A Comparison between the Iteration Methods and Adomian Decomposition Method for Solving Nonlinear Combustion Equations

Transcription

1 Applied Mathematial Sienes, Vol. 7, 0, no. 5, 5-9 HIKARI Ltd, A Comparison between the Iteration Methods and Adomian Deomposition Method or Solving Nonlinear Combustion Equations aramust Juntarakod Department o Mehanial and Aerospae Engineering King Mongkut s Universit o Tehnolog North Bangkok 58 raharat Road, Wongsawang, Bangsue, Bangkok 0800, Thailand paramust_kmitnb@hotmail.om Thanakom Soontornhainaksaeng Energ Management Sstems and Monitoring & Veriiation Control, Researh & Development Center Siene and Tehnolog Researh Institute Department o Mehanial and Aerospae Engineering King Mongkut s Universit o Tehnolog North Bangkok 58 raharat Road, Wongsawang, Bangsue, Bangkok 0800, Thailand tss@kmutnb.a.th Copright 0 aramust Juntarakod and Thanakom Soontornhainaksaeng. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in an medium, provided the original work is properl ited. Abstrat In this paper, we present a omparative stud between Newton-Raphson method, Householder s iteration method, Adomian deomposition method and High order iteration method applied to the solution o nonlinear hdroarbon ombustion equations or investigation o equilibrium omposition in ombustion produts. For this purpose, the paper produes a step o numerial or alulating the omposition o the hdroarbon ombustion, whih is based on the development

2 aramust Juntarakod and Thanakom Soontornhainaksaeng o a model using speies hemial equilibrium onsiderations. The non-linear sstem, whih is obtained rom the seven non-linear equilibrium equations and the our linear atom balane equations, is onverted to a non-linear sstem. This sstem is solved using the all iteration Method or non-linear sstems, with no onversion problems and small omputational ost. The program is applied suessull or hdroarbon uel, whih tpiall represents the gasoline uel. The omparison gives similar results and supplies quantitativel reliable results. Kewords: Newton Raphson method, Adomian deomposition method, Householder s iteration method, High order iteration method and Combustion. Introdution Man mathematial modeling whih explain natural phenomena are usuall ormulated in term o nonlinear dierential equations, both ordinar and partial. Solving nonlinear equations is one o the most important problems in numerial analsis. Muh attention has given to develop several iterative methods or solving nonlinear equations, see [-] and the reerenes therein. Newton Raphson method is the most popular and powerul method in solving non-linear ombustion equations. For example, Rakopoulos[7]. also used this method to solve equations o hemial equilibrium ombustion model in speies diesel alone ombustion produts, Masood and Ishrat [8] used this method to solve equations o low-temperature ombustion model with onl 0 produts speies, Rashidi [8] ombined Newton Raphson with suessive substitution methods to solve ombustion problem o hdroarbon uel with 8 produts speies and onluded that the ombined method gives ver ast and reliable onvergene during iteration proess. The purpose o this researh is to present a new reliable modiiation o Newton Raphson method, Adomian deomposition method, Householder s iteration method, High order iteration method or solving the nonlinear equation. For appling the method to ombustion equation, dierential operators will be established step o singular equation. Moreover, some ombustion equations o gasoline uel are illustrative or demonstrating the advantage o the omparison. roblem ormations The hdroarbon uel is to be speiied in terms o the C, H, O and N atoms in the uel (Gasoline, CH 7 7). The other parameters that need to be speiied are equivalene ratio and temperature. For the alulation o equilibrium onstant, the data or onstants is onsidered rom Grill [9]. The molar-air uel ratio is alulated rom the number o arbon, hdrogen, nitrogen and oxgen atoms present in the uel the data or onstants is onsidered rom Turn [0].

