Consecutive-Parallel Reactions in Nonisothermal Polymeric Catalytic Membrane Reactors

2094 Ind. Eng. Chem. Res. 2006, 45, 2094-2107 Consecutive-Parallel Reactions in Nonisothermal Polymeric Catalytic Membrane Reactors José M. Sousa, and Adélio Mendes*, Departamento de Química, UniVersidade de Trás-os-Montes e Alto Douro, Apartado 202, 5001-911 Vila-Real Codex, Portugal, and LEPAEsDepartamento de Engenharia Química, Faculdade de Engenharia, UniVersidade do Porto, Rua Roberto Frias, 4200-465 Porto, Portugal This work reports the development of a nonisothermal and nonadiabatic pseudo-homogeneous model to study a completely back-mixed membrane reactor with a polymeric catalytic membrane, for conducting the consecutive hydrogenation of propyne to propene and then to propane. The performance of the reactor is analyzed in terms of the propyne concentration in the permeate stream (the only outlet stream from the reactor), the conversion of propyne and hydrogen, and the selectivity and overall yield to the intermediate product propene. The operating and system parameters considered are the Thiele modulus, the dimensionless contact time, the Stanton number, and the effective hydrogen sorption and diffusion coefficients. To define the regions where the catalytic membrane reactor may perform better than a conventional reactor, a comparison between both reactors is made. For the range of parameter values considered, the reactor model in this study demonstrates that the catalytic membrane reactor performs better than the conventional catalytic reactor in some regions of the Thiele modulus parametric space, for medium to high Stanton number values and for the total flowthrough configuration (total permeation condition). Concerning the effective sorption and diffusion coefficients of hydrogen, they shall be higher than the ones of the hydrocarbons. 1. Introduction Theoretically speaking, the combination of a chemical reaction and separation modules in a single processing unitsa catalytic membrane reactorsshould have several advantages over the conventional arrangement of a chemical reactor followed by a separation unit. Among other advantages that can be explored in this new configuration, 1 two main ones can be identified. In one hand, membrane reactors can be used to increase the overall conversion above the theoretical thermodynamic value for equilibrium-limited reactions, by a selective product removal. 2,3 On the other hand, a segregated feed of the reactants can be used to improve the selectivity andor overall yield to an intermediate product in complex reactive systems andor to control the reactor temperature, thus improving operation safety. 4-7 Most of the potential applications for this new technology refer to the ensemble of processes conducted at high temperatures, from 300 to 1000 C. 1,8 As a consequence, only inorganic ceramic or metallic membranes, able to operate in such harsh conditions, can be considered. Nevertheless, beyond the applications in the field of biocatalysis, 1,9 catalytic polymeric membranes can also be integrated into membrane reactors to be used in specific areas where processes are conducted in mild conditions, namely, in fine chemical synthesis 10,11 and partial hydrogenation of alkynes andor dienes to alkenes, 12-14 among others. 15-17 The focus of this study is on the performance of a catalytic polymeric membrane reactor, by comparing it with a conventional reactor, when conducting a consecutive-parallel reaction system given by Reaction 1: A + B f C Reaction 2: B + C f D Many commercially important chemicals are intermediate * To whom correspondence should be addressed. Tel.: +351 22 5081695. Fax: +351 22 5081449. E-mail: mendes@fe.up.pt. Universidade de Trás-os-Montes e Alto Douro. Universidade do Porto. products in consecutive-parallel reactions, which, in this case, is product C. The interest in using catalytic membrane reactors to study its overall yield andor selectivity has been increased among the scientific community, according to the number and diversity of papers available on the open literature. 1,4,7,18-21 Some of such studies focus on the improvements that can be achieved with a distributed feed of a reactant alongside a tubular reactor, 4,19,20 while others analyze the impact of a strategy based in a segregated feed of the reactants on the overall yield to the intermediate (desired) product. 7 All these works consider inorganic membranes, catalytic 7,18 or inert. 4,19-21 They report modeling, 4,7 experimental, 18,21 or both modeling and experimental works. 19,20 Another important class of consecutive-parallel reactions that have been studied in polymeric catalytic membrane reactors are selective hydrogenations. 12-14,22-24 For example, the selective hydrogenation of impurities such as propyne and propadiene in an industrial propene stream is an important reaction in the petrochemical industry. 12,25 As a monomer for the industrial production of polypropylene, the purified propene stream should contain <10 ppm of propadiene and <5 ppm of propyne. 12 On the other hand, typical industrial propene streams produced by steam cracking contain about 5% of such species, 25 which, ideally, should be removed selectively. Among the methods known for removal of alkynes and dienes, catalytic hydrogenation is the most elegant one. Under adequate conditions, it reduces the content of such species (they may even be completely removed), avoiding, however, the deeper hydrogenation to the correspondent alkane or other possible reactions. As a result, the overall yield to the olefins may even be improved. Moreover, selective hydrogenation is a relatively simple process to implement, is efficient, and is easy to operate. The main objective of this work is to analyze in which conditions a catalytic polymeric membrane reactor can take advantage of its effective diffusivity and sorption selectivities to improve its performance over that obtained in a conventional reactor, considering the consecutive-parallel hydrogenation 10.1021ie050650h CCC: $33.50 2006 American Chemical Society Published on Web 02212006

Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2095 Figure 1. Schematic diagram of the catalytic membrane reactor. propyne f propene f propane. More specifically, can one lower the concentration of propyne in the outlet stream under levels obtained using a conventional catalytic reactor? To answer this question, we developed a comprehensive model of a completely back-mixed catalytic membrane reactor. The mathematical model is nonisothermal and nonadiabatic, and it is based on the model of the solution-diffusion for the transport through the membrane. A set of simulation results are then provided which illustrate some key points about the use of this membrane reactor. In addition to the analysis of the concentration of propyne on the permeate stream, also discussed is the possibility of enhancing the selectivity and overall yield to the desirable intermediate product (propene) and the conversion of the main reactants propyne and hydrogen. In principle, the conversion of propyne and the selectivity to propene can benefit from a catalytic membrane reactor. Ideally, the concentration of the intermediate desired product (propene) in the reaction medium should be as low as possible, to avoid its hydrogenation to propane. This can be achieved, for example, by enhancing its diffusivity in the membrane andor lowering its concentration at the catalyst surface. On the other hand, it is important to hydrogenate the contaminant propyne to a very low level. So, its diffusivity should be as low as possible (to increase its residence time), and the concentration in the reaction medium should be as high as possible. However, as the hydrocarbons propyne and propene are chemically alike, it is expected that their diffusivity and sorption coefficients are close. Relative to the remaining reactant, hydrogen, the desirable effective values for the diffusivity and sorption coefficients are dependent on the reaction considered. On one hand, hydrogen must react with propyne, to eliminate it. On the other hand, it is important to avoid the hydrogenation of propene. Considering that hydrogen has a positive order on both hydrogenations, its concentration at the catalyst sites should be high for the first reaction (hydrogenation of propyne) and low for the second reaction (hydrogenation of propene). Thus, the desirable values for effective diffusivity and sorption of hydrogen must be balanced and found from a complete numerical analysis. 2. Development of the Membrane Reactor Model The catalytic membrane reactor considered in this study has the general features of the one depicted in Figure 1. It consists of retentate and permeate perfectly mixed chambers, separated by a flat membrane with surface area A m and thickness δ. The perfectly mixed flow pattern assumption considerably simplifies the analysis, allowing an easier assessment of the impact of the different diffusivity and sorption selectivities on the reactor performance. If tubular membranes with plug-flow pattern were considered, the complexities introduced by concentration gradients along the length of the membrane would make the analysis much more complex. The reaction studied is of the type consecutive-parallel and irreversible, describing the hydrogenation of propyne, Reaction 1: A + B f C Reaction 2: B + C f D with A ) propyne, B ) hydrogen, C ) propene, and D ) propane. The model proposed for this reactor is based on the following additional main assumptions: (1) steady-state and nonisothermal conditions; (2) negligible film transport resistance for mass and energy; (3) negligible drop in the total pressure for the retentate and permeate sides; (4) Fickian transport through the membrane thickness; (5) sorption equilibrium between the bulk gas phase and the membrane surface described by Henry s law; (6) constant sorption and diffusion coefficients; 39-41 (7) unitary activity coefficients; (8) homogeneous distribution of the catalytic nanoparticles throughout the membrane; (9) reaction occurring only on the catalyst surface; (10) concentration of the reaction components on the catalyst surface and in the surrounding polymer matrix being equal (any relationship can be considered in principle, but this one simplifies the original problem without compromising the main conclusions). In principle, a change of the sorption coefficient of a given species in the membrane can lead to a change of the corresponding adsorption coefficient at the catalyst surface. However, we do not consider such an influence. The steadystate mass and energy balance equations are presented in the following sections. 2.1. Mass Balance for the Membrane. d 2 c 2 i D + i ν ij k j (T)f j (c i ) ) 0 i ) A, B, C, D (1) dz 2 j)1 where i refers to the ith component and j refers to the jth reaction, D is the effective diffusivity, c is the concentration inside the membrane (sorbed phase), and z is the spatial coordinate perpendicular to the membrane surface. ν is the stoichiometric coefficient, taken negative for reactants, positive for reaction products, and null for the components that do not take part in the reaction. k(t) is the reaction rate constant based on the conditions inside the membrane (temperature and concentration). f is the local reaction rate function, which is given by the following rate expressions: f 1 (c i ) ) c A c B i ) A, B, C (2) f 2 (c i ) ) c B i ) B, C, D (3) The justification for this proposal of rate expressions is provided below (Section 3.2). The reaction rate constants are assumed to follow the Arrhenius temperature dependence, (4) k 1 (T) ) k 0 1 exp( - E 1 RT) ) k 1 (T ref [ ) exp - E 1 R( 1 T - 1 T ref )] k 2 (T) ) k 0 2 exp( - E 2 RT) ) k 2 (T ref [ ) exp - E 2 R( 1 T - 1 T ref )] (5) where k 0 j and E j are the preexponential reaction rate constant and the activation energy for reaction j, respectively. R is the gas constant, and the subscript ref refers to the reference component or conditions. 2.2. Energy Balance for the Membrane. d 2 4 T dc λ + e ( i C dz 2 p,i (T)D i i)1 dz) dt 2 dz + (- H r j )k j (T)f j (c i ) ) 0 j)1 i ) A, B, C, D (6)

