2064
Ind. Eng. Chem. Res. 1988, 27, 2064-2070
Effect of Temperature on Equilibrium Shift in Reactors with a Permselective Wall Krishna Mohan and Rakesh Govind* Department of Chemical Engineering, University of Cincinnati, Cincinnati, Ohio 45221
The purpose of this paper is to consider the well-agitated continuous stirred tank membrane reactor (CSTMR) and the plug flow membrane reactor (PFMR) from the standpoint of equilibrium shift in reversible reactions. The general behavior of both reactor types is discussed, and comparisons are made of their performance. For CSTMR, analytical criteria have been developed to predict the stability of the steady state. Results on simulation studies of ethylbenzene dehydrogenation in a PFMR are also presented. Selective permeation through a semipermeable membrane has become an accepted chemical engineering process for separating fluid mixtures. In recent years, considerable attention has focused on combining the unit operations of separation and reaction to provide purer products and/or shifting of the thermodynamic equilibria. The use of membranes to remove products from the reaction zone has obvious industrial interest as it provides for lower reaction temperatures, smaller reactors, and reduced downstream separation costs. In the past, applications of combined reaction/separation processes using membranes (membrane reactors) could be found mainly in biological systems. More recently however, new technologies are emerging where membranes are used to immobilize inorganic catalysts (reactive membranes) and/or selectively remove components from the reaction zone. In this study, the membrane is considered to be a nonreactive permselective barrier which separates products from the reaction mixture. A limited number of experimental and theoretical studies on the membrane reactor (nonreactive membranes) can be found in the published literature. A review of existing research in this area is presented in Mohan and Govind (1988). Note that most of the previous studies have been limited to isothermal systems. The purpose of this paper is to consider the membrane reactor from the standpoint of equilibrium shift in reversible reactions. Since most industrial reactions are accompanied by heat effects, this study focuses on how reaction temperature influences the performance of a continuous stirred tank membrane reactor (CSTMR) and a plug flow membrane reactor (PFMR). In the analysis, an elementary A + B reaction is used, and both exothermic and endothermic situations are considered. The study is concluded with simulation results for a reaction of industrial interest.
Model Formulation Consider the membrane reactors (CSTMR and PFR) shown in Figure 1. Since the membrane is more permeable to product B than reactant A, the reversible reaction is not limited by product buildup, and conversions exceeding the equilibrium value can be achieved. The theoretical analyses of the simultaneous reaction and permeation system are based on the following assumptions: (1)the membrane is homogeneous; (2) the permeability of each gas component is the same as that of the pure gas; (3) the reaction mechanism is reversible and elementary; (4)the pressure drop in the feed flow path
* To whom correspondence concerning this article should be addressed. 0888-5885/88/2627-2064$01.50/0
Table I. Membrane Reactor Parameters and Associated Dimensionless Grouus physical dimensions properties operating variables membrane area permeability (PE) reaction side pressure rate constants permeation side pressure reaction vol membrane heat of reaction feed flow rate thickness specific heat feed temp activation energies feed composition dimensionless groups Damkohler number, Da# a reaction volume, l/feed flow rate, measure of residence time rate ratio, 6 a l/membrane thickness, permeability, permeation area, l/forward rate constant pressure ratio, P, = permeation side pressure/reaction side pressure = 0.0 (for system studied) = P E J P E(A ~ + B;B is the fastest permselectivity, a Pas) heat generation index, /3 0: heat of reaction, l/specific heat heat conduction index, $ a overall heat-transfer coeff for heat loss through membrane = 0.0 (for adiabatic operation)
is negligible; (5) the heat- and mass-transfer resistances, aside from the permeation process itself, me negligible; (6) the reactor operates adiabatically unless otherwise specified. Some of the above assumptions may not be valid in all applications of the membrane reactor. Coupling effects in gas mixtures are known to have a significant impact on actual permeabilities (Fang et al., 1975), and for the PFMR, the axial pressure drop can be significant at high Reynolds numbers. Moreover, nonideal conditions (incomplete mixing in the CSTMR; radical and axial gradients of temperature and concentration in the PFMR) will exist in actual operation for both types of membrane reactors. These and other problems deserve further study and will be the subject of future publications. However, for the purposes of this study, the above-listed assumptions are necessary to preserve generality of the conclusions. To facilitate parametric analysis, the various design, operating, and physical property parameters are characterized in dimensionless groups. The different parameters and associated dimensionless groups have been presented in Table I. The Damkohler number, Da,is a measure of the maximum forward reaction rate and is proportional to the reaction volume. The permselectivity, ai,denotes the ratio of permeation rate of component i to the permeation rate of the fastest gas. The pressure ratio, P,, is the ratio of the permeation side pressure to the reaction side pressure. The rate ratio, 6, measures the maximum 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2065 li0=1.0
permeate
**
*
1
r
O"
ic
0.6
constant stirred tank membrane reactor (CSTMR)
feed Fif Xf Tf
1-
X
6 3
.'