3 A omparison between the iteration methods 7 Logi o the numerial simulation o ombustion equation This proess based on onservation o mass that was modiied orm Turn [0], where the ollowing equation represents the hemial reation with the relevant speies involved whih was added gasoline supplied as given below, εφ CHON α β γ δ CO HO N O CO 5 0.O 0.79N H H O OH NO N () 0.0 Where φ is equivalent ratio, ε = and i are mole rations α 0.5β 0.5γ There is the onservation o atoms and rom mixture equation, so atom balaning an be written, C εφα = ( ) () 5 H εφβ = ( 7 9) () O εφγ 0. = ( ) () N εφδ.58 = ( 0 ) (5) Also the onstraint that the mole rations add up to unit must be satisied, = 0 () The expression or atom balane o eah equation an be eliminated b dividing Eq. () b Eq. (). The equation an be written as next equation. 7 9 d( 5) = 0 (7) Likewise the Eq. () and Eq. (), the equation an be written as [ d( 5)] = 0 (8) Likewise the Eq. () and Eq. (), the equation an be written as 0 [ d( 5)] = 0 (9) Equations o onstant value or a simple studied an be deined as, d = β, d = γ and d δ = α α εφα α εφα The above equations, Eq. () through Eq. (9) have 0 unknowns (,,..., ), thereore in order to solve or these 0 unknowns other more equations are needed whih ma be derived rom the onsideration o equilibrium among produts. The equilibrium onstant an be related to the partial pressure o the reatants and produts. CO CO O (0) H O HO () H O OH ()

4 8 aramust Juntarakod and Thanakom Soontornhainaksaeng H H O O ) () N N (5) O N NO () And the partial pressure o a omponent is deined relative to the total pressure and the mole ration, thus the equilibrium onstant an be rewritten as, (Equilibrium onstant is the ratio o mole.) CO O 5 K = 5, where = CO K (7) K H O =, where = K H O (8) K = OH 9 = 9 =, where = K H O (9) K = H 7 K 7, where H (0) K 5 O 8 K5 8 5, where 5 O () K N K, where N () NO 0 K7 0 = 7, where 7 = K7 () O N The equilibrium onstants are treated as a untion o gas temperature. It is the highest possible temperature that an be ahieved during ombustion. For omputation o temperature, it is assumed that no heat will be transerred through linder walls, that is heat must be zero and all energ transerred to engine work and exhaust produt. Values or K have been tabulated or several reations at various temperatures. These values ome rom the above relations, whih are determined b statistiall thermodnamis. Equilibrium onstant in Eq. (7) through Eq. () are urve itted to the Thermohemial Table, Grill [9] and their expressions are o the rom, ΔaT ΔaT ΔaT ΔaT 5 Δa K = exp Δa ( lnt ) Δa7 () 0 T

5 A omparison between the iteration methods 9 Where, T is in Kelvin (Temperature). For the range o 000 T 000K, oeiients or alulating the equilibrium onstant K, Substitution o the equilibrium onstants rom Eq. () and the urve-it oeiient onstants into Eq (0) through Eq. () and rearranging to express mole rations o all speies in terms o,, 5 and the mole rations oco, H, O and N respetivel. The expressions resulted are our nonlinear equations in our unknowns are as ollows, (,,, ) = 0, Where =,,, (5) 5 So, the sstems are (,, 5, ) = () (,, 5, ) = ( ) d( 5 5) (7) / / / (,, 5, ) = ( 5 ) d( 5 5) (8) (,,, ) = ( ) d ( ) (9) Modiied iteration method or Non-Linear Combustion Equation. Newton Raphson method The given equations an be solved b using the Newton-Raphson method. The general orm o Newton-Raphson method or nonlinear sstems is ( xk ) YK = YK,Where k=iteration (0) ' ( x ) k Eah o these ma be expanded in Talor expansion o ( x ) to a higher order or Newton- Raphson method, Δ Δ Δ 5 Δ 0, =,,, () 5 Funtions are evaluated rom the solution o interested untions (Eq- through Eq-9). = (,,,) The independent set o derivatives is obtained b solution o matrix equation that results rom dierentiating with respet to mole ration. The above an be arranged as set o linear equations in the matrix orm, [ A][ Z] [ B] = 0. Where [ A] =,[ Z] = [ Δ] and [ B] = [ i ], (Improving the derivative term o the sstem sees in the appendix A)

