Ind. Eng. Chem. Res. 1991,30, 286-291
286
m = number of active sites involved in the rate-controlling step of the main reaction M = methanol mx = m-xylene ox = o-xylene P,I, PJ, ... = coke precursors PT, PM,P,,, Pox, P,, = partial pressures of different components, atm P, = partial pressure of ith component, atm px = p-xylene r M ,rpx,rmx,rox = rates, mol/(h.g of catalyst) T = absolute temperature, K T = toluene t = time, h W = weight of the catalyst, g Greek Letter \zI = deactivation function defined by (2) Registry No. Toluene, 108-88-3; methanol, 67-56-1; p-xylene, 106-42-3; o-xylene, 95-47-6 m-xylene, 108-38-3: nitrogen, 7727-37-9 hydrogen, 1333-74-0: coke, 7440-44-0.
Literature Cited Bhat, S. G. T. Selectivity for Xylene Isomers in the Reaction of Alkylation of Toluene with Methanol on Zeolite Catalysts. J . Catal. 1982, 7*5,196-199. Bhat, S. G. T. Alkylation of Toluene with Methanol over L-Type Zeolite Catalysts. Indian J . Technol. 1983, 21, 65-69. Bhat, Y. S.; Halgeri, A. B.; Prasada Rao, T. S. R. Kinetics of Toluene Alkylation with Methanol on HZSM-8 Zeolite Catalysts. Ind. Eng. Chem. Res. 1989, 28, 890-894. Cavallaro, S.; Pino, L.; Tsiakaras, P.; Giordano, N.; Rao, B. S. S. Alkylation of Toluene with Methanol. 111: Para-selectivity on Modified ZSM-5 Zeolites. Zeolites 1987, 7, 408-411. Chen, N. Y.; Garwood, W. E. Some Catalytic Properties of ZSM-5, a New Shape Selective Zeolite. J . Catal. 1978, 52, 453-458. Corella, J.; AsCa, J. M. Kinetic Equations of Mechanistic Type with Non-separable Variables for Catalyst Deactivation. Ind. Eng. Chem. Process Des. Deu. 1982, 21(1), 55-61. Coughlan, B.; Carroll, W. M.; Nunan, J. Alkylation Reactions over Ion-exchanged Molecular Sieve Zeolite Catalysts. Parts I, 11, J . Chem. Soc., Faraday Trans. 1983, 79(2), 281-296. (Chem. Abstr. 1983, 98, 574, 178425d). Engelhardt, J.; Szanyi, J.; Valyon, J. Alkylation of Toluene with MeOH on Commercial X Zeolite in different Alkali Cation Forms. J . Catal. 1987, 107, 296-306. Gnep, N. S.; Martin de Armando, M. L.; Marcilly, C.; Ha, B. H.; Guisuet, M. Catalyst Deactiuation; Delmon, B., Froment, G. F.,
Eds.; Amsterdam: Elsevier, 1980; p 79. Hsu, Y. S.; Lee, T. Y.; Hu, H. C. Isomerization of Ethyl-benzene and M-xylene on Zeolites. Ind. Eng. Chen. Res. 1988,27, 942-947. Hughes, Ronald. Deactivation of Catalysts; Academic Press: London, 1984. Itoh, H.; Miyamoto, A.; Murakani, Y. Mechanism of the Side-chain Alkylation of Toluene with MeOH. J . Catal. 1980,64,284-294. Kaeding, W. W.; Chu, C.; Young, L. B.; Weinstein, B.; Butter, S. A. Selective Alkylation of Toluene with Methanol to Produce Paraxylene. J . Catal. 