Models for droplet heating and evaporation: application to simulation of the autoignition process in diesel engines - PDF

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Zapło/Igitio oe for ropet heatig a evaporatio: appiatio Sergei SAZHIN* Sergey AYNOV ariiu KISYADI Cyri CUA orga HEIKAL Shoo of Egieerig, Fauty of Siee a Egieerig, Uiverity of Brighto, UK oe for ropet heatig
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Zapło/Igitio oe for ropet heatig a evaporatio: appiatio Sergei SAZHIN* Sergey AYNOV ariiu KISYADI Cyri CUA orga HEIKAL Shoo of Egieerig, Fauty of Siee a Egieerig, Uiverity of Brighto, UK oe for ropet heatig a evaporatio: appiatio to imuatio of the autoigitio proe i iee egie A review of iqui a ga phae moe for fue ropet heatig a evaporatio, uitabe for impemetatio ito CFD oe with a view to moeig the proee i Diee egie, i preete. o eribe propertie of hot Diee fue pray, ew ub-moe for pray break-up, ropet heatig a evaporatio a She autoigitio were impemete ito the KIVA II CFD oe. hi utomie verio of the KIVA II oe wa ue to imuate autoigitio i Diee egie. he reut of imuatio were how to be oitet with i-houe experimeta ata referrig to autoigitio timig at i-yier preure from 5 to 9 Pa a a ijetio preure of 6 Pa. he autoigitio eay time i how to be eitive to the hoie of iqui phae moe for ropet heatig, but ot eitive to the hoie of ga phae moe. It i reommee that the effetive therma outivity iqui phae moe a the ga phae moe takig ito aout the effet of fiite thike of the therma bouary ayer are ue for the imuatio of the igitio proe i Diee egie. Some reet o-traitioa eveopmet of the moe referrig to ropet heatig a evaporatio are iue. hee are the moe bae o the oupe outio for the iqui a ga phae, the kieti moe of evaporatio, a the yami eompoitio tehique for the outio of oriary ifferetia equatio eribig ropet heatig a evaporatio a the igitio of the fue vapour/air mixture. Key wor: Diee egie, autoigitio, ropet heatig, ovetio, raiatio INODUCION o imuate autoigitio i a iret ijetio Diee egie oe ee to take ito aout a umber of proee, iuig the yami a break-up of pray, heatig a evaporatio of iiviua ropet a autoigitio of the fue vapour/air mixture i reaiti three imeioa eoure [,]. Whe moeig thee iiviua proee, it i eetia to fi a ompromie betwee the auray of the moe a their CPU effiiey. I pratie, thi e to moeig bae o over-impiti ub-moe of thee proee [3,4]. For exampe, the etaie hemia mehaim of the autoigitio proe wa repae by reue hemia moe, oe of the mot popuar of whih eem to be the She moe [5-]. Dropet heatig wa typiay moee bae o the aumptio that the therma outivity of ropet i ifiitey high [3,4]. I mot ae, iuig the appiatio of reue hemia moe, thee impifiatio are at preet uavoiabe. I ome ae, however, there eem to be room for oierabe ireae i the auray of the pray autoigitio moe without ubtatia CPU peaty. Bertoi a igiaio [] were perhap the firt to raw attetio to the fat that the auray of preitio, by omputatioa fui yami (CFD oe KIVA II, of the heatig, evaporatio a ombutio of Diee fue pray a be ireae if the aumptio of ifiitey high therma outivity of iqui i reaxe. hey uggete that the umeria outio of the heat outio equatio iie the ropet hou be ae to the outio of other equatio i thi oe. Athough thi approah i expete to ireae the auray of CFD preitio, the aitioa omputatioa ot might be too high for pratia appiatio. A umber of impifie moe of ropet heatig a