PROPT Denbigh's System of Reactions
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This page is part of the PROPT Manual. See PROPT Manual. |
Dynamic Optimization of Batch Reactors Using Adaptive Stochastic Algorithms 1997, Eugenio F. Carrasco, Julio R. Banga
Case Study I: Denbigh's System of Reactions
Problem description
This optimal control problem is based on the system of chemical reactions initially considered by Denbigh (1958), which was also studied by Aris (1960) and more recently by Luus (1994):
A + B -> X
A + X -> P
X -> Y
X -> Q
where X is an intermediate, Y is the desired product, and P and Q are waste products. This system is described by the following differential equations:
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx_2}{dt} = k_1*x_1-k_3+k_4*x_2 }
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx_3}{dt} = k_3*x_2 }
where x1 = [A][B], x2 = [X] and x3 = [Y]. The initial condition is
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_0) = [1 \ 0 \ 0]' }
The rate constants are given by
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_i = k_{i0}*exp(-\frac{E_i}{R*T}), i=1,2,3,4 }
where the values of ki0 and Ei are given by Luus (1994).
The optimal control problem is to find T(t) (the temperature of the reactor as a function of time) so that the yield of Y is maximized at the end of the given batch time t_f. Therefore, the performance index to be maximized is
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = x_3(t_f) }
where the batch time t_f is specified as 1000 s. The constraints on the control variable (reactor temperature) are
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 273 <= T <= 415 }
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.Problem setup
toms tSolve the problem, using a successively larger number collocation points
for n=[25 70]
p = tomPhase('p', t, 0, 1000, n);
setPhase(p);
tomStates x1 x2 x3
tomControls T
% Initial guess
if n==25
x0 = {icollocate({
x1 == 1-t/1000;
x2 == 0.15
x3 == 0.66*t/1000
})
collocate(T==273*(t<100)+415*(t>=100))};
else
x0 = {icollocate({
x1 == x1_init
x2 == x2_init
x3 == x3_init
})
collocate(T==T_init)};
end
% Box constraints
cbox = {
0 <= icollocate(x1) <= 1
0 <= icollocate(x2) <= 1
0 <= icollocate(x3) <= 1
273 <= collocate(T) <= 415};
% Boundary constraints
cbnd = initial({x1 == 1; x2 == 0
x3 == 0});
% Various constants and expressions
ki0 = [1e3; 1e7; 10; 1e-3];
Ei = [3000; 6000; 3000; 0];
ki4 = ki0(4)*exp(-Ei(4)./T);
ki3 = ki0(3)*exp(-Ei(3)./T);
ki2 = ki0(2)*exp(-Ei(2)./T);
ki1 = ki0(1)*exp(-Ei(1)./T);
% ODEs and path constraints
ceq = collocate({
dot(x1) == -ki1.*x1-ki2.*x1
dot(x2) == ki1.*x1-(ki3+ki4).*x2
dot(x3) == ki3.*x2});
% Objective
objective = -final(x3);Solve the problem
options = struct;
options.name = 'Denbigh System';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
x1_init = subs(x1,solution);
x2_init = subs(x2,solution);
x3_init = subs(x3,solution);
T_init = subs(T,solution);Problem type appears to be: lpcon
Time for symbolic processing: 0.18833 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Denbigh System f_k -0.633847592419794490
sum(|constr|) 0.000000569070071430
f(x_k) + sum(|constr|) -0.633847023349723090
f(x_0) -0.660000000000000140
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 14 ConJacEv 14 Iter 11 MinorIter 1028
CPU time: 0.046800 sec. Elapsed time: 0.048000 sec.
Problem type appears to be: lpcon
Time for symbolic processing: 0.18641 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Denbigh System f_k -0.633520858635081010
sum(|constr|) 0.000526177451145388
f(x_k) + sum(|constr|) -0.632994681183935650
f(x_0) -0.633848214286006680
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 20 ConJacEv 20 Iter 13 MinorIter 233
CPU time: 0.156001 sec. Elapsed time: 0.161000 sec.
end
t = collocate(subs(t,solution));
x1 = collocate(x1_init);
x2 = collocate(x2_init);
x3 = collocate(x3_init);
T = collocate(T_init);Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Denbigh System state variables');
subplot(2,1,2)
plot(t,T,'+-');
legend('T');
title('Denbigh System control');
