unique. Tradition-ally, pc(S) is determined under quasi-static or steadystate conditions. We have implemented a model byintegrating a non-equilibrium capillary pressure.The equilibrium capillary pressure is determinedunder quasi-static or steady state coefficient. Undertransient conditions, the dynamic capillary pressurepc

dyn prevails resulting in a dynamics relationship

p Scdyn ( ), so mathematically we have:

p p p f Snw w c� � � ( ) (9)

p p p SS

tnw w ceq� � ��( ) �

�

�(10)

p p pS

tp Snw w c

dynceq� � ��

�

�( ) (11)

p SSc � � �

�

��

�

��56 75 10 1

10623. ( ) exp

.(12)

Where:

��

� �

� �

� �

�

��

�

��

w d

wK

p

g

2

(13)

� factor is a phenomenological coefficient thattakes positive values and depends on water saturation:

� �( ) ( )Sp

SS

cstatic

B�d

d(14)

For the relative permeability, we write:

k Srl �2 (15)

k Srg � �( )1 2 (16)

For the flux of bound water:

l l l

rl

cVk k

p��

�� 1

��

� � � �

l

rl

lgg

l

k kp g( )

(17)

For the transport of the vapor phase, we use the“Dusty gas model”; this expression considers thatvapor water and air mobility depends on pressureand concentration gradient of the gaseous phase:

vg

v vg eq rg

ggg

gg

eqVk k

p D C��

�� (18)

ag

a ag eq rg

ggg

gg

eqVk k

p D C��

�� (19)

With these parameters, the perturbations in theconvective and diffusive dusty transport are consid-ered. The diffusion-sorption model described thebound water migration. A phenomenological ap-proach can explain water flows to the form dis-cussed extensively in literature:

J D Wb s b b�� (20)

Below FSP, the moisture is considered boundto the cell wall and, therefore, bound water diffu-sion can be considered the predominant mass trans-fer mechanism.

Hygroscopic equilibrium

The mass fraction of water vapor C is definedas:

C �vapour mass

humid air(21)

The equilibrium moisture content is expressedby

W W T aweq � ( , ) (22)

aw HR� (23)

Now we can write macroscopic balances formass:

For the air mass and water vapor:

�

�a

ag

atV� �� ( ) 0 (24)

�

�

W

tV V Jb

sl l v

gl� ��

��

� !�

10( ) (25)

An energy balance allows us to write:

�

�Cp

T

t�

(26)

� � � � � �[( )] l l l b l ag

a a vg

v vV Cp J Cp V Cp V Cp T

� �� ( ) ( ) ( ) ( ) �b b b v v b bV h h K h h K T 0

The specific heat of wood, expressed by:

Cp Cp Cp Cp Cps s l b l v v a a� � � � �( ) (27)


Equation for the mass balance in the dryer

The conservation equations in the dryer cham-ber are:

d

d

a

ch

ach pump

chaatm leak

cht

q

V

q

V�� (28)

For the dry-air

d

d

v

ch

vch pump cond

chvatm leak

chm

cht

q q

V

q

VF

A

V��

�� (29)

Boundary conditions

During drying, the pressure on the surfaces isin equilibrium with the chamber pressure pch. Re-calling that one of the primary variables is the aver-aged air density, a Dirichlet boundary condition isimposed for the air mass flux. The Dirichlet bound-ary condition has been modified to form an appro-priate non-linear equation for this primary variable.For moisture fluxes, equilibrium between water va-por at surface and the vapor pressure in the cham-ber was established. About dryer chamber, a mix-ture water-vapor/dry-air was considered. This mix-ture depends on dryer (chamber) temperature. Withrespect to energy fluxes, temperature at the surfaceis in equilibrium with the chamber temperature.Then, boundary conditions are written as:

p pag

ach� For dry air (30)

F hw vg

vdryer� � � �n ( ) For water vapor (31)