evaporatio, takig ito aout the effet of temperature graiet iie ropet a reiruatio withi them, have bee uggete [-]. Some of thee moe were impemete ito umeria oe foue o imuatig ropet ovetive a raiative heatig, evaporatio a the igitio of a fue vapour/air mixture [8-]. I thee paper, the reut of impemetatio of the aaytia outio of the heat outio equatio iie the ropet, for otat ovetio heat trafer oeffiiet h, ito a zero-imeioa umeria oe, were reporte. hi oe wa the appie to the umeria moeig of fue ropet heatig a evaporatio i oitio reevat to Diee egie [9]. It took ito aout the oupig betwee the iqui a ga phae a eribe the autoigitio proe bae o the the above metioe She moe. he effet of the temperature graiet iie ropet wa ivetigate by omparig the effetive therma outivity (EC moe (ee [5] a the ifiite therma outivity (IC moe, both of whih were impemete ito the zero-imeioa oe. he approah bae o the aaytia outio of the heat trafer equatio iie ropet wa how to be more CPU effetive a aurate tha that bae o the umeria outio of the iretie heat outio equatio iie the ropet [8], a more aurate tha the outio bae o the paraboi temperature profie moe [8]. he reativey ma otributio of therma raiatio to ropet heatig a evaporatio aowe the author to take it ito aout uig a impifie moe, whih oe ot oier the variatio i raiatio aborptio iie ropet [-3] (a more 46 7-SC-47 (P7-C47 7-SC SILNIKI SPALINOWE/Combutio Egie oe for ropet heatig a evaporatio: appiatio Zapło/Igitio etaie aayi of therma raiatio aborptio i iue i [4-6]. It wa poite out that i the abee of breakup, the ifuee of the temperature graiet iie ropet o ropet evaporatio uer reaiti Diee egie oitio wa geeray ma (e tha about 5%. I the preee of the break-up proe, however, the temperature graiet iie the ropet e to a igifiat ereae i evaporatio time. It wa reommee that the effet of the temperature graiet iie the ropet hou be take ito aout i omputatioa fui yami oe eribig ropet break-up, heatig a evaporatio proee, a the igitio of the evaporate fue/air mixture. I thi paper, the reety eveope moe of ropet heatig a evaporatio are reviewe. he the reut of impemetatio of thee moe ito the KIVA II CFD oe are iue. he reut of imuatio of the autoigitio proe i a eviromet typia for Diee egie, uig the utomie verio of thi oe, are iue a ompare with i-houe experimeta ata. Fiay, ome reet o-traitioa approahe to moeig thee proee are briefy ummarie. LIQUID PHASE ODELS he iqui phae moe atuay ue i CFD oe a the oe whih a potetiay be ue are IC (ifiite therma outivity a EC (effetive therma outivity moe. he IC moe are bae o the eergy baae equatio of the ropet a a whoe. he outio to thi equatio a be preete a []: 3ht = = + g ( g exp, ( ρ a g are the iitia ropet temperature a ambiet ga temperature repetivey, a ρ are iqui peifi heat apaity a eity repetivey. Dropet temperature oe ot epe o the itae from the ropet etre i thi ae. Aumig that the proe i pheriay ymmetri, the ropet traiet heatig i fiite iqui therma outivity moe i eribe by the foowig equatio [4]: K = + P( t,, ( t K = k /( ρ i the iqui therma iffuivity, k i the iqui therma outivity, aume to be otat, i peifie at the iitia momet of time a t= = (, P take ito aout the raiative heatig of a ropet. he bouary oitio at = foow from the probem ymmetry / =. Aumig that the ropet i heate by = ovetio from the urrouig ga, a ooe ue to evaporatio, the bouary oitio at the ropet urfae a be writte a: h ( g = ρ L + k =, t