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Simulation of Convective Drying of a Porous Medium with Boundary Conditions Provided by CFD
0263–8762/06/$30.00+0.00 # 2006 Institution of Chemical Engineers Trans IChemE, Part A, February 2006 Chemical Engineering Research and Design, 84(A2): 113– 123
www.icheme.org/journals doi: 10.1205/cherd.05047
SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM WITH BOUNDARY CONDITIONS PROVIDED BY CFD A. ERRIGUIBLE, P. BERNADA , F. COUTURE and M. ROQUES Laboratoire de Thermique Energe´tique et Proce´de´s, Pau, France
T
he description of the transfer between a porous medium and its surroundings is commonly made using transfer coefficients which are theoretically well described only under boundary layer hypothesis. Solving Navier –Stokes equations in the surroundings of the product in order to get information about the boundary conditions avoids the classical using of these transfer coefficients. After a brief description of the model, a simulation of convective drying of a rectangular piece of porous medium is proposed using a coupling method between a porous medium code and a CFD software. The analysis of the interfacial transfer coupled with the analysis of the temperature and moisture content profiles show the influence of the edges. We show that the global kinetics cannot reflect the local surface transfer phenomena: even at the so-called constant rate period heat is not entirely used for water evaporation with a portion for temperature increase within the product. Keywords: conjugate problem; modelling; coupling method; heat and mass transfer; fluent.
INTRODUCTION
about the boundary conditions for the transport equations in the medium and solve the complete drying problem. This article deals with an analysis of the boundary conditions during convective drying simulated with a coupling method between a porous medium and its surrounding. The coupling of the model associated to the porous medium with the one devoted to the external flow is performed by writing the boundary conditions at the interface of the porous medium/environment.
Heat and mass transfer at the frontier of a multiphase medium and its surroundings are only theoretically well described under boundary layer hypothesis. In the other cases, theoretical study of interfacial transfer is very difficult due to the geometry of the physical frontier between the material and its surroundings, or due to the process by itself: vacuum drying, high temperature drying, steam drying and so on. In these cases transfer coefficients cannot be defined rigorously. For example, the basic convective drying 2D configuration sketched in Figure 1, which represents the air velocity field around the porous medium during the drying, could not be simulated by using heat and mass transfer coefficients as boundary layer does not exist. Indeed, one of the boundary layer hypothesis implies that the thickness d of the gradient zone must be much smaller than the characteristic length L of the sample. The velocity field in Figure 1 shows that this classical assumption (Whitaker, 1977a; Slattery, 1999) is not verified. Moreover, boundary layer does not exist at the leading and vanishing edge. Hence it is necessary to solve Navier– Stokes equations in the surroundings of the product in order to get information
MODELLING Heat and Mass Transports Within the Porous Medium and the Environment The mathematical description of the physics involved during the drying of a hygroscopic capillary porous medium has been discussed by numerous authors. Most of them adopt a continuum approach based on Whitaker’s theory (Whitaker, 1977b, 1984), where the macroscopic partial differential equations are achieved by volume averaging the microscopic conservation laws. The value of any physical quantity at a point in space is given by its average value on the averaging volume (AV) centred at this point. The phase average is defined by ð 1 F dv (1) F ¼ v v
Correspondence to: Dr P. Bernada, Laboratoire de Thermique Energe´tique et Proce´de´s, EA 1932, 3 rue Jules Ferry, Bp 7511, 64075, PAU CEDEX, France. E-mail: [email protected]
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Figure 1. Air velocity field around porous medium during convective drying. This figure is available in colour via www.icheme.org/journals.
and the intrinsic phase average i by ð 1 i F ¼ F dv vi v
(2)
Let us define the density and the mass averaged velocity of the mixture in the usual manner: X r¼ ri (4) i
Within each AV, the porous medium consists of a continuous rigid solid phase which contains bound water (i.e., the product is assumed to be hygroscopic), an incompressible liquid phase (free water), a continuous gas phase which is assumed to be a perfect mixture of vapour and dry air, considered as ideal gases. The hypothesis and the way to obtain the governing equations for heat, mass and momentum transport during drying have been well established and are not discussed here (Couture, 1995). The equations which are solved in the porous medium are the mass conservation of water and dry air, the energy conservation and the momentum conservation (generalized Darcy’s laws and Fick’s laws). For the environment, the external flow of moist air which consists of two species (dry air and water vapour, respectively noted a and v) is considered as unsteady and incompressible. The conservation equations to be solved are the species, momentum and energy conservation equations. How to obtain the complete differential set of the necessary conservation laws for the porous medium and its surrounding is not detailed here (Erriguible, 2004). Only the final system is given in the following part.
rV ¼
X
ri V i
To complete the model, one relation must be added. The incompressibility of the air flow leads to: rV¼0
(6)
Momentum Conservation By considering the environment as a Newtonian fluid, the momentum conservation equation for the moist air is commonly described by @ðrVÞ þ r ðrV VÞ ¼ rg rP þ mDV @t 1 þ m r ðr V Þ 3
(7)
The diffusion for each species is modelled according to Fick’s laws.
ri ðVi VÞ ¼ rDr NAVIER – STOKES MODELLING FOR THE ENVIRONMENT
(5)
i
ri r
for i ¼ (a, v)
(8)
Species Conservation
Energy Conservation
Mass conservation is written for the two species (dry air and vapour):
The energy conservation equation is written in terms of enthalpy to be consistent with the one associated to the porous medium. By assuming that the viscous dissipation and the total derivative of pressure are negligible and by considering that there is no volumetric heat source, the
@r i þ r r iV i ¼ 0 @t
for i ¼ (a, v)
(3)
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A2): 113– 123
SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM energy equation is resumed to @
! X ri hi þr ri V i hi þ q ¼ 0 @t i
P
i
with
r v r g
(20)
D eff ¼ DB
(21)
v ¼
(9)
The heat flux q is expressed by the Fourier’s law, q ¼ lrT
115
(10)
and the enthalpies are given by, ha ¼ Cpa T T ref hv ¼ DHvref þ Cpv T T ref
(11) (12)
In this equation, B is a diagonal tensor which reduces the coefficient of vapour diffusion in the air D in order to take into account the anisotropic material resistance to the diffusion transport. g is the barycentric velThe velocity of the gas phase V ocity expressed by, g ¼ r g V v r gg V gv V a aþr
AVERAGED EQUATIONS FOR THE POROUS MEDIUM
and is commonly described by the generalized Darcy’s law
The complete set of the conservation laws for the mass, for the momentum transport and for energy is summarized below.
