Multicomponent Mass Transport in Chemical Vapor Infiltration

Ind. Eng. Chem. Res. 1996, 35, 1275-1287

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Multicomponent Mass Transport in Chemical Vapor Infiltration John Y. Ofori and Stratis V. Sotirchos* Department of Chemical Engineering, University of Rochester, Rochester, New York 14627

The effects of multicomponent mass transport on the predictions of chemical vapor infiltration models are investigated. The dusty-gas model is used as basis for describing multicomponent transport in the pore space of the preforms. Results are presented over a broad range of conditions extending from the bulk to the Knudsen diffusion regime using the full dusty-gas model and a number of variants. These variants are obtained by simplifying the full form of the dusty-gas model and retaining or omitting the viscous flow terms. The presented results demonstrate the importance of using the complete form of the dusty-gas model in most applications and elucidate the implications of some of the simplifying assumptions that are commonly made in describing gaseous mass transport in chemical vapor infiltration. Introduction Gas-solid reactions taking place in porous solids, both catalytic and noncatalytic, are in practice carried out under significant diffusional limitations in the pore space, with much different concentration profiles and mass transport fluxes among the species present in the gas phase. The effects of mass transport limitations are usually more pronounced when the solid undergoes transformation in the course of the reaction since in such a case the resistance to transport of the gaseous species in the porous solid evolves both spatially and temporally. Chemical vapor infiltration (CVI), a process in which chemical vapor deposition (CVD) reactions are used to deposit solid material within the interior of a porous substrate, is a typical example of a process involving solid transformation. When the intrinsic kinetics of the gas-solid reaction is complexsas is the case with the CVD reactions in CVIsinformation on the concentrations of a large number of gaseous species in the pore space of the reacting solid is needed for describing the overall reaction process. As a result, the use of a multicomponent flux model that offers a satisfactory description of the interplay of mass transport fluxes, partial pressures, and partial pressure gradients in the pore space is of paramount importance if the the overall process model is to have useful predictive capabilities. A multicomponent flux model for transport in a porous solid must be capable of representing accurately the behavior of the individual species in the multicomponent mixture, accounting for all mechanisms and interactions that contribute significantly to the mass transport fluxes. It should cover the whole range of diffusion from ordinary (concentration-driven) diffusion (or bulk diffusion) to Knudsen diffusion and viscous (pressure-driven) flow, and it should account for the influence of the structure of the porous solid on the transport process. In chemical vapor infiltration, solid material (ceramic or carbon in typical applications) is deposited on the interior surface of a porous preform from a mixture of gaseous reactants via a mechanism involving several homogeneous and heterogeneous reactions. CVI is usually carried out isothermally at high temperatures (over 1000 °C) and subambient pressures (see Naslain (1992) and references therein), but processes that operate at atmospheric or higher pressures have also been * To whom correspondence should be addressed.

0888-5885/96/2635-1275$12.00/0

developed (Besmann et al., 1991) or proposed (Sotirchos, 1991). The initial average pore size of porous media used as CVI substrates varies from a few tens of microns for woven substrates used in the fabrication of ceramic matrix composites to a few microns or less for substrates obtained from prepregs (resin-impregnated fibrous structures) used for making carbon matrix composites (Savage, 1993). Because of solid deposition, the average pore size changes significantly during chemical vapor infiltration, varying from its initial value to almost zero at the point of complete infiltration. Since the pressure of operation and the average pore size determine in which regime (bulk, transition, or Knudsen) diffusion and flow (Knudsen or viscous) take place, a CVI model of general applicability should be capable of describing diffusion and flow in any regime. Most of the CVI models that have been presented in the literature employ simple flux models, frequently of the type of Fick’s law, to describe the diffusive (concentration-driven) transport in the pore space, and focus only on the species that appear in the kinetics of the deposition reaction. With the exception of CVI processes in which a pressure gradient is intentionally applied across the thickness of the preform (forced-flow CVI or pulse CVI), the contribution of viscous flow to transport is usually omitted even though deposition reactions that proceed through gas phase decomposition reactions, such as the deposition of SiC from CH3SiCl3, are accompanied by an increase in the number of molecules present in the gas phase and, hence, pressure buildup within the pore space of the preform. In his analysis of convection and diffusion in pore networks, Sotirchos (1989) found that viscous flow and the interactions of the fluxes of different species have strong qualitative and quantitative effects on the predicted fluxes. Similar conclusions were reached from a comparison of chemical infiltration results obtained using Fick’s law with results obtained using a multicomponent flux model (Sotirchos, 1991). The importance of describing the multicomponent interactions during gaseous transport in porous media and including the viscous flow contribution to the fluxes has been studied by several other investigators in various contexts. A few examples: Feng et al. (1974) studied the effects of including multicomponent interactions in a flux model in order to satisfactorily predict experimentally observed values of effective diffusivities in catalyst pellets. The effects of multicomponent interactions in catalytic reactions involving porous media were studied by Kaza and Jackson (1980). Sotir© 1996 American Chemical Society

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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996

chos and Amundson (1984a,b) examined the effect of multicomponent diffusion on the pseudosteady-state and transient behavior of porous char particles reacting in an oxygen-containing environment. Sloot et al. (1992) addressed the importance of multicomponent effects on transport during catalysis in membrane systems at high temperature. In all these studies, the general conclusion has been that, unless the chemical reactants exist in very low concentrations, multicomponent transport has strong effects on the predicted behavior of the system, both of quantitative and of qualitative nature. The dusty-gas model (Mason et al., 1967; Mason and Malinauskas, 1983) is the most frequently used flux model for multicomponent gaseous transport in porous media. Application of this model to porous solids under various circumstances is treated at length in monographs by Jackson (1977), Cunningam and Williams (1980), and Mason and Malinauskas (1983). Its formulation is based on describing the solid as a matrix of rigid spheres (the dust), uniformly distributed at the molecular scale and fixed in space, and applying the Stefan-Maxwell equations (Bird et al., 1960) to the (n + 1)-component gas mixture (n gases and the dust). Three parameters depending on the structure of the porous medium are needed for application of the dustygas model: one for bulk diffusion, one for Knudsen diffusion and flow, and a third for viscous flow. The form of these parameters is known for the simple pore structure of uniform in radius, noninteracting, infinitely long pores, but rigorous treatment of other structures (Feng and Stewart, 1973) leads to a very complex set of equations that requires an enormous computational effort for application to mixtures of more than a few species. Using eigenvector-eigenvalue decomposition of the dusty-gas model equations, Sotirchos (1989) showed that if consistent methods are used to determine the three parameters at the limits of bulk diffusion, Knudsen diffusion and flow, and viscous flow, the complex model that results from the rigorous treatment of the dusty-gas model yields results that are very close to those predicted by the simple three-parameter model for most pore structures of practical interest. A detailed investigation of multicomponent and viscous flow effects during isothermal, isobaric chemical vapor infiltration over a broad range of operating conditions is presented in this study. Based on the conclusions of Sotirchos (1989), the three-parameter dusty-gas model is employed to describe multicomponent interaction within the densifying preforms, assuming a representation of randomly overlapping long pores for the pore space. A number of simplified variants of the dusty-gas model are formulated by retaining or omitting viscous transport and all or some of the multicomponent interactions from the full form of the dusty-gas model. These flux models are used in a general transport and reaction model to describe chemical vapor infiltration when SiC deposition from methyltrichlorosilane is employed as the densification reaction, and the effects of the various simplifications of the dusty-gas models on the predictions of the overall model are investigated.