2096 Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 where λ e is an effective thermal conductivity that depends on both thermal conductivities of the solid and sorbed species, H r is the reaction enthalpy, and C p is the heat capacity in the sorbed phase. Following the assumption of negligible external transport limitations at the membrane surface (assumption 2), the boundary conditions for the mass and energy balances are 7 z ) 0, c i ) S i p i R, and T ) T R z ) δ, c i ) S i p i P, and T ) T P (7) where the superscripts R and P refer to the retentate and permeate stream conditions, respectively. S is the sorption coefficient, p is the partial pressure in the bulk gas phase, and δ is the membrane thickness. 2.3. Partial and Total Mass Balances for the Retentate Side. Q F F p i RT ) Q R R p i F RT - dc i R Am D i i ) A, B, C, D (8) dz z)0 Q F P F ) Q RT F R P R RT R - Am i)1 where Q is the volumetric flow rate, P is the total pressure, and A m is the membrane surface area. The superscript F refers to the feed stream conditions. 2.4. Energy Balance for the Retentate Side. 4 Q F p F F i H i i)1 RT F The first and second terms of eq 10 account for the enthalpy of the gas phase in the feed and retentate streams; the third term accounts for the enthalpy transported by the species that come in or out of the membrane; the fourth term accounts for the heat exchange by conduction between the bulk gas and the membrane surface; the fifth term accounts for the sorption enthalpy; and the last term accounts for the heat transfer between the bulk gas and a heat exchanger with a coolant temperature T ext. H is the enthalpy for the gas phase, and H s is the sorption enthalpy. U t,r and A t,r are the external overall coefficient and area of heat transfer, respectively, for the retentate side. 2.5. Partial and Total Mass Balances for the Permeate Side. 2.6. Energy Balance for the Permeate Side. 4 4 Q R p R R i H i ) i)1 RT R A m λ e dt dz z)0 dc i D i dz z)0 4 dc i - A m H i D i - i)1 dz z)0 i ) A, B, C, D (9) 4 dc i - A m (- H s i )D i + i)1 dz z)0 U t,r A t,r (T R - T ext ) i ) A, B, C, D (10) Q P P p i RT + dc i P Am D i ) 0 i ) A, B, C, D (11) dz z)δ Q P P P 4 dc i + RT P Am D i i)1 dz z)δ 4 Q P p P P i H i i)1 RT P ) 0 i ) A, B, C, D (12) 4 dc i + A m dt H i D i + A m λ e + i)1 dz z)δ dz z)δ 4 dc i A m (- H s i )D i + U t,p A t,p (T P - T ext ) ) 0 (13) i)1 dz z)δ The meaning of the various terms in eq 13 is identical to the corresponding ones in eq 10. 2.7. Dimensionless Equations. The model variables were made dimensionless with respect to the feed conditions (Q F, P F, and T F ), to component A (D A, S A, and C p,a ), and to the membrane thickness, δ. The external and feed temperatures were considered to be equal. The reference temperature was set to 298 K. It seems reasonable to consider that the temperature is uniform for the entire catalytic membrane reactor, in view of the usually low membrane thickness (few hundred microns, normally) and the perfectly mixed flow pattern assumption for both chambers. According to this hypothesis, the energy balances for the retentate and permeate chambers, eqs 10 and 13, respectively, can be simplified to a single global energy balance, eq 23. Changing for dimensionless variables and introducing suitable dimensionless parameters, eqs 1-13 become as follows, 1 d 2 T* Pe H where dζ 2 + ( i)1 d2 c 2 i D + i dζ 2 φ2 ν ij κ j (T*)f j (c i ) ) 0 (14) j)1 f 1 (c i ) ) c A c B f 2 (c i ) ) c B (15) (16) k 1 (T*) ) exp[γ(1-1t*)] (17) k 2 (T*) ) R r exp[r E γ(1-1t*)] (18) 4 dc C i p,i D i dζ) dt* dζ - ζ ) 0, c i ) S i p i R* and T* ) T R* Q F* p i F* T F* Q F* P F* T F* φ 2 β Pe H (κ 1 f 1 + R H κ 2 f 2 ) ) 0 (19) ζ ) 1, c i ) S i p i P* and T* ) T P* (20) ) Q R* R* p i - Γ T R* φ + 1 D dc i i (21) dζ ζ)0 ) Q P R* - Γ T R* 4 φ + 1 i)1 dc i D i dζ ζ)0 4 4 Q F* p F* i C p,i - Q R* p R* i C p,i + i)1 i)1 Γ 4 dc C i p,i D i T φ + 1i)1 dζ R* + ζ)0 1 φ + 1 Γ (22) Pe H( dt* dζ ζ)0 - dt* dζ ζ)1) - St(TR* - T F* ) ) 0 (23) Q P* p i P* T R* + Γ φ + 1 D i dc i Q P* P P* + Γ 4 dc i D i T R* φ + 1i)1 dζ ζ)1 dζ ζ)1 ) 0 (24) ) 0 (25)

Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2097 c i ) c i C ref, p i ) p i P ref, P* ) PP ref, D i ) D i D ref, C ref ) S ref P ref, Q* ) QQ ref, R E ) E 2 E 1, S i ) S i S ref, φ is the Thiele modulus referred to the first reaction and at the reference temperature (ratio between a characteristic intramembrane diffusion time, for the reference component, and a characteristic reaction time); Γ is the dimensionless contact time referred to species A (ratio between the maximum possible flux across the membrane for the reference component, that is, permeation of pure species against null permeate pressure, and its molar feed flow rate); R r is the ratio of the reaction rate constants at the reference temperature; γ is the Arrhenius number based on the first reaction; R H is the ratio of the heats of reaction; R E is the ratio of the activation energies; Pe H is the modified heat Peclet number; β is the Prater number based on the first reaction; φ is the ratio of the reactants composition in the feed stream (referred to the first reaction); and St is the Stanton number. The remaining symbols are reported in the nomenclature. The performances of the catalytic membrane reactor (CMR) and the conventional catalytic reactor (CSTR) are not directly comparable, as the reactors are different. However, we may state that a CMR with a nonpermselective membrane and operating in the total flow-through configuration (total permeation condition) is loosely equivalent to a CSTR. This equivalence is discussed below in Section 6.1, where the results of simulating both reactors are shown. The dimensionless equations for the CSTR are presented below: Q F* P F* where T* ) TT ref, R H ) H r 2 H r 1, C p,i ) C p,i C p,ref, 12 D ref ], Γ ) Am RT ref S ref D ref, F δq ref y A ζ ) zδ, φ ) δ[ C ref k 1 (T ref ) St ) Ut,G A t,g RT ref Q ref P ref C p,ref, φ ) y B F y A F, R r ) 1 C ref k 2 (T ref ) k 1 (T ref ), Pe H ) C ref C p,ref D ref, β ) H r 1C ref D ref, γ ) E λ e λ e T 1 (RT ref ) ref Q F* F* p i Q O* O* p i - T F* T O* - QO* P O* T F* T O* 2 + Da ν ij κ j (T*)f j (p O* i ) ) 0 (26) j)1 - Da(κ 1 (T*)f 1 (p O* i ) + κ 2 (T*)f 2 (p i O* )) ) 0 (27) 4 4 Q F* p F* i C p,i - Q O* p O* i C p,i - DaB(κ 1 (T*)f 1 (p O* i ) + i)1 i)1 R H κ 2 (T*)f 2 (p O* i )) - Da(k 1 (T*)f 1 (p O i ) C 1* p + k 2 (T*)f 2 (p i O* ) C p 2* ) - St(T O* - T F* ) ) 0 (28) f 1 (p i O* ) ) p A O* p B O* C 1* p ) C p,c C 2* p ) C p,d f 2 (p i O* ) ) p B O* - C p,b - C p,a - C p,c - C p,b Da ) P ref VRT ref k 1 (T ref ), B ) H r 1 Q ref T ref C p,ref (29) (30) (31) (32) Table 1. List of the Base Set Parameter Values Used in the Simulations D i ) 1 Pe H ) 0.05 γ ) 20 S i ) 1 B ) -9.096 φ ) 1.5 C p,a ) 1 Γ [0.01 - TPC] R E ) 1.5 C p,b ) 0.469 St [0.1-100] P P* ) 0.5 C p,c ) 1.054 φ [10-2 - 10 3 ] R r ) 0.001 C p,d ) 1.214 Da [10-2 - 10 5 ] R H ) 0.753 y F A ) 0.0465 y F B ) 0.0698 y F C ) 0.8837 Da is the Damköhler number (ratio between the rate of the first reaction at the reference temperature and the feed flow rate to the reactor), and B is the adiabatic temperature rise. 26 The superscript O refers to the reactor exit conditions. 