1981, 67, 159-174. Kuester, J. I.; Mize, J. H. Optimization Techniques with Fortran; McGraw-Hill: New York, 1973; p p 240-250. Lacroix, C.; Eluzarche, A.; Kiennemann, A.: Boyer, A. Promotion Role of Some Metals (Cu,Ag) in the Side Chain Alkylation of Toluene by Methanol. Zeolites 1984, 4, 109-111. Meshram, N. R.; Kulkarni, Suneeta, B.; Ratnasamy, P. Transalkylation of Toluene with C9 Aromatic hydrocarbons over ZSM-5 Zeolites. J . Chem. Technol. Biotechnol. 1984, 34A, 119-126. Minachev, Kh.; Garanin, V. Molecular Sieve Zeolite-11. Adu. Chem. Ser. 1971, 102, 441-450. Rawtani, A. V.; Rao, M. S.; Gokhale, K. V. G. K. Synthesis of ZSM-5 Zeolite Using Silica from Rice-Husk Ash. Ind. Eng. Chem. Res. 1989, 28, 1411-1414. Streitwieser, A., Jr.; Reif, L. Mechanism of Transalkylation of Ethylbenzene with Gallium Bromide-hydrogen Bromide. J . Am. Chem. SOC. 1964,86, 1988-1993. Wang, Q.; Meng, Y.; Han, Q.; Miao, G.; Liu, N. Pulse Catalytic Reaction Kinetics of Toluene on HCeY Zeolite, Gaodeng Xuexiao Huaxue Xuebao 1986, 7(10), 912-916 (in Chinese) (from Chem. Abstr. 1987, 106, 215824q). Wei, J. A Mathematical Theory of Enhanced Para-Selectivity in Molecular Sieve Catalysts. J . Catal. 1982, 76, 433-439. Yashima, T.; Keiichi, S.; Tomoki; Nobuyoshi, H. Alkylation on Synthetic Zeolites. 111: Alkylation of Toluene with Methanol and Formaldehyde on Alkali Cation Exchanged Zeolites. J . Catal. 1972,26, 303-312. Young, L. B.; Butter, S. A,; Kaeding, W. W. Shape Selective Reactions with Zeolite Catalysts. 111. Selectivity in Xylene Isomerization, Toluene-methanol Alkylation, and Toluene Disproportionation over ZSM-5 Zeolite Catalysts. J . Catal. 1982, 76, 418-432. Zeng, Z.; Zheng, C. Shape-selective Catalysis of ZSM-5 Zeolites Modified with Alkali Compounds. Caodeng Xueico Huaxu Xuebao 1987,8(2),97-102 (in Chinese) (from Chem. Abstr. 1987, 107, 60974h). Zielinski, S.; Sarbak, Z. Alkylation of Toluene with Methanol over Rare Earth Forms of Na-X, FAU and Na-Y, FAU. React. Kinet. Catal. Lett. 1981, 16(2-3), 119-122.
Receiued for review March 12, 1990 Accepted August 16, 1990
Optimization of Catalyst Distribution in a Tubular Reactor Chengjun Du and Richard Turton* Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506
-
-
Optimization of catalyst distribution in a single tubular reactor for the simple series reaction A B C in which each reaction step requires a specific catalyst has been carried out. Optimal catalyst distribution and maximum final mole fraction of product C are presented graphically for some simple cases with elementary reaction kinetics and reactor conditions. T h e methodology for solving such problems, illustrated in this work, can be used by a process engineer in retrofitting old facilities.
