Design of Membranal Dehydrogenation Reactors: The Fast Reaction

Design of Membranal Dehydrogenation Reactors: The Fast Reaction Asymptote. Moshe Sheintuch ... Publication Date (Web): February 12, 1998. Copyright ...
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Ind. Eng. Chem. Res. 1998, 37, 807-814

807

Design of Membranal Dehydrogenation Reactors: The Fast Reaction Asymptote Moshe Sheintuch Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, Israel 32000

This study presents a systematic analysis of certain design aspects of a packed-bed catalytic membrane reactor for simple and fast dehydrogenation reactions. For sufficiently fast reactions, local chemical equilibrium can be assumed everywhere and conversion is transport limited. Under these assumptions, in a plug-flow isothermal reactor with a permselective membrane, analytical results can be derived showing the conversion dependence on reactor length and equilibrium coefficient. These results suggest that significant conversions should be expected in such a reactor with a reasonable membrane area-to-feed flow rate ratio (e.g., at 450 °C during butane dehydrogenation, with membrane flux of 1 cm2/cm3/min). High pressures lead to lower equilibrium conversions and to higher diffusive fluxes, resulting in a marginal change in overall conversion. Shell-side hydrogen partial pressure affects the conversion significantly, and the shell-to-tube flow rate ratio should be sufficiently large (10 to 100). Catalyst loading should be optimized to decrease length and improve selectivity. Ceramic Knudsen-selective membranes yield poor conversions. A preliminary analysis of an adiabatic reactor in which the diffusing hydrogen is combusted to supply the dehydrogenation enthalpy is also presented. These conclusions are contrasted with experimental observations obtained during isobutane dehydrogenation in a Pd membrane reactor. Introduction It is a great pleasure to contribute this article to a special issue dedicated to the 70th birthday of Prof. Westerterp. I find the topic of a membrane reactor appropriate because Prof. Westerterp has been one of the leaders into research of bifunctional reactors and because I conducted my first kinetic study (methylation of TPA) in his ‘laboratory’, as a student from the Technion with a summer job in a plant in Delfzijl, Holland, that (years later) I found had been managed at the time by Prof. Westerterp. Intensive research in the past two decades into membrane reactors for dehydrogenation and other equilibrium-controlled reactions (e.g., steam reforming) has shown significant gains in conversion, due to the separation of the product and shifting of the equilibrium (see recent reviews by Hsieh, 1991; Shu et al., 1991; Saracco and Specchia, 1994; Armor, 1995; Kikuchi, 1997). The membranes used were typically in one of two forms: dense and selective (typically Pd and its alloys), which allow for the transport of one (hydrogen) component; or porous and partially selective (made of ceramics of small pore size as low as 40 Å), which provide only partially selective transport, based on molecular weight (Knudsen diffusion). Recent works attempted to combine the mechanical strength of ceramics with the selective transport of Pd by depositing a thin Pd film on porous supports, using techniques like electrolysis plating, chemical vapor deposition, sputtering, dip coating, and sol-gel methods (see description in Kikuchi, 1997). Other membranes that show high selectivity for hydrogen transport include silica and zeolitic films (Jensen and Coker, 1996). We focus attention on dehydrogenation reactions because they seem to offer the best opportunity for commercial application of hydrogen permselective membrane reactors.

A systematic analysis of membrane reactors is necessary to compare and optimize the performance of different designs. Such an analysis is justified when the problem can be stated in a general way with few parameters. Although many studies have successfully simulated the observed behavior in various membrane reactors and using independent kinetic and transport information, the applied models were too specific and complex to allow a general analysis. Hsieh (1991) lists several models for membrane reactors that were developed for specific applications. Most of these models deal with dehydrogenation of cyclohexane, for which an almost complete conversion has been achieved experimentally without significant rates of undesired reactions or of deactivation (Itoh et al., 1985; Mohan and Govind, 1986; Mohan and Govind, 1988; Sun and Khang, 1988). The models commonly assume plug flow and isothermal conditions on the tube and shell sides. Other groups have considered the dehydrogenation of 1-butene in a dense-membrane reactor (Itoh and Govind, 1989), of n-butane in a porous-membrane reactor (Yogeshwar et al., 1993), or of isobutane in a Pd membrane reactor (Sheintuch and Dessau, 1996). Tsotsis et al. (1992) simulated the behavior of a packed-bed membrane reactor during ethane dehydrogenation and methane steam reforming. Ethane dehydrogenation in a permselective membrane reactor was studied and simulated by Gobina et al. (1995a); the same group also compared dense and microporous membrane reactors for ethylbenzene dehydrogenation (1995b). Carbon dioxide methanation in a reactor with a water vapor permselective membrane was studied and simulated by Ohya et al. (1997). Several groups have studied various design alternatives and considered certain modeling aspects using, again, specific reaction kinetics, usually of the massaction type. Many groups have compared the perfor-

