10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 21 TA Tutorial on BVPs FEMLAB BVPs What methods to use for each situation 1D o Finite Differences o ODE s 2nd order Two 1st order 1st order Shooting 2D o Finite Differences o Method of Lines Stiff o Non uniform grid o Finite Element o Finite Volume x x Figure 1 Boundary layer Capture the fast portion of reaction with adaptive time stepping Rxn fast 3D Figure 2 Adaptive time stepping Generally use commercial code Coding Boundary Conditions Linear A x b 0 state variables at grid points 0 0 0 BC x1 j 1 1 1 4 1 3 x1 1 x i j x N x N y 0 3 1 j 4 2 j 3 j 2 x 1 j x1 1 1 x1 2 1 d dx 1 x1 1 0 3 x 2 1 x 3 1 for d dx 0 1 j 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 Nonlinear use fsolve BC x1 j 1 x1 j 1 0 set to zero Non Uniform Grid Throw out all original equations polynomial between three points f x xk k 0 i x k N p x fiLi x Li x x k i x f x f1 df dx df dx Figure 3 A graph of a polynomial fit of three points x x 2 x x3 x x1 x x3 x x1 x x 2 f2 f3 x1 x 2 x1 x3 x 2 x1 x 2 x3 x3 x1 x3 x 2 differentiate the above expression f x and evaluate at x2 x2 f1 x2 d2 f dx 2 x2 x 2 x3 x1 x2 x1 x3 f2 2 x 2 x3 x1 x2 x1 f3 x3 x1 x3 x 2 x2 x1 x2 x3 2 f3 2 f1 2 f2 x1 x2 x1 x3 x2 x1 x2 x3 x3 x1 x3 x2 MatLAB Equation nonuniform example huge jump Figure 4 Graph of a function that is steep in the beginning 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 21 Page 2 of 4 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 Scaling vz 2C C A 2C A D 2A k C AC B C AB K 2 z z y Z z Y y y b z L L b C A DL 2 C A D 2 C A L 2 2 2 v L Z v b Y Z z vz z R C A 2C A 10 3 dy 2 10 3 z 2 Z Y 1 so if z 10 choose y 10 2 Figure 5 Diagram of pipe flow with reaction and diffusion CAo divide into 100 Figure 6 Non uniform grid y b Figure 7 Division of problem into and b regions 2 C A 9 C A 10 z z 2 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 21 Page 3 of 4 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 Problem 1 Using FEMLAB space dimension axial symmetry 2D axis grid setting 18 rmin 0 01 rmax 0 06 zmin 1 zmax 20 0 03 Subdomain settings Figure 8 Diagram of FEMLAB Example r 0 001 r r 0 05 2 Boundary settings mesh mode puts in finite elements Solve the problem DONE IN UNDER 3 MINUTES 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 21 Page 4 of 4 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
View Full Document
Unlocking...