g ¼ V
k krg g : rP g r gg g mg
The solid phase being rigid, we have l ¼ rl r ll V l
(13)
k krl l : rP l r ll g ml
(24)
in which the capillary pressure Pc is introduced as a function in term of moisture content (see appendix A),
The averaged mass conservation of the dry air gives @ra þ r ra V a ¼ 0 @t
(23)
For the liquid, the expression of the flux is also derivated from Darcy’s law,
Mass Conservation @rs ¼0 @t
(22)
(14)
l g P l ¼ P g Pc
(25)
For the water, the general equation of mass conservation is obtained by summation of the conservation equations of vapour (v), free water (l) and bound water (b) and written as follows @W 1 l þr r Vl þ rv Vv þ rb Vb ¼ 0 (15) @t r s l
For bound water, the flux is modelled by a classical phenomenological law (Perre´ and Degiovanni, 1990; Couture et al., 1995)
with the moisture content dry basis W defined by the relation:
Energy Conservation
W¼
r l þ r b þ r v r s
(16)
rb Vb ¼ rs Db :rWb
(26)
The assumption of the local thermal equilibrium between the solid, the gas and the liquid phases (Whitaker, 1991; Quintard and Whitaker, 1993) involves, s g l T s ¼ T g ¼ T l ¼ T
(27)
Momentum Transport Conservation The fluxes of each component are summarized below. For the solid, we have s ¼ 0 r ss V
Then the energy conservation is reduced to a unique equation n @ rh a h g þ r g V g lV l þ r r ga V a v v hv þ r l l hl þ rb Vb hb @t o (28) leff : rT ¼ 0
(17)
For the gas, the Fick’s law gives us for the two components (vapour and dry air) g
r g D rv rv Vv ¼ r v V g g eff
(18)
g þ r g D rv ra Va ¼ r a V g g eff
(19)
with s g g l rh ¼ r s h s þ r a h a þ r v h v þ r l h l þ r b h b
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enthalpies being expressed as follows: g h a ¼ Cpa T Tref l h l ¼ Cpl T Tref s h s ¼ Cps T Tref
(30) (31) (32)
l h l
h b ¼ DHb g h v ¼ DHvref þ Cpv T Tref
(33) (34)
This set of equations is closed by adding two assumptions. . The gaseous phase is considered as an ideal gas. It ensures g mi P i g r i ¼ for i ¼ (a, v) (35) RT and g g g P g ¼ P a þ P v (36)
r gg
¼
r ga
þ
r gv
Moreover the local thermal equilibrium at the interface involves the continuity of temperature and vapour mass fraction:
r gv rv ¼ r gg rg T ¼ T
(42) (43)
The no-slip condition implies that the tangential velocity of the humid air in the environment is equal to zero Vt¼0
(44)
with t the tangential vector to the porous medium. Moreover, the total pressure is continue at the interface (Ene and Sanchez-Palencia, 1975; Prat, 1989): g
P ¼ P g
(45)
(37)
. local thermodynamical equilibrium implies g P v ¼ aw Pvsat
APPLICATION TO CONVECTIVE DRYING (38)
Pvsat is the saturation vapour pressure. aw is the water activity equal to one when the water is free (nonhygroscopic region) and lower than one (see Appendix A), depending on temperature and moisture content when the water is bound to the solid phase (hygroscopic region). Boundary Conditions Between the Two Domains At the interface, we are dealing with a coupled problem between the heat and mass transfer within the external flow and within the porous medium. As discussed in Masmoudi and Prat (Masmoudi and Prat, 1991), we can make use of local heat and mass fluxes under the assumption that a typical averaging volume size is much smaller than the external flow characteristics length scales. The conservative form of the previous equations enables us to write rigorously the boundary conditions (Slattery, 1999; Erriguible, 2004; Erriguible et al., 2005). The natural conditions are the continuity of the heat and species fluxes in particular for dry air and water: g þ r g D rv n ¼ ra Va n r a V (39) g g eff l þ rb Vb þ r v V g r g D rv n r ll V g g eff (40) ¼ rv Vv n ¼ Fm 0 1 g þ r g D rv h g þ r a V g a g eff B C g B C g r g D rv h þ r l V l h l C n B r v V g g l v lA eff @ þrb Vb h b l eff :rT ¼ rv Vv hv lrT n ¼ Fh (41) with Fm and Fh respectively the evaporation flux and total heat flux at the interface.
Before we present the two dimensional simulation of the convective drying of the porous medium, the method of coupling between the two codes which was validated for one dimensional configuration in a previous work (Erriguible et al., 2003) is detailed. Numerical Implementation of the Coupling Method The porous medium code is a 2D Fortran program previously developed (Couture et al., 1995) and the code used to solve Navier– Stokes equations is the commercial software FLUENT 6. The coupling method consists of executing the porous medium code for one time step and successively executing the CFD code for the same time step. An explicit treatment for the boundary conditions is justified by the use of small time steps (0.1 s). An interface was made in ‘script’ language to make the link between the two codes. The spatial discretisation of the porous medium equations is performed by the finite element method. The finite volume one is preferred for the Navier– Stokes equations. In the first part, we described the conservation equations in the two domains in order to determine the boundary conditions between the two media. In this part we explain how the boundary conditions are allocated to the different conservation equations in order to simulate the conjugate heat and mass transfer problem. Boundary conditions allocated to the surrounding modelling Navier–Stokes equations are solved sequentially with FLUENT 6 by using the same time step as the one used in porous medium code. The numerical results provided by the porous medium code allow to allocate the mass flux, the temperature and mass fraction vapour in such a way that boundary conditions (39), (42), (43) and (44) are imposed. The momentum equations (7) are solved by setting the gas velocity vector at the interface between the product and its surrounding. The value of tangential velocity is given by the no-slip condition at the interface (44).
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A2): 113– 123
SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM The normal velocity is expressed by using the total flux continuity defined by the sum of equations (39) and (40): 1 g) n V n ¼ (rll V gg V l þ rb V b þ r r
(46)
Note that the continuity of the water flux (40) is ensured during the resolution of the porous medium equation. The difference between the total flux continuity (46) and the water flux continuity (40) imposes the dry air flux continuity. Since the velocities obtained to the first step may not satisfy the continuity equation (6) locally, a poisson-type equation for the pressure correction is derived from the continuity equation and the linearized momentum equations (SIMPLE algorithm). This pressure correction is then solved to obtain the necessary corrections to the pressure and velocity fields and the face mass fluxes such that continuity is satisfied. The mass conservation equation (3) is solved using the continuity of the mass fraction of the species at the interface (42). The energy equation (9) is solved using the continuity of the temperature at the interface (43).