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808 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

mance of reactors with cocurrent versus countercurrent flow. Sloot et al. (1990) and Veldsink et al. (1992) analyzed the behavior of a membrane reactor with a separated feed of reactants. Koukou et al. (1996) studied dispersion effects in porous membrane reactors, using the cyclohexane dehydrogenation kinetics, and suggested that radial effects are important in high-permeability membranes. Gobina et al. (1995a) concluded that radial dispersion effects are negligible for the tube dimension used experimentally. Reo et al. (1997), using mass-action kinetics, demonstrated that the yield of a porous membrane reactor offers little or no advantage over a diluted plug-flow reactor, and their simulations exhibit the existence of asymptotic solutions of the type analyzed here. Other studies considered adiabatic operation in which the transported hydrogen reacts with oxygen on the shell side to supply the necessary enthalpy; these studies will be reviewed later. In this study we analyze the design of a packed-tube and shell membrane reactor for the following simple dehydrogenation reaction:

B f C + H2

(1)

capitalizing on its fast reaction asymptote, a feature that was largely ignored in previous studies. If the reaction is sufficiently fast so that in the absence of side reactions local equilibrium is achieved for most of the reactor, then the conversion shifts only because of hydrogen separation. The principal problem in the case of selective hydrogen transport can then be stated with one variable, the ratio of accumulating flux to fluid flow rate, and one parameter, the equilibrium coefficient. Analytical results can be derived and analyzed then, and several design parameters and alternatives are considered: (i) The effect of membrane selectivity is analyzed to compare hydrogen-selective and partially (Knudsen)-selective membranes and determine the largest attainable conversion in the latter; obviously, the maximal conversion is an upper limit, and finite velocity reactions will show poorer conversions. Our results conclusively show that porous membrane reactors yield poor conversions. (ii) Reactors with fast reactions can be considered as a perturbation of the instantaneous reaction case. Certain general conclusions concerning optimal catalyst loadings for maintaining high selectivity in membrane reactors can then be derived. (iii) Isothermal design is compared with adiabatic operation in which the transported hydrogen reacts with oxygen on the shell side to supply the necessary enthalpy. A general analysis of simplified kinetics, like the one presented here, can give only partial results, and a detailed model is required for meaningful optimization. Actual kinetic models of dehydrogenation reactions are typically more complex than the usual mass-action kinetics and should account for role of hydrogen in maintaining catalytic activity or for the occurrence of undesired reactions (Sheintuch and Dessau, 1996). Some of our observations of isobutane dehydrogenation were modeled by a rate expression that accounts for the accelerating role of hydrogen pressure. Model Derivation The steady-state balances over the three species in a plug-flow reactor that are subject to exchange of material between the tube and shell take the following form:

dFi dt2 ) airFπ -Qi∆(Pyi)nπdt dz 4

(2)

