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Analysis of a carbon membrane reactor: from atomistic simulations of single-file diffusion to reactor design
Chemical Engineering Science 59 (2004) 4739 – 4746 www.elsevier.com/locate/ces
Analysis of a carbon membrane reactor: from atomistic simulations of single-file diffusion to reactor design Moshe Sheintuch∗ , Irena Efremenko Department of Chemical Engineering, Technion, Technion city, Haifa 32000, Israel Received 29 February 2004
Abstract We study the dehydrogenation of iso-butane in a membrane reactor with carbon membranes; these are molecular sieves with pores of molecular dimensions. To provide information on the transport laws of the various components, for reactor design purposes, we studied the separation in a membrane module either by maintaining the shell-side under lower pressure (or under vacuum) or by sweeping it with an inert diluent stream. The purpose of this work is to derive multi-component flux expressions for single-file diffusion in these two modes of separation, using molecular mechanics calculations of the thermodynamics of molecular adsorption into, diffusion within and desorption from the pores, and apply them for reactor design. In the process we calibrate these expressions with integral measurements of separation in a membrane module. Good predictions of reactor performance are obtained for a reaction coupled with separation by sweeping the hydrogen with nitrogen but poor predictions were achieved for a reaction coupled with vacuum-driven separation. Reactor performance in the former mode is better due to excellent transport selectivity, which we attribute to mutual blocking of counter-diffusion by nitrogen and hydrocarbons. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Membranes; Molecular sieves; Single-file diffusion; Molecular mechanics; Reaction engineering; Mass transfer
1. Introduction In a search for a highly selective, yet relatively inexpensive, membrane that may be employed in membrane reactors for separation of hydrogen simultaneously with equilibrium-limited reactions, we study the dehydrogenation of iso-butane in a membrane reactor with carbon membranes. Carbon membranes are molecular sieves that incorporate pores of molecular dimensions so that steric and energetic effects are important for describing transport processes. Membrane reactor design requires information on the transport laws of the various components. Separation or reaction in a membrane reactor can be tested and operated by one of the three modes for a driving force for separation: The shell-side can be either maintained under ∗ Corresponding author. Tel.: +972 4 8292823; fax: +972 4 8295672.
E-mail address: [email protected] (M. Sheintuch). 0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.09.016
lower pressure (or under vacuum) or swept with an inert diluent stream or maintained under reaction conditions that consume the diffusing component (as in hydrogen oxidation). As we show below, the first two modes in a molecular sieve membrane yield markedly different behaviors, and it is not clear which information from separation studies should be applied for reactor design purposes, and what are the underlying transport laws. Most previous studies employed simple linear transport laws, which may lead to significant errors. The purpose of this work is to derive multi-component transport expressions, based on calculated thermodynamics of molecular adsorption into and diffusion within the pores and a probability analysis of transport, calibrate them with independent results of transport, and apply them for reactor design by comparing the predictions with experimental results. We now review the main experimental observations of multi-component transport through the carbon membrane
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module, which incorporated 100 fibers each of 100 m diameter, obtained in the first part of this effort (Sznejer et al., 2004). The fluxes of pure components were studied under a pressure gradient and were found to be largely linear with the driving force. Nitrogen was used as a sweeping gas in the study of mixtures. In the temperature range of 25–400 ◦ C, for a series of light hydrocarbons along with their mixtures with hydrogen, we found that counter-diffusion of nitrogen, that flows on one side of the membrane, and C2 –C4 alkanes with hydrogen flowing on the other side, significantly inhibits the fluxes (and permeabilities) of both nitrogen and alkanes, while hydrogen flux is only slightly diminished (Fig. 1). This is a significant and desired result from reactor design perspective, since we are interested in a membrane that is selective to hydrogen transport. The membrane selectivity, the ratio of hydrogen to hydrocarbon permeabilities, may reach 100–1000 in propane or in (normal or iso-) butane mixtures with hydrogen. A lower selectivity value was found from single-component systems than from multicomponent measurements. This difference led us to the conclusion that transport is dominated by single-file diffusion. While we report permeabilities of the components involved in the multi-components transport in Fig. 1, it is not evident that these parameters should be constant and that the flux of component i should be linear with its concentration. Previous experimental studies reported mainly on the low (room) temperature separation selectivities of hydrogen–alkane mixtures in carbon membranes and were motivated by separation technologies. The permeabilities of pure gases at 296 K increased from H2 to CH4 to C2 H6 and then decreased as the molecular weight of the hydrocarbon chain increased. In a mixture, the permeabilities of the most weakly adsorbed components are drastically reduced in the presence of higher molecular weight hydrocarbons (Rao and Sircar, 1993; Sircar et al., 1999). In the temperature range of 25–120 ◦ C Kusuki et al. (1997) found that the permeability of light alkanes decreased with the kinetic diameter of the gas and increased with the temperature. Tanihara et al. (1999) found a selectivity factor of 540 in a mixture of H2 /CH4 at 50 ◦ C in a carbon hollow fiber membrane. The transport and separation phenomena in carbon membranes remain largely unclear. The transport regimes important for our case like pore diffusion, capillary condensation and molecular sieving cannot be reduced to a simple rate expression and various approaches have been applied to predict fluxes and selectivities; among them non-equilibrium molecular dynamics of diffusion in well-defined pore, (e.g. see Kaganov and Sheintuch (2003) who studied compositions relevant to these study; Mao and Sinnott (2000), Xu et al. (1999), and references therein). However, such computations are still limited to small ‘samples’, typically with one species, and the results cannot be extended to more realistic situations. To account for the experimentally determined purecomponent transport we have used molecular mechanics simulations to find the energetics of adsorption (insertion
of a molecule into the pore), diffusion (translation within the pore) and desorption (removal) of individual gases in nanopores modeled by carbon nanotubes (Sznejer et al., 2004). In pores that are extremely small, relative to the size of a certain molecule, adsorption and diffusion are activated while desorption is non-activated; in practice, this domain is extremely narrow and the probability of the molecule entering such pores is very small, therefore this domain can be ignored. While ignoring entropy effects, for most of the relevant domains, adsorption is typically not activated while desorption is activated and presents the rate limiting step with activation energy determined from Ed = (−Hads ). The molecular transport proceeds essentially by a single-file diffusion mechanism. A rate expression for a single-species transport, from a reservoir at high pressure on the left (Pil ) to a low one on the right (Pir ), in a molecular-sieve carbon membrane was derived by a mean-field approach. The overall transport frequency (fi ) kai (Pil − Pir ) = Ki (Pil + Pir ) + 2 fi (Ki Pil + 1)(Ki Pir + 1) + , kti /kdi Ki =
kai , kdi
(1)
shows non-linear dependence on the driving force (ka , kt , kd are the corresponding adsorption, transport and diffusion frequencies, and i denotes the component). Here, we update our estimates of ka , kd , based on entropy calculation, and extend the analysis to the case of two-species transport in a multi-pore system. For reactor design purposes we need to develop a theory of multi-component single-file transport. In the next section we extend the analysis of pure-component single-file transport to co- and counter-diffusion in two component systems. Especially, we want to find out whether both species diffuse or just one, and if so which one? An analysis based on a single-size pore still cannot explain the dramatic change in permeabilities measured by counter-diffusion and those measured as single component (Fig. 1c). We have to consider a system with the measured pore size distribution (Fig. 2a), in which most of the pore volume is associated with pores that are 0.6–1.0 nm in diameter. In Section 3 the same analysis is repeated for a family of parallel pores. We will not consider here the transport in pore networks. In Section 4 we test the model against the performance of a membrane reactor packed with a chromia alumina catalyst (see Sznejer and Sheintuch(2004) for details). Our choice of model reaction, the dehydrogenation of iso-butane, stems from the commercial interest in this design as an alternative to the periodically regenerated fixed-bed approach. This reaction, which requires high temperatures, had been tested in a Pd-membrane reactor, achieving 78% conversion at 500 ◦ C (Sheintuch and Dessau, 1996). Only few applications of molecular sieve membranes for membrane reactors
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Fig. 1. Hydrogen (a), nitrogen (b) and alkane (c) permeabilities (cm/min atm) as a function of hydrocarbon size at various module temperatures; alkane permeabilities under a pressure gradient are also presented in (c).
