Matlab: Nonlinear Algebraic Systems

1. Matlab: Nonlinear Algebraic Systems Nonlinear algebraic equation solvers Exothermic chemical reactor example

2. 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)

3. Exothermic Chemical Reactor Example Steady-state model Parameter values k0 = 3.493x107 h-1, E = 11843 kcal/kmol (-DH) = 5960 kcal/kmol, rCp = 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:

4. 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

5. Create m-file cstr.m