Numerical Methods for Chemical Engineers

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1 Numeril Methods for Chemil Engineers Chpter 4: System of Liner Algebri Eqution Shrudin Hron Pge 4 -

2 System of Liner Algebri Equtions This hpter dels with the se of determining the vlues,,, n tht simultneously stisfy set of equtions: n n = b n n = b.... n + n nn n = b n where the s re onstnt oeffiients, b s re onstnts nd n is the number of equtions. Pge 4 -

3 Gols nd Objetives Be ble to solve problems involving liner lgebri equtions Appreite the usge of liner lgebri equtions in ny field of engineering Mstering severl tehniques nd their relibility Nïve Guss elimintion Guss-Jordn elimintion LU deomposition Guss Siedel Be ble to use progrm to suessfully solve systems of liner lgebri equtions Pge 4 -

4 Nïve Guss Elimintion A systemti tehnique use to solve liner lgebri equtions simultneously with two steps: Forwrd elimintion - the equtions were mnipulted to eliminte ll the elements below the min digonl of mtri A b Bk Substitution - the elimintion step result in one eqution with one unknown. - the eqution ould be solved diretly nd the result bksubstituted into one of the originl equtions to solve the remining unknown. Pge 4-4

5 The proedure of Nïve Guss Elimintion Representing the liner lgebri equtions in n ugmented mtri form. n n n n n n n n = n n n = n n = n n = : : : : : : n + n + n nn n = n n n n n n nn 4 Lbel s row or eqn. (. n - ( - ( - ( - (n Pge 4-5

6 Steps of Nïve Guss Elimintion (e. unknowns in equtions. A Forwrd elimintion To eliminte the first unknown,, from the seond through the nth row/eqn. - row/eqn ( is lled the pivot eqution, nd is lled pivot element. ( ( ( ( ( ( (i ( / ( (ii ( ( (' (iii ( / (b (iv ( (b (' the prime ' indites tht the elements hve been modified. Pge 4-6

7 Steps of Nïve Guss Elimintion (e. unknowns in equtions. b To eliminte the seond unknown,, from the third through the nth row/eqn. - row/eqn (' is lled the pivot eqution. ( ( ( (i (' ' /' (' (ii (' (' ('' ( ( ( the double prime '' indites tht the elements hve been modified twie. Pge 4-7

8 Steps of Nïve Guss Elimintion (e. unknowns in equtions. B Bk Substitution From eqution ('' : ['' '' ] '' = '' = '' /'' the result n be bk-substituted into eqn (' nd ( to solve for nd. = ( = ( / / From eqution (' nd ( Pge 4-8

9 Nïve Guss Elimintion (ssignment in lss Use Nïve Guss elimintion to solve the following equtions. + = = = Pge 4-9

10 Guss-Jordn Elimintion Steps of Guss-Jordn Elimintion (e. unknowns in equtions. Chnge the vlue of to nd eliminte the other elements in the first olumn. ( ( ( ( ( ( (i ( / (' (ii (' (' (iii ( (' (' (iv (' ('b (v ( ('b (' the prime ' indites tht the elements hve been modified. Pge 4 -

11 Steps of Guss-Jordn Elimintion (e. unknowns in equtions. b Chnge the vlue of ' to nd eliminte the other elements in the seond olumn. ( ( ( ( ( ( (i (' /' ('' (ii ('' ' ('' (iii (' ('' ('' (iv ('' ' (''b (v (' (''b ('' the double prime '' indites tht the elements hve been modified twie. Pge 4 -

12 Steps of Guss-Jordn Elimintion (e. unknowns in equtions. Chnge the vlue of '' to nd eliminte the other elements in the third olumn. ( ( ( ( ( ( (i ('' /'' ( (ii ( '' ( (iii ('' ( ( (iv ( '' ( b (v ('' ( b ( the triple prime indites tht the elements hve been modified three times. Pge 4 -

13 Steps of Guss-Jordn Elimintion (e. unknowns in equtions. d The vlue of the unknowns n be determined diretly without the bk substitution step s in the nïve guss elimintion. ( ( ( = = = = Pge 4 -

14 Guss Seidel Method The most ommonly used itertive method for liner equtions solving. The liner equtions were derived so tht the first eqution n be solved for, the seond n be solved for nd the third n be solved for n n = n n = n n = : : : : : : n + n + n nn n = n = L n n = L n n = L n n n = n n n nn L nm m Pge 4-4

15 Step for Guss Seidel = = ( = ( First Itertion = ( = ( = ( Seond Itertion ε,i = j i i j i j % < ε s onvergene riteri where j nd j - re the present nd previous itertions Pge 4-5

16 Guss Seidel (ssignment in lss Use the Guss Seidel method to solve the following equtions (ε s = 5 % = = = Pge 4-6

17 Guss Seidel ssignment in lss Figure. shows hemil proess onsists of retors linked by pipes. The mss flowrte of hemil (g/s through eh pipe is equl to its onentrtion in eh retor, (g/m multiplied by the volume flowrte (m /s of the pipe. Assume the system is t stedy stte, so tht the trnsfer into eh retor will blne the trnsfer out. Develop mss-blne equtions for the retors, nd solve the equtions simultneously for the unknown onentrtions (,, using Guss-Siedel method with ε s = 5%. (4 m /s( (5 m /s( g/s (7 m /s( g/s R ( m /s( R (5 m /s( ( m /s( ( m /s( (5 m /s( Figure. 5 g/s R (7 m /s( Pge 4-7

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