Ind. Eng. Chem. Res. 2007, 46, 2295-2306
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Mass Transfer through a Dense, Polymeric, Catalytic Membrane Layer with Dispersed Catalyst† Endre Nagy* UniVersity of Pannonia, Research Institute of Chemical and Process Engineering, P.O. Box 158, 8201 Veszpre´ m, Hungary
The mass-transfer rates, which enter the catalyst membrane layer, accompanied by first-order, irreversible reactions, were investigated. Depending on the catalyst particle size, heterogeneous and pseudo-homogeneous models were developed to describe the mass transport through the catalytic membrane layer. It was assumed that the chemical reaction occurs inside the catalyst particles. Using a simple physical model for the distribution of the catalyst particles in the membrane layer and for the mass transport into it, explicit mathematical equations were developed for the prediction of the mass-transfer rates, as a function of the physical and chemical parameters. Besides the reaction rate, the mass-transfer rate is strongly influenced by the membrane properties, such as the size of the catalyst particles, the catalyst phase holdup, the distance of the first catalyst particle from the membrane interface, in the case of the heterogeneous model, as well as the diffusion coefficients in the membrane phase and in the catalyst particles, the membrane thickness, etc. By decreasing the particle size and the distance of the first particle from the membrane interface, the mass-transfer rate can be significantly improved. On the other hand, the low diffusion coefficient in the catalytic particles, which is very often the case, can strongly lower the mass-transfer rate, as well as the effect of the chemical reaction on it. The pseudo-homogeneous model was recommended for fine (submicrometer-sized) catalyst particles, whereas the heterogeneous model is recommended for larger particles (approximately several micrometers in size). The simple, mathematical equations that were developed can be applied very easily for the prediction of the mass-transfer rate in the case of a catalytic membrane layer for both models that have been investigated. Introduction The catalytic membrane reactor, as a promising novel technology, is widely recommended for application to heterogeneous reactions. Several reactions have been investigated by means of this process, such as dehydrogenation of alkanes to alkenes and partial oxidation reactions using inorganic or organic peroxides, as well as hydrogenations, hydration, etc. In regard to a catalytic membrane reactor for performing these reactions, one can use an intrinsically catalytic membrane (e.g., zeolite or metallic membranes) or a membrane that has been made catalytic via the dispersion, impregnation, etc. of catalytically active particles, as metallic complexes, metallic clusters1 or activated carbon, zeolite particles,2 etc., throughout the dense, polymeric, or inorganic membrane layers.3 Champagnie et al.4 studied ethane dehydrogenation using a commercial alumina membrane impregnated with a platinum catalyst. Liu et al.5 investigated the selective hydrogenation of propadiene and propyne using a catalytic cellulose acetate hollow fiber membrane that contained palladium catalyst particles. They proved that the selective hydrogenation can be accomplished more effectively by the permeated hydrogen than by the hydrogen that was premixed in the gas phase. With the controlled diffusion and solubility of reactants in the catalytic membrane phase, their optimal concentrations can easily be realized in the catalyst particles. The partial oxidation of alkenes1,6 or alkanes7,8 is also an important field of the catalytic membrane reactors. Vankelecom et al.1 used an organometallic complex of manganese for partial oxidation reactions, where the catalyst was incorporated * To whom correspondence should be addressed. Tel.: +36-88-624040. Fax: +36-88-624 038. E-mail:
[email protected]. † This paper was presented at the “Advanced Membrane Technology III: Membrane Engineering for Process Intensification” Conference, June 11-15, 2006, Cetraro, Italy.
