Backward Monte Carlo Simulations in Radiative Heat Transfer

Transcription

1 Backward Monte Caro Simuations in Radiative Heat Transfer Michae F. Modest Department of Mechanica and Nucear Engineering Penn State University University Park, PA 82 emai: August 29, 2 Abstract Standard Monte Caro method trace photon bundes in a forward direction, and may become extremey inefficient when radiation onto a sma spot and/or onto a sma direction cone is desired. Backward tracing of photon bundes is known to aeviate this probem if the source of radiation is arge, but may aso fai if the radiation source is coimated and/or very sma. In this paper various impementations of the backward Monte Caro method are discussed, aowing efficient Monte Caro simuations for probems with arbitrary radiation sources, incuding sma coimated beams, point sources, etc., in media of arbitrary optica thickness. Nomencature A = area, m 2 I = radiative intensity, W/m 2 sr = geometric ength, m ˆn = unit surface norma N = number of photon bundes q = radiative heat fux, W/m 2 r = position vector R = random number ŝ = unit direction vector S = radiative source, W/m 3 sr T = temperature, K V = Voume, m 3 Greek Symbos = extinction coefficient, cm = surface emittance

2 = waveength, m = scattering phase function = absorption coefficient, cm = soid ange, sr s = scattering coefficient, cm = incidence anges, rad Subscripts b = backbody emission j = path identifier n = bunde identifier w = wa = spectra Introduction The standard Monte Caro method for radiative heat transfer, as presented in various textbooks and review artices [ 3] is a forward method, i.e., a photon bunde is emitted and its progress is then foowed unti it is absorbed or unti it eaves the system. The method can easiy simuate probems of great compexity and, for the majority of probems where overa knowedge of the radiation fied is desired, the method is reasonaby efficient. However, if ony the radiative intensity hitting a sma spot and/or over a sma range of soid anges is required, the method can become terriby inefficient. Consider, for exampe, a sma detector (maybe mm mm in size) with a sma fied of view (capturing ony photons hitting it from within a sma cone of soid anges) monitoring the radiation from a arge furnace fied with an absorbing, emitting and scattering medium. In a standard Monte Caro simuation one woud emit many photon bundes within the furnace, and woud trace the path of each of these photons, even though ony the tiniest of fractions wi hit the detector. It may take many biion bundes before a statisticay meaningfu resut is achieved at the same time cacuating the intensity fied everywhere (and without need): ceary a very wastefu procedure. Obviousy, it woud be much more desirabe if one coud just trace those photon bundes that eventuay hit the detector. This idea of a backward tracing soution, sometimes aso known as reverse Monte Caro has been appied by severa investigators [ ]. A of these investigations have been somewhat imited in scope, ooking at ight penetration through nonemitting oceans and atmospheres [ ], computer graphics [7, 8], refecting boundaries [9], and emitting media []. A the aforementioned papers have deat with arge ight sources (in voume and/or soid ange range), making a backward simuation straightforward. It is the purpose of the present study to give a comprehensive formuation for backward Monte Caro simuations, capabe of treating emitting, absorbing and anisotropicay scattering media, media with diffuse or coimated irradiation (with arge or sma footprints), media with point or ine sources, etc. In addition, the method wi be described in terms of standard ray tracing (bundes of fixed energy) as we as using energy partition (bundes attenuated by absorption) [] (aso caed absorption suppression by Waters and Buckius [3]). 2

