Diffusive Plus Convective Mass Transport Through Catalytic Membrane Layer with Dispersed Nanometer-Sized Catalyst

Mass transfer rates across catalytic membrane interfaces accompanied by first-order, irreversible reactions have been investigated. The catalyst particles impregnated in the membrane matrix are assumed to be very fine, nanometer-sized particles which are uniformly distributed in the structure of the membrane layer. Pseudo-homogeneous models have been developed to describe mass transport through this catalytic membrane layer. The models developed include the mass transport into and inside the catalytic part icles as well as through the membrane layer taking into account convective and diffusive flows, so it is also valid in the limiting cases namely without convective flow (Pe=0), or with very large convective flow (Pe >> 1). The models describe two operating modes (with and without sweep phase on the permeate side of the catalytic membrane layer) and apply two different boundary conditions for the feed boundary layer. One of the boundary conditions approaches the diffusive flow by the Fickian one assuming linear concentration distribution while the other one solves exactly the mass transport in the feed boundary layer. The different model results obtained are compared to each other proving the importance of the carefully decision of the operating modes and boundary conditions. The mathematical model has been verified by means of experimental data taken from the literature.


Introduction
The catalytic memb rane reactor as a pro mising novel tech no log y is widely reco mmend ed fo r carry ing out heterogeneous reactions. A number of reactions have been in v es tig ated b y mean s o f th is p ro ces s , s u ch as dehydrogenation of alkanes to alkenes, partial o xidation reactions using inorganic or organic pero xides, as well as p artial hy d rog en atio ns , h yd ratio n , etc. As catalytic memb rane reactors for these reactions, intrinsically catalyt ic memb ranes can be used (e.g. zeolite or metallic memb ranes) or me mb ranes that have been made catalytic by dispersion or imp regnation of catalytically act ive part icles such as metallic co mplexes, metallic clusters or activated carbon, zeo lite p art icles, et c. throughout dense po ly meric-or inorganic membrane layers [1]. In the majority of the above experiments, the reactants are separated from each other by the catalytic membrane layer. In this case the reactants are absorbed into the catalyt ic memb rane matrix and then trans po rted by d iffus io n (and co nv ectio n ) fro m the memb rane interface into catalyst particles where they react. Mass transport limitation can be experienced with this method, which can also reduce selectivity. The application of a sweep gas on the permeate side dilutes the permeating component, thus increasing the chemical reaction gradient and the driving force for permeation (e.g. see Westermann and Melin [2]). At the present time, the use of a flow-through catalytic membrane layer is recommended more frequently for catalytic reactions [2]. If the reactant mixtu re is forced to flo w through the pores of a membrane which has been impregnated with catalyst, the intensive contact allows fo r h igh catalytic activity with negligib le diffusive mass transport resistance. By means of convective flow the desired concentration level of reactants can be maintained and side reactions can often be avoided (see review by Julbe et al. [3]). When describing catalytic processes in a memb rane reactor, therefore, the effect of convective flow should also be taken into account. Yamada et al. [4] reported isomerizat ion of 1-butene as the first application of a catalytic membrane as a flo w-through reactor. Th is method has been used for a nu mber of gas-phase and liquid-phase catalytic reactions such as VOC decomposition [5], photocatalytic oxidation [6], part ial oxidation [7], part ial hydrogenation [8][9][10] and hydrogenation of nitrate in water [11].
Fro m a chemical engineering point of view, it is important to predict the mass transfer rate of the reactant entering the memb rane layer fro m the upstream phase, and also to predict the downstream mass transfer rate on the permeate side of the catalytic memb rane as a function of the physico-chemical parameters. If this transfer (permeat ion) rate is known as a function of the reaction rate constant, it can be substituted into the boundary conditions of the differential mass balance equations for the upstream and/or the downstream phases. Basically, in order to describe the mass transfer rate, a heterogeneous model can be used for larger particles and/or a pseudo-homogeneous one for very fine catalyst particles [12]. Both approaches, namely the heterogeneous model fo r larger catalyst particles and the homogeneous one for submicron part icles, have been applied for mass transfer through a catalytic membrane layer. Nagy [12] has analysed diffusive mass transport through a membrane layer with dispersed catalyst particles. It was shown that both the heterogeneous and the pseudo-homogeneous models give practically the same results in the very fine, sub-micro meter particle size range. The convective velocity was not taken into account in Nagy's model [12] cited. Recently Nagy analysis the effect of the convective velocity on the enzyme catalysed reaction [13] as well as summarizes the most important mass transport equations of a memb rane layer taking also into account the simultaneous effect of the convective and diffusive flows [14,15]. These papers extend previous investigations by including the effect of convective flow, applying two different operating modes, namely with and without sweep phase on the permeate side as well as two different models, namely an approaching and the exact models. Mathematical equations have been developed to describe the simultaneous effect of diffusive flow and convective flow and this paper analyses mass transport and concentration distribution by applying the model developed.
The pseudo-homogeneous model will be presented in detail, assuming that the catalyst particles are in the nanometer-sized range, as this is the case in most catalytic memb rane reactors.
The main purpose of this paper is to present the various steps of the mathemat ical solutions, as well as to study the effect of mass transport parameters of the catalyt ic memb rane layer on the mass transfer rates. At the end of this paper, the predicted data are co mpared with measured ones taken fro m the literature. The method presented in this paper can also be applied to higher order chemical reactions.

