Ind. Eng. Chem. Res. 1990,29, 232-238
232
PROCESS ENGINEERING AND DESIGN A Catalytic Membrane Reactor: Its Performance in Comparison with Other Types of Reactors Yi-Ming Sunt and Soon-Jai Khang* Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221
This paper analyzes the performance of a catalytic membrane reactor (CMR) in comparison with three different reactor types: the inert membrane reactor with catalyst pellets placed on the feed side of the membrane (IMRCF), the plug flow reactor packed with catalyst pellets (PFR), and the mixed flow reactor in which catalysts are well-mixed with reactants (MFR). Three general categories of reactions are considered: (1)the volume is increased after reaction (gas-phase reaction with A n > 0), ( 2 ) the volume remains constant after reaction (liquid-phase reaction, or gas-phase reaction with A n = 01, and (3) the volume is decreased after reaction (gas-phase reaction with A n < 0). For category 1, the performances of PFR and MFR operated a t the feed-side pressure ( P ) are better than those of CMR or IMRCF. Between two membrane reactors, the performance of C d R is slightly better than that of IMRCF a t a longer diffusional space time (low pressure drop across the membrane). For category 2, both types of membrane reactors (CMR and IMRCF) perform better than the traditional PFR and MFR because of the equilibrium shift induced by a selective product separation. For category 3, the performance of IMRCF a t a longer diffusional space time is better than any other reactors in the studied case. The catalytic membrane reactor is not suitable for this category due to the undesirable equilibrium effect induced by the pressure variation.
In recent years, membrane reactors have been of considerable interest for many gas-phase catalytic reaction processes. Kameyama et al. (1981) demonstrated the application of a membrane reactor to the production of hydrogen from hydrogen sulfide with a microporous glass membrane adjacent to the reaction zone (the catalysts were in the feed-side stream of the membrane reactor). In a similar manner, Shinji et al. (1982) and itoh et al. (1983) reported experimental and theoretical studies of using the microporous glass membrane reactor for the dehydrogenation of cyclohexane, and Itoh et al. (1984) studied its use for the decompositionof hydrogen iodide. Itoh (1987) used a palladium membrane, which selectively permeates hydrogen only, for the dehydrogenationof cyclohexane and claimed almost a 100% conversion of cyclohexane. The previous studies by Sun (1987) and Sun and Khang (1988) also demonstrated the use of a platinum-impregnated Vycor glass membrane for dehydrogenationof cyclohexane in a catalytic membrane reactor; in this case, the microporous glass membrane was used as a separation medium as well as a catalyst. The purpose of the membrane in a reactor is to provide a separation effect while the reaction is proceeding. It is possible to selectively and continuously remove some species (preferably products) and retain some others (preferably reactants). Thus, the reaction conversion may be improved for the equilibrium-limited reversible reactions. Membrane reactors can be categorized into two different types: one in which the reaction zone is adjacent to the
* To whom all correspondence should be addressed. +Currentaddress: Ciba-Geigy Corp., 444 Saw Mill River Rd, Ardsley, NY 10502.
membrane (inert membrane reactor with catalyst of the feed side, IMRCF for short) (Itoh et al., 1983, 1984) and one in which the reaction zone is in the membrane (catalytic membrane reactor, CMR for short) (Sun, 1987). Both types of membrane reactors have been shown to be capable of achieving conversions above the equilibrium conversion based on the feed conditions. The total conversion of a dehydrogenation reaction from the CMR was found to be higher than that from the IMRCF for operations at relatively high space times; however, the situation might be reversed for operations at relatively low space times. It was also found that the performance of a CMR is more favorable for a reaction with a net increase of the total number of moles as the reaction proceeds (An > 0). However, the application of these two types of membrane reactors to the reactions with no change (An = 0) and a net decrase (An < 0) of the total number of moles as the reactions proceed has not yet been well investigated. The purpose of this paper is to further analyze the performance of the catalytic membrane reactor in comparison with the inert membrane reactor and also two other traditional ideal flow reactors-the plug flow reactor (PFR) and the mixed flow reactor (MFR). The comparison is made based on the same amount of catalyst and is made by using a well-established single-cell model (Itoh, 1987). The relationships among key dimensionless groups in the governing equations are discussed in detail in the following sections.
