A Monte Carlo Radiation Model for Simulating Rarefied Multiphase Plume Flows

Transcription

1 38th AIAA Thermophyscs Conference 6-9 June 005, Toronto, Ontaro Canada AIAA A Monte Carlo Radaton Model for Smulatng Rarefed Multphase Plume Flows Jonathan M. Burt * and Ian D. Boyd Department of Aerospace Engneerng Unversty of Mchgan, Ann Arbor, MI A Monte Carlo ray trace radaton model s presented for the determnaton of radatve propertes of Al O 3 partcles n the hgh alttude plume of a sold propellant rocket. A polydsperse dstrbuton of non-gray partcles s modeled as an emttng, absorbng and scatterng medum of arbtrary optcal thckness. Strong two-way couplng s allowed between radaton and flowfeld calculatons, where the gas s smulated usng the drect smulaton Monte Carlo method and partcle phase propertes are determned usng a smlar Lagrangan approach. Effects of ansotropc scatterng and nozzle searchlght emsson are consdered, and a procedure s descrbed for the calculaton of spectral radance. The model s appled to the smulaton and radaton analyss of the freely expandng plume from a subscale sold rocket motor, and varous flowfeld propertes are presented and dscussed. I. Introducton HE analyss and predcton of radaton sgnatures from sold propellant rocket plumes has been the subject of T extensve study over the past several decades. 1 Much work has focused on propertes of mcron-scale Al O 3 partcles whch account for up to 30% of the mass flow through the nozzle, and whch tend to domnate IR radatve propertes wthn the plume. Characterstcs of the partcle phase are extremely complex, and are generally ether poorly understood or dffcult to model wthn a numercal smulaton. Of partcular nterest here are the partcle phase propertes n plume flows at very hgh alttudes, where nteractons between the partcles and gas are both mportant and not well suted to tradtonal smulaton technques. In a typcal hgh alttude exhaust flow from a sold rocket motor (SRM), lqud alumnum droplets are formed along the propellant gran surface, and undergo a complcated process of combuston, vaporzaton, agglomeraton, breakup, and crystallzaton as they are forced by the expandng gas through the combuston chamber and nozzle. At the nozzle ext, the partcle phase conssts of a polydsperse dstrbuton of sphercal Al O 3 partcles, whch range n dameter from about 0.1 to 10 µm dependng on the sze of the nozzle. Larger partcles exst manly n lqud form wthn the nozzle and tend to develop greater temperature and velocty lags relatve to the gas. The smallest partcles are rapdly accelerated and cooled by the surroundng gas, and wll have fully soldfed nto some combnaton of metastable gamma phase and stable alpha phase polycrystallne structures as they pass through the nozzle ext plane. Wthn the plume nearfeld regon, the crystallzaton process becomes ncreasngly mportant for partcles of ntermedate sze, for whch radatve and convectve coolng are balanced by a heat release durng the phase transton process. The nteracton between the gas and partcles n ths regon s both sgnfcant and complex, as the gas develops a hgh degree of thermal nonequlbrum durng the rapd expanson downstream of the nozzle ext. Due to large partcle phase velocty and temperature lags, the nterphase transfer of momentum and energy may greatly affect partcle propertes, whle the consderable partcle mass fracton allows bulk gas propertes to be nfluenced by gas-partcle collsons. Further downstream, nterphase momentum and energy exchange become neglgble as the gas contnues to expand. The largest partcles may begn to soldfy far downstream of the nozzle, where radatve heat transfer from and between partcles tends to domnate the partcle phase energy balance. Dependng on the SRM sze and gran composton, the farfeld plume regons wll lkely have some ntermedate optcal thckness, so that long-range radatve energy exchange wthn the partcle phase may sgnfcantly nfluence partcle temperatures. * Graduate student, AIAA student member. Professor, AIAA assocate fellow. 1 Copyrght 005 by Jonathan M. Burt and Ian D. Boyd. Publshed by the, Inc., wth permsson.