4
k ! j
0.2
of membrane reactors.
X = (l/xf)[xf - x + 6Da(l - a)(x - x 2 ) ]
- material balance
TI 600
,
,
Ti
1000
energy balance
,
,
,
,
1400
la00
conversion
Figure 2. Effect of permeation rate on the performance of a CSTMR for a exothermic reaction. 6 = 2.0
lIB=-O.5
(2)
'\
(3)
or
x
T*(l + $) - 6
\
temperature, T (K)
CSTMR Equations x f - x + 6Da(l - a ) ( x - x 2 ) = f A / T * material balance
X =
611.7
6=0.0+
,'
151,
200
( x ) ( D a )expIy(1 - 1/T*)I KA1 - x ) e x p W - E,/T*)I (1)
+ ~4 - T * ( l + $) = 0
1 I
Reaction Rate
3
2.5
bproducts
~~~~
Table 11. Model Equations
/
y.6.0
K r =I80.275 a~ 0'0
I
00
fA
lie-1.25
I
plug flow membrane reactor (PFMR)
f~ =
.
t t t t t t t
z=oljj
Types
.'
.
r'
permeate
A2
Figure 1.
Da#=l.O Eri2.0
1
0'4
,
ll0=-1.00
'.
D a # = 1.0 E, = .5 Kr I.5
2.5
6 6
I
= 0.5
energy balance (4)
PXf
PFMR Equations dx
dZ
F
dF/dZ = -6Da[ax
[
dT* = 1 Da@$ dZ F
+ 1- x ] -
(5)
overall material balance (6)
+(* 2)] T -
///
material balance for A
bDa(1 - a ) ( x - x 2 ) -
energy balance (7)
at Z = 0 T* = Tf/To; F = 1.0; x = xf
permeation rate of the fastest gas to the maximum reaction rate. The heat generation index, @,is proportional to the heat of reaction and measures the maximum (complete conversion) temperature rise (or fall) in the reactor. For adiabatic operation, J, is zero since it is a measure of the external heat input. The conversion, X,is the main index used to measure the performance of the membrane reactor. The general dimensionless equations governing the steady-state behavior of a membrane reactor are shown in Table 11. For a CSTMR, the material and energy balance equations reduce to a set of algebraic equations. For a PFMR, a pair of ordinary differential equations have to be solved to determine temperature and composition profiles. Note that the model equations have been developed for a pressure ratio, P,,of zero (vacuum or very low pressures on the permeation side). The effect of finite pressure ratios will be featured in the discussion on the example reaction.
Continuous Stirred Tank Membrane Reactor In Figures 2 and 3, the conversion, X ,is plotted as a function of temperature for exothermic and endothermic reactions, respectively. The sigmoidal curves obtained by solving eq 2 and 4 represent the material balance, while the straight lines obtained from eq 3 represent the energy balance. The intersection of the energy balance line with
00
- material balance .. .
energy balance
i
200
600
1000
1400 TI
1800
temperature, T (K)
Figure 3. Effect of permeation rate on the performance of a CSTMR for a endothermic reaction.
