36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to

Transcription

1 ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to nd roots to systems o equatons? Up to ths pont, we have dscussed how to nd the soluton to sngle non-lnear equaton (Newton Raphson Method) systems o lnear equatons (Lnear Algebra) However, requently n chemcal engneerng, we need to nd the soluton to a system o non-lnear equatons. Ths orces us to combne the Newton-Raphson method wth Lnear Algebra. The technque we wll dscuss n ths secton s Multvarate Newton Raphson Method. 6. Multvarate Newton-Raphson - Theory Recall that when we wanted to nd the soluton to a sngle non-lnear equaton o the orm () (.) The bass o the Newton-Raphson method lay n the act that we can appromate the dervatve o () numercally. ( ) d() d () ( ) (.) leadng to the teratve ormula () ( ) (.5) Now consder the system o n non-lnear equatons and n unknowns.

2 ChE Lecture Notes - D. Keer, 5/9/98... n n (,,,...n, n) (,,,...n, n) (,,,..., ) (,,,...n, n) (,,,..., ) n n n n (6.) Ths s the most general orm o the problems that ace chemcal engneers and encompass lnear equatons (when all the unctons,, are lnear) and sngle equatons (when n ). One technque used to solve ths problem s called the multvarate Newton Raphson Method (MNRM). The dea ollows rom the sngle-varable case. The basc dea agan stems rom the act that the total dervatve o a uncton,, s d d d (6.) or the case o two varables or d n d (6.) or the n varable case. We can dscretze ths epresson and wrte: () () ({ } ) { } n () () ( ) ( ) (6.4) where the s the nde over unctons, the s the nde over varables and the superscrpt n parentheses stands or the teraton. Note that we have only one varable (n), ths reduces to the Newton-Raphson method that we have already learned. Now consder that we have n equatons so that to n. We have a system o equatons whch we can wrte as: () ({ } ) { } n () () () ( ) ( ) (6.5)

3 ChE Lecture Notes - D. Keer, 5/9/98 Just as n the sngle varable case, we want our net teraton to take us to the root so we assume (). We can then wrte ths system n matr notaton as: ( ) that { } J δ R (6.6) wherer s called the resdual vector at the k th teraton and s dened as ( k) R ({ } ) (6.7) wherej s called the Jacoban matr at the k th teratonand s dened as and where ( J ), (6.8) δ (k ) (6.9) so that the new guess or the s (k ) δ (6.) The algorthm or solvng the multvarate Newton-Raphson ollows analogously rom the sngle varable NRM. The steps are as ollows:. Make an ntal guess or. calculate the Jacoban and the Resdual.. Solve equaton Calculate new rom equaton I the soluton has not converged, loop back to step. The multvarate Newton-Raphson Method suers rom the same short-comngs as the sngle-varable Newton-Raphson Method. () You need a good ntal guess. () You don t get quadratc convergence untl you are close to the soluton. () I the partal dervatves are zero, the method blows up. I the partal dervatves are close to zero, the method may not converge.

4 ChE Lecture Notes - D. Keer, 5/9/98 6. Multvarate Newton-Raphson - Problems As wth all the numercal technques we have been usng, t s necessary to practce wth them beore we become procent wth them. Eample One: ( (,, ) ) ( ) ( ) 4 ( ) ( ) (6.) The vs plot o the ollowng unctons s gven below. The rst uncton s that o a crcle, centered at the orgn wth radus. The second uncton s a parabola. We see there are two and only two solutons to ths systems o equatons

5 ChE Lecture Notes - D. Keer, 5/9/98 In order to use MNRM, we must rst determne the unctonal orm o the partal dervatves ( J) ( J), ( J) ( J),,, Then ollowng the algorthm outlned above: Step One. Make an ntal guess. One o the solutons looks to be at. and.. Step Two. Usng that ntal guess, calculate the resdual and the Jacoban. ( 4 and R () J ) Step Three. Solve J δ R (Usng Lnear Algebra) () δ.. Step Four. Calculate new values or va equaton (6.) ().9.8 Step Fve. Loop back to Step. and repeat untl converged. Here are what urther teratons yeld teraton J R δ

6 ChE Lecture Notes - D. Keer, 5/9/ e.79e.559e e e e e e - 8 The other root n to the problem s located at by symmetry. Eample Two. Lnear systems are a subset o non-lnear systems. The multvarate Newton-Raphson solve lnear systems eactly n one teraton, ust as was the case n the sngle-varable problem. Consder the system o lnear equatons: ( (,, ) 5 ) ( ) ( ) 4 ( ) ( ) (6.)

