MATHEMATICAL MODELING OF METHANE COMBUSTION

Transcription

1 MATHEMATICAL MODELING OF METHANE COMBUSTION Mlada KOZUBKOVÁ 1, Jaroslav KRUTIL 2, Maran BOJKO 3, Václav NEVRLÝ 4 Research artcle Abstract: Key words: The paper presents the process of the creaton of the mathematcal model of methane turbulent combuston usng ANSYS FLUENT 13.0 software. The decommssoned mathematcal model for speces transfer wth chemcal reacton s descrbed, where burnng s based on stochometrc equatons of perfect combuston. Work also analyzes the approprateness of models dealng wth the knetcs of burnng and descrbes ther mutual comparson. ANSYS FLUENT, methane, numercal modelng, chemcal reactons, combuston. Introducton The bass of the combuston process s a burnng fuel. Burnng s a physcal and chemcal process whch combnes a combustble matter and oxdzer, whle chemcal reacton occurs, accompaned by heat generaton chemcally bound n the fuel and lghtng effect. Ths lght effect s a result of product temperature whch reached the vsble spectrum. Therefore we talk about burnng. (Kozubková and Krutl, 2012). Three basc factors of burnng process are requred: combustble materal (sold, gaseous, lqud fuel), oxdzer (usually oxygen), ntalzaton source wth suffcent energy and temperature (flame, hot surfaces, sparks). Materals and methods Prologue The problem defnton of mathematcal modelng of turbulent combuston s a very complex and lengthy process. Mathematcal model of mass, momentum and heat transfer would nclude the followng equaton (Shabanan et al., 2012): the contnuty equaton, equatons of moton, energy equaton. Boundary condtons and physcal propertes of these models can be defned as ether constant or temperature dependent. When modelng chemcal reactons, the model wll be expanded by the followng equatons: equaton of heat transfer ncludng member characterzng the heat generated by chemcal reactons, equaton for the mass fractons of speces wth chemcal reactons. However, only mathematcal model dealng wth burnng of gaseous mxture s analyzed n ths paper. Ths burnng s presented by one stochometrc equaton and s called the perfect combuston. The chemcal equaton descrbng the perfect methane combuston (oxdaton) has the followng form (Bebčák et al., 2009): CH4 2O2 CO2 2H2O (1) In the ANSYS FLUENT 13.0 program there are several approaches to the modellng of chemcal reacton n gases (Rchardson and Chen, 2012). To compare the knetcs of combuston, a model based on speces transport and chemcal reacton was chosen. Ths model s based on the soluton of transport equatons for speces mass fractons wth the reacton mechansm of chemcal reactons. 1 VŠB - Techncal Unversty of Ostrava, Faculty of Mechancal Engneerng, Department of Hydromechancs and Hydraulc Equpment, Ostrava, Czech Republc, mlada.kozubkova@vsb.cz 2 VŠB - Techncal Unversty of Ostrava, Faculty of Mechancal Engneerng, Department of Hydromechancs and Hydraulc Equpment, Ostrava, Czech Republc, jaroslav.krutl@vsb.cz 3 VŠB - Techncal Unversty of Ostrava, Faculty of Mechancal Engneerng, Department of Hydromechancs and Hydraulc Equpment, Ostrava, Czech Republc, maran.bojko@vsb.cz 4 VŠB - Techncal Unversty of Ostrava, Faculty of Safety Engneerng, Department of Fre Protecton, Ostrava, Czech Republc, vaclav.nevrly@vsb.cz 22