Models a. Catalytic Membrane Reactor. A schematic diagram of the annular catalytic membrane reactor is shown in Figure la. The feed-side (upstream) pressure, Pf, is higher than the permeate-side (downstream) pressure, Pp. 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 233 In the feed-side chamber (shell side)
mrmeate Side
(2)
In the permeate-side chamber (tube side) 0,
ap
xt
yI I
+
= 0 ( j = 1, 2,
QP0yj0- Qfij
Direction of Diffusion P (in lhn Membrane,)
..., n)
rD
(3) The dimensionless forms of eq 1-3 are written as follows: Catalyst
(b)
Bed
$; = x,
t Q,
t
'f
5
'Pi
+; = y;Pr
t '
j
yi
Figure 1. Schematic diagrams of (a) catalytic membrane reactor, (b)inert membrane reactor with catalysts on the feed side, (c) plug flow reactor, and (d) mixed flow reactor.
The total molar flow rate and the composition of each input stream, as well as Ppand Pf,are maintained constant during the operation. A single or multiple reaction may take place in the catalytic membrane, and the reaction rate expression of each reaction is denoted by ri = k i f i ( P ) , where i, k , and P a r e , respectively, the reaction number, the reaction rate constant, and the vector expression for the partial pressures of all the components. It is assumed that the system is isothermal and the interfacial mass-transfer resistance between the gas phase and the surface of the catalytic membrane is negligible compared to the internal mass-transfer resistance in the membrane. The membrane is assumed to have a microporous structure, and thus the gas diffusion through the membrane is assumed to follow the Knudsen diffusion behavior. I t is also assumed that the contents in the feed-side chamber and the permeate-side chamber are well mixed. The model is called a single-cell model because no longitudinal variation is considered. The steady-state equations for the system are given are follows. In the catalytic membrane
+ C u i j k i f i ( P=) 0
( j = 1, 2 ,
..., n ) I _
where qj, aj,4, and 8 are, respectively, the dimensionless partial pressure of component j , the ratio of diffusivities between component j and component 1or the ideal separation factor between the two components, the dimensionless radius, and the dimensionless molar flow rate. The dimensionless diffusional space time, 4, and the Thiele modulus, 9,are defined as 2rLDiPf (7) 4= 6R TQp
b. Inert Membrane Reactor with Catalysts on the Feed Side. For an inert membrane reactor with catalysts on the feed side, as shown in Figure lb, the governing equations based on the differential material balance in the direction of diffusion and the material balance for each component in the feed-side and the permeate-side chambers are derived assuming a well-mixed condition in each chamber. In the membrane Dj 1 E ;.[
at r = rf
P, = y;Pp
at r = rp
d
rd r ] = O ( j = 1, 2,
P; = x;Pf
..., n)
(9)
at r = rf
P, = yipp
at r = r p In the feed-side chamber (shell side)
..
