Reduced Order Models for Droplet Evaporation with Internal Circulation Introduction Numerical Formulation

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1 educed Ode Model o Doplet Evapoation with Intenal Ciculation K. Koenlein,. Kazakov, and F.L. Dye Mechanical and eopace Engineeing, Pinceton Univeity, Pinceton, NJ Intoduction Intenal ciculation geneated by elative ga/dop velocitie ha a igniicant and welldocumented eect on evapoation and combution o liquid doplet (Law, 198; Faeth, 1983; Dwye, 1989). Lage eynold numbe have typically been o inteet becaue o elevance to pay. Intenal ciculatoy motion in mall (~0 µm diamete) doplet ae quickly damped by vicoity but can contibute igniicantly to oveall behavio due to augmented heat and ma tane duing thei hot evapoative hitoie. Even when no elative ga/dop motion i initially peent, intenal liquid motion can be impated duing doplet geneation. Moeove, intenal motion alo can aie om luid intabilitie a a eult o uace tenion gadient induced by cicumeential uace tempeatue vaiation, and o multi-component doplet, cicumeential uace compoition. Thee have been eveal ecent numeical and expeimental tudie uthe exploing thee phenomena (Niazmand et al., 1994; Shih and Megaidi, 1995; Dwye and Shaw, 001; Savino and Fico, 004), but none have addeed condition that include local ga phae heat eleae (combution). In the lage (~1 mm diamete) doplet typically invetigated in micogavity eeach, even unde quiecent ga/dop condition, intenal motion have been obeved to peit ove the entie doplet buning lietime (Choi and Dye, 001). The eect o intenal liquid phae motion on ma and enegy tane have been included in numeical pheo-ymmetic modeling by utilizing an eective liquid phae Lewi numbe which wa much lage than that aociated with the themophyical popetie o the liquid component (Machee and Dye, 1997). Moe ecently, imila modiication wee equied to model alling dop expeimental eult o ethanol doplet combution (Kazakov et al. (003). To include liquid phae intenal motion in a ully coupled, dicete manne eult in a computationally contained omulation that likely will equie impliication in othe ubmodel component o ga phae tanpot, combution chemity, ooting, adiation coupling, and phae tanomation. The long tem goal addeed to which the peent eot contibute i to develop a moe igoou educed-dimenional (e.g., quai-one-dimenional) omulation that ucceully emulate the impotant eect o multi-dimenional intenal liquid phae motion. In ode to moe peciely adde pheoymmetic micogavity combution and to include low ga/dop convection, it i impotant to account o intenal phae liquid motion moe igoouly. The calibation and validation o any educed model equie availability o detailed (theoetical and/o expeimental) data. In the cae o intenal liquid motion inide the doplet, the diect expeimental data ae vey cace at bet, and clealy inuicient in tem o the eolution equied. Hee we develop a new detailed two-dimenional computational model that can be applied to geneate the equied calibation inomation o both liquid doplet evapoation and combution cae. The peented detailed two-dimenional computational model ully eolve the low ield o iolated doplet, and account o vaiable themo-phyical popetie and ee uace bounday condition. icou and themo-capillay eect ae alo included o that the eult can be applied to condition impotant unde micogavity iolated doplet combution condition with and without low ga/dop convection. Numeical Fomulation The computational ytem wa omulated uing a it-ode inite volume method. The equation o conevation o ma, momentum and enegy wee initially expeed a: 1

