J. Basic. Appl. Sci. Res., (5)480-486, 01 01, TextRoad Publicatio ISSN 090-4304 Joural of Basic ad Applied Scietific Research www.textroad.com Modificatio of Arrheius Model for Numerical Modellig of Turbulet Flames Ghodrat Ghassabi a, Ali Saeedi b*, Mohammad Moghima c a,b,c Departmet of mechaical Egieerig, Ferdowsi Uiversity of Mashhad, Ira ABSTRACT For fiite rate modelig of reactios, two major models are Arrheius model ad Eddy-dissipatio model. Arrheius model is used to simulate lamiar flames ad Eddy-dissipatio for turbulece reactios. Arrheius model is iaccurate for simulatio of turbulet combustio because of igorig turbulet fluctuatios. I the preset study, the Arrheius model has bee modified usig a sie fuctio to calculate the effect of temperature fluctuatios o reactio rate. Simulatio of combustio has bee doe by Sprit CFD code for the predictio of temperature distributio, rate of reactio ad CO mass fractio. Compariso of results of modified Arrheius model with the eddy-dissipatio model ad Experimetal data show that the modified Arrheius model has good agreemet with the eddy-dissipatio model ad has a qualitative tred same as experimetal data. Keywords: Eddy-dissipatio, Fluctuatio, Reactio rate, Sie fuctio 1. INTRODUCTION Combustio plays a importat role i may idustrial applicatios because it is the mai source of producig power ad eergy. Also from a evirometal poit of view, emissio of pollutats due to combustio causes sigificat health problems [1-3]. Therefore, study of combustio is a importat issue for may ivestigators. Sice experimetal ivestigatio of combustio is expesive, umerical simulatio has bee used for decades. Numerical methods have become powerful tools to simulate complex combustio processes ad to uderstad the physics ivolved [4]. For simulatio of combustio, Arrheius model, the eddy-dissipatio model, ad the PDF model are widely used i CFD. The PDF model could be able to simulate detailed fiite-rate kietics. However, it eeds a large computer memory ad computatio time. Therefore, the model is used oly for simple flows [5]. Oe of the most practical models is the eddy-dissipatio model, because it is easy to implemet ad its results are acceptable for premixed ad o-premixed flames [6]. I the eddy-dissipatio model, every reactio is the same ad the model oly takes ito accout the turbulet rate. Therefore, the model should be used oly for oe-step global reactio ad it caot predict radical species. Multi-step chemical mechaisms are based o Arrheius rates, ad the Arrheius model computes rate of reactio usig Arrheius expressios. The Arrheius model is exact for lamiar flames, but is iaccurate for turbulet flames, because this model igores turbulet fluctuatios that are effective o rate of reactio, temperature, ad cocetratio of pollutats [7]. Shag et al. [8] ivestigated Effects of gas temperature fluctuatio o the istataeous char reactio of pulverized coal particle. Their results show that the gas temperature fluctuatio leads to both faster char reactio ad particles size reductio. I the other research[9], they show that the particle istataeous temperature with the gas temperature fluctuatio is differet from that without the gas temperature fluctuatio. Temperature fluctuatios have a sigificat role i NO formatio. The recet study idicates that thermal NOx productio doubles for every 90 K temperature icrease whe the temperature is about 00 K. Zhag ad Zhag [10] studied effects of gas temperature fluctuatio o the NO release from coal particle durig char combustio. Their results represet that NO formatio durig the char combustio is further icreased by the icrease i the fluctuatio amplitude of the gas temperature. Also, their ivestigatios o Istataeous de-volatilizatio of coal particles i a hot gas show that the particles lose their mass at a higher rate with the gas temperature fluctuatio tha without the gas temperature fluctuatio [11]. Therefore, the Arrheius model eeds some correctios for turbulet flames. Our aim i this paper is the modificatio of the Arrheius model for simulatio of turbulet combustio flames. For this purpose, effects of temperature fluctuatios are cosidered by usig a time-varyig fuctio. Also, to ivestigate the accuracy of the modified model, Temperature distributio of this model compares with eddy-dissipatio model ad experimetal data. The results show that the modified Arrheius model has good agreemet with eddy-dissipatio model.. Mathematical modelig The mathematical modelig is the based o work of referece [1]. I this sectio, oly umerical combustio modelig will be determied, because this is the mai work of the preset study. *Correspodig Author: Ali Saeedi, Departmet of mechaical Egieerig, Ferdowsi Uiversity of Mashhad, Ira. Tel: +98 511 8763304, Email: Ali_sg@yahoo.com 480
