EFFICIENT ALGORITHM FOR OPTIMIZATION OF A MULTIBED ADIABATIC REACTOR SEQUENCE MARSHALL D. RAFAL AND JOSHUA S. DRANOFF Department of Chemical Engineering, Northestern University, Evanston, Ill.
The problem of optimizing the design of a sequence of adiabatic packed-bed reactors has been reviewed, and a practical and efficient digital computation algorithm for the solution of this problem has been developed. Based on a combination of analytic criteria for the optimum and a dynamic programming formulation, this algorithm becomes highly efficient as the number of reactors increases-reaching an effective upper limit in the significant computations required after about five stages. proach to a realistic system, the water-gas shift reaction, is illustrated.
HE
recent development of optimization techniques for large
T and complex engineering systems has made possible for
the first time a unified attack on certain chemical engineering design problems. Nowhere is this better illustrated than in the work of Aris (7) on the optimal design of chemical reactors. By applying the principles of dynamic programming, Aris has analyzed the problem of the optimization of several typical reactor configurations. He has outlined the direct optimization by standard dynamic programming techniques, and developed analytical criteria which lead to alternative graphical determinations of optimal designs in some cases. However, neither Aris nor other workers in this field have indicated the most practical scheme of attack for any specific reactor design problem. I n view of the importance of such calculations, it was decided to investigate a typical problem in some detail, with emphasis on the determination of an efficient approach to solution. The problem considered was the optimal arrangement of a multibed, adiabatic, catalytic reactor sequence. Such systems present an interesting and nontrivial design problem, and they are of commercial interest in several practical processes, including the water-gas shift reaction
CO
+ HzO $ COz + Hz
and the oxidation of sulfur dioxide 2
so2 + 0
2
i3 2 SO8
This paper briefly reviews the nature of this optimization problem, and its solution by dynamic programming tech-
c 0 ._ +
-rz t
e, 0
c
0
0
I
'
' I Tern p e ra t u r e
Figure 1. Concentration temperature plane
----
-.-.-.
Adiabatic paths loci of constant rate Locus of maximum rate
The application of this ap-
niques, as outlined by Ark. The development of an efficient digital computation algorithm combining analytic optimum criteria and a dynamic programming framework is then discussed, and the results of several sample calculations are presented. The Problem
The general problem under consideration is the optimal arrangement of a series of k e d - b e d adiabatic reactors and intercoolers used to carry out a reversible exothermic reaction between fixed composition limits. The design is to be optimal in the sense of maximizing profit. For such reactions this demands a careful selection of operating conditions in order to balance the kinetic and thermodynamic factors involved. This point may be more clearly visualized by reference to Figure 1, which is a plot of the product concentration (mole fraction)-temperature plane. The solid curves represent lines of constant reaction rate, including the zero rate or equilibrium curve. The latter exhibits the familiar decrease in equilibrium concentration as temperature increases, characteristic of such reactions, while the other curves are clearly double-valued with respect to temperature. A locus of maximum rate can be constructed through this family of curves, as indicated. The dashed lines shown represent typical adiabatic paths which might be followed by a reacting mixture in a single batch or plug flow reactor. T o minimize the total reactor volume required to effect the desired conversion, one would obviously wish to operate along the maximum rate locus. However, it is virtually impossible to maintain any real reactor along this curve because of the very special schedule of heat removal which would be required. Furthermore, in actual practice a large fixedbed reactor will behave very nearly adiabatically as long as external cooling is used. Thus, one is practically constrained to carry out reaction along paths closely approximating the adiabatic curves shown on Figure 1. A reasonably close approximation to the maximum rate path can be achieved by carrying out the reaction over a short adiabatic path which crosses the former, cooling the resultant stream to a lower temperature in an interstage cooler, reacting further in a second adiabatic reactor, and so on. Such a sequence can approximate the maximum rate path as closely as desired. I t is clear, however, that there will be an optimum arrangement somewhere between one large reactor and the infinite number of differential reactors and intercoolers required to constrain the operation along the maximum rate locus. VOL. 5
NO. 2
APRIL
1966
129
Frequently less than four or five will yield optimum return, T h e nature of the system concerned is shown in Figure 2 for a three-stage sequence. T h e optimization problem a t hand may now be stated more specifically: What is the optimal arrangement of reactors and intercoolers to make a given feed stream react to a desired final composition? For any fixed number of stages this reduces to the determination of the inlet and outlet temperatures for each reactor. Clearly, the optimal number of stages must also be determined. The criterion function for this problem is the total profit of the operation, based on product and feed values and the costs of equipment. As long as the number of stages remains small, the principal cost item will be the reactor-Le., the catalyst inventory (volume) requirement and the materials of construction. I n general the intercooler hardware and operating costs will not be comparable until the number of reactors becomes large, and these are neglected in the present case. This admittedly represents a special case and specific problems may deviate from this assumption. However, the techniques discussed below may be easily modified to apply to other more general situations. Before the actual optimization procedure is considered, the system will be defined in more quantitative fashion. T h e basic equations which describe a fixed-bed, plug-flow, adiabatic reactor are given below. The first is the standard differential material balance and the second is an accompanying energy balance.
