Multiphysics modeling of vacuum drying of wood

Multiphysics modeling of vacuum drying of wood

Applied Mathematical Modelling 35 (2011) 5006–5016 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.el...

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Applied Mathematical Modelling 35 (2011) 5006–5016

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage:

Multiphysics modeling of vacuum drying of wood Sadoth sandoval Torres a,⇑,1, Wahbi Jomaa a, Jean-Rodolphe Puiggali a, Stavros Avramidis b a b

Université Bordeaux 1, Laboratoire TREFLE site ENSAM, Esplanade des Arts et Métiers, 33405 Talence Cedex, France The Univ. of British Columbia, Dept. of Wood Science, 2424 Main Mall, Vancouver, BC, Canada V6T 1Z4.

a r t i c l e

i n f o

Article history: Received 29 July 2010 Received in revised form 1 April 2011 Accepted 5 April 2011 Available online 19 April 2011 Keywords: Vacuum drying Drying physics Coupled model Comsol multiphysicsÓ

a b s t r a c t Drying of porous media is characterized by the invasion of a gaseous phase replacing the evaporating liquid. Vacuum drying is an alternative method to alleviate discoloration for oakwood, so description of its underlying physics is important to understand this process. In this work, a coupled modeling is proposed to describe vacuum drying of oakwood at lab scale. This model describes the physics of wood-water relations and interactions with the vacuum dryer. Results provided important information about liquid and gas phase transport in wood. Water vapor and air dynamics in the chamber were simulated linking large scale (dryer) and macroscale (wood) changes during drying. We analyses results at 60– 100 bar and 250–300 mbar both at 70 °C. The phenomenological one-dimensional drying model is solved by using the COMSOL’s coefficient form and an unsymmetric-pattern multifrontal method. Good agreement was obtained for these drying conditions. The numerical results and experimental measures provide some confidence in the proposed model. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Modeling of wood drying has been the topic of much research works over the years. Many publications cover a wide range of applications, including the derivation of the macroscopic equations, the development of analytical and numerical solutions, the determination of the physical and mechanical characterization of the medium being dried, and the experiments carried out on both laboratory and industrial scales. Nowadays, advances in software engineering results in ever-increasing computational power, and thus numerical simulation have fast become a very powerful tool to study and optimizes drying operation [1]. It is nowadays well accepted that vacuum drying of wood offers reduced drying times and higher end-product quality compared with others conventional drying operations [2]. The reduction of the boiling point of water at low pressure facilitates an important overpressure to enhance moisture migration. Is well know drying is a critical step in manufacturing timber products and is one of the most pressing issues in wood industry since there is a growing emphasis on high quality dried lumber because customers demand timber products which are defect free [3]. Then, the aim of industrial drying is to accelerate the natural drying process and to take advantage of dry wood’s attributes, while minimizing some of the negative impacts [4]. Advantages of vacuum drying include the reduced drying times, recovery of water vapor, and a higher end-product quality [5]. Operating at low pressures the boiling point of water is reduced, which in turn enables an important overpressure that drives moisture efficiently [1,6,7]. For some wood species like oak that cannot be dried at high temperature conditions, vacuum drying offers the possibility to avoid collapse and discoloration [7–9]. As a consequence of these benefits, research in ⇑ Corresponding author. 1

E-mail address: [email protected] (S.s. Torres). CONACYT-Fellow Université Bordeaux 1, Moving to Instituto Politécnico Nacional, México (Contratación por Excelencia CIIDIR oaxaca).

