I n d . Eng. Chem. Res. 1987, 26, 167-170
167
Optimal Catalyst Activity Profiles in Pellets. 6. Optimization of the Isothermal Fixed-Bed Reactor with Multiple Zones The outlet conversion of an isothermal, heterogeneous plug-flow reactor, packed with multiple zones of nonuniformly active catalysts, where a bimolecular Langmuir-Hinshelwood reaction occurs, is optimized. In each zone, a different Dirac delta activity distribution within the pellet is permitted. T h e optimization variables are catalyst location within the pellets and the length of each zone. The total amount of active catalyst is maintained fixed. I t is found that, by use of only two or three zones (i.e., a reasonable situation for practical purposes), the theoretical optimal reactor performance is closely approached. Significant improvements over other activity distributions previously examined are obtained. The optimal activity distribution in a single catalyst pellet where an isothermal, bimolecular, Langmuir-Hinshelwood reaction occurs is given by a Dirac delta function (Morbidelli et al., 1982). This implies that the active catalyst should be deposited a t a specific location within the support, given by the position for which the reaction rate function exhibits a maximum. It has also been shown that the optimal Dirac delta distribution can be approximated well in practice by a thin band of active catalyst deposited around the optimal location. Recently, Morbidelli et al. (1986a,b) considered the same optimization problem with respect to the performance of an isothermal fixed-bed reactor. It was shown that the optimal activity distribution which maximizes the reactor's outlet conversion for a fixed amount of catalyst is again a Dirac delta distribution. However, the optimal location of the catalyst depends on the bulk reactant concentration; therefore, it changes along the reactor axis. Since this is cumbersome to apply in practice, the suboptimum case where all pellets have the same Dirac delta distribution was also considered. Significant improvements were observed when comparing the performance of this latter reactor with one packed with uniformly active or externally coated catalyst particles. However, the performance of this suboptimal reactor is inferior to that of the rigorous optimal reactor (where each pellet has a different activity distribution), particularly for large Damkohler-number values (Morbidelli et al., 1986b). Therefore, improved suboptimal reactors need to be explored to approach the rigorous optimal reactor performance more closely, while still remaining easy to implement in practice. In particular, this work considers a reactor packed in a series of zones, each characterized by catalyst pellets having a different Dirac delta distribution. In the limiting situation where the number of zones is equal to either one or infinity, this reduces to the previously examined suboptimum and rigorous optimum cases, respectively (Morbidelli et al., 1986a,b). In performing the optimization, besides varying the catalyst distribution in each zone, the relative length of each zone is altered too, but as a constraint, the total length of the reactor (hence the total amount of catalyst used) is kept constant. The same Dirac delta distribution function was found to yield optimal performance also for the more realistic case of a nonisothermal catalyst pellet (Morbidelli et al., 1985). A t least in principle this allows us to extend the results reported below to cases including thermal effects. Basic Equations The model equations for an isothermal, heterogeneous plug-flow reactor with nonuniformly active catalyst pellets were derived by Morbidelli et al. (1986a). When their notation is adopted and the activity distribution function for the ith zone is defined as a,(s), the fluid-phase mass balance for the multiple-zone reactor can be written as 0888-5885/87/2626-0167$01.50/0
with the initial conditions (IC) y = l y=o Assuming the following normalized Dirac delta catalyst activity distribution for the generic ith zone
the catalyst pellet mass balance can be solved analytically (Morbidelli et al., 1986a) leading to yi = iii
+ @$(I&)
(3) where iii = u(Q. Upon substituting eq 2 into eq l a , we obtain the following set of d ordinary differential equations (ODES), where d represents the total number of zones
(4) where yo&< y < yI and i = 1, ......, d. The IC for each equation in (4)is given by the following continuity condition at the interface between two adjacent zones (5) 7L-1 = &-l + = no, + P f ( i i 0 , ) = yo, where the subscripts Oi and i denote the conditions at the inlet and outlet of the ith zone, respectively. The reactor axis, y, within each zone varies from zero to the zone length, y,. In order to keep the total amount of catalyst constant, the total length of the reactor must be kept fixed, so that the following constraint must be satisfied in the optimization procedure
