Ind. Eng. Chem. Res. 2000, 39, 1443-1452
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Estimation of Effective Diffusivity in Drying of Heterogeneous Porous Media Maria A. Silva,* Piet J. A. M. Kerkhof,† and W. Jan Coumans Laboratory of Separation Processes and Transport PhenomenasTUE, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
This work presents a new method to evaluate the effective diffusivity in porous solids, considering the mass-transfer mechanisms occurring in the pores and on the surface. The binary friction model was used to evaluate the effective gas-phase diffusivity and the Maxwell-Stefan surface diffusion equation to the transport of bound water. Different expressions for the effective diffusivity for porous solids were derived considering parallel diffusion, series diffusion, and the Clausius-Mossotti model. The numerical results were compared to experimental data obtained for Son clay. In the pore diffusivity, when both are present, capillary flow showed dominance over gas diffusion. For the entire moisture content range, only a combination of the models presented good agreement with experimental data, showing changes in the interdependence of mechanisms during the drying process. Introduction In industrial practice, there are many examples of the use of mixtures of different particles, or more precisely, multicomponent porous solids. Recently, Kroes et al.1 and van der Sanden et al.2 started the investigation of effective diffusivity for multicomponent porous solids as a function of the effective diffusivity of each component, on a mixture of clays and on FCC catalysts (mixture of clay, zeolite Y, alumina, and silica), respectively. Nevertheless, if different types of particles are mixed, it is very important to know the contribution of each mechanism of mass transfer, to separate the effects due to the particle structure from those due to the composition of the mixture, and thus to be able to determine the diffusion coefficient of the mixture as a function of the diffusion coefficient for each mechanism involved. In drying calculations it is common to determine the effective diffusivity directly from the drying curves (simplified method, regular regime method) or to compare numerical values of mean moisture content to the experimental ones (numerical method). Zogzas et al.3 produced a review about such methods. In all of these methods the effective diffusivity is a lumped parameter resulting from the contribution of all mass transport mechanisms occurring in the different phases. Ketelaars4 and Kroes5 showed such mechanisms as gas diffusion, capillary flow, and bound water migration. A porous medium can be defined as a solid matrix with a set of pores, and the solid matrix can be microporous or not. During the drying of a porous medium, we are considering that there are gas and liquid phases present in the pores and adsorbed water (bound water) on the solid surface, as done previously by some authors.4-7 A representative volume of the porous medium described above is shown in Figure 1. * Corresponding author. Current address: Thermo Fluid Dynamics DepartmentsFEQ, State University of Campinas (Unicamp), P.O. Box 6066, 13081-970 Campinas-SP, Brazil. E-mail:
[email protected]. Fax: +55 (0) 19 788-3910. Phone: +55 (0) 19 788-3923. † E-mail:
[email protected]. Fax: +31 (0)40 2439303.
Figure 1. Schematic picture of a control volume of a partially saturated porous medium. White is the gas phase, gray is solids, and black is water.
For each phase the conservation equations can be written. The molar flux vector N1 needs to be specified through the momentum transfer equation and/or the transport properties for the materials involved in the study. In drying of porous media, the gas-phase transport has been treated as a combination of convective and diffusive flows. Using the volume-average method,9,10 some authors7,8,10 have represented the convective flow from Darcy’s law and have used a gas effective diffusion coefficient tensor expressed as a result of molecular diffusion and gas-phase dispersion.7,10 However, for small pores in which slip flow occurs, Darcy’s law cannot be applied. In such pores the momentum transfer between species and the wall becomes of the same order of magnitude as that between the moving species. In the case of drying we generally have a mixture of vapor and air, and in small pores the superposition of convective and diffusive flow breaks down.11 Under these conditions, momentum-balance equations for each species lead to pore-averaged flux equations such as those in the binary friction model (BFM).11
10.1021/ie990563n CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000
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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
Figure 2. Overview of the various volumes considered in the partially saturated porous material.
The migration of bound water was considered by Stanish et al.6 and Perre´ et al.7 as a molecular diffusion process, whose flux is directly proportional to the gradient in the chemical potential of the bound molecules. Krishna12 described the multicomponent surface or micropore diffusion of adsorbed species with a modified Maxwell-Stefan equation. We are going to develop a procedure to obtain the effective diffusivity of water in drying of porous solids considering pore and surface diffusion. Pore diffusivity will be obtained through the mass transports occurring in the pores: capillary flow and gas-phase transport. The gas-phase transport will be described by the BFM. The bound water, that is, the adsorbed water on the surface and in the micropores of the solid matrix, will be treated as a particular case of surface or micropore diffusion. This text presents a procedure to obtain the contribution of each mechanism to the diffusion coefficient of single-component solids, which can be extended to multicomponent solids.
Figure 3. Overview of transport mechanisms in pores and in solids: (a) parallel capillary liquid flow and vapor phase transport; (b) capillary liquid flow and vapor transport in series; (c) microporous transport in the solid phase.
