126
Ind. Eng. Chem. Process Des. Dev. 1984, 2 3 , 126-131
Control Optimization of Tubular Catalytic Reactors with Catalyst Decay Guldo Buzzi-Ferrarls, Edoardo Facchi, Pi0 Forzatti, and Enrico Tronconi Dipartimento di Chimica Industriale ed lngegneria Chimica del Politecnico, Piazza Leonard0 da Vinci, 32, 20 133 Milano, Italy
A general, approximate approach to the control optimization problem is presented, based on direct numerical search techniques. When compared to previous methods adopting numerical implementations of variational principles, this approach shows greater flexibility and computational efficiency, as demonstrated by a few detailed examples. Advantages are more evident when realistic, large dimensional problems are treated. The degree of approximation involved can be made as small as desired, more rigorous solutions requiring more conputational effort.
Introduction Catalyst deactivation is a common problem in industrial practice and has received both experimental and theoretical consideration in the past (Delmon and Froment, 1980). A related aspect of special interest is the control optimization of chemical reactors experiencing catalyst decay. Much effort has been devoted so far to investigating situations where analytical expressions for optimal policies may be derived the works of Chou et al. (1967), Szepe and Levenspiel (1968), Ogunye and Ray (1968), Crowe (1970), Gruyaert and Crowe (1974, 1976), and Reiff (1981) are concerned with optimization of ideal, isothermal reactors under the simplifying assumptions of concentration-independent deactivation kinetics and separable kinetic equations, with the rate of reaction depending on a single rate constant; reactor temperature is the only control variable considered, and the yield of a reaction product is chosen as the function to maximize. A few studies have been published in which more realistic situations are addressed. Among these, Ogunye and Ray (1971) have presented a general optimization procedure based on the extension of Pontryagin’s maximum principle made by Degtyarev and Sirazetdinov (1967) to treat cases where a whole set of control variables, u ( z , t ) ,is to be optimized with respect to both space and time coordinates. Controls may include a number of reactor state variables; the optimal initial catalyst distribution may be also considered, and it is possible to introduce operating costs as well as product values into the objective function. Although the generality of this approach is valuable and promising for applications to industrial problems, the computational algorithm proposed appears involved and subject to unnecessary limitations, mainly related to the numerical implementation of its mathematical background. In this paper, an approximate procedure for the control optimization of tubular reactors with catalyst decay is presented which retains and extends the generality of Ogunye and Ray’s approach, as also the optimal time interval of catalyst operation can be sought. However, superior computational flexibility and efficiency are expected from adopting direct optimum search techniques. After formulating the optimization problem and discussing the two approaches, a few numerical examples particularly selected for the purpose of comparison will be worked out. Attention will also be given to the question of how critical is the degree of approximation associated with the method proposed. Statement of the Optimization Problem Two assumptions are usually made: a plug flow chemical reactor model is considered, and a quasi-steady-state 0196-4305/84/ 1 123-0126$01.50/0
approximation is introduced due to the time scale for catalyst decay being in most cases much longer than the reactor time scale. Under these assumptions the general analytical statement of the variational problem is the following. Find the vectors of controls u , v , w which maximize the objective function
F =
s’ 1’ 0
0
G (x,y , u, v, w ;7) dzdt
(1)
subject to the following constraints axi az = f , (x,y ,
4;
ay1 _ - g; (x, Y , u ) ; at
xi (0,t) = ui ( t )
(i = 1, ..., s)
(2)
Y;(Z,O) = w,(z)
0’ = 1, ...)4 ) (0 I2
I1; 0 It I1) (3)
with z axial1 dimensionless coordinate and t = 817, where 0 is the time on stream and 7 is the total time of catalyst operation. Here x i represents a generic reactor state variable, namely temperature, pressure, extents of reaction; several catalyst activities y, are considered in the most general form of the problem. F is usually intended to represent the average profit over an operation cycle.