7 ChE Lecture Notes - D. Keer, 5/9/98 In order to use MNRM, we must rst determne the unctonal orm o the partal dervatves ( J) 5 ( J), ( J) ( J),,, Then ollowng the algorthm outlned above: Step One. Make an ntal guess. One o the solutons looks to be at. and.. Step Two. Usng that ntal guess, calculate the resdual and the Jacoban. ( 5 8 and R () J ) Step Three. Solve J δ R (Usng Lnear Algebra) () δ Step Four. Calculate new values or va equaton (6.) () A second teraton wll show that ths s the eact soluton. 6. Multvarate Newton-Raphson - MATLAB Very quckly problems lke ths become too dcult to solve by hand. We rely on sotware lke MATLAB to solve these problems or us. MATLAB does not necessarly use a multvarate Newton-Raphson method to solve a system o non-lnear equatons but t uses some smlar numercal technque. Ths alternate technque appromates the partal dervatves, so that the user only has to enter the unctonal orm o the unctons and not that o the partal dervatves. 7

8 ChE Lecture Notes - D. Keer, 5/9/98 On the webste, you can download a routne called syseqn.m. Ths routne wll allow you to solve a system o non-lnear algebrac equatons. The descrpton or how to use the le can be obtaned by openng MATLAB, movng to the drectory where you have downloaded the syseqn.m le, and typng help syseqn Ths yelds: syseqn(o) syseqn solves a system o nonlnear algebrac equatons. The system o equatons are stored n the le syseqnnput.m The ntal guesses are gven as a vector n o. The soluton s wrtten to the screen and to the le "syseqn.out" Author: Davd Keer Date: October, 998 The routne uses the MATLAB nstrnsc uncton zero to solve the system o equatons. At the MATLAB command lne, the routne s nvoke by typng, or eample, syseqn() you had one equaton and you wanted your ntal guess to be o. Or, you had a system o three non-lnear equaton, you could nvoke syseqn.m by typng syseqn([;4;6]) you had one equaton and you wanted your ntal guess to be, o, 6. o,, 4 o, The algebrac equatons are entered n the le syseqnnput.m Ths le can be as smple as uncton [] syseqnnput() ^-; to solve or the roots o -. The syseqnnput.m can also be much more complcated. Two more nvolved eamples o the syseqnnput.m or (a) solvng a system o mass balances n an etractor, and (b) determnng the chemcal equlbra compostons and temperature n a vessel where multple reactons are occurng n a non-adabatc and non-sothermal envronment. In the latter case, you wll see that the syseqnnput.m le can be very complcated. It doesn t matter how complcated t appears, so long as, at the end o the le, you have evaluated the uncton o nterest. Eample: Consder the ollowng problem rom chemcal engneerng. You want to use an etracton process to remove a contamnant rom a eed stream. The process dagram looks lke ths: 8

9 ChE Lecture Notes - D. Keer, 5/9/98 Etract, E Solvent, S mass transer Feed, F Ranate, R The data you are gven s F mol / hr F,b F,c F,..9. S S,b S, 5mol / hr S,c R? R,??. E? E,c E,?.? You have s unknown varables. You need s equatons to solve these. You have three mass balances: F S R E F S F,c R E S, E, You have two constrants on the compostons E,c You have one separaton rato: K.68 9

10 ChE Lecture Notes - D. Keer, 5/9/98 These are your s equatons. The second, thrd, and sth equatons are non-lnear. You need to use a technque lke multvarate Newton-Raphson to solve ths problem. In order to use MNRM to solve ths system o non-lnear equatons, rearrange the equatons so that the rght hand sde s zero F F S E,c S F,c S,.68 R R E E E, MATLAB can solve ths problem easly, usng the syseqn.m routne provded on the web-ste. You don t alter the syseqn.m le. You smply type the equatons nto the syseqnnput.m le and then nvoke the syseqn.m routne wth reasonable ntal guesses. Ths routne can then be used to do parameter studes.. Look at derent compostons n the ranate as a uncton o derent eed rates or solvent rates or eed or solvent compostons. (In order to change parameters, lke F, you only have to change the value o F n etractnput.m.). Look at derent cases. Perhaps you want to specy the Ranate composton and determne the eedrate. (In order to change whch unctons are varables, you smply make sure that you assgn your ntal guesses to the approprate varables at the begnnng o the routne.. Look at any mass balance system. Ths code gves the skeleton or solvng any system o nonlnear equatons. All that s requred s changng the nput le. Wth a lttle study o the program, you wll be able to mody t and use t n all your classes. Usng ths code solves the above eample n a racton o a second: F mol / hr F,b F,c F,..9. S S,b S, 5mol / hr S,c R 94.74mol / hr R, E 55.6mol / hr E, E,c