into a polydimethylsiloxane membrane. The catalytic membrane was tested for 3-penten-2-ol epoxidation, using hydrogen peroxide. Yawalkar et al.6 theoretically studied the alkene epoxidation reaction in a catalytic membrane reactor that contained dispersed particles. They concluded that the epoxidation reaction can be regarded as a pseudo-first-order reaction for H2O2 using an organophillic polymer membrane, in that the alkene concentration is much higher than that of the peroxide. The decomposition of the peroxide can be drastically reduced with the membrane-occluded catalyst.1 Langhendries et al.8 investigated the oxidation of cyclohexane, cyclodecane, and n-dodecane using t-butylhydroperoxide as an oxidant and zeolite-encaged iron phthalocyanine as a catalyst embedded in a hydrophobic polymeric membrane matrix. Their experimental results demonstrated the advantages of the catalytic membrane reactor, as compared to the conventional methods. In all the aforementioned experiments, the reactants are separated from each other by the catalytic membrane layer. Let us examine only the catalytic processes in this paper when the reagents diffuse in the catalyst particles and when they react on its internal interface, as in the case of, for example, activated carbon, zeolite, etc. The reactants, which are fed on the two sides of the catalytic membrane layer, are first absorbed in the polymer membrane matrix and then they diffuse from the membrane interface into the catalyst particles, where they react. An important question arises in regard to how the diffusional mass transport, through the membrane layer, accompanied by chemical reaction, can be described. From a chemical engineering point of view, it would be important to predict the masstransfer rate of the reactant passing into the membrane layer from the upstream phase, and also to predict the downstream mass-transfer rate on the other side of the catalytic membrane, as a function of the physicochemical parameters. If this transfer (permeation) rate is known, as a function of the reaction rate
10.1021/ie060965c CCC: $37.00 © 2007 American Chemical Society Published on Web 01/09/2007
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constant, it can be replaced into the boundary conditions of the differential mass balance equations given for the upstream and/ or downstream phases. Such mass-transfer equations are not found in the literature yet. Basically, to describe the masstransfer rate, a heterogeneous model (for larger particles) and/ or a pseudo-homogeneous model (for very fine catalyst particles) may be applied. Langhendries and Baron9 used two-dimensional and one-dimensional diffusion models for the zeolite-polymer composite structure. They give numerical results for the concentration distribution inside the membrane layer at different values of the diffusion coefficient in both phases. The effect of the chemical reaction was not taken into account in their paper. Other theoretical works10,11 also gave numerical results for the mass transport and concentration distribution. Jawalkar et al.6 applied the heterogeneous model for the catalytic membrane layer. This model was applied to describe the gas absorption process, in the gas-liquid boundary layer, in the liquid phase that contains a dispersed third phase.12-14 Mehra12 gave the numerical solution of the problem, whereas Nagy and Hadik13 and Nagy14 developed its analytical solution for the case of firstorder and zero-order reactions,15,16 and they expressed the masstransfer rate on the gas/liquid interface in explicit mathematical forms. When the catalyst particles are very fine, they can be regarded as points without measures and the much simpler pseudo-homogeneous model17 may be used. Applying both approaches for the mass transfer through a catalytic membrane layer that contains dispersed catalyst particles, explicit mathematical equations for describing the mass-transfer rates on both sides of the membrane layer have been developed for mass transfer, accompanied by a pseudo-first-order, irreversible reaction. The main purpose of this paper is to show the steps of the solutions, as well as to study the effect of the physical and chemical parameters of the catalytic membrane layer on the mass-transfer rates. The methods used also can be applied for higher-order reactions. Theoretical Background For the description of the diffusive mass transport, in a catalytic membrane reactor, generally two mathematical models can be generally used, depending on the particle size: a heterogeneous model for larger particles and a pseudohomogeneous model for very fine particles. In both cases, the internal mass transport should be taken into account, whereas the heterogeneous model also considers the outlet stream of the unreacted portion of the reactant from the catalyst particles. In the case of the homogeneous model, the entire amount of reactant that is transported in the particle will be reacted: there is no outlet stream of it. Heterogeneous Model. Primarily, depending also on the membrane thickness, when particles are falling into the micrometer size regime, the internal mass-transport mechanism, inside of catalyst particles, must be taken into account. A simple physical model could be applied for the description the process in this case, as it is schematically illustrated in Figure 1. The gas (or liquid) reactant initially enters the catalytic membrane layer and then diffuses to the first catalytic particle, perpendicular to the membrane interface. The chemical reaction, which is namely a first-order, irreversible chemical reaction, occurs only in the catalyst particles. It is assumed that the concentration of the organic reactant should be much higher in a hydrophobic polymer membrane than that of the reactant investigated, e.g., peroxides, oxygen or hydrogen, etc. The unreacted reactant then diffuses through the first catalytic particle to its other side and
Figure 1. Schematic diagram of a catalytic membrane layer filled by dispersed catalytic particles (dark squares) with the diffusion path perpendicular to the membrane interface. (HC here denotes hydrocarbon.)