Previous Work Among the methods for solving variational problems, the extension of Pontryagin’smaximum principle proposed by Degtyarev and Sirazetdinov (1967) has received considerable attention in the past. However, this method requires introduction of two sets of adjoint equations and leads to a two-point boundary value problem whose numerical solution has been shown to be intrinsically unstable (Resenbrock and Storey, 1966). In order to avoid this drawback, use of the Hamiltonian function has been suggested to calculate the gradient of the decisional functions, so that it becomes possible to integrate both the state and the adjoint equations along the directions which make them stable, respectively (Rosenbrock and Storey, 1966). Following these lines, a computational procedure was proposed by Ogunye and Ray (1971). A similar method had been previously employed by Jackson (1967). A number of disadvantages associated with this approach can be pointed out, however. (a) It is well-known that the efficiency of the gradient method can be very low in the neighborhood of the optimum, even when the number of decision variables considered is small. Since discretization of the controls tendentially results in a very large number of variables, the gradient method is expected to work even less efficiently when extended to variational 0 1983
American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
problems. In fact, Jackson (1967) reported convergence problems. Application of a conjugate gradient search procedure of the Fletcher-Reeves type (Lasdom et al., 1967) could possibly improve the efficiency, but only when the gradient direction can be computed with a high degree of accuracy. Once again, however, the large number of optimization variables resulting from discretizing the problem would require far too many univariate searches, thus making this approach impractical too. Besides, derivatives of the Hamiltonian function are obtained in this case from numerical integration of the two sets of equations. Therefore the gradient cannot be estimated so accurately to make the use of conjugate directions truly advantageous. (b) Introduction of the adjoint equations results in doubling the dimensions of the original problem. (c) The adjoint equations contain derivatives bearing no physical meaning, such as dgk/ax,, which may be expected to be ill conditioned in some cases, for example if gk 0: xia with cy < 1and xiapproaches zero. (d) The total operation time of the catalyst, T , must be assigned a priori. (e) Functional forms corresponding to controls that can be conveniently realized in practice may not be assigned to the control policies a priori. On the basis of the remarks listed above, strong and perhaps unnecessary limitations seem to affect numerical implementations of the maximum principle based on the gradient techniques.
An Approximate Approach An approximate approach to the solution of optimal control problems for tubular chemical reactors with catalyst decay is presented in this section. The same procedure has been successfully applied to optimize temperature profiles of tubular flow reactors (Rosenbrock and Storey, 1966), temperature controls for isothermal batch reactors (Millman and Katz, 1967), and batch distillation operations (Buzzi Ferraris et al., 1969). The procedure is as follows. (a) Analytically determined expressions containing unknown adaptive parameters B are provided for the controls u , v , w . u= u
(2,t;
B)
v = v (t;B) w = w ( 2 ; 8)
(4)
The vector B may include the total time of catalyst operation, 7,as well. (b) Initial guess values are assigned to the parameters. (c) State equations (2) and (3) are integrated, and the value of the objective function F , eq 1,is computed. F is thus a function of the parameters 8. (d) B is modified, and the procedure is iterated until those values of the parameters which make F a maximum are found. The search for the optimal parameters BOPT can be conveniently carried out by a general direct optimization program, similar to those available for nonlinear regression packages. Use of such programs is fully compatible with the existence of linear or nonlinear constraints on the parameters, so that constraints on the control policies can be easily introduced. Optimization with respect to the total time of operation can be handled as well, T being regarded as an ordinary decision variable. Furthermore, no adjoint equations need to be introduced, so that only (s + q ) differential equations are integrated a t each step, with a considerable saving of computer time. The possibility of adopting a priori assigned functional forms for the control policies is also obviously granted by the method proposed. The significant advantages listed above, however, may seem to be gained at the expense of a serious compromise,
127