0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.04.011

S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

Nomenclature A

total surface area of the board (m2)




water activity (–),


specific heat of wood (J/kg K) heat of desorption (J kg1) intrinsic permeability (m2) 1e16 vapor concentration (–) chamber condensation specific heat (J/kg K) diffusion coefficient (m2/s) equilibrium mass flux (kg/m2 s) fiber saturation point gravity force (m s1) relative humidity (%) latent heat of vapourisation at the reference temperature bound water flux (kg/m2 s) phase change rate of water pressure (Pa) capillary pressure (Pa) ideal gas constant, 8.314 472 J K1 mol1 saturation (–) solid surface temperature (K or °C) time (s) velocity (m s1) moisture content (dry basis)

rhb k ¼ C ch cond Cp D eq Fm fsp G HR hv J K P Pc R S s surf T t V W

1 if W > Wpsf expðAB100W Þ if W < Wpsf A ¼ 2:51e  4T 2  0:1780T þ 35:719 B ¼ 9:475e  4T þ 1:133

Greek Symbols q density (kg/m3) e porosity (–) l dynamic viscosity (kg/m s) r gradient operator r divergente operator k thermal conductivity (Wm1 K1) 0.386 W + 0.137 Subscripts and Superscripts v vapor f fluid w water Sat saturation b bound water l liquid i species g gas a dry air  average rl relative to liquid Input values for all model parameters Parameter Expression qa 1.2 ql 1000



qs e Ma Mv Wpsf R k

lg ll A1 A2 B1 B2 Cps Cpl Cpa Cpv hvo Vch Qpump Qlack Qcond Tini Patm qv Tatm Wini r Text HRi Db Deq

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750 0.5 29e3 18e3 0.40 8.314 1e18 1.82e5 1e3 10886.472 0 7600.705 6.697 1400 4187 1000 1840 2401e3 0.0504 2.72e3 1.778e6 0 273.15 + 25 101325 0.8 298.15 0.9 0.035 300 0.3 1e10 1e7

this field is currently receiving significant worldwide attention, especially for high quality hardwoods that are difficult to dry conventionally. According to Turner [10], in drying modeling, various approaches in the computational solution strategy can be adopted. For example, simple and efficient methods that use the assumption of constant physical parameters can be used to solve specific situations, including kiln sizing and the global effect of the product size. Among these methods, the dimensionless drying curves proposed by Van Meel [11] or the simple analytical solutions described by Crank [12], but these methods fails to provide information about the drying physics and phases motilities. A second set of computational strategies can be used that try to be more realistic by using a more suitable set of simplifying assumptions. All models based on the concept of ‘‘drying front’’ belong to this category [13]. Finally, all computational models based on a complete numerical solution of the nonlinear conservation laws constitute the third category. In this work we are interested to model vacuum drying using he macroscopic conservation equations that govern the heat- and mass-transfer phenomena that arise in wood. This description has been well established and understood by Whitaker [14–16]. In this work a one-dimensional problem is solved in COMSOL multiphysics 3.5aÓ. This package provides a number of application modes that consist of predefined templates and user interfaces already set up with equations and variables for specific areas of physics. The objective of this work was to solve a comprehensive mathematical model by using comsol multiphysics for the analysis of vacuum drying physics of European oakwood (Quercus pedonculata). We need to capture the physics of vacuum drying at 1D because of we are interested in the transport of colouring compounds in wood in one direction. So chemical compounds transport in wood with nondestructive methods are needed, Furthermore, liquid or vapor transfer cannot be distinguished in a real drying situation since there is no available technology to do so. More details about the facility and chemistry of discoloration can be founded in Sandoval et al. [9]. Our project considers three axes of research: the color of wood, the chemistry of wood, and the drying physics of plain vacuum drying as a new alternative to alleviate discoloration.