( j = 1, 2,
..., n) (10)
In the permeate-side chamber (tube side)
\
(11
P; = xjPf
4=1 at 4 = 0
at
Q,Oy,O
-
Qfij+
= 0 ( j = 1, 2,
..., n )
234 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 There is no reaction term in eq 9; thus, the dimensionless pressure gradient is constant throughout the membrane. Thus, the solution of eq 9 can be combined with eq 10 and 11 and can be rewritten in dimensionless forms as Xjo
- 8fXj -
Further, the diffusional space time, 4, in eq Ti can be rearranged as follows: d = maximum forward diffusion rate/
m
Lyj@(Xj
- Yjp,) + 8CUijKi&(X) = 0 i=l
+
x j o - efxj a j $ ( x j - yjPJ = 0
( j = 1, 2,
..., n )
c. Plug Flow Reactor. For a constant-pressure plug flow reactor filled with V , volume of catalysts (as shown in Figure IC), the governing equations are obtained by taking the material balance for component j in a differential element of volume d V dV
E(-vij)kifi(P) i=l
--
dz
m
- dC(-~ij)~i&(lC/) ( j = 1,2, ..., n )
QP
As shown above, 4 carries a similar physical meaning as that of 8, a dimensionless space time (Ishikawa et al., 1986). To differentiate these two dimensionless space times, we named 0 the “dimensionless reactional space time” and 4 the “dimensionless diffusional space time”. Note that both space times were defined with the condition that only a single component was present in the system and no reverse reaction nor backward diffusion existed. Dividing eq 14 by eq 7, the following equation is derived: 0
-=
4
It should be noted that there are no diffusion terms in eq 15 and only bulk flow exists in the PFR. If we replace the differential volume dV by V , dz, eq 15 can be written in dimensionless form as d(efxj)
1
across membrane/thickness of membrane) - (21)
(13)
where 8 is the dimensionless reactional space time defined as
-d(Qfxj) -
A,D,(maximum possible concentration difference
( j = 1, 2. ...) n ) (12)
(16)
i=l
where z is the dimensionless length of the reactor. Equation 16 is a first-order differential equation, and the initial condition depends on the feed composition and the input molar flow rate. The intermediate dimensionless molar flow rate (ef)and molar fractions of component j ( x j ) are given by
kl Vc6P?1-lRT 2aLD1
(22)
where V , is the volume of the catalytic membrane, which can be written as V , = 7r(rf - r,?)L. By substituting this form of V , into eq 22, we have
-8 - kl(rf2 - r,2)6PP-lRT 4
201 r,262klPP-1RT(rf/rP)* - 1 = @(ena- 1)/26 (23) Dl 26
and as the membrane thickness becomes very small, the following ratio reaches unity: lim (eza - 1)/26 = 1 6-4
Therefore, a very thin membrane the ratio 8/4 becomes the square of the Thiele modulus, a: 8 - = a2 (for 6 0 or a very thin membrane) (24) 4 Equation 24 is the exact relationship among 8, 4, and 9 if the membrane has a flat geometry. Thus, from eq 20, 21, and 24, the physical meaning of @ can be defined as CP = (maximum forward reaction rate/ maximum forward diffusion rate)’I2 (25)
-
d. Mixed Flow Reactor. For a mixed flow reactor filled with V , volume of catalysts (shown in Figure Id), the governing equations are derived from the material balance for component j over the reactor as a whole: m
. I
QPxj0 - Qfxj = V,C(-uij)kifi(P) ( j = 1, 2, ..., n ) i=l
The dimensionless form for eq 18 is m
(18)
From eq 21, the diffusional space time can also be rearranged as follows: ( P f / R T )- ( P p / R T ) 1 4 = 27rLrp1
-+
rP6
QP
P p / R T 1 permeation rate e. Relationships among 4, 8, and a. The relationships = + among 4, 8, and and their physical meanings are dis- - - - - - - , prL~2 rp6 QP input flow rate cussed in this section. (See eq 7,8, and 14 for definitions.) P,27rLr,,D1- P d R T 1 = cut + Pr4 (26) The relationship between 8 and 7 (real space time) and the relationship between 4 and the “cut” or “fraction recovery” r d QP (the ratio of permeate flow rate to the input flow rate in Therefore, the cut is related to the dimensionless diffua membrane separator) (Hwang and Kammermeyer, 1984) sional space time as follows: are also presented here. In the conventional of chemical reaction engineering, eq cut = $(l- P,) (27) 14 contains the form of dimensionless reactional space Note that the relationship is valid only when a single time. The physical meaning of 8 can be expressed as component is present without reaction. Nonetheless, eq follows: 27 approximates the relationship between cut and 4, and 8 = maximum forward reaction rate assuming a it also suggests that 4 has to be smaller than 1/(1- PJ &th-order reaction/input flow rate (20) for a stable operation under a chosen set of operating
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 235
0.8
0.6
1
0.4
1 /
/
1
'0.4
1 I
Sun(1987)
Calculated
0.2
0
0.0 1 0.0
/
0.2
0.4
0.6
0.0
1 .o
0.8
7 (SI Figure 2. Effect of space time ( 7 ) on the total conversion (XT) in the PFR [cyclohexane dehydrogenation, T = 574 K,Pf = 99.5 kPa, and xo = (0.5, 0.0,0.5)].