the S-shaped material balance curve for the operating Da and 6 gives the steady-state conditions in the CSTMR. For exothermic reactions (Figure 2), conversion initially increases with temperature and then decreases, as backreaction is favored at high temperatures. Therefore, there is an optimum temperature where conversion is maximized. For fixed temperatures, there is an optimum rate ratio above which conversion decreases. The membrane being permeable to both the gases, an increase in permeation rate is accompanied by a considerable loss of reactant. This loss negates the effect of equilibrium shift and conversion decreases. The plateaus D-D' and E-E ', at high rate ratios, correspond to situations when there is no reject stream (permeate flow rate = feed flow rate). Therefore, there is a temperature range for which the CSTMR composition is constant. Note that, if the membrane is permeable only to the product (CYA = o), there is no optimum permeation rate and complete conversions can be obtained in the membrane reactor. The feed temperature, Tf,and slope, (1+ J , ) / @ , of the energy balance line are important parameters in determining the performance of the membrane reactor. Since there is an optimum temperature below which conversion decreases, the intersection of the two curves (energy and material) should be near the maximum. External heat removal may be needed to adjust the slope of the energy balance line. For high permeation rates (6 > 1.7), as
2066 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 Table 111. Stability Analysis for a CSTMR Transient Equations dxt/dh = T~*[x, - ~t 6Da(l - ( Y ) ( x-~ ~ t -2 fA/Tt*l = fl(xt,Tt*) (8)
+
- Exothermc. Er
E 2.0, Kr = 80.0, B = 0.8 Endothermic, Er = 0.5. Kr = 0.5, B = -0.8
06 Da#=l 0 ~
6
3
4.0
y=60 GA
= 0 275
0 4 1-
Let b = x - x,;
Linearized Equations c = T * - Tt*
db/dh = a11b
+~
1 2 ~ ;
+
dc/dh = ~ 2 1 b
~
2
~ (IO)
2
where
1
Routh-Hurwitz Criterion
200
600
1000
1400
1800
feedtemperature. Tt (K)
all + a22 < 0 6Da(l - a)(l- 2x) < 3 + #
1
+ ,[(gx-gT) T
Figure 4. Effect of permeation rate on the performance of a PFMR. - 41
(12)
a11a22 - a12(321 > 0 6Da(l - a ) [ T * ( l- 2x){$J- 2T*(1 + 4) + &TI - (X - x2)gXl> [ T * + gxl[$J- 2T*(1 + #)I + @[gTT*-gx(x - x31 (13) where gx = afA/dx
gT =
afA/aT*
maximum conversion extends over a range of operating temperatures, adiabatic operation may be sufficient. Multiple steady states, a common phenomenon associated with a CSTR (Levenspiel, 1972; Denbigh and Turner, 1972; Bilous and Amundson, 1955), is also evident in membrane reactors. For the energy balance line, TIC,the middle state B is unstable, as any small perturbation in temperature will cause the reactor to move to one of the other stable states. Much of the previous analysis can be extended to endothermic reactions, the performance of which is represented in Figure 3. However, now there is no optimum operating temperature and there is no possibility of multiple steady states. Reactant loss by permeation is still a factor, and there is an optimum rate ratio, 6, which maximizes conversion. Once again, note that, at high permeation rates, there is an entire range of temperatures at which the conversion is constant. Stability. For exothermic reactions, multiple steady states could exist in a CSTMR (see Figure 2). Extending the well-known results of stability analysis on CSTRs we can conclude the following: (a) A necessary but not sufficient condition for stability is that the heat removal curve should have a greater slope than the heat generation curve at the point of intersection (Levenspiel, 1972; Denbigh and Turner, 1972). (b) In certain circumstances of nonadiabatic operation, it is possible for the reactor to show oscillatory instability at an intersection such as A and C (Boreskov and Slinko, 1961; Gilles and Hofmann, 1961; Denbigh and Turner, 1972). To obtain the necessary and sufficient conditions for stability of the CSTMR, the methodology developed by Bilous and Amundson (1955) can be applied. This technique involves rewriting the steady-state equations (eq 1 and 3) in their transient form and linearizing them about the steady-state position by using first-order Taylor series. Stability may be expected only if the characteristic equation associated with the linearized equations has roots
whose real parts are negative. The necessary and sufficient conditions for negative roots are known as the RouthHurwitz criterion (Bilous and Amundson, 1955; Denn, 1975). In Table 111, the transient equations governing the behavior of a CSTMR (eq 8 and g), the linearized governing equations (eq 10 and 11) and the two necessary and sufficient conditions for stability are shown (eq 12 and 13). Note that the criteria for stability or instability are given in terms of steady-state values, and no transients need be calculated. Thus, stability (or instability) can be determined from calculations made for steady-state design.