T Tchamber surf" For energy (32)[plain vacuum drying]

Materials

Samples of European oakwood belonging tothe Fagaceae family were cut from a freshly felled100-year-old Quecus pedonculata tree from a forestin Pessac-Toctoucau, in France. The selected loghad a central pith and normal growth ring patternand there was no apparent compression wood pres-ent identified by colour. The experimental setup is avacuum chamber where pressure is regulated be-tween two values (pmin, pmax). The chamber wasbuilt in glass; one balance is kept inside the cham-ber in order to log the mass variation of the sample.A thermometer gives the dryer temperature. Theheating source is an electrical resistance which tem-perature is controlled with the help of a PID con-troller. Experiments are performed on Oakwooddisks (7 cm diameter and 2.5 cm height). We call

this drying method “plain vacuum drying” since wedo not use a fluid as a drying agent, so only thepump aspiration accelerates mass flux. Pressure inthe chamber is controlled at 60–100 mbar. Temper-ature inside the wood sample is obtained at two dif-ferent positions. More details are published inSandoval et al.11 The parameters considered in thisstudy are limited by practical implications. The dry-ing temperature should be lower than 80 °C toavoid degradation of wood.

Fig. 2 shows the geometry considered in ourequations system 1D. Heat source is an aluminumplate; such plate is heated by an electrical resis-tance.

Results and discussion

To solve the equations, the commercial solverCOMSOL Multiphysics©13 was used. The partialdifferential equations (material scale) were writtenin the general form and by using an unsymmetric--pattern multifrontal method. The two ordinary dif-ferential equations (dryer scale) were introduced byconsidering a pump aspiration of 0.0027 m3 s–1 (thereal situation). To add a space-independent equationsuch as an ODE, a Global equation format was cho-sen. As the time derivative of a state variable (den-sity of air and water vapor) appears, the state vari-able needs an initial condition; for this reason theinitial condition for pressure was the atmosphericpressure. For solving time-dependent variables, thebackward differentiation formula (BDF) was ap-plied. The discretization of the time-dependent PDEvariables is:

0� � �L U U U t N U tF( , , , ) ( , ) # (33)

0� M U t( , ) (34)

This is often referred to as the method of lines.Before solving this system, the algorithm eliminatesthe Lagrange multipliers #. If the constraints 0 = Mare linear and time independent and if the constraintforce Jacobian NF is constant then the algorithm


F i g . 2 – Adopted configuration to model vacuum drying

also eliminates the constraints from the system.Otherwise, it keeps the constraints, leading to a dif-ferential-algebraic system.

The boundary conditions link variables at dryerscale (air and vapor in the chamber) and variablesat material scale (mass of air, moisture content, en-ergy). The general form provides a computationalframework specialized for highly nonlinear prob-lems. Direct solvers for sparse matrices involvemuch more complicated algorithms than for densematrices. The main complication is due to the needfor efficient handling of the fill-in in the factors Land U. In COMSOL®, a typical sparse solver con-sists of four distinct steps:

1. An ordering step that reorders the rows andcolumns such that the factors suffer little fill, or thatthe matrix has special structure such as block trian-gular form.

2. A symbolic factorization that determines thenonzero structures of the factors and creates suit-able data structures for the factors.

3. Numerical factorization that computes the Land U factors.

4. A solve step that performs forward and backsubstitution using the factors. For the most generalunsymmetric systems, the solver may combinesteps 2 and 3 (e.g. SuperLU) or even combine steps1, 2 and 3 (e.g. UMFPACK) so that the numericalvalues also play a role in determining the elimina-tion order. The direct (UMFPACK) solver was usedin this simulation.

As we have indicated, the solid phase was con-sidered as rigid and the intrinsic permeability k con-stant. We have solved this problem in 1D.

