A HYBRID METHOD FOR THE OPTIMAL CONTROL OF CHEMICAL PROCESSES 1998, E F Carrasco, J R Banga
Case Study II: Plug-Flow Tubular Reactor
This case study considers a plug-flow reactor as studied by Reddy and Husain, Luus and Mekarapiruk and Luus. The objective is to maximize the normalized concentration of the desired product.
Find u(t) to maximize
J = x1(tf) |
subject to:
| = (1−x1)*k1−x1*k2 |
| = 300*((1−x1)*k1−x1*k2)−u*(x2−290) |
where x1 denotes the normalized concentration of de desired product, and x2 is the temperature. The initial conditions are:
x(t0) = [0 380]′ |
The rate constants are given by:
k1 = 1.7536e5*exp(− |
| ) |
k2 = 2.4885e10*exp(− |
| ) |
where the final time t_f = 5 min. The constraint on the control variable (the normalized coolant flow rate) is:
0 <= u <= 0.5 |
In addition, there is an upper path constraint on the temperature:
x2(t) <= 460 |
Reference: [11]
toms t p = tomPhase('p', t, 0, 5, 30); setPhase(p); tomStates x1 x2 tomControls u % Initial guess x0 = {icollocate({x1 == 0.6*t/5 x2 == 380}) collocate(u == 0.25)}; % Box constraints cbox = {0 <= icollocate(x1) <= 10 100 <= icollocate(x2) <= 460 0 <= collocate(u) <= 0.5}; % Boundary constraints cbnd = initial({x1 == 0; x2 == 380}); % ODEs and path constraints k1 = 1.7536e5*exp(-1.1374e4/1.9872./x2); k2 = 2.4885e10*exp(-2.2748e4/1.9872./x2); ceq = collocate({ dot(x1) == (1-x1).*k1-x1.*k2 dot(x2) == 300*((1-x1).*k1-x1.*k2)-u.*(x2-290)}); % Objective objective = -final(x1);
options = struct; options.name = 'Plug Flow Reactor'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t = subs(collocate(t),solution); x1 = subs(collocate(x1),solution); x2 = subs(collocate(x2),solution); u = subs(collocate(u),solution);
Problem type appears to be: lpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Plug Flow Reactor f_k -0.677125737913556680 sum(|constr|) 0.000031724693791993 f(x_k) + sum(|constr|) -0.677094013219764700 f(x_0) -0.599999999999999200 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 346 ConJacEv 346 Iter 112 MinorIter 325 CPU time: 0.796875 sec. Elapsed time: 0.813000 sec.
subplot(2,1,1) plot(t,x1,'*-',t,x2/100,'*-'); legend('x1','x2/100'); title('Plug Flow Reactor state variables'); subplot(2,1,2) plot(t,u,'+-'); legend('u'); title('Plug Flow Reactor control');