Plug Flow Membrane Reactor In Figure 4, conversion, X , is plotted as a function of feed temperature, Tf, for a PFMR. The behavior of a PFMR is qualitatively similar to that of a CSTMR. For exothermic reactions, there is an optimum feed temperature which maximizes conversion as high temperatures favor back-reaction. For endothermic reactions, the feed temperature should be as high as possible since both reactiort rate and thermodynamic equilibria are favored at high temperatures. Reactant loss by permeation is again an important factor in determining performance. For a fixed feed temperature, Tf,there is an optimum rate ratio, 6, which maximizes conversion. Stability. A PFMR will not exhibit unstable characteristics since by definition it does not allow back-mixing of heat or mass. After a change in operating conditions, the new steady state passes as a plug through the reactor and, in time (usually a period of the residence time), the new steady-state conditions are established (Westerterp et al., 1984). However, in situations of product recycle, heat exchange between product and feed, or even dispersive backmixing (Perlmutter, 1972), multiple steady states can arise. These situations are beyond the scope of the present study but will be addressed in future publications. Optimum Temperature Policy Once the operating temperature range of a given reaction system has been established, the next task is to determine the most favorable reaction temperature or temperature sequence with respect to conversion and reactor volume. For endothermic reactions, the greatest forward reaction rate is achieved at the highest temperature which is economically and technically permissible. For exothermic
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2067 Dalt = 1 Er=2
1.25 Er=2
Kr = B O 1=6
Tmax=1600(K) 1.o
aA10.1
I
I
810.8
0.1
K.=BO
t-
;
t temperature
,.' .75
I
:
,
.'
, '
,
/
/
/
X or TiTmax 5
.-_ 6-1.0 . 6 = 3.0
.25
11-
-
Conversion
-
6=7.0
0
0 0
25
5
.75
1.0
dimensionless length, 2
Figure 5. Optimum temperature profile in a PFMR.
reactions, a compromise must be found between the requirements of a high degree of conversion (low temperature) and those of a high reaction rate (high temperature). Note that for a CSTMR at high permeation rates, the above analysis may have to be modified. For an endothermic reaction at a fixed Da (reaction volume) and 6 (permeation rate), there may be a minimum operating temperature above which no increase in conversion is possible (Figure 3). Likewise, for an exothermic reaction, there may be an entire range of operating temperatures which maximize conversion (Figure 2). It is apparent that, for an exothermic reaction in a PFMR or a sequence of CSTMRs, there exists a temperature profile which maximizes conversion. In Figure 5, the optimum temperature and maximum conversion profile in a PFMR is shown for different permeation rates. With an increase in rate ratio, the optimum temperature profile shifts upwards. Since product is being selectively removed from the reactor, higher temperatures can be maintained without risking back-reaction. A substantial increase in conversion is possible. However, at high rate ratios, the conversion decreases due to reactant loss by permeation. Note that at high permeation rates reactor size can be substantially decreased. For 6 = 7.0 there is no reactant (or product) beyond the midway point in the reactor. In computing the optimum temperature profile, the PFMR is considered to be a finite number of CSTMRs in series. The optimum temperature in each CSTMR is calculated by using eq 2 and 3 (Table 11). Convergence to the actual PFMR profile is achieved by increasing the number of CSTMRs until there is no significant change in the profile.