2 For accurate predcton of the radaton sgnature from a hgh alttude SRM plume flow, several of these processes must be consdered, and the nonequlbrum nature of the gas requres a smulaton approach whch can overcome lmtatons of tradtonal computatonal flud dynamcs (CFD) technques. Varous complex radatve propertes of multphase partcles should be accounted for, ncludng a temperature, sze, and wavelength dependence n spectral emssvtes. Emsson, absorpton, and scatterng may all be mportant, and strong two-way couplng can exst between the radaton feld and partcle temperatures. Ths last property may produce large errors n the tradtonal post-processng approach to plume radaton analyss. Wth these goals n mnd, a new procedure s proposed for the smulaton and radaton analyss of hgh alttude SRM plume flows. Several steps n ths procedure have been extensvely documented n recent papers, so are only dscussed here n general terms. The gas s smulated usng the Drect Smulaton Monte Carlo (DSMC) method of Brd, 3 and the sold phase s modeled usng a smlar Lagrangan approach n whch representatve partcles are tracked through the computatonal grd. Momentum and energy transfer to a partcle from the surroundng gas s modeled through a Green s functon technque of Galls et al., 4 where the total force and heat transfer rate s calculated durng each tme step by summng contrbutons from all DSMC gas molecules assgned to the same grd cell. The recprocal momentum and energy transfer from the partcle phase to the gas s computed usng a probablstc approach, n whch ndvdual nterphase collsons are modeled as nvolvng ether specular reflecton or dffuse reflecton wth full thermal accommodaton. 5,6 Several addtonal capabltes and physcal models have been developed and mplemented n the DSMC code MONACO. 7 Mxed dscrete and contnuous partcle sze dstrbutons may be used, and consderaton s made for the effects of partcle rotaton, nonsphercal partcles, the breakup of lqud droplets and nonequlbrum crystallzaton. In addton, a seres of nterphase couplng parameters 8 are utlzed to ncrease computatonal effcency, through a process nvolvng the automatc determnaton of flowfeld regons where calculatons for momentum and energy transfer n ether drecton may be avoded wth neglgble mpact on bulk flow propertes. In the followng sectons, a partcle radaton model s presented for use wth the above procedures n the smulaton of a hgh alttude SRM plume flow. Frst, a detaled descrpton of the method s gven, ncludng dscusson of procedures related to emsson, absorpton, scatterng and two-way couplng between radaton and flowfeld characterstcs. In addton, technques are outlned for consderaton of nozzle searchlght emsson and the calculaton of spectral radance. The radaton model s then appled to the plume smulaton for a representatve subscale SRM exhaustng nto a vacuum. Smulaton results are dscussed, and senstvtes of radaton characterstcs to varous nput parameters are evaluated. In partcular, we consder the effects of nozzle searchlght emsson and ansotropc scatterng on spectral radance and the net radatve energy flux. II. Radaton Modelng Procedure The proposed partcle radaton model uses a Monte Carlo Ray Trace (MCRT) approach, where a Lagrangan representaton s used to track large groups of photons through the computatonal grd. 9,10 As a frst step, the porton of the spectrum of nterest for radatve heat transfer wavelengths of roughly 0.5 to 5 µm s dvded nto a seres of wave number bns. Gven N η dfferent bns, each of wdth η and centered at a wave number η, a large number N b of representatve energy bundles are generated once every several tme steps at randomly selected source partcles throughout the grd. Every bundle represents some quantty of radatve power, and an equal number of bundles s assgned to each of the N η bns. A newly generated bundle s gven a drecton of propagaton accordng to a randomly generated unt vector u. The bundle s also gven some ntal power P b, based on the assgned wave number bn and the propertes of the source partcle. Followng a correlaton of Reed and Cala 1 based on Me theory calculatons of Plass, 11 the band-averaged spectral emssvty of the source partcle at bn may be approxmated as ε = 4k(T p, η)r p η (1) where T p and R p are the temperature and radus of the partcle respectvely, and k s a value for the absorpton ndex of Al O 3 at temperature T p and wave number η. Due to both a lack of expermental data and an extreme senstvty of k for sold partcles on lattce defects and mpurtes, 1 we neglect any dependence of k on the partcle phase composton. By applyng Eq. (1) to Planck s blackbody functon, we can compute the source partcle emssve power P p, wthn the wave number range to whch the bundle s assgned: 3 P = 3π c hr Ω(T ) p, 0 p p ()

3 where 4 k(t, η ) η dη Ω (T) = (3) η exp(hc η/ k T) 1 0 B The symbol c o here s the speed of lght through a vacuum, h s Planck s constant, and k B s Boltzmann s constant. The ntal power of the bundle s then determned as P NN η p b = WpPp, Nb (4) where N p s the total number of representatve partcles n the grd and W p s the numercal weght of the source partcle. Values of Ω (T) are calculated at smulaton startup for each wave number bn, at temperatures T for whch expermentally determned k(t, η ) values are avalable. The evaluaton of Ω (T p ) n Eq. () s then performed through lnear nterpolaton to the partcle temperature. Once created, each energy bundle s tracked through the grd durng the current tme step untl t exts through an nflow, outflow, or absorbng wall boundary. As an energy bundle passes through a cell n whch partcles are located (or have been located durng prevous tme steps) a fracton of the assgned power wll be absorbed, and there s some probablty that the bundle wll be scattered. Absorpton and scatterng propertes n the cell for a gven wave number bn are determned by the band-averaged spectral absorptance α and scatterng coeffcent σ. Consder a smulaton nvolvng N spec dfferent partcle speces j, where speces are desgnated accordng to the partcle radus R j. In the cell of nterest, each speces has an average temperature T j and a number densty n j, where averagng s performed over a large number of tme steps for both T j and n j. By applyng Eq. (1) and Krchoff s law to the defnton of spectral absorptance, we can compute α as N spec 3 4 k(t, j )Rj j1 = α= πη η n j. (5) The value of k(t j, η ) for each speces s found by nterpolatng tabulated k values to the temperature T j. The correspondng scatterng coeffcent σ s also gven as a summaton over all partcle speces: σ=π N spec j1 = Θ,j R j n j (6) The symbol Θ,j n Eq. (6) s the scatterng effcency factor for wave number bn and partcle speces j. Values of Θ,j are calculated at smulaton startup, usng a frst-order Me theory approxmaton of Segel and Howell. 1 Assumng that n >>k, where n s the real part of the ndex of refracton for the partcle materal at wave number η, Θ,j may be gven as a functon of the nondmensonal parameter x,j =πη R j. Θ 8 = n 1 3 n + 4,j x,j 1 x,j 3 n + 5 n + (7) Whle Eq. (7) can be assumed accurate for Al O 3 when x,j <<1, t greatly over predcts Θ,j for larger values of x,j. To allow for use wth a wder range of x,j values, we mpose a lmtng condton Θ,j. Ths gves relatvely good agreement wth Me theory calculatons of Plass, 11 and avods the detaled calculatons requred to fnd an exact Me theory soluton. Note that we neglect here any dependence of n on the partcle temperature, followng observatons that n values for Al O 3 are nearly constant over a wde range of temperatures. 11 Whle experments have shown some ncrease n n wth partcle sze, 13 we neglect ths dependence as well due to a lack of avalable expermental data. 3