Boundary conditions allocated to the porous medium modelling Conservation equations in the porous medium are solved for one time step, by using the remaining boundary conditions (40), (41) and (45) calculated with the results provided by the CFD code. The mass conservation equation of water (15) is solved using the continuity of the evaporation flux Fm (40) defined by: Fm ¼ (rv Vv ) n ¼ rv Vg n rDrv n
(47)
To solve the mass conservation of the dry air (14), the mass concentration r ga at the boundary is calculated assuming the pressure continuity at the interface (45) and using the ideal gas law:
r ga ¼
Ma Pa Ma (P Pv ) ¼ RT RT
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The energy conservation equation (28) is solved using the continuity of the total heat flux Fh (41) deduced from the values given by the CFD code as follows: Fh ¼ Fm hv lrT n
(49)
Physical Configuration The porous medium chosen for the two dimensional simulation is pine wood as the measurable parameters are well known (Lartigue and Puiggali, 1987; Bonneau, 1991). The basic convective drying configuration is sketched in Figure 1 and could not be simulated by using heat and mass transfer coefficients as boundary layer does not exist. A sample of wood (length: 0.01 m, thickness: 0.01 m) is placed at the centre of the drying tunnel (length: 0.5 m, diameter: 0.1 m) as described in Figure 2 which represents the Fluent mesh. The environment and porous medium meshes, which are represented in Figures 2 and 3, are composed respectively by 1884 and 121 nodes. Note that near the interface between the product and its surrounding, the size of the cells must be small enough ( 0:0001 m) in order to compute fluxes with a great accuracy. The initial moisture content (W ¼ 1) and initial temperature (T¯ ¼ 258C) inside the porous medium are assumed to be uniform and the boundary conditions of the tunnel (Figure 2) are described in the following parts. The inlet velocity condition is given by, for x ¼ 0 V1 ¼ u1 ex
(50)
For the simulation, the conditions at the inlet of the dryer are constant. The velocity u1 is equal to 0.5 m s21, the temperature T1 is equal to 608C and the relative humidity HR1 is equal to 15%. The walls of the tunnel are assumed to be adiabatic and impermeable. It involves for y ¼ b, rT n ¼ 0
(51)
and for i ¼ (a, v), (48)
ri V i n ¼ 0
Figure 2. Physical configuration, mesh of the drying tunnel.
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Figure 3. Porous medium mesh.
The symmetry of the tunnel is described by the relations, for y ¼ 0, rw n ¼ 0 with w ¼ T, Vx , r
(53)
Vy ¼ 0
(54)
and
At the outflow, the flow is supposed to be fully developed and obey the following relations, for x ¼ L, rw n ¼ 0 with w ¼ T, V, r
(55)
Vy ¼ 0
(56)
and
RESULTS Figure 4 represents the convective drying kinetic of the sample, it means the averaged total mass flux versus the average moisture content. We can observe the classical characteristic phases of the process: the initial transient period [noted (1) in Figure 4], the constant rate period (2), the first falling rate period (3) and the second falling rate period (4). However, the analysis of mass transfer at each edge leads to large differences compared to the
Figure 4. Convective drying kinetic, averaged total mass flux versus averaged moisture content.
Figure 5. Moisture content, temperature and pressure profiles at t ¼ 0.1 h.
Figure 6. Evolution of averaged mass flux on each edge during drying.
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Figure 7. Moisture content, temperature and pressure profiles at t ¼ 0.4 h.
Figure 8. Moisture content, temperature and pressure profiles at t ¼ 0.7 h.
phenomena usually described in literature when heat and mass transfer coefficients are used. The first stage corresponds to the transient period during which the temperature within the product rises by conduction to reach a proximate value to the wet bulb temperature (328C) [Figure 5(b)]. At 0.1 h, as it is illustrated on moisture content profiles [Figure 5(a)], the humidity loss on the leading edge, and more particularly on the corners, is much more important that those at the upper and vanishing edges.
During this period, the capillary pressure is the driving force and the liquid flows to the surface. We can note in Figure 6 that the averaged mass fluxes on different edges increases to a maximal value on the leading edge and to quasi-constant values on the upper and vanishing edges. Although a classical constant rate period seems to exist if we observe the global drying kinetic (noted 2 in Figure 4), this period does not appear in the mass flux evolution on the leading edge (Figure 6). Indeed its value is so important
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Figure 9. Illustration of heat flux vectors in the product at (a) t ¼ 0.4 h and (b) at t ¼ 0.9 h.
that the product reaches locally the bound water saturation point [Figure 7(a)]. The temperature does not equilibrate to the wet bulb temperature and continues to increase [Figures 7(b) and 8(b)]. It induces the rise of the pressure as illustrated in Figures 7(c) and 8(c). It is important to remark that the temperature of product areas containing free water increases. This is due to a strong conductive heat flux from the leading edge to the vanishing edge. This phenomenon is emphasized in Figure 9(a) which represents the heat flux vectors 2 leffrT together with the level of temperature inside the product. One of the consequences of the heat conductive transport is the increase of the averaged mass fluxes in the upper and vanishing edges, phenomenon observable in Figure 6. Indeed, the mass flux Fm is defined by the expression: Fm ¼ rv Vg n rDrv n
(57)
domain i.e. when moisture content is lower than the bound water saturation. Until 1.4 h (Figure 11), two areas can be distinguished in the product . a zone near the surfaces in the hygroscopic field for which moisture content migrates in adsorbed form by diffusion-sorption and in gaseous form gas by diffusion; . an internal area where the free water is present. The mass transport is ensured by capillarity. From 2.1 h, the second falling rate period begins, all the product is in the hygroscopic field [Figure 12(a)]. When the moisture content in the centre of the product decreases sufficiently so that the reduction of the water activity compensates and outclasses the
In this relation, the calculation shows that the convective term can be neglected so the mass flux Fm can be described by: Fm rDrv n
(58)
During this period, the density of humid air r decreases slowly while the diffusion coefficient D is constant, so the term rv n is responsible of the mass flux augmentation and in particular the fast rise of the mass fraction of vapour v due to the rise of temperature. The evolution of local mass flux at the vanishing edge reported in Figure 10 is consistent with the averaged mass flux behaviour (Figure 6). However, we can observe a particular period during which the local mass flux intensifies promptly (noted (a) in Figure 10). This phenomenon is explained by a rise of temperature due to an important conductive heat flux [Figure 9(b)] the adjacent areas reach the hygroscopic
Figure 10. Evolution of mass flux on each node at the vanishing edge.
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121
Figure 11. Moisture content, temperature and pressure profiles at t ¼ 1.4 h.
Figure 12. Moisture content, temperature and pressure profiles at t ¼ 2.1 h.
increase of vapour saturation pressure with the temperature, the overpressure decreases and tends towards zero [Figure 13(c)]. The moisture content reaches gradually the moisture equilibrium provided by the sorption isotherm (0.032) [Figure 13(a)] and the temperature reaches the temperature of the drying environment (608C) [Figure 13(b)]. To conclude, although the global kinetic seems to reveal a constant period (Figure 4), the analysis of local
mass flux shows that this typical period does not exist. Indeed, during this period, the heat flux contribution of humid air is not entirely used to evaporate water at the surface. An important part of the heat flux is transmitted within the product by conduction and tends to intensify average mass flux of evaporation on each edge. We do not observe a period during which the field of temperature is established with a constant value equal to the wet bulb temperature.