pressures if there is no reaction in the system. Finally, the dimensionless reactional space time is related to the "real" space time by the following equation:
1
0
-
-
f ( R= Pi - P2Pa3/Kp
+
(294
category 2: An = 0
f ( R= PlP2 - P3P4/KP category 3: An < 0 f ( P )= P1P23- P 3 P 4 / K p
(2%)
(29c)
In order to make the comparison of different types of reactors under the same basis, the volume of catalysts (V,) was kept the same for all the reactors. The permeability of product (component 3) was assumed to be 6 times that of the reactant (component 1) in order to simulate an equilibrium shift to a higher conversion. The governing equations for the catalytic membrane reactor in their dimensionless forms, eq 4-6, were solved numerically with the rate equations of eq 29. Ideally, this type of boundary value problem could be converted to an initial value problem and easily solved by the shooting method (Gerald and Wheatley, 1984). However, this approach produced a numerically unstable solution because of the fast chemical reaction rates in eq 29 (Ferziger, 1981). In order to avoid this numerical stiffness, the above equations were converted into unsteady-state partial differential equations. Thus, an additional unsteady-state term (a partial derivative with respect to time) was added to the right-hand sides of eq 4-6, and the new partial differential equations were solved by a standard finite
I
Sun & Khang (1988)
Calculated
! Xr
0
5
10
15
20
25
30
35
40
45
Tb) Figure 3. Effect of space time ( 7 ) on the total conversion (X,)in the CMR [T= 573 K,Pf = 191 kPa, Pp = 99 P a , and xo = (0.49, 0.0,0.51)].
--
.
Sun & Khang (1988)
12OI \
I
m
101
klPP-'RT7 (28)
Simulations Although the models presented in the previous section could be applied for multiple reactions, only a single-reaction case was considered for the present simulation work. Three different categories of reactions were considered. The dehydrogenation of cyclohexane represented the reaction of category 1for An > 0,and two other hypothetical reactions represented the reactions of category 2 (A B C + D)for An = 0 and category 3 (A + 3B C + D) for An < 0. The rate expressions of them are as follows: category 1: An > 0
I
0
0 0
m
\
Calculated
-
- al
- - -
;-; *-:-
QP
t
--
~
-~
\
5
10
15
20
25
30
35
40
45
V S )
Figure 4. Effect of space time ( T ) on the total output flow rates (Qf, Q,) in the CMR [T = 573 K,Pf = 191 kPa, P,,= 99 kPa, and xo = (0.49, 0.0,0.51)].
difference method until a steady-state solution was obtained. This method was proved efficient and stable when proper initial conditions were used. Equations 12,13,16,and 19 were solved by the nonlinear equation solver ZSCNT and the differential equation routine DEGEAR from the International Mathematical & Statistical Libraries (IMSL, 1984).