4 When an energy bundle enters a gven cell, the total dstance D e to ext the cell along the ntal trajectory s determned along wth α and σ values correspondng to the assgned wave number bn. The dstance D s to a scatterng event s then determned by evaluatng the probablty P ns that the bundle wll not have been scattered after t has traveled a dstance D s, where dp d ns D = σ s P ns. (8) Solvng for D s and settng P ns equal to a random number R [0,1], we fnd 1 Ds = lnr. (9) σ If D s >D e then the partcle wll ext the cell along ts ntal trajectory. Otherwse the bundle wll be scattered. If scattered, the bundle s moved a dstance D s along the trajectory, after whch ts drecton s reassgned and the procedure s repeated. The ansotropc nature of the scatterng process s approxmated through the use of the Henyey Greensten scatterng phase functon 14 1 g Φθ ( ) = 4π 1+ g gcosθ ( ) 3/. (10) The free parameter g n Eq. (10) s the average cosne of the scatterng angle θ. We can recover the correspondng dstrbuton functon f(θ) = πφ(θ)snθ f we set θ ' ' ' R = π Φ( θ)snθ θ 0 d (11) for a random number R n between 0 and 1. Followng Eqs. (10) and (11), we determne θ through the formula 1 1 g θ= + g gr g 1. (1) cos g 1 The unt vector u * n the fnal drecton of propagaton s then calculated as * u u t1 t = cosθ+ sn θcosφ+ sn θsn φ (13) where the azmuthal angle φ s assgned a random value n [0,π], u s the ntal drecton, and the unt vectors t 1 and t are gven by t 1 = u ˆ u ˆ and t = t1 u. For convenence, ˆ s defned here as the unt vector along the x-coordnate axs. The procedure nvolvng the evaluaton of Eqs. (9), (1) and (13) s repeated untl the bundle exts the cell. Gven a total dstance D t whch the bundle has traveled through the cell, the assgned power P b s then reduced by a fracton 1 exp( α D t ) to account for the effect of partcle phase absorpton on the transmtted radaton ntensty. 9 As the bundle passes through the cell, the power Q abs absorbed by an ndvdual partcle of radus R p may be gven as 4

5 πr ε p Q = P 1 exp( α D ) abs b t αvcell (14) where V cell s the cell volume and P b s the ntal power assgned to the bundle on enterng the cell. It follows that the contrbuton of the bundle to the drecton-averaged radatve energy flux q for the correspondng wave number bn s q P 1 exp D. (15) b = α t αv cell In an optcally thn cell where α D t <<1 the evaluaton of Eq. (15) may result n a large subtractve cancellaton error. To correct for ths error, when α D t <10-5 we calculate q usng a lnearzed form of Eq. (15) PD = ( ) b t q (16) Vcell based on a Taylor expanson of the exponental term. Note that Eq. (16) gves an exact soluton for q n regons outsde the partcle doman where α s zero. Durng each tme step for whch energy bundles are tracked through the grd, energy flux values n every cell are determned by summng contrbutons q from all bundles whch pass through the cell. The resultng values are then averaged over a large number of tme steps to reduce statstcal scatter. Once the flowfeld has reached steady state condtons, averagng may be performed over all subsequent tme steps durng whch radaton calculatons are made. Otherwse a subrelaxaton technque of Sun and Boyd 15 s used for the tme-averagng procedure, so that ncreased weghtng may be appled to more recent tme steps. As dscussed above, strong two-way couplng may exst between flowfeld characterstcs and plume radaton. Radatve heat transfer can sgnfcantly affect partcle temperatures, and may ndrectly nfluence other propertes such as the partcle phase composton, materal densty, and the rates of momentum and energy transfer between the partcles and gas. To account for the effect of radatve emsson and absorpton on partcle temperatures, the temperature T p of every representatve partcle s modfed durng each tme step by t Tp = rad mc p p Q (17) where t s the tme step nterval, m p s the partcle mass, c p s the specfc heat of the partcle materal, and Q rad s the net rate of radatve heat transfer to the partcle. Note that the partcle temperature s assumed spatally unform, based on a low Bot number approxmaton whch follows from the small partcle sze and relatvely hgh thermal conductvty of Al O 3. The radatve heat transfer rate s calculated as Nη rad = π 3 p η p η π 0 Ω p = 1 Q 4 R k(t, ) q 8 c h (T ) (18) followng Krchoff s law and Eqs. (1) and (). The above symbol q s the tme-averaged and drecton-averaged energy flux wthn a wave number bn, for the cell n whch the partcle s located. Radatve emsson from wthn the nozzle has been found to sgnfcantly ncrease radance values near the ext plane. 16 Ths emsson s generated prmarly by the nner nozzle walls, and s generally termed searchlght emsson under past assumptons that ts domnant source s upstream of the throat. Dependng on the optcal thckness of the exhaust flow, emsson from partcles wthn the nozzle may also contrbute sgnfcantly. As searchlght emsson s expected to nfluence the temperature and phase composton of partcles n the plume, a coupled approach to radaton and flowfeld smulaton should nclude consderaton of ths effect. We account for searchlght emsson through the generaton of addtonal energy bundles along nflow boundares at the nozzle ext. Each nflow boundary on the ext plane s represented as a blackbody wall at some characterstc temperature T w. Along every cell face located on an nflow boundary, N f new bundles are generated durng each tme step for whch 5