where Fi is the molar flow of species i (B, C, or H), ai is the stoichiometric coefficient (-1, 1, and 1, respectively), r is the reaction rate, F is the catalyst density, dt is the tube diameter, z is the axial coordinate, P is the total pressure, and yi is the mole fraction of species i. Two regimes of transport are considered: selective permeability of hydrogen, as in a Pd tube, and partially (Knudsen) selective transport, like in a ceramic membrane. The flux of species i through the membrane is Qi(Pyi)n with n ) 0.5 or 1 for the two transport regimes just described. The square-root dependence of hydrogen flux through a Pd membrane on its partial pressure was verified in numerous studies (e.g., Kikuchu, 1997). The driving force in the case of Knudsen diffusion may be more complex than that shown in eq 2, but eq 2 is a good approximation. Plug-flow behavior is assumed on the shell and tube sides with negligible radial gradients. Concentrations and temperature gradients between the catalytic and gas phases and between the membrane and gas phase are also ignored. Isothermal conditions are assumed everywhere, except for one section in which we study the coupling of dehydrogenation and oxidation reactions in an adiabatic reactor. To reduce the number of parameters, the shell side is usually assumed to be free of any reactants and products, either by sweeping at a fast rate or by combustion with oxygen. In that case, fluxes are maximal and conversions are high. The effect of a finite shell gas flow rate is considered as one of the design alternatives. The reaction rate will be assumed to follow eq 3:

rF ) kF

PyB - P2yHyC/K

(3)

D(yB,yC,yH)

where K is the equilibrium coefficient and the denominator (D) accounts for inhibition due to adsorption of reactants, products, or coke precursors. We show that when the reaction is fast, the results are insensitive to the kinetic details and depend mainly on the equilibrium coefficient. Selective Transport. For this case, QB, QC ) 0 and n ) 0.5. Let us define a new dimensionless coordinate and rate coefficient as follows:

Z ) QHxPπdtz/F0

k′ ) kF P0.5dt/4QH

(4)

where F0 is the molar feed rate. Then, we rewrite eq 4 in the following form:

k′( yB - PyC yH/K) d(FB/F0) )dZ D(yB,yC,yH) d(FH/F0) k′(yB - PyC yH/K) ) dZ D(yB,yC,yH)

(x x ) yH -

PS y P HS

(5)

where PS and yHS are the shell-side pressure and hydrogen mole fraction, respectively. To reduce the number of parameters, we express the molar flows as a function of compositions. To show this, note that at any point

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 809

FC ) F0 - FB

FH ) F0 - FB - Ftrans

∑F ) 2F0 - FB - Ftrans

(6)

where Ftrans is the integral of transport flux up to that point. Because that value is as yet unknown, let us express it in terms of yH ) FH/∑F. Using the aforementioned relations, we find

F0

∑F ) (1 - yH)

FH )

F0 yH (1 - yH) FB )

F0 yB (1 - yH)

(7)

Substituting these relations to eq 5 yields two ordinary differential equations in two state variables:

df ) dZ

(

(

k′(1 - yH) (1 - f) 1 +

) )

yH yH K′ K′

D(yB,yC,yH)

( (x x ))

dyH df ) (1 - yH)2 dZ dZ

yH -

PS y P HS

(8)

(9)

Figure 1. Conversion profiles in a selective membrane reactor assuming local equilibrium with K′ ) 0.39 (line 1, corresponds to 550 °C in isobutane dehydrogenation), K′ ) 0.23 (1ine 2, 525 °C), K′ ) 0.124 (line 3, 500 °C), and K′ ) 0.034 (line 4, 450 °C). Lines 3a and 3b show the effect of shell-side flow rate with tube-to-shell flow rate ratios of 0.01 and 0.1, respectively, and K′ ) 0.124. Also shown are experimental conversions measured during isobutane dehydrogenation in a packed Pd membrane reactor at 500 °C (after Sheintuch and Dessau, 1996).

The degree of conversion is

1-f)

composition for a closed system, can be shown to be

yB FB ) F0 (1 - yH)

(10) xe )

The Fast Reaction Asymptote In the limit of a very fast reaction, conditions very close to equilibrium will be quickly established everywhere in the bed and the reacting composition will vary only in response to hydrogen transport. The following analysis shows that conversions can be increased by reducing the flow rate, and these conditions drive the local state even closer to equilibrium. Under this assumption, an analytical solution can be secured and the problem can be stated with only one parameter. Selective Transport under Local Equilibrium. Using the equilibrium relation and yC ) 1 - yH - yB, we find

yB )

yH(1 - yH)

K′ )

(K′ + yH)

K P

(11)