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ERRIGUIBLE et al. The basis of this method lies in the rigorous modelling of the boundary conditions between the two domains and the deduction of the fluxes by the resolution of the conservation equations in the environment. The simulation proposed in this paper illustrates the interest of our approach because the classical heat and mass transfer coefficients are not available for this configuration. The coupling method developed in this work allows us to know the complete distribution of mass fluxes at the interface. The analysis of local mass flux shows that the constant rate period, which seems to be revealed in the global kinetic, does not exist locally. Indeed, during this period, the heat flux contribution provided by the air flow is not entirely used to evaporate water at the surface. An important part of the heat flux is transmitted within the product by conduction and tends to intensify average mass flux of evaporation on each edge. NOMENCLATURE Cp D D eff Db ex ey Fh Fm g ha h b hl hs hv DHb DHvref n P Q RH T t t u V W
constant pressure heat capacity, J kg21 K21 diffusivity in the air, m2 s21 diffusion tensor of vsapour in porous medium, m2 s21 diffusion tensor of bound water in porous medium, m2 s21 unit vector in x direction unit vector in y direction total heat flux, W m22 evaporation flux, kg m22 s21 gravity vector, m s22 intrinsic averaged enthalpy of dry air, J kg21 intrinsic averaged enthalpy of bound water, J kg21 intrinsic averaged enthalpy of free water, J kg21 intrinsic averaged enthalpy of solid, J kg21 intrinsic averaged enthalpy of vapour, J kg21 heat of desorption, J kg21 latent heat of vaporization at the reference temperature T ref, J kg21 outer unit normal to the product pressure, Pa heat flux, W m22 external relative humidity, % temperature, K or 8C time, s unit tangential to the product velocity, m s21 velocity, m s21 moisture content (in dry basis)
Greek symbols l thermal conductivity, W m21 K21 l eff effective thermal conductivity tensor, W m21 K21 r density, kg m23 m viscosity, kg m21 s21 v mass fraction of vapour in the porous medium v mass fraction of vapour in the surroundings
Figure 13. Moisture content, temperature and pressure profiles at t ¼ 4.5 h.
CONCLUSION This article presents a method of resolution for problems of conjugate transfer between a porous medium and its surrounding. This approach is valid for a great number of processes and its major interest is to avoid the use of transfer coefficients which are theoretically unknown in the majority of the cases.
Subscripts a b c eq g l s sat v
dry air bound water capillary equilibrium gas liquid solid saturation vapour
Superscripts g gas l liquid s solid
Trans IChemE, Part A, Chemical Engineering Research and Design, 2006, 84(A2): 113– 123
SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM Mathematical operators r gradient operator r divergence operator second order tensor ¼
REFERENCES Bonneau, P., 1991, Mode´lisasion du se´chage d’un mate´riau he´te´roge`ne: application a` un bois de re´sineux, PhD thesis, Universite´ de Bordeaux I, France. Couture, F., 1995, Mode´lisation fine d’un proble`me de se´chage, de´veloppement d’outils adapte´s, PhD thesis, Universite´ de Bordeaux I, France. Couture, F., Fabrie, P. and Puiggali, J.R., 1995, An alternative choice for the drying variables leading to a mathematically and physically well described problem, Drying Technology, 13(3): 519–550. Ene, I.H. and Sanchez-Palencia, E., 1975, Equations et phe´nome`nes de surface pour l’e´coulement dans un mode`le de milieu poreux, Journal de Me´canique, 14(1): 73–108. Erriguible, A., 2004, Mode´lisation des transferts a` l’interface d’un milieu multiphasique et de son environnement, PhD thesis, Universite´ de Pau et des Pays de l’Adour, France. Erriguible, A., Bernada, P., Couture, F. and Roques, M.-A., 2005, Modelling of heat and mass transfer at the boundary between a porous medium and its surroundings, Drying Technology, 23(3): 455–472. Erriguible, A., Bernada, P., Couture, F. and Roques, M.-A., 2003, Theoretical study of the transfer at the boundary of a multiphase material and its surroundings during convective drying, Proceeding of the European Drying Symposium 2003, 1: 79–86. Lartigue, C. and Puiggali, J.R., 1987, Caracte´ristiques du pin des Landes ne´cessaires a` la compre´hension des phe´nome`nes de se´chage,
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Actes du second Coll. Sciences et Industrie du bois, Arbolor Ed, 2: 57–64. Masmoudi, W. and Prat, M., 1991, Heat and mass transfer between a porous medium and a parallel external flow. Application to drying of capillary porous materials, Int J Heat and Mass Transfer, 34(8): 1975–1989. Perre´, P. and Degiovanni, A., 1990, Simulations par volumes finis des transferts couple´s en milieu poreux anisotropes: se´chage du bois a` basse et a` haute temperature, International Journal of Heat and Mass Transfer, 33(11): 2463–2478. Prat, M., 1989, Mode´lisation des transferts en milieu poreux: changement d’e´chelle et conditions aux limites, PhD thesis, Institut National Polytechnique de Toulouse, France. Quintard, M. and Whitaker, S., 1993, One- and two-equation models for transient diffusion processes in two-phase system, Advances in Heat Transfer, 23: 369– 464. Slattery, J.C. 1999, Advanced Transport Phenomena (Cambridge Press, Cambridge, UK). Whitaker, S., 1977a, Fundamental Principles of Heat Transfer (Krieger Publishing Company, Malabar, Florida). Whitaker, S., 1977b, Simultaneous heat, mass and momentum transfer in porous media: a theory of drying, Advances in Heat Transfer, 13: 119 –203. Whitaker, S., 1984, Moisture transport mechanisms during the drying of granular porous media, Proceeding of the Fourth International Drying Symposium, 1: 31 –42. Whitaker, S., 1991, Improved constraints for the principle of local thermal equilibrium, Ind Eng Chem Res, 30: 983– 997. The manuscript was received 28 February 2005 and accepted for publication after revision 25 January 2006.