6 radaton calculatons are performed. Each bundle s randomly assgned to a wave number bn, and s gven an ntal drecton u such that u n f = cos θ= R 1/ where n f s the nward normal unt vector at the cell face, θ s the zenthal angle of u relatve to the face, and R s a random number n [0,1]. The ntal power P b s then determned from Eq. (19) as the product of the face area A f, a weghtng factor N η /N f, and the ntegral of Planck s functon over the correspondng wave number bn. N η dη (19) η 1 3 P c ha η b = π 0 f N η exp(hc /k T ) f 0 B w For calculatons of plume radance, one or more smulated radometers are postoned somewhere outsde the grd doman. Consder a sensor wth surface area A s, outward unt normal n s, and angular resoluton defned by the zenthal angle ω. When each energy bundle exts the grd, we determne whether t wll ntersect the sensor surface along a trajectory gven by the unt vector u for the bundle drecton. If an ntersecton occurs and the condton u n s cosω s met, then the power P b assgned to the bundle s added to the total absorbed power ΣP for the correspondng wave number bn durng the current tme step. An nstantaneous value of the band-averaged spectral radance I may then be calculated as the rato of ΣP to the product of the sensor area, sold angle for absorbed radaton, and wavelength range of the bn. We fnd where I ΣP = (0) π (1 cos ω )As λ η λ = 1 η 4 ( η ). Values of I for each bn are averaged over a large number of tme steps to reduce scatter, where samplng s begun only after the flowfeld has reached steady state condtons. Note that the rate of statstcal convergence n the resultng radance values wll be roughly equal for every bn. Whle a much faster convergence rate s possble f we apply a reverse Monte Carlo method nvolvng the generaton of addtonal energy bundles at the sensor, 14 the procedure proposed here allows for relatvely fast convergence wthout addng much to the complexty of the radaton model mplementaton. III. Plume Flow Test Case Followng mplementaton n the DSMC code MONACO, 7 the procedures descrbed above are appled to an axsymmetrc test case for the plume flow from a subscale SRM expellng nto a vacuum. Inflow and boundary condtons are dentcal to those descrbed n a prevous paper, 8 as are all physcal models used n the flowfeld smulaton. We use a rectangular grd geometry extendng from the nozzle ext plane 100 m downstream and 40 m radally outward. The nozzle ext dameter s 7.85 cm, and ext plane data for both the partcles and gas are taken from Anfmov et al. 17 based on smulated nozzle flow characterstcs for a Star-7 motor wth 30% partcle mass loadng. The gas s a mxture of N, H and CO, and the varable hard sphere (VHS) collson model 3 s used for collsons wthn the gas phase. At the ext plane, the gas s assgned a bulk speed of 3113 m/s, a temperature of 1433 K and a densty of kg/m 3, wth mole fractons of 0.38 for H and 0.31 for both N and CO. The partcle phase has a dscrete sze dstrbuton, wth seven dfferent speces rangng n dameter from 0.3 to 6 µm. Due to a lack of avalable flowfeld nformaton at the nozzle ext, partcle propertes gven by Anfmov et al. 17 6

7 are assumed unform over the ext plane. A rough estmate of the ntal sold mass fracton for each partcle sze s found by takng the dfference between the nucleaton temperature (1930 K) for homogeneous crystallzaton of Al O 3 and the assgned partcle temperature, then multplyng ths by the rato of the specfc heat to the latent heat of fuson for lqud Al O 3. Partcle propertes at the nozzle ext are gven n Table 1. The thermal accommodaton coeffcent on the partcle surface s set to 0.9, so that 90% of nterphase collsons nvolve dffuse reflecton wth full accommodaton to the partcle temperature, whle the remanng 10% nvolve specular reflecton. Interphase momentum and energy transfer s computed usng the two-way couplng approach descrbed above, and the crystallzaton of lqud Al O 3 droplets s consdered usng a model for nonequlbrum phase change. 8 The phase change model accounts for a temperature dependence n the crystallzaton rate and the assocated heat release, whle neglectng the gamma-to-alpha transton for sold Al O 3 and the densty varaton between dfferent phases. Values of the partcle absorpton ndex k are taken from expermental data of Konopka, Reed and Cala. 18 Based on ths data, we use 10 wave number bns correspondng to the md-ir wavelength range from 1.3 to 4.5 µm. Of the two SRM exhaust flows from whch partcles were collected and nvestgated by these authors, the second (rocket ) was found to gve k values more n lne wth other expermental data and correlatons n the lterature. 19 Values calculated from SEM measurements for the second flow are therefore used here. The real part n of the partcle ndex of refracton s computed as a functon of η through a correlaton gven by Duval et al., n = ( Tp) η η η 1/ (1) where η s n unts of µm -1 and the weak temperature dependence s neglected by assumng a partcle temperature T p of 000 K. Followng Reed et al., 16 we set the average cosne of the scatterng angle to g = 0.5 and use an effectve temperature of T w = 1300 K for searchlght emsson at the nozzle ext. The unstructured grd conssts of approxmately 40,000 trangular cells scaled roughly accordng to the local mean free path. So that calculatons may be performed n a reasonable amount of tme on avalable computng resources, cells are about 10 to 50 tmes larger than the mean free path n a small regon just beyond the nozzle ext. Whle ths s a volaton of the standard rule of thumb for cell sze n DSMC, we can tolerate the assocated errors wth an understandng that the smulaton performed here s ntended less to represent one partcular flow wth maxmum accuracy than to show the general radaton characterstcs of a typcal small-scale SRM plume flow at hgh alttude. About two mllon DSMC gas molecules and 00,000 representatve sold partcles are tracked through the grd durng a typcal tme step at steady state. Samplng s performed once every fve tme steps, when 1600 energy bundles are generated and moved through the grd. To reach the desred level of statstcal scatter n sampled radaton and bulk flow propertes, the smulaton s run for 100 hours on four processors n a 1.4 GHz AMD Athlon cluster. Calculatons are dvded roughly evenly across all processors through doman decomposton, and buffer arrays are used to allow energy bundles to rapdly move across multple task domans whle restrctng nformaton exchange between neghborng tasks. Selected smulaton results are shown n Fgs. 1 through 8. IV. Smulaton Results Mass densty contours for both partcles and gas are shown n Fg. 1. Note frst that partcles are only found n roughly half of the smulaton doman, as the maxmum dvergence angle for the partcle phase s restrcted by partcle mass. All partcles are gven ntal trajectores between 0 and 15 degrees off the centerlne at the nozzle ext plane, where the trajectory angle for each partcle scales lnearly wth dstance from the axs. In the plume nearfeld regon just beyond the nozzle ext, partcles are forced outward from the centerlne by the expandng gas. The radal acceleraton of an ndvdual partcle n ths regon wll vary as the nverse of the partcle dameter, so that n a gven radal plane smaller partcles wll be found over a range whch extends further from the axs. As shown n the top half of the fgure, ths results n a gradual decrease n partcle mass densty wth dstance from the axs due to the presence of a range of partcle szes. As the dmensons of the smulaton doman are several orders of magntude greater than the nozzle ext radus, partcle characterstcs observed n Fg. 1 reflect only trends n the farfeld regon, where momentum and energy transfer between the partcles and gas can be assumed neglgble. The partcles move along nearly straght trajectores far from the nozzle, so the contour lnes shown projectng from the nozzle ext are straght f we neglect effects of statstcal scatter and nterpolaton between cell centers. The contnuous reducton n partcle mass densty 7