Fig. 3 compares predicted and experimentalaverage moisture content in wood. These resultscorrespond to the vacuum drying at 70 °C and60–100 mbar of pressure. It can be observed thatthe model is able to predict correctly the averagedrying kinetic over the 26 hours of this experiment.Theoretical drying kinetics shows good agreementwith experimental data. Differences between exper-imental and simulated kinetics can be explained bya variation of values in permeability, static capillarypressure and transfer coefficients of water vapor,since they can vary from one sample to another.Fluctuations in mass measurements during the ini-tial heating period can be explained by local inter-nal moisture variations within the sample. In thesame figure, one can see a good transition betweencapillary phase and hygroscopic phase (at approxi-mately 0.4 of moisture content). The same figuredepicts the chamber pressure. When the vacuumpump starts, an internal pressure reduction is ob-served. The simulated pressure in the chamber iscompared with experimental measures. Comparison

provides a good confidence of equations and modelat lab scale (dryer). Two regimes exist during vac-uum drying; the first one is called the active re-gime, and the second one the passive regime.Model describes both active (pump aspiration) andpassive regimes (stopping the aspiration). Duringvacuum drying, pressure is controlled between twovalues: pmax and pmin, this fact allows us to obtain azigzag behavior in chamber pressure that is wellrepresented by simulation.

The moisture movement depends on the per-meability of the wood and the internal pressure gra-dient. The permeability of wood is a dominant fac-tor in controlling moisture movement because themechanism of moisture movement in wood is flowof water vapor through the cellular structure ofwood driven by the pressure gradient in the vacuumdrying at a temperature exceeding the boiling pointof water (at 60 and 100 mbar the boiling point ofwater is 35.9 and 45.3 °C respectively). Of course,intrinsic anatomical structures, content of extract-ives and specific gravity of wood have an influencein permeability.

It is generally admitted that drying time de-creases as temperature increase. In this paper, weconsider one temperature to facilitate the discus-sion. During the first regime, acceleration of massflux is due to pump action, whereas during the sec-ond period re-homogenization is due to the pumpstopping. A strong coupling between equations atlab scale and material scale exists. This coupling iswell-simulated by COMSOL by introducing theboundary conditions. The link between the twoscales evolves according to the chamber conditionsthrough the boundary conditions imposed at thesurface of the porous medium (large-scale tomacroscale) and by the total heat, vapor, air, and,eventually, liquid fluxes leaving the wood surface.A strong effect of the change of the external pres-sure on internal transfer is when, for instance, a liq-


F i g . 3 – Simulation. Vacuum drying kinetics and moisturecontent in the wood surface.

uid flow is driven by the internal overpressure(high-temperature configuration). The macroscalemodel must be able to capture such a coupling and,in turn, supply the chamber model with relevantvalues of the different fluxes leaving the board.

The drying regimes produce natural oscilla-tions at the surface temperature, since during activephase (pump aspiration) a shut of temperature ex-ists due to evaporation, and during the passivephase re-homogenization is developed due to equi-librium. The mass flow is more important duringthe vacuum pumping, so there is a drop in pressurein the enclosure and consequently faster evapora-tion on the surface of wood. Evaporation causes anincrease in the pressure chamber and a homogeni-zation of quantities in the material. This fact is visi-ble on the flux on the surface. For better explana-tion, in Fig. 4 we show the average moisture con-tent and moisture distribution at the surface. We canobserve a drop in the moisture content around thefiber saturation point (fsp), which indicates that freewater has been eliminated. After this point (fsp), themechanisms of mass transport are mainly due todiffusion, since the moisture evacuated in gaseousphase is more slow. Our proposed model describesphysically this transition (Fig. 4). A mass flux isimproved during vacuum drying, since the externalvacuum reduces the required temperature for evap-oration.

In order to estimate the error or deviation of ourmodel, the quadratic error was computed betweenexperimental and numerical data, according to:

SW W

W�

�$

%&

'

()

exp model

exp

2

(35)

Fig. 5 depicts these values. It should be ob-served that at 6 hours of drying and at the end, the

more important deviations are more visible. Thesedeviations are explained by the capillary pressureand permeability function, and by the changes inmoisture equilibrium respectively, because thedesorption isotherms could change due to biologi-cal variations and chemistry (extractives) in wood.