Qopodx = rdV or
dx = r(x, T ) dB
Po -
or
Equation 1 may be substituted into Equation 2 and the result integrated to relate temperature and concentration within any adiabatic reactor, given inlet conditions. For present purposes i t is assumed that the heat of reaction and specific heat of the reactant fluid are independent of temperature and composition. This makes the adiabatic temperatureconcentration relation linear
T = Tf
+ H(x-
~ 2 )
(3)
These assumptions will generally be close approximations to reality. They may be relaxed, if appropriate data are available, without significantly affecting the structure of the numerical calculations. Now the holding time in any reactor in a multibed sequence may be written in terms of a n integral by combining Equations 1 and 3:
(4)
ef thus depends on entrance and exit compositions, reactor entrance temperature, and the form of the specific rate equation Equation 4 applies to each reactor in the sequence. For the interstage coolers it is clear that I
130
I & E C PROCESS D E S I G N A N D DEVELOPMENT
(5) while xi =
(6)
%+I’
Equation 6 follows from the fact that there is no change in composition in the cooler. [The notation used here corresponds to that of Aris (7).] The profit function for this problem may be formulated as the gain in value of the process stream due to reaction minus the reactor costs. Since the latter will be proportional to the required volume or total holding time, this function can be written as in Equation 7 for an N-stage system. 3
(7) Here u represents the value of the process stream per mole of desired production, while Xv represents the reactor hardware plus catalyst cost per unit volume. T h e total profit, P.,,, may also be written in terms of individual stage profits in the following manner: N
N
--.,
r
where use has been made of Equations 4 and 6. T h e problem is thus to maximize Pay, given xl’ and x,,,, by proper choice of T,v, TAT-1, . . ., T I , and XN- 1, x,.+ 2, . . ., x l . Since the product, PoQou, is assumed constant, one may equally well seek the maximum of PN, defined as PN/PoQu. Consideration of the form of this function will verify that the maximization is equivalent to the minimization of the total nominal holding time for the sequence. Subsequent treatment will be concerned with this minimization. A word is perhaps appropriate regarding the fixing of xl’ and x.,. in this analysis. I t is felt that this will be representative of many industrial cases in which it is desired to produce a product of specified quality from a fixed feed stream. Problems will certainly arise in which both end compositions will not be fixed, as when one considers the optimization of a larger processing scheme in which such a multistage reactor must be integrated with other units. However, such generalized problems are beyond the scope of the present investigation.