6 0 aramust Juntarakod and Thanakom Soontornhainaksaeng. Householder s iteration method The given equations an be solved b using Householder s iteration method. The general orm o Householder s iteration method or nonlinear sstems is '' ( xk) ( xk) ( xk) Y = K Y K ' ' ( xk) ( xk), or '' ' ' ( x ) ( x ) ( x ) ( x ) ( x ) Y Y = 0 () ( k k k k ) k ( K K) Eah o these ma be expanded in Talor expansion o ( x ) to a higher order or Householder s iteration method, see Eq. (). Funtions are evaluated rom the solution o interested untions (Eq. () through Eq.(9)). '' ' = ( xk) ( xk) ( xk) ( xk). The independent set o derivatives is obtained b solution o matrix equation that results rom dierentiating with respet to mole ration. The above an be arranged as set o linear equations in the matrix orm, [ A][ Z] [ B] = 0. Where [ A] =,[Z]=[ Δ] '' ' [ B] = ( xk) ( xk) ( xk) ( xk) () (Improving the derivative term o the sstem sees in the appendix A and B). The Adomian Deomposition method The given equations an be solved b using Adomian Deomposition method. The general orm o Adomian Deomposition method or nonlinear sstems is '' '' ( xk) ( xk) ( xk) ( xk) ( xk) Y = K Y K ' ' '5 ( xk) ( xk) ( xk),or '' ' '' ' ( xk) ( xk) ( xk) ' ( xk) ( xk) ( xk) ( xk) ( xk) ( YK YK) = 0() Eah o these ma be expanded in Talor expansion o ( x ) to a higher order or Adomian Deomposition method, that is '' ' '' ' ( xk) ( xk) ( xk) = ( xk) ( xk) ( xk) ( xk) (5) Funtions are evaluated rom the solution o interested untions (Eq. () through Eq. (9)) see Eq. (). The independent set o derivatives is obtained b solution o matrix equation that results rom dierentiating with respet to mole ation. The above an be arranged as set o linear equations in the matrix orm, [ A][ Z] [ B] = 0. Where [ A] =,[Z]=[ Δ] and '' ' '' ' ( xk) ( xk) ( xk) [ B] = ( xk) ( xk) ( xk) ( xk) () (Improving the derivative term o the sstem sees in the appendix A and B)

7 A omparison between the iteration methods. High order iteration method The given equations an be solved b using High order iteration method. The general orm o High order iteration method or nonlinear sstems is '' ( xk) ( xk) ( xk) ( xk) ( xk) Y = K Y K ' ' ' ( xk) ( xk) ( xk), or ' '' ' ( xk) ( xk) ( xk) ' ( xk) ( xk) ( xk) ( xk) ( xk) ( YK YK) = 0(7) Eah o these ma be expanded in Talor expansion o ( x ) to a higher order or High order iteration method, see Eq. (). Funtions are evaluated rom the solution o interested untions Eq. () through Eq. (9). ' '' ' ( xk) ( xk) ( xk) = ( xk) ( xk) ( xk) ( xk) (8) =,,,. The independent set o derivatives is obtained b solution o matrix equation that results rom dierentiating with respet to mole ation. The above an be arranged as set o linear equations in the matrix orm, [ A][ Z] [ B] = 0. Where [ A] =,[Z]=[ Δ] and ' '' ' ( xk) ( xk) ( xk) [ B] = ( xk) ( xk) ( xk) ( xk) (9) (Improving the derivative term o the sstem sees in the appendix A and B).5 Jaobian All o the iteration method, the solution are ( xk Δ) 0, it an be arranged as o linear equations in the matrix rom, [ / ] [ Δ ] = { }. The oeiient matrix on the let-hand side is alled the Jaobian o the sstem, see Eq. (0) that ma be solved using b Gauss Elimination. The seond approximation is then { } = { } { δ} =,,5,. See appendix C. k k k 5 Δ 5 Δ = Δ 5 5 Δ 5 (0)