to solid deposition at the solid-gas interface. An energy balance is not required in this treatment since the rate of enthalpy change during the reaction is not enough to cause significant temperature gradients and the solid is assumed to be heated by a heat source placed around it (e.g., a furnace). The mass balance equations for the gaseous species in the porous medium, written in terms of partial pressures pi, have the form

∂(epi/RT) ∂t

∑F νiFRvF

(1)

where e is the accessible porosity of the porous medium, Ni is the molar mass transport flux of the ith species, R is the ideal gas constant, T is the temperature, and νiF is the stoichiometric coefficient of the ith species in the Fth reaction. The volumetric reaction rate RvF (per unit volume of porous medium) of reaction F is given by either of the equations (depending on whether the reaction is homogeneous or heterogeneous)

RvF ) eRF

(2a)

RvF ) SeRsF

(2b)

where RsF is the intrinsic rate of heterogeneous reaction F (per unit area), RF is the rate of homogeneous reaction F (per unit volume of gas phase), and Se is the accessible internal surface area (per unit volume of porous medium). The mass balance of the deposited solid is described by the equations

vS

∂ξ

∑F νSFRvF

(3a)

0 - 0

(3b)

) ∂t

0 ξ)

where ξ is the conversion (fraction of initial pore space filled with solid material), vS is the molar volume of the deposited solid, 0 is the initial porosity of the preform, νSF is the stoichiometric coefficient of the solid in reaction F, and is the total porosity of the structure. Multicomponent Transport Model. We use the dusty-gas model to describe the interaction of fluxes in the pore space. This model considers that the mass transport flux of each species in the porous medium, Ni, consists of additive diffusive and viscous contribuV tions, ND i and Ni . In other words, V Ni ) ND i + Ni

(4)

The equations for the diffusive fluxes are derived by applying the Stefan-Maxwell equations to an (n + 1) -component gaseous mixture, with species (n + 1) representing the immobile “dust” molecules. These equations have the form

Transport and Reaction Model Material Balance Equations. The mathematical model we use is based on the general model of Sotirchos (1991) for CVI under dynamic or pseudosteady-state conditions. It is formulated for the general case in which n gaseous reactants take part in m chemical reactions, with some heterogeneous reactions leading

+ ∇‚Ni )

1 RT

∇pi )

∑ j*i

D xjND i - xiNj

ND i +

Deij

DeKi

(5)

where xi is the mole fraction of the ith species. The effective binary diffusivity Dije for the gas pair (i,j) and the effective Knudsen diffusion coefficient DeKi for the

Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1277

ith gas species are given by the expressions

(p*p )(T*T ) S T ) D* ( ) S T* 3/2

Deij ) D* ij DeKi

1

1/2

Ki

2

(6a) (6b)

where D* ij is the binary diffusion coefficient for the (i,j) pair computed at reference pressure and temperature p* and T*, and D*Ki is the Knudsen diffusivity of the ith species in a capillary of reference radius r* at reference temperature T*. S1 and S2 are structural parameters used to express the effects of the structure of the porous medium on the binary and Knudsen diffusion processes, respectively. D’Arcy’s law is used to describe the relationship between viscous fluxes and total pressure gradient, viz.

NVi

Bepi ∇p )µRT

(7)

where Be is the effective permeability of the porous medium and µ is the viscosity of the gaseous mixture. If the effect of the composition of the gaseous mixture on the viscosity is not significant, µ can be obtained using the equation

(T*T )

1/2

µ ) µ*

(8)

with µ* being the gas mixture viscosity for some representative composition at the reference temperature T*. Using eq 4, eqs 5 and 7 may be combined into a single equation, and this is usually the way in which the three-parameter (S1, S2, and Be) dusty-gas model is presented in the literature (Jackson, 1977; Mason and Malinauskas, 1983; Sotirchos, 1989). With the aid of eqs 6a,b, eq 5 can be rewritten in vector-matrix form as

-

1 T 1/2 S ∇p ) BND RT* 1 T*

( )

(9)

where p and ND are the n-dimensional vectors of partial pressures and component diffusive fluxes respectively for all components, and B is an n × n matrix whose elements are given by

Bij )

Bii )

-pi , j*i D*ijp* pj

∑ j*i D*p* ij

1 S1 T + D*Ki S2 T*

(10a)

(10b)

Like the Stefan-Maxwell equations from which they had been obtained, the equations of the dusty gas-model for the diffusive fluxes are written “wrong-side out” (Bird et al., 1960). They are explicit in the partial pressure gradients, while, since the objective is to use them to find the divergences of the fluxes that appear in the mass balance equations for the gaseous species, one would prefer them to be explicit in the fluxes instead. The analytical inversion of eqs 5, and the subsequent differentiation of the inverted equations, is a formidable task even when only a few species are present in the multicomponent mixture. For this reason, despite their ability to offer a satisfactory representation of multicomponent interactions during gas-