8 wth downstream dstance s due to the dvergence of partcle trajectores, and the restrcted maxmum dvergence angle s shown n the fgure to result n centerlne values of the partcle phase mass fracton (the rato of partcle mass densty to total mass densty for both partcles and gas) whch ncrease wth dstance from the nozzle. At the pont on the centerlne 100 m downstream of the nozzle ext, the partcle mass fracton s about. tmes the ntal value of 0.3. Fgure shows contours of average temperature for partcles of dameter 0.4 µm and 4 µm. The temperature of 0.4 µm partcles s generally observed to decrease wth downstream dstance but ncrease wth dstance from the axs. The frst trend may be attrbuted to the domnance of radatve heat loss n the partcle energy balance through nearly the entre smulaton doman, whle the second trend s prmarly a result of the radal varaton n gas densty wthn the plume nearfeld regon. As gas streamlnes dverge n ths regon just downstream of the nozzle ext, the densty of the gas wll decrease more rapdly along streamlnes further from the centerlne. A lower gas densty corresponds to a reducton n the rate of convectve heat transfer from partcles to the surroundng gas, so we can expect a smaller temperature drop n the plume nearfeld for partcles whch pass through the nozzle ext plane further from the centerlne. Ths s lkely the domnant mechansm for the ncrease n partcle temperature wth dstance from the axs. Another possble contrbutor to ths trend s the fact that partcles further from the axs may absorb more radatve energy from nozzle searchlght emsson. The plume optcal thckness scales wth partcle phase mass densty, and ths densty s shown n Fg. 1 to decrease wth dstance from the axs, so a correspondng ncrease n the magntude of long range radatve ntensty from nozzle emsson should result n some ncrease n absorpton among partcles far from the centerlne. However, ths trend s found to have only a very small effect on partcle temperatures, and s countered by an ncrease n short range radatve heat transfer between partcles near the axs. We expect short range radatve transfer to be the domnant mechansm by whch radaton causes a radal varaton n partcle temperatures, so that the net effect of couplng the radaton model to the flowfeld smulaton should be a slght reducton n the temperature ncrease for 0.4 µm partcles wth dstance from the axs. Note that a sgnfcant temperature drop s observed for these partcles at ponts very close to the axs. Ths s lkely the result of an unphyscal decrease n convectve heat transfer between the partcles and gas, whch follows from a reducton n the average number of DSMC gas molecules n cells along the centerlne just downstream of the nozzle ext. Smlar trends as descrbed above are found wthn 50 m of the nozzle ext for 4 µm dameter partcles, as shown n the lower half of Fg.. Average temperatures for these partcles n the frst 50 m downstream of the nozzle are several hundred degrees hgher than for the 0.4 µm partcles. Ths may be explaned by the fact that the drop n partcle temperature wthn the nozzle and the nearfeld plume regons scales roughly wth the nverse of the partcle dameter, followng a balance between partcle heat capacty and the heat transfer rate. As all partcles enter the nozzle at smlar temperatures, we can generally expect an ncrease n temperature wth partcle sze throughout the plume. Radatve heat transfer s shown n the fgure to result n a contnuous downstream reducton n temperature for 4 µm partcles, up to a narrow regon about 50 m away from the nozzle where these partcles reach a temperature of 1930 K. In our model for nonequlbrum phase change, ths s specfed as the nucleaton temperature at whch a crystallzaton front forms unformly over the surface of an ntally lqud partcle. Once formed, ths front progresses toward the partcle center at a temperature dependent rate, so long as the partcle temperature remans below the equlbrum meltng temperature of 37 K. As the heat of formaton for lqud Al O 3 s released durng crystallzaton, partcles wll experence a rapd temperature ncrease at the ntaton of the phase change process. The rate of ths ncrease wll be reduced further downstream as the partcle temperature rses, so the crystallzaton front progresses more slowly, and as the front area decreases, so a smaller volumetrc phase change rate wll exst for a gven velocty of progresson toward the partcle center. Ultmately, as the partcle becomes completely soldfed, we expect the partcle temperature to agan begn to decrease wth downstream dstance due to radatve heat loss. For 4 µm partcles, the temperature reducton followng crystallzaton should occur beyond the downstream lmt of the smulaton doman, so ths trend s not shown n Fg.. Note that the phase change process for 0.4 µm partcles s completed prmarly wthn the nozzle, so no smlar temperature jump s shown n the upper half of the fgure. In Fg. 3, partcle temperature profles are shown along the radal plane 100 m downstream of the nozzle ext. Average temperatures for partcles of four dfferent szes are plotted as a functon of dstance from the axs, and for comparson, correspondng temperature profles are shown for a smulaton n whch the radaton model s dsabled. Frst consder the temperature profles for the latter case: For all partcle szes shown, temperatures are found to ncrease wth dstance from the axs due to the radal varaton n gas densty wthn the plume nearfeld, as dscussed above. The temperature s also found to generally ncrease wth partcle sze, so that the lowest 8