again enters the polymer membrane matrix, and so on (the route of this mass-transfer process is illustrated by an arrow denoted by J in Figure 1). This diffusion path exists only for the heterogeneous portion of the membrane interface (which is the projection of the cubic catalyst particle onto the membrane interface). There can be a portion of the membrane interface, that is the so-called homogeneous portion of the interface, where the diffusing reactant does not cross any catalyst particle (this mass stream is denoted by J° in Figure 1). This also affects the resultant mass-transfer rate. The assumed cubic6,12-14 catalyst particles are supposed to be uniformly distributed in the polymer membrane matrix. The diffusion path is assumed to be perpendicular to the membrane interface. The diffusion of the reagent investigated occurs alternately through the polymer matrix and the catalyst particle. In the present work, a onedimensional mathematical model has been developed to describe the mass transport through a catalytic membrane layer, taking into account the diffusion mass transport and the chemical reaction in the catalyst particles as well. The main assumptions of the model used are as follows: (i) the diffusion mass-transport process, in the catalytic membrane, is one-dimensional and steady state; (ii) the catalyst particles are cubic and uniformly distributed, and they have a uniform size; (iii) the diffusion of the reactant is perpendicular to the membrane interface and the lateral diffusion is negligible; (iv) the pseudo, first-order, irreversible, chemical reaction occurs only in the catalyst particles; (v) the diffusivities and the sorption coefficients are, in both the membrane matrix and the catalyst particles, constant and both can be very different in both phases; (vi) the distance of the first particle from the membrane interface may vary, depending on the preparation of the catalytic membrane layer, and its value is larger than zero (that is, there is no direct contact of the reactant with the catalytic particle at the membrane interface); (vii) the external mass-transfer resistances should be taken into account; and (viii) Henry’s law, with constant solubility, is valid for the sorption equilibrium between the reactant and the membrane phases and between the membrane and the catalyst phases. For the description of this transport process, the catalyst membrane layer should be divided into 2N + 1 sublayers, perpendicular to the membrane interface. Namely, N sublayers for catalyst particles located perpendicular to the membrane interface, N + 1 sublayers for the polymer membrane matrix between particles (∆X) and between the first particle (X1) and the last particle (1 - X/N) and the membrane interfaces (see Figure 2b). To obtain a mathematical expression for the masstransfer rates, a differential mass-balance equation should be given for each sublayer. Thus, one can obtain a differential equation system that contains 2N + 1 second-order differential equations. This equation system, with suitable boundary conditions, can be solved analytically. which is also demonstrated in
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2297
Thus, one can obtain 2N + 1 algebraic equations with twice as many parameters (Ai, Bi (i ) 1, 2, 3, ..., N + 1), as well as Ei and Fi (i ) 1, 2, 3, ..., N)), which are to be determined. Their values can be determined by means of suitable boundary conditions at the external interfaces of the membrane, at X ) 0 and X ) 1, as well as at the internal interfaces of every segment in the membrane matrix, at Xi and X/i , with i ) 1, 2, 3, ..., N. The effect of the external mass-transfer resistances (expressed by β°1 and β°2; see Figure 1; for notations, see Figure 2a as well) should be taken into account in the external boundary conditions):
At X ) 0, then:
this paper. The number of particles (N) and the distance between them (∆X) can be calculated from the particle size (dp) and the catalyst phase holdup () (see the Appendix). The distance of the first particle from the membrane interface, X1 (X1 ) x1/δ), can be regulated by the preparation method of the catalytic membrane layer. The value of X h 1, which is the distance of the first particle from the middle particle line of model B (see Figure A1 of the Appendix; two particle distribution models are analyzed in detail in the Appendix, but only model B, being the more realistic, is investigated in detail in this paper) is determined by X1, , and dp. The important notations used for the following mathematical equations are given in Figures 2a and 2b. The differential mass-balance equations for the sections of the polymer membrane phase and for that of the catalyst particles can be given, in dimensionless form, as follows, respectively: 2
At X ) 1, then:
1
i
i+1,
)
X/N e X e 1 (1)
( )
(
kδ2 C )0 Dp p dX2
d for Xi e X e Xi + δ
)
(2)
The dimensionless membrane concentration can be obtained easily using the following expression:
C)
wF c1oM
where w is the concentration of reactant (in units of kg/kg) in the membrane and F is the average density of the membrane (in units of kg/m3). The solutions of the aforementioned differential equations, for the ith sections, are well-known, namely,
C ) AiX + Bi
(for 1 e i e N + 1)
Cp ) Ei exp(λX) + Fi exp(-λX)
(for 1 e i e N)
(3) (4)
λ)
x
(6)
at X ) Xi: DAi ) Dpλ(Ei exp(λXi) - Fi exp(-λXi)) (8) as well as for the other side of the catalyst particles, namely at X ) X/i :
d ) X/i : δ (Ai+1X/i + Bi+1)H ) Ei exp(λX/i ) + Fi exp(-λX/i ) (9)
at X ) Xi +
d ) X/i : δ DAi+1 ) Dpλ(Ei exp(λX/i ) - Fi exp(-λX/i )) (10)
at X ) Xi +
Equations 7 and 9 express that there is an equilibrium on the sublayer interfaces, whereas eqs 8 and 9 indicate that there is no accumulation or source at the internal interfaces. Thus, an algebraic equation system with 2(N + 1) equations can be obtained that can be solved analytically with a traditional method, using the Cramer rules. The solution is briefly discussed in the Appendix. As a result of this solution, the mass-transfer rate on the upstream side of the membrane interface, relative to its heterogeneous component (which is the projection of the cubic catalyst particle onto the membrane interface), can be given as follows:
J ) βmHmc°1(1 - TC°2) where
βm ) β°mH
[
] ( )
Un(N+1) + HHm(β°m/β°2) UN+1 + HHm(β°m/β°2)
(11)
N
∏ i)1
Uni Rni Ui Ri
(12)
as well as
T)
kδ Dp
)
at X ) Xi: (AiXi + Bi)H ) Ei exp(λXi) + Fi exp(-λXi) (7)
with 2
(
AN+1 + BN+1 - C°2 Hm dC ≡ -β°mAN+1 ) -D dX
The boundary conditions for the internal interfaces of the sublayers are also well-known12-14 (see Figure 2b):
and
d2Cp
)
A1 dC ) -D ≡ -β°mA1 (5) Hm dX
β°2(C2 - C°2) ≡ β°2
DdC )0 δ2 dX2
(for 0 e X e X , X + δd e X e X
(
β°1(C°1 - C1) ≡ β°1 C°1 -
Figure 2. (a) Concentration distribution in the membrane layer and (b) a particle line perpendicular to the membrane interface with notations.
1 N
(Un(N+1) + HHm(β°m/β°2))
(UniRni) ∏ i)1
(13)
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with
[ [
Table 1. List of Variables That Should Be Applied for Calculation of the Mass-Transfer Rates
] ]
ξnN RnN
Un(N+1) ) H(1 - X/N) +
(14)
Rni )
ξN U(N+1) ) H(1 - X/N) + RN
(15)
tanh(Had) ξni 1 + κ tanh(Had) ξni ) + 1 Uni ) H∆X + Uni Uniκ Rni
for i ) 1, ..., N
Ri )
as well as
Un1 ) 1 Rnl ) κ tanh(Had) ξn1 ) 1
(
HC ) Cp
U1 ) HX1 + HHm
X/N ) X1 + (N - 1)∆X + N HmCj ) C/j
(δd)
(for j ) 1, 2)
The values of Uni, Rni, and ξni, as well as those of Ui, Ri, and ξi, should be calculated from sublayer to sublayer, that is, from 1 to N (Rni, ξni, Ri, ξi) or N + 1 (Uni, Ui), from the equations given in Table 1. It may also be important to know the portion of the reactant that reacts in the catalytic membrane layer during its diffusion, or if there is an unreacted portion of the diffused reactant that passes on the downstream side of the membrane into the continuous phase. This outlet mass-transfer rate, for the heterogeneous component of the membrane interface, at X ) 1 can be given as follows:
[
Jout ) βoutHmc°1 1 -
N
∏ i)1
with N
∏ i)1
βout ) β°mH
(
(
tanh(Had) ξi 1 + κ tanh(Had) ξi ) + 1 Ui ) H∆X + Ui Uiκ Ri
as well as
D ) β°m δ
C°2
cosh(Had) -
cosh(Had) -
ξi Ri
sinh(Had)
ξi Ri
sinh(Had)
UN+1 + HHm(β°m/β°2)
)
]
)
(16)
J° ) β°sumHmc°1(1 - C°2)
)
β°m β°1
with κ )
λDd D