eous diffusion in porous media, the use of the full dustygas model equations in gas-solid reactions studies is usually avoided, and simplified versions that are easier to handlesat least in a first analysissare employed instead. One of the most common approaches is to either neglect the off-diagonal components and, in this way, obtain a readily invertible matrix, or employ a Fick’s law-type equation for each species using an approximate pseudobinary effective diffusivity. Computational Procedure. The model equations given in the transport and reaction model are discretized in space, for one-dimensional transport, using B-spline collocation (De Boor, 1978), generating a set of algebraic and ordinary differential equations. The discretization procedure involves approximating the profiles of partial pressure and conversion with piecewise continuous cubic polynomials over a set of subintervals defined by a chosen sequence of breakpoints. The polynomial approximations of the partial pressure profiles are then required to be continuous and have continuous first derivatives at these breakpoints, and satisfy the mass balance equations at two collocation points in each subinterval and the boundary conditions at the boundaries. The approximation of the conversion profile is required to be continuous and have a continuous first derivative at the breakpoints, and satisfy the solid mass balance at the collocation points and the boundaries. The resulting system of algebraic and ordinary differential equations is integrated in time using a Geartype solver (Gear, 1971). This procedure requires that the divergence of the fluxes be obtained at every spatial and temporal point in the determination of the solution. To do this we utilize a numerical scheme formulated by Sotirchos (1991) which avoids the analytical inversion and subsequent differentiation of the dusty-gas model equations by using numerical inversion of the differentiated form of eqs 9 to obtain the divergences of the fluxes. Since this computational scheme does not involve any simplification of the dusty-gas model or of the mass balance equations, it can be applied to any problem involving transport and reaction in porous media regardless of how many reactive species and how many reactions it involves and what the kinetics of these reactions are. Sotirchos (1991) gives further details of the computational procedure employed. Results and Discussion The results we discuss in this section and use to derive conclusions on the effects of multicomponent mass transport on the predictions of CVI models are obtained for the case where SiC is deposited through decomposition of methyltrichlorosilane (MTS), according to the overall (macroscopic) reaction: H2

CH3Cl3Si 98 SiC + 3HCl

(11)

On the basis of the results of Brennfleck et al. (1984), we assume first order kinetics, viz.

Rs ) ωks0 exp(-E/RT)pMTS

(12)

with

ks0 ) 2.290 × 10-2 kmol/(m2‚s‚atm); E ) 120 × 106 J/kmol ω is a dimensionless factor used to vary the rate of the

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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 Table 1. Equations for Structural and Transport Properties (Sotirchos and Yu, 1985; Sotirchos, 1991)

(

r ) r0

e ) ) 0(1 - ξ) S1 ) /ηb S2 ) r/ηKr*

Be )

reaction from that given by Brennfleck et al. (1984), the latter obtained for ω ) 1. We envisage a slab-shaped preform of thickness 2a placed in the reaction atmosphere, and we use a onedimensional representation of the problem across the thickness of the preform (Figure 1). The pore structure of the preform is assumed to consist of randomly overlapping cylindrical pores of uniform size. The equations giving the evolution of e, Se, S1, S2, and Be with the conversion are given in Table 1, along with the values of other parameters used in the computations. All tortuosity factors, ηb, ηK, and ηv, were taken to be equal to 3. Burganos and Sotirchos (1989) and Tomadakis and Sotirchos (1993) showed that ηb and ηK are smaller than 3 at high porosities, but our study concentrates on the intermediate to low porosity region. The composition of the gas phase on both sides of the preform is assumed to be identical. For negligible mass transfer resistance from the bulk of the gas phase to the external surface of the preform, the boundary conditions at the surface of the preform are written

pi|z)a ) pib

(13)

where subscript “b” refers to the bulk phase around the preform. The standard no-flux condition at the symmetry plane of the preform provides the second set of boundary conditions:

Ni|z)0 ) 0

(14)

The initial conversion and gas partial pressures inside the preform are assumed to be known functions of z:

ξ|t)0 ) ξ(z)

(15a)

pi|t)0 ) pi(z)

(15b)

The time variable given in the results section is scaled so that τ ) 10 corresponds to complete plugging of a capillary of radius r0 ) 20 µm at the ambient conditions used in each case. This leads to the following definition for τ:

τ)

10νSiCks0 exp(-E/RTb)(pMTS)b t r0

(16)

vSiC is the molar volume of SiC. It should be noted that even though the complete transient model has been used to get the densification results we present, essentially

1/2

r2 8ηv(r*)2

Se ) S ) -

Figure 1. Schematic diagram of slab-shaped preform undergoing densification by CVI.

)

ln(1 - ) ln(1 - 0)

2(1 - ) ln(1 - ) r

the same results would have been obtained if a pseudosteady-state model, employing the steady-state form of the mass balances for the gas-phase species, had been used in the computations. The reason is that the time constants associated with changes in partial pressure profiles of the gas-phase species are much smaller than that characterizing changes of the solid density profile (solid mass balance). As a result, with the exception of the very early period of the densification process during which the partial pressure profiles are moving toward steady state, the mass balance equations for the gases are at steady state for all practical purposes. Variants of the dusty-gas model are used to investigate the importance of multicomponent effects in the MTS-SiC CVI system over various operating conditions. In reference to matrix B (eqs 10a,b), we consider the following cases: Case 1: All cross-diffusion terms are retained (full B-matrix). Case 2: All cross-diffusion terms are ignored (diagonal B-matrix, Bij ) 0, j * i). Case 3: MTS cross-diffusion interactions are ignored (Bij ) 0, i or j ) 1, j * i). Case 4: HCl cross-diffusion interactions are ignored (Bij ) 0, i or j ) 2, j * i). Case 5: H2 cross-diffusion interactions are ignored (Bij ) 0, i or j ) 3, j * i). These five cases are examined both in the presence and absence of viscous terms. We will use V after the number of each case to indicate retention of the viscous terms, whereas use of W will refer to the flux model used without viscous terms. For example, 3W will refer to the case where viscous terms and MTS cross-diffusion terms are omitted from the dusty-gas model equations. Effects in the Absence of Viscous Flow. We initially examine the effect of omitting cross-diffusion interactions from the flux model when viscous terms are not included in the dusty-gas model; that is, we compare the results obtained for cases 2W, 3W, 4W, and 5W with the predictions of the complete two-parameter dustygas model (case 1W). Figure 2 shows steady-state partial pressure profiles for the different cases for two fractional conversion profiles: zero conversion (ξ ) 0) everywhere in the preform and a parabolic with respect to distance profile (ξ ) 0.1 + 0.3(z/a)2), where z ) 0 refers to the center and z ) a refers to the surface of the preform. The results shown in the figure lead us to conclude that the most important cross-diffusion interactions are those of HCl with the other components of the mixture. They are followed in the order of decreased importance by those of MTS, whereas the omission or retention of the H2 cross-diffusion terms appears to have practically no effect on the predictions of the overall transport and reaction model.


Figure 2. Comparison of the MTS partial pressure profiles at steady state, without viscous flow. Cases 5W (not shown) and 1W coincide. a ) 0.01 m; xMTS ) 0.1; xH2 ) 0.9; r0 ) 20 µm; pb ) 1 atm; Tb ) 1300 K; ω ) 1.0.

Figure 3. Comparison of MTS partial pressure profiles at three reaction times, without viscous flow. Cases 5W (not shown) and 1W coincide. Reaction conditions same as Figure 2. t ) τ‚4583 s.