3 Theoretica Deveopment Simiar to the deveopment of Waters and Buckius [], we wi start with the principe of reciprocity described by Case []. Let I and I 2 be two different soutions to the radiative transfer equation for a specific medium, ŝ I jr ŝ S j r ŝ r I jr ŝ s r subject to the boundary condition I jr ŝ r ŝ ŝ d! j 2 () I jr w ŝ I w jr w ŝ j 2 (2) where r is a vector pointing to a ocation within the medium, ŝ is a unit direction vector at that point, S is the oca radiative source, is the extinction coefficient, s the scattering coefficient, is the scattering phase function, and denotes soid ange. The principe of reciprocity states that these two soutions are reated by the foowing identity: A ˆn" ŝ# $ 2 ŝ I ŝ ŝ I 2 ŝ&% I w r w r w I w r w r w ˆn ŝ(' da V )$ I 2r ŝ S r ŝ I r ŝ S 2r ŝ&% d dv (3) where A and V denote integration over encosure surface area and encosure voume, respectivey, and ˆn ŝ* indicates that the integration is over the hemisphere on a point on the surface pointing into the medium. I r ŝ S r In the ŝ Backward Monte Caro scheme, r ŝ the soution to [with specified interna source I 2 r ŝ and boundary intensity I w ] is found I from the soution to a much simper probem. In particuar, if we desire the soution to at ocation r i (say, a detector at the wa) into direction ŝ i (pointing out of the medium into the surface), we choose I 2 to be the soution to a coimated point source of unit strength ocated aso at r i, but pointing into the opposite direction, i+ ŝ Mathematicay, this can be expressed as 2 ŝ I w S 2 r w r ŝ,' r (' ŝ r i ŝ i where the' are Dirac-deta functions for voume and soid anges, defined as (a) (b) ' r.- / r r r i i r (5a) r V' r i r i dv (5b) 3

4 3 -s i da i s n s da r w r Figure : Typica ray path in a backward Monte Caro simuation. and simiary for soid ange. If the infinitesima cross-section of the source, norma to ŝ i, is da i, then this resuts in an I 2 intensity at r i of I 2 r i ŝ ' ŝ ŝ i da i + () As the I 2 ight beam traves through the absorbing and/or scattering medium, it wi be attenuated accordingy. Sticking Eqs. () into Eq. (3) yieds the desired intensity as I r i ŝ i A V ˆn" ŝ# I w r w ŝ I 2 r w ŝ ˆn ŝ d da S r ŝ I 2 r ŝ d dv+ (7) Whie the I 2 probem is much simper to sove than the I probem, it remains quite difficut if the medium scatters radiation, making a Monte Caro soution desirabe. Therefore, we wi approximate I as the statistica average over N distinct paths that a photon bunde emitted at r i into direction ŝ i traverses, as schematicay shown in Fig., or I r i ŝ i 2 N I N n nr i ŝ i (8) where the soution for each I n is found for its distinct statistica path (with absorption and scattering occurrences chosen exacty as in the forward Monte Caro method). Aong such a zig-zag path of tota ength from r i to r w, consisting of severa straight segments pointing aong a oca direction ŝ r, I 2 is nonzero ony over an infinitesima voume aong the path, dv da i, and an infinitesima soid ange centered around the oca direction vector ŝ ŝr + At its fina