Methods of Solution
Having now specified in some detail the problem under consideration, let us examine methods for obtaining the optimal policy. Clearly this is a natural problem for the application of dynamic programming, since it represents a multistage decision process with no feedback. Hence one can apply Bellman’s (2) oft-quoted principle of optimality directly to find the solution. The technique in such a case is to consider first the last stage of the process-i.e., stage 1 in Figure 2. Regardless of what precedes stage 1, the over-all system can be optimal only if stage 1 operates optimally for any feed supplied to it from previous stages. One may thus determine a n optimal policy-Le., a n inlet temperature-for the Iast stage for every possible feed composition and tabulate this information in some convenient form. Stages 1 and 2 may then be considered together and the optimal policy for the pair determined for all possible feed states to stage 2. This procedure is described precisely by the following symbolic representation of the principle of optimality :
Combining Equations 13, 15, and 16 yields
Maximum profit for two stages = max [profit for stage 2
+ (max profit for
stage 1 using feed produced by stage 2)]
(9)
T h e previously determined one-stage optima are used in this calculation. When the two-stage optima have been determined, a second convenient table may be constructed and the entire process repeated for a three-stage system. Equation 9 is again applied, with stages 1 and 2 grouped together as a single pseudo-stage whose optimal policies are known. This general procedure may be extended to any number of stages in the same manner. Much has been written about the advantage of such a dynamic programming attack on this type of problem compared to direct numerical search, and little need be added here. Suffice it to say that dynamic programming requires much less calculation to find the optimal policy in any specific case lvhile furnishing results useful for analysis of many other related and subcases. These remarks are especially apropos when the optimal policy requires only two decisions a t each stage-namely, the inlet temperature and exit composition (or exit temperature or stage holding time). At this point a calculation algorithm based on a direct application of dynamic programming as described above could be formulated. However, it is possible in the present case to proceed further by analytical techniques to obtain direct criteria for the optimal policy. The development shown below follows from the work done originally and independently by Aris ( 7 ) and Horn ( 3 ) . T h e approach is illustrated for a two-stage sequence for present convenience. The function to be maximized may be written explicitly from Equation 8:
x
1;
[I
Since
X?
- r(x, T I
+ H(x -
and x i ’ are considered known, and
1
dx
XI))
(10)
x2’ = xl,
The necessary conditions for P z to be maximized are found from the calculus.
O n applying these conditions the following results arise :
where
Similarly,
Finally,
r
\
l
Equations 13, 15, and 17 now represent criteria which must be satisfied by the optimal policy. The integrals, J , are to be evaluated along adiabatic paths. T h e rate match indicated in Equation 17 is possible because of the double valued nature of the rate function. This approach may be extended simply to the case of an lV-bed sequence with the results shown in Equations 18 and 19.
J(xi, T i ) = 0 ,
i
=
1, 2 , . . ., .4‘
(18)
A calculational algorithm based on these criteria can be and has been formulated ( 3 ) . This involves trial and error search over possible inlet conditions, as follows. An inlet temperature, T.V, is chosen and Equations 18 and 19 are applied successively until a value for X I ’ is determined. This process is then repeated until the latter agrees with the xi’ value fixed by the problem statement. Such calculations are best performed numerically because the analytical evaluation of integral J is, in general, impossible. Thus far two methods of solving the optimization problem have been discussed : (1) direct dynamic programming and ( 2 ) direct search based on analytical criteria for the optimal policy. These two approaches have distinct characteristics which must be considered in the choice of a working algorithm. Dynamic programming requires the calculation of many suboptimal cases in the solution of any specific problem. Although this may involve considerable computation, the information so generated can be utilized in the determination of optimal policies for related problems involving fewer stages or alternative feed conditions. The direct search technique, on the other hand, will lead after several trials to the desired optimal solution. Fewer calculations are required than by dynamic programming, but little if any uniformly spaced, systematic information regarding suboptimal cases is generated in this way. Moreover, this approach requires numerical evaluation of the integral J , which involves the temperature derivative of the rate function. If experimental rate data are to be used, they must be known with higher accuracy in order to permit such differentiation. Both of these techniques require an amount of calculation roughly proportional to the number of stages in the cascade under consideration. If one were faced with the solution of only one such optimization, clearly the direct search approach would be the logical choice. However, this is rarely the case in design considerations. More likely one would seek to investigate many problems clustered about some region and the type of information produced by a dynamic programming solution would be of most value. Because of this situation, it was decided to develop a third algorithm combining the best features of the two just described. This method is most clearly distinguished by the fact that it has an upper limit of significant computation with a n increase in the number of beds. This differs from the other two schemes, where computation is roughly proportional to the number of stages. The details of the resultant method are presented below. Since the ideas involved may be of use in handling similar problems, the structure of the algorithm is described in some detail. VOL. 5