Results and Discussion Catalytic Membrane Reactor with a Reaction of Category 1. The dehydrogenation of cyclohexane was used for category 1. Both experimental and simulation works were performed. The intrinsic reaction rate constant was measured independently by using a plug-flow reactor with crushed particles of catalytic membrane. Figure 2 shows the effect of space time on the total conversion in the PFR. For the CMR, the total conversion (X,)and the total output flow rates (Qf and Q,) from the simulation are plotted as a function of space time ( 7 ) in Figures 3 and 4, respectively, along with the previous experimental data for cyclohexane dehydrogenation (Itoh et al., 1985). The two horizontal dashed lines in Figure 3 show the equilibrium conversions without any preferential separation of product: the top line (Xe,f)is based on the feed-side composition and pressure ( P f ) and , the bottom line (Xa,*) is based on the permeate-side composition and pressure ( P ). Because the extent of the reaction depends on t i e mean residence time of the reactant(s), the total conversion increases with the space time as expected. At high space
236 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 1 .o
0.8
’
0.8
THIELE MODULU
0.6
20.4
0.2
Y.”
0.0
0.3
0.9
0.6
1.2
1.5
0.0 0.0
0.3
0.6
0.5)].
time, the total conversion (X,)becomes higher than the feed-side equilibrium conversion (Xes)because the product (hydrogen)preferentially diffuses through the membrane, thus shifting the equilibrium conversion to a higher value. The total conversion is also improved by a pressure variation effect within the catalytic membrane because the pressure in the membrane is lower than the feed pressure (Pf),and thus the actual equilibrium conversion of the reactant (cyclohexane dehydrogenation) inside the membrane is higher than the calculated equilibrium conversion based on the feed-side pressure. This type of apparent equilibrium shift does not occur at a low space time because the mean residence time of the reactant is too short and the mass-transfer resistance inside the catalytic membrane hinders the chemical reaction from reaching a new equilibrium throughout the catalytic membrane. In Figure 4, the flow rate of the permeate stream (8,) decreases slightly with increasing space time, while the flow rate of the reject stream (Qf, the output from the feed-side chamber) exhibits a considerable decrease with increasing space time. This is due to the constant total pressure difference applied across the membrane so that the permeation rate through the membrane remains relatively constant; hence, the flow rate of the permeate stream also remains relatively constant. As the space time increases, the flow rate of the reject stream decreases and eventually approaches zero, and all the reacting species permeate through the catalytic membrane. (This condition is called a total permeation or a 100% cut.) Thus, there exists a maximum space time allowed (for example, 7, = 43 s in Figure 4) for each set of operating pressures in the feedside and permeate-side chambers. Figure 5 shows the effect of different Thiele moduli on the performance of the catalytic membrane reactor. In this figure, every curve has an end point to its right. These end points correspond to the condition that the feed-side output flow rate (Qf) equals zero (refer to the total permeation condition in Figure 4). The locus of these end points is named the “total permeation line” (TPL). Maximum conversion is achieved at the total permeation condition because the reacting species have the highest contact time with the catalysts in the membrane. The total conversion increases with the dimensionless diffusional space time up to the total permeation condition. However, the total conversion is always lower than the permeate-side equilibrium conversion (the top dash line). When the Thiele modulus is smaller than 1.2, the catalytic membrane reactor is not capable of achieving the
1.2
1.5
Cb
@ Figure 5. Toal conversion (X,)vs the dimensionless diffusional space time (6)for different values of the Thiele modulus (a) in the CMR [T= 573 K, Pf = 191 kPa, Pp= 99 kPa, and xo = (0.5, 0.0,
0.9
Figure 6. Total conversion of the CMR, IMRCF, PFR, and MFR vs the dimensionless diffusional space time (6)for the m e of An > 0 [ n i: 3, Y = (-1, 1, 3), a = (1, 1, 6),X O = (0.5, 0, 0.5),Q, 10.0, Pf = 191 kPa, Pp = 95.5 kPa, and Kp = 4.471 X lo6 kPa31. 1 .o
t
1
0.0 0.0
0.2
0.6
0.4
0.8
1 .o
Cb Figure 7. Total conversion of the CMR, IMRCF, PFR, and MFR vs the dimensionless diffusional space time ($1 for the case of An = 0 [n = 4, Y = (-1, -3, 1, l),(Y (1,1, 1, 6),X O = (0.5, 0.5, 0, O), Q, = 10.0, PI = 500 kPa, Pp= 100 kPa, and Kp = 11.