5 JHH da i5 ŝ da r w ˆn, so that Eq. (7) simpifies to I n ŝ 78 r i ŝ i I w r w r w exp S r ŝ r 7 exp destination on the encosure surface, the beam of cross-section da i iuminates an area of ony where? < r9 d;: < r= d>: d (9) d> indicates integration aong the piecewise straight path, starting at r i, and I is the oca absorption coefficient. It is seen that n ŝ rw r i ŝ i consists of intensity emitted at the wa into the direction of (i.e., aong the path toward S r i ), attenuated by absorption ŝ r aong the path, and by emission aong the path due to the source, in the direction of (aso aong r the path toward r i ), and attenuated by absorption aong the path, between the point of emission,, and r i. This resut is intuitivey obvious since it is the same as the symboic soution to the standard radiative transfer equation (RTE) [], except that we here have a zig-zag path due to scattering and/or wa refection events. If we trace a photon bunde back toward its point of emission, aowing for intermediate refections from the encosure wa (as indicated in Fig. ), then, at the emission point r w, I w I b r w, where is the oca surface emittance (assumed to be diffuse here), and I b is the backbody intensity or Panck function. And, if the interna source of radiation is due to isotropic S r ŝ2 r r + emission, then, comparing the standard RTE [] with Eq. () we find I b Thus, I 78 ra db: r i ŝ i r w I b r w exp r9 ra 7C I b exp < r>9 d=d: d () where the subscript has been dropped since it is no onger needed. Equation () may be soved via a standard Monte Caro simuation or using the energy partitioning scheme E described by Modest [] and Waters and Buckius F [3]. For the standard method scattering engths are chosen as we as an absorption ength. The bunde is then traced backward from r i unattenuated F [i.e., the exponentia decay terms in Eq. () are dropped], unti the tota path ength equas or unti the emission ocation r w is reached (whichever comes first). Thus, I I K GHH r9 r9 d FML I b nr i ŝ i ra r9 d FMN + () r w I b r w I b If energy partitioning is used ony scattering engths are chosen and I n is found directy from Eq. (). Radiative Fuxes If radiative fux onto a surface at ocation r i over a finite range of soid anges is desired, the fux needs to be computed using the statistica data obtained for I n r i ŝ i. For 5

6 3 2 8 IRQ in S8 UTWVYX XZ\[ d d S8 IRQ in dtwvyx 2 d ]_^`2Omax a btcvdx 2 N max n I n ŝ n ŝ n (2) e e 2,TcVdX 2 e where TcVdX 2 the e directions ŝ n need to be picked uniformy from the interva max. The azimutha ange n is found in standard fashion from 2 Rf n, whie n is found from RO? dh ]_^g`2on exampe, for a detector ocated at r i with opening ange max one obtains POmax q det 2 2 dh btcvyx 2 btcvyx 2 n ]_^`2Omax max? XgZ\[ 2 XgZ\[ 2 n or nixgz\[ jk RO XZ9[ maxm (3) max RO Rf e e where and are random numbers picked uniformy from R. If the detector is of finite dimension, points distributed across the surface are chosen ike in a forward Monte Caro simuation. Coimated Irradiation Backward Monte Caro is extremey efficient if radiative fuxes onto a sma surface and/or over a sma soid ange range are needed. Conversey, forward Monte Caro is most efficient if the radiation source is confined to a sma voume and/or soid ange range. Both methods become extremey inefficient, or fai, if radiation from a sma source intercepted by a sma detector are needed. For coimated irradiation (and simiar probems) backward Monte Caro can be made efficient by separating intensity I into ŝn a direct d (coimated) ŝ s ŝ and a scattered part, as outined in Chapter of []. Thus, etting r, I r, I r, resuts in a direct component, attenuated by absorption and scattering, d ŝ2 I r, q co g' ŝ opqsr& r< r w ŝ rt s dsbu () which satisfies the RTE without the inscattering term. This eads to a source term in the RTE for the scattered part of the intensity, due to (first) scattering of the coimated beam, of S r ŝ2 s r q co r w exp 7 c s dc: r ŝ ŝ (5) where q co is the coimated fux entering the medium at r w, traveing a distance of c toward r in the direction of ŝ, and the scattering phase function r ŝ ŝ indicates the amount of coimated fux arriving at r from ŝ, being scattered into the direction of ŝ. Therefore, the diffuse component of the intensity at r i is found immediatey from Eq. (9) as I n S r ŝ9 78 < d>b: d r i ŝ i exp ()