Comparison of PFMR and CSTMR Levenspiel (1972) has shown that for endothermic reactions the best reactor type, that which minimizes residence time, is plug flow. However, for exothermic reactions he found that there are regions depending on feed temand heat of generation index, p, where one perature, Tf, reactor type is better than the other. The following analysis studies the influence of permeation rate on choice of membrane reactor. Although the results obtained are for a fixed value of permselectivity, CYA, the general behavior can be extended for other reactant permeabilities. Exothermic. In Figure 6 the difference in conversion between a PFMR and a CSTMR, AX, is plotted as a function of the rate ratio, 6, for different values of the heat
5
10
1.5
2.0
rate ratio, 6
Figure 6. Difference in conversion between PFMR and CSTMR as a function of permeation rate.
generation index, 6. For a given set of operating conditions, there may be a critical rate ratio, 6, beyond which a CSTMR is better. In a PFMR there is a high concentration of reactant at the feed end, so for low feed temperatures (low reaction rates) and/or high permeation rates (high 6), reactant loss may be severe. In a CSTMR, since a uniform reactor composition determines the permeation rate, reactant loss is not as detrimental. Consequently, with an increase in the rate ratio, 6, a situation may arise where the equilibrium shift in a CSTMR is better than that of a PFMR. As discussed above, the feed end is critical in determining the performance of the PFMR. At low feed temperatures, the PFMR is unable to take advantage of the high reactant concentrations, and conversion suffers. Since the operating temperature in a CSTMR is usually higher than the feed temperature, conditions may favor a greater forward reaction rate than the average rate obtained in a PFMR. Therefore, for a given permeation rate, 6, and heat generation index, 0,there may be a critical feed temperature above which a PFMR is better than a CSTMR. Isothermal operation is favored in a PFMR irrespective of the permeation rate. However, at low feed temperatures (low reaction rates in the critical feed end of the PFMR) and high permeation rates (high reactant loss), the difference in performance between the two reactor types is negligible. Endothermic. A PFMR always performs better than a CSTMR irrespective of the feed temperature, Tf, or the heat generation index, p. The operating temperature and reactant concentration in a CSTMR, usually being lower than the feed temperature and composition, favor a lower forward reaction rate than the average rate obtained in a PFMR. This together with low product permeation rates ensures an inferior performance.
Design Considerations A serious problem with membrane reactors (CSTMR and PFMR) is the loss of reactant by permeation. Results from previous isothermal studies (Mohan and Govind, 1986,1988)have shown that there is a limiting conversion (X,) that can be attained in such reactors. For nonisothermal systems, X , is dependent not only on membrane characteristics but also on feed temperatures. The dependence of X L on Tfwill be discussed in the section on example reaction. The use of permeate recycle to increase conversion has been discussed in Mohan and Govind (1988). Conversions
2068 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988
cannot be increased above the limiting value unless the permeability of reactant is greater than the product or the reactant permeability is in between that of the products (A = B C). For fixed feed temperatures, similar behavior is also observed for nonisothermal systems. As previously mentioned, at high permeation rates there may be no reject flow (Figures 2 , 3 , and 5). This suggests that, for membranes which have low (or zero) permselectivity for the reactant, a tremendous reduction in reactor volume (over impermeable wall reactor) can be made while achieving significant (or complete) conversion. However, for the cases shown in Figures 2, 3, and 5, at high permeation rates there is a decrease in conversion due to reactant loss. This suggests that the actual design of a membrane reactor may have to be made on the basis of economics: large reactors with high conversions (high Da and low 6) or small reactors with low conversions (low Da and high 6).
z=o ual gas
feed
+
Example Reaction In the following analysis, a reaction of industrial interest is considered. The ethylbenzene to styrene reaction
!-,
. Dermeation
- ,
- t u b e side
..
t.
separated aas
. . .-.. . . ........ .......
................... .................. ................... .................. ................... ..................