as the true optimal policies are in this case only approximated. This is only apparently so. Indeed, a more thorough discussion is necessary on the degree of approximation involved. When discretized and numerically implemented, the rigorous variational methods, such as those based on the maximum principle, reduce to a pointwise determination of the optimal control policies, the points usually being vertexes of a convenient grid in the ( 2 , t ) space. The approach presented in this work is able to produce the same results by simply regarding the values of the control policies a t each point as single optimization parameters. More generally, it is always possible to find a functional form able to approximate as closely as desired the mathematically optimum policies, provided that the number of optimization parameters is large enough. Therefore, the approach proposed does not result in principle in a lack of accuracy. However, it is worthwhile pointing out that with both methods the search for the exact solution to the variational problem may become very costly in terms of computational effort. The number of computations in fact grows more than linearly when the number of optimization variables is increased; convergence difficultiesare also more likely to occur, and the final results may be somewhat uncertain. Alternatively, the approach presented hereby allows one to obtain an approximate solution to the variational problem by assigning very simple forms to the controls. Due to the small number of parameters involved, optimization is very efficient in this case, and altogether more promising to yield a correct result when complex problems are treated; the more so, since in many instances the form of the best control policies can be qualitatively predicted on the grounds of physical reasoning. The flexibility of the proposed approach is evident. In particular, it may be a good compromise to adopt an approximate, easy-to-find solution as a starting point for a second more accurate optimization based on a large number of optimization parameters. Numerical Applications Three examples are worked out to illustrate the power and the efficiency of the approach described above. Since all these examples have been already treated in the literature, a comparison is possible with results derived by other methods. In all cases, a single catalytic activity has been considered, and the differential equation for catalyst decay, eq 3, has been numerically integrated during computations by a Euler method. A general purpose optimization program was employed, making use of several direct search procedures combined (Buzzi Ferraris, 1970). All the calculations were performed on a UNIVAC 1100 computer. Example 1. Fixed-Bed Isothermal Deactivating Catalytic Reactor. Optimal Temperature Policy with an Upper Constraint T + for a Simple Irreversible B. Consider reaction and deactivation Reaction A kinetics in the form
-
-rA = k g E I R TC A Y ; -dy/dt = ko/e-EIRTy
(y(0) = 1) (5)
For an isothermal, plug-flow reactor, it is well-known (Chou et al., 1967) that, if E’ > E, the optimum operating policy is such that the conversion of A is constant during the run, i.e. d(ky)/dt = 0 (6)
provided that the objective function is the average conversion over T and the total time on stream T is given.
128
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table I. Equations for Example 1 plug flow reactor model X , = 1- exp[-539.6 exp(-10 0 0 0 / R T )X 1 . 5 ~ 1 deactivation kinetics 1
30.1x
dY
= -1.36
x
y(O)= 1;
l o 5 exp(-23 (7
000/RT)y
= 30.1 days)
objective function F = JgX, d t = X , 720-
constraints T < BOOK
When an upper temperature constraint is assigned, the optimum temperature policy consists of a first section where temperature is still being raised to keep conversion constant until the upper bound is reached; then, isothermal operation follows until completion of the run (Crowe, 1970). By making use of a quasi-steady-state approximation, the equation for the PF model can be integrated analytically. The differential equation for the catalyst activity has been integrated numerically by the Euler method, time interval being divided into 100 steps. Table I summarizes the equations employed. While optimum value of conversion to be maintained (and, therefore, the initial temperature To,must be determined numerically, the exact analytical form of the optimum temperature policy is known in this case (Chou et al., 1967) T ( t )= 1 / ( A + B In (C- t / T ) ) (7) if T ( t )ITe then T = T ( t ) ;else T = Te. Therefore it can be assigned to implement the optimization method proposed in this work. For the data reported by Levenspiel and Sadana (1978), one obtains F = xA = 0.51479 (To= 721.6 K) Suppose now that the exact analytical form for the optimum temperature policy is unknown. It is always possible, though, to provide a general empirical expression for it. For the same problem, the time interval 0-7 has been divided into 10 subintervals; within each of them a parabolic interpolation has been employed, yielding a total number of 21 parameters. Results are f A = 0.51479 (To = 721.7 K) Conversion is actually constant as long as T 5 Te, in agreement with the analytically determined optimal policy. For this relatively simple problem, even a crude approximation of the optimal T ( t )policy be means of a single parabola over the whole interval 0-7 proved to be satisfactory. Only three optimization parameters and substantially less computing time are here needed to obtain the following results, very close to those presented above %A = 0.51475 (To = 724.2 K) Conversion ranges between 54.0% and 52.8% before the upper temperature bound is reached. The three temperature policies are compared in Figure 1. Computer time required for optimization was in the order of a few seconds in all cases. Example 2. Isothermal Reactor with Consecutive Reactions. In this example, the optimization of a deactivating isothermal plug flow reactor is considered for the reaction scheme A B C, when the reactor temperature is the control variable. As the objective function, the yield of B, which should be maximized, is chosen. No analytical solution to this problem is available. However, a numerical solution has been presented by Ogunye and
--
1
0
02
04
06
08
Figure 1. Optimal temperature policy for example 1: (-) analytical form, eq 7; (--) interpolation with 10 parabolas; (--) interpolation with 1 parabola.