3. Model Reaction System and Input Parameters Table 1 summarizes the default values and ranges of the parameters used in the simulations. Some of these values were calculated from literature data. Others were proposed based on qualitative knowledge of the reactions. In the following sections, we discuss our selection of such values. 3.1. Reaction System. The model reaction system simulated in this work corresponds to the hydrogenation of propyne to propene. The deeper hydrogenation of propene to propane is also considered, but other possible reactions (oligoisomerizations, for example) are ignored. 28 All reaction steps are assumed to be irreversible. 28 3.2. Reaction Kinetics. The simple power-law type kinetics for the reaction rate equations have been often proposed in the literature for catalytic reactions. However, these simple kinetic equations should be regarded rather as empirical correlations that predict the rates of reactions within the range of the experimental conditions. 4,29 For example, some published works report that the power-law type rate equations represent the experimental results better than the Langmuir-Hinshelwood- Hougen-Watson models for the same hydrogenation reactions as the one considered in the present study, even though they were carried out in packed-bed reactors (see refs 25, 28, and 30 as well as the references therein). Thus, we consider also power-law type kinetics in the present study. Although any type of reaction rate expression can be inserted into the catalytic membrane reactor model, this formulation considerably simplifies the problem, without compromising the main conclusions. We consider that the reaction rate of the first reaction depends on the concentration of both reactants, propyne and hydrogen, because their concentrations are set to close values in the present study (φ ) 1.5). For the second reaction, on the other hand, we consider zero order relative to the hydrocarbon. 25 As the concentration of propene is in very large excess relative to that of hydrogen and is almost constant during the reaction, we included it in the reaction rate constant. 25 Additionally, we consider that the reaction kinetics is kept constant over the entire range of simulated conditions. 4 We choose the activation energies based on experimental data, despite the fact that they show a large scattering. 25,28,30-32 We considered E 1 ) 50 kj mol -1 and E 2 ) 75 kj mol -1 (γ 20, R E ) 1.5). This ratio of activation energies is compatible with the decrease of the selectivity to the intermediate product with the increase of the reaction temperature. 30 Also, Jackson and Kelly 33 reported the same qualitative relation of activation energies in a propyne-to-propene hydrogenation study over platinumsilica catalysts. For reactions of this type, it is desirable that the ratio between the rates of the selective and unselective reactions (first and second hydrogenations, respectively), that is, the ratio k 1 (T)f 1 -

2098 Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 (c i )(k 2 (T)f 2 (c i )), should be as high as possible, to improve the selectivity to the intermediate product. From the work by Fajardo et al., 25 we considered a feed partial pressure of 5000 Pa for hydrogen and 95 000 Pa for propene and a temperature of 350 K. In these conditions, such a ratio took a value of 18 or 53, depending on the reaction-rate model considered. 25 The same ratio was also obtained with data from Godínez et al., 30 leading to values of 50-70. Thus, based on the feed composition and reference conditions considered in our study, we set the ratio of the reaction rates to 50. From this value, we calculated the ratio of the reaction rate constants, R r, by eqs 2-5, where the rate of reaction j is given by k j (T)f j (c i ). It should be emphasized that the present work intends to analyze the possible operating and system conditions where a catalytic polymeric membrane reactor can take advantage of its effective sorption and diffusivity selectivity to outperform a conventional catalytic reactor. Hence, only the ratios of the reaction rates and activation energies are truly necessary. A change of the respective values will change the values of conversion, selectivity, overall yield, and outlet composition. Yet, the influence of the different sorption and diffusion selectivities will be maintained, providing that the same relative trends of the reaction rates and activation energies are kept. 3.3. Membrane Features. We consider in the present work a hypothetical rubbery membrane filled with a nanosized catalyst. 14,34-36 The transport of a species through a homogeneous rubbery membrane can be described by the sorptiondiffusion model. 34-36 However, when solid particles (metallic clusters or zeolites, for example) are built-in in a rubbery membrane, the transport mechanism becomes more complex. 35,37,38,42 First, a decrease in the diffusivity due to the diffusion barrier created by the clusters, which act as inorganic fillers, should be considered. 35,37,38,42 On the other hand, the catalyst particles occluded in the membrane affect its overall sorption capacity, that is, the adsorption on the catalyst particles should also be considered in addition to the sorption on the polymeric phase. 37,42 To the best of our knowledge, sorption and diffusion coefficients in a nonporous polymeric catalytic membrane were not yet determined under the real conditions of sorption diffusionreaction. In few cases, only effective sorption and diffusion or permeation coefficients were reported. 13,34-36 Hence, we will also consider effective diffusion and sorption coefficients for the reaction species. We consider that all the hydrocarbons have identical effective diffusion coefficients in the membrane matrix. This assumption is based on the propenepropane diffusivity selectivities of 1.3 (measured by the authors on a PDMS homogeneous membrane) and 1.5 (reported for a polyethylene homogeneous membrane 43 ) and on the difficulty to find data of diffusivities for propyne. Concerning the effective sorption capacity, we also considered equal values for all hydrocarbons, based on the propenepropane selectivities of 0.90 (obtained by the authors in PDMS homogeneous membranes) and of 0.89 (reported for a polyethylene homogeneous membrane 43 ) and also on the difficulty to find data of sorption for propyne. Despite the fact that these selectivities are relative to homogeneous polymeric membranes, we consider that the presence of the catalyst built-in in the membrane affects the diffusivity and