-
Introduction This paper considers the simple series reaction A B C in which the first reaction step requires a catalyst and the second step requires a different catalyst. The question that this paper addresses is what is the optimal distribution of catalyst in a single reactor that optimizes the production of final product C during a single-pass operation? The production of C in a single reactor will in general be lower than in a two-reactor scheme in which the con-
-
-
-
ditions of each reactor may be adjusted to optimize both B and B C. Multithe reaction steps, namely, A ple-reactor schemes have been studied by a variety of workers (Aris, 1965; Chartrand and Crowe, 1969), and an excellent discussion of the various cases considered has been given by Doraiswamy and Sharma (1983). We concentrate rather on the case when only a single reactor is available for both the reaction steps. Such a case is usually suboptimal compared with a multiple-reactor configuration
0888-5885/91/2630-0286$02.50/0 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 287 but is nevertheless a useful and nontrivial problem since the process engineer may be faced with retrofitting an existing plant in which a single reactor is available and is without the funds to add a multiple-reactor scheme with associated equipment. An industrial example of this problem is retrofitting old acrylic acid facilities in which acrylic acid is formed by the partial oxidation of propylene via the intermediate acrolein. Originally this process was done in a single reactor with a single catalyst, but very specific catalysts are currently available for each of the two reaction steps and a mixture of these catalysts in a single reactor would improve the production of acrylic acid. The production of mixtures of ketones for the solvents industry is another example for using mixed catalysts in a single reactor. Although a continuously varying distribution of the two catalysts could be placed in the reactor, in practice the only realistic catalyst distribution would be to fill a fraction of the reactor with the first catalyst and the remainder of the reactor with the second catalyst. Thus the problem is reduced to finding the optimal value of z , the fraction of the reactor length that is filled with the first catalyst. In general the formulation of the problem will consist of steady-state mass and energy balances for each of the catalyst zones in the reactor and a description of the deactivation of each of the catalysts with time. Designating the first and second catalysts by 1 and 2 respectively we can write mass balance (Levenspiel, 1984; Froment and Bischoff, 1990)
1.02
ycl max
[-I
-
k l h
.. .
10
--
1/10
0.5-
1
... ....
i, , I /
--.o
0.0
0.01
0.1
/II,,
I”... ,
..
k2T
, .
0.0
10
1
....
100
[-I
Figure 1. Maximum mole fraction of C and optimal fraction of catalyst 1 for isothermal, first-order reactions.
simplify the problem, we make the following assumptions: (i) The reactor is operated isothermally. (ii) Catalyst activities do not change. (iii) Feed to the reactor contains only A. (iv) Both reactions (A B) and (B C) are first order; this means
-
-
-r, = klC,
(9)
= kzcb
(10)
-rb
Assumptions i and ii eliminate (3)-(8), the energy balance and catalyst deactivation. With assumption iv, we are able to integrate (1)and (2), the mass balance, to give the following results: Cal = CaOexp(-klrz) cbl
= C b m exp[-kg(l - 2)]
(11) (12)
In terms of conversions of A and B (Xaland X b l ) , and dimensionless parameters [ ( k 1 / k 2 )(,k 2 7 ) , and 21, (11)and (12) can be written as energy balance (Froment and Bischoff, 1990)
Xa1 x b l
= 1 - exp[-(k,/kz)(k2dz1
(13)
= 1 - exp[-(k27)(l - z ) ]
(14)
Assumption iii enables us to express the final mole fraction of C (Ycl) in the following form: ycl
catalyst deactivation (Levenspiel, 1984) (5)
t=0, a l = l (7) t=0, a2=1 (8) A complete analytical solution to this problem is very difficult since (1)-(8) are highly nonlinear and coupled. This paper presents the results for the following simple cases: (1)isothermal, first-order reactions; (2) isothermal, higher order reactions; (3) isothermal, first-order reactions, with catalyst deactivation; (4) first-order reactions with heat generation in an adiabatic reactor. 1. Isothermal, First-Order Reactions The main interest in this part is to see how the reaction kinetics affect the optimal distribution of catalysts. T o
=
XalXbl
(15)
From (13), (14), and (15),we see that the mole fraction of C (Ycl) and the conversions of A and B (Xaland X b l ) are functions of three parameters: the ratio of reaction rate constants ( k l / k 2 ) ,dimensionless rate constant ( k 2 7 ) , and the fraction of catalyst 1 ( z ) in the reactor. In this study, three levels of (k1/k2) have been chosen, (k1/k2)= 1,10, and 1/10. The optimal fractions of catalyst 1 (zOpt)a t different values of (1227) are found by using an optimization technique with (15) as the objective function. These results are plotted in Figure 1 with (k1/k2) as a parameter. It can be seen that, for (k1/k2)= 1,zOptis equal to 0.5 for any value of (kZ7);for ( k l / k z ) = 10, zOptbecomes less than 0.5 as ( k 2 7 ) increases; in contrast, for ( k l / k 2 ) = 1/10, zOpttends to be greater than 0.5 as (k27)increases. These results are expected, since a large reaction rate constant implies a fast reaction rate, and hence less catalyst is needed for that reaction. Figure 1 also shows the increase in maximum mole with increasing values of the ratio fraction of C ( Ycl),,, of (k1/k2). This is to be expected since, for a given value of ( k Z r ) the , final mole fraction of C will increase with an increase of B formed in zone 1of the reactor which in turn is directly proportional to k,. Hence for a given value of ( k 2 T ) , the value of Yclmaxwill increase with increasing
288 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 Table I. Expressions for Conversions of A and B and for Mole Fraction of C reactions
exuressions
A"=~-B=c k,
kz
+.