example, in the case of a polymer membrane filled with zeolite particles as a catalyst, the value of Dp is ∼4 orders of magnitude lower than that in the polymer matrix.4,6 The specific mass-transfer rate, relative to the total catalytic membrane interface (for details, see the Appendix), can be given as
Jave ) KJ2/3 + J°(1 - K2/3)
(20)
or that for the outlet mass-transfer rate can be given as
Javeout ) KJout2/3 + J°(1 - K2/3)
(21)
The value of the mass-transfer rate can be easily obtained for the homogeneous component of the interface, J°m, namely,
J°m ) β°mHmc°1(1 - C°2)
(22)
To calculate the enhancement during the mass transfer accompanied by chemical reaction, the physical mass-transfer rate, relative to the total membrane interface, should also be defined:
(17)
The physical mass-transfer rate for the heterogeneous component of the interface is as follows:
(18)
The physical mass-transfer coefficient, with external masstransfer resistances, for the portion of the membrane interface where there are particles in the diffusion path, taking into account the effect of the catalyst particles as well, can be given by the following equation:
β°sum )
for i ) 2, ..., N
1 [Hm/(β°1 + β°2)] + (1/β°m) + (Nd/Dp)[(1/H) - (Dp/D)] (19)
Depending on the value of the diffusion coefficient Dp, the solubility coefficient H (Cp ) CH), as well as the number of particles perpendicular to the interface (N), the value of the physical mass-transfer coefficient of the membrane with catalytic particles (β°sum) might be completely different from that of the membrane layer without catalyst particles, β°m ( β°m ) D/δ). For
J°ave ) KJ°2/3 + J°m(1 - K2/3)
(23)
The value of the factor K can be obtained from the distribution of the catalyst particles in the polymer membrane matrix. Its value, depending on the particles distribution in the membrane matrix (see Figure A1 in the Appendix), should be K ) 1 or K ) 1.8715 for models A and B, respectively. The value of J, in the case of model B, should be calculated by averaging two h 1 distance values mass-transfer rates, J1 and J2, using X1 and X from the membrane interface, respectively, as illustrated in Figure A1. In this case, the average value of the mass-transfer rate is J ) (J1 + J2)/2, because the same value of the catalytic particle interface belongs to both mass-transfer rates. During the calculation of the mass-transfer rates, using different values of the parameters, as a function of the reaction rate, only the more-realistic model B will be applied in this part of the paper. The two particle distribution modelssnamely, models A and Bsare briefly compared only in the Appendix. Although the enhancement, relative to the heterogeneous component of the membrane interface, as a function of reaction rate, is slightly higher in the case of model A, the average value of the masstransfer rate, relative to the total membrane interface (Jave), will
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2299
be slightly higher using model B, as a consequence of the higher value of the factor K in eq 20. Namely, the projection of the cubic catalyst particles onto the membrane interface is much higher in the case of model B. In regard to the tendency of the enhancement curves, as a function of Hap, there is no significant difference between the two models. Pseudo-homogeneous Model for Submicrometer-Sized Particles. As will be shown later, if the diffusion coefficient of the catalyst particles is much lower than that in the membrane structure, the size of particles should be chosen to be as low as possible. Thus, the application of nanosized catalyst particles is recommended. When the particle size is low enough, the simpler pseudo-homogeneous model could also be used to describe the catalytic mass transport inside the membrane. To do this, the mass-transfer rate into the catalyst particles must be defined first. The internal specific mass-transfer rate in spherical particles, for steady-state conditions and when the mass transport accompanied by first-order chemical reaction, can be given as follows:17
j ) βpC/p
(24)
rate increases. Thus,
(
(25)
x
(32)
with
βm ) β°mϑ × 1 + (β°mϑHm/β°2) tanh ϑ [1 + (β°mϑ) Hm /(β°1β°2)] tanh ϑ + β°mϑ[(Hm/β°1) + (Hm/β°2)] (33) 2
2
where
x
ωδ2βsum
(1 - )D
and
T) kR2 Dp
(31)
J ) βmHmc°1(1 - TC°2)