sorption of all hydrocarbons in the same way. In fact, Theis et al. 44 report a constant propene propane permselectivity of 1.1 for PDMS membranes, pure and built-in with different amounts of palladium nanoclusters. For hydrogen, we consider an effective diffusivity 10 times higher than the one for the hydrocarbons, based on a hydrogen propane diffusivity selectivity of 11 obtained by the authors in a homogeneous PDMS membrane. We assume also that the influence of the catalyst particles on the diffusivity of hydrogen is equivalent to the one considered for the hydrocarbons. For the sorption capacity, a hydrogenpropane selectivity of < 1 was obtained by the authors and by Merkel 45 for a homogeneous PDMS membrane. However, the hydrogen sorption capacity of a catalytic membrane built-in with palladium catalyst can increase strongly 46,47 (the other components sorb poorly on palladium). Because we are assuming equal concentration on the catalyst surface and in the surrounding polymer matrix, we may consider a wide range of values for the effective sorption coefficient of hydrogen. Thus, we consider values higher than, equal to, and lower than the effective sorption coefficient for the hydrocarbons. 3.4. Thermal Parameters. We obtained the heat capacities and heats of reaction from the literature at the reference temperature. 48 The molar heat capacity of each component is considered constant within the range of temperatures and pressures considered and independent of being in the sorbed or gas phases. A more exact dependence of the heat capacity with temperature could be considered, but this extra complexity would not change the main conclusions. Moreover, the operating temperature of the reactor considered in this study is close to the reference one. We also considered the sorption enthalpy constant with temperature. Accordingly, H i 0,s + (- H i s ) ) H i, where H i 0,s and H i are the enthalpies of component i in sorbed and gaseous phases, respectively, at the reference temperature. We arbitrated a value for the modified heat Peclet number in such a way that the temperature change across the membrane thickness is small. This is a reasonable assumption, since the thickness of these membranes is normally small (usually less than a few hundred microns). The ratio between the Prater number and the modified heat Peclet number was replaced by the adiabatic temperature rise (already defined in Section 2.7): βpe H ) H 1 r (C p,ref T ref ) ) B. In this way, parameter B directly reflects the heat of the first reaction. Moreover, this choice eliminates the need to assign a value to the reference diffusion coefficient (to set the Prater number). 3.5. Feed and Operation Conditions. The feed stream to the reactor is made up of propyne, propene, and hydrogen. The ratio between concentrations of hydrogen and propyne, φ, was set to 1.5. The concentration of propyne in a mixture of propyne and propene was set to a value of 5%. 25 The feed volumetric flow rate was varied over a wide range, by changing the dimensionless contact time parameter. The maximum value for this parameter is defined by the total permeation condition (TPC), that is, when the retentate volumetric flow rate is zero (total flow-through configuration). 4. Analysis Strategy As mentioned in the Introduction, it is crucial to determine what improvements can be achieved with our hypothetical membrane reactor (CMR) when compared with the more conventional counterpart. Conventional, in the context of this work, means a catalytic gas-phase reactor with perfectly mixed flow pattern (CSTR) fed with the same stream, with the same reactions taking place at the catalyst surface and described by equivalent kinetic equations. Given the rather large number of parameters that describe a catalytic membrane reactor, it is impractical to perform a complete parametric sensitivity analysis. However, the main

Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2099 Figure 2. Conversion of hydrogen (A) as a function of the Stanton number and Thiele modulus for the catalytic membrane reactor at the total permeation condition and (B) as a function of the Stanton number and the Damköhler number for the conventional catalytic reactor. The other parameters have the values defined in Table 1. objective of this work can be achieved by varying only a few key parameters. These include the effective sorption coefficient of hydrogen (all the other sorption and diffusion coefficients are fixed, as discussed in Section 3.3), the contact time, the Thiele modulus, and the Stanton number. The study will focus primarily on the analysis of the concentration of propyne in the outlet stream (for a total flow-through configuration), because, as was stated in the Introduction, this is the key variable in an industrial purified propene stream for the production of polypropylene. Additionally, we also perform the analysis of the selectivity and overall yield to the intermediate propene, as well as the conversion of propyne and hydrogen. The conversion of a component i in the membrane reactor is given by X i ) 1 - QR* p i R* T R* + Q P* p i P* T P* Q F* p i F* T F* (33) There are different ways to define the selectivity to an intermediate product. In our work, we consider the selectivity defined as the net moles of C produced per mole of A reacted 4 (this quantity is also known as relative yield 27 ). For the membrane reactor, it is given by σ C ) QR* p C R* T R* + Q P* p C P* T P* - Q F* p C F* T F* Q F* p A F* T F* - (Q R* p A R* T R* + Q P* p A P* T P* ) The overall yield to the species C is defined as the net moles of C produced per mole of A fed. For the membrane reactor, it is given by Y C ) QR* p C R* T R* + Q P* p C P* T P* - Q F* p C F* T F* (34) Q F* p A F* T F* ) X A σ C (35) A negative value for both σ C and Y C means that all of species A (propyne) and part of species C (propene) fed to the reactor are converted in the final reaction product D (propane). 5. Resolution of the Model Equations The general strategy considered in this work for solving the model equations is the same that was used before. 49 The steadystate eqs 14, 19, 21, 23, and 24 were transformed into pseudotransient ones, while eqs 22 and 25 were solved explicitly in order to calculate the volumetric flow rate. Equations 14 and 19 were subsequently transformed into a set of ordinary differential equations in time by a spatial discretization using orthogonal collocation 50 in a transformed mesh with 11 internal collocation points. 