104 Zopt
[-I
--------====. : \
0.5-
\
-~-
0 07
.\.-
n
m
1 1 1 2 2 1 2 2
\ \ \ \\
'-. --__ r
k,,C,,~
(m=Z)
or
kzrT (m=1) [-3
Figure 3. Maximum mole fraction of C for isothermal, higher order reactions for the case of (kll/kz2CaO) = 1, (k12CaO/k21)= 1, and (kiz/kzz) = 1.
values are plotted in Figure 3. Referring to Figure 2, for small values of (kz2Cao~) and ( k z l 7 ) , zOptis determined by the reaction order ( m )of B C, while for large values of (kzzCao7) and (kZ17), zopt depends on both reaction orders ( n , m). Figure 3 shows that Yclmax,for small values of (k2zCao~) and ( k Z 1 ~ )is, also determined by the reaction order (m)of B C; in contrast to Figure 2, for large values of (kzzCao~)and (k217), Yclmaxfor first-order reactions differs greatly from YCl- for second-order reactions. The values of Ycl for the intermediate cases tend to a single line a t large values of (kzZCa07) and (kZ17).
-
-
3. Isothermal, First-Order Reactions with Catalyst Deactivation In this section, the effect of catalyst deactivation on the optimal distribution of catalysts will be investigated. Assumption i in part 1 is still needed and eliminates (3) and (4),and assumptions iii and iv are kept for further simplification, while assumption ii is replaced by the following: (ii) Both catalyst 1 and 2 deactivations are concentration independent, i.e., independent deactivation (Levenspiel, 1984) and first order. The above assumption means a, =
exp[-kd,t]
(25)
exp[-kdzt] (26) where t is the time from the start of the operation and is large compared with the space time T. Since t >> r we can use a pseudo-steady-state assumption, and along with assumptions ii, iii, and iv, we are able a2 =
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 289 Table 11. Expressions for Average Conversions of A and B and for Average Mole Fraction of C
Xal = -JtdXaldt 1 = td
0.0 0.1
1 .o
0.0 100.0
10.0 k2T
[-I
td
Figure 4. Maximum mole fraction of C and optimal fraction of catalyst 1 for isothermal, first-order reactions with catalyst deactivation, for the case of ( k l / k z ) = 1 and ( k d l t d ) = 1. 11.0
1.07
T--.