Fig. 2. Pore size distribution (a) and ratio of adsorption–desorption frequencies for H2 , N2 , iso-butane and iso-butene molecules (b).
have been reported. A carbon membrane was used for cyclohexane dehydrogenation at 195 ◦ C (Itoh and Haraya, 2000). Other applications employed zeolite membrane (Casanave et al., 1995, 1999; Van de Graaf et al., 1999).
2. Theory of single-file transport Permeability will be determined by two main factors: the occupancy of the pore (largely determined by ka /kd ), and the transport within the pore. As explained in the Intro-
duction, in the present work we present the thermochemistry data (Table 1) retrieved from the full geometry optimization of the molecule-in-nanotube systems and of the separated components followed by frequency calculations (see Irikura (1998) for computational details) using molecular mechanics simulations. Application of this approach is justified by the absence of chemical interaction and existence of well-developed parameters (Universal Force Field) to describe the short- and long-range interactions between atoms. The most salient features of the pore–molecule interactions (non-activated diffusion, high adsorption energy and strong change in entropy) were checked by DFT calculations for smaller systems. While entropic effects are usually ignored in adsorption processes, we found its effect to be crucial in determining the transport regime at high temperatures. Sterically restricted situation of a molecule inside nanopore results in extremely strong decrease in the translational and rotational components of entropy change as compared to those of the separated components; restricted vibrations bring about a sharp increase in S in smallest pores. The geometrical picture ensuing from the molecular mechanics results is that of a straight pore of N sites, where each site may be occupied by one of the competing molecules freely moving from site to site and undergoing elastic collisions with each other and with pore walls. Molecular adsorption is controlled by the probability to enter the pore. Therefore, the adsorption, transport and diffusion constants are estimated as follows: if Ti,ads = Pi /(2Na mi kT )1/2 is the flux of incident molecules i, then kai = sT i,ads Na Ap , where s = exp(S/R) is the ‘sticking coefficient’, which characterizes the probability of a molecule to enter the pore of radius rp , mi (kg/mol) is the molecular mass, k is the Boltzmann constant, Na is the Avogadro’s number and Ap = rp2 is the area of the pore mouth. Desorption is controlled by the kinetic energy needed to overcome the pore–molecule interaction; its frequency is assumed to follow kd = (kT / h) exp(−Ed /RT ) with Ed = (−Hads ). Comparison of the frequencies characterizing the two √ steps kai /kdi = Pi Na Ap e−Ga /RT / 2mi Na kT (kT / h) (Ga = Hads − T S is the change in Gibbs free energy)
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Table 1 Calculated pore–molecule interaction energies (H , kJ/mol) and entropy changes (S, J/mol K) at 627 K Pore
H2
D, A˚
H
S
H
S
H
S
H
S
5.40 6.26 7.00 7.82 8.60 9.39 10.96 15.66
−2.64 −23.50 −20.74 −16.62 −14.13 −12.67 −10.91 −9.96
−113.16 −79.4 −35.26 −20.81 −61.37 −56.87 −89.82 −94.17
167.64 −6.43 −35.78 −33.36 −26.37 −22.93 −18.15 −16.57
−169.62 −164.21 −91.56 −44.85 −118.47 −82.68 −20.32 −122.13
−51.72 −70.42 −99.85 −88.61 −78.33
−173.97 −195.81 −155.77 −159.51 −160.96
102.98 −90.81 −112.30 −74.96 −38.36
−256.57 −202.35 −153.01 −189.6 −234.51
N2
i-C4 H8
for H2 , N2 and C4 hydrocarbons at atmospheric pressure shows that due to the strong entropic effects at the range of temperatures of interest ka /kd is extremely small (Fig. 2b), implying small occupancy of the pore. This is not so at lower temperatures and/or for other components, as we show elsewhere. Therefore, below we consider both lowand high-occupancy situations. Diffusion inside a pore is usually described as a randommotion hopping of a molecule. Assuming the diffusivity to be constant along the pore, the transport frequency is approximated as kt = (kT / h) exp(−Et /RT )/L, where L is pore length and hopping distance. For most conditions Et is very small (Sznejer et al., 2004) and it is not clear whether random-motion or ballistic-motion analysis applies. Comparison of kdi and kti , shows that for pores that are 1000 sites long (a fraction of 1 m), kdi ?kti except for small molecules or for large pores (Sznejer et al., 2004). Non-activated molecular diffusion inside carbon single-wall nanotubes was numerically shown to lead to extremely high macroscopic diffusivities and several experiments pointing to that effect were also reported (Lee and Sinnott, 2004; Chen and Sholl, 2004). We should also point out that the process of single-file diffusion is known to have different characteristics than ordinary diffusion and the mean square displacement is proportional to the square root of time r 2 ∼ 2t 1/2 . The probability (p(x, t)) of finding a particle at position x after time t, when the particle was located at x = 0 at t =0, follows a Gaussian distribution with (Kollmann, 2003) p(x, t)=(4Ds t)−1/2 exp[−x 2 /Ds t]. Deterministic models were derived mainly in the context of diffusion in zeolites, which typically form regular arrays of pores (Coppens et al., 1998; Nedea et al., 2002). Such an approach does not apply for carbon sheets, which have continuous, smooth walls of uniform composition that results in unique non-activated molecular diffusion within the pores (Sznejer et al., 2004). Deterministic models cannot predict the case of single-file counter-diffusion; evidently, in the latter case the two components will exert mutual inhibition of diffusion. We need to derive a simple flux expression to be used in the reactor design section. We proceed assuming that diffusion is governed by random-motion. We briefly review the derivation of Eq. (1). If the flux is linear with the gradient
i-C4 H10
of occupancy then, at steady state, the three steps operate at equal rates and for the case of a single component, kai Pil (1 − 1 ) − kdi 1 = kt (1 − N ) = −(kai Pir (1 − N ) − kd N )
(2)
(where 1 is the probability of occupation at site 1), which can be solved to yield Eq. (1). It can be further reduced in the following cases: (i) In the case of low-occupancy (kai , kti >kdi ) fi = (kai /kdi )kti (Pil − Pir )
(3)
and the flux is linear with driving force (i.e., permeability is constant). (ii) When desorption is limiting (kdi >kai , kt ), the pore is saturated with adsorbents and Eq. (2) leads to fi = kdi [Pl /(Pl + Pr ) − Pr /(Pl + Pr )] = kdi (Pl − Pr )/(Pl + Pr ).
