CH E 374 Computational Methods in Engineering Fall 2007

Transcription

1 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 5. The data o the varato of the rato of stagato pressure to statc pressure (r ) wth Mach umber ( M ) for the flow through a duct are as follows: 5 M r Ft a fourth-degree polyomal to the data. SOLUTION 5 Let r= a+ am + am + am + am 5 ; =,,...,5 () Where Equato () ca be expressed a matrx form as [ B] a = r () M M M M a r a r M M M M [ B] M, M a a M M = =, r = r M a M M M r M a 5 M5 M5 M 5 5 r5 Usg the data M = 0.,0.,0.6,0.8,.0 r =.05,.,.,.55,.9 Equato () ca be solved to fd a. Matlab scrpt

2 CH E 7 Computatoal Methods Egeerg % problem 5 % set the data M = [ ]; r = [ ]; Fall 007 % coeffcet matrx B = [ M() M()^ M()^ M()^; M() M()^ M()^ M()^; M() M()^ M()^ M()^; M() M()^ M()^ M()^; M(5) M(5)^ M(5)^ M(5)^] %rhs vector r = r'; % Soluto a = B\r Sample Soluto >> probset >> B =.0000e e e e e e e e e e e e e e e e e e e e e e e e e+000 a =.000e e e e e+000

3 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 6. The power developed by a hydraulc mpulse turbe ( P ) by chagg the pestock dameter ( D) s foud to be as follows: D ( m ) P ( MW) Ft a cubc polyomal to the data. I ths case we are gog to use polyft (ths s a least squares mplemetato by matlab) % problem # 6 % data D = [ ] P = [ ] c = polyft ( D, P, ) c = D D -70.8D The kematc vscosty of SAE 0 ol wth varato temperature was foud to be as follows: Temperature, T ( C ) Vscosty μ (m /s).5x0-5.5x0 - x0-5x0-5 x0-5.x0-5 6x0-6 Develop a relatoshp betwee the two parameters of the form ( b/ T) μ = ae Soluto ( b / T ) μ = ae () Take log for both sdes b l μ = l a + () T

4 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto Usg y = l μ, c = l a ad x = T we ca express equato () as y = c+ bx () The sum of squares = ( S = y c bx Mmzg the sum of squares ) () S ( y c bx) ( ) = 0 c = (5) S = ( y c bx) ( x) = 0 b = (6) Equato (5) ad (6) ca be expressed as Or ( y c bx) = 0 (7) = ( xy cx bx = ) = 0 (8) ( c ) + x b = y (9) = = x c + x b = yx = = = Soluto of equato (9) ad (0) s: (0) x y yx = = = b = ( ) x = x y x yx = = = = c= y = ( ) x = For the gve data, we fd (for = 7) () ()

5 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 7 7 x =.999, y = = = 7 7 xy= , x =.0786 = = Gvg b=.565, c = 0.89 a = exp(c) 8. The thermal coductvty of ro (k) s foud to vary wth temperature (T) as follows: T ( K ) W k cm K Determe a relatoshp the form Soluto a Tk = b Ths ca be wrtte learzed form as lt + al k = l b a Tk = b where a ad b are costats. let x = lt y = l k ad c = l b We ca use the procedure to solve ths as a lear equato {} Alteratvely, we ca go ahead ad make drect lear ft of l T vs l k whch should be lear % problem # 8 T = [ ]; k = [ ]; c = polyft ( log(t), log(k),) a = (-/c()) b = exp(a*c()) c = a =.885 b = Tk =

6 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 9. Derve the Lagrage terpolato polyomal that passes through the followg pots x y x =,,,,0; =,...,5 y = 5, 0,,,9; =,...,5 Lagrage 5 5 x x j yx ( ) = y j, j x x = = j ; = 5 ( x+ )( x+ )( x+ ) x = 5 ( + )( + )( + )( ) + 0 ( x+ )( x+ )( x+ ) x + ( + )( + )( + )( ) ( x+ )( x+ )( x+ ) x + ( + )( + ) + )( ) ( x+ )( x+ )( x+ )( x+ ) + 9 ()()()() 5 y( x) = ( x+ )( x+ )( x+ ) x+ ( x+ )( x+ )( x+ ) x ( x+ )( x+ )( x+ ) x + ( x+ )( x+ )( x+ )( x+ ) 8 To evaluate the polyomal at a value, say x = y(.5) = (0.5)( 0.5)(.5)(.5) + (.5)( 0.5)(.5)(.5) (.5)(0.5)( 0.5)(.5) + (.5)(0.5)( 0.5)(.5) 8 y(.5) =.85 0 The varato of heat trasfer coeffcet per ut area (q) durg the bolg of water uder pressure (p) has bee foud to be as follows: q, ( MW / m ) p ( MPa )

7 CH E 7 Computatoal Methods Egeerg Fall 007 Develop a sutable polyomal relato betwee q ad p. Soluto Lets make a plot of the data order to see f we ca get a glmpse of the ature of the relatoshp. % problem 0 q = [ ] p = [ ] plot(p,q,'*') xlabel ('p') ylabel ('q') Sample Soluto.5 q p From the above graph, we ca see that a quadratc fucto wll ft the data adequately. Usg polyft (aga) C = polyft(p,q,) q = p p from the plot below, t s clear that a d order polyomal s suffcet. 7

8 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto.5 =.5 q.5 = = p. The heat trasfer coeffcet (h) a forced covecto heat trasfer cross-flow past a cylder at room temperature s foud to vary wth the velocty of the flud (v) flowg past the cylder as follows: v ( m/ s ) 6 8 W h m. K Ft a lear equato betwee h ad v Soluto Lear relatoshp y = a0 + ax Where x= v ad y = h Usg x =.0,.0, 6.0, 8.0 ad y = 6000., 0000., 000.,5000. a ad a ca be foud from the followg equatos 0 8

9 a a DEPARTMENT OF CHEMICAL & MATERIALS ENGINEERING CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto yx xyx = = = = 0 = x x = = y x x y = = = = x x = = We obta x y x y x = 0.0, = 000.0, = 5 0, = 0.0 a = 500.0, a = MATLAB soluto { f you wat to skp the had calculatos above ft a frst order polyomal} %%% problem x = [ ]; y = [ ]; a = polyft(x,y,) a =.5000e e+00. The drag coeffcet ( C) wth Reyolds umber (Re) for a smooth sphere s foud to vary accordg to the followg data: Re C Develop a sutable relatoshp betwee Re ad C. Soluto Ths data does t look lear. Lets plot t % problem % let us plot the data to see the tred Re = [ ]; C = [ ]; hold off fgure 9

10 plot (Re,C); plot (log(re), log(c)); DEPARTMENT OF CHEMICAL & MATERIALS ENGINEERING CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto

11 CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto The data looks more lke a expoetal decay Assume the relatoshp as Where a d b are costats. Learzg () we obta C = ae br () l C = l a br () Settg y = l C, d = l a c = a ad x = R, we obta y = d + cx () We ow proceed to mmze the sum of squares ad obta the followg relatoshp d=.008, c = a=exp(d) = exp(.008)=7.696 b=-c = thus C = R e