10.5
'
2
0.0 0.1
0.0
1 .o
10 k2T
100
[-I
Figure 5. Maximum mole fraction of C and optimal fraction of catalyst 1 for isothermal, first-order reactions with catalyst deactivation, for the case of ( k l / k 2 ) = 1 and ( k d l / k d 2 ) = 2.
to integrate (1)and (2) to obtain analytical expressions for the conversions of A and B (Xal and Xbl) and the mole fraction of C (Ycl) a t any lifetime of catalysts. Thus, Xa1 = 1 - exp[-kl/k2)(k27)z exp(-(kdl/kd2)(kdlt)I xbl
= 1 - exp[-(kzr)(l - z ) exp(-k&t)] ycl
=
Ycl = -JLdYcldt 1 =
XalXbl
(27) (28) (29)
Supposing that the space time (7) in the reactor is kept constant during the life of the catalysts, then performing the integration of (271429) over the catalyst regeneration life ( t d ) , we obtain expfessions for the average conversions of A and B (Xal and Xbl), and the average mole fraction of C (Ycl), which are summarized in Table 11. Equations 30-32 show that the average conversions of A and B, (Xaland 8 b 1 ) and the average mole fraction of C ( Ycl) are determined by the deactivation rate constant ratio (kdl/k&) and dimensionless deactivation time (kd2td), in addition to the reaction rate constant ratio (kl/kz), dimensionless rate constant ( k 2 7 ) , and fraction of catalyst 1 ( 2 ) . In this section, we study the effects of two parameters, ( k d l / k d Z ) and (kdstd), on the optimal fraction of catalyst 1 (topt). A. Effect of the Ratio of Deactivation Rate Constants. ( k l / k 2 )and (kd2td) are set to be 1. Three levels of (kdl/kd2) (1, 1/10, 2) were chosen for study. For each of the above three levels of (kdl/kd2), the optimal fractions of catalyst 1 (zopt) a t various (k27) were found with a golden-section technique, (32) being the objective function. The results are plotted in Figure 4. When ( k d l / k d 2 ) = 1, the same amounts of catalysts 1 and 2 are needed in the reactor for any value of (kz7);when (kdl/kd2) = 1/10, more catalyst 2 is needed in the reactor than catalyst 1, as the (k27)increases; while for ( k d l / k d 2 ) = 2, the opposite is true. These results are expected, since a catalyst with a large
X1{l- eXP[-kl/k2)(kzT)z eXP(-(kdi/kdz)(kdztd)E)1J
x
11 - exP[-(kzT)(l - 2) exP(-(kd2td)F)ll dE (32)
deactivation constant will deactivate faster, and the amount of this catalyst needs to be increased to keep a desired average reaction rate. Figure 4 also shows that, a t the same value of (k27), a larger value of ( k d l / h d 2 ) corresponds to a smaller value of Yclmax. The reason is as follows: a large value of means a large value of (kdltd) under the specified conditions of (k1/h2) = 1 and (kd2td) = 1,which implies a faster deactivation of catalyst 1, and hence, more catalyst 1 is needed in the reactor, as indicated in Figure 4. This will also result in a lower overall reaction rate in the reactor; therefore, the average maximum final mole fraction of C ( Ycl ,I will be reduced. B. Effect of Dimensionless Deactivation Time. (hl/h2) and ( h d l / h d 2 ) are set to be 1 and 2, respectively. Three levels of (hd2td) (1,1/2, 2) are chosen for study. The results are presented in Figure 5 . It can be seen that the optimal fraction of catalyst 1 (zOpt)at the same (k27) tends to increase as dimensionless deactivation time (kd2td) increases from 1 / 2 to 2. This is reasonable because the values of the parameters chosen for this figure are ( k 1 / k 2 ) = 1 and ( k d l / k d 2 ) = 2, which means that both reactions have the same reaction rate constant, but catalyst 1 deactivates faster than catalyst 2. This implies that the average activity of catalyst 1 becomes increasingly smaller than that of catalyst 2 as the dimensionless deactivation time (kd2td) increases; therefore more catalyst 1 is needed in the reactor. Figure 5 also shows the similar trends of Ycl versus (k27)to those in Figure 4,except with (kd2td) as parameter instead of ( h d l l h d 2 ) . That is, the average mole fraction of C ( Ycl,,I will decrease with increasing dimensionless deactivation time (kd2td).An explanation similar to that given above also applies here. That is, a large (katd) means a large (hdltd) also, under the specified conditions of (kl/k2) = 1 and ( k d l / h d z ) = 2. Large (kdltd) and (kd2td) indicate fast deactivation of catalysts 1 and 2. T h u t t h e average reaction rate in the reactor will be low, and Ycl mar will be reduced. 