(4)
Note, that when Pr = 0 (i.e., under vacuum or high flow) the flux is fi = kdi . We can describe a probabilistic approach that confirms the mean-field result (Eq. (4)): Occupancy is high and desorption can occur on the left or right, and it will be followed instantaneously by adsorption, either on left or right with a probability that is determined by the ratio of ka Pl or ka Pr . If desorption occurs on the left and adsorption on the right, than each molecule moves one step to the left and we denote it as left-ward motion. The frequencies of such left-ward and right-ward motions are Ml , Mr . Now, Ml,r = (frequency of left or right desorption)×(probability of occupation on opposite side). The net flux (fi ), number of transported molecules/time, is f =Mr −Ml . The frequency of desorption, either on left or right, is kdi and the probability of occupation by adsorption on left is Pl /(Pl +Pr ). Substitution yields Eq. (4). 2.1. Co-diffusion Consider a membrane separating two components, indexed 1 and 2, mixed on the left at pressure P1l and P2l from the mixture on the right with pressures P1r and P2r ; both species can penetrate and diffuse. That corresponds to a diffusion experiment under a partial pressure gradient. In either
M. Sheintuch, I. Efremenko / Chemical Engineering Science 59 (2004) 4739 – 4746
low- or high-occupancy case the time-averaged occupation probability is space-independent and equal to probability of adsorption by species 1 on the left side
1 = l = N = S1 = (ka1 Pl1 )/(ka1 Pl1 + ka2 Pl2 ),
(5)
where S1 is the adsorption selectivity. In the low-occupancy case the two components do not interact and the flux is as above. In the desorption-limited case, four events may occur: desorption on the right followed by adsorption on the left (contributing to rightward motion) or on the right (stagnant situation) or desorption on the left followed by adsorption on right (leftward motion) or left (stagnant). The frequency of desorption, either on left or right is kdl (1 ) or kdr (N ), and it depends on the probability of site i occupation by species 1 ( ). The probability of adsorption on left or right is the ratio of adsorption on that side to the total adsorption rate on both sides qal = kal Pl /(kal Pl + kar Pr ), kal Pl = kal Pll + ka2 P2l , kar Pr = kal Plr + ka2 P2r .
qar = 1 − qal , (6)
Let the net flux be from left to right. Now, the desorption on left and right are identical, kdl = kdr = 1 kd1 + (1 − 1 )kd2 = S1 /kd1 + (1 − S1 )/kd2
(7)
and f1 = S1 (Mr − Ml ) since only a S1 fraction of the flux is of species 1. Now, Mr = kdr qal , Ml = kdl qar but recall that kdl = kdr = kd and f1 = S1 (Mr − Ml ) = S1 kd (qal − qar ) kd1 ka1 P1l + kd2 ka2 P2l ka1 P1l = ka1 P1l + ka2 P2l ka1 P1l + ka2 P2l ka1 (P1l − P1r ) + ka2 (P2l − P2r ) × ka1 (P1l + P1r ) + ka2 (P2l + P2r )
(8)
while f2 = (1 − S1 )(Mr − Ml ) is similar. For multi-component desorption-limited system, and when the right side is kept at P1l = P2l = 0, the equation above is reduced to fi = Si kd , where Si is the selectivity of adsorption (Eq. (5)), which determines the selectivity of transport, while kd is the desorption constant averaged according to the selectivity. When only one of the components is desorption-limited the same equation still applies. 2.2. Counter-diffusion Consider a membrane separating two components, one on the left at pressure P1l and another one on the right at pressure P2r ; these pressures are constant, and the pressure of the second component is nil. In the low-occupancy case the pores are only rarely occupied and the problem is reduced to finding the probability of a molecule of species 1 escaping from the left to the right (or, similarly for species 2, in the opposite direction) while not encountering molecules of the other species. This will depend now on the transport time,
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which in turn depends on the transport regime (random- or ballistic-motion), and on the pore length. We show elsewhere (as should be intuitively evident) that for sufficiently long pores the two species will block each other and the flux will diminish considerably. In the case of desorption-limited transport we assume all sites to be occupied. In this case the counter-diffusion regime is intriguing since one of the components will conquer the pore and the selectivity will be either 0 or 1. Four events may occur: desorption on the right followed by adsorption on the left (contributing to rightward motion) or on the right (stagnant situation) or desorption on the left followed by adsorption on right (leftward motion) or left (stagnant). The frequency of desorption, either on left or right is kdl (1 ) or kdr (N ) and the probability of occupation by adsorption on left is ka1 P1l /(ka1 P1l + ka2 P2r ), etc. Hence 1 , 1 + ka2 P2r /ka1 P1l ka2 P2r /ka1 P1l Ml = kdl . 1 + ka2 P2r /ka1 P1l Mr = kdr
(9)
A simple analysis is the following: Assume that section (of length ) of the pore is occupied by species 1 and the rest (on the right) by species 2. Then, kal = ka1 , kar = ka2 , and kdr P1l − kdl ka2 P2r /ka1 d = M r − Ml = dt P1l + ka2 P2r /ka1 ( − )P1l kd2 ka2 P2r = kd1 ; = ; = . (1 + )P1l kd1 ka1 P1l
(10)
Thus, species 1 will conquer the pore if > or K1 P1l > K2 P2r . Eventually the pore is occupied by species 1, except for the pore end. The total flux in a long pore is established to be kdr − kdl f = Mr − Ml = 1+ kdl (kdr P1l /kdi − ka2 P2r /ka1 ) = . (11) P1l + ka2 P2r /ka1 3. Calibration Let us consider now the transport in a set of parallel pores, whose size is defined by a certain distribution (Fig. 2a). We will take a heuristic approach, which incorporates the results of multi-component transport with the atomistic understanding outlined above. We consider here only one of the reaction modes described above, in which the tube-side is kept under a nitrogen flow at atmospheric pressure while the reaction is maintained on the shell-side and we have varying concentration of iso-butane, iso-butene, hydrogen and some nitrogen. We denote the transport law of species i in mode j (j = CD for counter-diffusion; the case of j =v (tube-side under vacuum) is considered elsewhere) by Ti,j . For simplicity we assume that the alkane and olefin have identical properties, as supported by the data in Table 1 and Fig. 2b. The transport law in the absence of reaction under counter-diffusion
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where kH,I , kH C,III are the average permeabilities of hydrogen over region I or of hydrocarbon over region III. We want to relate these expressions to the flux of individual components in the absence of reaction that was measured experimentally. Note, that under counter-diffusion of nitrogen and hydrocarbon, in the absence of reaction, the flow of each components (TN,CD0 , TH C,CD0 ) is similar to that under reaction conditions, while the hydrogen flux can be expressed as TH,CD = aI TH,0 PH /PH,0 + TH C,CD SH . Note, that non-linearity in the flux-driving force relation is introduced only by domain (II). If our assumptions are valid, then for the range of experimental conditions employed (temperature, pore sizes) such a region does not exist and exploit of constant permeabilities is justified.
4. Application
Fig. 3. Schematic representation of the transport behavior in nanopores of various sizes.
