0098-1354/94 $6.00+0.00
Pergamon Press Ltd
Computers chem. Engng, Vol. 18, Suppl., pp. S415-S419, 1994
Printed in Great Britain
OPTIMIZATION OF DISCRETE CHARGE BATCH REACTORS
V. S. VASSILIADIS, C. C. PANTELIDESl and R. W. H. SARGENT
Centre for Process Systems Engineering, Imperial College,
London SW12BY, U.K.
ABSTRACT
A special class of multistage optimal control problems arises in the case of batch reactors fed
by discrete instantaneous additions of raw material. The underlying differential-algebraic equations (DAEs) remain unchanged throughout the time horizon of interest, but the instantaneous
additions of material are impulsive inputs that result in discontinuities in the state variables.
Optimization parameters can include, among others, the amount of each charge and the reaction
stage duration following each addition.
In this paper, the problem of optimizing the operation of discrete charge batch reactors is addressed within a general framework that is independent of kinetic mechanisms and nonidealities.
No approximation need be involved as the solution procedure is based on a general purpose robust
algorithm for dynamic optimization problems.
KEYWORDS
Discrete charges; Batch reactors; Multistage optimal control; Differential-algebraic equations.
INTRODUCTION
Recently Levien (1992) proposed a methodology based on kinetic insight that optimizes the product distribution obtained from competing chemical reactions taking place in parallel. Kokossis
and Floudas (1990) have considered the same problem as a case study for their superstructure
approach for reactor networks based on continuous stirred tank reactor (CSTR) modules. This
results in mixed integer nonlinear programming problems (MINLPs). An alternative approach
is that of Achenie and Biegler (1988) who examined a continuous parameter superstructure for
reactor networks resulting in optimal control problems. Unlike the Floudas and Kokossis method,
this method cannot account for instrumentation costs arising from existence or non-existence of
units.
In any case, the optimization of a single batch reactor, or a fixed batch reactor train, does not
require integer decisions. Therefore, as in the work of Achenie and Biegler, an optimal control
model is the natural choice. In this way, any type of kinetics, non-idealities both in mixing and
operation, as well as more detailed models involving partial differential equation models can be
tackled very effectively.
1 Author
to whom all correspondence should be addressed.
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FORMULATION
The formulation of discrete-charge batch reactor problems is demonstrated by an example.
Here, the Trambouze reaction scheme as tested by Levien (1992) and Kokossis and Floudas (1990)
is examined in detail. The reaction scheme is shown in Figure 1, and the reaction kinetics lead
to the following differential equations:
(1)
where Cj denotes the molar concentration of component j, and k. are reaction rate constants,
kl 0.2 min-I, k2 25 mol m-3, and k3 = OAx10-3 mol-I min-I.
=
=
The reactor is loaded initially with pure A, which corresponds to the initial condition:
CA(O)
= 1000.0molm-3 ;
CR(O)
= 0.Omolm-3 ;
Cs(O)
= O.Omolm-3;
CT(O)
= 0.Omolm-3 :
(2)
Fig. 1. 1'rambouze reaction scheme
Further amounts of fresh feed may be added instantaneously during the reaction. If we denote
the volume of the i-th such additional amount by d;, the following instantaneous mass balances
hold at the time of addition, t;:
V;CS(tt)='V;-ICS(t;);
V;Cr(tt)=V;-ICr(t;);
V;=V;-I+d;;
(3)
N
i
= 1,2"., ,N; Ld; = 0,1
;=1
where V; denotes the volumetric holdup in the reactor following the ith addition, and CAO the
feed concentration of A (1000 mol m- 3 ). Here it is assumed that the first feed injection occurs at
time 0 (i.e. tt 0) and that the reactor is initially empty (Vo 0).
=
=
The desired product is R. Two different objective functions were maximized separately, namely
the overall fractional yield 4>, and the overall product yield y, respectively defined as:
(4)
An alternative way of operating the reactor is to have a. continuous feed of flow rate u instead of
discrete charges. In this case, the model changes to:
veA == -UCA + uCAO - l ' [kl C A
veR = -u Cn + l' kl C,4
ves = -tLCS + l' k2
ver = -u Cr + If k3 C~
+ k2 + k3C~J
V==u
0:$ u(t):$ 10;
V(t,) = 0.1;
CA(O) = 1000.0
Cn(O) = 0.0
Cs(O) = 0.0
Cr(O) = 0.0
"(0) = Vo
0:$ t :$ t,
(5)
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In this continuous case, the feed rate u was treated as a continuous control variable, represented
by piecewise linear approXimations over 10 elements of variable length. The initial volume in
the reactor was set to 10-5m 3. The control profile was bounded between 0.0 m 3 /min and 10.0
m 3 /min, and was initially set to a uniform value of 0.01 m 3 /min. The cOntrol element sizes were
bounded between 2.0 min and 20 min, initialized as 3.0 min each. However, for the optimization
of the overall product yield, the lower bound of the penultimate control element was set to 0.3
min in order to allow a steep jump in the feed level.
In all cases, the optimization of the overall fractional yield and that of the overall product yield
result in the same optimal final time and control profile up to the point where the reactor volume
reaches 0.1 m 3 • The optimization of the overall fractional yield termjnates the reaction at that
•point, whereas the optimization .of the overall product yield allows it to proceed further up to the
point where the reactant concentration reaches zero, without any further addition of material.