Saeedi et al., 01 Combustio modelig The eergy additio due to combustio is determied i cosideratio of two-step, irreversible, global reactio followig fiite rate chemistry as: C16H 34 4.5 O 3.76N 16CO17H O9. 1N (1) CO 0.5 O CO The Arrheius model ad the eddy-dissipatio model are used for combustio modelig. Rate of reactio i the eddy-dissipatio model is cotrolled by turbulet mixig ad is determied as followed[13]: (3) R j A mi[ S F m F, m O ] k The Arrheius model computes rate of reactio usig Arrheius expressios. Rate of reactio is described at the mea temperature ad obtaied usig followig the equatio: E ( ) exp( ) C C (4) R T A F O RT I this paper, Temperature fluctuates with time aroud some mea represeted by the form [14]: T (5) 1 a f ( t ) T Where a is the amplitude of the fluctuatio ad f(t) is some time-varyig fuctio i which: 1 (6) 1 f ( t) 1 ad T Tdt 0 T(t) ca be cosidered to be composed of T T (t) where T' is the fluctuatig compoet aroud the mea. Istataeous rate of reactio is: E ( ) exp( ) C C (7) R T A F O RT Dividig the two expressios (4) ad (7), oe obtais: R ( T ) E T (8) exp{( )[ 1 ( )]} R ( T ) R T T Obviously, the, for small fluctuatios: T a f ( t) (9) 1 ( ) a f ( t) T [1 a f ( t)] The expressio for the mea rate is writte as: R ( T ) 1 R ( T ) 1 E (10) dt exp( a f ( t )) dt 0 0 R ( T ) R ( T ) RT 1 E [1 a 0 RT But recall: 0 f ( t) dt 0 f ( t ) 1 E ( RT a f ( t ))...] dt If the fluctuatios are cosidered siusoidal, the [14]: 1 E R( T ) R( T)[1 ( 4 RT a ) ] 3. Numerical Procedure Sprit CFD code, that is a early versio of Fluet [15], was used to simulate the processes iside the furace [16]. The gas coservatio equatios were solved usig a cotrol-volume based computatioal procedure. The power law scheme was used to discretize the covective terms. The flow field pressure liked equatios were solved by the SIMPLE algorithm. The set of algebraic equatios were solved sequetially with the lie-by-lie method which is the combiatio of Gauss-Seidel method ad the tridiagoal-matrix algorithm. The covergece criterio was determied by the requiremet that the maximum value of the ormalized residuals of ay equatio must be less tha 1 10-5. A umerical mesh of 100 36 grid odes was used after several experimets, which showed that further refiemet i either directio did ot chage the result () (11) (1) 481
J. Basic. Appl. Sci. Res., (5)480-486, 01 (maximum differece i velocity ad other scalar fuctios i the carrier phase) by more tha %. The grid spacig i axial ad radial directios were chaged smoothly to miimize the deterioratio of the formal accuracy of the discretizatio scheme due to variable grid spacig ad i such a way higher cocetratio of odes occur ear the ilet. 4. RESULTS AND DISCUSSION Numerical simulatios are performed for a cylidrical combustor with the iteral diameter 0 cm ad legth 50 cm. Figure 1 shows furace schematic plae. The furace boudary coditios are give i table 1. The fuel spray is cosidered to cosist of a fiite size rage, with the size distributio specified by the Rosi Rammler fuctio. Figure 1. Schematic of the combustor Table1- Boudary coditios Velocity of Ilet Air(m/s) Swirl Number of Ilet Air Fuel Mass Flow(Kg/s) Equivalece Ratio Miimum diameter of fuel droplets Maximum diameter of fuel droplets Ilet Air Temperature(K) Chamber Wall Temperature(K) Fuel Spray Temperature(K) 3.45 0.5 5.0*10-4 0.44 [10-50] µm [60-10] µm 600 1300 333 To establish the accuracy of the preset study, a compariso of the axial temperature distributio betwee umerical modelig result ad experimetal data uder a similar coditio is preseted i Figure. It is observed that umerical results are i good qualitative agreemet with the experimetal data tred [17]. Also, this CFD code has bee validated i the previous works [4, 1, 16]. Figure. The compariso of axial temperature distributio betwee umerical result ad experimetal data 48