e 0
E
024
0 0242
_. c
0,2400
02
04
06
08 t
1
Figure 2. Optimal temperature policy for example 2. Table 11. Equations for Example 2 plug flow reactor model k , = 2.02 x 10l8exp(-l/T') k , = 5.47 x 10" exp(-0.67/T')
X,
= 1 - exp(-k,y)
XB = k , / ( k 2 - - k , )[exp(-k,y) - e x ~ ( - k , ~ +) l i l - exP(-k,Y)l deactivation kinetics d y / d t = -6.35 x lo6 exp(-1/3T')y2
( y ( 0 )=
1) objective function F = J:, ( X , - X , ) d t constraintsu T' G 0.0246 a
T' is a dimensionless temperature = T R / E .
Ray (1971) based on a gradient method. The state equations governing the extents of reaction and the catalyst decay have been taken from their paper along with the numerical values of all the constants. Again, numerical integration with respect to time was necessary for the decay equation only, since analytical solutions to the reactor equations are possible. The set of equations is reported in Table 11. The functional form of the temperature policy proposed was again a parabolic interpolation of 21 points, as already explained in example 1. Two different isothermal initial temperature profiles have been considered; the final results of the optimization procedure were the same for both cases. The optimum temperature profile found well reproduces the optimal profile presented by Ogunye and Ray (see Figure 2). The value of the
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
objective function is F = 0.5560, which compares with 0.5485 as obtained by Ogunye and Ray. Computer time required was only 15 s in this case. Example 3. Industrial Vinyl Chloride Monomer Reactor. Ogunye and Ray (1970) have studied the optimization problem of an industrial vinyl chloride monomer reactor. The nonisothermal reactor consists of several hundred tubes of small diameter packed with catalyst, and it is surrounded by a high-pressure cooling jacket.. The catalyst (mercuric chloride on activated carbon) experiences decay as a result of sublimation due to reactor hot spots. Ogunye and Ray provide expressions both for reaction and catalyst deactivation kinetics; their plug-flow reactor model consists of three differential equations representing the material, enthalpy, and momentum balances along the reactor axial coordinate. For the optimization problem, an objective function is introduced that contains both product values and catalyst costs, and represents the profit averaged over the period of operation. Control variables considered are the coolant temperature U, ( t ) ,the inlet temperature V2(t)and pressure V3(t),the feed rate to the system, ~ ( t )the , molar ratio of HC1 to C2H2,f ( t ) , and finally, the initial catalyst activity distribution, w(z). The authors used the gradient-based computational algorithm previously illustrated to solve the optimization problem for several subcases; in each of them, only some of the control variables above were released. In all cases the total time of operation T was fixed. It was our goal to apply the approximate method developed in this work to the same problem for the purpose of comparison. The whole set of equations used for computations is listed in Table 111. The constants were taken from Ogunye and Ray’s paper. However, due to insurmountable uncertainties on the numerical values of some of these constants (Ray, 1982), we found it impossible to reproduce exactly the results reported in the paper. Therefore, a quantitative comparison with the optimal conditions calculated by Ogunye and Ray is not significant. Yet, we believe that the two approaches can be at least qualitatively compared on the basis of their performances, as discussed below. Since the reactor model may not be integrated analytically, a special numerical routine based on Runge-Kutta methods was chosen to solve the set of three differential equations. Self-adjusting step length and variable method order are among the features included in this routine; in fact, owing