49 The time integration routine LSODA 51 was then used to integrate the resulting set of equations until a steady-state solution was reached. The corresponding equations of the fixed bed reactor, eqs 26-28, were solved using the same strategy. 6. Results and Discussion 6.1. Catalytic Membrane Reactor with a Nonpermselective Membrane. As mentioned in Section 2.7, we may assume that the results of the variables under study obtained for the conventional catalytic reactor (CSTR) are loosely equivalent to the ones obtained for the catalytic membrane reactor (CMR) operating at the total permeation condition and with a catalytic membrane not showing any diffusivity and sorption selectivities. We may conclude that such an equivalence is realized by comparing the simulation results of the hydrogen conversion (Figure 2), selectivity to propene (Figure 3), and reactor temperature (Figure 4) for both reactors, as a function of the Thiele modulus (for the CMR) or the Damköhler number (for the CSTR) and Stanton number parameters. From this point on, we consider this CMR as the reference reactor, which will be named throughout the text as equivalent conventional catalytic reactor or ECSTR.

2100 Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 Figure 3. Selectivity to propene (A) as a function of the Stanton number and the nthiele modulus for the catalytic membrane reactor at the total permeation condition and (B) as a function of the Stanton number and the Damköhler number for the conventional catalytic reactor. The other parameters have the values defined in Table 1. Figure 4. Temperature of the reactor (A) as a function of the Stanton number and the Thiele modulus for the catalytic membrane reactor at the total permeation condition and (B) as a function of the Stanton number and the Damköhler number for the conventional catalytic reactor. The other parameters have the values defined in Table 1. The analysis of the results presented in Figures 2-4 for the membrane reactor (left half), as well as the corresponding ones for the propyne conversion and propene overall yield (not shown here), allows us to define the most suitable regions for operating the membrane reactor. According to the specifications of the outlet stream in terms of the concentration of propene, it is desirable to maximize the conversion of propyne and minimize that of propene, that is, maximize the selectivity to the intermediate product. Therefore, we selected the parametric space of intermediate-to-high Thiele modulus (10-1000) to ensure significant reactants conversion and intermediate-to-high Stanton number (10-100) values. We selected a minimum value of 10 for the Stanton number to assess the influence of some temperature rise in the performance of the reactor. In fact, for the reaction system considered in this study, an increase in the catalyst temperature has a detrimental effect on the selectivity of the partial hydrogenation product (see Figures 3 and 4). This is typical, for example, in most hydrocarbon partial hydrogenation and oxidation systems. This trend is due in part to the relative values of the activation energies. That is, the activation

Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2101 Figure 5. (A) Conversion of propyne and (B) temperature of the reactor as a function of the dimensionless contact time and the Thiele modulus for the catalytic membrane reactor, with St ) 10. The other parameters have the values defined in Table 1. Figure 6. Molar fraction of propyne in the (A) retentate and (B) permeate streams as a function of the dimensionless contact time and the Thiele modulus for the catalytic membrane reactor, with St ) 10. The other parameters have the values defined in Table 1. energy of the species C produced in reaction 1 (propyne hydrogenation) is less than the activation energy of the species C consumed in reaction 2 (propene hydrogenation). As the reaction temperature increases, the rate of reaction 2 increases more than the rate of reaction 1, because of such relative activation energies. Anyway, the operating reactor temperature is not far from the feed temperature in the selected parametric region. The discontinuities evidenced in Figures 2-4, along the Stanton number parameter (for the lower values) and along the Thiele modulusdamköhler number (for intermediate values), represent a jump from the lower-temperature steady state to the higher-temperature steady state. This jump occurs because of the existence of a region of multiple steady states, 27,52 typical of exothermic reactions. However, the analysis of this operating region is out of the scope of the study. Moreover, the Thiele modulus and Stanton number parametric space chosen to perform the analysis is far from the region where multiple steady states occur. The results shown in Figures 2A-4A were obtained in the total permeation condition. Nevertheless, a scanning of the dimensionless contact time parameter should also be performed, to identify its optimal operating range. Following this, we show in Figure 5 the propyne conversion (left half) and the reactor

2102 Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 Figure 7. Ratio between the quantities (A) conversion of propyne and hydrogen, (B) molar fraction of propyne in the permeate stream, (C) selectivity to propene, and (D) overall yield to propene in the CMR and in the ECSTR, as a function of the Thiele modulus and for different Stanton number values, for D B ) 10 and Γ corresponding to the total permeation condition. The other parameters have the values defined in Table 1. temperature (right half) and in Figure 6 the molar fraction of propyne in the retentate (left half) and permeate (right half) streams, all as a function of the Thiele modulus and dimensionless contact time and for a medium Stanton number value (St ) 10). The results presented previously reveal several features, which are primarily a consequence of one or more of the following three factors: Factor I: Influence of the Dimensionless Contact Time. The retentate flow rate decreases with an increase of the dimensionless contact time. Thus, the relative permeant flux across the membrane increases according to the balance, directly affecting the retentate composition: the concentration of the reaction products increases, while the concentration of the main reactants (propyne and hydrogen) decreases. Factor II: Influence of the Catalytic Activity. The reaction rate depends directly on the Thiele modulus