9 temperatures are observed for the smallest partcles shown, whle the hghest temperatures occur for the largest partcles. An mportant excepton to ths trend s observed for 0.6 µm dameter partcles, whch experence temperatures sgnfcantly hgher than those of the much larger 4 µm partcles. Ths can be explaned by characterstcs of the crystallzaton process. Whle 4 µm partcles never undergo suffcent convectve coolng to reach the nucleaton temperature of 1930 K at whch crystallzaton may begn, the 0.4 µm and 0.6 µm partcles pass through the nozzle ext plane below the meltng temperature and n a state of partal soldfcaton. The entre phase transton process for 0.4 µm partcles takes place wthn and just beyond the nozzle, where convectve heat loss domnates the partcle energy balance and allows for temperatures several hundred degrees below the meltng temperature of 37 K. In contrast, the phase change process for 0.6 µm partcles occurs at a slower rate, and contnues well beyond the nearfeld plume regon where the assocated heat release may be balanced by convectve heat transfer to the gas. As a result, the temperature of these partcles approaches a farfeld lmt whch s sgnfcantly greater than that of both larger and smaller partcles. In comparng temperature profles between the cases wth and wthout actvaton of the radaton model, we fnd that radatve heat loss accounts for a roughly 10 K decrease n temperatures for 0.4 µm partcles and a 00 K decrease for 0.6 µm partcles. Among 4 µm partcles however, radatve heat transfer s found to ncrease temperatures 100 m downstream of the nozzle by about 40 K. As above, ths trend may be explaned by the phase change process: Convectve coolng wthn the nozzle and plume nearfeld regons s not suffcent to reduce the temperature of 4 µm partcles to below the nucleaton temperature, but the addton of radatve coolng results n farfeld partcle temperatures at whch crystallzaton fronts may form. Because the rate of heat addton for a partcle undergong crystallzaton s far greater than the rate of radatve heat loss, these partcles wll experence a rapd temperature ncrease as shown n Fg.. For 6 µm partcles, a comparson of temperature profles shows an even more complcated trend. Radaton s found to unformly reduce partcle temperatures, but the magntude of ths reducton vares greatly wth dstance from the axs. Ths trend may be explaned as follows: Radal varaton n nearfeld convectve heat transfer results n an ncrease n temperature for these partcles wth dstance from the axs, as descrbed above. Ths means that partcles further from the axs must experence radatve heat loss over a longer tme perod before ther temperatures have been suffcently reduced for crystallzaton to begn. The temperature rse assocated wth crystallzaton therefore occurs further downstream for partcles at a greater dstance from the axs. At the radal plane 100 m downstream of the axs, 6 µm partcles close to the centerlne have begun the phase change process, whle the fracton of partcles on whch crystallzaton fronts have formed s generally found to decrease wth radal dstance. Ths reducton n lqud mass fracton wth radal dstance corresponds to the temperature drop observed n Fg. 3 for 6 µm partcles far from the axs. Fgure 4 shows the centerlne varaton n the magntude of mean convectve and radatve heat transfer rates, as calculated per partcle and averaged over all partcle szes. Both rates are negatve through the entre smulaton doman, so that partcles throughout the plume are losng energy to ther surroundngs through both convectve and radatve heat transfer. The radatve heat transfer rate s shown to be relatvely constant wth downstream dstance, due to an overall gradual varaton n partcle temperatures and the fact that radatve absorpton s found to have a comparatvely small effect. Most of the varaton n radatve transfer observed on the plot s the result of statstcal scatter, due to the small number of representatve partcles whch pass through cells borderng the axs. However, a sgnfcant ncrease n the radatve transfer rate s found about 50 m downstream of the nozzle ext. Ths may be attrbuted to a temperature jump assocated wth the onset of phase change n 4 µm partcles, whch account for about 60% of the total Al O 3 mass wthn the plume. In contrast to the relatvely unform rate of radatve heat transfer, the mean convectve heat transfer rate s shown to decrease rapdly wth downstream dstance. The spatal varaton n convectve heat transfer s found to occur manly as a result of a downstream decrease n gas densty. As the gas approaches a free molecular state far from the nozzle, the gas densty wll decrease along the centerlne as the nverse square of the dstance from some pont near the nozzle ext. The farfeld convectve heat transfer rate wll therefore have the same nverse square of dstance varaton, as s shown n Fg. 4. A convenent defnton of the plume nearfeld regon n a freely expandng SRM plume flow s the range beyond the nozzle ext where the partcle energy balance s domnated by convectve heat transfer to the gas. By ths defnton, the nearfeld regon extends along the centerlne about 1 m (or 13 nozzle ext dameters) downstream of the nozzle ext, beyond whch radatve emsson becomes the domnant mechansm for energy transfer between a partcle and ts surroundngs. In Fg. 5, contours are shown for the net drecton-averaged radatve energy flux. The energy flux s greatest at the nozzle ext, due to the correspondng maxmum n partcle mass densty and a reducton n ntensty of searchlght emsson wth dstance from the nozzle. Partcles are modeled through Eq. (1) as volumetrc emtters, 1 so 9