Adding eqs 5, using the relations in eqs 7 and 11, and assuming yHS ) 0 (i.e., infinite flow rate on the shell side), we can write

(

)

yH yH d + ) -xyH dZ K′ + yH 1 - yH

(12)

After integration, eq 12 yields (x2 ) yH)

Z/2 ) G(xe) - G(x) G(x) )

(13)

x 1 1+x x + + ln + 2 4 1 - x 2K′ + 2x2 2 - 2x x 1 tan-1 (14) x x 2 K′ K′

where the initial composition, which is the equilibrium

x

1 x1 + 1/K′ + 1

(15)

The reactor length required for complete conversion (L) corresponds to yH ) x ) 0, and because G(0) ) 0, that reactor length is G(xe), which can be expressed in terms of the inlet equilibrium conversion (f ) fe) as follows:

G(xe) )

1 + f 1 1 + xf/(1 + f) + ln + f 4 1 - xf/(1 + f)

x

1 2

1 x1/f 2 - 1 tan-1 x1/f - 1 (16) 2 Figure 1 portrays the conversion dependence on Z for K′ values that correspond to isobutane dehydrogenation at 550 (line 1), 525 (line 2), 500 (line 3), and (line 4) 450 °C. Note that complete conversion at temperatures above ∼450 °C can be achieved with Z < 10. Recall that Z is the dimensionless ratio of hydrogen transfer in the whole reactor to the feed flow rate. The required space times, λ ) dtZ/(4QH), are reasonable for a Pd-selective membrane for which the hydrogen flux (QH) is ∼1 cm3/ min/cm2 at atmospheric pressure and dt (the tube diameter) is several millimeters. Note that for a moderate shift from equilibrium, the line is linear; so, an experimental plot of the distance from equilibrium versus inverse flow rate should yield a straight line. Complete conversion is achieved at a finite length (due to the square-root dependence of flux), so we can compute that length dependence on temperature and pressure and study its asymptotes. Increasing pressure results in higher fluxes but lower equilibrium conversions, and it leads to some decline in complete conversion length, but in the range 5-20 atm its effect is marginal. Pressure effects should be studied, however, for the case of no sweeping flow (i.e., atmospheric hydrogen pressure) as a means of obtaining high-grade

810 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

hydrogen. The gain achieved by increasing temperatures above 500 °C is marginal, whereas below 400 °C, the lengths become extremely high (see Figure 1). We derive now asymptotic solutions for very low or very high temperatures and pressures. For high temperatures, K′ . 1, fe approaches unity, and

L ) 2G(1) f x2 + ln(1 + x2) ∼ 2.295

(17)

Similarly, for very low pressures, K . 1, and the length required for complete conversion, L, is ∼2.295. The corresponding dimensional length, l, is 2.295 F0/ (QHπdtxP). At very low temperatures, K′ , 1, fe f xK′, and from eq 16

(

)

1 K′-1/2 L ) G(xK′) ) K′-1/4 + π 2 2 2

(18a)

As K′ f 0, the last term becomes dominant. At very high pressures, the same asymptote holds, but K′ ) K/P and L ∼ P1/2. Therefore, the dimensional asymptote is now

l)

F0

(18b)

2xKQHdt

Finite Shell Flow Rates. The case of very high flow rates, where the hydrogen shell partial pressure is nil, represents the asymptotic solution analyzed in the previous cases. We would expect that with decreasing shell flow rates, the conversion will decline to the equilibrium value determined by overall mass balance. The experimental results (Sheintuch and Dessau, 1996) show, however, that maintaining a certain small level of hydrogen is important for obtaining high activity and slow aging. Although the model described later does not account for these effects, it helps to assess the effect of shell flow rate on total conversion and find conditions at which the hydrogen pressure in the tube will be small while the resulting conversion and required reactor length are reasonable. Note that unlike in the case of infinite shell flow rate, in this case, the conversion profile in the tube approaches its equilibrium value asymptotically (i.e., an infinite length is required). The shell-side hydrogen pressure can be determined by writing its balance, adding it with the hydrogen tubeside balance, and finding an invariant relation between the two (see Sheintuch and Dessau, 1996, for derivation). For cocurrent flow, with a hydrogen-free shellside stream at the inlet, we find, even in the absence of equilibrium, that