2. Mathematical formulation The approach proposed by Whitaker [14–16], Carbonell and Whitaker [17] and Perré [18] was followed in this work. Mass and energy conservation relations are written for liquid water, water vapor and for the gaseous mixture (water vapor + air). A

S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016


set of macroscopic equations are obtained, where the flux of the different components are described using different transport coefficients and driving forces. Dryer scale and material scale are coupled by means of boundary conditions. Specifically, the porous medium (wood sample) consists of a continuous rigid solid phase which contains bound water, liquid phase (free water), a continuous gas phase which is assumed to be a perfect mixture of vapor and dry air, considered as ideal gases. For the mathematical description of the transport phenomena in the porous media, we adopt a continuum approach, where the macroscopic partial differential equations are achieved by volume averaging of the microscopic conservation laws. The method of volume averaging is a technique used to rigorously derive continuous macroscopic equations from the description of the problem at a microscopic scale for multiphase systems. This method allows to change the scale of description of the problem. The value of any physical quantity at a point in space is given by its average value on the averaging volume (AV) centered at this point [18]. The phase average is defined by

Z ¼1 w w  dV V v


and the intrinsic phase average i by

Z i ¼ 1 w w  dV: V i vi


Two problems related to wood drying simulations are usually the approximations made to the complete models and the question of their validity [19]. Additional assumptions are required for proper evaluation of all simplifications. Then, the next hypothesis must be established in this work: European Oakwood have a very low permeability, this fact allows us to consider the gravitational effects are negligible. We consider that temperature and pressure in the dryer are homogeneous Vapor is an ideal gas both in the material and in the dryer. Thermodynamic equilibrium, so average temperatures for each phase are the same, and partial vapor pressure is at the equilibrium.  No chemical reaction or shrinking occurs during wood drying.  Lack of heat and mass losses assuming ideal isolation [20].    

Then, we can write:

T v ¼ T f ¼ T Thermodynamic equilibrium; Pv ¼ aw  Psat v Vapor pressure; where W, is the moisture content in wood:

mass of water : mass of dry wood


We know that saturation S is the volume fractional of free water within the pores, then:

Volume of free water : Volume of Porous


Bound water content Wb is defined as:

Wb ¼

mass of bound water : mass of dry-wood


During drying, free water is the fist to be evaporated, in this way saturation tends to zero. When saturation is zero, bound water is in its maximal concentration. In wood technology this point is referred as the fiber saturation point (fsp). For European oakwood (Quercus pedonculae) this point is 0.4 approximately [21]. Porosity can be defined as follow:

Volume of pores : Total volume


Compressibility effects in the liquid phase are neglected.

qll ¼ ql ¼ cste:


Gas phase is considered as an air/water vapor ideal mixture, then:

q gi ¼

mi P gi

Pour i ¼ aðairÞ or v ðvaporÞ; RT Pgg ¼ P ga þ Pgv ;

q gg ¼ q ga þ q gv :

ð8Þ ð9Þ ð10Þ


S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

3. Transport of the liquid phase in wood Free water transport is explained by Darcy’s law. The velocities of the gaseous and liquid phases are, respectively, expressed using the generalized Darcy’s:

k krl V l ¼  ¼ ¼  ðrPll Þ;


k krg V g ¼  ¼ ¼  ðrPgg Þ:




Capillary pressure is defined as:

Pll ¼ P gg  Pc;

  1:062 ðPaÞ: Pc ¼ 56:75  103 ð1  SÞ exp S

ð13Þ ð14Þ

Capillary pressure is a driving force during capillary phase (free water evacuation) [3]. In fact above the fps only capillary and bulk flow may be considered imperative [19] We can write the flux of free water as:

k krl

k krl

ql V l ¼ ql ¼ ¼  rPc  ql ¼ ¼ ðrPgg Þ: l1 ll


3.1. Transport of the vapor phase in wood We consider water vapor and air mobility depends on pressure and concentration gradients of the gaseous phase. Then, for the mass flux of air and water vapor we write:

k  k

¼eq ¼rg

q gv V v ¼ q gv


 gg D rC;  rPgg  q


 gg D rC;  rPgg  q



k  k

q ga V a ¼ q ga

¼eq ¼rg



where k and D are the equivalent permeability and the diffusion coefficient respectively; with these parameters the per¼eq ¼eq turbations in the convective and diffusive dusty transport are considered [21–23]. The thermophysical parameter are obtained from Hernandez [21] The diffusion-sorption model described the bound water migration. A phenomenological approach can explain water flows to the form discussed extensively in literature [23]:

 s D rW b  q  s D rT: J b ¼ q ¼b



Below FSP, the moisture is considered bound to the cell wall and, therefore, bound water diffusion can be considered to be the predominant mass transfer mechanism [3]. In this stage gas pressure and diffusion are the most important mechanisms. The mass fraction of water vapor C in wood is defined by:

vapour mass : air humide mass


The Equilibrium moisture content is expressed by

W eq ¼ WðT; awÞ;


aw ¼ HR: Equilibrium moisture content is currently expressed by desorption isotherms of wood (Fig. 1). This figure describes the relationship between the equilibrium moisture content and water activity (relative humidity in the dryer). Averaged equations for the mass, momentum, and heat transports obtained from the liquid phase. The equations for the mass conservation are: For the air and vapor (gaseous phase):

 g  a @q  a V a ¼ 0; þr q @t



S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

0.4 0.35

Equilibrium Moisture Content (kg.water/kg.dry wood)



30°C 40°C




60°C 70°C

0.15 0.1 0.05 0 0











Relative humidity (%) Fig. 1. Desorption isotherm of European Oakwood (Hernandez, [21]).

 g  v @q v V v ¼ K þ Kb: þr q @t


For the liquid phase:

 l  l @q  l V l ¼ K: þr q @t


For the bound water:

b @q þ r  J b ¼ K b ; @t


where K and Kb are the phase change flux for the liquid water and vapor water respectively. During drying, free water (liquid water), water vapor and bound water evaporates and leave the board, so the global balance is expressed by:

   @W 1 þr ql V l þ q gv V l þ Jb ¼ 0: @t qs


And for energy, we have:


  @T   ga V a Cpa þ q  gv V v Cpv Þ rT  qb V b  rhb þ ðhv KÞ þ ðhv þ hb ÞK b  r  ðk  rTÞ ¼ 0: þ ðql V l Cpl þ J b Cpl þ q @t


The specific heat of wood, described by:

qCp ¼ q s Cps þ ðq l þ q b ÞCpl þ q v Cpv þ q a Cpa :


3.2. Equation for mass balance in the dryer The air, vapor, and water mass balance equations are derived by assuming that the pressure and temperature fields within the chamber are homogeneous. Using a control volume that includes the surfaces of the wood sample, together with the mass fluxes, and vacuum pumping, the following balance equations for the vacuum chamber are obtained: The conservation equations in the dryer chamber are:

qpump dqch qleak a þ qatm for the dry-air; ¼ qch a a dt V ch V ch dqch v ¼ qch qpump þ qcond þ qatm qleak þ F A For the water-vapor: m v v V V ch dt V ch ch

ð28Þ ð29Þ

Pressure in the chamber is computed according to the ideal gas law. 4. Boundary conditions The pressure at the external drying surfaces is fixed at the atmospheric pressure Pch (Pressure of the chamber). Recalling that one of the primary variables used for the computations is the averaged air density we impose the Dirichlet boundary


S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

Fig. 2. Configuration.

condition for the air flux. This Dirichlet boundary condition has been modified to form an appropriate non-linear equation for this primary variable. For moisture fluxes, we consider equilibrium between water vapor at the wood surface and the vapor pressure in the chamber. Water vapor depends on pressure since to compute vapor density the ideal gas law is used. In the chamber atmosphere, we established that a mixture water–vapor/dry-air exists, that depends on dryer atmosphere temperature. For energy fluxes we establish that temperature in the wood surface is in equilibrium with the chamber temperature [1,7,10,24].

