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methodto solvethe followine I (a)UsetheGauss-Seidel untilthepercentrelativeerror falls below s, : 5olc: identify 126 Of the following threesetsof linearequations, the set(s)that you could not solve using an iterativemethod such as Gauss-Seidel Show using any number of iterations that is necessarythat your solution does not converge Clearly stateyour convergence criteria (how you know ir is n()1 converglng)
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(a) Repeat but use overrelaxation with I : 12 Usethe Gauss-Seidel method to solve the followine untilthe percentrelative error falls below e" - 57o:
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Repeat Prob 122but useJacobiiteration Thefollowing systemof equationsis designedto de(the ine concentrations c's in g/m3)in a series coupled of asa function of the amount of mass inout to each Itheright-hand sidesin g/day): r'::3800 t- 3c:'r * l8r2 - 6cr : 1200 c2't l2ca: 2350 Ithis problem with the Gauss-Seidel method to Usethe Gauss-Seidel method (a) without relaxation (b)withrelaxation(,)": 12) to solve the following sysof reanangethe t0 a tolerance s : 57o If necessary, to achievec'onvergence 2 " t- 6 r ; - " r 3 : - 3 8 1 - 3 r 1- t : l ' 7 t , : - 3 4
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nonlinear 127 Determine the solution of the simultaneous equatlons l':--l-+x+u/) )*5x-r':-r2 Use the Newton-Raphson methodand ernployinitial guesses : of r : _t' 12 nonlinear 128 Determinethe solution of the simultaneous equatlons:
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)il:,r2 (a) Graphically (b) Successive substitution using initial guesses r : of (c) puesses : r,r:
ITERATIVE METHODS
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ccs -
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FIGURE P129
129 Figure P129 depicts a chemical exchange process consistingof a seriesof reactors which a gasflowing from in left to right is passed over a liquid flowing from right to left The transferof a chemicalfrom the gasinto the liquid occurs at a ratethat is proportionalto the differencebetweenthe gas and liquid concentrations each reactorAt steadystate,a in massbalancefor the first reactor can be written for the sas as Qccco- Qcccr+ D(cuand for the liquid as Qrctz - Qrcu I D(ccr - crr) : 0 ccr):0
derivatives,which result in a system of linear algebraic equationsUse the Causs-Seidelmethod to solve for the temperatures the nodesin Fig Pl210 of
25"C
25"C
100'c
where Q" and Q, are the gas and liquid flow rates, respectively, andD : the gas-liquidexchange rateSimilar balances can be written for the other reactors Solvefor the concentrationsgiventhefollowingvalues: l, D : 08, Q6 :2, Q t: c c o : 1 0 0c m : 7 0 , 1210 The steady-statedistribution of temperatureon a heated plate can be modeled by the Laplace equation:
T,,,
100"c
o:***
75C
If the plate is represented a seriesof nodes(Fig P I 2 I 0), by centered finite differences can be substituted for the second
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13 througtrto oescribe-techniques to fit curves to such data to esti-ates In addi-rion, ri:" you rnay require a sirn, lftaln ilterm"diute plified version of a complicared funcrion One *uy to ooil;;;;oure values of the : ,::, funct'ion a numberof discretevalue alon^g at s the range inr"r"r, rnn a simplerfunction of i: may be derived to fir thesevaluesBoth of theseapplicationsare knownascurre trtting ,,;;
Dara often is given discrere for varues arong mayrequire esrimalesat pointsbetween discrete the values " "11111111; chapters "o*,11you
other'on basis rhearnount errorassqciared the of of ;rh ;;; ;ii;;;ffi;;;;,,*'r,, exhibitssignificant a degree error "scatter,,, ,t u,"gfi, t"jrr" , of or ,t singre curve that
represents generaltrend of the data the Because any individualdadpoint *"V l i""rcr we make no efiort to ln^tersect every point Ratherthe curve is de_ srgnedto follow the pattern of the points takenas a
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