and
Hap )
dC | dX X)1
The mass-transfer rate on the upstream side of the membrane can be given as follows:
ϑ)
)
(30)
at X ) 1: β°2(C2 - C°2) ) -D
where
Dp Hap βp ) -1 R tanh(Hap)
dC | dX X)0
at X ) 0: β°1(1 - C1) ) -D
1 cosh ϑ[1 + (β°mϑHm tanh ϑ/β°2)]
(34)
Similarly, the mass-transfer rate for the downstream side of the membrane, at X ) 1, is
The external mass-transfer resistance through the catalyst particle is dependent on the diffusion boundary layer thickness (δp). According to Figure A1 in the Appendix, the value of δp could be estimated from the distance of particles from each other. Namely, its value is limited by the neighboring particles; thus, the value of β°p will be slightly higher than that which follows from the well-known equation of 2 ) β°pdp/D, where the value of δp is supposed to be infinite. Thus, one can obtain18
2D D β°p ) + dp δp
(26)
where
(
〈
Jout ) βoutHmc°1 1 - cosh ϑ tanh ϑ +
〉)
β°mϑHm C°2 β°1
(35)
with
βout )
(
β°mϑ × cosh ϑ 1
)
tanh ϑ{1 + [(β°mϑ) /(β°1β°2)]Hm2} + [(Hm/β°1) + (Hm/β°2)]β°mϑ (36) 2
Results and Discussion
h - dp 2
δp )
(27)
From eqs 24 and 26, one can obtain, for the mass-transfer rate with the overall mass-transfer resistance,
[
j ) βsumc°1C ) c°1
C (1/β°p) + (1/Hβp)
]
(28)
The differential mass-balance equation for the polymer membrane phase, for steady state, can be given as follows:18
( )
ωδ2 d2C β C)0 2 1 - sum dX
D
(29)
For the sake of generalization, in the boundary conditions, one should take into account the external mass-transfer resistance on both sides of the membrane, although it should be noted that the role of the β°2 will gradually diminish as the reaction
The effect of the particle size, the Hatta number of particles, and the catalyst phase holdup on the mass-transfer rate will be briefly discussed, because of their importance, applying both models that have been presented. A practical examplesnamely, mass transfer during the oxidation of alkenes to epoxides in a polymer membrane reactor with dispersed zeolite catalyst particlessthen will be shown by means of the models that have been developed. The heterogeneous model is recommended to use for the micrometer-sized catalyst particles, whereas the pseudo-homogeneous model is recommended in the case of submicrometer particles. Heterogeneous Model. The effect of the chemical reaction on the specific mass-transfer rate can be predicted either for the heterogeneous component (where there are catalyst particles in the diffusion path perpendicular to the membrane interface) of the catalytic membrane interface using eq 11, or for the total membrane interface using eq 20. Similarly, the outlet masstransfer rates can be calculated using eqs 16 and 21. The physical mass-transfer rate, in the presence of catalyst particles (eq 19),
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Figure 3. Effect of the chemical reaction rate, Hap, on enhancement in the heterogeneous portion of the membrane interface at different values of the external mass-transfer coefficient. (H ) Hm ) 1; D ) Dd ) 1 × 10-10 m2/s; β°1 ) β°2; and C°2 ) 0.)
Figure 5. Effect of the catalyst particle size on the mass-transfer rate related to the total membrane interface, as a function of reaction rate. (Hm ) H ) 1; D ) Dd ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.)
membrane layer is high.3,6 When the mass transport is accompanied by a chemical reaction, the role of the external masstransfer resistance on the upstream side then could significantly increase as the reaction rate increases. Thus, the effect of the external mass-transfer coefficient should be carefully studied before being neglected. As can be seen in these figures, the enhancement tends toward a limiting value at higher Ha values. At these high reaction rates, Had > 10-30, depending on the values of β°1 and X1, the mass-transfer resistance is determined by the resistance from the bulk inlet phase to the first catalyst particle in the membrane; that is,
If Had f ∞, then
Figure 4. Effect of the distance of the first catalyst particle in the diffusion path from the membrane interface on the mass-transfer rate. (Hm ) H ) 1; Dp ) D ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.)