System Description First we want to illustrate the system considered and the various modes of transport. In Figure 1 we have given a sketch of a typical control volume of a partially saturated porous medium. In Figure 2 we indicate the associated volume fractions. We have a volume fraction of solids s, which includes the micropores and small mesopores (typically 0.023. The effective diffusivity calculated by the models presented in the previous section and the experimental values are shown in Figure 9. Thus, Figure 9 presents the curves obtained with the parallel and Clausius-Mossotti models, considering for each one only continuous capillary flow (eq 23) or a combination of continuous/discontinuous capillary flow (eq 25). Figure 9 shows that there is good agreement between the experimental data and continuous capillary flow models for 0.03 < X < 0.06. For X > 0.09, the best agreement is for the combination parallel/series models. That indicates that there is a transition region for 0.06 < X < 0.09, where there is a decrease of the parallel contribution. This means that there is no occurrence of “necks” in the smaller pores, but there is in the bigger ones. Figure 9 also shows that for X < 0.03 there is a gap between the experimental and numerical results; nev-
a1 ) activity of species 1 a ) water activity ) relative humidity b ) fraction of discontinuous capillary flow b* ) fraction of dead end and isolated pores B0 ) intrinsic permeability (m2) B0g ) gas permeability (m2) B0l ) liquid permeability (m2) c ) phase molar concentration (kmol m-3) C ) total molar concentration based on the total volume (kmol m-3) d ) density (kg m-3) dM ) molar density (kmol m-3) dm ) molecular diameter (m) dp ) pore diameter (m) D ) Fickian diffusion coefficient (m2 s-1) } ) Maxwell-Stefan diffusion coefficient (m2 s-1) fim ) modified wall friction factor g ) gravitation vector (m s-2) k ) Boltzmann constant (J K-1) M ) molecular mass (kg kmol-1) N ) molar flux (kmol m-2 s-1) NK ) Knudsen number P ) pressure (Pa) Pc ) capillary pressure (Pa) R ) gas constant (8.314 J mol-1 K-1) S ) saturation degree (m3water m-3pores) T ) temperature (K) v ) mass-averaged velocity (m s-1) x ) mole fraction X ) moisture content, dry basis (kgwater kgds-1)
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1451
suggested by Krishna,40 and, subsequently, to get an expression for the effective gas diffusivity.
Greek Letters ) volume fraction φ ) mass fraction γ ) activity coefficient Γ ) thermodynamic correction factor κ ) fractional viscous contribution (s) λ ) mean free path length (m) µ ) dynamic viscosity (Pa s) θ ) coverage F ) bulk density (kgds mtotal-3) τ ) tortuosity
-ct∇T,Px1 )
∑ j)1
xjNi - xiNj Pt^eij
+
Pt Di,K 0e
(
- ∇T,Pxi +
xi
)
∇TPt ) RT
Pt
x
8RT πM1
Pt}ij
+ fim
RT Ni Pt
fim ) (Di,K + B0g/κi)-1
(A.10)
For a binary mixture and in the absence of pressure gradients, we have
-ct∇T,Px1 )
N1 - x1(N1 + N2) + f1mN1 (A.11) }12
(A.2) and substituting eq A.10 into eq A.11, we have
-ct∇T,Px1 ) (A.3)
For a binary mixture, we have
(
∑ j)1
xjNi - xiNj
where
and
p dp τ2 3
n
(A.9)
where
p ^ij τ2
(A.8)
Binary Friction Model. Equation A.9 can be found in work by Kerkhof.11,24
RT Ni
(A.1)
D1,K 0e )
(A.7)
P2 1 1 ) + g Dp Pt}12 D1,K 0
n
^eij )
(A.6)
Considering component 2 as a stagnant gas, it is found that
∇TPt ) RT
Pt
(A.5)
D1,K 0e
1 + x1(1 + N2/N1) 1 1 ) + 1,K g } Dp D0 12
For isothermal gases, assuming ideal gas behavior and no external forces acting on species i, we have the following equations, according to the models discussed before. Dusty-Gas Model. Equation A.1 can be found, for example, in Mason et al.,14 Jackson,19 and Froment and Bischoff.20
)
N1
or for a straight cylindrical pore,
Appendix A. Derivation of the Gas Diffusivity in a Straight Cylindrical Pore
(
+
1 - x1(1 + N2/N1) 1 1 ) + 1,K g,eff e Dp }12 D0e
ds ) dry solid e ) effective eff ) effective exp ) experimental g ) gas K ) Knudsen l ) liquid m ) monolayer mix ) mixture p ) pore P ) constant pressure s ) solid sat ) saturated t ) total T ) constant temperature
- ∇T,Pxi +
}e12
For porous media,
Subscripts and Superscripts
xi
N1 - x1(N1 + N2)
)
x1 - ∇T,Px1 + ∇TPt ) Pt N1 - x1(N1 + N2) RT N1 RT + (A.4) Pt D1,K Pt}e12 0e If we consider that there are no pressure gradients, the molar flux is only due to the concentration gradients and it is possible to compare with Fick’s law, as
N1 - x1(N1 + N2) N1 + 1,K (A.12) }12 D + B0g/κ1
Comparing to Fick’s law, we get
1 - x1(1 + N2/N1) 1 1 ) + 1,K g }12 Dp D + B0g/κ1
(A.13)
Now, considering component 2 as a stagnant gas, it is found that
P2 1 1 ) + 1,K g P } Dp D + B0g/κ1 t 12
(A.14)
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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
Appendix B. Parameters of the Equations in Table 3 ca1 ) 5.052 632 94 ca2 ) 0.813 154 05 ca3 ) -0.002 154 79 A1 ) 1.961 659 A2 ) -7.397 54
A3 ) -8.568 A4 ) -2.399 × 10-4 A5 ) -1.112 × 10-3 A6 ) 16 337 672 A7 ) 8 683 448
A8 ) 1 828 570 A9 ) 190 213 A10 ) 9843 A11 ) 209.6 A12 ) 7.014
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Received for review July 29, 1999 Revised manuscript received February 8, 2000 Accepted February 11, 2000 IE990563N