Theory
In this section different mathemat ical models will be shown and solved in order to describe the mass transport through catalytic memb rane layer with forced flow through it. It is assumed that the catalyst particles are very fine particles with size less than 1 μm. Thus, the so called pseudo-homogeneous model [12,14,15] was applied for description of the mass transport through the catalytic me mb rane layer. The catalyst particle can be porous one (e.g. zeo lite part icles), or dense one without diffusivity inside it (e.g. metal clusters). Accordingly, d ifferent mass transfer rate equations can be defined between the memb rane and particles as will be shown in this section.
In order to increase the efficiency of the catalytic memb rane, the size of particles chosen should be as low as possible [12]. The use of nanometer-sized catalyst particles is thus recommended. The differential mass balance equation for diffusive and convective flows in the catalytic membrane layer, for unsteady-state, can be as: with init ial and boundary conditions as: if t = 0, y > 0 then C=C o if t > 0, y = 0 then C=C * if t > 0, y = δ then C=C o The functions Q provides the specific source term induced by the mass transport into the catalytic nanoparticles distributed uniformly in the catalytic memb rane layer. The mass transfer rate into the catalytic particle can depend on the external mass transfer resistance around the particles as well as on the internal transport accompanied by chemical reaction. Note that in the case of nanometer-sized part icles the diffusion time, t D , (t D =R 2 /D p ) can be very small, thus, it can easily be much shorter than the reaction t ime, t r (t r =1/k 1 for first-order reaction). Th is fact should be taken into account to define the Q value. E.g. for d p =100 n m and D p =1 x10 -12 m 2 /s, the diffusion time is equal to 0.01 s. In the slow reaction rate regime, , the saturated concentration, C * , can exist throughout the particle, i.e. if t r < 9 s according to the above example. In this case the so called effect iveness factor is considered to be unit and accordingly the internal mass transport can be regarded to be instantaneous.

Mass Trans port into the Dis persed Nanometer-Sized Catal yst Particles
Two important cases will be discussed, namely instantaneous and finite internal mass transport rates. Both cases can be important when chemical react ion takes place inside of the particles or on the particle surface. As chemical reaction, the first-order one will be discussed. Several non-linear reactions can be approached by first-order one dividing the membrane layer into thin sub-layers as will be shown in the Appendix. Accordingly these reactions can be handled as first-order ones.