APPENDIX A Physical Properties of the Porous Medium: Pinewood 1 r s k Cps Wps aw
Porosity Solid density (kg m23) Intrinsic permeability (m2) Solid heat capacity (J kg21 K21) Moisture content of fibre saturation point Water activity
0.615 476 4 . 10216 1400 0.3 1 if W . Wps exp ( AB100W þ C)
if W Wps
5 2
A ¼ 2:86 10 T 1:07 102 T þ 10:24 B ¼ 5:41 104 T þ 1:01 2
C ¼ 4:97 106 T 2:67 103 T þ 0:35 Deff
Effective diffusivity of vapour in the medium (m2 s21)
1:81 B(m) 8:92 105 TP g g
B(m) ¼ 16 2 21
Db
Bound water diffusivity (m s
l
Effective thermal conductivity (W m21 K21)
Pc
)
0 if W . Wps or
1 r s
" # 2590:1 1046:63 exp 11:954 T W T 12:35
6 104 r s 1þW 1þrv 0:166 rv ¼ 0.15
if W . Wps
rv ¼ 0:44W
if W Wps
c ¼ 0.093
d ¼ 1.400
Capillary pressure (Pa)
(aS exp ( bS) þ c(1 S)Sd )(1 2:79 103 (T 273:16)) 105 Pa a ¼ 1.937
b ¼ 3.785
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www.icheme.org/journals doi: 10.1205/cherd.05047
SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM WITH BOUNDARY CONDITIONS PROVIDED BY CFD A. ERRIGUIBLE, P. BERNADA , F. COUTURE and M. ROQUES Laboratoire de Thermique Energe´tique et Proce´de´s, Pau, France
T
he description of the transfer between a porous medium and its surroundings is commonly made using transfer coefficients which are theoretically well described only under boundary layer hypothesis. Solving Navier –Stokes equations in the surroundings of the product in order to get information about the boundary conditions avoids the classical using of these transfer coefficients. After a brief description of the model, a simulation of convective drying of a rectangular piece of porous medium is proposed using a coupling method between a porous medium code and a CFD software. The analysis of the interfacial transfer coupled with the analysis of the temperature and moisture content profiles show the influence of the edges. We show that the global kinetics cannot reflect the local surface transfer phenomena: even at the so-called constant rate period heat is not entirely used for water evaporation with a portion for temperature increase within the product. Keywords: conjugate problem; modelling; coupling method; heat and mass transfer; fluent.
INTRODUCTION
about the boundary conditions for the transport equations in the medium and solve the complete drying problem. This article deals with an analysis of the boundary conditions during convective drying simulated with a coupling method between a porous medium and its surrounding. The coupling of the model associated to the porous medium with the one devoted to the external flow is performed by writing the boundary conditions at the interface of the porous medium/environment.
Heat and mass transfer at the frontier of a multiphase medium and its surroundings are only theoretically well described under boundary layer hypothesis. In the other cases, theoretical study of interfacial transfer is very difficult due to the geometry of the physical frontier between the material and its surroundings, or due to the process by itself: vacuum drying, high temperature drying, steam drying and so on. In these cases transfer coefficients cannot be defined rigorously. For example, the basic convective drying 2D configuration sketched in Figure 1, which represents the air velocity field around the porous medium during the drying, could not be simulated by using heat and mass transfer coefficients as boundary layer does not exist. Indeed, one of the boundary layer hypothesis implies that the thickness d of the gradient zone must be much smaller than the characteristic length L of the sample. The velocity field in Figure 1 shows that this classical assumption (Whitaker, 1977a; Slattery, 1999) is not verified. Moreover, boundary layer does not exist at the leading and vanishing edge. Hence it is necessary to solve Navier– Stokes equations in the surroundings of the product in order to get information
MODELLING Heat and Mass Transports Within the Porous Medium and the Environment The mathematical description of the physics involved during the drying of a hygroscopic capillary porous medium has been discussed by numerous authors. Most of them adopt a continuum approach based on Whitaker’s theory (Whitaker, 1977b, 1984), where the macroscopic partial differential equations are achieved by volume averaging the microscopic conservation laws. The value of any physical quantity at a point in space is given by its average value on the averaging volume (AV) centred at this point. The phase average is defined by ð 1 F dv (1) F ¼ v v
Correspondence to: Dr P. Bernada, Laboratoire de Thermique Energe´tique et Proce´de´s, EA 1932, 3 rue Jules Ferry, Bp 7511, 64075, PAU CEDEX, France. E-mail: [email protected]
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Figure 1. Air velocity field around porous medium during convective drying. This figure is available in colour via www.icheme.org/journals.
and the intrinsic phase average i by ð 1 i F ¼ F dv vi v
(2)
Let us define the density and the mass averaged velocity of the mixture in the usual manner: X r¼ ri (4) i
Within each AV, the porous medium consists of a continuous rigid solid phase which contains bound water (i.e., the product is assumed to be hygroscopic), an incompressible liquid phase (free water), a continuous gas phase which is assumed to be a perfect mixture of vapour and dry air, considered as ideal gases. The hypothesis and the way to obtain the governing equations for heat, mass and momentum transport during drying have been well established and are not discussed here (Couture, 1995). The equations which are solved in the porous medium are the mass conservation of water and dry air, the energy conservation and the momentum conservation (generalized Darcy’s laws and Fick’s laws). For the environment, the external flow of moist air which consists of two species (dry air and water vapour, respectively noted a and v) is considered as unsteady and incompressible. The conservation equations to be solved are the species, momentum and energy conservation equations. How to obtain the complete differential set of the necessary conservation laws for the porous medium and its surrounding is not detailed here (Erriguible, 2004). Only the final system is given in the following part.
rV ¼
X
ri V i
To complete the model, one relation must be added. The incompressibility of the air flow leads to: rV¼0
(6)
Momentum Conservation By considering the environment as a Newtonian fluid, the momentum conservation equation for the moist air is commonly described by @ðrVÞ þ r ðrV VÞ ¼ rg rP þ mDV @t 1 þ m r ðr V Þ 3
(7)
The diffusion for each species is modelled according to Fick’s laws.
ri ðVi VÞ ¼ rDr NAVIER – STOKES MODELLING FOR THE ENVIRONMENT
(5)
i
ri r
for i ¼ (a, v)
(8)
Species Conservation
Energy Conservation
Mass conservation is written for the two species (dry air and vapour):
The energy conservation equation is written in terms of enthalpy to be consistent with the one associated to the porous medium. By assuming that the viscous dissipation and the total derivative of pressure are negligible and by considering that there is no volumetric heat source, the
@r i þ r r iV i ¼ 0 @t
for i ¼ (a, v)
(3)
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SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM energy equation is resumed to @
! X ri hi þr ri V i hi þ q ¼ 0 @t i
P
i
with
r v r g
(20)
D eff ¼ DB
(21)
v ¼
(9)
The heat flux q is expressed by the Fourier’s law, q ¼ lrT
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(10)
and the enthalpies are given by, ha ¼ Cpa T T ref hv ¼ DHvref þ Cpv T T ref
(11) (12)
In this equation, B is a diagonal tensor which reduces the coefficient of vapour diffusion in the air D in order to take into account the anisotropic material resistance to the diffusion transport. g is the barycentric velThe velocity of the gas phase V ocity expressed by, g ¼ r g V v r gg V gv V a aþr
AVERAGED EQUATIONS FOR THE POROUS MEDIUM
and is commonly described by the generalized Darcy’s law
The complete set of the conservation laws for the mass, for the momentum transport and for energy is summarized below.
