Calculation of Combustion Products Using 6 Species Equilibrium Model

Transcription

1 SWU Egieerig Joural (017) 1(1), วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 Calculatio of Combustio Products Usig 6 Species Equilibrium Model Sompop Jarugthammachote Eergy System Research Group, Faculty of Egieerig at Sriracha, Kasetsart Uiversity Sriracha Campus, Choburi, 030, sompop.j@ku.th ABSTRACT A equilibrium model for hydrocarbo combustio calculatio is developed. 6 major products i combustio gas are cosidered i this model. It gives more accurate result tha that from low temperature combustio model ad it is more simple tha 10 species equilibrium model. I the developmet of the model, Elemetal balace equatios ad equilibrium costat equatios are combied to form oly oe equatio requirig umerical solvig. Newto- Raphso techique with Laguerre- Samuelso iitializatio method is applied to fid the solutio. The simulatio result from the model shows that this model ca be used to provide the iitial values for other models, for example, 10 species model, without temperature ad pressure limitatios. If the mole fractios of major gas species are eeded to kow with high accuracy, the model ca be employed with temperature ad pressure coditios. Keyword: Equilibrium costat, Combustio model, Laguerre-Samuelso iitializatio 1. Itroductio Combustio of fuel i automobiles ad power geeratio systems are major sources of pollutio emissio. The kowledge of the exact gas compositio i exhaust emissio is a importat data for estimatio of a egie performace ad pollutio cotrol i power geeratio system. For combustio aalysis, the thermodyamic equilibrium model has bee ofte used to fid the gas compositio [1,]. However, for detailed aalysis ad high accurate result, it ivolves a large umber of species. The calculatio, therefore, highly cosumes computatioal time ad computer resources. I cylider combustio, Heywood [ 3] ad Ferguso [ 4] suggest that the aalysis should be doe with 10 gas species i exhaust gas for high temperature combustio. To fid 10 costituets, a famous method developed by Olikara ad Borma [5] is frequetly employed. I this method, mole fractios of some gas species are expressed as fuctios of mole fractios of CO, H, O, ad N. Thus, oly four equatios are solved by umerical method. The iitial values of these four gas species are required i umerical solvig ad a low temperature model is ofte used for this purpose [6]. I some studies, researchers usually applied a assumptio of complete combustio of fuel i which the combustio gas cosists of CO, H O ad N (for example [7]). This approach may ot suitable for some applicatios, requirig fairly or high accurate result, because more tha these 3 species ca be detected i combustio gas with sigificat cocetratio. Eve through the combustio takes place with 100 percet theoretical air, the dissociatio ca cause geeratio of other gas products. For example, CO i combustio gas at high temperature ca dissociate ito CO ad O. For H O, it ca

2 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 95 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 geerate H ad O as well as other gas species which are formed by dissociatio accompay with chemical reactios. I fuel lea combustio ( also referred as lea mixture combustio i this paper), oly N, H O, CO, ad O are cosidered to be foud i combustio gas [8-11]. The umber of moles of these species ca be obtaied by usig elemetal balace. For fuel rich combustio, the gas species preseted i combustio gas are N, H O, CO, CO ad H [10,11]. The elemetal balace method ad a equatio of equilibrium costat are itroduced to fid the umber of moles of these products. This simplify method, discussed above, is also kow as low temperature model [6, 1]. However, the simplify method may lack i precisio, because at higher temperatures ( greater tha about,00 K), these major species dissociate ad react to form additioal species i sigificat amouts [3]. As foud i the above discussios, 10 gas species model is required low temperature model or assumptio of complete combustio as iitial values for umerical solvig. Moreover, it also eeds to fid 4 4 Jacobia matrix [4,6,13] i which the derivatives of reduced equatios with respect to the umber of moles of CO, H, O ad N are foud ad these are quite complicated. I calculatio, the Jacobia matrix also highly cosumes computer resources. For the simplify method or low temperature model, it is simple ad cosumes less computer resources ad computatioal time. However, the result has low accuracy as well. To compromise betwee these two methods, 10 gas species model ad low temperature model, a model predictig 6 compositios i combustio gas is developed i this study. CO, H O, N, CO, H, ad O are cosidered i the preset model without eglectig i some coditios as treated i low temperature model. The model ca be used to fid the mole fractios of 6 gas species for hydrocarbo combustio.6 equatios for calculatio the umber of moles of 6 gas species are reduced to oe equatio which is required a umerical method to fid the solutio. This fial equatio is simple to solve by Newto- Raphso method because of o Jacobia matrix ad easily to fid its derivative. The calculatio result of this model is, the, verified with the result from CEA program, developed by NASA [ 14]. Moreover, the model is used to fid the mole fractios of combustio products at high temperature ad pressure. Its results are, the, compared with the results from CEA program calculatig mole fractios of 10 gas species. The compariso is doe oly 6 major products i order to show the performace of the model for high temperature ad pressure applicatios.. Method The problem cosidered i this research is combustio of hydrocarbo fuel i air. The fuel is defied as C X H Y O Z ad air is assumed that it cotais 79% by mole of N ad 1% by mole of O. I this work, oly 6 major gas species i combustio gas are focused. There are CO, H O, N, CO, H, ad O. The combustio reactio for burig oe mole of hydrocarbo fuel ca be writte as: 3.76 CXHYOZ a O N CO CO CO H O H O N CO HO H O N (1) It ca be observed that there are 6 ukow, eeded to fid the values, CO,,,,, CO HO H O ad N Therefore, it requires 6 equatios. The first 3 equatios ca be formed by elemetal balaces, as show i the followig. Carbo balace: X () CO CO