L i the atet heat of evaporatio, h i the ovetio heat trafer oeffiiet eribig heat obtaie by ropet from ga (ue to heatig of fue vapour thi heat i e tha the heat ot by ga. Effet of weig are igore at thi tage. he geera aaytia outio of Equatio (, takig ito aout the hage i ropet raiu ue to evaporatio, wou be a iffiut tak. hi ou be oieraby impifie if we take ito aout that thi outio i ue i the umeria aayi whe the time tep i ma. I thi ae we a aume that the ropet raiu i otat, but the effet of evaporatio a be take ito aout by repaig g with the effetive temperature [7]: eff = g + ρ L / h. t he outio to Equatio ( for h=ot a be preete a [4]: i ( eff λ µ = + q exp( κ λ = λ V t i λ µ ( τ λ [ ] exp κ λ ( t τ τ i [ + p λ exp( κ λ t]i λ V τ = κ λ (3 SILNIKI SPALINOWE/Combutio Egie 7-SC 7-SC-47 (P7-C47 47 Zapło/Igitio oe for ropet heatig a evaporatio: appiatio h eff ( t µ =, = k V ( ( / V ( / q, i the ropet iitia temperature itributio, =, κ = k /( ρ, λ are outio to the equatio: o λ + h i λ = λ, h = hr k, V =.5( i λ / λ, V = i( /, P( t, p = V ( ( / V. If we take ito aout the itributio of aborptio of therma raiatio iie ropet, uig the impifie moe uggete i [4], we a preet thi term a: λ 3π P( = w( r, λ Qa Bλ ( ext λ ρ, (4 λ λ B ( r = /, = 5 πλ B λ i the Pak futio: ( ext λ ext [ exp( C /( λext ], C C = W µm 4 /m, C =.439 x 4 µm K. λ i the waveegth i µm. It wa aume that the extera therma raiatio i that of a bak boy. w r, i the ormaie petra power of raiatio per uit voume aborbe iie the ropet. Uig the DP approximatio (ee [4] a a umber of aitioa aumptio a impifiatio, the foowig expreio were obtaie [4]: ( λ w µ * Θ( r ( / r [.6( µ µ / ] + γ ( µ + γ = (5 γ *, µ = µ =, τ = a λ = 4πκ λ / λ, r = (.5 / τ (.6 /, Θ( x = whe x , a Θ( x = whe x ; or [ ξ ( τ τ ] ( / ξ [ exp( ξτ ], 3 ξ τ exp w = (6 τ ( ξτ + 3 τ =, ξ = /( + µ. Equatio (5 wa ue whe τ . 5, otherwie Equatio (6 wa ue. A a λ i Equatio (4 λ a λ eribe the petra rage of therma raiatio whih otribute to ropet heatig. It wa aume that λ =.5 µm, λ =6 µm, a wa auate a [7]: 48 7-SC-47 (P7-C47 7-SC SILNIKI SPALINOWE/Combutio Egie oe for ropet heatig a evaporatio: appiatio Zapło/Igitio = + λ λm., (7 ( λ λ +. m =.46, λ m =3.4 µm. hi moe wa geeraie i [5] to the ae of aymmetri iumiatio of ropet. Further refiemet of thi moe were reporte i [6]. Igorig the epeee of the itributio of therma raiatio aborptio o a t, we a preet the expreio for P( a [-3]: 3σ b 4 P( = a, (8 ext ρ we aume that the raiatio temperature θ i equa to the extera temperature ext. For ow uphur ESSO AF 33 iee fue, it wa fou that the bet approximatio for a a b i the rage 5 5 µ m a ext 3 K i provie by the futio []: a= ( ext / +.8 ( ext /, b= ( ext / ( ext /. he geeraiatio of the above moe to the ae of movig ropet (EC moe a be ahieve by repaig the therma outivity of iqui k by the effetive therma outivity k eff = χ k, the oeffiiet χ varie from about (at ropet Peet umber to.7 (at ropet Peet umber 5 [5]. he variatio of a parameter with temperature a time were aoute for whe aaytia outio (3 wa iorporate ito a umeria oe [4,8,9]. he moe uggete i [] i bae o a a priori aumptio about the temperature profie iie the ropet i the form: (r,t= (t+[ (t (t](/, (t a (t are the temperature i the etre (= a o the urfae (= of the ropet repetivey. he bouary oitio at = i atifie. Subtitutio of Equatio (9 ito the bouary oitio at = give []: (9 =ζ( g - /+ρ Lr /(k, ( t ζ =.5 Nu k g / k, Nu= h /k g i the Nuet umber. Equatio (6 hou atify the equatio of therma baae, whih i obtaie from itegratio of Equatio ( aog the raiu []: ρ 3 t = h( g + ρ t [ L ( ], ( = 3 3 (. From the efiitio of a Eq.