to the strong dependence of the catalyst activity on temperature, the optimal solution is sensitive to local inaccuracies in the numerical computation of temperature and conversion profiles along the reactor axis. Several approaches corresponding to different levels of complexity were attempted, thus covering a wide range of possibilities. Results are illustrated in the following. (a) As a basis for comparison,the objective function was first calculated corresponding to the nominal design conditions reported by Ogunye and Ray (See Table 111). For this unoptimized situation, F = 0.388. (b) The controls ~ ( t )U,(t), , V2(t),V3(t)were then simultaneouslyoptimized for a uniform catalyst distribution, w(z) = 1;T = f were kept at unity. The simplest functional form was proposed for the control policies at this stage, namely they were assumed to be constant with time. Two upper bounds for the feed flow rate and the inlet pressure V , had to be introduced, while two lower bounds were adopted for the coolant temperature U, and the feed inlet temperature V , (see Table 111). Calculations showed that the optimal policies are the extreme ones, yielding a value of F = 0.622. We believe
129
Z
Figure 3. Example 3. Conversion profiles with pure catalyst loading, w(z) = 1; optimal controls as in point b.
L
Figure 4. Example 3. Activity profiles with pure catalyst loading, w(z) = 1; optimal controls as in point b.
this to be a significant result, as it provides a reasonable amount of information about the optimal operating conditions of the reactor at a minimal expense of computational effort, only four being the number of optimization parameters involved. Although such very simple policies may be regarded as a starting point for a further refinement of the optimum search, they could be satisfactory and convenient solutions to the optimum problem in many practical cases as well. The trends shown by the optimal controls can be rationalized as follows. (1)The rate of reaction depends almost quadratically on the pressure; on the other hand, no terms related to compression costs appear in the objective function. This takes care of the optimal V , being located on the upper bound. (2) The coolant and inlet temperatures (U,, V,) tend to their lower bounds because these conditions prevent hot spot formation and slow down catalyst decay; again, no terms related to cooling or preheating costs are included in the objective function. (3) As for the tendency of x to reach the upper bound, it seems that either the reactor is oversized or the total time of operation is too short, as indicated by Figure 3 and 4 showing the development of conversion and activity profiles with time. The optimal solutions presented by Ogunye and Ray do not point clearly to such trends in the controls, so that their discretized variational method is suspected to meet difficulties in the optimum search when dealing with large dimensional, constrained problems. (c) The optimization of point (b) was repeated including a uniform initial catalyst activity among the parameters to be determined. Also was allowed to vary. Calculations gave a value of the objective function F = 0.790 for an initial activity w = 0.376 and an optimal x = 1.095. All the other controls reached the values indicated in case (b). As expected, inclusion of the optimal catalyst distribution problem leads to a signficant improvement. Notice also
130
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Table 111. Equations for Example 3 plug flow reactor model k , = exp(20.849 - 3019.628/T) k , = exp(7.34776 - 3875.189lT) P Z k , Y (1 - x ) ( t r= (1 + t - x)[(E - x ) ( k , + P ) + k z l PM = 26.0 + 36.5 t; [glg-mol] Cp = 0.469 + 0.311t; - 0 . 1 8 ~[kcal/m3 K ]
v, =
0.08205 T X (1+ PMxP
t;
- X)
r
A
4601
l
x
e;
-
380-
E
8
300
[mlh 1
L 02
0
04
06
08
1
t
ax PMr az - 1 O O O x
Figure 6. Example 3, point e; optimal policies for controls U,,V,.