might be very different from that of a pure polymer membrane (eq 22), because of the possible difference in the diffusion coefficients in the membrane and catalyst particles. Parameters that affect the mass-transfer rate at the membrane interface are physical parameters (such as the diffusion coefficients D and Dp), external mass-transfer coefficients (β°1 and β°2), masstransfer coefficients of the polymer membrane (β°m) and that of the cubic particles (β°p) (β°p ) Dp/d), the membrane thickness (δ), the catalyst phase holdup (), the size of the catalyst particles (dp) (from , dp, and δ, the distance between particles (∆x) and the number of particles in the diffusion path perpendicular to the interface (N) can be calculated), the value of x1 (the distance of the first particles in the diffusion path from the upstream membrane interface), and the chemical reaction rate constant (k). All these parameters can be measured or predicted. From technological points of view, the enhancement, relative to the total membrane interface (Jave/J°ave), is a most interesting quantity. Despite that fact, for the sake of better understanding, the effect of the external mass-transfer coefficient, and that of the X1 value (X1 ) x1/δ), on the enhancement in the heterogeneous component of the membrane interface, or, rather, on the value of βm/β°sum are plotted in Figures 3 and 4, as a function of the chemical reaction rate (that is, as a function of Had ( Had )
xkd2/Dp). Their effect is essentially important in the case of a
membrane reactor. The external mass-transfer resistance is very often negligible in the case of a thin, polymer membrane layer, especially if the mass-transfer resistance of the catalytic
βm 1 ) β°sum Hmβ°m + X1 β°1
(37)
In our case, δ ) 30 µm and x1 ) 1 µm when β°1 f ∞; the value of βm/β°sum then tends to be equal to 30. Decreasing the value of β°1/β°sum, the maximum value of the enhancement also significantly decreases. Figure 3 clearly shows the important role of the external mass-transfer coefficient. From these results, it can be stated that the negligibility of the external mass-transfer coefficient should be carefully investigated before any decision. In contrast to that mentioned previously, the effect of the downstream side mass-transfer coefficient decreases significantly as the reaction rate increases. The value of X1 is also primarily important, because it could basically alter the mass-transfer rate, as it also follows from eq 37. The distance of the first particle from the membrane interface, X1, can be influenced by the preparation method of the catalytic membrane layer. Several methods may be applied for the preparation of catalytic membrane layers, e.g., impregnation, suspension of particles in the casting solution, etc.3 It is important to choose the value of X1 to be as small as possible. In the case of X1 ) 0, the curve of βm/β°sum vs Had, when the external mass-transfer resistance is negligible (β°1 f ∞), is the same as that in the case of a homogeneous, irreversible reaction. Increasing the value of X1, from 1 µm up to 5 µm, the maximum values of the enhancement decrease from 30 down to 6, in our case. The size of the catalyst particle has also a significant effect on enhancement (see Figure 5; for all figures hereafter presented in this paper, the enhancement is relative to the total membrane interface). At a given value of catalyst holdup , the size of the catalyst particles increases the distance between the particles behind each other, that is, the value of ∆X. Furthermore, it
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2301
Figure 6. Effect of the ratio of the diffusion coefficients on the masstransfer rate, as a function of Hap for the heterogeneous model (continuous lines) and the pseudo-homogeneous model (dotted lines). (Hm ) H ) 1; D ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.)
Figure 9. Enhancement as a function of reaction rate through application of the heterogeneous model (continuous lines; Jave/J°ave; d ) 1.26dp) and the pseudo-homogeneous model (dotted lines; βm/β°m). (H ) Hm ) 1; Dd ) D ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.)
Figure 7. Enhancement as a function of the reaction rate at different catalyst phase holdup () for the heterogeneous model (continuous lines) and the pseudo-homogeneous model (dotted lines). (H ) Hm ) 1; Dd ) D ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.) Figure 10. Effect of the catalyst particle size on the mass-transfer rate for the heterogeneous model (continuous lines) as well as the pseudohomogeneous model (dotted lines) (βm/β°m). (H ) Hm ) 1; Dd ) D ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.) Table 2. Parameter Values Used for Simulation of the Mass-Transfer Rate for the Oxidation of Alkenes to Epoxides
Figure 8. Change in the outlet mass-transfer rate, relative to the upstream side mass-transfer rate, as a function of the reaction rate at different values. (The dotted lines were obtained by the pseudo-homogeneous model; H ) Hm ) 1; D ) Dd ) 1 × 10-10 m2/s; β°1 ) β°2 f ∞; and C°2 ) 0.)
