Matlab: Nonlinear Algebraic Systems
1. Nonlinear algebraic equation solvers
2. Exothermic chemical reactor example
Matlab: Nonlinear Algebraic Equations
fzero – scalar nonlinear equation solver» Syntax: x = fzero(‘fun’,xo)
– ‘fun’ is the name of the user provided Matlab m-file function (fun.m) that evaluates and returns the LHS of f(x) = 0
– xo is an initial guess for the solution of f(x) = 0
– Discussed last lecture
fsolve – multivariable nonlinear equation solver» Function for solving system of nonlinear algebraic
equations
» Syntax: x = fsolve(‘fun’,xo)– Same syntax as fzero, but x is a vector of variables and the function,
‘fun’, returns a vector of equation values, f(x)
» Part of the Matlab Optimization toolbox
» Multiple algorithms available in options settings (e.g. trust-region dogleg, Gauss-Newton, Levenberg-Marquardt)
Exothermic Chemical Reactor Example
Steady-state model
Parameter values» k0 = 3.493x107 h-1, E = 11843 kcal/kmol
» (-H) = 5960 kcal/kmol, Cp = 500 kcal/m3/K
» UA = 150 kcal/h/K, R = 1.987 kcal/kmol/K
» V = 1 m3, q =1 m3/h,
» CAf = 10 kmol/m3, Tf = 298 K, Tj = 298 K.
Problem» Find all steady-state points:
0
0
0 ( ) exp( / )
0 ( ) ( ) exp( / ) ( )Af A A
p f A j
q C C Vk E RT C
qC T T H Vk E RT C UA T T
),( TCA
x = fsolve('cstr',xo,options) 'cstr' – name of the Matlab m-file function (cstr.m) for
the CSTR model xo – initial guess of steady-state solution: xo = [CA T] ' options – Matlab structure of optimization parameter
values created with the optimset function
Example usage
>> xo = [10 300]';
>> x = fsolve('cstr',xo,optimset('Display','iter'))
Solution of CSTR Model with fsolve
Create m-file cstr.m
function f = cstr(x)
ko = 3.493e7;E = 11843;H = -5960;rhoCp = 500;UA = 150;R = 1.987;V = 1;q = 1;Caf = 10;Tf = 298;Tj = 298;
Ca = x(1);T = x(2);
f(1) = q*(Caf - Ca) - V*ko*exp(-E/R/T)*Ca;f(2) = rhoCp*q*(Tf - T) + -H*V*ko*exp(-E/R/T)*Ca + UA*(Tj-T);
f=f';
Single Steady-State Solution
>> xo = [10 300]';>> x = fsolve('cstr',xo,optimset('Display','iter'))
Norm of First-order Trust-region Iteration Func-count f(x) step optimality radius 0 3 1.29531e+007 1.76e+006 1 1 6 8.99169e+006 1 1.52e+006 1 2 9 1.91379e+006 2.5 7.71e+005 2.5 3 12 574729 6.25 6.2e+005 6.25 4 15 5605.19 2.90576 7.34e+004 6.25 5 18 0.602702 0.317716 776 7.26 6 21 7.59906e-009 0.00336439 0.0871 7.26 7 24 2.98612e-022 3.77868e-007 1.73e-008 7.26Optimization terminated: first-order optimality is less than options.TolFun.
x =
8.5637 311.1702