(

)

F0 yH f+ FOS 1 - yH yHS ) F0 yH 1+ f+ FOS 1 - yH

(

)

(19)

where FOS is the shell-side flow rate at the feed. Now, yHS(yH,f) should be substituted into the respective reactor model. For small yHS, the denominator is about unity, and when equilibrium is assumed, the relation f ) K′/(K′ + yH) can be applied. Figure 1 (lines 3a,b) presents the conversion dependence on length in a reactor with selective transport and inlet tube-to-shell flow rates ratios, (F0/FOS), of 0.01 and 0.1 (K′ ) 0.124).

Figure 2. Conversion dependence on Z (a) and on k′Z (b) in a selective membrane reactor assuming mass-action kinetics with K′ ) 0.39 and reaction velocities of k′ ) 1, 5, and 25 [and 100 or 1000 in (b)]; because k′Z is proportional to catalyst weight, it represents the rate of undesired reactions.

The results in Figure 1 suggest that for conversions >0.7, this ratio should be kept significantly below 0.1. Selective Transport with Mass-Action Kinetics. To find the effect of a finite reaction velocity, we integrated eqs 8 and 9 for various k′ values without catalytic inhibition (D ) 1 or mass-action kinetics) and K′ ) 0.39 (or 550 °C in isobutane dehydrogenation). Figure 2(a) presents the conversion profiles showing that with k′ ) 25, equilibrium is established almost at the inlet, and when this entrance region is ignored, the solution is similar to that already derived for instantaneous reactions. The solution for k′ ) 5 deviates from the asymptote only at the inlet. For low reaction velocities, transport is not limiting (yH ∼ 0), and the conversion changes exponentially (for this mass-action kinetics). We also plot in Figure 2(b) the conversion versus k′Z for increasing values of k′ and observe that for small k′, the conversion climbs monotonically to unity, whereas with fast reactions, the closed-system equilibrium is established quickly; the system will eventually climb to high conversions with k′Z of the order k′. Apparently, the system exhibits two asymptotes: at small k′Z (typically k′Z < 1) it behaves like a closed system, whereas at high k′Z it approaches the instantaneous reaction solution described earlier. To analyze the nature of the equations in the general case and show its asymptotes, we rewrite eqs 8 and 9 in terms of conversion (f) and h ) f - yH/(1 - yH), the difference between the equilibrium value of the two variables, to find

(

)

df f-h 1 1-f+f ≡ F (20a) ) d(k′Z) 1 + f - h K′(1 + f - h)

x1 +f -f -h h ≡ G

dh 1 ) d(k′Z) k′

(20b)

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 811

Equation 20a is simply eq 8 expressed in terms of the new variables, whereas eq 20b was obtained by adding eqs 8 and 9. For large k′, eqs 20a and b have two length scales, because F and G are of the order unity. For large k′Z, F f 0, the dynamics are dictated by eq 20b, and the conversion depends mainly on Z. For small k′Z, h ∼ 0 and the conversion is dictated by eq 20a and it depends on k′Z. Perturbation solutions around these two asymptotes can be derived, but will not be presented here. We discuss now the effect of catalyst load (i.e., k′) on selectivity. Although increasing k′ has a favorable effect on conversion, it will have a detrimental effect on selectivity. The main undesired reactions during dehydrogenation are cracking or hydrogenolysis and isomerization reactions. Both the reactant and product, in the simple B f C + H2 reaction, may undergo these reactions, although at a different rate. If we assume equal rates of reactant and product cracking or isomerization, then rate will be proportional to the catalyst weight (i.e., k′Z), but independent of conversion. We further assume that side reactions are still sufficiently slow so as not to affect the primary reaction. Conversion dependence on k′Z (Figure 2b) demonstrates that for a specified conversion, poorer slectivities can be expected with increasing catalyst loading. This effect becomes significant at high conversions. Partially Selective Transport under Local Equilibrium. We examine now the relation between membrane selectivity and conversion to determine whether desired conversions can be achieved in partially selective (ceramic) membranes. The conversion definition now is based on the orefin flow on the tube side and does not account for olefin loss by transport from the tube to the shell side. With such a definition, complete conversions cannot be achieved and, in fact, there exists an optimal length that yields maximal conversions. (Many other studies have defined a conversion based on the total olefin flow in the shell and tube side, and compared it with equilibrium values based on tube-side conditions. As recently noted by Reo et al. (1997), such reactors offer no advantage over diluted PFRs). We simulated the reactor for a partially selective membrane, assuming that local equilibrium is maintained everywhere and that n ) 1 (eqs 1-2) and QB ) QC ) mQH, where m is the selectivity parameter and the permeability difference between B and C is ignored. These assumptions are justified when the selectivity is based on molecular weight, as in a Knudsen diffusion regime. We also assume that the shell side is evacuated, so that backdiffusion of the inert on the shell side can be ignored. Defining now the length coordinate as Z ) QHPπdtz/F0, we combine balances (eq 2) to eliminate the rate term in the following form:

(

)

d FB + FC ) -m(yB + yC) ) -m(1 - yH) (21a) dZ F0

(

)

d FB + FH ) -(myB + yH) ) dZ F0

(

- m + (1 - m)yH +

)

mK′(1 - yH) (21b) K′ + yH

where the equality in eq 21b is obtained by imposing the equilibrium relation (eq 11). Under these assump-

Figure 3. Conversion profiles in a partially selective (Knudsen) membrane reactor assuming local equilibrium with K′ ) 0.124 (500 °C in isobutane dehydrogenation) and transport selectivities of m ) 0 (upper line) up to m ) 0.17, in steps of 0.01, as well as m ) 0.185; in isobutane dehydrogenation, m ) 29-0.5 ) 0.185, suggesting poor conversions.

tions the three molar flow rate can be expressed as

FH ) σyH F0

(22)

yH(1 - yH) FB )σ F0 K′ + yH

(23)

f≡σ

K′(1 - yH) K′ + yH

(24)

where σ is the dimensionless total flow:

σ)

∑F F0

(25)

Equations 21a,b can be rearranged as two differential equations in either yH and σ

dyH yH(1 - m)(yH + K′)2(1 - yH) ) dZ (1 + K′)(K′ + y 2)σ H

dyH dσ σ ) -m + (26) dZ (1 - yH) dZ or in yH and f

K′yH(1 - m)(yH + K′)(1 - yH)2 dyH )dZ (1 + K′)(K′ + yH2)f df f dyH mK′(1 - yH) )(27) dZ yH + K′ (yH + K′) dZ with yH(0) ) xe2 (eq 15), σ(0) ) (1 - yH)-1, and f (0) ) K′/(K′ + yH). Figure 3 presents the conversion for m ) 0 (selective transport, line 1), 0.01, 0.02...up to 0.17 as well as at 0.185 (with K′ of isobutane dehydrogenation at 500 °C). With selective transport, a complete conversion may be achieved (although now at an infinite length because n

812 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

) 1), whereas for a partially selective membrane, the conversion passes through a maximum before declining or is declining monotonically. For many practical situations, the expected yield is worse than that at equilibrium. In a Knudsen diffusion regime, the hydrogen-to-butane diffusivity ratio is m ) (29)-0.5 ) 0.185, suggesting that a ceramic membrane will yield very poor conversions. Coupling of Oxidation and Dehydrogenation Reactions in a Selective Membrane. We consider now the design of an overall adiabatic reactor in which oxygen is supplied to the shell side to react with hydrogen to supply the enthalpy necessary for the endothermic dehydrogenation reaction. Detailed models of such reactors using cyclohexane dehydrogenation kinetics were presented by Itoh (1990, 1995) using a one-dimensional model; to reduce the number of parameters they assumed, in several simulations, infinite heat-transfer coefficients between the tube and the shell sides. Experimental results and corresponding simulations were reported by Itoh and Wu (1997). The simplest model to be considered makes the following assumptions: (a) heat transfer resistance is negligible on both sides of the membrane (as in Itoh, 1990); (b) the dehydrogenation reaction is in local equilibrium, and (c) the oxidation reaction is instantaneous, which is a good approximation for the high temperatures assumed in this study. These assumptions imply that temperatures of the shell and tube streams are identical at every position and the temperature varies in response to the hydrogen transport rate. The system is described then by eq 2 and eq 28:

(F0CPO + FOSCPS)

dT ) dz

QH(PyH)1/2πdt(-∆H)ox -

dFB (-∆H)de (28) dz

where CPS and CpO are the average molar heat capacities of the two streams, which flow at inlet molar rates of FO and FOS, and the (-∆H)i are the reaction enthalpies of oxidation (i ) ox) and dehydrogenation. Substituting the dimensionless length coordinate (eq 4) yields

〈CP〉

(-∆H)de dFB dT ) y (-∆H)ox dZ x H FO dZ FS (29) 〈CP〉 ≡ CPO + CPS F0

Using the relation FB/FO ) 1 - f ) yH/(K′ + yH) (eq 10), we can integrate eqs 29 and 12 to determine the concentration profile. The average heat capacities and reaction enthalpies are assumed to be constant within the temperature range employed in the simulation. Now, when local equilibrium is assumed, the temperature at the inlet is discontinuous, dropping from Tin to TO as a result of the enthalpy adsorbed in establishing the equilibrium conversion at T0. This drop is described by

(F0 - FBO)(-∆H)de ) 〈CP〉(T0 - Tin) ) F0 K′ (-∆H)de (30) K′ + yHe where FBO, yHe, and K′ are the reactant molar flow rate,

Figure 4. Performance of an adiabatic reactor: conversion and temperature profiles in a selective membrane reactor assuming local equilibrium with combustion of the diffusing hydrogen (∆Tad ) 0, 44, 96, 150, and 200 °C in lines 1 through 5, respectively; Tin ) 773 - ∆Tad).

hydrogen concentration, and equilibrium coefficient at To (i.e., after equilibrium was established), respectively. Beyond that point, hydrogen diffuses out of the tube and the oxidation enthalpy is contributed to the stream. Equation 29 applies then and, after integration, we find

〈CP〉(T - T0) )

(FBe - F0)(-∆H)de + F0

∫xyH dZ

(-∆H)ox

(31)

Noting that yH1/2 is described by eq 12 because equilibrium is assumed, we find that:

y

y

∫xyH dZ ) 1 -HyH + K′ +H yH - 1

-

(32)

Adding eqs 30-32 and rearrangement yields:

〈CP〉(T - Tin) )

K′ [(-∆H)de + (-∆H)ox] K′ + yH yH (-∆H)ox (33) 1 - yH

where K′ is temperature dependent. Equation 33 provides an implicit relation between conversion and temperature that cannot be used directly to simplify the integration. The temperature and conversion profiles can be determined by deriving dT/dZ from eq 33 and integrating it along with the mass balance. The adiabatic temperature rise is obtained at complete conversion (i.e., yH ) 0) yielding ∆Tad ) ((-∆H)ox +

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 813

(-∆H)de)〈CP〉, which is positive for most dehydrogenation reactions. The adiabatic temperature rise can be controlled by changing the shell flow rate (i.e., 〈CP〉). The effect of ∆Tad on the reaction profile is presented in Figure 4 (∆Tad ) 0, 44, 96, 150, and 200 °C in lines 1 through 5, respectively) using an inlet temperature so that the complete conversion temperature is 773 K; K′ ) 0.124 exp[14 790(1/T - 1/773)] is the temperaturedependent equilibrium coefficient of isobutane dehydrogenation and the ratio of the enthalpies [(-∆H)ox/(∆H)de ) -2.3] is that of isobutene dehydrogenation to hydrogen oxidation reactions. The sum of these enthalpies is 38 720 cal/g mol and thus an adiabatic temperature rise of 110 °C is obtained with a shell-to-tube flow rate ratio of ∼50.

the shell-to-tube flow rate ratio should be sufficiently large (10 to 100). Increasing the temperature beyond a certain limit has a negligible effect on the required length (there exists an asymptotic solution) but is likely to have a detrimental effect on selectivity and aging. Increasing pressure has a negligible effect on yield. This conclusion agrees with experimental results. Pressurized processes can be used, however, for producing highpurity hydrogen. Catalyst loading should be optimized to decrease length and improve selectivity. Ceramic Knudsen-selective membranes yield poor yields because a significant fraction of the orefins is diffusing to the shell side and wasted. Preliminary analysis of an adiabatic reactor in which the diffusing hydrogen is combusted to supply the dehydrogenation enthalpy, is also presented.