2 Mathematcal model of speces transfer wth chemcal reacton ANSYS FLUENT calculates wth "tmeaveragng _ values of the speces local mass fractons" Y t. They are descrbed by smlar balancng equaton as n the case of energy equaton, whch have ths shape n conservatve form (Kozubková, 2003): Y uy J, RS (2) t x x _ where ρ s densty, u s tme-averagng component of flow velocty. On the rght sde R s the rate producton of speces through chemcal reactons and S s the rate of the producton ncrease of dstrbuted phase. The mentoned equaton (2) s vald for N-1 speces, where N s the total number of speces phase n the system. Dstrbuton of speces can be carred n dfferent assumptons. Usually the dstrbuton can be dstngushed for lamnar and turbulent flow (Kozubková, 2003). In the case of lamnar flow n equaton (2), J, represents the dffuson flux of speces and s defned as: Y J, D m, (3) x where D,m s the mass dffuson coeffcent for speces n the mxture. In turbulent flows, the mass dffuson for speces s expressed n the followng form: t Y J Sc t x j (4) where Sc t s the turbulent Schmdt number ( Sc t t, where μ t s the turbulent vscosty and Dt D t s the thermal dffusvty, The default Sc t s 0,7). It s mportant to say that the mass dffuson coeffcents (for multcomponent mxtures) are calculated usng the knetc theory (Šrůtek, 2009). Models descrbng the rate of speces producton The reacton rates that appear as source term n equaton for speces transfer are computed for lamnar flow usng Arrhenus expresson, for turbulent flow they are modeled n accordance to the work of Magnussen and Hjertager and are called the eddy-dsspaton model (Ansys, Inc, 2011a). Constant actvaton energy and pre-exponental factor have sgnfcant effects on the results. In the specalzed lterature there exst many varants of these constants, for example Zambon Chellah, Pur-Seshadr, Andersen et al, Bbrzyck-Ponsot etc. In ths case, the constants of one-equaton model by Zambon Chellah are used (Kozubková and Krutl, 2012): Pre-exponental factor: 1, [cm 3.mol -2.s -1 ], Actvaton energy: [cal.mol -1 ]. For the solvng of producton rate of speces through chemcal reactons, ANSYS FLUENT defnes these models (Ansys, Inc, 2011a): Lamnar fnte-rate model - the effects of turbulent fluctuatons are gnored and reacton rates are determned by the Arrhenus knetc expresson. Eddy-Dsspaton model - reacton rates are assumed to be controlled by the turbulence, so expensve Arrhenus chemcal knetc calculatons can be avoded. The model s computatonally cheap, but for realstc results, only one or two step heat-release mechansms should be used. Fnte-rate/Eddy-Dsspaton model - combnaton of the two prevous models. Eddy-Dsspaton-Concept model (EDC model) - ths model ncludes a very detaled knetcs of combuston n the flame. Because the specfc turbulent task was tested, the solvng of the frst model (lamnar model) was no longer consdered. Eddy-Dsspaton model Whle the chemcal reacton proceeds rapdly, the total reacton rate s controlled by turbulent mxng. Bascally, there are two basc types of reactons, wth the premxed and non-premxed reactants. ANSYS FLUENT provdes a turbulence-chemstry nteracton model based on the Magnussen and Hjertager work (called the eddy-dsspaton model). The average rate of producton of speces n reacton k s gven by the smaller of the two expressons below: N R Y R mn, mn R M ka, k1 k R Rk, M, R Y P P, kab k N j, km j j (5) where Y P s the mass fracton of any product speces (P), Y R s the mass fracton of a partcular reactant (R), A s an emprcal constant (equal to 4) and B s 23