@,-lmL-J
d
ZY, = 1
(6)
r=l
Optimization Technique u < 8. In this region, reactor steady-state uniqueness is guaranteed (Morbidelli et al., 1986a), and eq 4 can be directly integrated to give (7) Day, = F&; GO,;P,) where F is defined in the Nomenclature section. The reactor outlet conversion X can be computed to be X=l-yd (8) For a given set of u and Da values, X can be maximized with respect to @, (which fixes catalyst location S,) and y, by solving eq 6 together with
-ax- a@,
= -ax "'
a@,
= - ax ... = -ax ... = - ax ... = -ax a@d
aY1
ay,
ayd-1
-0 (9)
0 1987 American Chemical Society
168 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987
a*. Here, ignition to the high conversion branch occurs,
0 01 -
--
0 I0
and the particle concentration jumps down to the value E*. Since the middle branch is unstable, it is not examined in the sequel. Because of possible reactor multiplicity, the outlet conversion X is no longer a smooth single-valued function of p, and yi. As a result, analytical solution for large d becomes cumbersome. Therefore, the optimization problem was solved numerically by using a version of the quasi-Newton optimization technique. In the case of a reactor packed with all identical catalyst pellets (i.e., d = l),the optimization problem can be fully solved by utilizing the master plot of the optimal P value as a function of cr and Da. (See Figure 4 of Morbidelli et al., 198613). This same result can be readily extended to the multiple-zone reactor through the introduction of the following new dimensionless quantities, made dimensionless via the inlet concentrations of each zone
IO
D A M K O H L E R NUMBER , DO
OB
X
i 0
06
for i = 1
v)
ti
w
d , where i- 1
z
8 t
-
yi = L i / L
04
for i = 1
02
-
Li = [ L - CLjIAi
Lo
E
0
j=O
06AiS1 (d - 1) and d-1 Yd
0001
-
0.01
= 1- c y , i=l
01
I O
DAMKOHLER N U M B E R , DO
Figure 1. Comparison of reactor outlet conversion X for different number of zones, d as a function of Damkohler number, Da. (a) u = 5 , (b) u = 20.
where 0 6 yi 6 1 and 0 S pi are the feasible regions. When a solution inside the feasible region of pi and yi values cannot be found, then the solutions along the bounds of the variables are checked to find the maximal conversion. It can be shown that for a reactor composed of d zones, the optimal configuration is given by yi = l / d (loa) u. = a . aoi= aoj 1 1 (lob) ai = 1/(u2iioi) for all i and j from which Pi (i.e., the optimal catalyst locations, ai) can then be readily calculated by using eq 3, 5, and 7 . In Figure la, the optimal conversion values thus obtained are shown as a function of Da for u = 5 and for d = 1, 2, 3, and a. It is clear that for any given Da, as d increases, the reactor conversion increases to approach the rigorous optimal solution corresponding to d = m. u > 8. In this case, the reactor may exhibit multiple solutions and/or ignition phenomena. Thus, eq 7 is modified in order to account for the possibility of ignition along the reactor (Morbidelli et al., 1986a), leading to Dayi = F(iii; iii*; Pi) + F(iii*; aOi;Pi) (11) The second term of eq 11represents the first part of the packed-bed reactor which operates on the low conversion branch and therefore ends a t the lower bifurcation point,
Thus for d = 2
y1 = A,
y2 = 1 - A1
and for d = 3 y2 = (1 - A,)A, = A1 y3 = 1 - A, - A 2 ( l - A,)
Y1
The optimization variables are P’s and A’s. For the last zone, the optimal P d can be found readily from the above-mentioned master plot by using u = Ud and Da = Dad. The results for the optimal conversion values are shown in Figure 1b as a function of Da for cr = 20 and for d = 1, 2, 3, and a.
Results and Discussion The optimal /3 and y values are shown in Figure 2 as a function of Da, for d = 2 and u = 5 and 20. In Figure 2a, when Da > 0.58, the analytic solution, eq 10, gives an unrealistic negative value for the optimal in the second reactor zone; thus, PP,opt is set equal to 0, and the optimization is carried out analytically with respect to the remaining parameters, p1 and yl. As Da 03, y1 0 asymptotically, while /31,0ptremains at a constant value and is equal to 0. For u = 20 (Figure 2b), because of the effect of multiplicity, y1 and p z exhibit a discontinuity at Da = 0.205. A similar procedure as described above was used to obtain the portions of the curves for Da > 0.28 when P2,0pt= 0. A reactor packed with three zones of different catalyst particles (i.e., d = 3) has also been examined in detail. Since the obtained optimal pi and yi values exhibit the same qualitative behavior as shown in
- -
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 169
1
06
x -
Q I
I
-
DAMKOHLER NUMBER, Do
Figure 4. Comparison of the performance of the optimal CSTR with optimal plug-flow reactors (d = 1 and =I, as a function of Damkohler number, Da. (a) u = 5, (b) u = 20.