( it foow []: t ( = +.ζg / ψ +.ζρ L /( kψ, ( t SILNIKI SPALINOWE/Combutio Egie 7-SC 7-SC-47 (P7-C47 49 Zapło/Igitio oe for ropet heatig a evaporatio: appiatio ψ= +. ζ. Equatio ( a ( give the fu outio of the probem of ropet heatig i the preee of evaporatio. A how i [], i the abee of evaporatio, thi moe give reaoaby aurate reut ue the time i oe to zero. I the ame paper a orretio to thi moe wa uggete whih give aurate reut over the whoe time perio tartig with t=. A aitioa verio of thi moe, takig ito aout the effet of evaporatio, i iue i [3]. hi moe wa bae o the aumptio that = ot. he -quare aw of ropet evaporatio wa moifie to take ito aout ropet heatig. he moifie form of the -quare aw, ae the eipti aw, wa bae o the eipti preetatio of the pot of ropet raiu veru time. he impemetatio of the paraboi moe ito the reearh verio of the VECIS oe of iaro Coutig Egieer wa bae o Equatio (9, i auate from Equatio ( without evaporatio term [8]. he vaue of i the atter equatio i take from the previou time tep. he paraboi moe moifiatio of ropet heatig wa iorporate ito the ODE for ropet temperature a raiu. A foow from VECIS omputatio for reaiti Diee egie parameter, the paraboi moe preit the iitia rate of ireae of ropet urfae temperature to be ighty e whe ompare with the preitio of the iotherma moe. he above metioe paraboi moe i muh imper tha the umeria outio of Equatio ( a the rigorou aaytia outio of thi equatio (outio (3. he vaue of h are otroe by the oitio i the ga phae, iue briefy i the ext etio. Variou approximatio for h are uuay eribe i term of the orrepoig approximatio for the Nuet umber Nu = h / k. Dropet heatig, eribe i the previou etio, i aompaie by ropet evaporatio, eribe by the foowig equatio: t k pg g g =, (3 ρ Sh k g i the ga therma outivity, pg i the ga peifi heat apaity at otat preure, Sh i the Sherwoo umber. he umeria agorithm, bae o the aaytia outio of heat outio equatio (Equatio (3 a (3, a the impifie therma raiatio aborptio moe, wa ue to eveop a zero-imeioa oe [8,9]. hi oe took ito aout the temperature graiet iie the ropet, the oupig betwee iqui a ga phae a eribe the autoigitio proe bae o the She moe. It wa ue to tuy the effet of temperature graiet iie fue ropet o ropet evaporatio, break-up a the igitio of fue vapour/air mixture. he preitio of the oe were vaiate agait experimeta ata pubihe by Bearii et a [9], Nomura et a [3], a aabe et a [3]. I the abee of break-up, the ifuee of temperature graiet o ropet evaporatio i reaiti Diee egie oitio wa geeray ma (-3%. I the preee of the break-up proe, however, the temperature graiet iie ropet ou ea to a igifiat ereae i the evaporatio time uer the ame oitio. hi wa attribute to the fat that the temperature graiet iie ropet ea to a ubtatia ireae i ropet urfae temperature at the iitia tage of heatig. hi ireae, i tur, ea to a ereae i the ropet urfae teio a threho raii of the utabe ropet, aumig that bag a trippig break-up are the omiat mehaim of ropet break-up. Eve i the abee of break-up, the effet of temperature graiet iie ropet e to a otieabe (up to about % ereae of the tota igitio eay time (ompriig the phyia a hemia igitio eay. I the preee of break-up, thi effet wa ehae ubtatiay, more tha havig the tota igitio