aP
-az_ - - V,(4.6782 x
+
1.1844 X lO-'x) [atm]
aT [26.0r_ _
1 1 0 5 9 . 2 ( T - V,)] [K 1 CpV, ( 1 + E - x )
az
deactivation kinetics
a v i a t = --5 x
l o 5exp[-7045.8/2']
y(z,O)= w(z) objective function F=J':x
041 I
{ ~ ( l , t ) 0- . 2 [ 1 - x ( l , t ) ] - 0.2[6 - ~ ( l , t ) ] 0 . 4 ~ ( 1 , t-) 0 . 0 5 x ( l , t ) } d t - O . ~ ~ ' , W ( Zdz)
I 02
0
04
06
1
08 t
Figure 7. Example 3, point e; optimal policies for control V,, f .
constraints
xQ
7000 kg/m2 h
V, 2 423 K V, < 2 atm U , > 300 K
t>O nominal conditions (for case a)
x = 5400 kg/mZh ; V, = 1.5 atm; V, = 423 K ; U , = 353 K; t; = 1
-4 5
C
K=294 0 0
02
04
06
08
1
2
P
Figure 8. Example 3, point e; optimal catalyst distribution.
3
e
i
Y
L"
0 x
I
U m
m -c
I
Z
Figure 5. Example 3, point d; optimal catalyst distribution.
that the method allows optimization with respect to an average initial catalyst activity, a form of activity distribution which can be possibly realized in practice by diluting the catalyst. (d) Next, the problem of optimal catalyst distribution was again considered with the remaining control variables fiied at the same values as in point (c). A set of 21 points linearly interpolated was selected to represent w ( z ) in a continuous form. The optimal initial activity profile resulting from calculations is shown in Figure 5. Correspondingly, F = 0.833. The shape of the w ( z ) curve is qualitatively similar to the optimal profile presented by Ogunye and Ray (1970) (see Figures 8 and 9 in their paper;
a value of F = 0.560 results from our calculations in this case). Particularly, the minimum is located in the region of the reactor where hot spots arise. However, no reduction in the activity is evident near the exit of the reactor, as presented instead by Ogunye and Ray's solution. (e) Finally, an optimization was performed including the operation time T as well. Simple linear expressions were assigned to V&), V3(t),5 ( t ) ;relying on the previous results, U,(t) was allowed to deviate in two steps from 300 K only after 90% of the total operation time had elapsed. x was kept at 7000 kg/m3 h while w ( z ) was interpolated by means of adjacent parabolas. (See Figure 6,7, and 8.) A total of 15 optimization parameters was then required. Starting point for the optimization were the results of point (f) above. Calculations yielded F = 0.895, corresponding to an optimal T = 2.94, almost three times longer than the time interval of operation assumed by Ogunye and Ray. These results can be certainly further refined, provided that more complex expressions for the controls are chosen. Given the approximate nature of the method proposed, however, the present study was intentionally confined to relatively simple forms of control policies allowing easy practical implementations. On the other hand, results offered by a more rigorous approach applied to the same problem do not seem superior.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 131
Conclusions Although approximate in principle, the optimization approach proposed in the present work is well suited to handling control optimization problems of tubular reactors with catalyst decay, the computational efficiency being superior to that of rigorous methods, aa shown by the three constrained optimization problems solved in the previous section. For example 1,at least the form of the analytical solution to the problem is known: by comparison, the proposed approach gives a very well-approximated solution even when simple expressions are assigned to the control policies. Example 2 proves that, for still relatively easy problems, the method reproduces the results obtained by more cumbersome numerical techniques based on rigorous variational principles. In example 3, a complex problem, closer