10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #35: Problem Solving Summary and Review.
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 Æ Rewrite equations in standard form
• Algebraic equations o Linear o Non-linear
• Differential eq uations o ODE
ƒ Initial Value Problems
ƒ 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 Æ Run Computer Æ 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
∂
∂
sensitive to 1% error, ill-conditioning
M·x ≈ b Æ use SVD
• once you get a solution, do Taylor expansion, linear equations, check condition number cond(M)
• Metropolis Monte Carlo
• Gillespie Kinetic Monte Carlo
Y data
(x) Y model
(x , θ ,q) find θ best min χ
2 ( θ ) is best
) small enough? not going to adjust
χ
2 = 14
χ
2 = 5
θ
2
χ
2 = 2
θ
1
θ
( θ best
1 best ± δθ
1
, θ
2 best ± δθ
2
)
Figure 1.
Diagram showing search for best θ .
Start with messy equations……… differential equations
1) Discretize Æ F(x) = 0
2) Taylor series Æ 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].