eay. hi reutio of the tota igitio eay time i uertoo to be a ombie effet of the ifuee of ireae ropet urfae temperature o the hemia igitio eay, a the ifuee of thi temperature o ropet evaporatio (i the preee of break-up proee. It wa reommee that the effet of temperature graiet iie ropet are take ito aout i omputatioa fui yami oe whih eribe ropet break-up a evaporatio proee, a the igitio of the evaporate fue/air mixture. GAS PHASE ODELS he ifferee betwee variou ga moe i eetiay eribe i term of ifferet approximatio of Nuet a Sherwoo umber. hee approximatio are briefy ummarie beow [9]. oe : ( + B 3 (.3e / Pr / 3 =, Sh ( + B (.3e / / + S Nu + B =, 5 7-SC-47 (P7-C47 7-SC SILNIKI SPALINOWE/Combutio Egie oe for ropet heatig a evaporatio: appiatio Zapło/Igitio B i the Spaig ma umber, Shmit umber repetivey. e, Pr, S are eyo (bae o ropet iameter, Prat a oe : ( + B 3 (.3e / Pr / 3 =, Sh ( + B (.3e / / + S Nu + oe : B B i the Spaig heat trafer umber. =, ( + B / / 3 / 3 = (.3e Pr / F( B, Sh ( + B ( +.3e S / / F( B Nu + B F( B, ( + B.7, = ( + B,. B, =, oe 3: ( + B Nu + / 3.77 = ( + [( e Pr max[, e ] ], B ( + B Sh + / 3.77 = ( + [( e max[, e ] ]. B oe 4: ( + B Nu + S / 3.77 = ( + [( e Pr max[, e ] ]/[ F( B ]] B ( + B Sh + S, / 3.77 = ( + [( e max[, e ] ]/[ F( B B ]] B oe 5: ( + B Nu +.76 e B. / / 3 / 3 = ( Pr / F( B, Sh ( + B (.76 e / + S / F( B oe 6: Nu +.57 e ( + B /.7 F Pr / 3 =, Sh = e ( + B /.7 S / 3 =, pf ( g Q = L B F, Q i the heat rate uppie to the ropet by ovetio, L Q ropet heatig. Note that i the origia efiitio of B, the effet of therma raiatio wa iorporate. F Q i the heat rate pet o L oe -5 are bae o the ombiatio of fittig experimeta ata a theoretia aayi of the proee (emitheoretia moe. oe 6 i bae o orreatio whih are iferre merey from the aayi of experimeta ata. SILNIKI SPALINOWE/Combutio Egie 7-SC 7-SC-47 (P7-C47 5 Zapło/Igitio oe for ropet heatig a evaporatio: appiatio Fig. Pot of a veru time for the iitia ga temperature g = 88 K, ga preure p g =3 Pa, ropet temperature = 3 K, raiu =5 µm a reative veoity v = m/. he overa voume of ijete iqui fue wa take equa to mm 3, a the voume of air, the fue wa ijete, wa take equa to 883 mm 3, ext = K. he reut were obtaie bae o the effetive therma outivity (EC moe, the aaytia outio of the heat outio equatio, a uig eve ga phae moe. he iqui a ga moe eribe above were tete for the vaue of parameter typia for Diee egie. A thee moe were impemete ito the above metioe zero imeioa i-houe CFD oe, takig ito aout oupig betwee ga a iqui phae. At firt the iqui moe wa fixe a the effet of ga moe o ropet heatig a evaporatio wa ivetigate [9]. he break-up proee a hemia reatio i the ga phae were igore at thi tage. he temperature graiet iie ropet a reiruatio withi ropet were take ito aout bae o the EC moe. he phyia propertie of fue (exept raiatio propertie were take a for -oeae (ee Appeix A i [9]. he raiatio effet wa eribe by Equatio (8. Dropet were ijete at room temperature ( =3 K ito air at a temperature of 88 K a preure of 3 Pa. he overa voume of ijete iqui fue wa take to be equa to mm 3, a the voume of air, the fue wa ijete, wa take to be 883 mm 3. I thi ae, provie that a fue i evaporate without ombutio, the fue vapour/air mixture i expete to beome oe to toihiometri [9]. he raiatio temperature wa aume equa to K. he iitia ropet raiu a reative veoity were aume to be equa to 5 µm a m/, repetivey. he pot of ropet raiu a