................... .................. ...................d
l...... . +
,
catalyst
u
;
membrane
Figure 7. Scheme of a catalytic membrane reactor. Table IV. Model Equations for a Catalytic PFMR ep = (1/ T,*)o.5
temperature correction for permeability
gi = bDaai(xi - yiP,)ep
permeation flow rate
Reaction Side, 2 > 0 ((+) for Products; (-) for Reactants) hli = dFi/dZ = &rA- gi material balance (14)
-dT,* - dZ
N
@rA - (Tr* - 1)Cgi- $(Tr* - Tp*) - $wCTp* ,=1
is a typical example of endothermic dehydrogenation. In industrial practice, the process is typically operated with a steam-to-ethylbenzene mole ratio of 10-15, at about atmospheric pressure, with operating temperatures of about 850-950 K. Conversions of 40-60% are achieved. Since this reaction needs high temperatures to obtain reasonable conversion, there may be great economic advantages to using a membrane reactor. Being a preliminary analysis, the side reactions to toluene and benzene are neglected. Kinetic and equilibrium data on this reaction were obtained from Wenner and Dybdal(l948) and Smith (1970). The choice of such a reaction was dictated by the availability of suitable membranes. Polymeric membranes cannot be used at the high temperatures typically needed for most industrial reactions. Inorganic membranes, in which Knudsen diffusion is the mode of transport, are useful in situations where there is a great difference in molecular weight between the reactant(s) and product(s). Consequently, dehydrogenation reactions have been the focus of a large number of investigations (literature review in Mohan and Govind (1988)). In this study, two membranes are studied: (a) porous Vycor glass and (b) a hypothetical membrane permeable only to hydrogen. The general behavior of reactors with the hypothetical membrane can be applied to palladium membrane reactors. Palladium has the unique property of allowing only hydrogen to permeate. Experimental analysis of cyclohexane dehydrogenation using such a membrane was conducted by Itoh (1987). He found that complete conversions could be attained by increasing the reactant residence time (high Da). Since the reaction is endothermic, the CSTMR is not considered viable. In the previous analysis, we had assumed P, = 0.0 and neglected effects of the permeation side. In actual practice, the PFMR will resemble a shell and tube heat exchanger with reaction occurring on one side and product permeating to the other. A scheme of the double tubular plug flow membrane reactor modeled in this study is shown in Figure 7. To facilitate external heat transfer when needed, the catalyst should be packed on the shell side outside the porous tubes. The model equations which govern a double tubular membrane reactor are given in Table IV. Note that, for systems where there is a significant difference in the
gas
- Ta*)l - ... -
N
[CFiI i-1
i=l
Permeation Side, Z > 0 ((+) for Cocurrent; (-) for Countercurrent)
hpi= dQi/dZ = &gi
(16)
material balance
dTp*
-- dZ
N
N
i-1
i-1
[&4(Tr* - 1)Cgi+ $4Tr* - Tp*)I - (Tp* - 1)ChziI N
-
... -
[ C Qil i-1
Tp* dCPp -CPp d Z
energy balance (17)
i = 1,2,,..,Nwhere N is the total number of components (reactants and products)
specific heats of the different components (reactants, products, inerts), it is important to account for the change in specific heat along the length of the membrane reactor. Details on the methodology used to solve the ordinary differential equations are given in Mohan and Govind (1988). Kinetic data and rate expression, along with relevant physical property parameters, are given in Table V. Effect of Inerts. In current practice, a large amount of steam is fed along with the reactant to reduce the temperature drop in the adiabatic fixed bed reactor. If the amount of steam needed can be reduced, there may be economic advantages to using such reactors. In Figure 8, the effect of inert (steam) in the feed is shown as a function of permeation rate. In the rate ratio range 0.5-1.25, the porous Vycor glass reactor with a steam-to-ethylbenzene mole ratio of 1.0 performs better than an adiabatic reactor with larger amounts of steam in the feed. As we have previously indicated, reactant loss is a major problem in membrane reactors. Consequently, for the Vycor glass reactor, there is an optimum rate ratio which maximizes conversion.
Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2069 - Vycor glass membrane
Table V. Parameters Used in Study Kinetic Data CaHlo + Hz + CaHa (A + B C) AH = 30000 cal/mol K E = 0.0017 exp(-AH(l/T - 1/675)/1.987) EA = 20000 cal/mol
[
. . .. .
membrane permeable only to H2
+
X
.-
0.6
2 z
s Specific Heat Data C,(ethylbenzene) = 70 cal/(mol.K) C (styrene) = 65 cal/(mol.K)