of nitrogen with hydrocarbons, or of nitrogen with a (1:1) mixture of hydrogen and hydrocarbons (Fig. 1a–c), are denoted as Ti,CD0 . The frequency fi [molecules/pore × time] is converted to a measured flux as Ti = fi /Na [moles/unit area × time] for a single pore and as Ti = an fi (rp )/Na , for a family of pores, where an is the fraction of pore of radius rp and r is pore density. When operating under counter-diffusion by nitrogen, we can divide the pore size spectrum into several domains (see Fig. 3): (I) In this domain pores are narrow and transfer hydrogen molecules only; since their flux is not limited by desorption (Fig. 2b), it is proportional to the partial pressure gradient of hydrogen, fH,I ∼ PH ; (N). In this domain hydrogen and nitrogen are undergoing counter-diffusion, and we assume net transport of nitrogen, fN,N ∼ PN ; this is the only domain where nitrogen can diffuse since its molecules are small enough to allow for transport; (II) In this domain the hydrocarbons are desorption-limited and their flux is fH C,II =(1−SH )kd,H C ; hydrogen is trapped between the hydrocarbon molecules and fH,II = kd,H C PH , where SH is the adsorption selectivity (Eq. (6)), which is largely dependent on partial pressures and the molecular size. (III) In this domain of even larger pores the hydrocarbons are adsorption-limited and meet counter-diffusion of nitrogen; consequently their flux is small and we set, fH C,III ∼ 0; fN,III ∼ 0; i.e., we assume mutual blocking effect. The total fluxes in the counter-diffusion domain are TH,CD = [aI kH,I PH + aII kd,H C SH ]/Na , TH C,CD = [aII kd,H C ](1 − SH )/Na = (1 − SH )TH C,CD0 , TN,CD = aN kN,N PN = TN,CD0 ,
(12)
The experimental system, described in detail in Sznejer and Sheintuch(2004), incorporated a 30 cm hollow fiber carbon membrane module (Carbon Membranes LTD., Temed industrial park, Israel), composed of 100 fibers each of 100 m in diameter with a 10 m thick wall. The calculated membrane area of this module is about 150 cm2 . The module was modified by casting it in a larger tube (of 7 mm i.d.) to allow for a free space (of 2 mm) between the tube and the membrane module, into which the catalyst pellets were loaded. In modeling the system, we assume isothermal isobaric conditions, with plug flow on both sides of the membrane and ignore radial gradients (see discussion of radial gradients, in Sznejer and Sheintuch(2004) and references therein,). The change in the partial molar flows (Fj , mol/min) of component j (j = H2 , i-C4 H10 , i = −C4 H8 , N2 ) in the shell (reactive)-side or in the tube ( j ) is due to the reaction and the separation through the membrane: dFj = j R(p)c A − sT j (p, P ), dz Fj i PT , pi = PS , Pi = F j
d j = sT j (p, P ), dz (13)
where R is the reaction rate [mol/g min], c is the catalyst density [g/cm3 ], A is the cross section of the catalyst side [cm2 ] and s [cm] is the perimeter, j are the stoichiometric coefficients and PT , PS denote the pressures on the tube and shell-sides, respectively (1 atm). In the inner membrane side no reaction occurs and the change in molar flow is only due to the separation through the membrane. Tj is the general non-linear function describing the flux of component j in terms of pi , Pi . Partial pressures are calculated by the ratio of the single-component flow by the total flow. Reaction performance was studied at 450 and 500 ◦ C. The experimental data in a simple PFR (no sweeping on the tube side) were
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not linear with gradient, but yet it is very small in comparison with the other fluxes (Fig. 1c). Consequently we can limit the analysis to a linear dependence of Tj =kj (pj −Pj ) on the corresponding partial pressure gradient (where kj is the permeability for component i). Membrane permeabilities were determined in independent separation experiments (Fig. 1) and were extrapolated to the desired temperature. Model predictions, of the membrane reactor in the sweeping gas mode, agree well with experimental results at 500 ◦ C but underestimate the results at 450 ◦ C (Fig. 4a,b). Conversions are significantly higher than those in a PFR (Fig. 4a), but could be improved significantly by finding a better catalyst or increasing catalyst loading in the reactor (Fig. 4a, dashed-dotted line represents tenfold increase in loading or activity). The effect of nitrogen stream flow rate on conversion is significant (Fig. 4c), suggesting improved hydrogen separation; the model underestimates this trend. The discrepancy between measurements and model predictions can be explained by the inaccuracy of the transport model and the variation in membrane constants in different membrane units.
5. Concluding remarks
Fig. 4. Comparison of experimentally measured conversion at 500 ◦ C (a) and 450 ◦ C (b) and model predictions (broken lines) as a function of iso-butene feed rate ( N2 = 0.025 gmol/min.). Fig. (a) presents also the conversion in PFR and in a membrane reactor with a tenfold better catalyst (dash–dotted line). Fig. (c) presents observed and simulated effect of nitrogen flow rate at the two temperatures.
fitted with a reaction rate expression of the form − RiC4 =
C1 (piC4 − (PS pi=C4 pH /Ke RT )) (1 + C2 pi=C4 )
(mol/h g), (14)
where Ke is the equilibrium constant and (C1 , C2 ) are (6.085×10−4 , 17.645) at 450 ◦ C and (4.312×10−3 , 11.103) at 500 ◦ C. Fig. 4 presents the simulated conversion showing that equilibrium has not been achieved for the range of feed rates studies. The conversion achieved in the counter-diffusion operation, with counter-current flow of nitrogen, is the highest amongst all modes with a maximum of 85% at 500 ◦ C (Fig. 4). The significant conversion improvement is partially due to nitrogen transport and dilution. To simulate this behavior, we note that hydrogen and nitrogen fluxes can be expressed by a constant permeability while that of the hydrocarbon is
This research tests a membrane-reactor equipped with a molecular-sieve carbon membrane using iso-butane dehydrogenation as a model reaction. The novelty of this study is in the proposed application at relatively high temperatures; very few studies tested carbon (or molecular sieve) membrane reactors. We apply the multi-scale approach, which incorporates molecular mechanics calculations, derivation of approximate flux expressions for single- and multicomponent systems, calibration of these expressions and simulation of the behavior in a membrane reactor. We cannot yet predict membrane performance from basic principles but we can use them to derive transport laws and calibrate them with appropriate experiments as demonstrated here. The shell-side can be either maintained under lower pressure (or under vacuum) or swept with an inert diluent stream. The suggested approach works well for the latter situation. Also, the conversions achieved in the counter-diffusion operation method are higher than in all other modes (up to 85% at 500 ◦ C) but this improvement partially results from the nitrogen transport and dilution. Even if nitrogen permeation could be avoided, the dilution of hydrogen on the permeate side will reduce its economic contribution. Better economics should be achieved when operating under a pressure gradient, so that hydrogen can be collected at least at atmospheric pressure. Transport experiments conducted under a pressure gradient showed poor selectivity due to hydrocarbon transport (Sznejer et al., 2004). The conversions obtained in the vacuum mode show only modest gains above the ones received in the PFR. Simulations that used the experimentally calibrated transport expressions showed poor agreement with experiments in this case.
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Acknowledgements Work supported by the Israeli Ministry of Science and by the Ministry of the Immigrant Absorption. References Casanave, D., Giroir-Fendler, A., Sanches, J., Loutaty, R., Dalmon, J.-A., 1995. Control of transport properties with a microporous membrane reactor to enhance yields in dehydrogenation reactions. Catalysis Today 25, 309–314. Casanave, D., Ciavarella, P., Fiaty, K., Dalmon, J.A., 1999. Zeolite membrane reactor for isobutane dehydrogenation: experimental results and theoretical modeling. Chemical Engineering Science 54, 2807– 2815. Chen, H., Sholl, D.S., 2004. Rapid diffusion of CH4 /H2 mixtures in single walled carbon nanotubes. Journal of American Chemical Society 126, 7778. Coppens, M.O., Bell, A., Chakraborty, A.K, 1998. Effect of topology and molecular occupancy on self-diffusion in lattice models of zeolites–Monte Carlo simulations. Chemical Engineering Science 53, 2053. Itoh, N., Haraya, K., 2000. A carbon membrane reactor. Catalysis Today 56, 103–111. Kaganov, I.V., Sheintuch, M., 2003. Nonequilibrium molecular dynamics simulation of gas-mixture transport in carbon. Physics Review E 68, 046701. Kollmann, M., 2003. Single-file diffusion of atomic and colloidal systems: asymptotic laws. Physical Review Letters 90, 180602. Kusuki, Y., Shimazaki, H., Tanihara, N., Nakanishi, S., Yoshinaga, T., 1997. Gas permeation and characterization of asymmetric polyimide hollow fiber membrane. Journal of Membrane Science 134, 245.