2 ρ ud + ρd + u ρd = 0 u ρud = Pnd µ ( u) ρctd + u ρctd = λ( T ) nd whee in addition to explicit coupling o the enegy equation and the low via convection thee i an implicit coupling via the evaluation o tanpot paamete and liquid denity. Liquid denity, vicoity, themal conductivity, and uace tenion wee evaluated uing expeimentally baed coelation o peciic liquid taken om the compilation o Daubet and Danne (1989). The popetie o n-decane wee utilized o the eult peented hee. We ued a olution pocedue baed upon the SIMPLE algoithm outlined by Patanka (1980). Poion equation with modiication to account o vaiation in liquid denity with tempeatue wa ued to olve o the peue ield. peue coecto equation wa applied to enoce local ma conevation, and taggeed gid wee employed o both adial and angula velocitie. The above conevation equation wee adapted to an axiymmetic pheical ytem, allowing o dynamic bounday motion. The uace o each contol volume wee allowed to move to ix the oute-mot contol volume bounday on the egeing liquid uace, with the containt that the elative pacing between the inne node emain contant. In ode to inceae computational accuacy in the vicinity o the uace, a well a to povide moe accuate gadient evaluation in all cell, the contol volume wee contained to contain equal volume. The chaacteitic diuive length ued in detemining gadient wee deined baed upon the centoid o each contol volume. Since the value o each popety identiied with a paticula contol volume i intended to epeent an aveage value o that popety within the gid cell, the centoid mot accuately epeent the location with that value o the popety. In ode to imulate a combution envionment in the uounding ga-phae, a ive-equation algebaic model wa ued, baed upon a deivation imila to that o the claic d -law (Tun, 000). Sepaate contant (non-unity) value o the inne and oute low egion Lewi numbe, along with aveaged value o the emaining paamete, and an ininite value o the Damköhle numbe wee aumed. The ou equation o conevation o uel ma, oxidize ma, enegy at the lame heet and enegy at the liquid-olid inteace can be expeed a: m& 1 = [ ρd ( ) (, )] oxidize ln 1+ ν ρd uel ln 1 YF / 4π D uel ln( 1 YF, ) = 1 1 D ln 1+ ν c exp λ 4π T T exp λ 4π oxidize ( ) T T 1 exp λ 4π nd c p / hc = exp λ 4π c T m& m T T 1 exp 1 λ h ( T ) c g ρ & & = + + g p λ 4π liquid 4π 4π The lat equation i detemine the uace ma action o uel, Y F,, a a unction o vapo peue and and an aumed mean molecula weight o the pot-combution gae. Both the patial peue and latent heat o vapoization wee evaluated a unction o uace tempeatue with coelation om Daubet and Danne (1989). Thi ytem wa olved iteatively o uace tempeatue and the Stean

3 lux at each time tep. The Stean lux, o, moe accuately, the Stean lux pe unit uace aea, wa then conveted to a ma vapoization ate pe unit uace aea, poviding the ma ink tem o each uace location and the tempeatue to be ued a the enegy ouce. The intantaneou uace tempeatue wa ued to calculate the intantaneou local uace tenion, and uace tenion gadient detemined the uace momentum ouce in the angula momentum equation. Thee ive equation equie ou paamete, ρd uel, ρd oxidize (1+ν -1 ), c p /λ, and c p, which ae not known a pioi, but could be povided with igniicant numeical analyi. We calibated thee paamete though one-dimenional doplet combution imulation peomed with an in-houe code (Cho et al., 199) that ha been utilized extenively (e.g. Choi and Dye, 001; Kazakov et al., 003). Calculation wee peomed uing n-decane popetie om Daubet and Danne (1989) and the n- decane chemical kinetic model o Zhao et al. (005). The pedicted value o buning ate, doplet uace tempeatue, doplet tando atio and peak lame tempeatue wee then ued to evaluate the above paamete. Peliminay Computational eult Simulation wee peomed o a numbe o cae involving n-decane doplet. Doplet with Peclet numbe (baed upon initial uace velocity) o 0,, and 4 wee imulated at initial doplet adii o 10 µm, 100 µm, and 1 mm. Thee Peclet numbe wee elected baed upon the chaacteitic Peclet numbe ound in conventional pay combution and doplet ize chaacteitic o both pay and undamental iolated doplet combution tudie. ample meh deployment ued in the imulation i peented in Fig. 1. The total numbe o gid cell ued in the imulation wa 1105, 45 in adial and 5 in angula diection, epectively. ingle computational un took appoximately 3 hou on an thlon XP 800+ Linux woktation. The incompeible olution o ybczynki and Hadamad (Happel and Benne, 1965) o a pheical liquid doplet in a ga moving with contant ee-team velocity wa applied a an initial condition (Fig. ). In all cae, the initial velocity ield decayed quite apidly to imple expanding doplet due to vicou diipation and uace oce om themal gadient. te le than.5% o the total buning time o the doplet, all obeved low in the ytem conited o the eect o expanion om the penetating themal wave (Fig 3). Without a peitent tangential diving oce povided at the doplet uace, in the om o an ga/dop elative velocity o olutal Maangoni intabilitie in multi-component doplet o example, all intenal motion wee quickly damped. ample doplet buning hitoy i peented in Figue 4. The tando atio o the lame i accuately epoduced duing the it 75% o the buning hitoy in compaion with the onedimenional imulation, paticulaly conideing the implicity o the uace model. expected, the apid inceae in lame tando atio nea bunout i not epoduced becaue the uace model contain no chemical inluence. Thu, a the doplet begin to undego extinction-type behavio, the imple uace heating model no longe tack the decay in chemical conveion. The model alo epoduce an initial heating tanient ollowed by an appoach to a contant lope in the diamete quaed time hitoy poile, a dicued by Law (198). the ytem emain conductively limited duing the entie buning hitoy (Figue 5), the lope o the diamete-quaed cuve continuouly inceae with time. In the vey beginning o the buning hitoy (Figue 6), one can ee a mall inceae in diamete aociated with the heating o the outemot laye o liquid beoe thee i uicient vapo peue to uppot apid combution. Summay Intenal ciculation geneated by elative ga/dop velocitie, doplet deployment, o Maangoni eect have igniicant eect on doplet buning popetie in both pactical and undamental doplet combution. Modeling thee eect unde combution condition with ealitic popetie i impotant to intepeting undamental micogavity combution obevation a well a developing ub-model 3