8 aramust Juntarakod and Thanakom Soontornhainaksaeng 5 Numerial experiments We present ombustion equations o gasoline uel to illustrate the eiien method proposed in this paper. We ompare the method with Newton-Raphson method (NRM), Householder s iteration method (HoIM), Adomian Deomposition method (ADM) and High order iteration Method (HIM). The results were ompared the number o iteration (NI), obtained solution (i) [denote b mole ration o ombustion produts] or emissions. Numerial omputations have been arried out b using the sotware MATLAB. The results are presented in Tables and or ompared with, varied temperature K and ixed equivalene ratio is 0.8. The results are presented in Figure and or ompared with, varied equivalene ratio , ixed temperature 000 and 500 K,and the uel is Gasoline ( CH 7 7). Table The solution obtained or the dierene method and Number o iteration o Newton-Raphson method and Adomian Deomposition method Method i 500 K 000 K 500 K 000 K NRM NI= ADM NI=88 H CO N O H O CO H O OH NO N H CO N O H O CO H O OH NO N

9 A omparison between the iteration methods Table The solution obtained or the dierene method and Number o iteration o Householder s iteration Method and High order iteration Method. Method i 500 K 000 K 500 K 000 K H CO N O HoIM H O CO NI= H O OH NO N H CO N O HIM H O CO NI=97 H O OH NO N Conlusions The main goal o this work is to ondut a omparative stud between the iteration method and the Adomian deomposition method. The our methods are powerul and eiient methods that both give approximations o higher aura and losed orm solutions i existing. The omparison gives similar results and supplies quantitativel reliable results. The results show that the rate o onvergene o Newton-Raphson method, Householder s iteration method, Adomian deomposition method and High order iteration method are aster o number o iteration, respetivel. Aknowledgements This researh was supported b Energ Management Sstems and Monitoring & Veriiation Control, Researh & Development Center [ES-MVC]. King Mongkut s Universit o Tehnolog North Bangkok, Thailand.

10 aramust Juntarakod and Thanakom Soontornhainaksaeng Figure : Combustion produts or (C 7 H 7 0.O 0.79N ) Equivalene ratio , T = 000 K, Newton-Raphson method (NRM) Figure : Combustion produts or (C 7 H 7 0.O 0.79N ) Equivalene ratio , T = 500 K, Newton-Raphson method (NRM)

11 A omparison between the iteration methods 5 Reerenes [] C.C. Mei, F. Gao, A Few Numerial Methods or Solving Nonlinear Equations, International Mathematial Forum, (008), 7. [] F. Saleki, R. Ezzati, On the Constrution o New Iterative Methods with Fourth-Order Convergene b Combining revious Methods, International Mathematial Forum, (0), 9. [] R.U. Muhammad, Three-Step Iterative Method with Sixth Order Convergene or Solving Nonlinear Equations, Int. Journal o Math. Analsis, (00), 59. []. ue-on, N. Viriapong, Modiied Adomian Deomposition Method or Solving artiular Third-Order Ordinar Dierential Equations, Applied Mathematial Sienes, (0), pd [5] A.A. Hemeda, New Iterative Method: Appliation to n th -Order Integro Dierential Equations, International Mathema tial Forum,7(0), 7. [] C.D. Rakopoulos, D.T. Hountalas, E.I. Tzanos, G.N. Taklis, A ast algorithm or alulating the omposition o diesel ombustion produts using speies hemial equilibrium sheme, Advanes in Engineering Sotware, 9(99), [7] M. Masood, M.M. Ishrat, Computer simulation o hdrogen diesel dual uel exhaust gas emissions with experimental veriiation, Fuel 87(008), [8] M. Rashidi, Calulation o equilibrium omposition in ombustion produts, Chem. Eng. Tehnol 0(997), [9] M. Grill, M. Chiodi. H. Berner, M. Bargende. Calulating the Thermodnami roperties o Burnt Gas and Vapor Fuel or User-Deined, Fuels, MTZ 05(007), [0] S.R. Turns, An Introdution to Combustion, Conepts and Appliations, MGraw- Hill, New York, 99.