Pga ¼ Pchamber a

For dry air;


qgv ¼ qchamber For water vapor; v


T chamber  T surf


For energy½plain vacuum drying:

We remind to the reader, this vacuum drying do not use steam. In this work we study a plain vacuum drying because of we have explored this method to alleviate discoloration in wood. In another paper we have published results of plain vacuum drying conditions on color alteration [9]. Fig. 2 shows the geometry considered in our equations system 1D. 5. Materials and methods The experimental setup is a vacuum chamber where pressure is regulated between two values (Pmin, Pmax). The chamber is built in glass; one balance is kept inside the chamber in order to log the mass variation of the sample. A thermometer gives the dryer temperature. The heating source is an electrical resistance which temperature is controlled with the help of a PID controller. Experiments are performed on Oakwood disks (7 cm diameter and 2.5 cm height). The conductive heat source is homogenized by an aluminum plate. Pressure in the chamber is controlled at different intervals (60–100, 150–200, and 250–300 mbar). Temperature inside the wood sample is obtained at two different positions. In this paper we compare simulations results from experimental data at 70 °C and 60–100 mbar to facilitate discussion. Details about the facility can be found in Sandoval et al. [9]. 6. Results and discussion As mentioned, to solve the set of equations, we have used the commercial solver COMSOL Multiphysics 3.5aÓ. COMSOL is advanced software for modeling and simulating any physical process described by partial derivatives equations. The set of equations above introduced, were solved with their relative initial and boundary conditions, COMSOL offers solvers with a very high level of performance. Globally, COMSOL offers three possibilities to write the equations: (1) by using a template (Fick’s law, Fourier’s Law), (2) by using the coefficient form (for mildly nonlinear problems), and (3) by using the general form (for most nonlinear problems). We have written differential equations in the coefficient form and by using an unsymmetric-pattern multifrontal method. We have used a direct solver for sparse matrices (UMFPACK), which involve much more complicated algorithms than for dense matrices. The main complication is due to the need for efficient handling the fill-in in the factors L and U. A typical sparse solver consists of four distinct steps: 1. An ordering step that reorders the rows and columns such that the factors suffer little fill, or that the matrix has special structure such as block triangular form. 2. An analysis step or symbolic factorization that determines the nonzero structures of the factors and creates suitable data structures for the factors. 3. Numerical factorization that computes the L and U factors. 4. A solve step that performs forward and backward substitution using the factors.


S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

In this work we use the UMFPACK solver which considers all these 4 steps. We have used the Arbitrary Lagrange–Eulerian (ALE) formulation. The sparse matrix A factorized by UMFPACK can be real or complex, square or rectangular, and singular or nonsingular (or any combination). The mesh consists of 135 elements (1-D), time stepping is 0.1 (0 to 20 s of solution), 5 (20 to 100 s of solution, 100 (100 to 10000 s of solution), and 10 (10000 to 100000 s of solution). Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to insure that the results are grid independent. We have established a maximum element size of 2e4. For solving time-dependent variables, we use the backward differentiation formula (BDF). Relative tolerance was set to 1e3 whereas absolute tolerance was set to 1e4. The simulations were performed using a Hewlett Packard PC with Intel Core 2 duo, at 1.83 GHz, with 2046 MB of RAM, running under Windows XP. We have written the partial differential equations (material scale) in the general form. The two ordinary differential equations (dryer scale) were introduced by considering a pump aspiration of 0.0027 m3/s (the real situation). To add a spaceindependent equation such as an ODE, we have chosen a global equation format. As the time derivative of a state variable (density of air and water vapor) appears, the state variable needs an initial condition; in this problem we consider chamber pressure begins at atmospheric pressure. The total drying times and the maximum internal temperatures are important parameters in industrial drying. In particular, the former has to do with the efficiency of drying and the latter with the quality of the final product. Fig. 3 shows predicted and experimental results for the vacuum drying at 70 °C and 60–100 mbar of pressure. One can observe that the model is able to predict the kinetics of drying. In the same figure one can see that pressures have a good agreement. The differences can be explained by variation of values in permeability, capillary pressure and transfer coefficients of water vapor, since they can vary (wood heterogeneity). For this experiment we can see that the final moisture content of wood is around 8% and it is reached at approximately 28 hours. Fig. 4 shows the experimental drying kinetics at 70 °C and 250–300 mbar and simulation results. In both experimental observation and simulation one can see a more important drying time compared to drying time at 60–100 mbar. This may be explained by a lower overpressure generated between the heart and its wood surface. It is well known that