conversion higher than the feed-side conversion (the bottom dash line even at the total permeation condition). When the Thiele modulus is larger than 1.2, the apparent equilibrium shift exists and exhibits the conversion higher than the feed-side conversion because of the selective separation of products in the feed-side chamber and the pressure variation effect in the catalytic membrane. However, if the Thiele modulus is very large (>50, in this case), the reaction is already very close to the “local equilibrium” throughout the catalytic membrane, and the total conversion shows no further improvements. As illustrated by this simulation, there are two conflicting design criteria for CMR on one hand, the Thiele modulus should be large for selective separation (high mass-transfer resistance); on the other hand, the Thiele modulus should be small for an effective use of the catalyst (low mass-transfer resistance). This is very different from the traditional design criteria for the membrane separator and for the heterogeneous catalyst. Thus, increasing the Thiele modulus to an extremely large value (by increasing the internal mass-transfer resistance, the intrinsic reaction rate, or the catalytic membrane thickness) may not be favorable for the performance of the catalytic membrane reactor. Conversions in CMR, IMRCF, PFR, and MFR. Figures 6-8 show the total conversion-vsthe dimensionless diffusional space time for three categories of reactions in each type of reactor under similar operating conditions.
Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 237 1 .o
0.8
1
PFR (PP)
o.20.0 0.0 ~
MFR 0.3( P P I
0.6
0.9
1.2
1.5
@ Figure 8. Total conversion of the CMR, IMRCF, PFR, and MFR vs the dimensionless diffusional space time (g) for the case of An < 0 [ n = 4,u = (-1, -3, 1, l),a = (1, 1, 1, 6), xo = (0.1, 0.8, 0, O.l), = 10.0, Pr = 200 kPa, P, = 100 kPa, and K , = 1.213 X 10" kPa4].
The total conversion is determined by the total consumption of reactant 1 in each reaction. The operating pressures of the PFR and the MFR are respectively assumed to be at Pfand Pp.It should be noted that the total conversion of a PFR or a MFR will approach the original equilibrium conversion (Xe,for Xe,p)at its own operating pressure (Pfor P,),and thus, no equilibrium shift can exist. For the reactions of category 1 (An > 0 ) shown in Figure 6 , the conversions in both membrane reactors (CMR and IMRCF) could reach higher than the feed-side conversion (Xe,3but remain still lower than the permeate-side conversion (Xe,p).In this case, the equilibrium shift induced by the pressure variation effect is more important than that induced by the selective product separation effect. Thus, the performance of a PFR or a MFR operated at a lower pressure (such as Pp)will be better than that of a CMR or an IMRCF over the entire range of space times. The total conversion of the CMR is higher than that of the IMRCF at relatively high space times (4 > 0.51, in this case). This effect is due to the pressure variation rather than the selective separation by the membrane. For the reactions of category 2 (An = 0) shown in Figure 7, the equilibrium conversion at Pf is identical with that at Pp;thus, there is an effect on conversion due to the pressure variation. The total conversion in both the CMR and the IMRCF could reach beyond the equilibrium conversion due to the selective separation effect in the membrane. The performance of the IMRCF is superior to that of the CMR due to the selective product separation from actual reaction zones in the catalytic membrane. For the reaction of category 3 (An < 0 ) shown in Figure 8, the equilibrium conversion at Pp( X e d is lower than that at Pf ( X e d . In this case, the pressure variation is disadvantageous to the total conversion in the CMR. While the equilibrium shift induced by selective product separation is significant to the performance of the IMRCF, this improvement is offset by the pressure variation in the membrane of the CMR.
