Ind. Eng. Chem. Res. 2004, 43, 309-314
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Optimal Feed Distribution in a Packed-Bed Membrane Reactor: The Case of Methanol Oxidative Dehydrogenation Victor Diakov and Arvind Varma* Department of Chemical Engineering and Center for Molecularly Engineered Materials, University of Notre Dame, Notre Dame, Indiana 46556
For catalytic methanol oxidative dehydrogenation to formaldehyde, the packed-bed membrane reactor (PBMR) productivity is enhanced by modifying the oxygen feed distribution. An experimentally validated PBMR model is used for this purpose. The optimal membrane feed distribution is calculated by both conventional Euler-Lagrange and local optimization approaches. The latter offers a significant reduction in the number of equations to be solved, allowing one to identify the important factors influencing the optimal distribution profile. Both approaches produce identical distribution patterns and demonstrate the superiority of variable versus uniform permeability membranes. The calculated optimal permeability profiles provide higher oxygen feed at the reactor entrance, where elevated methanol concentration allows for increased reaction rate, as compared to regions downstream. The analysis confirms that the variation in the reaction rates, caused by consumption and/or dilution of the directly fed reactant (methanol), is the most significant factor leading to the nonuniformity in optimal feed distribution of the other reactant (oxygen) along the reactor. It is shown that, as compared to a uniform membrane, optimizing the feed distribution may result in a 2-3-fold increase in the overall reactor productivity. 1. Introduction Efforts to improve the packed-bed reactor performance by distributing reactant feed to the catalyst bed1 via inert inorganic membranes are supported in various publications, where the advantages of packed-bed membrane reactors (PBMRs) over conventional fixed-bed reactors (FBR) are demonstrated.2,3 When compared with the FBR at identical feed,4 equal conversion,5 or optimal feed conditions,6,7 the PBMR performance is often superior to that of the FBR. However, the benefits of a membrane-distributed feed to catalytic packed-bed reactors are not fully explored. Thus far, little attention has been devoted to the experimental means of tailoring optimal membrane permeation patterns and even less to their quantitative assessment.8 As demonstrated experimentally for methane coupling9 and for oxidative dehydrogenation of butane,10 membranes with nonuniform permeation patterns can further enhance the packed-bed reactor performance. Clearly, any membrane reactor can potentially benefit from the adjustment of the membrane permeation pattern to the kinetic parameters of the reaction system. Recently, our efforts have been focused on the quantitative analysis and modeling of PBMR and FBR for the industrially important reaction of methanol oxidative dehydrogenation over a Fe-Mo oxide catalyst governed by the consecutive reaction network
CH3OH + O2 9 8 CH2O + H2O + 1/2O2 9 8 CO + 2H2O 1 2 The plug-flow PBMR model, employing experimentally determined reaction kinetics, was found to predict the reactor performance accurately.11,12 * To whom correspondence should be addressed. Tel.: 1-574631-6491. Fax: 1-574-631-8366. E-mail:
[email protected].
Figure 1. Schematic diagram of the PBMR.
The schematic diagram of a packed-bed reactor with membrane-distributed addition of a reactant is shown in Figure 1. The PBMR can operate in two modes: with either the oxygen (PBMR-O) or methanol (PBMR-M) fed through the membrane and the coreactant directly over the catalyst. In our case, as is typically expected for partial oxidation reactions, the PBMR-O performance is superior to that of the FBR and PBMR-M.7,12 In the present work, we investigate the further optimization of PBMR-O productivity by modifying the membrane permeation pattern. The plug-flow PBMR model, experimentally validated for ethylene epoxidation13 and methanol partial oxidation12 reaction networks, allows one to determine the optimal permeation pattern by solving the corresponding Euler-Lagrange equations. In the present work, we calculate the optimal membrane feed distribution and compare the performances of membrane reactors with optimal versus uniform permeation patterns. 2. Nonuniform Membrane Reactor Model The details of the experimental setup for methanol oxidative dehydrogenation over a Fe-Mo oxide catalyst in a PBMR are given elsewhere.11 In this configuration, the overall feed via the membrane is fixed. Therefore, we focus on relative changes of the permeability distribution along the membrane rather than on the absolute permeability values.