g ¼ V
k krg g : rP g r gg g mg
The solid phase being rigid, we have l ¼ rl r ll V l
(13)
k krl l : rP l r ll g ml
(24)
in which the capillary pressure Pc is introduced as a function in term of moisture content (see appendix A),
The averaged mass conservation of the dry air gives @ra þ r ra V a ¼ 0 @t
(23)
For the liquid, the expression of the flux is also derivated from Darcy’s law,
Mass Conservation @rs ¼0 @t
(22)
(14)
l g P l ¼ P g Pc
(25)
For the water, the general equation of mass conservation is obtained by summation of the conservation equations of vapour (v), free water (l) and bound water (b) and written as follows @W 1 l þr r Vl þ rv Vv þ rb Vb ¼ 0 (15) @t r s l
For bound water, the flux is modelled by a classical phenomenological law (Perre´ and Degiovanni, 1990; Couture et al., 1995)
with the moisture content dry basis W defined by the relation:
Energy Conservation
W¼
r l þ r b þ r v r s
(16)
rb Vb ¼ rs Db :rWb
(26)
The assumption of the local thermal equilibrium between the solid, the gas and the liquid phases (Whitaker, 1991; Quintard and Whitaker, 1993) involves, s g l T s ¼ T g ¼ T l ¼ T
(27)
Momentum Transport Conservation The fluxes of each component are summarized below. For the solid, we have s ¼ 0 r ss V
Then the energy conservation is reduced to a unique equation n @ rh a h g þ r g V g lV l þ r r ga V a v v hv þ r l l hl þ rb Vb hb @t o (28) leff : rT ¼ 0
(17)
For the gas, the Fick’s law gives us for the two components (vapour and dry air) g
r g D rv rv Vv ¼ r v V g g eff
(18)
g þ r g D rv ra Va ¼ r a V g g eff
(19)
with s g g l rh ¼ r s h s þ r a h a þ r v h v þ r l h l þ r b h b
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enthalpies being expressed as follows: g h a ¼ Cpa T Tref l h l ¼ Cpl T Tref s h s ¼ Cps T Tref
(30) (31) (32)
l h l
h b ¼ DHb g h v ¼ DHvref þ Cpv T Tref
(33) (34)
This set of equations is closed by adding two assumptions. . The gaseous phase is considered as an ideal gas. It ensures g mi P i g r i ¼ for i ¼ (a, v) (35) RT and g g g P g ¼ P a þ P v (36)
r gg
¼
r ga
þ
r gv
Moreover the local thermal equilibrium at the interface involves the continuity of temperature and vapour mass fraction:
r gv rv ¼ r gg rg T ¼ T
(42) (43)
The no-slip condition implies that the tangential velocity of the humid air in the environment is equal to zero Vt¼0
(44)
with t the tangential vector to the porous medium. Moreover, the total pressure is continue at the interface (Ene and Sanchez-Palencia, 1975; Prat, 1989): g
P ¼ P g
(45)
(37)
. local thermodynamical equilibrium implies g P v ¼ aw Pvsat
APPLICATION TO CONVECTIVE DRYING (38)
Pvsat is the saturation vapour pressure. aw is the water activity equal to one when the water is free (nonhygroscopic region) and lower than one (see Appendix A), depending on temperature and moisture content when the water is bound to the solid phase (hygroscopic region). Boundary Conditions Between the Two Domains At the interface, we are dealing with a coupled problem between the heat and mass transfer within the external flow and within the porous medium. As discussed in Masmoudi and Prat (Masmoudi and Prat, 1991), we can make use of local heat and mass fluxes under the assumption that a typical averaging volume size is much smaller than the external flow characteristics length scales. The conservative form of the previous equations enables us to write rigorously the boundary conditions (Slattery, 1999; Erriguible, 2004; Erriguible et al., 2005). The natural conditions are the continuity of the heat and species fluxes in particular for dry air and water: g þ r g D rv n ¼ ra Va n r a V (39) g g eff l þ rb Vb þ r v V g r g D rv n r ll V g g eff (40) ¼ rv Vv n ¼ Fm 0 1 g þ r g D rv h g þ r a V g a g eff B C g B C g r g D rv h þ r l V l h l C n B r v V g g l v lA eff @ þrb Vb h b l eff :rT ¼ rv Vv hv lrT n ¼ Fh (41) with Fm and Fh respectively the evaporation flux and total heat flux at the interface.
Before we present the two dimensional simulation of the convective drying of the porous medium, the method of coupling between the two codes which was validated for one dimensional configuration in a previous work (Erriguible et al., 2003) is detailed. Numerical Implementation of the Coupling Method The porous medium code is a 2D Fortran program previously developed (Couture et al., 1995) and the code used to solve Navier– Stokes equations is the commercial software FLUENT 6. The coupling method consists of executing the porous medium code for one time step and successively executing the CFD code for the same time step. An explicit treatment for the boundary conditions is justified by the use of small time steps (0.1 s). An interface was made in ‘script’ language to make the link between the two codes. The spatial discretisation of the porous medium equations is performed by the finite element method. The finite volume one is preferred for the Navier– Stokes equations. In the first part, we described the conservation equations in the two domains in order to determine the boundary conditions between the two media. In this part we explain how the boundary conditions are allocated to the different conservation equations in order to simulate the conjugate heat and mass transfer problem. Boundary conditions allocated to the surrounding modelling Navier–Stokes equations are solved sequentially with FLUENT 6 by using the same time step as the one used in porous medium code. The numerical results provided by the porous medium code allow to allocate the mass flux, the temperature and mass fraction vapour in such a way that boundary conditions (39), (42), (43) and (44) are imposed. The momentum equations (7) are solved by setting the gas velocity vector at the interface between the product and its surrounding. The value of tangential velocity is given by the no-slip condition at the interface (44).
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SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM The normal velocity is expressed by using the total flux continuity defined by the sum of equations (39) and (40): 1 g) n V n ¼ (rll V gg V l þ rb V b þ r r
(46)
Note that the continuity of the water flux (40) is ensured during the resolution of the porous medium equation. The difference between the total flux continuity (46) and the water flux continuity (40) imposes the dry air flux continuity. Since the velocities obtained to the first step may not satisfy the continuity equation (6) locally, a poisson-type equation for the pressure correction is derived from the continuity equation and the linearized momentum equations (SIMPLE algorithm). This pressure correction is then solved to obtain the necessary corrections to the pressure and velocity fields and the face mass fluxes such that continuity is satisfied. The mass conservation equation (3) is solved using the continuity of the mass fraction of the species at the interface (42). The energy equation (9) is solved using the continuity of the temperature at the interface (43).