Conclusions Table I summarizes the results of the present study. Three general categories of reactions are considered: (1) the volume is increased after reaction (gas-phase reaction with An > 0), (2) the volume remains constant after reaction (most liquid-phase reaction, or gas-phase reaction with An = 0), and (3) the volume is decreased after reaction (gas-phase reaction with An < 0). For category 1, the performances of the PFR and the MFR operated a t the
Table I. Reactor Selection Guide in the Order of Preference category low space time high space time An > 0 1. PFR (at P,) 1. PFR (at P,) 2. MFR (at $ 1 2. MFR (at Pp) 3. CMR 3. PFR (at prP 4. IMRCF 4. IMRCF 5. MFR (at Pr) 5. PFR (at Pr) 6. CMR 6. MFR (at Pt) 1. IMRCF An = 0 1. PFR (at Pr) 2. IMRCF 2. CMR 3. PFR (at Pr) 3. MFR (at Pr) 4. CMR 4. MFR (at Pr) 1. IMRCF An < 0 1. PFR (at Pr) 2. PFR (at Pr) 2. IMRCF 3. MFR (at Pr) 3. MFR (at Pr) 4. PFR (at P ) 4. CMR 5. MFR (at 8,) 5. PFR (at P ) 6. MFR (at 8,) 6. CMR permeate-side pressure are better than those of the CMR or the IMRCF. Between two membrane reactors, the performance of the CMR is slightly better than that of the IMRCF at a longer diffusional space time (low pressure drop across the membrane). For category 2, both types of membrane reactors perform better than the traditional PFR and MFR. Between two membrane reactors, the performance of the IMRCF is better than that of the CMR at a longer space time. For category 3, the performance of the IMRCF is better than any other reactors. The CMR is not suitable for this category due to the undesirable equilibrium effect induced by the pressure variation in the membrane.
Nomenclature
Dj= effective diffusivity for component j in the membrane, m2/s
F o = total input volume flow rate, QoRT/P,m3/s f i = reaction rate function for reaction i, (kPa)@i
fi = dimensionless reaction rate function for reaction i K , = equilibrium constant in partial pressure units, units depend on stoichiometry ki= reaction rate constant of reaction i, km~l/(m~l~.~.(kPa)@i) L = length of the membrane tube, m m = total number of reactions, dimensionless n = total number of components, dimensionless An = net change of number of moles, dimensionless P = total pressure, kPa Pf = total pressure on the feed side of a membrane reactor, kPa Pj = partial pressure of component j , kPa Pp = total pressure on the permeate side of a membrane reactor, kPa P, = ratio of the permeate-side pressure to the feed-side pressure, P,/Pf, dimensionless P = vector form for the partial pressures of all the Components Q = total molar flow rate, kmol/s R = universal gas constant, 8.314 kJ/(kmol.K) r = radius, m ri = reaction rate expression for reaction i, kmol/(m3.s) rm = logarithmic mean radius, (rf - r,)/ln (rf/rp), m T = operating temperature, K V, = total volume of catalysts (or catalyticmembrane), =s($ - r,2)L, m3 Xe,f= equilibrium conversion based on the feed composition and pressure P f , dimensionless Xe,, = equilibrium conversion based on the feed composition and pressure PPIdimensionless XT = total conversion, dimensionless xi = molar fraction of component j in the feed-side chamber of membrane units in the CMR and IMRCF or molar
238 Ind. Eng. Chem. Res., Vol. 29, No. 2, 1990 fraction of component j in the PFR and the MFR, dimensionless x = vector form for the molar fractions of all the components y, = molar fraction of component j in the permeate-side chamber of t h e membrane units in t h e CMR and t h e IMRCF, dimensionless z = dimensionless reactor length or volume