3 96 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 Hydroge balace: Oxyge balace: Nitroge balace: Y (3) HO H Z (4) CO CO HO O 3.76a N (5) For more equatios, the equilibrium costat equatios are used. Two chemical reactios ivolvig the combustio reactio are utilized, there are: Water gas shift reactio: CO HO CO H (6) Oxidatio of carbo mooxide: CO O CO (7) The equilibrium costats correspodig to reactios (6) ad (7), respectively, are: K K CO H (8) 1 COH O CO tot CO O (9) P where tot is the total umber of moles i combustio gas ad P idicates the pressure of combustio reactio. To fid the equilibrium costats, K 1 ad K, Eq. (10) is recommeded to used [15]. l K RT o G T (10) o where R is the uiversal gas costat ad G T is the stadard Gibbs of reactio which ca be calculated by [16]:, (11) G h Ts o o o T i f T T PR o The stadard ethalpy of formatio, h f, T, ad The stadard of etropy, s o T, at specific temperature T ca be obtaied from Ref [17]. However, fidig the equilibrium costats followig the above method, it is for geeral use, but it cosists of at least 4 steps to be doe. Ferguso [4] has show the equilibrium costats as a fuctio of temperature which were obtaied from cure fittig by usig JANAF table. The fitted equatio for K 1 is: l K1.743 T / T /1000 T / (1) I order to fid the equilibrium costat for reactio (9), the data from Ref. [4] is modified ad it gives: - log K l( T / 1000) 5 ( ) / T T T -3-8 (13) To reduce the umber of equatios used to fid the mole fractios of gas species i combustio gas, a ew parameter is itroduced. It is a ratio of mole of CO to mole of CO i combustio gas, as: CR CO (14) CO I fact, there are may ways to reduce Eqs. (-5) ad (8-9). However, as observed from Eq. (1), the gas species that compose of carbo, C, are oly CO ad CO. Therefore, CR is defied as that it is expected to be able to represet the umbers of moles of CO ad CO. I the ext step, the umbers

4 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 97 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 of moles of other gas species are defied i terms of CR. From Eq. (3), H ca be writte i terms of H O as: Y HO H (15) Substitutig Eq. (1) ito Eq. (0), it results: Z K CR 1 CR 1 CR K 1 1 P ( Y) CR1 CR KP 1 X K CR 1 CR K 1 CR 1 P 4X Y 3.76 a /K CR 1 CR1 1 () From Eqs. (8), (14), ad (15), H O ca be expressed i terms of CR as: Eq. () is rearraged ad it, the, shows a fourth order polyomial equatio or quartic equatio i CR as: HO Y KCR 1 (16) Accordig to Eqs. () ad (14), CO ad CO ca be writte, respectively, as: where DCR 4 3 ACR BCR CCR E 0 (3) For O CO CR X CR 1 X CO CR 1, Eq. (9) is employed ad it is foud that: (17) (18) tot CR K (19) O P where K P is K P. For the last equatio which has ot bee used, Eq. (4), Eqs. (16-19) are substituted ito this equatio ad it gives: tot Y X CR Z (KCR) CR 1 CR K 1 P Based o the defiitio of tot (0), it is fially foud that: P P P P 1 1 A K1K X Z, KK 1 X Z B, KP X Z Y K Z 4X Y K C, Z X Y 3.76a K Z Y 3.76a K D, Z X Y /3.76a E Z Y 7.5a (4) To Fid the umbers of moles of all gas species, the fuel ad the amout of air per mole of fuel are firstly defied i order to kow the chemical elemets itroduced to combustio system. The combustio pressure ad temperature are chose. The equilibrium costats ca be estimated from Eqs. ( 1) ad ( 13). CR i Eq. ( 3) is solved. The umbers of moles of gas species ca be foud by usig Eqs. (14-19). For N, it ca be calculated from Eq. (5). tot Y (3.76 a) CR KP CR K X 1 P (1)