4. First-Order Reactions with Heat Generation In this part, the main objective is to study the effect of the heat of reaction on the optimal distribution of catalysts. Assumption ii in part 1is still needed to eliminate (5)-(8), and assumptions iii and iv are kept for further simplification, while assumption i is replaced by the following one: (i) The reactor is operated adiabatically, the specific heat of the reactants and products is the same and constant (CJ, and the heat of reaction (-AHa and -AHb) is temperature-independent. We further simplify the analysis by assuming that the molar volume of the reactants does not change, which will
290 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
not be strictly true for nonisothermal operation. Under the restriction of dilute systems or small temperature changes through the reactor, this assumption should not' cause major error. With the above assumptions, we are able to formulate the problem in terms of seven dimensionless parameters: ratio of heats of reaction ( m a / m b ) , dimensionless heat of reaction B -,C ( - m b / c , T o ) , ratio of activation energies (,?$/E2),dimensionless activation energy (E2/RT0),ratio of reaction rate constants a t the inlet conditions (klTO/ k2TO), dimensionless reaction rate constant a t the inlet conditions ( h 2 T o ~and ) , fraction of catalyst 1 (2). The equations plus the initial conditions are as follows:
-d X-a di
- z ( k i ~ ~ / k z ~ o ) ( k z T -~ ~Xa) )(lX exp[(El/E2)(E2/RT0)(1- 1/(Tz/5"0))1 (33) Tz/TO
=
(ma/mb)(-mb/cpTO)xa
+1
1 .o :'
: /
[-I o.5
.......
_--,..."
- 0.5
""
(-AH
-
/CpTO)
0.0 1 .o
. . . . . 3.0 rO.0
0.0i
Figure 6. Maximum final mole fraction of C and optimal fraction of catalyst 1 for first-order reactions in an adiabatic reactor, for the case of (m,/Mb)= 1, (&/E2) = 1, (E2/RTo)= 1, and ( k l m / k m ) = 1. 1.o
1.01
di (1 - ~ ) ( k * T 0 ~ )-( 1x b ) exp[(EZ/RTo)(l - 1/(7'-z/T0))1 (35)
+ Tm/TO = (ma/mb)(-mb/cpTO)xftl +1
Tl-z/TO = Tm/TO
(-mb/cpTO)xalxb
ycl
=
XalXbl
1.o
'c1 max
(34)
-dXb- -
/-
Yct max
Zopt
[-I
[-I
~-0.5
(36)
(37) (38)
{=O,
xa=o
(39)
{=o,
x b = o
(40)
where { is the normalized axial distance in the tubular reactor. Equations 33 and 35 are the mass balances, and (34), (361, and (37) are the energy balances. Numerical integration of (33) using (34) and the initial condition (39) will give the conversion of A (Xal)and the temperature of reactants a t the end of zone 1 of the reactor (7"). Numerical integration of (35) using (36) and (37) as well as the initial condition (40) Will give the conversion of B ( x b 1 ) a t the exit of the reactor. With the numerical results (Xal, X b l ) , the final mole fraction of C (Ycl) can be found from (38). Careful examination of these equations will show that the conversions of A and B (Xal and X b l ) and the final mole fraction of C ( Ycl)are determined by seven parameters: ( A H a / A H b ) , ( - A H b / c p T o ) , (Ei/Ez), ( E Z / R T O ) , (klTO/kSTO),( & T O T ) , and z. Thus the problem becomes more complicated. In this paper, only two parameters, ( - A H b / C p T O ) and ( E 2 / R T 0 )are , studied. A. Effect O f ( - A H b / C p T o ) . By letting ( A H a / A H b ) , & / E 2 ) , (E2/RT0)and (klTOlk2TO) equal 1, we are able to find the optimal fraction of catalyst 1 (zopt) for each of three levels of ( - M b / c p T O ) (0, 1,3) at various ( k , T ) , with (38) being the objective function. The results are presented in Figure 6. Referring to Figure 6, when ( - m b / c p T o ) = 0, the problem is reduced to the isothermal one, where the optimal fraction of catalyst 1 (zOpt)is 0.5 for any values of &TOT); when ( - A H b / C p T O ) = 1 or 3, zap! becomes larger than 0.5 as (k,s) increases. The reason is that zone 2 will be a t higher temperatures than zone 1 when (--mb) > 0. Thus, with (-AH), E , and km being equal and greater than 0 for both reactions, the reaction rates in zone 2 will be much higher than those in zone 1. Hence, less catalyst is needed in zone 2; i.e., zOpt> 0.5.