10.1021/ie0208624 CCC: $27.50 © 2004 American Chemical Society Published on Web 08/07/2003
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t0 dv + η(s) ) {r1(a,c) - r2(b,c)} ds 2pT
The species mass balances for a steady-state plugflow PBMR model,12 modified to account for the nonuniform permeation pattern, are given by
d(vyi) ds
[∑ ]( ) Nr
)
Fij
j)1
t0
CT
+ η(s) xi; i ) 1, ..., Ns
(1)
where yi and xi are the mole fractions of species i inside the reactor and membrane streams, respectively, and η(s) characterizes the local membrane permeability. These equations describe a packed-bed reactor with a thick (∼1 mm) macroporous (0.2 µm pore diameter) membrane. The macroporosity provides essentially viscous flow in the pores, while longer pores (thick membrane) inhibit backdiffusion. In this manner, the composition of the permeate equals that of the membrane feed (Figure 1). The case of thin and/or microporous membranes (not considered here), where both permeating flux and its composition depend on the fluid-phase concentrations in the reactor, requires a more complex approach. Equation 1 does not account for radial and axial mixing, as well as intrapellet mass- and heattransfer limitations, thus describing chemical transformations in thin (e10 pellet diameters), long (g50 pellet diameters) catalyst layers operating isothermally under kinetic control. Note that model predictions were validated experimentally for a range of feed compositions, flow rates, and temperature for a packed-bed reactor with a uniform membrane.12 In vector form, eq 1 yields
d(vy) ) g(y) + η(s) x ds
[ ] Nr
gi )
t0
Fij ∑ C j)1
(3)
T
represents the dimensionless formation rate of species i. In the case of a uniformly distributed membrane flux, η ) q, the volumetric feed fraction permeating the membrane, transforming expressions (1) and (2) to the form presented earlier.12 For the methanol partial oxidation reaction, the rate expressions for reactions 1 and 2
c r1(a,c) ) 0.0475e-17630(1/T-1/485)a0.5 K+c r2(b,c) )
2.91e-8710(1/T-1/485)bc [1 + 3.89e-5920(1/T-1/485)c]2
(4)
are incorporated in the following mass balance equations for partial pressures of methanol (a), formaldehyde (b), and oxygen (c) and the flow rate (v):
d(va) ) -r1(a,c) t0 + η(s) xa pT ds d(vb) ) {r1(a,c) - r2(b,c)}t0 + η(s) xb pT ds t0 d(vc) ) {r1(a,c) - r2(b,c)} + η(s) xc pT ds 2
While in the original kinetics12 K ) 0, in eq 4, to facilitate the numerical solution, a virtual oxygen concentration dependence was introduced. For K ) 10-7 bar, the modification is insignificant at any detectable oxygen concentration level, yet it makes the numerical simulation easier by avoiding the necessity to force r1 ) 0 at c ) 0, which yields a discontinuity in r1. Because there are two reactions involved, only two variables in eq 5 are independent. We choose to express c and v through a and b, leading to
d(va) ) -r1(a,c) t0 + η(s) xa pT ds d(vb) ) {r1(a,c) - r2(b,c)}t0 + η(s) xb pT ds v ) vf + [vf(af + bf/2) - v(a + b/2)]/ pT + (1 + xa + xb/2)χ(s) vc ) v(a + b/2) + vf(cf - af - bf/2) + (xc - xa - xb/2)χ(s) pT (6) where
χ(s) )
(2)
where
(5)
∫0sη(τ)dτ
(7)
and the subscript f refers to feed values. The last two expressions in eq 6 relate the oxygen flux (vc) and the overall flow rate (v) to the fluxes of methanol (fa ) va) and formaldehyde (fb ) vb) and the integral membrane permeability [χ(s)], allowing one to express species concentrations and reaction rates in terms of fa, fb, and χ(s):
r1(a,c) ) r1[fa/v(fa,fb,χ),c(fa,fb,χ)] ) r1(fa,fb,χ) r2(b,c) ) r2[fb/v(fa,fb,χ),c(fa,fb,χ)] ) r2(fa,fb,χ)
(8)
From now on, we focus our attention on the PBMR-O because this configuration provides superior performance.7 In this case, there is no oxygen or formaldehyde fed directly to the catalyst bed (cf ) bf ) 0), and no methanol or formaldehyde fed via the membrane (xa ) xb ) 0). Then, the differential equations in eq 6 do not include η(s), which facilitates their numerical integration. 3. Calculation of the Optimal Membrane Permeation Pattern We now use the PBMR model to optimize formaldehyde production
∫01φ ds
(9)
φ ) r1 - r2
(10)
I) where
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along with constraints arising from eq 6:
h1 ) h2 )
d(fa) + r1(fa,fb,χ) t0 ) 0 ds
d(fb) - {r1(fa,fb,χ) - r2(fa,fb,χ)}t0 ) 0 ds
(11)
For the Lagrangian defined as
Λ ) (r1 - r2)t0 + λ1(s) h1 + λ2(s) h2
(12)
the corresponding Euler-Lagrange equations are14
[ ] [ ]
∂Λ d ∂Λ )0 ∂fa ds ∂f′a
Figure 2. Optimal membrane permeation profile for q ) 0.05, 235 °C, 10% methanol, 5% oxygen feed, and 100 sccm overall feed flow rate.