Lee, K.-H., Sinnott, S.B., 2004. Computational studies of non-equilibrium molecular transport through carbon nanotubes. Journal of Physical Chemistry B ACS ASAP. Mao, Z., Sinnott, S.B., 2000. A computational study of molecular diffusion and dynamic flow through carbon nanotubes. Journal of Physical Chemistry 104, 4618. Nedea, S.V., Jansen, A.P.J., Lukkien, J.J., Hilbers, P.A.J., 2002. Steady state properties of single-file systems with conversions. Physics Review E 65, 066701. Rao, M.B., Sircar, S., 1993. Nanoporous carbon membranes for separation of gas mixtures by selective surface flow. Journal of Membrane Science 85, 253. Sheintuch, M., Dessau, R.M., 1996. Observations modeling and optimization of yield selectivity and activity during dehydrogenation of isobutane and propane in a Pd membrane reactor. Chemical Engineering Science 51, 535–547. Sircar, S., Rao, M.B., Thaeron, C.M.A., 1999. Selective surface flow membrane for gas separation. Separation Science Technology 34, 2081. Sznejer, G.A., Sheintuch, M., 2004. Application of carbon membrane reactor for dehydrogenation reaction. Chemical Engineering Science 59, 2013–2021. Sznejer, G.A., Efremenko, I., Sheintuch, M., 2004. Carbon membranes for high-temperature gas separation: Experiment and theory. A.I.Ch.E. Journal 50, 596–610. Tanihara, N., Shimazaki, H., Hirayama, Y., Nakanishi, S., Yoshinaga, T., Kusuki, Y., 1999. Gas permeation properties of asymmetric carbon hollow fiber membranes prepared from asymmetric polyimide hollow fiber. Journal of Membrane Science 169, 179. Van de Graaf, J.M., Zwiep, M., Kapteijn, F., Moulijn, J.A., 1999. Application of a zeolite membrane reactor in the metathesis of propene. Chemical Engineering Science 54, 1441–1445. Xu, L., Tsotsis, T.T., Sahimi, M., 1999. Nonequilibrium molecular dynamics simulation of transport and separation of gases in carbon nanopores. I. Basic results. Journal of Chemical Physics 111, 3252.
Analysis of a carbon membrane reactor: from atomistic simulations of single-file diffusion to reactor design Moshe Sheintuch∗ , Irena Efremenko Department of Chemical Engineering, Technion, Technion city, Haifa 32000, Israel Received 29 February 2004
Abstract We study the dehydrogenation of iso-butane in a membrane reactor with carbon membranes; these are molecular sieves with pores of molecular dimensions. To provide information on the transport laws of the various components, for reactor design purposes, we studied the separation in a membrane module either by maintaining the shell-side under lower pressure (or under vacuum) or by sweeping it with an inert diluent stream. The purpose of this work is to derive multi-component flux expressions for single-file diffusion in these two modes of separation, using molecular mechanics calculations of the thermodynamics of molecular adsorption into, diffusion within and desorption from the pores, and apply them for reactor design. In the process we calibrate these expressions with integral measurements of separation in a membrane module. Good predictions of reactor performance are obtained for a reaction coupled with separation by sweeping the hydrogen with nitrogen but poor predictions were achieved for a reaction coupled with vacuum-driven separation. Reactor performance in the former mode is better due to excellent transport selectivity, which we attribute to mutual blocking of counter-diffusion by nitrogen and hydrocarbons. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Membranes; Molecular sieves; Single-file diffusion; Molecular mechanics; Reaction engineering; Mass transfer
1. Introduction In a search for a highly selective, yet relatively inexpensive, membrane that may be employed in membrane reactors for separation of hydrogen simultaneously with equilibrium-limited reactions, we study the dehydrogenation of iso-butane in a membrane reactor with carbon membranes. Carbon membranes are molecular sieves that incorporate pores of molecular dimensions so that steric and energetic effects are important for describing transport processes. Membrane reactor design requires information on the transport laws of the various components. Separation or reaction in a membrane reactor can be tested and operated by one of the three modes for a driving force for separation: The shell-side can be either maintained under ∗ Corresponding author. Tel.: +972 4 8292823; fax: +972 4 8295672.
E-mail address: [email protected] (M. Sheintuch). 0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.09.016
lower pressure (or under vacuum) or swept with an inert diluent stream or maintained under reaction conditions that consume the diffusing component (as in hydrogen oxidation). As we show below, the first two modes in a molecular sieve membrane yield markedly different behaviors, and it is not clear which information from separation studies should be applied for reactor design purposes, and what are the underlying transport laws. Most previous studies employed simple linear transport laws, which may lead to significant errors. The purpose of this work is to derive multi-component transport expressions, based on calculated thermodynamics of molecular adsorption into and diffusion within the pores and a probability analysis of transport, calibrate them with independent results of transport, and apply them for reactor design by comparing the predictions with experimental results. We now review the main experimental observations of multi-component transport through the carbon membrane
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module, which incorporated 100 fibers each of 100 m diameter, obtained in the first part of this effort (Sznejer et al., 2004). The fluxes of pure components were studied under a pressure gradient and were found to be largely linear with the driving force. Nitrogen was used as a sweeping gas in the study of mixtures. In the temperature range of 25–400 ◦ C, for a series of light hydrocarbons along with their mixtures with hydrogen, we found that counter-diffusion of nitrogen, that flows on one side of the membrane, and C2 –C4 alkanes with hydrogen flowing on the other side, significantly inhibits the fluxes (and permeabilities) of both nitrogen and alkanes, while hydrogen flux is only slightly diminished (Fig. 1). This is a significant and desired result from reactor design perspective, since we are interested in a membrane that is selective to hydrogen transport. The membrane selectivity, the ratio of hydrogen to hydrocarbon permeabilities, may reach 100–1000 in propane or in (normal or iso-) butane mixtures with hydrogen. A lower selectivity value was found from single-component systems than from multicomponent measurements. This difference led us to the conclusion that transport is dominated by single-file diffusion. While we report permeabilities of the components involved in the multi-components transport in Fig. 1, it is not evident that these parameters should be constant and that the flux of component i should be linear with its concentration. Previous experimental studies reported mainly on the low (room) temperature separation selectivities of hydrogen–alkane mixtures in carbon membranes and were motivated by separation technologies. The permeabilities of pure gases at 296 K increased from H2 to CH4 to C2 H6 and then decreased as the molecular weight of the hydrocarbon chain increased. In a mixture, the permeabilities of the most weakly adsorbed components are drastically reduced in the presence of higher molecular weight hydrocarbons (Rao and Sircar, 1993; Sircar et al., 1999). In the temperature range of 25–120 ◦ C Kusuki et al. (1997) found that the permeability of light alkanes decreased with the kinetic diameter of the gas and increased with the temperature. Tanihara et al. (1999) found a selectivity factor of 540 in a mixture of H2 /CH4 at 50 ◦ C in a carbon hollow fiber membrane. The transport and separation phenomena in carbon membranes remain largely unclear. The transport regimes important for our case like pore diffusion, capillary condensation and molecular sieving cannot be reduced to a simple rate expression and various approaches have been applied to predict fluxes and selectivities; among them non-equilibrium molecular dynamics of diffusion in well-defined pore, (e.g. see Kaganov and Sheintuch (2003) who studied compositions relevant to these study; Mao and Sinnott (2000), Xu et al. (1999), and references therein). However, such computations are still limited to small ‘samples’, typically with one species, and the results cannot be extended to more realistic situations. To account for the experimentally determined purecomponent transport we have used molecular mechanics simulations to find the energetics of adsorption (insertion