Discussion The fast reaction asymptote should be considered as a kinetics-independent upper limit of the membrane reactor performance. The analysis of a specific reaction will require detailed kinetic expressions and the analysis cannot be generalized to a large class of reactions. A detailed analysis of isobutane dehydrogenation, that was carried out in a membrane reactor made of a Pd/ Ru (or Pd/Ag) tube packed with a supported Pt catalyst, was conducted by Sheintuch and Dessau (1996). Experimental points, obtained at 500 °C with 0.75 or 1 g of catalyst, are plotted as squares or diamonds in Figure 1 versus the dimensionless length (Z) using the definition in eq 4 with the independently measured QH ) 52 cc/min (data from beds no. 1 or 9, Figure 1, in Sheintuch and Dessau, 1996; F0/FOS varied between 0.04 and 0.004). The attained yields are significantly below the fast-reaction solution (compare line 3). The yield tapered off at low flow rates reaching 0.76 at Z ) 16 or Z ) 52. The yields were limited at low feed rates by suppressed catalyst activity in the absence of hydrogen. We considered three models for parameter estimation. Mass action kinetics (D ) 1) and orefin inhibition (D ) PC ) Pf) models failed to represent the data adequately, whereas a model that also accounts for inhibition by a coke precursor (D ) Pf + k1(PyH2)0.5 + k2Pf/yH) showed satisfactory agreement. The suggested form of inhibition in the latter model increases dramatically with declining hydrogen pressure, in agreement with observations, and it can be justified by assuming that the coke precursor is destroyed by reaction with hydrogen. The degrees of cracking and isomerization were adequately described by a single-parameter rate expression that assumes that the main and side reactions occur on the same sites. The model was optimized to determine the feed and shell flow rates that maximize the yield.

Nomenclature ai ) stoichiometric coefficients B, C ) components in the reaction Cp ) average molar heat capacity D ) denominator of rate expression dt ) tube diameter Fi ) molar flow rate of species i F0, FOS ) molar feed flow rates on tube and shell side, respectively f ) conversion K, K′ ) dimensional and dimensionless equilibrium coefficients, respectively k, k′ ) dimensional and dimensionless kinetic parameters, respectively l, L ) dimensional and dimensionless reactor length, respectively, required for complete conversion m ) membrane selectivity parameter P, PS ) tube- and shell-side pressures, respectively Qi ) membrane flux of species i under pressure gradient of 1 atm q, qS ) tube and shell flow rates, respectively qmax ) maximal volumetric hydrogen flow through the membrane r ) reaction rate T, T0 ) bed temperature and its inlet value yi ) mole fractions of respective species y3, yN ) mole fractions of cracking and isomerisation products, respectively z, Z ) dimensional and dimensionless reactor coordinate, respectively F ) catalyst density (-∆H)ox, (-∆H)de ) oxidation and dehydrogenation reaction enthalpies, respectively Subscripts B, C, H ) of the respective species HS ) of hydrogen on shell-side

Conclusions

Literature Cited

With sufficiently fast reactions, where local equilibrium can be assumed and conversion is transport limited, significant conversions can be achieved in a selective Pd membrane reactor with a reasonable membrane area-to-feed flow rate ratio (e.g., 1 cm2/cm3/min at 450 °C during butane dehydrogenation). Under these assumptions, in a plug-flow isothermal reactor free of mass transfer resistances, the conversion depends only on the length and equilibrium coefficient. Shell-side hydrogen partial pressure affects the conversion significantly, even with simple mass action kinetics, and

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Received for review May 9, 1997 Revised manuscript received August 27, 1997 Accepted October 23, 1997 IE9703301