Saeedi et al., 01 Figures 3 represet cotour of O reactio rate for three models. It is see that oxyge ad fuel bur as soo as they eter the combustor for the eddy- dissipatio model. Also, the modified Arrheius model ca predict this behavior very close to the eddy-dissipatio model result. However, reactats bur slowly for the Arrheius model. The reaso is the Arrheius model igores turbulet fluctuatios that its maximum is i the jet outlet. Figure 3. The compariso cotour of O reactio rate at the ilet Figure 4 compares the predictios of radial temperature distributio for the Arrheius model, the modified Arrheius model, ad the eddy-dissipatio model at the ilet. The results show that tred ad behavior of the modified Arrheius model is the same as the eddy-dissipatio model. However, the behavior of the Arrheius model is very differet from the eddy-dissipatio model. Figure 4. The compariso of radial temperature distributio at the ilet 483
J. Basic. Appl. Sci. Res., (5)480-486, 01 Figure 5 demostrates the effect of the ilet turbulece kietic eergy o computed maximum temperature iside the combustor for the Arrheius model, the modified Arrheius model ad the eddy-dissipatio model. I this figure, it is observed that the modified Arrheius model results have bee affected by turbulece kietic eergy. Also, these results have a qualitative tred same as the eddy-dissipatio model. The turbulece kietic eergy has a sigificat effect o temperature because of it causes ehacemet of the mixig rate ad therefore a better combustio. Never ca the Arrheius model predict the effect of turbulece kietic eergy o combustio. Figure 5. Effect of the turbulece kietic eergy o maximum temperature iside the combustor A compariso of the axial temperature distributio betwee the modified Arrheius model ad the eddydissipatio model is preseted i Figure 6. It ca be see that the modified Arrheius model has good agreemet with the eddy-dissipatio model. Figure 6. The compariso of predictio of axial temperature distributio Figure 7 displays the effect of equivalece ratio o the output temperature ad CO mass fractio for the modified Arrheius model ad the eddy-dissipatio model. It ca be see that by icreasig equivalece ratio temperature reaches a peak ad the decreases for both models. Also, CO mass fractio decreases to a miimum ad the icreases for the modified Arrheius model. However, CO mass fractio is the first costat ad the icreases for the eddy-dissipatio model. This icoherece ca be attributed to this reaso that the model caot predict radicals such as CO mass fractio. 484
Saeedi et al., 01 Figure 7 The compariso of output temperature ad CO mass fractio uder differet equivalece ratios 5. Coclusio I this paper, effects of turbulet fluctuatios are cosidered by usig a time-varyig fuctio for Arrheius model. The Sprit CFD code is used to predict temperature distributio ad CO mass fractio. The results of modified Arrheius model are compared with eddy-dissipatio model ad Experimetal data. The followig coclusios are take from the aalysis of the results: The temperature fluctuatios have the sigificat role i the rate of reactio. The modified Arrheius model has good agreemet with the eddy-dissipatio model ad has a qualitative tred same as the experimetal data. The modified Arrheius model is better tha the eddy-dissipatio model i predictig the mass fractio of carbo mooxide. REFERENCES [1] Curtis, L., et al., 006. Adverse health effects of outdoor air pollutats. Eviromet Iteratioal,3(6): p. 815-830. [] Ramaatha, V. ad Y. Feg, 009. Air pollutio, greehouse gases ad climate chage: Global ad regioal perspectives. Atmospheric Eviromet,43(1): p. 37-50. [3] Kampa, M. ad E. Castaas, 008.Huma health effects of air pollutio. Evirometal Pollutio,151(): p. 36-367. [4] Ghasemi, A., A. Saeedi, ad M. Moghima, 011.Applicatio of Multi-Objective Optimizatio for Pollutats Emissio Cotrol i a Oil-Fired Furace. Joural of the Chiese Society of Mechaical Egieers,3(4): p. 303-31. [5] Zhou, L.X., et al., 000.Secod-order momet turbulece-chemistry models for simulatig NOx formatio i gas combustio. Fuel,79(11): p. 189-1301. [6] Brik, A., et al.,000. Possibilities ad limitatios of the Eddy Break-Up Model. Combustio ad Flame,13(1-): p. 75-79. [7] Veyate, D., Vervisch, L., 00. Turbulet combustio modelig. Progress i Eergy ad Combustio Sciece, 8(3): p. 193 66. [8] Shag, Q., J. Zhag, ad L. Zhou, 005. Effects of gas temperature fluctuatio o the istataeous char reactio of pulverized coal particle. Fuel,84(16): p. 071-079. 485
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Saeedi et al., 01