of a molecule into the pore), diffusion (translation within the pore) and desorption (removal) of individual gases in nanopores modeled by carbon nanotubes (Sznejer et al., 2004). In pores that are extremely small, relative to the size of a certain molecule, adsorption and diffusion are activated while desorption is non-activated; in practice, this domain is extremely narrow and the probability of the molecule entering such pores is very small, therefore this domain can be ignored. While ignoring entropy effects, for most of the relevant domains, adsorption is typically not activated while desorption is activated and presents the rate limiting step with activation energy determined from Ed = (−Hads ). The molecular transport proceeds essentially by a single-file diffusion mechanism. A rate expression for a single-species transport, from a reservoir at high pressure on the left (Pil ) to a low one on the right (Pir ), in a molecular-sieve carbon membrane was derived by a mean-field approach. The overall transport frequency (fi ) kai (Pil − Pir ) = Ki (Pil + Pir ) + 2 fi (Ki Pil + 1)(Ki Pir + 1) + , kti /kdi Ki =
kai , kdi
(1)
shows non-linear dependence on the driving force (ka , kt , kd are the corresponding adsorption, transport and diffusion frequencies, and i denotes the component). Here, we update our estimates of ka , kd , based on entropy calculation, and extend the analysis to the case of two-species transport in a multi-pore system. For reactor design purposes we need to develop a theory of multi-component single-file transport. In the next section we extend the analysis of pure-component single-file transport to co- and counter-diffusion in two component systems. Especially, we want to find out whether both species diffuse or just one, and if so which one? An analysis based on a single-size pore still cannot explain the dramatic change in permeabilities measured by counter-diffusion and those measured as single component (Fig. 1c). We have to consider a system with the measured pore size distribution (Fig. 2a), in which most of the pore volume is associated with pores that are 0.6–1.0 nm in diameter. In Section 3 the same analysis is repeated for a family of parallel pores. We will not consider here the transport in pore networks. In Section 4 we test the model against the performance of a membrane reactor packed with a chromia alumina catalyst (see Sznejer and Sheintuch(2004) for details). Our choice of model reaction, the dehydrogenation of iso-butane, stems from the commercial interest in this design as an alternative to the periodically regenerated fixed-bed approach. This reaction, which requires high temperatures, had been tested in a Pd-membrane reactor, achieving 78% conversion at 500 ◦ C (Sheintuch and Dessau, 1996). Only few applications of molecular sieve membranes for membrane reactors
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Fig. 1. Hydrogen (a), nitrogen (b) and alkane (c) permeabilities (cm/min atm) as a function of hydrocarbon size at various module temperatures; alkane permeabilities under a pressure gradient are also presented in (c).
Fig. 2. Pore size distribution (a) and ratio of adsorption–desorption frequencies for H2 , N2 , iso-butane and iso-butene molecules (b).
have been reported. A carbon membrane was used for cyclohexane dehydrogenation at 195 ◦ C (Itoh and Haraya, 2000). Other applications employed zeolite membrane (Casanave et al., 1995, 1999; Van de Graaf et al., 1999).
2. Theory of single-file transport Permeability will be determined by two main factors: the occupancy of the pore (largely determined by ka /kd ), and the transport within the pore. As explained in the Intro-
duction, in the present work we present the thermochemistry data (Table 1) retrieved from the full geometry optimization of the molecule-in-nanotube systems and of the separated components followed by frequency calculations (see Irikura (1998) for computational details) using molecular mechanics simulations. Application of this approach is justified by the absence of chemical interaction and existence of well-developed parameters (Universal Force Field) to describe the short- and long-range interactions between atoms. The most salient features of the pore–molecule interactions (non-activated diffusion, high adsorption energy and strong change in entropy) were checked by DFT calculations for smaller systems. While entropic effects are usually ignored in adsorption processes, we found its effect to be crucial in determining the transport regime at high temperatures. Sterically restricted situation of a molecule inside nanopore results in extremely strong decrease in the translational and rotational components of entropy change as compared to those of the separated components; restricted vibrations bring about a sharp increase in S in smallest pores. The geometrical picture ensuing from the molecular mechanics results is that of a straight pore of N sites, where each site may be occupied by one of the competing molecules freely moving from site to site and undergoing elastic collisions with each other and with pore walls. Molecular adsorption is controlled by the probability to enter the pore. Therefore, the adsorption, transport and diffusion constants are estimated as follows: if Ti,ads = Pi /(2Na mi kT )1/2 is the flux of incident molecules i, then kai = sT i,ads Na Ap , where s = exp(S/R) is the ‘sticking coefficient’, which characterizes the probability of a molecule to enter the pore of radius rp , mi (kg/mol) is the molecular mass, k is the Boltzmann constant, Na is the Avogadro’s number and Ap = rp2 is the area of the pore mouth. Desorption is controlled by the kinetic energy needed to overcome the pore–molecule interaction; its frequency is assumed to follow kd = (kT / h) exp(−Ed /RT ) with Ed = (−Hads ). Comparison of the frequencies characterizing the two √ steps kai /kdi = Pi Na Ap e−Ga /RT / 2mi Na kT (kT / h) (Ga = Hads − T S is the change in Gibbs free energy)
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Table 1 Calculated pore–molecule interaction energies (H , kJ/mol) and entropy changes (S, J/mol K) at 627 K Pore
H2
D, A˚
H
S
H
S
H
S
H
S
5.40 6.26 7.00 7.82 8.60 9.39 10.96 15.66
−2.64 −23.50 −20.74 −16.62 −14.13 −12.67 −10.91 −9.96
−113.16 −79.4 −35.26 −20.81 −61.37 −56.87 −89.82 −94.17
167.64 −6.43 −35.78 −33.36 −26.37 −22.93 −18.15 −16.57
−169.62 −164.21 −91.56 −44.85 −118.47 −82.68 −20.32 −122.13
−51.72 −70.42 −99.85 −88.61 −78.33
−173.97 −195.81 −155.77 −159.51 −160.96
102.98 −90.81 −112.30 −74.96 −38.36
−256.57 −202.35 −153.01 −189.6 −234.51
N2
i-C4 H8
for H2 , N2 and C4 hydrocarbons at atmospheric pressure shows that due to the strong entropic effects at the range of temperatures of interest ka /kd is extremely small (Fig. 2b), implying small occupancy of the pore. This is not so at lower temperatures and/or for other components, as we show elsewhere. Therefore, below we consider both lowand high-occupancy situations. Diffusion inside a pore is usually described as a randommotion hopping of a molecule. Assuming the diffusivity to be constant along the pore, the transport frequency is approximated as kt = (kT / h) exp(−Et /RT )/L, where L is pore length and hopping distance. For most conditions Et is very small (Sznejer et al., 2004) and it is not clear whether random-motion or ballistic-motion analysis applies. Comparison of kdi and kti , shows that for pores that are 1000 sites long (a fraction of 1 m), kdi ?kti except for small molecules or for large pores (Sznejer et al., 2004). Non-activated molecular diffusion inside carbon single-wall nanotubes was numerically shown to lead to extremely high macroscopic diffusivities and several experiments pointing to that effect were also reported (Lee and Sinnott, 2004; Chen and Sholl, 2004). We should also point out that the process of single-file diffusion is known to have different characteristics than ordinary diffusion and the mean square displacement is proportional to the square root of time r 2 ∼ 2t 1/2 . The probability (p(x, t)) of finding a particle at position x after time t, when the particle was located at x = 0 at t =0, follows a Gaussian distribution with (Kollmann, 2003) p(x, t)=(4Ds t)−1/2 exp[−x 2 /Ds t]. Deterministic models were derived mainly in the context of diffusion in zeolites, which typically form regular arrays of pores (Coppens et al., 1998; Nedea et al., 2002). Such an approach does not apply for carbon sheets, which have continuous, smooth walls of uniform composition that results in unique non-activated molecular diffusion within the pores (Sznejer et al., 2004). Deterministic models cannot predict the case of single-file counter-diffusion; evidently, in the latter case the two components will exert mutual inhibition of diffusion. We need to derive a simple flux expression to be used in the reactor design section. We proceed assuming that diffusion is governed by random-motion. We briefly review the derivation of Eq. (1). If the flux is linear with the gradient
i-C4 H10
of occupancy then, at steady state, the three steps operate at equal rates and for the case of a single component, kai Pil (1 − 1 ) − kdi 1 = kt (1 − N ) = −(kai Pir (1 − N ) − kd N )
(2)
(where 1 is the probability of occupation at site 1), which can be solved to yield Eq. (1). It can be further reduced in the following cases: (i) In the case of low-occupancy (kai , kti >kdi ) fi = (kai /kdi )kti (Pil − Pir )
(3)
and the flux is linear with driving force (i.e., permeability is constant). (ii) When desorption is limiting (kdi >kai , kt ), the pore is saturated with adsorbents and Eq. (2) leads to fi = kdi [Pl /(Pl + Pr ) − Pr /(Pl + Pr )] = kdi (Pl − Pr )/(Pl + Pr ).
