10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 35 Problem Solving Summary and Review Problem Solving Well Posed Problems In reality you define the problem Example At Exxon Professor Green was told that the lubricant gums up in the engine Had to take ill posed problem and transform to a well posed problem Could solve kinetics Could solve thermodynamics RECOGNIZE WHAT TYPE OF PROBLEM Rewrite equations in standard form Algebraic equations o Linear o Non linear Differential eq uations o ODE Initial Value Problems Boundary Value Problems o PDE Optimization Stochastic Simulations Estimate SOLUTION REALITY CHECK Set constraints for optimization i e least squares Good initial guess At least think about UNITS Write some MATLAB Run Computer SOLUTION OR Error or warning message Check if solution works is reasonable to spend as much time checking solution as obtaining it o e g have two different programs written by two different people How important is it that you are right Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Even if solution works does not guarantee solution will happen Sensitivity Analyses check for ill conditioning Numbers from numerical solution will not match experiment because input parameters have uncertainty Usually at least 1 error in measurements If solution sensitive to 1 error ill conditioning input parameters M x b use SVD once you get a solution do Taylor expansion linear equations check condition number cond M Models v Data Stochastics Metropolis Monte Carlo Gillespie Kinetic Monte Carlo Ydata x Ymodel x q min 2 find best 2 14 is 2 best small enough not going to adjust 2 5 2 2 2 1 best 1best 1 2best 2 Figure 1 Diagram showing search for best Start with messy equations differential equations 1 Discretize F x 0 2 Taylor series Linearize 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 35 Page 2 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY J x F Solve linear equations Then may discretize a different way to see whether we get the same answer Newton s method has best convergence close to minimum Computer Programming Key to MATLAB Reusability avoid writing too much code HEADER to function program inputs outputs function description what it does 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 35 Page 3 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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