Catalyst Mixing

Second-order sensitivities of general dynamic systems with application to optimal control problems. 1999, Vassilios S. Vassiliadis, Eva Balsa Canto, Julio R. Banga

Case Study 6.2: Catalyst mixing

Contents

Problem formulation

This problem considers a plug-flow reactor, packed with two catalysts, involving the reactions

S1 <-> S2 -> S3

The optimal mixing policy of the two catalysts has to be determined in order to maximize the production of species S3. This dynamic optimization problem was originally proposed by Gunn and Thomas (1965), and subsequently considered by Logsdon (1990) and Vassiliadis (1993). The mathematical formulation is

Maximize:

$$ J = 1 - x_1(t_f) - x_2(t_f) $$

subject to:

$$ \frac{dx_1}{dt} = u*(10*x_2-x_1) $$

$$ \frac{dx_2}{dt} = u*(x_1-10*x_2)-(1-u)*x_2 $$

$$ 0 <= u <= 1 $$

$$ x(t_0) = [1 \ 0]'$$

$$ t_f = 1 $$

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 30);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
% Note: The guess for tf must appear in the list before expression involving t.
x0 = {icollocate({
    x1 == 1-0.085*t
    x2 == 0.05*t
    })
    collocate(u==1-t)};

% Box constraints
cbox = {0.9 <= icollocate(x1) <= 1
    0 <= icollocate(x2) <= 0.1
    0 <= collocate(u)   <= 1};

% Boundary constraints
cbnd = {initial({x1 == 1; x2 == 0})
    final({x1 <= 0.95})};

% ODEs and path constraints
ceq = collocate({
    dot(x1) == u.*(10*x2-x1)
    dot(x2) == u.*(x1-10*x2)-(1-u).*x2});

% Objective
objective = -1+final(x1)+final(x2);

Solve the problem

options = struct;
options.name = 'Catalyst Mixing';
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
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Catalyst Mixing                f_k      -0.048059280695325612
                                       sum(|constr|)      0.000000452031697998
                              f(x_k) + sum(|constr|)     -0.048058828663627616
                                              f(x_0)      0.964999999999998080

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   66 ConJacEv   66 Iter   43 MinorIter  247
CPU time: 0.156250 sec. Elapsed time: 0.156000 sec.

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Catalyst Mixing state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Catalyst Mixing control');