0.9 Experimental 70°C 60-100mbar





P exp


P model



0.4 400

0.3 0.2

Chamber Pressure (mbar)

Moisture Content (kg. water/kg. dry-wood)



0.1 0

0 0






Time (sec) Fig. 3. Comparison between vacuum drying kinetics: experimental and model. 70 °C and 60–100 mbar.



Moisture Content (kg. water/kg. dry-wood)

0.8 0.7





Pressure Experimental



700 600

0.5 500 0.4








0 0





0 200000

Time (sec) Fig. 4. Simulation results versus experimental kinetics. 70 °C 250–300 mbar.

Chamber pressure (mbar)




S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

overpressure generated in wood accelerates the mass transfer in the material. Increase of permeability decreases the temperature gradients, but increases the diffusive coefficients. The relative change of temperature gradients at high temperatures is small so that the effect of decreasing the permeability on the diffusive coefficient prevails and drying rates are reduced. At lower temperatures (our case), the relative change of the temperature gradients becomes higher, and this counterbalances the increase of the diffusive coefficients. The temperature curves are not smooth, and this can be explained by both the functions utilized for calculation and aspiration/no aspiration dynamics of the pump. Fig. 5 shows experimental and simulated temperatures during drying at 60–100 mbar and 250–300 mbar at 70 °C. Figure show a classic behavior in temperature profiles, since wood is heated by contact (heat transfer by conduction). Oscillations are due to natural perturbations in the cooling/heating phenomena. The pump works at two regimes: passive and active (on and off). The external vacuum reduces the required temperature for evaporation. The effective vapor transport increases with decreasing distance from the surface. The concentration of water vapor in the chamber is one of the variables to solve. Given the strong coupling in our system of equations, it is interesting to see the behavior of the vapor concentration over time for the two cases discussed here (twolevel pressure). We can see in Fig. 6 how the vapor concentration is more important during the capillary phase (before fiber saturation point). At the hygroscopic phase of drying vapor concentration is lower, due to a decrease in the mass flux, because of free water has been evacuated, and vapor and bound water requires more energy to be evaporated. When comparing the mass fluxes, it can be identified two phases or regimes of drying (one active and other one passive). Fig. 7 shows such fluxes, for a vacuum drying experiment at 70 °C and 250–300 mbar of pressure. The drying phases produces natural oscillations at the surface temperature, since during active phase (pump aspiration) a temperature shut exist due to evaporation, and during the passive phase a re-homogenization is developed. Then, during the active phase mass flow is more important due to the operation of the pump, so there is a pressure drop in the enclosure and consequently faster evaporation in the surface of the material. Compared with the passive regime, during the active regime the evaporation is less intense since the pump is stopped, what causes an increase in the pressure chamber and a moisture homogenization, visible at the surface.

50 45

Temperature (°C)

40 35 30

Tsurf 70°C 250mbar model

25 Tsurf 70°C 250mbar experimental

20 15

Tsurf 70°C 60mbar model


Tsurf 60mbar experimental

5 0 0










Time (sec) Fig. 5. Experimental temperature and simulation results. 60 mbar and 250 mbar.

0,25 Vapour concentration_60-100mbar



Vapour concetration_250-300mbar

0,25 0,15

0,2 0,15


0,1 0,05 0,05 0

Concentration_250-300mbar (kg. Water vapour/kg. Dry-air)

Concentration_60-100mbar (kg. Water vapour/kg. Dry-air)


0 0






Time (sec)

Fig. 6. Concentration of water vapor in the chamber. Simulations at 60–100 mbar et 250–300 mbar.