and on the concentration of the main reactants. Yet, the concentration of the reactants is also a function of the Thiele modulus, that is, the reactants are consumed in a rate depending on the catalytic activity. Thus, the effective change of the reaction rate across the membrane depends directly on the balance between these two opposite trends. The catalytic activity influences also the fraction of the membrane thickness effectively used in the chemical reaction. Because of the increasing depletion of the main reactant hydrogen with the increase of the Thiele modulus, the reaction takes place in an effectively narrower and narrower fraction of the catalytic membrane. As a consequence, there is an enrichment of the retentate stream in relation to the reaction products, with the corresponding depletion of the main reactants. Factor III: Relative Extension of Reactions 1 and 2. If the main reactants hydrogen and propyne exist in the reaction medium in relative excess, that is, if the reaction occurs in a kinetic-controlled regime, the extension of each reaction depends essentially on the kinetic parameters. As the operation regime changes from kinetic- to diffusion-controlled, the competition between both reactions for the limiting reactant (hydrogen) becomes more and more important. Because of the different sensitivity of the selective (reaction 1) and unselective (reaction 2) reactions concerning the reactants concentration, the result of such a competition is favorable to the unselective reaction, as its reaction rate depends only on the concentration of hydrogen, while the selective reaction rate depends on the concentrations of propyne and hydrogen. Figure 6A shows that the propyne molar fraction in the retentate stream is a monotonically decreasing function, either with the dimensionless contact time for a fixed Thiele modulus value (primarily a consequence of factor I) or with the Thiele modulus value for a fixed dimensionless contact time (primarily a consequence of factor II). Concerning the propyne molar fraction in the permeate stream, it shows a much more complex dependence on the dimensionless contact time, for a fixed Thiele modulus value (Figure 6B). This dependence is the one expected for the lower Thiele modulus values, due primarily to factor I. That is, a decrease of the main reactants concentration on the retentate stream, for a kinetic-controlled regime, leads to a decrease of the driving force accountable for the transport across the membrane and, consequently, to a decrease of its concentration on the permeate stream. When increasing the Thiele modulus value, the propyne concentration in the permeate stream decreases with the dimensionless contact time until it reaches a minimum, due essentially to factors I and II. After this turning point, it increases continuously until the dimensionless contact time reaches the total permeation condition, now due primarily to factor III, even though there is also a secondary influence of factors I and II. The influence of factor III extends progressively as the Thiele modulus increases, leading to a decrease of the turning point corresponding to the minimum molar fraction of propyne. Figure 6B shows also a decreasing trend of the propyne molar fraction with the Thiele modulus, for a fixed dimensionless contact time, until a minimum value. Such behavior is now due essentially to factor II. The subsequent increase is a consequence of the increasing importance of factor III. This trend can be

Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2103 Figure 8. Ratio between the quantities (A) conversion of propyne and hydrogen, (B) molar fraction of propyne in the permeate stream, (C) selectivity to propene, and (D) overall yield to propene in the CMR and in the ECSTR, as a function of the Thiele modulus and for different Stanton number values, for D B ) 10, S B ) 10, and Γ corresponding to the total permeation condition. The other parameters have the values defined in Table 1. also realized in Figure 5A for the total permeation condition. Even though the conversion depends simultaneously on the upstream (retentate) and downstream (permeate) flow rates, the influence of the upstream rate is higher. It should be emphasized that a nonzero retentate flow rate means that a fraction of the propyne and hydrogen fed to the reactor is lost, which leads to a decrease of its conversion. The maximum attainable propyne conversion approaches a level that depends on the relative extent of the selective reaction and on the feed composition ratio (φ), while for hydrogen, the maximum attainable conversion is 100% (Figure 2A), due essentially to factors II and III. We can see from Figure 6 that the retentate stream is much richer in propyne than the permeate stream, on one hand, and that the level of propyne in the retentate or permeate streams changes considerably. A decision about the conditions to operate the reactor must be taken based on the specification of the outlet streams. 6.2. Catalytic Membrane Reactor with a Permselective Membrane. To define the membrane characteristics range as well as the operating conditions where the CMR outperforms the CSTR, a criterion based on the ratio between the propyne molar fractions in the outlet streams of both reactors (permeate stream for the CMR) will be considered. The performance of the CMR improves relative to the ECSTR (equivalent to the CSTR) if, for a certain region of the Thiele modulus-stanton number plane, such a ratio is <1. Additionally, equivalent ratios for the selectivity and overall yield to the intermediate product propene and for the conversion of the main reactants propyne and hydrogen are also presented. In these last cases, the membrane reactor performs better if the respective ratios are >1. The range of the effective values for the dimensionless sorption and diffusion coefficients considered in the simulations was stated in Section 3.3. 