Boundary conditions allocated to the porous medium modelling Conservation equations in the porous medium are solved for one time step, by using the remaining boundary conditions (40), (41) and (45) calculated with the results provided by the CFD code. The mass conservation equation of water (15) is solved using the continuity of the evaporation flux Fm (40) defined by: Fm ¼ (rv Vv ) n ¼ rv Vg n rDrv n
(47)
To solve the mass conservation of the dry air (14), the mass concentration r ga at the boundary is calculated assuming the pressure continuity at the interface (45) and using the ideal gas law:
r ga ¼
Ma Pa Ma (P Pv ) ¼ RT RT
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The energy conservation equation (28) is solved using the continuity of the total heat flux Fh (41) deduced from the values given by the CFD code as follows: Fh ¼ Fm hv lrT n
(49)
Physical Configuration The porous medium chosen for the two dimensional simulation is pine wood as the measurable parameters are well known (Lartigue and Puiggali, 1987; Bonneau, 1991). The basic convective drying configuration is sketched in Figure 1 and could not be simulated by using heat and mass transfer coefficients as boundary layer does not exist. A sample of wood (length: 0.01 m, thickness: 0.01 m) is placed at the centre of the drying tunnel (length: 0.5 m, diameter: 0.1 m) as described in Figure 2 which represents the Fluent mesh. The environment and porous medium meshes, which are represented in Figures 2 and 3, are composed respectively by 1884 and 121 nodes. Note that near the interface between the product and its surrounding, the size of the cells must be small enough ( 0:0001 m) in order to compute fluxes with a great accuracy. The initial moisture content (W ¼ 1) and initial temperature (T¯ ¼ 258C) inside the porous medium are assumed to be uniform and the boundary conditions of the tunnel (Figure 2) are described in the following parts. The inlet velocity condition is given by, for x ¼ 0 V1 ¼ u1 ex
(50)
For the simulation, the conditions at the inlet of the dryer are constant. The velocity u1 is equal to 0.5 m s21, the temperature T1 is equal to 608C and the relative humidity HR1 is equal to 15%. The walls of the tunnel are assumed to be adiabatic and impermeable. It involves for y ¼ b, rT n ¼ 0
(51)
and for i ¼ (a, v), (48)
ri V i n ¼ 0
Figure 2. Physical configuration, mesh of the drying tunnel.
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Figure 3. Porous medium mesh.
The symmetry of the tunnel is described by the relations, for y ¼ 0, rw n ¼ 0 with w ¼ T, Vx , r
(53)
Vy ¼ 0
(54)
and
At the outflow, the flow is supposed to be fully developed and obey the following relations, for x ¼ L, rw n ¼ 0 with w ¼ T, V, r
(55)
Vy ¼ 0
(56)
and
RESULTS Figure 4 represents the convective drying kinetic of the sample, it means the averaged total mass flux versus the average moisture content. We can observe the classical characteristic phases of the process: the initial transient period [noted (1) in Figure 4], the constant rate period (2), the first falling rate period (3) and the second falling rate period (4). However, the analysis of mass transfer at each edge leads to large differences compared to the
Figure 4. Convective drying kinetic, averaged total mass flux versus averaged moisture content.
Figure 5. Moisture content, temperature and pressure profiles at t ¼ 0.1 h.
Figure 6. Evolution of averaged mass flux on each edge during drying.
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Figure 7. Moisture content, temperature and pressure profiles at t ¼ 0.4 h.
Figure 8. Moisture content, temperature and pressure profiles at t ¼ 0.7 h.
phenomena usually described in literature when heat and mass transfer coefficients are used. The first stage corresponds to the transient period during which the temperature within the product rises by conduction to reach a proximate value to the wet bulb temperature (328C) [Figure 5(b)]. At 0.1 h, as it is illustrated on moisture content profiles [Figure 5(a)], the humidity loss on the leading edge, and more particularly on the corners, is much more important that those at the upper and vanishing edges.
During this period, the capillary pressure is the driving force and the liquid flows to the surface. We can note in Figure 6 that the averaged mass fluxes on different edges increases to a maximal value on the leading edge and to quasi-constant values on the upper and vanishing edges. Although a classical constant rate period seems to exist if we observe the global drying kinetic (noted 2 in Figure 4), this period does not appear in the mass flux evolution on the leading edge (Figure 6). Indeed its value is so important
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Figure 9. Illustration of heat flux vectors in the product at (a) t ¼ 0.4 h and (b) at t ¼ 0.9 h.
that the product reaches locally the bound water saturation point [Figure 7(a)]. The temperature does not equilibrate to the wet bulb temperature and continues to increase [Figures 7(b) and 8(b)]. It induces the rise of the pressure as illustrated in Figures 7(c) and 8(c). It is important to remark that the temperature of product areas containing free water increases. This is due to a strong conductive heat flux from the leading edge to the vanishing edge. This phenomenon is emphasized in Figure 9(a) which represents the heat flux vectors 2 leffrT together with the level of temperature inside the product. One of the consequences of the heat conductive transport is the increase of the averaged mass fluxes in the upper and vanishing edges, phenomenon observable in Figure 6. Indeed, the mass flux Fm is defined by the expression: Fm ¼ rv Vg n rDrv n
(57)
domain i.e. when moisture content is lower than the bound water saturation. Until 1.4 h (Figure 11), two areas can be distinguished in the product . a zone near the surfaces in the hygroscopic field for which moisture content migrates in adsorbed form by diffusion-sorption and in gaseous form gas by diffusion; . an internal area where the free water is present. The mass transport is ensured by capillarity. From 2.1 h, the second falling rate period begins, all the product is in the hygroscopic field [Figure 12(a)]. When the moisture content in the centre of the product decreases sufficiently so that the reduction of the water activity compensates and outclasses the
In this relation, the calculation shows that the convective term can be neglected so the mass flux Fm can be described by: Fm rDrv n
(58)
During this period, the density of humid air r decreases slowly while the diffusion coefficient D is constant, so the term rv n is responsible of the mass flux augmentation and in particular the fast rise of the mass fraction of vapour v due to the rise of temperature. The evolution of local mass flux at the vanishing edge reported in Figure 10 is consistent with the averaged mass flux behaviour (Figure 6). However, we can observe a particular period during which the local mass flux intensifies promptly (noted (a) in Figure 10). This phenomenon is explained by a rise of temperature due to an important conductive heat flux [Figure 9(b)] the adjacent areas reach the hygroscopic
Figure 10. Evolution of mass flux on each node at the vanishing edge.
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SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM
121
Figure 11. Moisture content, temperature and pressure profiles at t ¼ 1.4 h.
Figure 12. Moisture content, temperature and pressure profiles at t ¼ 2.1 h.
increase of vapour saturation pressure with the temperature, the overpressure decreases and tends towards zero [Figure 13(c)]. The moisture content reaches gradually the moisture equilibrium provided by the sorption isotherm (0.032) [Figure 13(a)] and the temperature reaches the temperature of the drying environment (608C) [Figure 13(b)]. To conclude, although the global kinetic seems to reveal a constant period (Figure 4), the analysis of local
mass flux shows that this typical period does not exist. Indeed, during this period, the heat flux contribution of humid air is not entirely used to evaporate water at the surface. An important part of the heat flux is transmitted within the product by conduction and tends to intensify average mass flux of evaporation on each edge. We do not observe a period during which the field of temperature is established with a constant value equal to the wet bulb temperature.