Mass Transport Rates with Instantaneous Internal
Mass Transport The chemical reaction takes place on the internal interface of the catalyst particles with the reaction rate: In Eq. (2) the external mass transfer resistance is neglected, thus the interface concentration of the catalytic particles is equal to the "bulk" one in the membrane matrix.
In the case of inorganic catalyst particles, as zeo lite, the Lang muir-Hinshelwood kinetics the most co mmon ly used kinetic exp ression to explain the heterogeneous catalytic process [16]. Assuming that the react ion of a co mponent occurs in a simple unimo lecular elementary reaction step and that the kinetics are first order with respect to the surface concentration of the adsorbed reactant, the reaction rate can be expressed as [16]: This reaction is often approximated to first-order kinetics for condition KC << 1 or to zero-order kinetics for condition KC >> 1. Similar kinetic expression can be applied for biochemical reactions according to the Michaelis-Menten or Monod kinetics: Replacing Eqs. (3) into Eq. (1), the d ifferential equation obtained for steady-state case can only be solved by analytical approach (this will be shown in Appendix) or numerically. How the Eqs. (3) can be linearized is given by Eq. (A1) in the APPENDIX.

Mass Transport with Finite Internal Mass Transport
It the reaction rate is fast then there can be strong concentration gradient inside of the particle, thus the internal transport should also be taken into account. The internal specific mass transfer rate in spherical particles, J p , for steady-state conditions and when mass transport is accompanied by a first-order chemical reaction, can be given according to Bird et al. [17] as follows: The external mass transfer resistance through the catalyst particle depends on the thickness of the diffusion boundary layer, δ p . The value of δ p can be estimated fro m the distance between particles [12]. As this value is limited by neighboring particles, the value o f β p will be so mewhat higher than that calculated from the well known equation, name ly 2 = β p d p / D p , where the value of δ p is assumed to be infinite. Th is results in: Fro m Eqs. (5) and (6), for the mass transfer rate with overall mass transfer resistance with H p = C p /C: Thus, the value of Q will be as: where ω is the catalyst surface per memb rane volume, m 2 /m 3

Reaction Occurs on the Outer Interface of the Catalyt ic Particles
It often might occur that the chemical react ion takes place on the interface of the particles, e.g. in cases of metallic clusters, the diffusion inside the dense particles are negligibly. Assuming the Henry's sorption isotherm of the reacting component onto the spherical catalytic surface (CH f = q f ), applying Dd C/dr = k f H f C boundary condition at the catalyst's interface, at r = R, the Φ reaction modulus can be given for first-order reaction, as follows[see Eq. (11) for Φ]: where k f is the interface reaction rate constant. The above model is obviously a simp lified one.

Mass Trans port in the Catalytic Membrane Layer
Taking into account Eqs. (2) or (8) as source term, one can get simple first-order kinetics. The differential mass balance equation for the polymeric or macroporous ceramic catalyt ic memb rane layer, fo r steady-state, taking both diffusive and convective flo w into account, can be given, according to Eq.
(1), as: where υ denotes the convective velocity, D is the diffusion coefficient of the membrane, and δ is the membrane thickness. The membrane concentration, C is given here in a unit of measure of g mol/ m 3 . This can be easily obtained by means of the usually applied in the e.g. g/g unit of measure with the equation of C = wρ/M, where w concentration in kg/kg, ρ -membrane density, kg/m 3 , M-mo lar weight, kg/mo l.
The general solution of Eq. (13) is well known [14], so the concentration distribution in the catalytic memb rane layer can be given as follows:

Pẽ
The inlet and the outlet mass transfer rate can easily be expressed by means of Eq. (14). The overall in let mass transfer rate, namely the sum of the diffusive and convective mass transfer rates, is given by: The outlet mass transfer rate is obtained in a similar way to Eq. (15) fo r Y=1: The value of parameters T and S can be determined fro m the boundary conditions which can vary according to the operating conditions used. Assuming that the external mass transfer resistance on the feed side, namely the diffusive resistance, is not negligible, the simultaneous effect of the flu id boundary layer should also be taken into account. Basically, the effect of the external, diffusive mass transfer resistance, in presence of convective flow can be described by two models: (i) the diffusive mass flow is regarded to be constant through the boundary layer (this diffusive flow is called as Fickian one in this paper), accordingly the sum of the diffusive and convective flows varies in the boundary layer (Model A) or (ii) that it varies in the boundary layer, according to the change of the convective mass flow due to the curvature of the concentration in the boundary layer (Model B). This latter one should be regarded as exact solution, namely the su m of the d iffusive and convective flows is constant throughout the boundary layer. According to the mass transfer conditions of the permeate phase, namely there is a sweep phase or there is not sweep phase, two operating modes will be distinguished according to Figs. 1a and 1b. The essential d ifference between the models is that there is a sweep phase that can remove the transported component fro m the downstream side providing the low concentration of the reacted component and its high d iffusive mass transfer rate (d C/d Y > 0; Models A 1 and B 1 ). There is no sweep phase, thus the outlet phase is moving by convective flow fro m the memb rane due to the lower pressure on the permeate side and there is no diffusive flow, on the outlet membrane interface (d C/d Y = 0; Models A 2 and B 2 ).

Mass Transport Models with Fickian Diffusive Flow in the Boundary Layer (Approaching Solution, Models A)
The simultaneous effect of the memb rane and the boundary layers, on the mass transport, is taken into account. In presence of the convective flo w, the overall mass transfer rate will be the sum of the d iffusive and convective flows. Regarding the effect of the boundary layers on both sides of the membrane, the boundary conditions can be different on the feed side and permeate side, depending on the operating mode. Note that the Fickian diffusive flow along the diffusion path[as it is given in Eqs. (17) and (18)] assumed that the concentration distribution is linear, the concentration gradient is constant, in the boundary layer, independently of the presence of convective flow. Accordingly, the sum of the diffusive and the convective flow will change throughout the boundary layer due to the concentration change. In the reality, the concentration curve will be concave one due to the convective velocity thus, the sum of the diffusive and the convective flo w remains constant as a function of the local coordinate in the boundary layer.
Model A 1 (dC/dY>0 at Y=1). In this case, due to the effect of the sweeping phase, the external mass transfer resistance β is gradually dimin ished as the catalytic reaction rate increases. The concentration distribution in the catalytic memb rane, when applying a sweep phase on the two sides of the memb rane, is illustrated in Figure 1a. On the upper part of the catalytic membrane layer, in Fig. 1a, the fine catalyst particles are illustrated with black dots. It is assumed that these particles are homogeneously distributed in the memb rane matrix. Due to sweeping phase, the concentration of the bulk phase on the permeate side may be lower than that on the membrane interface. The value o f o L C here denotes the liquid or gas phase concentration on the bulk phases (see Fig. A1), on both sides of the catalytic membrane layer. The boundary conditions can be given for that case as: Boundary conditions given by Eqs. (17) and (18) [14].
Model A 2 (dC/dY=0 at Y=1). For the convective flow catalytic membrane reactor operating in another mode, for instance in dead-end mode as in Figure 1b, the boundary condition on the permeate side of the memb rane should be changed. In this case the concentration of the permeate phase does not change during its transport from the outer memb rane interface, and consequently dC/dY = 0. If there is no sweeping phase on the downstream side then the correct boundary conditions will be as[the value of J is defined by Eq. (15)]: After solution one can get as: Thus, the physical mass transfer rate, through the boundary layer, can be obtained, by means of Eq. (33), taking into account both the diffusive and the convective flows[see Eq. (15)] as follows: Knowing the mass transfer rate into the boundary layer (Eq. 33) and the memb rane layer[Eq. (22) for Model A1 and Eq. (39) for Model B 1 ], applying the well-known resistance-in-series model, the overall mass transfer rate can be given for the above case, as well.