∂Λ d ∂Λ )0 ∂fb ds ∂f′b ∂Λ d ∂Λ )0 ∂χ ds ∂η
[ ]
(13)
The substitution of eq 11 into eq 12 and then eq 12 into eq 13, with subsequent redefinition of Lagrange multipliers
µ1 ) 1 + λ1 - λ2 µ2 ) λ2 - 1
(14)
leads to the following expressions:
(
)
(
)
∂r1 ∂r1 ∂r2 ∂r2 dµ1 ) µ1t0 + µt ds ∂fa ∂fb ∂fa ∂fb 2 0 ∂r2 dµ2 ∂r1 µt + µt ) ds ∂fb 1 0 ∂fb 2 0 -µ1
∂r1 ∂r2 ) µ2 ∂χ ∂χ
(15)
Note that eqs 11 and 15 together provide a complete set to solve for the five unknowns: fa, fb, χ, µ1, and µ2. These ordinary differential equations were integrated numerically with initial conditions at s ) 0 and χ ) 0, component fluxes dictated by feed composition, and arbitrary initial conditions for Lagrange multipliers µ, further determined by maximizing the formaldehyde flux at the reactor exit (s ) 1). As can be seen from eq 15, increasing µ1 and µ2 by the same factor does not affect the resulting concentration and permeation profiles. Therefore, when the maximal formaldehyde yield is searched for, the initial value of µ2(s)0) ) µ2,f was varied, while keeping µ1,f ) 1. The numerical solver was built in Matlab and verified using the standard procedure described by Roache.15 The numerical deviation did not exceed 0.01% for formaldehyde yield and permeability distribution. Over a wide range of reactor operating conditions, it was determined that the oxygen concentration along the reactor in the optimized case is essentially zero. For example, for the conditions corresponding to Figure 2, c(s) < 2 × 10-7. The resulting optimal membrane feed distribution profile is presented in Figure 2 and exhibits a linearly decreasing permeability along the reactor.
This decrease results from the fact that more oxygen is needed to support reaction 1 at the reactor entrance, where the methanol concentration is higher. The linearity of this dependence is a specific result of the limit of small q and dilute solutions and is explained below. Because the Lagrange multipliers are additional parameters not related directly to reactor operating conditions, no estimates of the accuracy of their numerical calculation were made. 4. Local Membrane Reactor Optimization 4.1. Analysis of Limiting Cases and Analytical Solutions. The optimal permeation pattern can be obtained analytically in the limit K f 0 (eq 4). In this case, the kinetics of methanol oxidative dehydrogenation provide maximal formaldehyde production at oxygen concentrations approaching zero when (i) the second reaction rate is negligible and (ii) there is still enough oxygen to support the first reaction; i.e., the oxygen supply via the membrane equals its consumption by the first reaction. By satisfaction of these conditions, the oxygen feed is distributed in a manner that locally optimizes formaldehyde production along the reactor. With this, setting c ) 0 and r2 ) 0 along the reactor, eq 5 becomes
dfa ) -r1t0 ds dfc r1 ) - t0 + xCη(s) pT ) 0 ds 2 r1 dv ) t + η(s) ds 2pT 0
(16)
After accounting for the concentration dependence of the reaction rate with K ) 0, we have
η(s) )
γ a1/2 2xC pT
d(va) ) -γa1/2 ds dv 1 + xC 1/2 ) γa ds 2xC pT
(17)
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I)
∫01φ(y1,...,yN ) ds s
(21)
with constraints
dyi ) ψi(y1,...,yNs,η) ds i ) 1, ..., Ns
(22)
and initial conditions
yi(s)0) ) yi0 i ) 1, ..., Ns Figure 3. Optimal membrane permeation profile for q ) 0.9, 235 °C, 10% methanol, 5% oxygen feed, and 138 sccm overall feed flow rate.