decreases the number of particles located in the membrane layer, perpendicular to the membrane interface (N) and increases the value of X h 1 in model B (see Figure A1 in the Appendix) applied here. These parameters directly alter the mass-transfer rate. Thus, the mass-transfer rate decreases as the particle size increases. At low reaction rates, this change in enhancement may even be significant. The effect of the size was calculated over a wide range of particle sizes, namely, in the size range of 0.01-10 µm. In a limiting case, the enhancement is determined here also by the maximal value of the physical mass-transfer rate. Taking into account the values of parameters used, the maximal value of the enhancement is equal to 12.7, as it is plotted in Figure 5.
parameter
value
Dm Dp δ Hm H k′ C°Alkene k β°1 (f ∞)
8.5 × 10-10 m2/s 3.4 × 10-14 m2/s 100 µm 0.05 2 0.01 1/s 1500 gmol/m3 k′Alkenem β°2 (f ∞)
At very low particle sizes, especially in the nanosize range, the enhancement can approximately reach its maximum value, even at low Ha values. The number of particles in the diffusion path also increases rapidly as the particle size decreases. For example, if dp ) 40 nm, then N will be equal to 320 in the diffusion path for the case of ) 0.1, δ ) 30 µm, and X1 ) 1 µm. This means that one should produce catalyst particles in the polymer membrane layer that are as fine as possible. Applying nanoparticles as a catalyst, the mass-transfer rate could be much higher than that in the micrometer-sized particles. The ratio of the diffusion coefficients (Dp/D) is also an important factor (see Figure 6). Very often, the diffusion coefficient in the catalyst particle is much less than that in the polymer membrane layer.3 For example, the Dp value is 4 orders of magnitude less in zeolite particles used as catalytic particles
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Figure 11. (a) Mass-transfer rate as a function of the catalyst particle size, in the case of the oxidation of alkenes to epoxides in catalytic membrane reactor, using data given in Table 2 (continuous lines represent data from the heterogeneous model, and dotted lines represent data from the pseudohomogeneous model). (b) Outlet mass-transfer rates as a function of the particle size during the oxidation of alkenes (continuous lines represent data from the heterogeneous model; dotted lines represent data from the pseudo-homogeneous model).
than that of an organophillic polydimethylsiloxane membrane layer3,6 filled with zeolite particles and used as a catalytic membrane layer. This large difference in diffusion coefficients significantly changes the diffusion rate through the membrane layer in the presence of zeolite catalyst particles, as can also be concluded from eq 19 for physical mass transport. A 4-orderof-magnitude difference in the diffusion coefficients means 104 times larger mass-transfer resistance in the same length of the diffusion path. This effect is illustrated in Figure 6 at four different Dp/D values, in the cases of the heterogeneous model (continuous lines) and of the pseudo-homogeneous model (dotted lines). Note that the value of the physical mass-transfer rate, J°ave, becomes substantially lower as the Dp/D value decreases. Despite this fact, the curves are shifted to the righthand side when the Dp/D value decreases. This means that the effect of the chemical reaction also decreases as the Dp/D value decreases. On the other hand, it should also be noted that the Hap value is dependent on both the reaction rate constant and
particle diffusion coefficient; namely, Hap ) xkR2/Dp (the size of the cubic particle is d ) 1.24dp). The maximum value of Jave/J°ave also changes as the Dp/D value changes, because the value of J°ave decreases as the value of Dp decreases (see eq 23). For example, at Dp/D ) 10-4, the value of Jave/J°ave is very low, at even a very high Hap value. The limiting values of the enhancement, with increasing value of the reaction rate, are high because of the very low x1 value. Comparison of the two models will be made in the next section. The effect of the catalyst phase holdup is illustrated in Figure 7 at different reaction rates by the heterogeneous model
(continuous lines). As expected, the mass-transfer rate increases monotonously as a function of the catalyst phase holdup . As the value of increases, both the distance between catalyst particles (∆X) and the value of X h 1 (see Figure A1 in the Appendix) decreases. Note that the so-called homogeneous portion of the membrane interface also decreases gradually as the catalyst holdup increases. At ≈ 0.39, this portion of the interface will be practically equal to zero, which means that the second term of eqs 20 or 23 will be zero, namely, 1 1.87152/3 ≈ 0, applying model B. In reality, the value should be