Model B 1 (dC/dY>0 at Y=1):
The overall in let mass transfer rate, applying Eqs. (22) and (33) is, for first-order chemical reaction in the memb rane layer, as follows: The value of β is defined by Eq. (23). Note that the mass transfer resistance in the boundary layer of the permeate side is not involved in Eq. (35). It can be g iven, but that case is not discussed here because its complexity and of its lesser importance due to the chemical reaction.
Model B 2 (dC/dY=0 at Y=1). Look at first the concentration distribution and the mass transfer rate in the memb rane layer fo r that case, namely when dC/d Y = 0 at Y=1. Assuming that there is no mass transfer resistance in the feed phase, thus, at Y=0 then Applying to it Eq.
(25) as boundary condition for the permeate side, and using Eq. (14), one can get as [13]: with Φ and Θ defined after Eqs. (11) and (13). Note that Eq. (38) does not involve the external mass transfer resistances. The inlet mass transfer rate, namely the sum of the diffusive and the convective flows, can be given as: The overall mass transfer rate, applying Eqs. (33) and (39), will be as:

Intrinsically Catalytic Membrane Reactor
In this case, only the value of Ф differs fro m that of the catalytic memb rane layer with dispersed catalyst particles. In the case of a first-order reaction, the value of Ф can be expressed for an intrinsic memb rane layer with the following simp le equation: The mass transfer rate o r the concentration profile can be estimated in a similar way to that for a memb rane layer with dispersed catalyst particles. This case is not discussed in detail in this paper. The effect on the mass transfer rate and concentration distribution in the membrane reactor is basically the same as that obtained by the model developed for dispersed catalyst particles.

Results and Discussion
Two important transport models for convection flow catalytic membrane layer are presented in this paper. The difference between them is determined by the flow conditions on the permeate side, namely the permeated component is transported by a sweep phase fro m the memb rane interface (d C/d Y>0 at Y=1, Models A 1 and B 1 ) or there is no sweep phase (dC/dY=0 at Y=1, Models A 2 and B 2 ) on the permeate side. Obviously, these models can give essentially different in let mass transfer rate. On the other hand, two important cases are also discussed regarding the external mass transfer resistance, namely modeled it by the so called Fickian diffusion flow (Models A 1 and A 2 ) and by the so called exact model, where the diffusion flow permanently increases due to the decreasing convective flow (their sum should be constant throughout the boundary layer) on the diffusion path in the boundary layer (Models B 1 and B 2 ). Note that the application of the Fickian diffusion flow for the inlet and the outlet boundary layers[Eqs. (17) and (18)] is an approximation. The question is that its application can be acceptable and under what conditions. Typical concentration distribution curves of a catalyt ic membrane layer are shown for Model A 1 (Fig. 2) and for Model B 1 (Fig. 3).