4 appopiate o pactical combution ituation involving pay. quai-one-dimenional epeentation o liquid phae intenal ciculation eect that accuately epoduce the multidimenional cae would pemit incluion intenal ciculation eect in model with detailed deciption o ga phae tanpot, chemical kinetic, ooting, and adiation. In ode to calibate, uch a model, a two-dimenional tet poblem ha been developed. The aumption o developing the model wee dicued, and ome initial eult baed upon calculation have been peented, and calibation o a one-dimenional epeentation o the eult i undeway. cknowledgement Thi wok wa uppoted unde NS Gant NCC3-735 and NNC0466. eeence Choi, M.Y., and Dye, F.L. (001) Micogavity Combution: Fie in Fee Fall (Howad o, ed.) cademic Pe. Daubet, T.E., and Danne,.P. (1989), Phyical and Themodynamic Popetie o Pue Chemical, Hemiphee. Dwye, H.., (1989) Pog. Enegy Combut. Sci. 15, 131. Dwye, H.., and Shaw, B.D. (001) Combut. Sci. Tech. 16, 331. Faeth, G.M., (1983) Pog. Enegy Combut. Sci. 9, 1. Happel, J and Benne, H. (1965) Low eynold Numbe Hydodynamic. Pentice-Hall Kazakov,., Conley, J., and Dye, F.L. Combut. Flame Combut. Flame 134, 301. Law, C.K. (198) Pog. Enegy Combut. Sci. 8, 171. Machee,.J., and Dye, F.L. (1997) Combut. Sci. Tech. 14, 371. Niazmand, H., Shaw, B.D., Dwye, H.., and haon, I. (1994) Combut. Sci. Tech 103, 19. Patanka, S.. (1980) Numeical Heat Tane and Fluid Flow, McGaw-Hill. Savino,. and Fico, S. (004) Phy. Fluid 16, Shih,.T. and Megaidi, C.M. (1996) Int. J. Heat Ma Tane 39, 47. Siignano, W.. (1983) Pog. Enegy Combut. Sci. 9, 91. Tun, S.. (000) n Intoduction to Combution: Concept and pplication. McGaw-Hill. Zhao, Z., Li, J., Kazakov,., Zeppiei, S.P., and Dye, F.L. (005) Combut. Sci. Tech. 177, 89. Figue 1. Low-eolution example o meh deployment ued in two-dimenional imulation (a ine gid ued in actual poduction un i not hown o viual claity). The cicle indicate the location o centoid within each gid cell. Figue. Initial condition o velocity ield om the analytical olution o ybczynki and Hadamad ( = 1 mm, Pe = 4). 4

5 Figue 3. Flow ield coeponding to Figue ate 0.1 econd (.5% total buning hitoy) Figue 4. Buning hitoy o 1 mm initial adiu doplet with initial Peclet numbe o 4. Figue 5. Suace tempeatue and doplet aveage tempeatue hitoy o 1 mm initial adiu doplet with initial Peclet numbe o 4 Figue 6. Buning hitoy o 1 mm initial adiu doplet with initial Peclet numbe o 4. Note initial inceae in doplet diamete aociated with themal expanion. 5