6.2.1. Higher Hydrogen Diffusion Coefficient. We are going to study in this section the case where the effective diffusion coefficient of hydrogen is 10 times higher than that of the hydrocarbons, with equal effective sorption coefficients for all components. In this way, we are analyzing only the influence of a higher hydrogen diffusion coefficient. Figure 7 shows the simulation results for this case. Globally, these results show that there is a region on the Thiele modulus-stanton number parametric space where the CMR performs better than the ECSTR, in terms of the propyne molar fraction in the outlet stream (Figure 7B). Moreover, the selectivity (Figure 7C) and the overall yield (Figure 7D) to the intermediate product are also improved, in the entire and in part of the parametric region, respectively. These results are the conjoint expression of the above-mentioned factors II and III and of factor IV, described next. Factor IV: Diffusion Rate of the Main Reactants Hydrogen and Propyne. An increase of the hydrogen diffusivity (corresponding to an increase of the respective permeability) leads to a faster transport of this component through the membrane. As a consequence, the concentration of hydrogen decreases, either in the retentate stream or in the reaction medium. The global effect of this decrease of the hydrogen concentration is a decrease of the global reaction rate and an enhancement of the selective reaction (the two reactions show different sensitivities toward a change in the hydrogen concentrationssee factor III description). A more detailed analysis of Figure 7 reveals several more features. Let us consider first the region of the lower Thiele modulus values, where the conversion of hydrogen and propyne is lower in the CMR than in the ECSTR (Figure 7A). This trend is due essentially to the influence of factor IV. However, the influence of the higher hydrogen diffusion coefficient is more marked for hydrogen conversion than for propyne conversion (Figure 7A), resulting in a more selective production of the intermediate product propene (Figure 7C). The molar fraction of propyne in the permeate stream is also penalized in this region, essentially because of the lower propyne conversion (Figure 7B). By increasing the Thiele modulus, different trends can be observed. The hydrogen conversion in the CMR increases

2104 Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 Figure 9. Ratio between the quantities (A) conversion of propyne and hydrogen, (B) molar fraction of propyne in the permeate stream, (C) selectivity to propene, and (D) overall yield to propene in the CMR and in the ECSTR, as a function of the Thiele modulus and for different Stanton number values, for D B ) 10, S B ) 0.1, and Γ corresponding to the total permeation condition. The other parameters have the values defined in Table 1. continuously until a value equal to the one attained in the ECSTR (Figure 7A). This result was expected and can be explained by the combined influence of factors II and IV. On the other hand, the conversion of propyne in the CMR increases until a value higher than the one attained in the ECSTR (Figure 7A), for intermediate Thiele modulus values. This evolution is also a consequence of factors II and IV. Increasing the Thiele modulus more leads to a slow decrease of the conversion of propyne, because of the competition between both selective and unselective reactions, as pointed out in factor III. The propyne molar fraction in the outlet stream directly reflects this evolution of the propyne conversion, Figure 7B. The selectivity to the intermediate product propene is always improved in the CMR, because the higher hydrogen diffusivity is more detrimental for hydrogen conversion than for propyne conversion, due to factors II and IV. The trend shown in Figure 7C results from the balance between the evolution of the conversions of propyne and hydrogen. That is, the relative propyne conversion grows faster than the relative hydrogen conversion for the lower Thiele modulus values, causing the selectivity to increase. Above a given Thiele modulus value, the relative hydrogen conversion grows faster than the relative propyne conversion and the selectivity decreases. The minimum propyne molar fraction in the permeate stream is favored by the lower temperatures (Figure 7B), because the ratio of the activation energies favors the unselective reaction, as was already referred in Section 6.1. However, there is a region where the higher temperatures (for the range considered) are more favorable for the propyne conversion, as can be concluded from Figure 7A for the different Stanton number values and, consequently, for the propyne molar fraction in the outlet stream (Figure 7B). This is not an inconsistency, but the effect of factors II and IV, associated with the increase of both reaction rates with the temperature. That is, a higher hydrogen diffusion coefficient leads to a decrease of its concentration inside the membrane (factor IV) and, consequently, to a decrease of the global reaction rate. This negative effect can be partially canceled out by increasing the temperature. As the Thiele modulus approaches its upper limit, the reaction tends to occur in an infinitesimal fraction of the membrane thickness at the retentate surface. In this region, the advantage of the diffusion selectivity of hydrogen vanishes completely. As a result, the CMR has no advantage over the ECSTR when φ f. 6.2.2. Higher Hydrogen Diffusion and Sorption Coefficients. We are going to study in this section the case where both effective diffusion and sorption coefficients of hydrogen are 10 times higher than those of the hydrocarbons. Figure 8 shows the simulation results for this case. Globally, these results show that the propyne conversion reached in the CMR is never penalized in the entire Thiele modulusstanton number parametric space. The propyne molar fraction in the outlet stream is also enhanced for the same parametric region, though more marked for the lower Thiele modulus values. Concerning the selectivity and overall yield to the intermediate product, they are considerably penalized for the lower-to-medium Thiele modulus values and slightly favored in the region of medium-to-high Thiele modulus values. Additionally, the conversion of hydrogen attained in the CMR is improved for the lower-to-medium Thiele modulus values. For the higher Thiele moduli region, the hydrogen conversion reaches the ECSTR value, as in the case discussed in the previous section. These global trends are a consequence of the conjoint influence of factors II, III, and IV and of factor V, described next. Factor V: Sorption Coefficient of the Main Reactant Hydrogen. An increase of the hydrogen sorption capacity by the membrane (corresponding to an increase of the respective permeability) leads not only to a faster transport from the retentate to the permeate chambers but also to an increased intramembrane concentration. Because this last factor is much more important than the increase of the permeability, there is