IMRCF = inert membrane reactor with catalyst particles packed on t h e feed side of t h e membrane reactor MFR = mixed flow reactor PFR = plug flow reactor TPL = total permeation line
Greek Symbols aj = diffusivitiy ratio or ideal separation factor for component j t o component 1, Dj/Dl, dimensionless a = vector form for the diffwivity ratios of all the components p, = forward reaction order of reaction i, dimensionless 6 = constant, In (rf/rp), dimensionless e = dimensionless molar flow rate, Q/Qp 8 = dimensionless reactional space time, (klPpV,)/Qp K , = ratio of reaction rate constants, (ki/kl)PF+1,dimensionless vL, = stoichiometric coefficient of component j in reaction i, dimensionless v = vector form for t h e stoichiometric coefficients of all t h e components in one reaction F = dimensionless radius, In (r/rJ/ln (rf/rp) 'T = space time, =Vc/Fo,s = Thiele modulus, r6(RTklPfi-1/D1)1/2, dimensionless 4 = dimensionless diffusional space time, (27rLD,Pf)/(GRTQP) I/. = dimensionless total pressure, P / P , I/.j = dimensionless partial pressure of j , P,/Pf 9 = vector form for the dimensionless partial pressures of all t h e components
Ferziger, J. H. Numerical Methods for Engineering Applications; John Wiley and Sons: New York, 1981. Gerald, C. F.; Wheatley, P. 0. Applied Numerical Analysis; Addison-Wesley: Reading, MA, 1984. Hwang, S. T.; Kammermayer, K. Membrane in Separations. Reprint; Krieger: Malabar, FL, 1984. IMSL. IMSL Library Reference Manual, 9.2 ed.; IMSL: Houston, TX, 1984. Ishikawa, H.; Nishida, H.; Hikita, H. Theoretical Analysis of a New Separation Process Utilizing Extraction and Enzyme Reaction. J. Chem. Eng. Jpn. 1986,19, 149. Itoh, N. Membrane Reactor Using Palladium. AZChE J . 1987,33, 1576. Itoh, N.; Shindo, Y.; Hakuta, T.; Yoshitome, H. Enhanced Catalytic Decomposition of HI by Using a Microporous Membrane. Int. J. Hydrogen Eng. 1984,9, 835. Itoh, N.; Shindo, Y.; Haraya, K.; Obata, K.; Hakuta, T.; Yoshitome, H. Simulation of a Reaction Accompanied by Separation. Znt. Chem. Eng. (Engl. Trawl.) 1985,25,138; Kagaku Kogaku Ronbunshu 1983, 9, 572. Kameyama, T.; Dokiya, M.; Fujishige, M.; Yokokawa, H.; Fukuda, K. Possibility for Effective Production of Hydrogen from Hydrogen Sulfide by Means of a Porous Vycor Glass Membrane. Znd. Eng. Chem. Fundam. 1981,20, 97. Shinji, 0.;Misono, M.; Yoneda, Y. The Dehydrogenation of Cyclohexane by the Use of a Porous-glass Reactor. Bull. Chem. SOC. Jpn. 1982,55, 2760. Sun. Y. M. Catalvtic Membrane Reactor-Modeline and Exoerimenta of a Micioporous Membrane Separation-Reaction Sy&em. Ph.D. Dissertation, University of Cincinnati, Cincinnati, OH, 1987. Sun, Y. M.; Khang, S. J. Catalytic Membrane for Simultaneous Chemical Reaction and Separation Applied to a Dehydrogenation Reaction. Ind. Eng. Chem. Res. 1988, 27, 1136.
Superscript o = refers to t h e input conditions Subscripts e = refers t o t h e equilibrium condition f = refers t o t h e feed or t h e feed-side conditions i = reaction number j = component number k = component number p = refers t o the permeate-side conditions Abbreviations CMR = catalytic membrane reactor
Literature Cited
Received for review February 22, 1989 Revised manuscript received September 5, 1989 Accepted September 26, 1989