RESULTS
Vassiliadis (1993) considers the optimal solution of a special class of multistage optimal control
problems of the form:
(6)
subject to:
j<k)(X(t),X(t), u(t), t) = 0;
t E [tk-h tk];
k
= 1,2, ... , N
(7)
where k denotes the stage number. It should be noted that the set of variables in the problem remains the same from one stage to the next, but the equations may vary. General initial conditions
of the form:
C(x(to),x(to), u(to), to) = 0
(8)
as well as junction conditions:
(9)
and end-point constraints:
(10)
can also be accommodated. The algorithm has been implemented in a general computer code,
DAEOPT (Vassiliadis, 1992). It can easily be seen that the example of the Trambouze reaction
scheme presented in the previous section is a special case of this type of problem.
Three discrete charge schemes were examined, with 2, 5, and 10 charges respectively. The optimal
values of the optimization parameters are listed in Table 1. Figures 2(a), 2(b), and 2( c) depict the
trajectories for C A and C R for each charge scheme respectively. The results of the continuous feed
scheme are depicted in Figures 3(a) and 3(b) showing the flowrate and state profiles respectively.
Several optimization related indices arc listed ill Table 2. Reported there are the CPU time in
seconds on a SUN SPARC IPX workstation, the average cost ratio of a sensitivity integration to a
state-only integration, the number of quadratic programming subproblems (QPs) solved, and the
number of line searches (LSs) required within the sequential quadratic programming algorithm
used. The integration tolerance used to control the local error of the trajectory at each integration
step was set to 10- 7 • The optimization tolerance used to control the satisfaction of constraints,
bounds, and optimality with respect to the Lagrangian function was set to 10- 5 •
CONCLUSIONS
In this work the optimal design and operation of semibatch reactors is considered and modelled
as a special class of multistage optimal control problem. This approach glla~antees optimality (at
least locally) and allows the description of sllch systems in a.ll detail required.
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Table 1. Discrete charges-Optimal solution characteristics
DISCRETE CHARGE SCHEME
Optimal Reaction Stage Durations (min)
Kokoui& 8 Flouda&
Thi6 Wor*
Levien
Number of charges:
2
4.98329
3.61465
1
2
3
4
5
6
7
8
9
10
5
4.99042
3.59945
2.99764
2.60574
2.29831
10
5.00099
3.62580
3.00489
2.59224
2.27831
2.07455
1.88947
1.71083
1.59454
1.49857
2
4.891
3.601
10
4.981
3.601
3.007
2.607
2.310
2.079
1.893
1.739
1.609
1.497
10
6.691
3.504
2.352
1.767
1.413
1.178
1.009
0.883
0.784
0.706
3.372
3.754
5.055
6.563
8.258
10.133
12.188
14.421
16.833
19.423
10.546
9.593
9.857
9.942
9.979
10.000
10.011
10.019
10.024
10.028
Optimal Charge LeveJs (%~
47.33
52.67
1
2
3
4
5
6
7
8
9
10
12.47
13.95
18.66
24.43
30.48
3.371
3.818
5.107
6.645
8.230
10.17
12.28
14.26
16.85
19.27
.(7.320
52.679
Table 2. Optimization performance indices
Number of charges:
~opt
tf
CPU
Sens. CPU
-sr.
#QPs
# LSs
Yopt
t/
2
0.44751
8.59795
23.6
4.9
6
8
42.7636
11.96403
DISCRETE CHARGE SCHEME
Levien
This Work
2
10
5
10 continuous
0.49906 0.447513 0.4895
0.47620
0.48954
8.491 25.323
16.4915
25.2702
37.1713
302.3
640.2
154.3
9.2
8.8
12.2
17
30
10
33
49
17
46.3276
47.0672
45.2233
42.40899
20.41069 29.90683
K okossis
Floudas
10
0.4872
20.287
134.7 (VAX 3200)
fj
.0.-. . . . . .
~
~
(a) 2 Charges
(b) 5 Charges
(c) 10 Cha.rges
Fig. 2. Concentration profiles of A and R for discrete charge schemes
_ ....-
European Symposium on Computer Aided Process Engineering-3
....,-r--.----..----.---,,.....
--
8419
.
(a) Feed fiowrate
u(t)
Fig. 3. Optimal profiles for continuous feed case
The results obtained for the Trambouze reaction example considered, agree very well with the
exact results given in Levien (1992).
REFERENCES
Achenie, L. K. E., and L. T. Biegler (1990). A Superstructure Based Approach to Chemical
Reactor Network Synthesis. Comput. Chem. Engng, 14 (1), 23-40.
Kokossis, A. C., and C. A. Floudas (1990). Optimization of Complex Reactor Networks-I.
Isothermal Operation. Chem. Eng. Sci., 45 (3),595-614.
Levien, K. L. (1992). Maximizing the Product Distribution in Batch Reactors: Reactions in
Parallel. Chem. Eng. Sci., 47 (7), 1751-1760.
Vassiliadis, V. S. (1992). DAEOPT, User's Manual, llersion 1.0. Centre for Proc. Syst. Eng.,
Imperial College, London, U.K.
Vassiliadis, V. S. (1993). Computational Solution of Dynamic Optimization Problems with General Differential-Algebraic Constraints. Ph.D. Thesis, University of London, U.K.