to real industrial situations, is treated. A range of solutions with different degrees of approximation is given to illustrate the flexibility of the method, which easily allows identification and rationalization of trends in the optimal control policies. The possibility of solving the optimal catalyst distribution problem in terms of practically plausible distributions (e.g., one up to a few “slices” with uniform initial catalyst activities) should be emphasized. Also, inclusion of the time of operation among the decision variables is of particular significance. The features discussed should make this approach specially attractive for industrial applications involving large dimensional control problems. Extensions to design problems (e.g., the selection of the optimal length of the catalyst bed) can be immediately accomplished along the same lines. Acknowledgment This work was supported by Italian Consiglio Nazionale delle Ricerche, Roma (Progetto finalizzato Chimica Fine e Secondaria). Nomenclature C = heat capacity, example 3 $E‘ = activation energies for reaction and deactivation rate constants in example 1 F = objective function, eq 1 f, = rate of the ith chemical reaction, eq 2 G = integrand of objective function g, = rate of the jth decay reaction, eq 3 ko, k,,’ = rate constants for reaction and deactivation rate expressions in example 1 P = pressure PM = molecular weight, example 3
q = number of decay equations R = gas constant r = rate of reaction s = number of state equations T = temperature 7+ = upper temperature constraint in example 1 T’= dimensionless temperature in example 2 To= initial temperature in example 1 t = dimensionless time = O / T U, = coolant temperature, example 3 u = vector of control variables VI = velocity of flowing mixture, example 3 v ( t )= vector of inlet controls Vz,V, = inlet temperature and pressure, example 3 w (t)= vector of initial controls; initial catalyst activity along the reactor length x = vector of reactor state variables XA = conversion of reactant A y = vector of catalyst activities z = axial dimensionless reactor coordinate
Greek Letters
B = vector of adaptive parameters, eq 4 0 = time
E
T
= molar feed ratio of HCl to C2H2,example 3
= total time of catalyst operation
x = mass velocity of the feed, example 3 Literature Cited Buui-Ferraris, 0.; Antolinl, G.; Donati. 0. lng. Chim. Ital. 1969, 5 , 1. Buzzi-Ferraris, G. Working party on routine computer programs and the use of computers in chemical engineering, Florence, 1970. Chou, A.; Ray, W. H.; Arls. R. Trans. Inst. Chem. Eng. 1967, 45, T153. Crowe, C. M. Can. J. Chem. Eng. 1970, 4 8 , 576. Delmon, B.; Froment, G. F., Ed. ”Catalyst Deactivation”; Eisevier: Amsterdam, 1980. Degtyarev, G. L.; Sirazetdinov, T. K. Autom. Remote Control 1967, 28, 1842. Gruyaert. F.; Crowe, C. M. AIChE J. 1974, 20,1124. Gruyaert. F.; Crowe, C. M. AIChE J. 1978, 22,985. Jackson, R. Trans. Inst. Chem. Eng. 1967, 45, T160. Lasdon, L. S.;Mitter, S.K.; Warren, A. D. I€€€ Trans. Control 1967, AC-12, 2, 132. Levenspiel, 0.; Sadana, A. Chem. Eng. Scl. 1978, 33,1393. Millman, M. C.; Katz, S., Ind. Eng. Chem. Process Des. Dev. 1967, 6 . 447. Ogunye, A. F.; Ray, W. H. Trans. Inst. Chem. Eng. 1968, 46, T225. Ogunye, A. F.; Ray, W. M. Ind. Eng. Chem. Process Des. D e v . 1970, 9 , 619. Ogunye, A. F.; Ray, W. M. AIChE J. 1971, 17, 43. Ray, W. M., University of Wisconsin, Madison, personal communication, 1982. Reiff, F. K. Ind. Eng. Chem. ProcessDes. Dev. 1981, 20,558. Rosenbrock, H. H., Storey, C. “Computational Techniques for Chemical Engineers”, Pergamon Press; Oxford 1966; p 228. Szepe, S.; Levenspiel, C. Chem. Eng. Scl. 1968, 23,881.
Receiued for review July 6 , 1982 Revised manuscript receiued March 8, 1983 Accepted March 25, 1983