Models for Membrane Reactor wi th S weep
The concentration was predicted by Eqs. (19) and (20) as well as by Eq. (21). The two models g ive significantly different concentration distribution. Accordingly the concentration gradient, and thus, the overall mass transfer rates will be d ifferent. The difference between Models A 1 and B 1 at e.g. Φ =0.01 is caused by the curvature of the of the concentration distribution in the boundary layer (not shown here) due to its convective velocity, namely Pe L1 =1 for Model B 1 . It is interesting to note that the inlet concentration increases with the increase of Φ value for the exact solution, i.e. for Model B 1 . Let us look at the inlet mass transfer rates of the two models as a function of the reaction rates. Fig. 4 shows it when the Fickian diffusion flow is applied for the boundary layers (Model A 1 ). J o represents the physical mass transfer rate into the catalytic membrane, this being the sum of the diffusive and convective flows at Ф=0. As can be seen, the tendency of the curves is different in the reaction ranges Pe < 1 and Pe > 1. In the first case, the value of J/J o increases with increasing value of Ф. In the range o f, Pe > 1, however, the mass transfer rate decreases as a function of Ф. Perfectly other trend is shown by the exact model in presence of sweep phase on the permeate side (Model B 1 , Fig. 5). As it is expected, the mass transfer rate g radually increases with the increase of the reaction rate. On the other hand, the effect of the reaction rate decreases gradually with the increase of the Peclet nu mber (note here also Pe L1 =Pe because Obviously, the two models gives the same mass transfer rate when there is no convective flow in the boundary layer (at Pe=0.01 it is practically true), but the difference strongly increases with the increase of the Pe-nu mber. The ratio of the mass transfer rates is plotted in Fig. 6, namely the approaching solution (Model A 1 ) is related to the exact one, (Model B 1 , i.e. J Model B1 /J Model A1 is plotted, noted by B 1 /A 1 in the axis of ord inate) as a function of the memb rane Peclet nu mber, at different values of the in let o 1 β mass transfer coefficient. As can be seen there exists an essential difference between the two models. This difference can be very large with increasing value of the Peclet number (note here Pe L1 =Pe). The difference between the two models can only be neglected at low values of Pe-nu mber. That means that the Model A 1 can be applied in very limited cases, only.  Here the diffusive flow in the catalytic memb rane layer is equal to zero, thus is can not have any influence on the concentration distribution in the catalytic memb rane layer. Against that there is difference between the models. The Model B gives somewhat higher memb rane concentration. With increasing value of the Φ reaction modulus, the concentration decreases, and due to it, the difference between the models also decreases. It can be stated the difference between the models is much less than that in the case when dC/dY>0 at Y=1. On the other hand, the average value of the membrane concentration can be higher here comparing that to the case of the models with dC/d Y>0 at Y=1. Accordingly the reaction rate can also be higher in this case. This can be important especially when the reaction rate constant is low, e.g. in the cases of bioreactions.
The overall mass transfer rates can also differ fro m each other (not shown here).  Fig. 2  The presence of a sweep phase can strongly affect the outlet concentration and thus, the diffusive outlet flow on the permeate side of the catalytic membrane layer. Accordingly, the sum of the diffusive and the convective flows should depend on the operating conditions, namely there is a sweep phase (dC/d Y>0 at Y=1) or there is not a sweep phase (dC/d Y=0 at Y=1). Thus, it seems to be interesting to investigate how the ratio of the mass transfer rates can depend on the values of Φ and Pe (Fig. 9) applying the exact model, namely Model B. As can be seen the ratio of the mass transfer rates, namely J Model B1 /J Model B2 can essentially differ fro m unity. Th is ratio gradually tends to unit with increasing reaction rate due to the lo w concentration values of reactant in the membrane layer. Otherwise, it strongly depends on Peclet number at lower values of reaction modulus. This dependence is decreasing with increasing convective velocity. Accordingly, it can be predicted by the models presented which one is more advantageous to application for a given reaction system.

Case Study
The catalytic membrane reactor in a flo w-through mode also appears to be a promising process for industrial application [2]. A special case will be shown to demonstrate the role of the convective velocity in the membrane reactors. In this examp le, the memb rane operates in dead-end mode and no separation procedure is performed. The task of the memb rane is to provide for intensive contact between reactant and catalyst, combined with a short contact time and a narrow residence time distribution. Ilinitch et al. [11] have measured the reduction of aqueous nitrates using mono-and bimetallic pallad iu m-copper catalysts impregnated in γ-Al 2 O 3 support layers. The metal content was kept between 1.7 and around 7 w% with a particle size belo w 3-5 n m. The memb rane layer is p laced in a tank with stirring to circulate all feed solution through this membrane. The concentration of the reactant passing the membrane can be much lower than that in the feed phase. This value depends on the convective stream and the chemical reaction rate. The authors measured the nitrate conversion in three different modes, i.e. without convective flo w and for two different values of convective flow with a surface velocity of 7 x 10 -6 and 16.5 x 10 -6 m/s (see Figure 12 in Ilinitch et al. [11]). For evaluation of the experimental results, the model A has been used with 0 o 2 = β . It is easy to see that the outlet concentration is equal to the "bulk" concentration behind the memb rane layer, as illustrated in Fig. 1b The data used for calculation are listed in Table 1.
Taking the diffusion stream through the membrane into account, the value of D t represents the residence time of the reaction solution as given by the following equation: Performing the calculation using data from Table 1, the value of D t obtained was 400 min. Table 1. Values of parameters used for calculation of the concentration change in the membrane layer [11,15] V= 85 x 10 -6 m 3 δ=4,6 The concentration of the reactant solution obtained was about 0.4 for the case of the diffusion driven mode (Pe=0) with Ф=1, whereas the value is around 0.08 for the case of the convective flow mode with Pe=10 at 3 t / t D = . These data are in line with the measured values, as can be seen in Fig. 10. The points represent the measured data whilst the continuous lines indicate the calculated values. It should be noted that the overall first-order kinetics was assumed for the nitrate-ion in the calculat ion. The H 2 concentration was kept constant during the reaction. The Ф values should be estimated for calcu lation of the conversion vs. time function. The value was obtained by fitting the measured conversion data fro m diffusion-driven flo w (Pe=0). The continuous line for Pe=0 in Figure 10 was obtained using an estimated value of Ф=1.8. This value was then used for calculation of the curves for Pe=3.5 and 8. The continuous lines obtained by simu lation are plotted together with the measured points. The calculated data for Pe=3.5 are slightly lo wer than the measured values, whereas the data obtained for Pe=8 are in surprisingly good agreement with the measured points. The good agreement between the measured and the calculated data proves that the model developed is suitable for estimating mass transport and conversion in the presence of both convective and diffusive flows.