k2J
[-I
Figure 7. Maximum final mole fraction of C and optimal fraction of catalyst 1 for first-order reactions in an adiabatic reactor, for the case of (AHa/AHb)= 1, (.&/E2)= 1, (-fi&/cpTo) = 1, and (klm/ k n o ) = 1.
Figure 6 also shows that a large value of (-AHb/CpTo) corresponds to a large value of Ycl a t the same (kZT07). This result is expected since a large ( - n H b / C p T O ) means large amounts of heat generated, hence the temperature in the reactor will be high, and the reaction rate will be fast. Thus more product C will be produced. B. Effect of (E2/RT0).By letting (iw,/mb), (El/ E 2 ) ,( - m b / c p T o ) , and (k,m/k2m) equal 1, we are able to find the optimal fraction of catalyst 1 (zopJ for each of two levels of (E2/RT0)(1.5, 3.0) at various (kn07).The results are presented in Figure 7, where isothermal plots are also drawn as a reference. Figure 7 shows trends of zOptand YClm versus ( k m ~ )similar to those in Figure 6. The same explanation as the above applies here well, except that (E2/RT0),whose value indicates the temperature sensitivity of the reaction rate constant, replaces ( - m b / c p T o ) , whose value indicates the amount of heat generated in the reaction. Concluding Remarks General equations are presented for describing the series reaction A B -,C occurring in a single plug flow reactor for the case when each reaction step requires a separate catalyst, which may deactivate. The study of this kind of problem will help in retrofitting some older industrial facilities. Expressions for the optimal distribution of catalysts in a single reactor are difficult to obtain in closed form, but the results of numerical optimization for several simple cases are given graphically. The number of parameters to describe even the simple cases considered here is large, making convenient graphical presentation difficult
-
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 291 for all but the most elementary reaction kinetics and reactor conditions. Although the problem of optimizing catalyst distributions in a single reactor involves many parameters, the formulation of the problem, as presented here, is relatively straightforward and the methodology for solving such problems can be implemented easily on the computer. Depending on the actual requirement, a real problem in practice may be simplified to the simple cases studied in this work or be solved by use of the methodology illustrated in this work.
Acknowledgment We acknowledge that this work was carried out under a grant from the National Science Foundation (CBT8657548) and support from Union Carbide Corp.