The behavior seen in Figure 2 can be explained on the basis of the magnitude of the mass transport fluxes of the three species in the pore space of the densifying medium. Since only one reaction is considered in the particular application of the mathematical model we discuss here and the results shown in Figure 2 refer to the steady-state behavior of the system, the mass balance equation for the gaseous species can be simplified to

∇‚Ni ) νiRv

(17)

Integrating this equation using the symmetry boundary condition at the centerline of the preform gives that the ratio of the fluxes of any two species is equal to the ratio of their generalized stoichiometric coefficients. From the stoichiometry of the overall reaction (eq 11) it follows that the steady-state flux of HCl is 3 times larger in magnitude than the flux of MTS, while the steady-state flux of H2 is zero. Therefore, the omission of the crossdiffusion terms of HCl has the strongest effect on the behavior of the MTS profile because HCl has the largest, in absolute value, molar flux. On the other hand, the omission or retention of the H2 terms has practically no effect because the molar steady-state flux of this species is zero. Since HCl is transported toward the external surface of the preform, its flux has a negative effect on the inward flux of MTS, and as result, the omission of its cross-diffusion terms leads to an increase in the MTS concentration in the interior of the preform. It should be noted that the MTS profiles for cases 2W, 3W, and 4W are very close to each other because in all three cases the main cross-diffusion interaction for MTS, that with HCl, is ignored. Figure 3 presents partial pressure profiles for MTS obtained at three time instants during densification by CVI of a preform with initial uniform porosity of 50%. The fractional conversion profiles at the same time instants are shown in Figure 4. Since the conversion profile changes much slower than the partial pressure profile, the transport and reaction of the gaseous species can be described satisfactorily by the steady-state form

Figure 4. Comparison of conversion profiles at three reaction times, without viscous flow. Cases 5W (not shown) and 1W coincide. Reaction conditions same as Figure 2. t ) τ‚4583 s.

of eq 1. In agreement with the results of Figure 2, it is seen that the MTS profiles for cases 1W and 5W in Figure 3 are almost identical. Also, the differences in the predictions of the model for cases 2W, 3W, and 4W are very small, and this in turn leads to small differences in the predicted ξ profiles in Figure 4. The fractional conversion at the external surface of the preform is the same for all cases since it depends only on the ambient concentration of MTS and the temperature. It is interesting to observe that, at τ ) 8, the MTS profiles of cases 1W and 5W are similar, especially at the center of the preform, to those for cases 2W4W. This is a coincidence brought about by the fact that the increase in the mass transport flux of MTS in cases 2W-4W caused by the omission of the cross-diffusion HCl terms (primarily) is offset by the stronger diffusional limitations for the same cases caused of the higher conversion levels.

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Figure 5. Comparison of HCl partial pressure profiles at three reaction times, without viscous flow. Cases 5W (not shown) and 1W coincide. Reaction conditions same as Figure 2. t ) τ‚4583 s.

Figure 6. Comparison of H2 partial pressure profiles at steady state, without viscous flow. Cases 5W (not shown) and 1W coincide. Reaction conditions same as Figure 2.

The HCl profiles that correspond to Figures 2 and 3 behave in a fashion similar to the MTS profiles. As seen in Figure 5, which presents the HCl concentration profiles that correspond to Figure 3, the HCl profiles for cases 1W and 5W are almost identical and predict lower concentrations at all positions than the HCl profiles for cases 2W-4W, which again differ insignificantly from each other. It is surprising that omission of the HCl or MTS cross-diffusion terms leads to an increase in the concentration of HCl in the interior of the preform. The MTS flux opposes that of HCl, and as a result, ignoring the interaction of these two fluxes should, in principle, decrease the transport resistance that the HCl encounters as it diffuses toward the external surface of the preform and reduce its concentration in the interior. To understand the reasons for this behavior, it is helpful to consider the specific form that the dusty-gas model equations take for the case considered in Figures 2-5 for MTS and HCl. For HCl, we have

of reaction and, therefore, to larger, in magnitude, MTS and HCl fluxes than in cases 1W and 5W. Since the HCl concentration is zero at the external surface, the omission or retention of the cross-diffusion terms (last term in eq 18) has no effect on the partial pressure gradient of HCl there. Therefore, a larger gradient of partial pressure of HCl must prevail in the vicinity of the surface in cases 2W-4W in order to allow removal of the larger quantities of HCl from the interior. H2 concentration profiles are shown in Figure 6 for steady-state reaction and the same conversion profiles as in Figure 2. It is seen that the concentration of H2 is uniform when all cross-diffusion (2W) or the H2 (5W) interactions are ignored. However, significant variation from the external surface concentration is observed when either the MTS or the HCl cross-diffusion terms are neglected. Specifically, concentrations higher than the external surface value are predicted in the absence of HCl interactions (4W), and the opposite situation holds when the MTS terms are dropped. The behavior of the H2 concentration profile can be explained easily on the basis of the equation obtained for the pressure gradient of H2 from the dusty-gas model under pseudosteady-state conditions, which has the form

-

(

)

xH2 xMTS 1 1 ∇pHCl ) NHCl e + e + e RT DH2,HCl DMTS,HCl DK,HCl NMTS (18) xHCl e DHCl,MTS

and an analogous equation is obtained for MTS by interchanging subscripts MTS and HCl in the above equation. To derive eq 18, we have ignored the viscous contribution to the flux and used the fact that the total flux of H2 is almost zero under pseudosteady-state conditions. It follows from eq 18 and its counterpart for MTS that, for fixed values of fluxes and mole fractions, omission of the MTS or HCl cross-diffusion terms should lead to a decrease in the magnitude of the partial pressure gradient for both MTS and HCl. We remarked that this result is consistent with the behavior of MTS but not with that of HCl. The reason for this apparent discrepancy is that the concentration of HCl at the external surface of the preform is zero. The higher concentration of MTS in the interior of the preform in cases 2W-4W leads to higher overall rate

-

(

)

NMTS NHCl 1 ∇pH2 ) -xH2 e + e RT DH2,MTS DH2,HCl

(19)

It is clear from this equation that neglecting all interactions or the H2 interactions only would lead to zero pressure gradient and that, because of the different direction of the MTS and HCl fluxes, omission of the MTS or HCl terms would produce opposite effects. Since NHCl ) -3NMTS and the binary diffusivities of the (H2, MTS) and (H2, HCl) pairs do not differ by more than a factor of 3, eq 19 predicts lower concentrations in the interior than at the external surface when the complete form of the dusty-gas model with no viscous flow (case 1W) is used. Total pressure profiles for the case of Figures 2, 5, and 6 are shown in Figure 7. It is seen there that pressure differences in excess of 10% between the external surface and the preform center can arise in


Figure 7. Comparison of total pressure profiles at steady state, without viscous flow. Reaction conditions same as Figure 2.