12 aramust Juntarakod and Thanakom Soontornhainaksaeng Appendix A. Equations and derivative equations o Eq. (), Eq. (7). Equation- ' 7 ( x) = 5 5 ' ( x) = 7 ( x ) = ' 5 x ' ( ) = / '' 7 ( x) = / / 5 5 / / / '' ( x) 7 / / '' 5 '' ( x) = / / ( x ) = 7 5/ 5/ ( x) = 5/ 5/ = 5 5 5/ 5/ 5/ ( x) = 5/ 5/ 8 8 ' Equation-7 ' 5 ( x) = ( x ) = d ( ) ' 5 x ' ( ) = '' '' 5 ( x) = / / / '' 5 d d '' ( x) = / / 5 ( x) = 5/ 5/ 5/ d ( x) = 5/ 5/ 8 8 d '' 5 ( x) = / / / '' 5 '' ( x) = / / 5 ( x) = 5/ 5/ 5/ d ( x) = 5/ 5/ 8 8

13 A omparison between the iteration methods 7 Appendix A. Equations and derivative equations o Eq. (8) and Eq. (9). ' 7 ( x) Equation ' ( x) 7 d 5 ( x ) = d ( ) ' 5 x ' ( ) = '' 7 ( x) = / = = 5 5 / / / '' ( x) 7 d 5 / / / '' 5 '' ( x) = / 7 ( x) = 5/ 8 = 5 5 5/ 5/ 5/ 8 8 ( x ) = d 5/ 5/ 5/ ( x) = 5/ Equation-9 / ' 7 ( x) = ' 7 5 ( x) = ( x ) = d ( ) ' 5 ( x ) = d ' '' 7 ( x) = / / '' 7 5 ( x) = / / / '' 5 '' d 7 ( x) = 5/ 5/ ( x) = 5/ 5/ 5/ d Appendix B. The sstem o nonlinear equations with our unknowns. Newton Raphson method [ B ] = F,[ B ] = F,[ B ] = F,[ B ] = F N N N N

14 8 aramust Juntarakod and Thanakom Soontornhainaksaeng Householder s iteration method '' ' '' ' '' ' [ B] = F N F N ( x) ( x) ( x ) ( x ) ( x ) ( x ) [ B ] F F ( x ) ( x ) ( x ) ( x ) '' ' '' ' = N N [ B ] F F ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) '' ' '' ' '' ' = N N [ B ] F F ( x ) ( x ) ( x ) ( x ) '' ' '' ' = N N Adomian deomposition Method '' ' '' ' '' ' B = F F ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) [ ] N N ( ) '' ' '' ' '' ' FN ( ( x) ( x) ( x) ( x) ( x) ( x) ) '' ' '' ' [ B] = F N F N ( ( x ) ( x ) ( x ) ( x )) '' ' '' ' FN ( ( x) ( x) ( x) ( x) ) '' ' '' ' '' ' [ B] = F N F N ( ( x) ( x) ( x ) ( x ) ( x ) ( x )) '' ' '' ' '' ' FN ( ( x) ( x) ( x) ( x) ( x) ( x) ) '' ' '' ' [ B] = F N F N ( ( x) ( x ) ( x) ( x )) '' FN ( ( x ' '' ' ) ( x) ( x) ( x) ) High order iteration Method '' ' '' ' '' ' N N ( ) [ B ] = F F ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) F x x x x x x ' ' ' N ' ' F '' ' '' ' N ( x) ( x) ( x) ( x) [ B] = F N F N ( ( x ) ( x ) ( x ) ( x )) '' ' '' ' '' ' N N ( ) [ B ] = F F ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) F x x x x x x ' ' ' N '' ' '' ' [ B] = F N F N ( ( x ) ( x ) ( x ) ( x )) ' ' F N ( x) ( x) ( x) ( x)

15 A omparison between the iteration methods 9 Appendix C The element o the Jaobian matri This set o linear equations an then be solved or,, 5, and iterative proedures undertaken until the orretions are less than a speiied tolerane. For onveniene, deining ollowing partial derivatives o i i =,, 7,8,9,0, Di = D =,,5, i So that D7 D8 D9 D D D D D D / 0 7 D5 D 5 A D D 0 A D D D D D A D 5 5 A D D D 7 9 A 0 A D D d D 9 A d D d 5 5 A D D D 7 9 A D 0 A D D D 8 D D d D 9 0 A D d D d A D D 9 A D D 0 A D d D 0 A d D d A Reeived: September, 0