5 98 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ Numerical Solvig As see from Eq. (3), CR, which is eeded to fid the value, is preseted i polyomial form. For quartic equatio, the aalytical method is available i some refereces [18, 19]. However, it is quite complicated. The umerical method is aother way applied to fid the roots. Newto- Raphso method is a umerical techique, ofte used to fid the root i oliear equatio. It is, the, selected to solve Eq. (3) i this study. Based o the Newto- Raphso method, if Eq. ( 3) is called f, roots of this equatio ca be foud by: k 1 k f j j f (5) where superscript k is the umber of iteratios ad subscript j represets the umber of roots. f expresses the derivative of f with respect to CR. I this umerical method, the iitial value adopted to fid the solutio to Eq. ( 5) ca be simply obtaied from may cocepts. I this study, a iitializatio method, called Laguerre- Samuelso (LS) [0-], is proposed. This method shows that for the polyomial equatio 1 ax 0 ax 1... a 1xa 0, the roots of equatio are bouded by a 1 1 xlow, high a 1 a (6) 1 However, it is possible that there are 4 roots for Eq. (3) ad i some cases, the umber of roots for quartic equatio is less tha 4, because there are repeated roots. I the ext sectio, the root of Eq. (3), which relates to the physical meaig of combustio process cosidered, is ivestigated ad the Laguerre- Samuelso iitializatio is additioally discussed. 4. Results ad Discussio I the first part, the solutio behavior of Eq. (3) ad LS iitializatio method are ivestigated. Ethae (C H 6 ) is utilized as fuel i this sectio. The combustio is assumed to occur at pressure ad temperature of 1 atm ad 1800K, respectively. Both rich ad lea mixture coditios are explored. Before ivestigatio of characteristic of fuctio f is coducted, a simple verify of the developed mole is doe i order to make sure that the computer code is correctly writte followig the approach discussed i the previous sectio ad it gives reasoable result. First, the equilibrium costats are computed ad their values are validated with the data from Ref [3]. Next, the mole fractios of products i combustio gas are foud at stoichiometric combustio at low temperature, 1600K, for various hydrocarbo fuels. Had calculatio usig elemetal balace is doe with oly 3 gas species, CO, H O, ad N. The mole fractios of these gas species from the model ad elemetal balace method are compared. From the ivestigatios, they show that the equilibrium costats from Eqs. (1) ad (13) match with that from [0] at various temperatures. For mole fractios of CO, H O, ad N, the model provides a slightly differet value from elemetal balace method. However, it is impossible to have the same mole fractio i each gas species from both approaches, because of differet umber of gas species cosidered i the calculatio. Nevertheless, the slightly differet results i all tests ca be implied that the ivestigated model was correctly developed followig the method previously discussed.

6 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 99 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 After simply checkig the model, it is operated to study the behavior of fuctio f. Fig. 1 depicts the variatio of f at differet CR for rich mixture with a 3. (see Eq. (1)). For lea mixture, the behavior of f is quit the same as that foud i rich mixture coditio. From the figure, it expresses that the roots of Eq. (3) are both egative ad positive values. However, the egative real root has o meaig i practice. Oly positive real root is cosidered. Fig. 1. The variatio of f versus CR Fig. magifies the value of f for i which the root correspodig to 0.05 CR 0.6 Fig.. The positive real root of f correspodig to the combustio coditio the combustio coditio is foud. As see from Fig., the root is 0.558, associated with the rich combustio coditio, ad it is the maximum amog all roots. This is also observed i lea mixture case. LS bouds the roots of cosidered case as ad The lower boud, egative value, is useless for this study. From the ivestigatio result, it leads to the summary that oly the upper boud or higher value obtaied from LS method is used as iitial value to fid the root of Eq. (3). 4.1 Validatio To show that the calculatio result from the preset model is correct, the CEA program developed by NASA is applied to validate the result. The compariso is doe with the same coditio ad it focuses oly 6 products i combustio gas. Methae (CH 4 ) is used as fuel ad the combustio pressure is 1 atm. Figs. 3(a) ad 3( b) show the validatio result for rich mixture combustio with equivalece ratio, ( ), of 1.. Both figures preset the mole fractio of gas species at differet temperatures. The symbols represet the results obtaied from the preset model while the results give by CEA program are expressed as lies. Figs. 3(a) ad 3(b) illustrate that the mole fractios of gas species obtaied from preset model match with that from CEA program with very small differet values. However, it is worth to poit out here that the compariso has to be doe with the same sigificat umbers of mole fractio for each gas species. A high value of relative percetage differece ca be foud i compariso of gas species havig very low fractio i combustio gas if the compariso is doe without fixig sigificat umber. For example, CEA program was set to show the result oly 5 decimals ad it gives mole fractio of O at 000K as while the preset model show that it is