urfae temperature veru time for ga phae oe -6 are how i Fig.. A foow from thi figure, the effet of the hoie of ga moe o time evoutio of i reativey ma, epeiay at the very iitia tage of heatig. However, the effet o time evoutio of i eary viibe. he ifferee i the evaporatio time preite by variou moe a reah amot %. If we aume that oe 6 i the mot aurate, a the oe bae merey o experimeta ata, the we a oue that the mot aurate emi-theoretia moe i oe 4 a the eat aurate i oe, ue i [8]. hi i geeray ofirme by the reut of ompario for other vaue of parameter [9]. Our ompario of preitio of iqui phae moe wa bae o oe 4 for the ga phae. he mai avatage of oe 4 ompare with oe 6 i that the former take ito aout the ueryig phyi more auratey tha the atter [9]. I otrat to the ae without therma raiatio, the maxima o the temperature pot a be ee. A poite out i [3,33], i the abee of raiatio, the ropet temperature approahe ome equiibrium or wet-bub' temperature, at 5 7-SC-47 (P7-C47 7-SC SILNIKI SPALINOWE/Combutio Egie oe for ropet heatig a evaporatio: appiatio Zapło/Igitio whih a of the heat omig to the ropet urfae from the ga i pet o evaporatio (atet heat, a the et heat peetratig to the iqui phae beome zero. I the preee of raiatio, however, the ropet urfae temperature otiue to rie above that wet-bub temperature, ue to raiatio eergy uppie to the ropet iterior. A the ropet urfae temperature rie, the heat omig to the ropet urfae by ovetio ereae, but the heat pet i evaporatio ireae. A a reut, the iretio of the et heat fux iie the ropet beome egative (heat fow from ropet etre to ropet urfae. Durig the proe of evaporatio, the tota raiative power aborbe by the ropet (ropet voume time P( ereae approximatey proportioay to.6. he power ot by the ropet urig the outio heat trafer from the ropet etre to the ropet urfae i approximatey proportioa to ropet urfae area ivie by ropet raiu (proportioa to. Hee, at a ertai ropet raiu, the power ot by the ropet urig the outio heat trafer beome equa to the raiative power aborbe by the ropet. hi orrepo to the maximum ropet average temperature. he momet thi happe i expete to be oe to the momet whe the maximum ropet urfae temperature i reahe. Uig the ame iput parameter a i Fig., the pot of ropet raiu a urfae temperature veru time were auate for ga phae oe 4 a IC a EC iqui phae moe [9]. he evaporatio time preite by both iqui phae moe ture out to be oe. However, a otieabe ifferee i temperature wa oberve at the iitia tage of ropet heatig a evaporatio. hi e to a rather trog epeee of the preite tota autoigitio eay o the hoie of iqui phae moe. hi i reate to trog o-iear epeee of the hemia part of thi eay o ga temperature i the viiity of ropet. Ao, the hoie of a iqui phae moe i importat whe ropet break-up proee are take ito aout [8,9]. Severa iqui a ga phae moe were ue for moeig ropet heatig a evaporatio, together with the autoigitio of the mixture of air a fue vapour proue by evaporatig ropet [8,9]. he hemia part of the autoigitio proe wa moee bae o the She moe i the form uggete i []. It wa poite out that the tota igitio eay (phyia a hemia eay epe weaky o the hoie of the ga phae moe. It epeee o the hoie of the iqui phae moe ture out to be trog, i agreemet with [8]. I the preee of ropet break-up proee, the evaporatio time a the tota igitio eay epe both o the hoie of ga a iqui phae moe [8,9]. he
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