10 the magntude of radatve energy flux should scale roughly wth the local partcle mass densty. As ths densty decreases wth downstream dstance due to the dvergence of partcle trajectores, a contnuous reducton n radatve energy flux wll occur n the axal drecton. A reducton n energy flux s also observed n the radal drecton, partcularly outsde the regon where partcles are found. Ths follows prmarly from the fact that n an axsymmetrc smulaton the rato of cell volume to the projected area on the grd wll scale wth dstance from the axs. The average number of energy bundles whch pass through a cell s proportonal to the projected area of that cell, and the contrbuton of each bundle to the energy flux s proportonal to the nverse of the cell volume. We therefore expect the net energy flux through cells outsde the partcle doman to scale approxmately wth the nverse of the dstance from the axs, as s shown n the fgure. Fgure 6 shows a comparson of net radatve energy flux contours between smulatons wth and wthout actvaton of the model for nozzle searchlght emsson. Due to the pont-source nature of ths effect, searchlght emsson s found to have relatvely lttle nfluence on energy flux magntudes far downstream of the nozzle. For clarty we therefore restrct the fgure to an axal range of 10 m from the nozzle ext. The presence of searchlght emsson results n a 100% ncrease n radatve energy flux at the pont along the centerlne 0.5 m from the nozzle ext. At ponts 1,, 5 and 10 m downstream, we fnd correspondng ncreases of about 45%, 13%, 11% and 9%, respectvely. An equvalent contour plot to evaluate the mpact of ansotropc scatterng s ncluded as Fg. 7. The top half of ths fgure shows contours of net drecton-averaged energy flux for the base smulaton, as descrbed above, where we apply the Henyey Greensten scatterng phase functon. The lower half of the fgure s taken from a smulaton where sotropc scatterng s used, so that each scattered energy bundle s randomly assgned a new drecton of propagaton wth no dependence on the ntal drecton. We fnd lttle f any measurable effect of the scatterng model on energy flux contours, as the small dfferences observed between the upper and lower halves of Fg. 7 are prmarly a result of statstcal fluxuatons due to the probablstc nature of the radaton model. Spectral radance s calculated at a smulated radometer centered at a pont 0 cm upstream and 4 cm radally outward from the ntersecton of the central axs wth the nozzle ext plane. The sensor has a crcular face of area 50 cm and a surface normal vector nclned 4 from the axs. The angular resoluton s 4, so that the boundary of the concal vewng area ntersects the grd plane along a lne whch s parallel to the axs. Values of spectral radance are plotted n Fg. 8, where the sold lne denotes the base smulaton and dashed lnes correspond to smulatons for whch ether searchlght emsson or the ansotropc scatterng model s dsabled. All three lnes generally follow the gray body-lke trend descrbed by Reed and Cala. 1 As expected, we fnd that searchlght emsson produces an ncrease n radance values upstream of the nozzle. However, the magntude of ths ncrease s very small compared to the ncrease n radatve energy flux near the nozzle shown n Fg. 6. As dscussed by Reed et al., 16 the relatvely small dependence of measured radance on searchlght emsson may be attrbuted to a strong preference for forward scatterng among Al O 3 partcles. Any nozzle emsson absorbed by the sensor must be scattered off one or more partcles at a large angle, for whch the scatterng effcency s very low. Searchlght emsson wll therefore have a far smaller effect on radance upstream of the nozzle than on radance measured at a pont downstream of the ext plane. If the angular dependence n the scatterng model s dsabled, then we expect to fnd a sgnfcant ncrease n radance upstream of the nozzle due to searchlght emsson. Ths trend s shown n Fg. 8, where a jump n spectral radance of up to 0% s found when scatterng s assumed to be sotropc. V. Conclusons A radaton model was presented for the Al O 3 partcle phase n SRM plume flows at hgh alttude. The model ncludes capabltes for strong two-way couplng between radaton and flowfeld calculatons, and accounts for the nfluence of emsson and absorpton on partcle temperatures. No lmtatons are mposed on optcal thckness of the flowfeld, and band-averaged partcle radatve propertes are allowed to vary as an arbtrary functon of both temperature and wavelength. Procedures were descrbed for the calculaton of spectral radance and consderaton of effects assocated wth nozzle searchlght emsson. Followng applcaton of the model n the axsymmetrc smulaton of a subscale plume flow, several radaton and flowfeld characterstcs were evaluated. A complex relatonshp was shown between partcle sze and temperature n farfeld plume regons, due n part to the nteracton between radatve heat loss and crystallzaton of lqud Al O 3. Searchlght emsson was found to sgnfcantly affect the radatve energy flux through a large regon beyond the nozzle ext, whle consderaton of the angular dependence n partcle scatterng results n an ntensty reducton upstream of the nozzle. Due to a lack of avalable expermental data for comparson, the overall accuracy of our radaton model n the smulaton of a hgh alttude SRM plume flow could not be quanttatvely evaluated. However, we expect that the sngle greatest error source s the selecton of approprate values for the partcle absorpton ndex, partcularly at low 10