S.s. Torres et al. / Applied Mathematical Modelling 35 (2011) 5006–5016

2,00E-03 1,80E-03

Flux (kg water/m2 sec)

1,60E-03 1,40E-03

Fm_model Fm_experimental

1,20E-03 1,00E-03 8,00E-04 6,00E-04 4,00E-04 2,00E-04 0,00E+00 2000









Time ( sec) Fig. 7. Masse flux. Vacuum drying at 70 °C and 250–300 mbar.

The air pressure increases in wood as the water content in the material decreases, air enter in the porous when the water is expulsed by evaporation. In the practice, these parameters are difficult to obtain because the disturbance, positioning and experimental facility of pressure sensors should be considered. In our experimental setting, energy source is a heat plate, and then heat is transfer meanly by conduction. During drying, the wood surface is constantly feed in water or moisture. The evaporation rate depends mainly on the level of pressure and temperature. During the capillary phase, the moisture transport of free water can be studied by the Darcys law. In the hygroscopic phase (below fsp), one can see a drop of moisture in the surface, because of vapor transport depends on vapor viscosity, desorption isotherms, and water bounded to the cellular walls of wood needs more energy. The system trend towards a thermodynamical equilibrium, so average moisture in wood will equality the moisture content of the surface. About vapor water transport, we wrote the diffusion coefficient as function that depends on temperature and pressure. Different values of the coefficient have been proposed in the literature, but it is difficult to have unanimity. In this work these parameters and the equilibrium isotherms have been extracted from another thesis realized at the Laboratoire TREFLE site ENSAM [21]. This model and numerical results provide more information about mobilities of phases in wood.

7. Conclusion We have proposed a numerical solution for a model of plain vacuum drying which appears satisfactory. A good agreement between experimental results and those of the simulation is assessed. It is interesting to see how this model allows distinguishing the phases of vacuum drying, it simulates fast drying phase (active phase) and the stage of homogenization (passive phase). The coupling between the wood material (product) and dryer (process) is also respected, this coupling is ensured by the boundary conditions imposed in our 1D simulations. These simulations are relevant because they represent quite well the experimental curves in terms of average kinetic, overall behavior of the dryer and liquid and vapor mobility. The simulation is complex because of we consider the dryer behavior and wood interaction. The simulation and experimental data have a good agreement, and provide information about physics of drying. Traditional two-phase flow models use an algebraic relationship between capillary pressure and saturation. This relationship is based on measurements made under static conditions. However, this static relationship is then used to model dynamic conditions, and evidence suggests that the assumption of equilibrium between capillary pressure and saturation may not be justified. Future research, must pay attention to capillary-hygroscopic transition, because of capillary pressure versus saturation curves are estimated from air–water interface curvature, but a more complex description must be added, since saturation and relative permeability are variables with a more complicated dependence on physical quantities like surface tensions, contact angles between phases, viscosities, pore structure and flow conditions. The next step for our research group is to develop a model by considering a dynamic capillary pressure. References [1] P. Perré, I.W. Turner, A dual-scale model for describing drier and porous medium interactions, AIChE J. 52 (9) (2006) 3109–3117. [2] B.J. Ressel, State-of-the-art on vacuum drying of timber, in: Proceedings of the Fourth International IUFRO Wood Drying Conference, Rotorua, New Zealand, 1994, pp. 255–262. [3] L. Cai, A. Koumoutsakos, S. Avramidis, on the optimization of a RF/V kiln drying schedule for thick western hemlock timbers, J. Inst. Wood Sci. 14 (6) (1998) 283–286. [4] J.W. Wallace, I.D. Hartley, S. Avramidis, L.C. Oliveira, Conventional kiln drying and equalization of Western hemlock (Tsuga heterophylla (Raf.)[Sarg]) to Japanese equilibrium moisture content, Holz als Roh – und Werkstoff 61 (2003) 257–263.


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