NO. 2
APRIL 1966
131
After careful consideration of the calculations to be performed and the previously described methods of solution, it was decided to construct an algorithm based essentially on the dynamic programming approach but utilizing a simple numerical search routine for single-bed optima and the rate match criterion of Equation 17 between stages 1 and 2. Furthermore, special care was taken to monitor and identify all calculations in order to avoid wasteful repetition of earlier calculations during the later stages of the program. Two important subprograms in this algorithm are the singlebed optimization program and the rate-matching program. These are described first. The function of the single-bed optimization routine is to determine the optimal policy for a one-bed reactor operating between any two specified concentration limits. With these limits fixed this amounts to a single variable search over reactor inlet temperature in order to find the minimum holding time (or reactor volume). There is a natural constraint on such a search because the reactor outlet temperature must lie below the equilibrium temperature corresponding to the outlet composition. The maximum temperature may be calculated from the equilibrium equations (Equations 20 and 22, with 7 = 0) if the composition is known. A trial outlet temperature is selected which is less than the maximum by some specified amount. The corresponding inlet temperature is then found using the energy balance, Equation 3, and the holding time is calculated by a Simpson's rule integration. This process is repeated with successively lower values of outlet temperature until the holding time passes through a minimum, a t which point the calculation is backed up two trials and the process is repeated using a smaller decrement until the temperature is determined within desired accuracy, This type of procedure was always rapidly convergent. The function of the rate-matching subprogram is to determine the temperature such that the rate of reaction a t the end of bed two is the same as that a t the inlet to bed one (see Figure 2). If the temperature and stream composition a t the inlet of bed one are known, the rate a t that point can be calculated by the rate equation, Equation 20. The rate-matching program then finds the higher temperature which yields the same rate of reaction a t the same composition. This is accomplished by direct search over temperature, proceeding with trial temperatures which are specified increments above T I . A
Figure 2.
Typical three-bed system
Temperature and concentration of feed to b e d i Temperature and concentration in effluent from b e d i ci'.
Ti, ci. Ti',
Ferd
2nd triol for x,
I
\
1st triol for XI/
procedure similar to that described above is used to converge to the required temperature, With these two subprograms in mind, one can consider the general flow of the calculations. Initially, the composition spread between known feed and desired product stream compositions is divided into a specified number of equal intervals. An optimal two-bed policy is then found between each of the intermediate compositions so formed and the fixed outlet composition. (Such a policy consists of the specification of T2, T2',and T I ,which fixes the system.) The two-bed optimization is performed utilizing the previously described subprograms. Initially, the concentration range between X I ' and the x 2 value in question is split into another equally spaced grid, as illustrated in Figure 3. Starting from the right (outlet) end, the first of these grid splits is considered. Bed one is optimized using the single-bed optimization routine, and bed two is subsequently optimized by means of the rate match routine. This procedure is repeated for each grid split and the minimum two-stage holding time is thus determined for each x z value. At this point the optimal policy decisions (temperatures) and the corresponding minimum holding times (maximum profits) are recorded. One may use as fine an initial concentration grid as desired for these calculations. A more efficient search (such as the Fibonacci technique) might have been used here, but a t a sacrifice of the symmetrically spaced data generated by the present technique. The significance of these data is discussed below. The process is made efficient by the recognition that many of the single-bed calculations required are duplicated a t several points. This is easily seen by considering, for example, a fictitious case in which the terminal concentration is zero while the inlet concentration, X Z , varies from 0 to 25 units. If the main concentration grid spacing is taken as a 5 while the intermediate grid also contains 5 points, the values of intermediate concentration which would be tested are those shown in the body of Table I. Single-bed optimizations would be run between each of these values and the exit concentration, which is zero. I t is clear from this table that a number of these single-bed cases have the same inlet composition; in fact, there are not 25 distinct cases in the 5 X 5 table, as might be expected, but only 14. If the optimal values of holding time and temperature for each case are stored, they may be used whenever repeated cases occur. The presence of such repetition may be signaled by associating with each intermediate concentration a key number (equal to the product of the indices) which locates the entries in Table I. This device permits substantial time savings in the determination of the two-bed optima. When the complete array of two-bed optimizations has been completed in this way, the three-bed system may be considered : Using the same set of intermediate concentrations as before, the system is considered as bed three plus a pseudo-
Table 1.