(4)
Note, that when Pr = 0 (i.e., under vacuum or high flow) the flux is fi = kdi . We can describe a probabilistic approach that confirms the mean-field result (Eq. (4)): Occupancy is high and desorption can occur on the left or right, and it will be followed instantaneously by adsorption, either on left or right with a probability that is determined by the ratio of ka Pl or ka Pr . If desorption occurs on the left and adsorption on the right, than each molecule moves one step to the left and we denote it as left-ward motion. The frequencies of such left-ward and right-ward motions are Ml , Mr . Now, Ml,r = (frequency of left or right desorption)×(probability of occupation on opposite side). The net flux (fi ), number of transported molecules/time, is f =Mr −Ml . The frequency of desorption, either on left or right, is kdi and the probability of occupation by adsorption on left is Pl /(Pl +Pr ). Substitution yields Eq. (4). 2.1. Co-diffusion Consider a membrane separating two components, indexed 1 and 2, mixed on the left at pressure P1l and P2l from the mixture on the right with pressures P1r and P2r ; both species can penetrate and diffuse. That corresponds to a diffusion experiment under a partial pressure gradient. In either
M. Sheintuch, I. Efremenko / Chemical Engineering Science 59 (2004) 4739 – 4746
low- or high-occupancy case the time-averaged occupation probability is space-independent and equal to probability of adsorption by species 1 on the left side
1 = l = N = S1 = (ka1 Pl1 )/(ka1 Pl1 + ka2 Pl2 ),
(5)
where S1 is the adsorption selectivity. In the low-occupancy case the two components do not interact and the flux is as above. In the desorption-limited case, four events may occur: desorption on the right followed by adsorption on the left (contributing to rightward motion) or on the right (stagnant situation) or desorption on the left followed by adsorption on right (leftward motion) or left (stagnant). The frequency of desorption, either on left or right is kdl (1 ) or kdr (N ), and it depends on the probability of site i occupation by species 1 ( ). The probability of adsorption on left or right is the ratio of adsorption on that side to the total adsorption rate on both sides qal = kal Pl /(kal Pl + kar Pr ), kal Pl = kal Pll + ka2 P2l , kar Pr = kal Plr + ka2 P2r .
qar = 1 − qal , (6)
Let the net flux be from left to right. Now, the desorption on left and right are identical, kdl = kdr = 1 kd1 + (1 − 1 )kd2 = S1 /kd1 + (1 − S1 )/kd2
(7)
and f1 = S1 (Mr − Ml ) since only a S1 fraction of the flux is of species 1. Now, Mr = kdr qal , Ml = kdl qar but recall that kdl = kdr = kd and f1 = S1 (Mr − Ml ) = S1 kd (qal − qar ) kd1 ka1 P1l + kd2 ka2 P2l ka1 P1l = ka1 P1l + ka2 P2l ka1 P1l + ka2 P2l ka1 (P1l − P1r ) + ka2 (P2l − P2r ) × ka1 (P1l + P1r ) + ka2 (P2l + P2r )
(8)
while f2 = (1 − S1 )(Mr − Ml ) is similar. For multi-component desorption-limited system, and when the right side is kept at P1l = P2l = 0, the equation above is reduced to fi = Si kd , where Si is the selectivity of adsorption (Eq. (5)), which determines the selectivity of transport, while kd is the desorption constant averaged according to the selectivity. When only one of the components is desorption-limited the same equation still applies. 2.2. Counter-diffusion Consider a membrane separating two components, one on the left at pressure P1l and another one on the right at pressure P2r ; these pressures are constant, and the pressure of the second component is nil. In the low-occupancy case the pores are only rarely occupied and the problem is reduced to finding the probability of a molecule of species 1 escaping from the left to the right (or, similarly for species 2, in the opposite direction) while not encountering molecules of the other species. This will depend now on the transport time,
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which in turn depends on the transport regime (random- or ballistic-motion), and on the pore length. We show elsewhere (as should be intuitively evident) that for sufficiently long pores the two species will block each other and the flux will diminish considerably. In the case of desorption-limited transport we assume all sites to be occupied. In this case the counter-diffusion regime is intriguing since one of the components will conquer the pore and the selectivity will be either 0 or 1. Four events may occur: desorption on the right followed by adsorption on the left (contributing to rightward motion) or on the right (stagnant situation) or desorption on the left followed by adsorption on right (leftward motion) or left (stagnant). The frequency of desorption, either on left or right is kdl (1 ) or kdr (N ) and the probability of occupation by adsorption on left is ka1 P1l /(ka1 P1l + ka2 P2r ), etc. Hence 1 , 1 + ka2 P2r /ka1 P1l ka2 P2r /ka1 P1l Ml = kdl . 1 + ka2 P2r /ka1 P1l Mr = kdr
(9)
A simple analysis is the following: Assume that section (of length ) of the pore is occupied by species 1 and the rest (on the right) by species 2. Then, kal = ka1 , kar = ka2 , and kdr P1l − kdl ka2 P2r /ka1 d = M r − Ml = dt P1l + ka2 P2r /ka1 ( − )P1l kd2 ka2 P2r = kd1 ; = ; = . (1 + )P1l kd1 ka1 P1l
(10)
Thus, species 1 will conquer the pore if > or K1 P1l > K2 P2r . Eventually the pore is occupied by species 1, except for the pore end. The total flux in a long pore is established to be kdr − kdl f = Mr − Ml = 1+ kdl (kdr P1l /kdi − ka2 P2r /ka1 ) = . (11) P1l + ka2 P2r /ka1 3. Calibration Let us consider now the transport in a set of parallel pores, whose size is defined by a certain distribution (Fig. 2a). We will take a heuristic approach, which incorporates the results of multi-component transport with the atomistic understanding outlined above. We consider here only one of the reaction modes described above, in which the tube-side is kept under a nitrogen flow at atmospheric pressure while the reaction is maintained on the shell-side and we have varying concentration of iso-butane, iso-butene, hydrogen and some nitrogen. We denote the transport law of species i in mode j (j = CD for counter-diffusion; the case of j =v (tube-side under vacuum) is considered elsewhere) by Ti,j . For simplicity we assume that the alkane and olefin have identical properties, as supported by the data in Table 1 and Fig. 2b. The transport law in the absence of reaction under counter-diffusion
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where kH,I , kH C,III are the average permeabilities of hydrogen over region I or of hydrocarbon over region III. We want to relate these expressions to the flux of individual components in the absence of reaction that was measured experimentally. Note, that under counter-diffusion of nitrogen and hydrocarbon, in the absence of reaction, the flow of each components (TN,CD0 , TH C,CD0 ) is similar to that under reaction conditions, while the hydrogen flux can be expressed as TH,CD = aI TH,0 PH /PH,0 + TH C,CD SH . Note, that non-linearity in the flux-driving force relation is introduced only by domain (II). If our assumptions are valid, then for the range of experimental conditions employed (temperature, pore sizes) such a region does not exist and exploit of constant permeabilities is justified.