During drying, the wood surface is constantlyfed in moisture, but free water is easier to evacuate.The evaporation rate depends mainly on the level ofpressure and temperature. By consequence, a moreimportant concentration of water vapor exists in thechamber during this capillary stage, neverthelessglobal pressure (vapor + air) is regulated betweenpmax and pmin. At the very beginning of the process,the chamber pressure equals the atmospheric pres-sure and the gas mixture consists mainly of air.

Fig. 6 displays the moisture profiles in woodduring vacuum drying; this figure displays moisturedistribution in the sample. Free water transport iswell identified. Two different distributions are visi-ble: the first one explained by capillary forces, andthe second by diffusional mechanisms. In the hy-


F i g . 4 – Average moisture content and moisture distribu-tion at the surface

F i g . 5 – Quadratic errors of the simulated drying kinetic

F i g . 6 – Moisture profiles during plain vacuum drying

groscopic stage, the thermodynamic equilibrium be-tween gas phase and bound water allows a less im-portant drying rate.

A parabolic distribution is displayed in the hy-groscopic stage. In the capillary zone, we found adifferent behavior, due to dynamic capillary pres-sure, since this term depends on the evolution ofsaturation

��

��

�

��

�

�

S

t

w(36)

Fig. 7 shows the mass flux leaving the surface.The pump operates at first to decrease the totalpressure, and then a flux of vapor coming from theboard increases the chamber pressure. The chamberdynamics are primarily driven by the fluxes leavingthe wood, the pump flow rate when switched on,and the eventual rate of condensation on the cham-ber walls. In practice and for the experimentalchamber under consideration herein, the chamberdynamics usually has a time constant much smallerthan the internal transfer arising within the wood.The dynamics and the convergence conditions ofthe wood result mainly from the rapid change of theboundary conditions. Vapor transport is driven bytemperature and pump aspiration. We have writtenthe diffusion coefficient as a function that dependson temperature and pressure.

A more important pressure is developed bythermal effect, since water vapor depends stronglyon temperature (according to gas law). Differentvalues of the coefficient have been proposed in theliterature, but it is difficult to have unanimity. Inthis work, these parameters have been extractedfrom Hernandez.12 Fig. 7 reveals passive and activeregimes during drying; an acceleration and re-ho-mogenization is visible. The model describes cor-rectly pump action in the wood surface. The pumpflux together with the chamber volume defines a

time constant for the pressure evolution when thepump is on (maximum allowable pressure attained),resulting in pressure-decreasing periods. The modelcan predict some of the more subtle mechanismsobserved in practice, such as the increase of massflux during active regime and moisture distributionin wood and on surface.

Conclusion

We have proposed a numerical solution for atwo-scale model for vacuum drying by consideringa dynamic capillary pressure, which appears satis-factory. Good agreement between experimental re-sults and those of the simulation is assessed. It is in-teresting to see how this model allows distinguish-ing of the phases of vacuum drying, it simulatesfast drying phase (active phase) and the stage of ho-mogenization (passive phase).

The coupling between the wood material(product) and dryer (process) is also respected. Wehave solved a model by considering a dynamic termfor capillary pressure. The critical point betweencapillary and hygroscopic phase is well identifiedand a drop in moisture content is visualized at thesurface. Simulations are relevant because they rep-resent quite well the experimental curves in termsof average kinetic, moisture in the surface, massfluxes and overall behavior of the dryer by consid-ering a dynamic term that depends on the evolutionof saturation.

The simulation is complex because we con-sider the dryer behavior and wood interaction. Thesimulation and experimental data are in good agree-ment, and provide information about the physics ofdrying. Future research must focus on capillary-hy-groscopic transition by considering physical quanti-ties like surface tensions, contact angles betweenphases, viscosities, pore structure and flow condi-tions.