1
I
JQ
1
- initial X-
qrid
triols f o r ' x , associated with 1st t r i o l for x z
secondary q r i d
Concentration grids for two-bed optimization Figure 3. calculations 132
Monitoring Repeated Cases
Product
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
5 4 1 5 4 2 10 8 3 15 12 4 20 16 5 25 20 a I. Intermediate grid index. is the same f o r all matching cases.
3
2
3 2 6 4 9 6 12 8 15 10 J . M a i n grid index.
7 1 2 3 4 5 Product I J
bed consisting of beds one and two combined. T h e third bed is optimized between the X Q and x 3 ’ limits in the grid, using the single-bed optimization routine. The optimum two-bed arrangement for the remaining composition change is then found from the previously determined table of two-bed optima. i5’here necessary, linear interpolation between previously calculated cases is utilized for the latter determination. Threebed optima and corresponding optimal policy decisions are thus determined over the desired array of concentrations. T h e rate match criterion is not used in dealing with the pseudobed system because interpolation for appropriate inlet temperaturrs \vould be necessary. However, the inlet temperature variation is so irregular that this could be accomplished only by use of a very fine concentration grid, which would significantly increase the computations. For subsequent optimization of larger systems the process is repeated using the previously determined (n - 1)-bed optima in the solution for the n-bed system, according to Equation 9. I n all such computations, the same concentration array is used. Inasmuch as the same single-bed optimizations must be computed for any ne\v bed added beyond the second (since the
concentration array is the same), the possibility of repeated calculations again arises. By keeping a record of the calculations already made, using the technique mentioned above, the repetition is avoided with a great saving in calculation time. This makes the extension of results to any number of beds beyond three a very simple and efficient process, and more than any other feature enhances the efficiency of the algorithm. I t permits the extension of the program to large numbers of reactor beds with little increase in computing time because the remaining effort consists largely of table look-ups and linear interpolations. The above discussion has presented the main features of the algorithm. I t i s essentially a dynamic program combined with a simple single-bed optimization search routine and a rate-matching routine based on analytic criteria for the optimum. T h e other features which greatly enhance its operation are the avoidance of repetitive calculation by careful monitoring of calculations to be performed and the use of linear interpolation (among previous values) to find optimal performance of pseudo-beds. Results and Discussion
Table II. Component
co co2
H20
HP
The performance of the algorithm was tested by carrying out calculations for a case approximating a real system. I n particular, attention was focused on the water gas shift reaction, for which sufficient kinetic and physical property data were found in the literature. T h e following rate and equilibrium expressions were obtained from a recent article by Moe (5),dealing with a low temperature (400’ to GOO0 F.) catalyst:
Feed and Product Stream Compositions Feed Mole Product Fraction M o l e Fraction 0,0463 0.0102 0,0520 0,0881 0.6566 0,6205 0.2451 0.2812
Table 111.