which admits an exact analytical solution in implicit form that is not reported here owing to complexity. Two limiting cases of flow distribution lead to explicit solutions, presented below. For small volumetric membrane flows (q , 1) and dilute solutions (af , 1), the flow rate does not vary along the reactor (v , 1). Then, eq 17 takes the form
η(s) )
γ
a1/2
2xC pT
da ) -γa1/2 ds
(18)
leading to a linearly decreasing optimal permeability profile
η(s) )
γ γ a 1/2 - s 2xC pT f 2
(
)
(19)
in accordance with the results of numerical integration of Euler-Lagrange equations presented in Figure 2. An opposite limit of large membrane flows (vf , 1) may also be considered. In this case, methanol concentration variation near the reactor inlet (small s) is caused mainly by dilution with permeate rather than by chemical consumption, and eq 17 becomes
η(s) )
γ a1/2 2xC pT
can be formulated as choosing η to maximize φ. Because the initial value of φ is fixed, this can be achieved by maximizing
dφ
Ns
)
ds
∂φ dyi
∑ i)1∂y
i
ds
(24)
which implies that Ns
∂φ ∂ψi
∑ i)1∂y
i
)0
(25)
∂η
The last expression, together with eqs 22 and 23, completes the set of Ns + 1 equations with initial conditions to determine the unknowns yi and η. This is a significant reduction as compared with the more rigorous approach using Euler-Lagrange equations described above (see section 3), which leads to 2Ns + 1 equations with an additional search for optimal initial values of Ns Lagrange multipliers. Note that the exact conditions, at which local optimization yields the global optimum, are not known. However, it may be easily shown that for Ns ) 1 and monotonic φ(y), local optimization provides the global optimum. Recall that Figures 2 and 3 present numerical integration of Euler-Lagrange equations corresponding to the two limiting cases discussed in section 4.1. The agreement between the numerical and analytical results demonstrates that for methanol oxidative dehydrogenation in a PBMR-O the local optimization approach leads to the true optimal permeation distribution. 5. Concluding Remarks
d(va) )0 ds dv 1 + xC 1/2 γa ) ds 2xC pT
(23)
(20)
leading to the asymptotic behavior χ3/2 ∼ s for small s, also obtained by integrating the Euler-Lagrange equations numerically (Figure 3). The above analysis confirms that the variation in the reaction rates, caused by consumption and/or dilution of the directly fed reactant (methanol, in our case), is the most significant factor leading to the nonuniformity in optimal feed distribution of the other reactant (oxygen) along the reactor. 4.2. Generalized Formulation. In more general terms, the local optimization approach for maximizing
For methanol oxidative dehydrogenation, the optimal membrane feed distribution pattern in a PBMR was identified. The productivity of the desired product formaldehyde was selected as the basis for optimization. Owing to the specific reaction kinetics, optimal operation of the catalyst bed is achieved when operated at oxygen concentrations close to zero, which minimizes the undesired reaction with no penalty on the rate of the desired one. The optimal distribution of the oxygen feed was determined following two approaches, which yield identical results. While the conventional EulerLagrange method requires numerical computation, local optimization is simpler and admits an exact analytical solution. The PBMR-O results presented here, as well as those reported previously,7 demonstrate that both uniform and optimal membranes enable the packed-bed reactor
Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 313 Q ) volumetric flow rate over the catalyst bed as a function of position, m3 s-1 r1 ) methanol to formaldehyde reaction rate, bar/s r2 ) formaldehyde to CO reaction rate, bar/s s ) z/L, dimensionless coordinate along the reactor t ) time/t0 t0 ) V/F, overall residence time of reactants in the FBR, s T ) temperature, K v ) Q/F, dimensionless volumetric flow rate (as a function of position) V ) catalyst bed volume, m3 x ) vector whose components represent the mole fractions of the feed components permeating from the membrane to the catalyst bed, considered constant along the reactor y ) vector whose components are the mole fractions of the reactants and products in the PBMR z ) distance from the reactor inlet, m Subscripts f ) feed value i ) chemical species j ) reaction number Greek Symbols Figure 4. Maximal formaldehyde productivity at 235 °C and 98% yield as a function of (a) fraction nitrogen fed with methanol (σ) and feed of 10% methanol and 5% oxygen and (b) feed of methanol with stoichiometric amounts of oxygen at σ ) 0.