4. Application
Fig. 3. Schematic representation of the transport behavior in nanopores of various sizes.
of nitrogen with hydrocarbons, or of nitrogen with a (1:1) mixture of hydrogen and hydrocarbons (Fig. 1a–c), are denoted as Ti,CD0 . The frequency fi [molecules/pore × time] is converted to a measured flux as Ti = fi /Na [moles/unit area × time] for a single pore and as Ti = an fi (rp )/Na , for a family of pores, where an is the fraction of pore of radius rp and r is pore density. When operating under counter-diffusion by nitrogen, we can divide the pore size spectrum into several domains (see Fig. 3): (I) In this domain pores are narrow and transfer hydrogen molecules only; since their flux is not limited by desorption (Fig. 2b), it is proportional to the partial pressure gradient of hydrogen, fH,I ∼ PH ; (N). In this domain hydrogen and nitrogen are undergoing counter-diffusion, and we assume net transport of nitrogen, fN,N ∼ PN ; this is the only domain where nitrogen can diffuse since its molecules are small enough to allow for transport; (II) In this domain the hydrocarbons are desorption-limited and their flux is fH C,II =(1−SH )kd,H C ; hydrogen is trapped between the hydrocarbon molecules and fH,II = kd,H C PH , where SH is the adsorption selectivity (Eq. (6)), which is largely dependent on partial pressures and the molecular size. (III) In this domain of even larger pores the hydrocarbons are adsorption-limited and meet counter-diffusion of nitrogen; consequently their flux is small and we set, fH C,III ∼ 0; fN,III ∼ 0; i.e., we assume mutual blocking effect. The total fluxes in the counter-diffusion domain are TH,CD = [aI kH,I PH + aII kd,H C SH ]/Na , TH C,CD = [aII kd,H C ](1 − SH )/Na = (1 − SH )TH C,CD0 , TN,CD = aN kN,N PN = TN,CD0 ,
(12)
The experimental system, described in detail in Sznejer and Sheintuch(2004), incorporated a 30 cm hollow fiber carbon membrane module (Carbon Membranes LTD., Temed industrial park, Israel), composed of 100 fibers each of 100 m in diameter with a 10 m thick wall. The calculated membrane area of this module is about 150 cm2 . The module was modified by casting it in a larger tube (of 7 mm i.d.) to allow for a free space (of 2 mm) between the tube and the membrane module, into which the catalyst pellets were loaded. In modeling the system, we assume isothermal isobaric conditions, with plug flow on both sides of the membrane and ignore radial gradients (see discussion of radial gradients, in Sznejer and Sheintuch(2004) and references therein,). The change in the partial molar flows (Fj , mol/min) of component j (j = H2 , i-C4 H10 , i = −C4 H8 , N2 ) in the shell (reactive)-side or in the tube ( j ) is due to the reaction and the separation through the membrane: dFj = j R(p)c A − sT j (p, P ), dz Fj i PT , pi = PS , Pi = F j
d j = sT j (p, P ), dz (13)
where R is the reaction rate [mol/g min], c is the catalyst density [g/cm3 ], A is the cross section of the catalyst side [cm2 ] and s [cm] is the perimeter, j are the stoichiometric coefficients and PT , PS denote the pressures on the tube and shell-sides, respectively (1 atm). In the inner membrane side no reaction occurs and the change in molar flow is only due to the separation through the membrane. Tj is the general non-linear function describing the flux of component j in terms of pi , Pi . Partial pressures are calculated by the ratio of the single-component flow by the total flow. Reaction performance was studied at 450 and 500 ◦ C. The experimental data in a simple PFR (no sweeping on the tube side) were
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not linear with gradient, but yet it is very small in comparison with the other fluxes (Fig. 1c). Consequently we can limit the analysis to a linear dependence of Tj =kj (pj −Pj ) on the corresponding partial pressure gradient (where kj is the permeability for component i). Membrane permeabilities were determined in independent separation experiments (Fig. 1) and were extrapolated to the desired temperature. Model predictions, of the membrane reactor in the sweeping gas mode, agree well with experimental results at 500 ◦ C but underestimate the results at 450 ◦ C (Fig. 4a,b). Conversions are significantly higher than those in a PFR (Fig. 4a), but could be improved significantly by finding a better catalyst or increasing catalyst loading in the reactor (Fig. 4a, dashed-dotted line represents tenfold increase in loading or activity). The effect of nitrogen stream flow rate on conversion is significant (Fig. 4c), suggesting improved hydrogen separation; the model underestimates this trend. The discrepancy between measurements and model predictions can be explained by the inaccuracy of the transport model and the variation in membrane constants in different membrane units.
5. Concluding remarks
Fig. 4. Comparison of experimentally measured conversion at 500 ◦ C (a) and 450 ◦ C (b) and model predictions (broken lines) as a function of iso-butene feed rate ( N2 = 0.025 gmol/min.). Fig. (a) presents also the conversion in PFR and in a membrane reactor with a tenfold better catalyst (dash–dotted line). Fig. (c) presents observed and simulated effect of nitrogen flow rate at the two temperatures.
fitted with a reaction rate expression of the form − RiC4 =
C1 (piC4 − (PS pi=C4 pH /Ke RT )) (1 + C2 pi=C4 )
(mol/h g), (14)
where Ke is the equilibrium constant and (C1 , C2 ) are (6.085×10−4 , 17.645) at 450 ◦ C and (4.312×10−3 , 11.103) at 500 ◦ C. Fig. 4 presents the simulated conversion showing that equilibrium has not been achieved for the range of feed rates studies. The conversion achieved in the counter-diffusion operation, with counter-current flow of nitrogen, is the highest amongst all modes with a maximum of 85% at 500 ◦ C (Fig. 4). The significant conversion improvement is partially due to nitrogen transport and dilution. To simulate this behavior, we note that hydrogen and nitrogen fluxes can be expressed by a constant permeability while that of the hydrocarbon is
This research tests a membrane-reactor equipped with a molecular-sieve carbon membrane using iso-butane dehydrogenation as a model reaction. The novelty of this study is in the proposed application at relatively high temperatures; very few studies tested carbon (or molecular sieve) membrane reactors. We apply the multi-scale approach, which incorporates molecular mechanics calculations, derivation of approximate flux expressions for single- and multicomponent systems, calibration of these expressions and simulation of the behavior in a membrane reactor. We cannot yet predict membrane performance from basic principles but we can use them to derive transport laws and calibrate them with appropriate experiments as demonstrated here. The shell-side can be either maintained under lower pressure (or under vacuum) or swept with an inert diluent stream. The suggested approach works well for the latter situation. Also, the conversions achieved in the counter-diffusion operation method are higher than in all other modes (up to 85% at 500 ◦ C) but this improvement partially results from the nitrogen transport and dilution. Even if nitrogen permeation could be avoided, the dilution of hydrogen on the permeate side will reduce its economic contribution. Better economics should be achieved when operating under a pressure gradient, so that hydrogen can be collected at least at atmospheric pressure. Transport experiments conducted under a pressure gradient showed poor selectivity due to hydrocarbon transport (Sznejer et al., 2004). The conversions obtained in the vacuum mode show only modest gains above the ones received in the PFR. Simulations that used the experimentally calibrated transport expressions showed poor agreement with experiments in this case.
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