One-, Two-, and Three-Bed Optimal Designs A. One-Bed Optimal Designsa
XI
Ti
T I’
0.0138 0.0174 0.0210 0.0247 0.0283 0.0319 0,0355 0,0391 0.0427 0.0463
865.8 879.3 884.8 886.3 885.8 883.2 880,7 876.2 871.7 867.1
872.4 892 4 904.4 912.4 918.4 922.4 926.4 928.4 930.4 932.3
B. X*I
X2
0.0138 0.0174 0.0210 0.0247 0.0283 0.0319 0.0355 0.0391 0.0427 0.0463
=
T2 896.8 930.3 972.8 1013 994.8 1023 1051 1077 1103 1128
XI
0.0120 0.0131 0.0146 0.0160 0.0156 0.0167 0.0178 0.0189 0.0200 0.0210
x3
0.0138 0.0174 0,0210 0,0247 0.0283 0.0319 0,0355 0.0391 0.0427 0.0463
xg’
=
x2
0.0113 0.0131 0.0146 0.0189 0.0192 0.0210 0.0203 0.0218 0.0232 0,0247
el x
Two-Bed Optimal Designsa Tz TI Ti
900.1 938.2 984.6 1029 1018 1051 1083 1114 1144 1174
855.1 861.2 868.6 873.9 872.6 876.6 878.7 880.7 882.8 884.8
858.4 866.4 876.4 884.4 882.4 888.4 892.4 896.4 900.4 904.4
C. Three-Bed Optimal Designsn T3 83
T3
882.6 930.1 969.6 1050 1068 1106 1099 1129 1159 1187
887.1 938.0 981.3 1060 1084 1125 1126 1160 1194 1226
704
0.7705 1.2191.527 1.761 1.950 2.110 2.250 2.376 2.490 2.597
0.4763 0.5281 0.5649 0.2960 0.3871 0.3569 0.4708 0.4367 0.4038 0.3728
81
e2
0,4472 0.6522 0,8772 1.062 1.012 1.144 1.254 1.354 1.444 1.527
0.3156 0.5280 0.5651 0.5607 0.7352 0.7086 0.6760 0.6410 0.6057 0.5713
el
+ e2
0.2675 0.6361 0.8571 1.296 1.325 1.442 1.395 1.481 1.557 1.623
el
el
+ e2
0.7628 1.180 1.442 1.623 1.754 1.852 1.930 1.995 2.050 2.098
+ o2 + e3 0.7439 1.164 1.422 1,592 1.712 1.799 1.866 1.918 1.960 1.996
xi’ = 0.0102 for all cases.
VOL. 5
NO. 2
APRIL
1966
133
1. Rate equation.
Y
2.
k =
=
k
[
XCOXHZO
xC0,xHz - __
K
--8820
Rate constant.
e16.1e
(21)
T 820
3. Equilibrium constant.
K
= e--4*33e
(22)
Other physical property data were obtained from standard tabulations (4). The resultant value of H was found ,to be
1810° F. Although the relations given in Equations 20 to 22 were for the low temperature range indicated, they were assumed valid over the entire range of this work. Thus, the numerical results obtained may not be valid for any real reactor system. They should, however, be of the correct order of magnitude for a realistic system and conclusions derived from their use should be generally correct. For purposes of illustration, the feed and product stream compositions listed in Table I1 were selected. The entries represent mole fractions and correspond to a total conversion of 78% based on the CO. This approximates a realistic case for this system (5). The results of actual optimization calculations for one-, two-, or three-bed reactor systems are presented in Table 111, A, B, and C, respectively. The minimum holding time, 8, for the entire system is shown in each case along with the corresponding inlet and outlet compositions and temperatures and individual stage holding times. For the three-bed case the intermediate temperature and the individual holding times for the combined pseudo-bed are not given explicitly. These might be approximated closely by interpolation among the values in the two-bed table. Alternatively, one might repeat the two-bed optimization with the exact inlet composition (xa' = xg) corresponding to the threebed optimal design. These tables might be used in the design of one-, two-, or three-bed optimal systems over the indicated composition ranges. In carrying out these computations, the reactor inlet compositions were divided into ten intervals. The intermediate grid corresponding to each of these points was further divided into tea intervals. The data shown here have been selected merely to illustrate the type of information which can be generated and are not to be construed as actual optimal designs for the real system involved. More complete data are presented elsewhere (6). From a study of the single-bed optimal designs generated in this work, some light has been shed on a point of generalization frequently of interest for such reactors. As exemplified by Moe ( 5 ) , attention is sometimes focused on determining the optimum approach of exit temperatures to the theoretical maximum limit given by the equilibrium curve a t the exit composition of interest. The present work has shown that there is no clearly defined optimum AT for any general system. Rather, for fixed outlet composition AT tends to increase as inlet composition increases, and decrease as inlet composition falls. In other words, it is better to operate closer to the equilibrium curve a t the end of the reactor than to cause the beginning of the reactor to operate a t very low temperatures. Furthermore, for a fixed conversion interval ( x i - x i ' ) , A T is higher a t the lower levels of composition (x). The interpretation of the actual results was not, however, the prime purpose of this work. Rather, major emphasis has been on the performance characteristics and efficiency of the algorithm. Table IV shows results of some tests of the sensitivity 134
