10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 6 Modern Methods for Solving Nonlinear Equations 1D Problem unknown T of reactor Qrxnexp Ea RT h T Ta c T4 Ta4 0 f x 0 heat of reaction Gain heat convection Lose heat f T radiation Lose heat 2 steady state temperatures Make a plot with MATLAB Figure 1 1D problem 0 T netheat m function qdot netheat T computes the net heating rate of a reactor qdot 0 at the steady state qdot Q exp Ea R T h T Ta c T 4 Ta 4 Q 2e 5 Ea 5000 R 1 987 h 3 Ta 300 c 1e 8 Tvec linspace 300 3000 qdot netheat Tvec plot Tvec qdot Figure 2 Professor Green modified variables Q and c until the plot looked like the one above Increased Q and decreased c To solve for steady state zeros a b f T 0 Figure 3 Have computer bracket in and find small range where plot goes from negative to positive 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 Bisection start a b such that f a 0 and f b 0 x Figure 4 Function must be continuous a b 2 if f x f a 0 a x else b x This is a problem of TOLERANCE if b a tol stop Types of tolerance Absolute tolerance atol has units if f x atol f Relative tolerance rtol if b a rtol a has to be BIG number while abs b a atolx x a b 2 if f x f a 0 a x else b x end In MATLAB bisect m function x bisect f a b atolx rtolx atolf solves f x 0 while abs b a atolx x 0 5 b a if feval f x feval f a 0 a x else b x end end Command Window x bisect netheat 300 2000 0 1 0 0 x 1 2373e 003 CHECK netheat 1237 1 0474 close Keep in mind never get actual solution but can come close We can change tolerances to improve results while abs b a atolx abs b a rtolx abs a x 0 5 b a AND must satisfy both conditions if abs feval f x atolf return if value becomes low enough return value x bisect netheat 300 2000 0 1 1e 2 0 5 x 1 2363e 003 looser tolerance gives less accurate answer Bisection cuts interval by 2 each time 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 6 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 Every time we cut 3 times we lose a sig fig In bisection time grows linearly with the number of significant figures a xtrue b xtrue xsoln b a 2 Newton s Method 1 D evaluates slope of f x next guess is the xnew that satisfies f xnew 0 for a line from f xguess with the slope at f xguess Figure 5 Newton s Method f x f x0 f x0 x x0 O x2 0 f xguess f xguess x xguess x new x guess guess f x guess f x For a good guess Newton s method doubles the number of significant figures after every iteration however we lose robustness if guess is poor If f xguess 0 doesn t work f x 0 Figure 6 NO intersection Another drawback is one needs a derivative of the function Secant Method same as Newton s but uses f x approximate f x k f x k 1 f approx x x k x k 1 Bisection method works only for 1D problems but Newton Secant can be used for problems with greater dimension 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 6 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 Broyden s Method Multi dimensional F x F x0 J x0 x x0 Method breaks down when J is singular f i x j xo j j xo x j f x 0 approx J B outer product is opposite of dot product B k 1 B k F1 x1 Outer Product F2 x1 F1 x 2 F2 x 2 F x k 1 x k 1 x k T x 2 F1 x3 F2 x3 Newton s Method Multi dimensional O F x0 J x0 x x0 J x F x0 LU B k x F LU LU k 1 without redoing factorization Done in detail in homework problem 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 6 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
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