to provide essentially 100% formaldehyde yields. The major advantage of optimal as compared to uniform membrane feed distribution is the intensification of catalyst productivity. For example, as presented in Figure 4, depending on the feed conditions, maximal formaldehyde productivity is 2-3 times higher for the case involving optimized feed distribution. The modification of stainless steel membranes, utilized in our experimental studies,12 aimed at tailoring the desired optimal permeation profile for methanol oxidative dehydrogenation, is underway. Acknowledgment We gratefully acknowledge financial support from the National Science Foundation (Grant CTS-9907321). Notation a ) methanol partial pressure over the catalyst bed, bar b ) formaldehyde partial pressure over the catalyst bed, bar c ) oxygen partial pressure over the catalyst bed, bar CT ) overall gas concentration, mol m-3 Da ) Damkohler number ) r1,f.t0/af fa ) va ) methanol flux fb ) vb ) formaldehyde flux f′i ) dfi/ds, i ) a, b F ) overall feed volumetric flow rate, m3 s-1 hi ) constraints (eqs 6), i ) 1, 2 g ) vector denoting rates of formation of species by reactions K ) constant in eq 4 L ) reactor length, m Nr ) number of reactions Ns ) number of species in the reaction mixture pT ) reactor pressure, bar q ) membrane feed volumetric flow rate/overall feed volumetric flow rate
γ ) Da × af1/2 ) constant in eqs 17-20 λi ) Lagrange multipliers, i ) 1, 2 µi ) modified Lagrange multipliers defined by eq 14, i ) 1, 2 χ(s) ) integral membrane permeability ) volumetric gas flow rate permeating the membrane portion between the reactor inlet and the point s/overall feed flow rate η(s) ) differential membrane permeability ) d{χ(s)}/ds Fij ) production rate of the ith species in the jth reaction, mol/m3‚s σ ) inert species partition, nitrogen fed with methanol/ overall nitrogen feed
Literature Cited (1) Reyes, S. C.; Iglesia, E.; Kelkar, C. P. Kinetic-Transport Models of Bimodal Reaction SequencessI. Homogeneous and Heterogeneous Pathways in Oxidative Coupling of Methane. Chem. Eng. Sci. 1993, 48, 2643-2661. (2) Dixon, A. G. Innovations in Catalytic Inorganic Membrane Reactors. Catalysis 1999, 14, 40-92. (3) Saracco, G.; Neomagus, H. W. J. P.; Versteeg, G. F.; van Swaaij, W. P. M. High-Temperature Membrane Reactors: Potential and Problems. Chem. Eng. Sci. 1999, 54, 1997-2017. (4) Tonkovich, A. L. Y.; Jime´nez, D. M.; Zilka, J. L.; Roberts, G. L. Inorganic Membrane Reactors for the Oxidative Coupling of Methane. Chem. Eng. Sci. 1996, 51, 3051-3056. (5) Pen˜a, M. A.; Carr, D. M.; Yeung, K. L.; Varma, A. Ethylene Epoxidation in a Catalytic Packed-Bed Membrane Reactor, Chem. Eng. Sci. 1998, 53, 3821-3834. (6) Dixon, A. G. Analysis of intermediate product yield in distributed-feed nonisothermal tubular membrane reactors. Catal. Today 2001, 67, 189-203. (7) Diakov, V. E.; Varma, A. Methanol Oxidative Dehydrogenation in a Catalytic Packed-Bed Membrane Reactor: Yield Optimization Experiments and Model. Chem. Eng. Sci. 2003, 58, 801-807. (8) Thomas, S.; Seidel-Morgenstern, A. Selectivity Improvement with Stagewise Adjusted Membrane Properties. 4th International Conference on Catalysis in Membrane Reactors, Book of Abstracts, University of Zaragoza: Spain, 2000; pp 29 and 30. (9) Lu, Y.; Dixon, A. G.; Moser, W. R.; Ma, Y. H. Oxidative Coupling of Methane in a Modified γ-alumina Membrane Reactor. Chem. Eng. Sci. 2000, 55, 4901-4912. (10) Ge, S. H.; Liu, C. H.; Wang, L. J. Oxidative dehydrogenation of butane using inert membrane reactor with a nonuniform permeation pattern. Chem. Eng. J. 2001, 84, 497-502.
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(11) Diakov, V. E.; Lafarga, D.; Varma, A. Methanol Oxidative Dehydrogenation in a Catalytic Packed-Bed Membrane Reactor. Catal. Today 2001, 67, 159-167. (12) Diakov, V. E.; Blackwell, B.; Varma, A. Methanol Oxidative Dehydrogenation in a Catalytic Packed-Bed Membrane Reactor: Experiments and Model. Chem. Eng. Sci. 2002, 57, 1563-1569. (13) Al-Juaied, M.; Lafarga, D.; Varma, A. Ethylene Epoxidation in a Catalytic Packed-Bed Membrane Reactor: Experiments and Model. Chem. Eng. Sci. 2001, 56, 395-402. (14) Schechter, R. S. The Variational Method in Engineering; McGraw-Hill: New York, 1967; p 53.
(15) Roache, P. J. Verification and Validation in Computational Science and Engineering; Hermosa Publishers: Albuquerque, NM, 1998; pp 112-121.
Received for review October 30, 2002 Revised manuscript received May 6, 2003 Accepted May 6, 2003 IE0208624