001
-
I
'
01
-
the optimal variables, Pi and yi. Due to multiplicity behavior of the reactor when Q > 8, the outlet conversion may exhibit discontinuities. In particular, it has been observed that small changes of Piand yi below their respective optimal values lead to larger conversion decreases than do identical changes of the same variables above their optimal values. However, the magnitude of such a difference decreases for increasing Da values, due to the averaging effect of the different particles located at different positions along the reactor length. These same features are also exhibited by the single-zone reactor, as discussed in detail elsewhere (Morbidelli et al., 1986b).
I
CSTR Model
IO
DAMKOHLER N U M B E R . DO
"""'I
"""'I
'
"""'
'
x
For completeness, we now analyze the case of a heterogeneous CSTR, packed with the nonuniformly active catalyst pellets as mentioned above. The fluid-phase mass balance is given by
X = 1 - y = Daf(u(s))
-
DAMKOHLER NUMBER, Do
Figure 2. Optimal pi and y1 values for the two-zone reactor as a function of Damkohler number, Da. (a) u = 5, (b) u = 20. (0)D = 5 ,
+'=07,
Da.03,
n=l
IY)
0
3
021 ' 0
,
,SI
,
,
j
021
-
I
z. 0
$ 2
10
d=l
1
0 2 1 3;
) , s p , 3j ;
j
02 04 0 6 08 10 ' 0 02 04 0 6 08 10
+'=03, n=I
0 06 4M
02
L o0 0
f
, ",
REACTOR A X I S , y
( b ) Cr = 2 0 , D o = O I (Y)
I
:m:H
0 2 04 0 6 08 10 ' 0
I
,Y, ,
0 2 04 06 08 I O
0 0 2 04 0 6 0 8 I O
0
0 2 04 06 0 8 I O
REACTOR A X I S , y
Figure 3. Sample calculations of the optimal catalyst location B along the reactor axis for d = 1, 2, 3, and a. (a) u = 5, (b) u = 20.
Figure 2, they are omitted here for brevity. Some sample calculations of the optimal catalyst location SLwithin the pellets along the reactor axis for d = l , 2, 3, and 00 are shown in parts a and b of Figure 3 for Q = 5 and 20, respectively. As expected on physical grounds, for increasing d values, the optimal reactor configuration evolves toward the rigorous optimum one achieved for d
-- a.
Finally, it is worthwhile to briefly address the issue of the optimal outlet-conversion sensitivity to the values of
(15)
while the solid-phase mass balance is again given by eq 3. It is easy to see that the maximum X value is obtained by selecting (i.e., the optimal location s) such that the rate of reaction, f ( u ) , is maximum. This occurs at ii = l / o . This is exactly the same result obtained earlier for the single-pellet or the infinite-zone plug-flow reactor (Morbidelli et al., 1982, 1986b), although the latter requires a changing S along the reactor length in order to attain the maximum. This solution holds only up to ii = (1/u) < y; for y S ( l / u ) , ii = y and Popt = 0. In Figure 4, the performance of the optimal CSTR is compared with that of infinite- (d = a) and single-zone (d = 1)plug-flow reactors. It may be seen that for Da C Da,, where Da, is the Damkohler number when ii = y = l / u , the CSTR yields the same conversion as the d = reactor; however, for larger Da values, the performance of the CSTR drops drastically, and eventually the outlet conversion becomes even lower than that obtained with the suboptimal single-zone reactor. As an alternative, when Da > Da,, the performance of the d = a reactor can be achieved by a CSTR operating at Dq, followed by a PFR (with S = 1)for the remaining (Da Da,) value.
Acknowledgment We are grateful to Prof. Rutherford Aris of the University of Minnesota for helpful suggestions and to the National Science Foundation (Grant CPE-8308326) for financial support of this work.