Figure 8. Comparison of MTS partial pressure profiles at steady state, with viscous flow included. Cases 4V (not shown) and 2V coincide. Reaction conditions same as Figure 2.

some cases. The behavior of the various pressure vs distance curves can be explained by deriving an equation for the total pressure gradient in the absence of viscous flow by adding the equations for the partial pressure gradients of the three species. When all terms are retained (case 1W), this equation is

-

NMTS NHCl 1 ∇p ) e + e RT DK,MTS DK,HCl

(20)

For the average pore size and total pressure used to obtain the results of Figure 7, the Knudsen diffusivities are much larger than the binary diffusivities, and thus, insignificant total pressure differences develop in the interior of the particles in case 1W. Omission of crossdiffusion leads to the appearance of two extra terms of opposite sign in the right-hand side of eq 20, one involving the flux of MTS and the other that of HCl. According to the results of Figure 7, these terms have the most effect when the HCl interactions are ignored. When the MTS interactions are neglected, the two terms

NMTS

(

xH2

xHCl

e DHCl,MTS

+

e DH 2,MTS

)

(

and NHCl

xMTS

e DMTS,HCl

)

are of comparable magnitude, and as a result, they can lead to zero total pressure gradient at some point in the preform. Effects with Viscous Flow. The main effect of the inclusion of viscous terms in the flux model is a more accurate prediction of the total pressure at any spatial position. The size and relative importance of the viscous flux in the presence of a pressure gradient (such as that induced by a chemical reaction that results in a change in the number of gas molecules) depends on the magnitude of the total mass transport flux and the relative magnitudes of the resistances for diffusive and viscous transport in the preform. The dependence of these resistances on pore size and pressure will be dealt with later. Figures 8-11 present results on the effects of the omission of cross-diffusion terms on partial pressure and solid fraction profiles when viscous flow is accounted

Figure 9. Comparison of HCl partial pressure profiles at steady state, with viscous flow included. Reaction conditions same as Figure 2.

for in the dusty-gas model. The origin of the effects of the omission of cross-diffusion terms on the predictions of the overall transport and reaction model is less clear when viscous fluxes are included since in this case the fluxes and pressure gradients of different species are linked to each other not only through the cross-diffusion terms but through the driving force for the viscous fluxes, the total pressure gradient, as well. It is seen in Figures 8 and 9 that cases 2V, 3V, and 4V do not lead to profiles for MTS and HCl similar to those for the corresponding cases without viscous flow (2W, 3W, and 4W) in Figures 2, 3, and 5. The differences between cases 1V and 5V are larger than those between cases 1W and 5W, but since the total flux of H2 under pseudosteady-state conditions is zero regardless of whether viscous effects are allowed for or not, these differences are still small and practically insignificant. The differences between the various cases are amplified

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Figure 10. Comparison of conversion profiles at three reaction times, with viscous flow included. Cases 4V (not shown) and 2V coincide. Reaction conditions same as Figure 2. t ) τ‚4583 s.

Figure 11. Comparison of H2 partial pressure profiles at steady state, with viscous flow included. Reaction conditions same as Figure 2.

as CVI takes place and densification progresses, and this may be seen in Figure 10, which presents conversion vs distance results at the same time instants as Figure 4 without viscous flow. The effects of the omission of cross-diffusion terms in the presence of viscous flow are shown more clearly in Figure 11, which presents H2 concentration profiles for all five cases. Comparison with Figure 7 shows that the H2 concentration for the cases without any crossdiffusion interactions (2V) and H2 interactions (5V) are not uniform as in cases 2W and 5W. This difference is because of the existence of a viscous flux directed toward the external surface of the particles, which requires that a nonzero H2 partial pressure gradient be present in order to make the total flux of H2 equal to zero. When viscous fluxes are included, the equation for the partial pressure gradient of H2 becomes

-

(

)

NMTS NHCl 1 NV ∇pH2 ) -xH2 e + e + e RT DK,H2 DH2,MTS DH2,HCl

(21)

For cases 2V and 5V, only the first term of the above equation is retained, but case 2V presents larger H2 pressure gradients because of having higher reaction rate (see Figure 8) and, hence, larger viscous flux. Equation 21 can be used to explain the behavior of the pH2 vs distance curves for the other cases as well. For instance, the largest H2 pressure gradients are obtained in case 3V (when the MTS terms are omitted) because the two terms left in the right-hand side of eq 21 are of the same signsthe total viscous flux, NV, has the same direction as the HCl flux. Total pressure profiles corresponding to Figures 8-11 are not shown since in all cases the predicted pressure in the interior of the preform is only slightly higher than that at the external surface. This finding is in sharp contrast with the behavior shown in Figure 7 for the total pressure in the absence of viscous flow. Comparison of Figures 2, 5, and 6 reveals that when all crossdiffusion terms are preserved, the inclusion of viscous terms has insignificant effects on the HCl and MTS profiles, affecting significantly only the H2 partial pressure. This result leads us to conclude that, for CVI

Figure 12. Comparison of MTS partial pressure profiles at steady state. Reaction conditions same as Figure 2.

systems which do not have externally imposed total pressure gradientssas in the case of pressure pulsings and include large concentrations of an inert species, it may be possible to omit the viscous terms from the dusty-gas model without introducing significant errors in the predicted concentrations for the species that participate in the gas-solid reaction. Comparison with the Fick’s Law Model. We now consider a simplified flux model of the Fick’s law form and compare its predictions with those of the complete dusty-gas model with or without viscous terms (1W and 1V) and of the dusty-gas model without cross-diffusion interactions (2W and 2V). Results are given in Figures 12 and 13 for the partial pressure profiles, in Figure 14 for the temporal evolution of the MTS flux at the external surface of the preform during densification, and in Figures 15 and 16 for the evolution of the average conversion.


Figure 13. Comparison of HCl partial pressure profiles at steady state. Reaction conditions same as Figure 2.

Figure 15. Comparison of average conversion of the preform during densification. Reaction conditions same as Figure 2. t ) τ‚4583 s.

The dusty-gas model without viscous terms is given by eq 9, which by ignoring the off-diagonal (crossdiffusion) terms in matrix B, may be written as

1 RT

∇pi ) Ni

(

∑ j*i

xj

1 +

Deij

DeKi

)

(23)

Equation 23 can in turn be expressed in the Fick’s law form of eq 22a by using the effective diffusivity

1 ) Dei

∑ j*i

xj Deij

xji ) xH2 +

Figure 14. Comparison of dimensionless MTS flux into the preform during densification. Reaction conditions same as Figure 2. t ) τ‚4583 s.