7 100 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ The relative percetage differece is 43%. If it is trucated to show oly 5 decimals, the compariso with the same sigificat umber gives zero relative percetage differece. Moreover, the 6 differece of both result, , is very low ad it does ot have sigificat effect to other mole fractios of gas species if it is trucated. each gas species, the symbols completely coicide with the lie. As observed from rich ad lea mixture cases, it ca be fially cocluded that the preset model developed i this study ca provide correct predicted result. Fig. 3. Variatio of gas species at differet temperatures for CH 4 combustio with 1. at P=1 atm, symbols are results from preset model ad lies are results from CEA (a) N, H O, CO, CO, ad H, (b) O For lea mixture with 0.9 the mole fractios of combustio gas compoets at differet temperatures are expressed i Figs. 4(a) ad 4(b). From the figures, they are clearly see that, for Fig. 4. Variatio of gas species at differet temperatures for CH 4 combustio with 0.9 at P=1 atm, symbols are results from preset model ad lies are results from CEA (a) N, H O, CO, ad O, (b) CO ad H It is worth to discuss here that, for rich mixture (Figs. 3(a) ad 3(b)), O caot be foud exhaust gas. However, whe temperature icreases, the dissociatio causes rise of O i combustio gas. Therefore, mole fractio of CO reduces cosequetly mole fractio of CO icreases. Due

8 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 101 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 to H ca react well with O, this results mole fractio of H O grows up, while mole fractio of H decreases. For lea mixture (Figs. 4(a) ad 4(b)), higher temperature makes O dissociate, the mole fractios of CO ad H O, the, reduce. This dissociatio also causes advet of H ad CO i combustio gas. 4. Compariso betwee 10 ad 6 species models As well kow that at high temperature, the compoets i combustio gas are dissociated ad the cocetratios of some products, such as OH, O, H, ad NO, icrease. Predictio of combustio products with cosiderig oly 6 gas species may give less accurate result whe combustio temperature grows up. I this sectio, CEA program is operated to fid mole fractio of 10 gas species. Its results are compared with predicted mole fractios of 6 species obtaied from the preset model. The cocetratios of oly 6 major compoets are compared betwee both models. The limitatio of the preset model is fially discussed. Fig. 5. The variatio of 6 major gas species at differet equivalece ratios for combustio of mixture of C 3 H 8 ad C 4 H 10 at P=1 atm (a) T=1600K, (b) T=000K, (c) T=400K, ad (d) T=600K