3 an emprcal constant (equal to 0.5), ρ s the densty of speces. The chemcal reacton rate s governed by the large-eddy mxng tme scale k/ε as n the eddy-breakup model of Spaldng. The process of chemcal reacton proceeds when the flow s turbulent (k/ε < 0) (Ansys, Inc, 2011a; Kozubková et al., 2008). Fnte-rate/Eddy-Dsspaton model Another turbulent model s a combned fnte-rate/eddy-dsspaton model. In ths model, the rate of reacton s determned by the Arrhenus and by eddy-dsspaton equaton. Local reacton rate s gven as the mnmum value from these two equatons. Although ANSYS FLUENT allows mult-step reacton mechansms for the eddy-dsspaton model and fnte-rate/ eddy-dsspaton model, these wll lkely produce ncorrect solutons. The reason s that mult-step chemcal mechansms are based on Arrhenus rates and these chemcal mechansms are dfferng for each reacton. In the eddy-dsspaton model, every reactons have the same rate and therefore the model wll be used only for one-step (reactant product), or two-step (reactant ntermedate product, ntermedate product product) global reactons (Ansys, Inc, 2011a; Kozubková et al., 2008). Eddy-Dsspaton-Concept (EDC model) In ths model, the mult-step chemcal knetcs mechansm s ncluded. Ths model assumes that reacton occurs n small turbulent structures, called the fne scales (Ansys, Inc, 2011a; Kozubková et al., 2011). Due to chemcal reacton for speces, the source term R ncluded n the equatons of energy s calculated usng the relaton (6), where Y s the mass fracton of speces, Y * s the mass fracton of speces for the fne scaled, C ξ s a volume fracton constant (equal 2,1377), C r s a tme scale constant (equal 0,4082), v s knematc vscosty (Ansys, Inc, 2011a; Kozubková et al., 2008). R * Y Y Cr 1,5 2 C k C 3 k 2 (6) If the chemcal reacton s too fast, then ths model uses the STIFF mechansm. It s the auxlary mechansm that ncludes a constant actvaton energy and pre-exponental factor. 0,033 m 0,003 m Results Numercal model of non-premxed methane combuston n the tube To solve ths case, the fnte element method s used. The geometry and grd are shown n Fg. 1, where the characterstc dmensons are ndcated. It can be seen very well that the grd model s made up exclusvely of rectangular cells and s composed from 4800 cells. For the smplfcaton, ths model s solved n 2D space as an axally symmetrcal model. The axs of symmetry s dentcal to the axs of the tube. Fg. 1 Geometry and grd of the mathematcal model Boundary condtons of the model are shown n Fg. 1. Fuel enters to area through the narrow slt (see Fg. 1) by the velocty v CH4 = 10 m.s -1. The fuel temperature at nput s 300 K. Oxdzer enters to tested area separately from the fuel. Composton of oxdzer s defned by N 2 = 77 %, O 2 = 23 %. The oxdzer temperature s 300 K and the velocty s v N2,O2 = 0,5 m.s -1. Graphcal evaluaton of results 2 m The am of the work was to create mathematcal models of gaseous fuel combuston and then to compare the results of three basc turbulent models contaned n the numercal software ANSYS FLUENT. The temperature felds and detecton of flame behavor durng combuston were of nterest. Fg. 2 shows the comparson of temperature felds n all three models. 1.80e e e e e e e e e e e e e e e e e e e e e+01 INLET FUEL INLET OXIDIZER AXIS WALL PRESSURE OUTLET B) Eddy-dsspaton C) Eddy-dsspaton concept Fg. 2 Temperature felds n solved models [K] 24

4 Fg. 3 shows a place where methane burns wth oxygen (an envelope of flame). The comparson of three models s ntroduced agan. 3.00e e e e e e e e e e e e e e e e e e e e e+00 Fg. 3 Heat of reacton of gas mxture [W] 1.00e e e e e e e e e e e e e e e e e e e e e e+00 B) Eddy-dsspaton C) Eddy-dsspaton concept B) Eddy-dsspaton C) Eddy-dsspaton concept Fg. 4 Decrease of a methane mass fracton nfluence by combuston gas mxture Process of decreasng the methane mass fracton by combuston of gaseous mxtures s shown on Fg. 4. Temperature profle s compared n concluson. Comparson s made n the axs of tube. We observe n Fg. 5, that the temperatures n all three models are almost dentcal. Temperature [ C] Fg. 5 Temperature profles dependng on the dstance (n axs of tube) Concluson The artcle s devoted to the possbltes of the mathematcal modelng of turbulent methane combuston. Introducton of modelng s focused on the solvng of equatons for speces transport wth chemcal reactons. The work also compares the possblty of usng a mathematcal model of turbulence wth respect to the three dfferent expresson rates of the producton of speces through chemcal reacton. From comparson of the results, t s evdent, that all three of these models acheved very smlar results (temperature feld), see Fg. 5. Models Fnte-Rate/Eddy-Dsspaton and Eddy- Dsspaton Concept dd not acqure maxmum values as hgh as that of Eddy-Dsspaton model. Ths means that t does not burn so ntensely. Fnally, we can say that much accuracy n model results has the value of actvaton energy and pre-exponental factor n Arrhenus expresson. Acknowledgments Fnte-rate/Eddy-dsspaton Eddy-dsspaton Eddy-dsspaton concept 0 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Dstance tube [m] Ths work was supported by the Mnstry of Educaton, Youth and Sports of the Czech Republc va the projects LD11012 and LD12020 (both n the frame of the COST CM0901 Acton). References Ansys, Inc (2011a). ANSYS FLUENT Theory Gude Ansys, Inc (2011b). ANSYS FLUENT User's Gude BEBČÁK, A., KONDERLA, I., DAMEC, J. (2009). Comparson of Effects of Varous Inert Gases on Explosve Range of Combustble Lqud. Transactons of the VŠB - Techncal Unversty of Ostrava, Safety Engneerng Seres. Ostrava, str ISSN (n Czech). BIRD, R. B., STEWART, W. E., LIGHTFOOT, N. N (2002). Transport Phenomena. 2 ed, Wley, 2002, 914 s. ISBN

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