In the Fick’s law treatment of the (MTS,HCl,H2) three-component system, each reactive component (MTS or HCl) is assumed to behave as in a binary mixture with the inert component (H2), with MTS and HCl not having any effect on each other. For transport without viscous flow, the Fick’s law model is written

Dei ∇p Ni ) RT i

(22a)

1 1 1 ) e + e e Di Di,H2 DKi

(22b)

where i in the case we consider stands for HCl or MTS. In the presence of viscous flow, eq 22a becomes

Ni ) -

(

)

Bpi 1 ∇p Dei ∇pi + RT µ

(22c)

xji

1 +

) DeKi

∑

j*i,H2

1 +

e Di,H 2

( )

xj

e Di,H 2

Deij

DeKi

(24a)

(24b)

The Fick’s law flux model (eq 22a) differs from the diagonal form of the dusty-gas model (eq 23) only in the form of Dei . It follows from eqs 22b and 24a that if xji > 1, the effective diffusivity in the diagonal dusty-gas model is smaller than that in the Fick’s law expression. Equation 24b gives that this happens for MTS when xHCl e e > (DMTS,HCl /DMTS,H )(1 - xH2), and for HCl when xMTS > 2 e e (DHCl,MTS/DHCl,H2)(1 - xH2). For 90% H2 in the gas phase and the reaction conditions that are typically used in SiC CVI, these inequalities require that the mole fraction of HCl be greater than about 0.02 and that of MTS be larger than about 0.01. Unless deposition occurs at a very high rate and very large concentration gradients develop within the densifying medium, the constraint on MTS is satisfied at all positions in the preform. On the other hand, unless the reaction proceeds at a very low rate and the concentration of HCl in the preform is kept at a low level, the constraint on HCl is expected to be satisfied away from the immediate neighborhood of the external surface where the HCl concentration is assumed to be zero. The behavior of the concentration profiles of MTS and HCl in Figures 12 and 13 is in agreement with these observations. For HCl, the diagonal dusty-gas model

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Figure 16. Comparison of average conversion of the preform during densification. Reaction conditions same as Figure 2, except xMTS ) 0.3; xH2 ) 0.7. t ) τ‚1527 s.

without viscous terms (2W) predicts higher diffusion resistance and, therefore, higher HCl concentrations in the interior than the corresponding Fick’s law model throughout the preform. In the case of MTS, the two profiles intersect each other close to the external surface indicating that the diffusion resistance predicted by the diagonal dusty-gas model is higher only away from the external surface of the preform. We saw in the previous section that allowing for viscous terms has no significant effects on the results obtained from the complete dusty-gas model, but affects considerably the predictions of its simplified versions. Figure 13 shows that the latter also happens for the Fick’s law flux model. Inclusion of viscous terms facilitates the outward movement of the HCl molecules, leading to smaller partial pressure gradients and HCl partial pressures. For the diagonal dusty-gas model, inclusion of viscous terms leads at the conditions of Figure 13 to HCl profiles that are in very good agreement with those of the complete dusty-gas model. However, as can be verified using the results of Figure 14, which presents the evolution of the MTS flux at the external surface in time, this does not happen for the MTS partial pressure profile. The only reactant that we consider in the present application of the model is MTS, and therefore, its flux at the external surface of the preform is a measure of the overall deposition rate in the interior. The results of Figure 14 show that the relative differences between the overall reaction rates that are predicted by the various models change both quantitatively and qualitatively as densification progresses. These changes result not only from the different multicomponent interactions of each model but also from the different evolution of the conversion profile in each case. Average conversion vs time profiles that correspond to the various cases of Figure 14 are shown in Figure 15. Relatively large differences are seen to exist among the predictions of different flux models. For the two extreme cases, the complete dusty-gas model and the model with Fick’s law and no viscous terms, the difference in the average conversion levels at large times is about 30%, and a much larger difference exists between

the reaction times needed, according to these models, to reach high average conversion levels. Since the effects of the multicomponent interactions become stronger as the mole fractions of the active species (reactants and products) in the reactive mixture increase, an increase in the concentration of MTS in the exterior of the preforms leads to larger differences among the different models. This behavior is seen clearly in the results of Figure 16, which presents results at the same reaction conditions as Figure 15 but with 30% MTS in the surrounding gas phase. About 80% difference in the average conversion level is exhibited by the two extreme cases of Figure 16, which are the same as the extreme cases of Figure 15. Comparison of the intermediate cases in the two figures shows that an increase in the concentration of MTS in the gas phase does not affect all curves in the same way, and it can therefore lead to changes in the qualitative differences among the various models as well. It should be noted that, at a given reaction time, the conversion at the external surface of the preform is the same for all models in each of Figures 15 and 16 (see Figures 4 and 10). Effects at Lower Pressures. Thus far, we have examined the effects of multicomponent interactions at a total pressure of 1 atm, where the primary mode of diffusion is bulk diffusion. In this section, we present results at a total pressure of 0.01 atm and use them to reach some conclusions on the effects of the pressure of operation on the importance of multicomponent interactions. Since operation at lower pressure leads to a decrease of the resistance for intraparticle mass transport, we have increased the preexponential factor of the reaction rate expression in order to obtain relative partial pressure gradients comparable to those seen at 1 atm. To do so, we used a reference case the simple flux model of Fick’s law without viscous terms. For this model, the relative intraparticle partial pressure gradients are determined by the value of the Thiele modulus, Φ2, defined as

Φ2 ) a2

RTbωks0 exp(-E/RTb)S e DMTS

(25)

where a is the half-thickness of the preform, Tb is the temperature in the bulk, and DeMTS is the effective diffusivity of MTS defined in equation (22b). Application of eq 25 to 0.01 and 1 atm total pressure gives that, for the parameter values used in the present study, ω ) 12.4 must be employed at 0.01 atm in order to obtain an initial Thiele modulus identical to that found at 1 atm for ω ) 1. Results for the variation of the MTS partial pressure and conversion profiles at 0.01 atm are shown in Figures 17 and 18, respectively. In contrast to the behavior seen at 1 atm, the retention or omission of viscous terms has practically no effect on the model results. Moreover, the differences between the complete dusty-gas model and the model without H2 interactions (cases 1V/1W and 5V/ 5W) are negligible, and the same is true for the differences among the predictions of the model for cases 2, 3, and 4. The differences between the two groups of curves (1 and 5) and (2, 3, and 4) decrease as the conversion increases, the average pore size decreases, and the diffusion process moves closer to the Knudsen diffusion regime, where each species diffuses independently of the other. On the basis of the results of Figures 17 and 18, it is reasonable to conclude that for infiltration at very low pressures, use of a simplified


Figure 17. Comparison of MTS partial pressure profiles at three reaction times during densification at low pressure. Cases 5V/W (not shown) and 1V/W coincide; cases 3V/W (not shown), and 4V/W (not shown) coincide with case 2V/W. Reaction conditions same as Figure 2 except pb ) 0.01 atm, ω ) 12.4. t ) τ‚36959 s.