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
of the time requirement (all times represent actual processing time on the digital computer of programs generated from the Fortran I1 system) to variations in the control parameters. The results apply to a two-bed optimization. As shown, the computation time varies directly with the number of intervals taken in the concentration grids, but only slightly with changes in the initial temperature increments used in search procedures of the single-bed optimization and rate matching routines. The final increments used in these programs corresponded to 2' and 1' F., respectively. The true efficiency of the algorithm is well shown by the data of Table V, where the times required for up to eight-bed design problems are indicated. These results indicate clearly that very little incremental time is necessary to optimize multibed systems beyond that required for three- or four-bed cases. This is a direct consequence of the care taken to avoid repetitive calculations, as pointed out earlier, and the use of linear 1)-bed optima during interpolation in searching out the (n an n-bed optimization study. This feature makes the algorithm especially attractive for problems involving four or more beds. The economies of this computation compared to a direct or dynamic program, where computation is roughly proportional to the number of stages, should be apparent. A number of important questions arise during the construction of such an algorithm which depend significantly on the particular system being considered. These are enumerated here as a guide for use in adapting this approach to other systems.
-
How sensitive is the rate match to temperature? This determines not only the fineness of the temperature intervals used in searching for matching rates, but also the advisability of using the rate match criterion (Equation 19) beyond the first two beds. How sensitive is a one-bed optimization to temperature? This point guides the selection of initial and final temperature steps used in converging to the optimum. Can linear interpolation be used to find holding times of pseudo-beds? The answer here depends on the nature of the result for the concentration array used. This decision influences the size of the concentration grid necessary for good accuracy within reasonable computing time. What primary and intermediate concentration grids should be used? This can best be answered by subjecting the program to various tests and observing the results with respect to computation time as well as accuracy.
Table IV.
NB" 2 2 2 2 2 2 a NB.
NJ 10 20 10 10
Computer Use and Program Parameters
iVI
Computer Time, Min.
INT AT, AT2 21 10 100 2.77 10 21 10 100 5.38 20 21 10 100 4.05 10 41 10 100 4.78 10 10 21 20 100 2.24 10 21 10 10 200 2.77 Number of beds. N J . Number of points i n concentration table. iVI. Number of intermediate concentrations. I N T , Number of points i n holding time integraiion. A T I . Initial temperature interoal associated w i t h one bed optimization, F. A T * . Initial temperature interval asJociated w i t h rate match, F. Table V. ivo. of Beds 2 3
5 8
10
Beds Optimized vs. Computer Time Total Computer Time, Min. 5.38 13.76 15.90 16.73
Several other related questions arise during the design of the algorithm, such as the optimum number of points to be used in the holding time integrations, which are best answered by simple experiment with the program. Careful attention to these and the four points enumerated above will repay Ihe designer of such a program in terms of final efficiency and reliability in his algorithm.