ACKNOWLEDGEMENTS

To Instituto Politécnico Nacional (contrataciónpor Excelencia), México and to CONACYT(Repatriación).

N o m e n c l a t u r e

* � velocity field, m s–1

k � permeability, m–2

p � pressure, mbar

S � saturation, –

r � radius of curvature, m

V � volume, m3


F i g . 7 – Mass fluxes during vacuum drying

fsp � fiber saturation point, –

V � average velocity field, m s–1

K � phase change rate of water, kg m–2 s

D � diffusion coefficient, m2 s–1

C � vapor concentration, –

R � ideal gas constant, J K–1 mol–1

w � moisture content, kg kg–1

T � temperature, K

Fm � mass flux, kg m–2 s–1

aw � water activity, –

HR � relative humidity, %

Cp � specific heat, J kg–1 K–1

t � time, s

A � area, m2

U � function

hv � latent heat of vaporization, kJ kg–1

J � bound water flux, kg m–2 s–1

G r e e k l e t t e r s

� viscosity, kg s–1 m–1

� density, kg m–3

� � variable, –

� � superficial tension, N m–1

� � solvent-particle contact angle, –

+ � porosity, –

� � phase, –

� � pore size distribution, –

, � apparent water saturation, –

�hb � heat of desorption, J kg–1

� � conductivity, W K–1 m–1

# � Lagrangian operator

� � damping coefficient

� � gradient

S u b s c r i p t s

i � specie

c � capillary

d � displacement

e � effective

r � irreductible water saturation/relative

b � bound water

g � gas

l � liquid

s � solid

surf � surface

nw � non wetting

w � wetting

v � vapor

eq � equivalent

pump� vacuum pump

cond � condensate

S u p e r s c r i p t s

g � gas phase

l � liquid phase

s � solid phase

dyn � dynamic

dryer� dryer

eq � equilibrium

ch � chamber

atm � atmosphere

exp � experimental

model� model

R e f e r e n c e s

1. Xu, L., Davies, S., Schofield, A. B., Weitz, D. A., PhysicalReview Letters 101 (2008) art. no. 09450.

2. Lehmann, P., Assouline, S., Or, D., Physical Review E 77,056309 (2008).

3. Chauvet, F., Duru, P., Geoffroy, S., Prat, M., Physical Re-view Letters 103 (12) (2009).

4. Hassanizadeh, S. M., Celia, M. A., Helge, K., Vadose ZoneJournal 1 (2002) 38.

5. Besserer, H., Hilfer, R., Old problems and new solutionsfor multiphase flow in porous media, In Porous media:Physics, models, simulation. Dmitrievsky, A., Panfilov, M.(Eds). World Scientific Publ. Co., Singapore. 2002.

6. Van Dijke, M. I. J., Sorbie, K. S., McDougall, S. R., Ad-vances in Water Resources 24 (2001) 365.

7. Ataie-Ashtiani, B., Majid Hassanizadeh, S., Celia, M. A.,Journal of Contaminant Hydrology 56 (2002) 175.

8. Slattery, J. C., Am. Inst. Chem. Eng. Journal 13 (1967)1066.

9. Whitaker, S., The Method of Volume Averaging. Theoryand Applications of Transport in Porous Media, KluwerAcademic Publishers, Dordrecht, The Netherlands, (1999).

10. Perré, P., Transport in Porous Media 66 (2007) 59.

11. Sandoval, T. S., Marc, F., Jomaa, W., Puiggali, J. R., WoodResearch Journal 54 (2009) 45.

12. J. M. Hernandez, Séchage du chêne. Caractérisation,séchage convectif et sous vide. PhD-Thesis UniversitéBordeaux 1, 1991.

13. COMSOL multiphysics©. Version 3.5a.


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