Figure 18. Comparison of conversion profiles at three reaction times during densification at low pressure. Cases 5V/W (not shown) and 1V/W coincide; cases 3V/W (not shown) and 4V/W (not shown) coincide with case 2V/W. Reaction conditions same as Figure 17. t ) τ‚36959 s.

model, such as Fick’s law without viscous transport, can satisfactorily describe the behavior of the reactive system. We can use the graph shown in Figure 19 to better understand the effects of decreasing pressure. This graph presents the variation with the pore size of the various transport coefficients that are encountered in the transport of MTS in an MTS-H2 mixture at 0.01 and 1 atm. Bep/µ is the transport coefficient for viscous flow, and (Bep/µ + DeK,MTS) the coefficient of mass transport of MTS under the influence of a total pressure gradient (that is, because of Knudsen flow and viscous flow). The other coefficients have the same definitions as in eq 22b. For 20 µm pores and at 1 atm pressure, the effective diffusion coefficient is practically identical

Figure 19. Variation of transport coefficients with pore radius at 0.01 and 1.0 atm pressures. T ) 1300 K; ) 0.5.

to the binary diffusion coefficient of MTS in H2, e , suggesting that the total resistance for difDMTS,H 2 fusive transport in the preform is primarily controlled by molecular diffusion, that is, molecule-molecule collisions. At 0.01 atm, the effective diffusion coefficient is very close to the Knudsen diffusion limit, and therefore, it is the frequency of molecule-wall collisions that determines the overall diffusion resistance. Since molecular diffusion is the source of cross-diffusion interactions in the dusty-gas model, one would expect the effects of the various simplifications of the dustygas model to diminish with decreasing contribution of molecular diffusion to the total diffusion resistance, that is, with decreasing pressure. The results of Figure 19 may also be used to explain the different effects of viscous flow at the two pressure levels. Since the Knudsen diffusion coefficient varies proportionally to the average pore size and the viscous transport coefficient varies proportionally to the pressure and to the square of the average pore size, the coefficient for pressure-driven flow, that is, the sum of these two coefficients, approaches the former at low pressures and small pore sizes and the latter in the large pore size and high pressure region. For 20 µ m pores, the coefficient for viscous flow is much larger than the Knudsen diffusivity at 1 atm, and the opposite is the case at 0.01 atm. As a result, neglecting the viscous flow terms has practically no effect on the transport process at 0.01 atm but affects significantly the predictions of some of the simplified flux models at 1 atm. As we pointed out during the discussion of the effects of multicomponent interactions in the presence of viscous flow, the consideration of viscous transport terms in the complete dusty-gas model does not produce noticeable changes in the predicted results at 1 atm. This behavior is a consequence of the fact that even though the viscous transport coefficient is much larger than the Knudsen diffusion coefficient, the latter is in turn much larger than the binary diffusion coefficient, and as a result, even without viscous flow, relatively small total pressure gradients maintain the net convective flow toward the external surface of the preforms (see Figure 7).

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Summary and Further Remarks A comprehensive investigation of the effects of multicomponent mass transport interactions on the predictions of chemical vapor infiltration models was carried out in this study. The dusty-gas model was used to describe the interplay of mass transport fluxes, partial pressure gradients, and partial pressures of all species present in the reacting gaseous mixture in the pore space of the densifying structure. The effects of multicomponent transport on the predictions of the overall chemical vapor infiltration model were examined by considering a number of variants of the dusty-gas model, obtained by omitting or retaining some of the crossdiffusion interactions and the viscous transport terms. The deposition of SiC through methyltrichlorosilane (MTS) decomposition under conditions of diffusiondriven mass transport (isobaric CVI) was used as a model system, and the chemical vapor infiltration model was employed to investigate the densification of preforms whose initial pore structure can be represented by a randomly overlapping population of cylindrical capillaries of uniform size. The obtained results revealed that significant differences may exist between the predictions of the transport and reaction model that uses the complete dusty-gas models and the models that are based on its simplified variants. These differences increase with increasing concentration of reactant (MTS) and increasing pressure of operation, a result of increasing mass diffusion interactions among different species. Even with only 10% MTS in the feed, differences as high as 30% may exist between the times needed to reach a certain densification level according to the complete dusty-gas model and according to some of the simplified models. For reactions at conditions that lead to transport in the Knudsen diffusion regime, such as small pores and low pressure, mass transport interactions among different species are essentially absent, and as a result, even a simple flux model of the Fick’s law type provides a satisfactory description of the infiltration process. However, preforms typically employed in CVI applications are characterized by average pore sizes as large as a few hundred microns, and therefore, very low pressures of operations are needed to move the diffusion process in the Knudsen regime. The inclusion of viscous terms was found to affect strongly the predictions of the simplified variants of the dusty-gas model and of the Fick’s law model but rather insignificantly the results of the dusty-gas model itself, even when under conditions where cross-diffusion interactions are important (large pores or high pressures). This behavior is a consequence of the fact that in the complete dusty-gas model pressure-driven flow is accounted for by both Knudsen flow and viscous flow and that even though the Knudsen diffusion coefficient of each species is smaller than the coefficient for viscous transport for large pores or high pressures, both are much larger than the molecular diffusion coefficients. In the absence of externally imposed pressure gradients on the preform, mass transport in the preform arises because of reaction-induced concentration gradients, and therefore, it is the molecular diffusion coefficients that determine the magnitude of the mass transport fluxes. It should be noted that when the mass transport fluxes in the preform are determined not by the reaction-induced concentration gradients but by externally imposed pressure gradientssas is the case in forcedflow CVI or pulse CVIsthe viscous terms must be included in the complete dusty-gas model in order to

Figure 20. Comparison of MTS partial pressure profiles at three time instants during instantaneous pressurization. Reaction conditions same as Figure 2. t ) τ′‚0.2273 s.