AH, J k
K
Pi p.\ QO
Conclusions
This paper has sought to describe in sufficient detail the construction of an algorithm for the problem a t hand. A careful combination of direct dynamic programming iechniques, analytical criteria, and methods for monitoring and avoiding repetitive calculations has resulted in a program of high efficiency. Extension of these approaches to other more general problems should be feasible and easily made.
r
T U
XU
V X
0 Po
Acknowledgment
Nomenclature = specific heat of reactant stream, energy/(mole =--- ‘ H R ,
0
lr Z
1
literature Cited
The authors acknowledge with thanks free computer time made available by the Northwestern University Computing Center and a n NDEA Title IV fellowship (to MR) which made this work possible.
Cp H
heat of reaction, energy/(mole of product formed) moles dx, [(volume of catalyst) (time) ( 0 F.) -I = specific rate constant, moles/(time) (volume of catalyst) = equilibrium constant for reaction = profit from ith stage, dollars = total profit from a n N-stage system, dollars = volumetric flow rate of feed a t standard conditions, volume/ time moles = rate of production of product, (time) (volume of catalyst) = temperature, ’ F. = volume of process stream, dollars/(mole of desired product) = cost of reactor, dollars/(volume of catalyst) = catalyst volume, volume = mole fraction of product = nominal reactor holding time, time = molar density of feed a t standard conditions, moles/ volume =
’ F.)
(1) Aris, Rutherford, “Optimal Design of Chemical Reactors,” Academic Press, h’ew York, 1961. (2) Bellman, Richard, “Dynamic Programming,” Princeton University Press, Princeton, N.J., 1957. (3) Horn, Frederick, 2.Eleklrochem. 6 5 , 295-303 (1961). (4) Hougen, 0. A., Watson, K. M., Rogatz, R. A., “Chemical
Process Principles,” Part I, 2nd ed., Wiley, New York, 1958. (5) Moe, J. M.,Chem. Eng. Progr. 5 8 , 33-6 (1962). (6) Rafal, Marshall, M. S. thesis, Northwestern University, Evanston, lll., 1964.
F,
RECEIVED for review February 12, 1965 ACCEPTED November 15, 1965
G P
INLET-GAS HUMIDIFICATION SYSTEM FOR AN ELECTROSTATIC PRECIPITATOR S EN
-I
C H I
MASUDA
,
Department of Electrical Engineering, University of Tokyo, Tokyo, J a p a n
T0SH I 0 0N I SH I A ND H I R0S H I
SA IT0
,
Onoda Cement Manufacturing Go., Tokyo, J a p a n
A system for rapid evaporation of mist was developed for inlet-gas humidification in an electrostatic precipitator, which can evaporate a large quantity of water in a small spray chamber in the temperature range of 150” to 200” C. An improved type of multiple twin-fluid nozzles, developed for this purpose, can produce extremely uniform and small particles of mist in any amount, with the aid of an equal magnitude of high fluid resistances inserted into its water branches. Confinement of mist within the narrow space of a spray chamber was made possible by the mist thermal repulsion action of the wall surface kept a t elevated temperature. The mist lifetime measured in an actual spray chamber can b e explained by the theory, if a new type of mist average diameter is utilized.
EVERSE
ionization has long caused major trouble in electro-
R static precipitation, but it can be prevented by reducing
the electrical resistivity of dust within the precipitator below a critical value of about 2 X 1010 -v 1 X 1011 ohm-cm. with the aid of inlet-gas humidification (78). The inlet gas may most economically be humidified by water atomization within a spray chamber installed immediately ahead of the precipitator. However, the temperature of the inlet gas in most cases is in such a low range (about 150’ to 200’ C.) that mist usually evaporates very slowly. As a result, mud or slurry often builds up inside the spray chamber be-
cause of the poor evaporation of mist, which may sometimes make necessary shutdown of the whole process. Complete evaporation may be possible, if the size of the spray chamber can be greatly enlarged or the mist particle size can be reduced to any desired amount. The present practice is to atomize water within a very large spray chamber, often larger than the precipitator itself, with the use of single-fluid nozzles, featured by very low operating cost as well as large mist size. This may be the most simple and economical solution in the case of large inlet-gas volume, although it can be applied only where adequate space is available for the installation of such a VOL. 5
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