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ERRIGUIBLE et al. The basis of this method lies in the rigorous modelling of the boundary conditions between the two domains and the deduction of the fluxes by the resolution of the conservation equations in the environment. The simulation proposed in this paper illustrates the interest of our approach because the classical heat and mass transfer coefficients are not available for this configuration. The coupling method developed in this work allows us to know the complete distribution of mass fluxes at the interface. The analysis of local mass flux shows that the constant rate period, which seems to be revealed in the global kinetic, does not exist locally. Indeed, during this period, the heat flux contribution provided by the air flow is not entirely used to evaporate water at the surface. An important part of the heat flux is transmitted within the product by conduction and tends to intensify average mass flux of evaporation on each edge. NOMENCLATURE Cp D D eff Db ex ey Fh Fm g ha h b hl hs hv DHb DHvref n P Q RH T t t u V W
constant pressure heat capacity, J kg21 K21 diffusivity in the air, m2 s21 diffusion tensor of vsapour in porous medium, m2 s21 diffusion tensor of bound water in porous medium, m2 s21 unit vector in x direction unit vector in y direction total heat flux, W m22 evaporation flux, kg m22 s21 gravity vector, m s22 intrinsic averaged enthalpy of dry air, J kg21 intrinsic averaged enthalpy of bound water, J kg21 intrinsic averaged enthalpy of free water, J kg21 intrinsic averaged enthalpy of solid, J kg21 intrinsic averaged enthalpy of vapour, J kg21 heat of desorption, J kg21 latent heat of vaporization at the reference temperature T ref, J kg21 outer unit normal to the product pressure, Pa heat flux, W m22 external relative humidity, % temperature, K or 8C time, s unit tangential to the product velocity, m s21 velocity, m s21 moisture content (in dry basis)
Greek symbols l thermal conductivity, W m21 K21 l eff effective thermal conductivity tensor, W m21 K21 r density, kg m23 m viscosity, kg m21 s21 v mass fraction of vapour in the porous medium v mass fraction of vapour in the surroundings
Figure 13. Moisture content, temperature and pressure profiles at t ¼ 4.5 h.
CONCLUSION This article presents a method of resolution for problems of conjugate transfer between a porous medium and its surrounding. This approach is valid for a great number of processes and its major interest is to avoid the use of transfer coefficients which are theoretically unknown in the majority of the cases.
Subscripts a b c eq g l s sat v
dry air bound water capillary equilibrium gas liquid solid saturation vapour
Superscripts g gas l liquid s solid
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SIMULATION OF CONVECTIVE DRYING OF A POROUS MEDIUM Mathematical operators r gradient operator r divergence operator second order tensor ¼
REFERENCES Bonneau, P., 1991, Mode´lisasion du se´chage d’un mate´riau he´te´roge`ne: application a` un bois de re´sineux, PhD thesis, Universite´ de Bordeaux I, France. Couture, F., 1995, Mode´lisation fine d’un proble`me de se´chage, de´veloppement d’outils adapte´s, PhD thesis, Universite´ de Bordeaux I, France. Couture, F., Fabrie, P. and Puiggali, J.R., 1995, An alternative choice for the drying variables leading to a mathematically and physically well described problem, Drying Technology, 13(3): 519–550. Ene, I.H. and Sanchez-Palencia, E., 1975, Equations et phe´nome`nes de surface pour l’e´coulement dans un mode`le de milieu poreux, Journal de Me´canique, 14(1): 73–108. Erriguible, A., 2004, Mode´lisation des transferts a` l’interface d’un milieu multiphasique et de son environnement, PhD thesis, Universite´ de Pau et des Pays de l’Adour, France. Erriguible, A., Bernada, P., Couture, F. and Roques, M.-A., 2005, Modelling of heat and mass transfer at the boundary between a porous medium and its surroundings, Drying Technology, 23(3): 455–472. Erriguible, A., Bernada, P., Couture, F. and Roques, M.-A., 2003, Theoretical study of the transfer at the boundary of a multiphase material and its surroundings during convective drying, Proceeding of the European Drying Symposium 2003, 1: 79–86. Lartigue, C. and Puiggali, J.R., 1987, Caracte´ristiques du pin des Landes ne´cessaires a` la compre´hension des phe´nome`nes de se´chage,
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Actes du second Coll. Sciences et Industrie du bois, Arbolor Ed, 2: 57–64. Masmoudi, W. and Prat, M., 1991, Heat and mass transfer between a porous medium and a parallel external flow. Application to drying of capillary porous materials, Int J Heat and Mass Transfer, 34(8): 1975–1989. Perre´, P. and Degiovanni, A., 1990, Simulations par volumes finis des transferts couple´s en milieu poreux anisotropes: se´chage du bois a` basse et a` haute temperature, International Journal of Heat and Mass Transfer, 33(11): 2463–2478. Prat, M., 1989, Mode´lisation des transferts en milieu poreux: changement d’e´chelle et conditions aux limites, PhD thesis, Institut National Polytechnique de Toulouse, France. Quintard, M. and Whitaker, S., 1993, One- and two-equation models for transient diffusion processes in two-phase system, Advances in Heat Transfer, 23: 369– 464. Slattery, J.C. 1999, Advanced Transport Phenomena (Cambridge Press, Cambridge, UK). Whitaker, S., 1977a, Fundamental Principles of Heat Transfer (Krieger Publishing Company, Malabar, Florida). Whitaker, S., 1977b, Simultaneous heat, mass and momentum transfer in porous media: a theory of drying, Advances in Heat Transfer, 13: 119 –203. Whitaker, S., 1984, Moisture transport mechanisms during the drying of granular porous media, Proceeding of the Fourth International Drying Symposium, 1: 31 –42. Whitaker, S., 1991, Improved constraints for the principle of local thermal equilibrium, Ind Eng Chem Res, 30: 983– 997. The manuscript was received 28 February 2005 and accepted for publication after revision 25 January 2006.
APPENDIX A Physical Properties of the Porous Medium: Pinewood 1 r s k Cps Wps aw
Porosity Solid density (kg m23) Intrinsic permeability (m2) Solid heat capacity (J kg21 K21) Moisture content of fibre saturation point Water activity
0.615 476 4 . 10216 1400 0.3 1 if W . Wps exp ( AB100W þ C)
if W Wps
5 2
A ¼ 2:86 10 T 1:07 102 T þ 10:24 B ¼ 5:41 104 T þ 1:01 2
C ¼ 4:97 106 T 2:67 103 T þ 0:35 Deff
Effective diffusivity of vapour in the medium (m2 s21)
1:81 B(m) 8:92 105 TP g g
B(m) ¼ 16 2 21
Db
Bound water diffusivity (m s
l
Effective thermal conductivity (W m21 K21)
Pc
)
0 if W . Wps or
1 r s
" # 2590:1 1046:63 exp 11:954 T W T 12:35
6 104 r s 1þW 1þrv 0:166 rv ¼ 0.15
if W . Wps
rv ¼ 0:44W
if W Wps
c ¼ 0.093
d ¼ 1.400
Capillary pressure (Pa)
(aS exp ( bS) þ c(1 S)Sd )(1 2:79 103 (T 273:16)) 105 Pa a ¼ 1.937
b ¼ 3.785
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