9 10 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ gas species focused i use of CEA program are CO, H O, N, CO, H, O, O, OH, H, ad NO. The fuel used i this ivestigatio is the mixture of propae (C 3 H 8 ) ad butae (C 4 H 10 ) at the ratio of 0. 6: There are 4 combustio temperatures iterested to observe the cocetratios of compositios, 1600, 000, 400, ad 600K ad the combustio is assumed to take place at 1 atm. The supply air is varied to geerate lea to rich fuel combustio coditios. This is preseted i terms of equivalece ratio,. The computatioal results from both programs are expressed i Figs. 5( a)- 5( d).from the figures, oly 6 major gas species ( from totally 10) obtaied from CEA program are used i this compariso. From the results, it ca be metioed that the mole fractios of 6 major gas species from the developed model are geerally agree with that calculated from CEA program i all cases of combustio temperatures. However, i Fig. 5(a), the preset model gives the mole fractio of O at lea fuel combustio, 1, a little higher tha that computed by CEA. It may be because of that O ca dissociate ad react to geerate other gas spices (O, OH, ad NO). As expressed i [3], NO ad OH ca be foud i lea mixture coditio eve low combustio temperature ( 1750K). Therefore, the mole fractio of O should be lower tha that predicted from the developed model i which O, OH, ad NO are ot cosidered. Aother poit, has to be discussed here, is that eve mole fractios from the model match with that from CEA program, they cotiuously diverse from CEA results as combustio temperature icreases. From Figs. 5(a)-5(d), the symbols, represetig the data from developed model, are completely overlap to the lies at lower temperature. The, they slightly shift up to locate above the lies whe combustio temperature raises. At T= 600K, the relative percetage differeces (RPD) are averagely higher tha 5%, except N case. Moreover, the higher RPD is foud i the case of O. It varies from % through the studied rage of. As it has bee discussed, the developed model does ot cosider some gas species which compose of oxyge. This leads to over predicted O i combustio gas. For T=400K, the RPD of all gas species, except O case, are usually lower that 5%. However, for CO i lea fuel combustio coditio, the RPD is quite high. For example, at 0.903, CEA program give the mole fractio of CO as while the developed model shows it as The RPD, therefore, is 8.8%. To show the effect of pressure o the predicted result, the combustio pressure is varied from 10 to 30 atm. The mole fractios of combustio products are calculated ad compared with that from CEA program. The computatioal results from both programs for pressure of 0 atm are show i Table 1. The RPD is also expressed i this table. For the case of 10 ad 30 atm, oly RPD is preseted i Table. From Table 1, it ca be foud that the highest RPD occurs i the compariso of O at stoichiometric combustio. Moreover, at 1, the RPD for the case of CO ad H are also high. These pheomea are also observed i combustio pressure of 10 ad 30 atm. At specific, the RPD for CO icreases as pressure icreases. For CO, H O, ad N, the RPD slightly reduces whe pressure icreases. As the pressure chage from 10 to 30 atm, the RPDs for all gas species, however, margially chage.

10 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 103 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 Table 1 Compariso of mole fractios of gas species obtaied from CEA program ad the preset model at 400 K, 0 atm xco xco xh xh O xn xo CEA Model RPD CEA Model RPD CEA Model RPD Table The relative percetage differeces for compariso of gas species at 10 ad 30 atm RPD P (atm) xco xco xh xh O xn xo To show the predictio ability of model i widely rage of, the developed model is performed to provide the mole fractios of 6 major gas species for 0.5. The combustio temperature ad pressure are 400K ad 10 atm, respectively, which are the upper accepted boud, previously discussed. The predicted results are compared with that from CEA program. There are, however, oly 6 species cosidered i the compariso. Fig. 6 shows the mole fractios of 6 major gas species, derived from both models. It is clearly see that mole fractios of all gas species preseted by the developed model almost completely cocur with these from CEA program. Fig. 6. Variatio of mole fractios of 6 major gas species for combustio of mixture of C 3 H 8 ad C 4 H 10 at 400 K, 10 atm for 0.5