Nomenclature A = species A ul = activity of catalyst 1 u2 = activity of catalyst 2 B = species B C = species C C, = concentration of A, mol/m3 Ch = concentration of B, mol/m3 C, = concentration of species j, mol/m3 Ck = concentration of species k , mol/m3 CaO= initial concentration of A, mol/m3 C,, = final concentration of A, mol/m3 cbl = final concentration of B, mol/m3 Cbm = concentration of B at the end of zone 1, mol/m3 C, = average specific heat of A, B, C, and I, J/(mol.K) C,, = specific heat of A, J/(mol.K) Cp,b = specific heat of B, J/(mol.K) C,,c = specific heat of C, J/(mol,K) Cp,i = specific heat of I, J/(mol.K) d, = inside diameter of reactor, m E , = activation energy for reaction A B, J/mol E2 = activation energy for reaction B C, J/mol Fa = flow rate of A, mol/s F b = flow rate of B, mol/s F, = flow rate of C, mol/s Fi = flow rate of inert, mol/s -AHa = heat of reaction for A -.B, based on A, J/mol A -mb = heat of reaction for B -. C, based on B, J/mol B I = species I (inert) k , = first-order reaction rate constant for A B, l / s k 2 = first-order reaction rate constant for B C,I/s k , , = first-order reaction rate constant for A B, l / s k12 = second-order reaction rate constant for A B, m3/ (mol4 kP1 = first-order reaction rate constant for B C, l / s k22 = second-order reaction rate constant for B C, m3/ (mold k d l = deactivation rate constant of catalyst 1, l/s k d 2 = deactivation rate constant of catalyst 2, l/s k l m = first-order reaction rate constant at the inlet condition
--
----
-
for A B, l/s k2T0= first-order reaction rate constant at the inlet condition for B C, l / s
L = total length of the reactor, m 1 = axial distance in a tubular reactor, m m = reaction order of reaction B C n = reaction order of reaction A B m’ = concentration dependency of catalyst 2 deactivation n’ = concentration dependency of catalyst 1 deactivation m” = deactivation order of catalyst 2 n” = deactivation order of catalyst 1 R = universal gas constant, J/(mol.K) -r, = rate of disappearance of A, mol/(m3.s) -rb = rate of disappearance of B, mol/(m3.s) T = temperature, K To = initial temperature, K T1 = final temperature, K T f = temperature of fluid (A, B, C, and I), K T , = temperature at the end of zone 1, K T, = temperature of surrounding, K T, = temperature at any position in zone 1, K TlWz = temperature a t any position in zone 2, K t = deactivation time of catalysts 1 and 2, s t d = regeneration life of catalysts 1 and 2, s U = overall heat-transfer coefficient based on the inside diameter of the reactor, W/(m2.s.K) X, = conversion of A at any position in zone 1 Xb = conversion of B at any position in zone 2 X,,= final conversion of A X b l = final conversion of B X,, = average final conversion of A over the regeneration life ( t d ) of catalysts 1 and 2 X b l = average final conversion of B over the regeneration life ( t d ) of catalysts 1 and 2 = final mole fraction of C Ycl = average final mole fraction of C over the regeneration life ( t d )of catalysts 1 and 2 = maximum final mole fraction of C Yclmax = average maximum final mole fraction of C over the regeneration life ( t d ) of catalysts 1 and 2 V = reactor volume, m3 z = fraction of catalyst 1 in a tubular reactor zOpt= optimal fraction of catalyst 1 in a tubular reaction
--
xcl
Greek Symbols {=
1/L, normalized axial distance in a tubular reactor
E = t / t d ,normalized deactivation time T
= space time, s
Literature Cited Ark, R. Introduction to the Analysis of Chemical Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1965. Chartrand, C.; Crowe, C. M. The Optimization of the Catalytic Oxidation of Sulphur Dioxide in a Multi-Bed Adiabatic Reactor. Can. J . Chem. Eng. 1969,47, 296-301. Doraiswamy, L. K.;Sharma, M. M. Heterogeneous Reactions: Analysis, Examples, and Reactor Design;Wiley: New York, 1983; Vol. 1, Chapter 12. Froment, G.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990; pp 334-337. Levenspiel, 0. The Chemical Reactor Omnibook; OSU Book Store Inc.: Corvallis, OR, 1984; p 3.1,31.1.
Received for review February 6 , 1990 Revised manuscript received July 16, 1990 Accepted August 21, 1990