11 temperatures where much of the partcle materal has soldfed. As dscussed by Reed and Cala, 1 the absorpton ndex of sold Al O 3 partcles s prmarly an extrnsc property, and s a strong functon of the concentraton of mpurtes wthn the lattce structure. As mpurty concentratons wll vary greatly between dfferent SRM exhaust flows, and between dfferent regons n the same flow, the dentfcaton of reasonably accurate values for the absorpton ndex of sold Al O 3 becomes extremely dffcult. Based on expermental measurements 18 we can expect these values to be accurate to about one order of magntude, so that radatve heat transfer rates and energy flux values far from the nozzle may vary on ths same scale. Another potental source of sgnfcant error s the dffculty n accurately determnng partcle propertes at the nozzle ext. As a result of complcated physcal processes whch occur wthn the combuston chamber and nozzle, such as partcle agglomeraton, breakup, combuston, and turbulent dsperson, the accurate numercal calculaton of partcle propertes at the ext plane may be very dffcult. Other dffcultes are assocated wth the expermental measurement of partcle phase characterstcs along the nozzle ext, and no suffcently detaled expermental data could be found n the open lterature. We must therefore rely on smplfed nozzle flow smulatons whch neglect many of the physcal phenomena expected n such flows, so that a loss of accuracy wll result n the calculaton of partcle propertes wthn the plume. Other possble sources of sgnfcant error n the radaton model nclude the lack of consderaton for IR absorpton and emsson nvolvng exhaust gas speces, approxmatons used to compute partcle emssvty and scatterng coeffcents, and an unphyscal reducton n farfeld energy fluxes and radatve heat transfer due to the truncaton of the smulaton doman. Whle the radaton model presented here was developed for applcaton to hgh alttude SRM plume flows, the above procedures may be appled to any smulaton nvolvng a Lagrangan representaton of mcron-scale Al O 3 partcles. Ths ncludes the smulaton of nternal and external SRM exhaust flows at lower alttudes, where the gas phase may be accurately modeled usng contnuum CFD methods. To the authors knowledge, ths s the frst mplementaton of a Monte Carlo ray trace model to allow for strong couplng between radaton and flowfeld characterstcs n a hgh speed emttng, absorbng and scatterng multphase medum of arbtrary optcal thckness. Wth suffcent modfcatons, we expect that the general procedure descrbed here may be appled to a varety of gas-partcle flows. Acknowledgments The authors gratefully acknowledge the Ar Force Research Laboratory at Edwards Ar Force Base for fnancal support of ths work, wth Dean Wadsworth and Tom Smth as techncal montors. References 1 Reed, R. A., and Cala, V. S., Revew of Alumnum Oxde Rocket Exhaust Partcles, AIAA Paper , Gesler, R. L., A Global Vew of the Use of Alumnum Fuel n Sold Rocket Motors, AIAA Paper , Brd, G. A., Molecular Gas Dynamcs and the Drect Smulaton of Gas Flows, Clarendon Press, Oxford, Galls, M. A., Torczynsk, J. R., and Rader, D. J., An approach for Smulatng the Transport of Sphercal Partcles n a Rarefed Gas Flow va the Drect Smulaton Monte Carlo Method, Physcs of Fluds, Vol. 13, No. 11, 001, pp Burt, J. M., and Boyd, I. D., Development of a Two-Way Coupled Model for Two Phase Rarefed Flows, AIAA Paper , Burt, J. M., and Boyd, I. D., Partcle Rotaton Effects n Rarefed Two Phase Plume Flows, 4th Internatonal Symposum on Rarefed Gas Dynamcs, Monopol, Italy, Detrch, S., and Boyd, I. D., Scalar and Parallel Optmzed Implementaton of the Drect Smulaton Monte Carlo Method, Journal of Computatonal Physcs, Vol. 16, 1996, pp Burt, J. M., and Boyd, I. D., Monte Carlo Smulaton of a Rarefed Multphase Plume Flow, AIAA Paper , Farmer, J. T., and Howell, J. R., Monte Carlo Predcton of Radatve Heat Transfer n Inhomogeneous, Ansotropc, Nongray Meda, Journal of Thermophyscs and Heat Transfer, Vol. 8, No. 1, 1994, pp Mahan, J. R., Radaton Heat Transfer: a Statstcal Approach, John Wley and Sons, New York, Plass, G. N., Temperature Dependence of the Me Scatterng and Absorpton Cross Sectons for Alumnum Oxde, Appled Optcs, Vol. 4, No. 1, 1965, pp Segel, R., and Howell, J. R., Thermal Radaton Heat Transfer, Hemsphere Publshng, Washngton, Laredo, D., and Netzer, D. W., The Domnant Effect of Alumna on Nearfeld Plume Radaton, Journal of Quanttatve Spectroscopy and Radatve Transfer, Vol. 50, No. 5, 1993, pp Everson, J., and Nelson, H. F., Development and Applcaton of a Reverse Monte Carlo Radatve Transfer Code for Rocket Plume Base Heatng, AIAA Paper , Sun, Q., and Boyd, I. D., Evaluaton of Macroscopc Propertes n the Drect Smulaton Monte Carlo Method, Journal of Thermophyscs and Heat Transfer, submtted for publcaton,

12 16 Reed, R. A., Beale, K. S., Neese, D. W., Sherrell, F. G., Roberds, D. W., and Olver, S. M., The Effect of Seachlght Emsson on Radaton from Sold Rocket Plumes, AIAA Paper 9-918, Anfmov, N. A., Karabadjak, G. F., Khmelnn, B. A., Plastnn, Y. A., and Rodnov, A. V., Analyss of Mechansms and Nature of Radaton from Alumnum Oxde n Dfferent Phase States n Sold Rocket Exhaust Plumes, AIAA Paper , Konopka, W. L., Reed, R. A., and Cala, V. S., Measurements of Infrared Optcal Propertes of AlO3 Rocket Partcles, AIAA Paper , Duval, R., Soufan, A., and Tane, J., Coupled Radaton and Turbulent Multphase Flow n an Alumnsed Sold Propellant Rocket Engne, Journal of Quanttatve Spectroscopy and Radatve Transfer, Vol. 84, No. 4, 004, pp Dameter, µm Table 1. Partcle propertes at the nozzle ext. Mass flux, kg/m s Temperature, K Speed, m/s Lqud mass fracton Fgure 1. Contours of mass densty for partcles and gas. Values are gven n kg/m 3. Fgure. Mean temperature contours for 0.4 µm and 4 µm dameter partcles. Values are n K. 1

13 Fgure 3. Profles of partcle temperature 100 m downstream of the nozzle ext. Fgure 4. Mean convectve and radatve heat transfer rates along the centerlne. Fgure 5. Contours of the net drecton-averaged radatve energy flux. Values are n W/m. Fgure 6. Comparson of the net radatve energy flux wth and wthout searchlght emsson. Values are n W/m. 13

14 Fgure 7. Energy flux contours for smulatons employng ansotropc (top) and sotropc scatterng models. Values are n W/m. Fgure 8. Spectral radance at a sensor 0 cm upstream of the nozzle ext. 14