11 104 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 Oly mole fractios of H O ad O i lea mixture rage show slightly differet values i compariso of data from both models. 5. Coclusio A simple approach, used to fid mole fractios of 6 major gas species i combustio gas, was developed i this study. The mole fractios obtaied from this model ca be employed as iitial values for other models, for example, 10 species model. Moreover, it ca also be used to fid mole fractios of major products i combustio gas. As foud that some combustio models are required iitial values which are mole fractio of some gas species, the developed model i this study ca serve for this propose. Due to the fact that the iitial values, which approach to the solutios, ca save computatioal time ad protect the calculatio from divergece, use of the developed model for iitializatio ca achieve the covergece to the solutio with less iteratio i calculatio. Moreover, the model developed i this study ca be used to fid mole fractios of 6 major combustio products with accurate result. However, if high accurate result is desired, the model should be performed with limitatio of temperature ad pressure ot higher tha 400K ad 10 atm, respectively. 6. Refereces [1] X. Li, J.R. Grace, A.P. Watkiso, C.J. Lim, ad A. Ergüdeler, Equilibrium modelig of gasificatio: a free eergy miimizatio approach ad its applicatio to a circulatig fluidized bed coal gasifier, Fuel; vol. 80, pp , 001. [] S. Jarugthammachote ad A. Dutta, Equilibrium modelig of gasificatio: Gibbs free eergy miimizatio approach ad its applicatio to spouted bed ad spout-fluid bed gasifiers, Eergy Coversio ad Maagemet, vol. 49, pp , 008. [3] Heywood, J. B., Iteral Combustio Egie Fudametals, McGrawHill, Sigapore, pp. 9 93, [4] Ferguso, C. R., Iteral Combustio Egies, Applied Thermoscieces, Joh Wiley ad Sos, New York, pp , [5] C. Olikara ad G. L. Borma, A computer program for calculatig properties of equilibrium combustio products with some applicatios to I. C. egies, SAE Techical, Paper o , [6] H. K. Kayadele ad Y. Üst, Predictio of equilibrium products ad thermodyamic properties i HO ijected combustio for C α H β O γ N δ type fuels, Fuel, vol. 113, pp , 013. [7] Y. Haseli, I. Dicer, ad G.F. Naterer, Uified approach to exergy efficiecy, evirometal impact ad sustaiable developmet for stadard thermodyamic cycles, Iteratioal Joural of Gree Eergy, vol. 5, pp , 008. [8] T. Sriivas, A. V. S. S. K. S. Gupta, B. V. Reddy ad P. K. Nag, Parametric aalysis of a coal based combied cycle power plat It. J. Eergy Res., vol. 30, pp , 006. [9] S.M. Aithal, Impact of EGR fractio o diesel egie performace cosiderig heat loss ad temperature- depedet properties of the workig fluid It. J. Eergy Res. Vol. 33, pp , 009.

12 วารสารว ศวกรรมศาสตร มหาว ทยาล ยศร นคร นทรว โรฒ 105 ป ท 1 ฉบ บท 1 เด อนมกราคม ม ถ นายน พ.ศ.560 [10] E. Abu-Nada, I. Al-Hiti, B. Akash, ad A. Al- Sarkhi, Thermodyamic aalysis of sparkigitio egie usig a gas mixture model for the workig fluid. It. J. Eergy Res., vol. 31, pp , 007. [11] A. E. Khalifa, M. A. Atar, ad M.S. Farag, Experimetal ad Theoretical Comparative Study of Performace ad Emissios for a Fuel Ijectio SI Egie with Two Octae Bleds, Arab. J. Sci. Eg., vol.40, pp , 015. [1] C.D. Rakopoulos, D.T. Houtalas, E.I. Tzaos, ad G.N. Taklis, A fast algorithm for calculatig the compositio of diesel combustio products usig 11 species chemical equilibrium scheme, Advaces i Egieerig Software, vol. 19, pp , [13] R. Ada, H.H. Masjuki, ad T.M.I. Mahlia, Mathematical modelig o the effect of equivalece ratio i emissio characteristics of compressio igitio egie with hydroge substitutio, Applied Mathematics ad Computatio, vol. 17, pp , 011. [14] S. Gordo ad B. J. MaBride, Computer program for calculatio of complex chemical equilibrium compositios with applicatios; I. aalysis NASA Referece Publicatio 1311, NASA, [15] S. Jarugthammachote ad A. Dutta, Thermodyamic equilibrium model ad secod law aalysis of a dowdraft waste gasifier, Eergy vol. 3, pp , 005. [ 16] Wark K., Advaced Thermodyamics for Egieers, McGrawHill, New York, pp. 413, [17] B.J. McBride, S., ad M.A. Reo, Coefficiets for calculatig thermodyamic ad trasport properties of idividual species, NASA Techical Memoradum 4513, NASA, [18] Abramowitz, M. I: Abramowitz, M. ad Stegu, I. A., editors, Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, Dover, New York, pp , 197. [19] E.L. Lee, Graphical Discussio of the Roots of a Quartic Equatio The America Mathematical Mothly, vol. 9, pp , 19. [0] K.A. Redish, O Laguerre's Method It. J. of Math. Edu. i Sci. ad Tech., vol. 5, pp , [1] Jese, S. T. The Laguerre Samuelso Iequality with Extesios ad Applicatios i Statistics ad Matrix Theory, MSc Thesis, Departmet of Mathematics ad Statistics, McGill Uiversity, [] U. K. Deiters ad R. Macías- Salias, The calculatio of desities from cubic equatios of state: revisited, Idustrial & Egieerig Chemistry Research, vol. 53, pp , 014. [3] Borgakke C. ad Sotag R.E. Fudametals of Thermodyamics, Joh Wiley & Sos, Sigapore, pp. 691, 014.