be able to obtain a satisfactory description of the process. The only exception is when the process occurs well inside the Knudsen diffusion regime, in which case the coefficient for viscous transport becomes much smaller than the Knudsen diffusion coefficients. Some results regarding this observation in the case of pulse CVI are shown in Figure 20, which presents MTS partial pressure profiles at three time instants during instantaneous pressurization of a preform from 0 to 1 atm at ξ ) 0 and for the same values of model parameters as in Figure 2. We see that at transient conditions (first two time instants) the predictions of the complete dusty-gas model with viscous flow differ more from those of its counterpart without viscous flow than from those of the diagonal dusty-gas model with viscous terms. However, when the system reaches pseudosteady state, τ ) 3 (also see Figure 2), viscous flow has practically no effects on the results of the complete dusty-gas model. Acknowledgment This work was supported by a grant from the National Science Foundation. Some of the work was done while S. V. Sotirchos was visiting the Laboratory for Thermostructural Composites at the University of Bordeaux (France). Nomenclature Symbols that do not appear here are defined in the text. a ) half-thickness of the preform (m) Be ) effective permeability of the porous medium (m2) Dei ) effective diffusion coefficient of species i (m2/s) Dij, Dije ) binary diffusivity of the (i,j) pair and effective binary diffusivity, respectively (m2 /s) e ) Knudsen diffusivity of species i in a pore and DKi, DKi effective Knudsen diffusivity D*K ) vector of reference Knudsen diffusivities of species in a pore for reference radius and temperature r* and T*, respectively E ) activation energy (J/kmol) ks0 ) preexponential factor of the reaction (kmol/m2‚atm‚s N ) vector of mass transport fluxes Ni ) mass transport flux of species i (kmol/(m2‚s))

Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1287 p ) total pressure of the mixture (atm) pi ) partial pressure of species i (atm) p ) vector of partial pressures r0 ) initial pore or capillary radius (m) r* ) reference capillary radius (m) R ) ideal gas law constant (J/(kmol‚K)) RF ) rate of homogeneous reaction F per unit volume (kmol/ (m3‚s)) RvF ) rate of reaction F per unit volume of porous medium (kmol/(m3‚s)) RsF ) rate of heterogeneous reaction F per unit surface (kmol/(m2‚s)) S ) internal surface area (m2/m3) t ) time (s) T ) temperature (K) vi ) molar volume of solid i (S or SiC) (m3/kmol) xi ) mole fraction of species i x ) vector of mole fractions z ) distance variable (m) Greek Letters ) porosity ηj ) tortuosity factor (see Table 1) µ ) viscosity of the gaseous mixture (kg/(m‚s)) νiF ) stoichiometric coefficient of species i in reaction F ξ ) conversion defined in eq 3b ω ) multiplicative term in pre-exponential factor (eq 12) τ ) dimensionless time used in isobaric long-time CVI analysis τ′ ) dimensionless time used in dynamic short-time CVI analysis Subscripts b ) denotes quantities in the gas or bulk phase around the preforms i ) denotes quantities referring to species i K ) refers to Knudsen quantities S ) denotes quantities referring to the solid product 0 ) denotes quantities referring to the unreacted preform or to t ) 0 Superscripts D ) refers to diffusive quantities or transport mechanism e ) denotes effective quantities V ) refers to viscous quantities or transport mechanism * ) denotes reference quantities Other MTS ) methyltrichlorosilane CVI ) chemical vapor infiltration

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Burganos, V. N.; Sotirchos, S. V. Analysis of Multicomponent Diffusion in Pore Networks. AIChE J. 1988, 34, 1106-1118. Burganos, V. N.; Sotirchos, S. V. Knudsen diffusion in parallel multidimensional or randomly oriented capillary structures. Chem. Eng. Sci. 1989, 44, 2451-2462. Cunningham, R. E.; Williams, R. J. Diffusion in Gases and Porous Media; Plenum: New York, 1980. De Boor, C. A Practical Guide to Splines; Springer-Verlag: New York, 1978. Feng, C.; Stewart, W. E. Practical models for isothermal diffusion and flow of gases in porous solids. Ind. Eng. Chem. Fundam. 1973, 12, 143-147. Feng, C. F.; Kostrov, V. V.; Stewart, W. E. Multicomponent Diffusion of Gases in Porous Solids. Models and Experiments. Ind. Eng. Chem. Fundam. 1974, 13, 5-9. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice Hall: Englewood Cliffs, NJ, 1971. Jackson, R. Transport in Porous Catalysts; Elsevier: New York, 1977. Kaza, K. R.; Jackson, R. Diffusion and reaction of multicomponent gas mixtures in isothermal porous pellets. Chem. Eng. Sci. 1980, 35, 1179-1187. Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 1973, 45, 574-588. Krishna, R. Problems and pitfalls in the use of the Fick Law formulation for intraparticle diffusion. Chem. Eng. Sci. 1993, 48, 845-861. Mason, E. A.; Malinauskas, A. P. Gas Transport in Porous Media: The Dusty-Gas Model; Elsevier: New York, 1983. Mason, E. A.; Malinauskas, A. P.; Evans, R. B., III. Flow and diffusion of gases in porous media. J. Chem. Phys. 1967, 46, 3199. Naslain, R. In Ceramic Matrix Composite; Warren, R., Ed.; Chapman and Hall: Glasgow, 1992; Chapter 8. Sloot, H. J.; Smolders, C. A.; van Swaiij, W. P. M.; Versteeg, G. F. High Temperature Membrane Reactor for Catalytic Gas-Solid Reactions. AIChE J. 1992, 38, 887-900. Sotirchos, S. V. Multicomponent Diffusion and Convection in Capillary Structures. AIChE J. 1989, 35, 1953-1961. Sotirchos, S. V. Dynamic modeling of chemical vapor infiltration. AIChE J. 1991, 37, 1365-1378. Sotirchos, S. V.; Amundson, N. R. Dynamic behavior of a porous char particle in an oxygen-containing environment, Part 1: Constant particle radius. AIChE J. 1984a, 30, 537-548. Sotirchos, S. V.; Amundson, N. R. Dynamic behavior of a porous char particle in an oxygen-containing environment, Part 2: transient analysis of a shrinking particle. AIChE J. 1984b, 30, 548-556. Sotirchos, S. V.; Yu, H. C. Mathematical modeling of gas-solid reactions with solid product. Chem. Eng. Sci. 1985, 40, 20392052. Stinton, D. P.; Besmann, T. M.; Lowden, R. A. Advanced Ceramics by Chemical Vapor Deposition techniques. Ceram. Bull. 1988, 67, 350-355. Tomadakis, M. M.; Sotirchos, S. V. Effects of fiber orientation and overlapping on Knudsen, Transition and ordinary regime diffusion in fibrous substrates. AIChE J. 1991, 37, 1175-1186. Tomadakis, M. M.; Sotirchos, S. V. Transport properties of random arrays of freely overlapping cylinders with various orientation distributions, J. Chem. Phys. 1993, 98, 616-626.

Received for review May 31, 1995 Accepted December 20, 1995X IE9503252

X Abstract published in Advance ACS Abstracts, March 1, 1996.

Multicomponent Mass Transport in Chemical Vapor Infiltration

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