Conclusions
A mathematical model has been developed in order to predict mass transport through a catalytic membrane reactor containing dispersed nanometer-sized catalyst particles using forced convective flow through the membrane layer, as well as for the case where the nanometer-sized catalyt ic particles are regularly dispersed in the membrane layer or the me mb rane intrinsically catalytic. Transport models developed include the mass transport into and inside the catalytic particles as well as through the catalytic membrane layer, taking into account both convective and diffusive flows. It has been shown that the two operating modes, namely with or without sweep phase on the permeate side can give essentially different in let mass transfer rates. On the other hand, the application of the Fickian d iffusive flo w in the feed boundary condition in presence of convective flow can lead significant error in the mass transfer rate predicted. This error quickly increases with increasing boundary layer's Peclet nu mber due to the increasing curvature of the concentration distribution. Accordingly the application of the mass transfer coefficient predicted by the dimensionless number fo r the feed boundary layer should be avoided in the presence of forced convective flow. The models presented describe mass transport in good agreement with the measured data, proving that it can be used to estimate the mass transport process for a catalytic memb rane reactor.  14), Θ =dimensionless quantity after Eq. (13), Subscripts f =interface L =liquid ov =overall mass transfer coefficient or rate p = catalyst particle 1,2 =continuous phases on both sides of membrane APPENDIX A second-order steady-state differential equation with variable (concentration dependent and/or local coordinate dependent) parameters and or in the case of nonlinear chemical reaction kinetics, e.g. Michaelis-Menten kinetics[Eq. (5)], can not be generally solved analytically. A numerical method or analytical approach can be recommended for its solution. Herewith we show a rather simp le analytical approach where the mass transfer rate or the concentration distribution on side the catalytic membrane can be expressed in closed, exp licit mathematical forms. For the solution the membrane layer should be divided into N sub-layers with constant parameters. This is illustrated in Fig.  A1. Figure A1. Important notations for the catalytic membrane divided into N sub-layer for linearization of e.g. Michaelis-Menten kinetics The linearized form of the M ichaelis-Menten kinetics can be as [13,14]: (A1) where the value of k i is the first-order reaction rate constant, i C denotes the average value of i C in the i th me mbrane sub-layer. Due to the unknown real value of i C a few iteration step is needed to get the exact value of C i . The equation for the i th sub-layer to be solved will be for steady-state as: The T i and S i parameters can be determined by suitable boundary conditions. Neglecting the external mass transfer resistances, the boundary conditions can be given as [14]: C o = T 1 + S 1 at Y=0 (A4) ( ) The above, well known boundary conditions serve 2N algebraic equations for 2N parameters to be determined. Details for the solution see e.g. Nagy's book [14]. The mass transfer rate at Y=0 can be g iven as: The value of T 1 and S 1 for determination of the mass transfer rate The concentration distribution and/or the outlet mass transfer rate can also be easily determined. Applying the boundary conditions the other T i and S i values (i=2,…,N) can be determined [14].
So me calculat ions for the Michaelis-Menten kinetics are presented in Nagy's book [14].