Nomenclature a = activity distribution function C = reactant concentration within the pellet Cf = reactant concentration in the external fluid phase Cp = reactant concentration at reactor inlet
Ind. Eng. Chem. Res. 1987, 26, 170-174
170
d = number of zones in the_reactor Da = Damkohler number, ( k / [ ( l+ ~ ) ~ ] ) ( L / v )-(tl)
De = effective diffusion coefficient
f ( u ) = dimensionless reaction rate function, [ u ( l
+
k = volume average reaction rate constant K = adsorption equilibrium constant L = reactor length L, = length of the ith zone n = integer characteristicof pellet shape, equals 0 for infinite slab, equals 1 for infinite cylinder, equals 2 for sphere R, = characteristic pellet dimension, half-thickness ( n = 0) or radius ( n = 1 and 2) s = x/R, S, = dimensionless location of the active catalyst within the pellet for the ith zone u =
cjcp
a[ = U ( S J a* = lower bifurcation point of the y vs. a curve from eq 3 E* = from eq 3, value of u where y ( u ) = -y(ii*) u = velocity of external fluid phase x = distance from pellet center X = reactor outlet conversion y = Z/I/ Y,= L l / L z = axial coordinate along the reactor Greek Symbols
P,
=
I$2/(n +
Y=
c,/c,o
6 = Dirac delta function e = bed void fraction u = dimensionless adsorption constant, KCP $* = Thiele modulus, k R P 2 / [ ( 1 u)*De] $ n ( ~ ) = {I- s for n = 0, In (l/s) for n = 1, and l / ( s - 1) for n=2
+
Subscripts Oi = value at the inlet of the ith zone i = value at the outlet of the ith zone
Literature Cited Morbidelli, M.; Servida, A.; Varma, A. Ind. Eng. Chem. Fundam. 1982,21,27a. Morbidelli, M.; Servida, A,; Carra, S.; Varma, A. Ind. Eng. Chem. Fundam. 1985,24, 116. Morbidelli, M.; Servida, A.; Carra, S.; Varma, A. Ind. Eng. Chem. Fundam. 1986b,25, 313. Morbidelli, M.; Servida, A.; Varma, A. Ind. Eng. Chem. Fundam. 1986a, 25, 307.
* Author to whom correspondence
should be addressed.
Cassian K. Lee, Massimo Morbidelli, Arvind Varma* Department of Chemical Engineering University of Notre Dame Notre Dame, Indiana 46556 Received for review January 21, 1986 Accepted August 14, 1986
l)l$n(SJ
Permeation Characteristics of Amino Acids through a Perfluorosulfonated Polymeric Membrane Permeation of amino acids through perfluorosulfonated cation-exchange membranes, such as “Nafion”, has been found to depend strongly on pH and therefore on the relative dominance of the cationic, zwitterionic, or anionic species in solution. For the acid form of the membrane, permeation of amino acids was favored below their isoelectric p H values, at which pHs the solution contained dominant quantities of cationic species. In contrast, for the sodium-salt form of Nafion, the permeation was favored above the isoelectric pH, i.e., where anionic species prevailed. Transport data for six amino acids are reported for Nafion-117. Perfluorosulfonated cation-exchange membranes, such as Nafion (Nafion is a registered trademark of E. I. du Pont de Nemours & Co.), hold promise for various electrolytic applications (Govindan, 1982; Yeo, 1982; Kipling, 1982). The dominant commercial use of Nafion-type membranes is in the manufacture of caustic soda and chlorine by the electrolysis of brine. In this application the membrane is in the sodium-salt form, through which only sodium ion diffusion takes place from the anolyte to the catholyte. Electrical neutrality on both sides of the membrane in the cell is preserved by compensating electrode reactions. Though cation transport through Nafion has been extensively studied, its physicochemical mechanism is not well understood. Nafion ionomers are copolymers of tetrafluoroethylene and a sulfonated fluoro polymer. The generic chemical structure is ( CF 2 CF2 )”(C FCFp,,,)
I
(OCF2CF )lOCF2CH2X
c F3 where X = S03F,S03H, S03Na, etc., and n, m, and 1 are the numbers of corresponding repeating units. Analytical studies of Nafion membranes by small-angle X-ray scattering have shown the existence of a hydrophilic sulfonated 0888-5885/87/2626-0170$01.50/0
cluster network imbedded in the hydrophobic fluoropolymer matrix (Grierke and Hsu, 1982). Cations are believed to move through these clusters which are connected by narrow channels, the exact mechanism of such transport being still unknown. The hydrophilicity of Nafon, which is due to the presence of sulfonate-exchange sites, causes considerable swelling of the membrane in polar solvents such as water. Depending on the membrane composition and the experimental conditions, 2&30% by weight of water is common. Most transport studies using Nafion membranes have exploited the selective permeation of cations by using electrolytic reactions such as the formation of sodium hydroxide from sodium chloride. A notable exception is the work of Chum et al. (1983). They reported their chance discovery of the permeation of some carboxylic acids due to a concentration gradient alone. They postulated that these acids permeated as cationic species. However, no convincing experimental evidence was presented in this very preliminary work. Sikdar (1985a) reported permeation data for nine carboxylic acids through Nafion-11‘7 membranes in the acid form and after treatment with alkali. He showed that the diffusion rate of acetic acid through the acid membrane decreased with increasing ionization of the carboxylic acids, i.e., with increasing pH. The acetic acid diffused through Na+-Nafion-117, however, 8 1987 American Chemical Society