7 with S from Eq. (5). As before, Eq. () may be soved using standard tracing [picking absorption ength F, and dropping the exponentia attenuation term in Eq. ()] or energy partitioning [using Eq. () as given]. Point and Line Source. Backward Monte Caro aso becomes inefficient if the radiation source comes from a very sma surface or voume and/or if the source is unidirectiona. The trick is again to break up intensity into a direct component (intensity coming directy from the source without scattering or wa refections), and a mutipy-scattered and refected part. Again, we et I d satisfy the radiative transfer equation without the inscattering term, or, ŝ I d r, ŝ S d r, ŝ) r I d r, ŝ (7) which has the simpe soution I d r, ŝ S d r, ŝovpqsrw rt r< s dsu ds (8) where the main integra is aong a straight path from the boundary of the medium to point r in the direction of ŝ. For exampe, if there is ony a simpe point source at r with tota strength Q, emitting isotropicay across a tiny voume' V, Eq. (8) becomes d ŝ2 Ux Q rx 2opqyrz I r, r r t r s dsu{' ŝ ŝ (9) where ŝ is a unit vector pointing from r toward r, and use has been made of the fact that ',' A' s ' ' 2 x rx s V (2) r where' is the soid ange, with which' V is seen from r. Equation (9) can be used to cacuate the direct contribution of Q hitting a detector, and it can be used to determine the source term for the RTE of the scattered radiation as ŝ s r S r, d ŝ ŝ d I r, r, ŝ) s r Q rx 2opq r r 2xr t s dsu ŝ + r, ŝ (2) r The rest of the soution proceeds as before, with I n r i ŝ i found from Eq. (). Sampe Cacuations e e As a first exampe we wi consider a one-dimensiona sab z L m of a gray, purey isotropicay scattering medium s m const, bounded at the topz by vacuum and at the bottom L z by a cod, back surface. Coimated irradiation of strength e re Q R + W is normay incident on this nonrefecting ayer, equay distributed over the disk m, as shown in Fig. 2. A sma detector 2 cm + m 2 cm in size, with an acceptance ange of max is ocated on the back surface at x x 2 y. The object is to determine the fux 7

8 Q R x z L detector Figure 2: One-dimensiona sab with normay incident coimated irradiation. incident on the detector for varying acceptance anges, comparing forward and backward Monte Caro impementations. In a Forward Monte Caro simuation emission points across the irradiation disk for N bundes are chosen, and Q5 emission is aways into the ŝ ˆk or z-direction. Each bunde carries an amount of energy of N and traves a distance of E s}[ RE (22) before being scattered into a new direction. For isentropic scattering the incident direction is irreevant and one may set the new direction to that given for isotropic emission. The L bunde * is then L traced aong as many scattering paths as needed, L unti it eaves the ayer z, or z. If the bunde strikes the bottom surface z, incidence ange (ŝ *~TcVYX 2 ˆk max?) and ocation (x y on detector?) are checked and a detector hit is recorded, if appropriate. Resuts are shown in Fig. 3. As the detector s acceptance ange increases, more photon bundes are captured. Obviousy, this resuts in a arger detector-absorbed fux. However, it aso increases the fraction of statisticay-meaningfu sampes, decreasing the variance of the resuts or the number of required photon bundes to achieve a given variance. Here a cacuations were carried out unti the variance fe beow 2% of the cacuated fux, and the necessary number of bundes is aso incuded in the figure. For the chosen variance about bundes are required for arge acceptance anges, rising to 52 for max. Resuts are difficut to obtain for L max. Simiar remarks can be made for detector area: as the detector area decreases, the necessary number of bundes increases. Modeing a more typica detector mm mm in size woud amost be impossibe. * R In a Backward Monte Caro simuation, since no direct radiation hits Q5 the detector x, the scattered irradiation is cacuated from Eqs. () and (5) with q co R 2 as n I r i ŝ i sq e E HR r d sz 2 R 2 (23) where consists of a number of straight-ine segments, for which d dz5 TWVYX and H is Heaviside s unit step function. Therefore, n 3 sq I r i ŝ i z 2j e E sz dz 3 e E ee Q sz j sz 2j 2 R 2 s 2 zj R 2 (2) s zj j z j where s zj TWVYX j is the z-component of the direction vector for the j-th segment, and z j and z 2j are the z-ocations between which the segment ies within the cyindrica coumn re R (note that 8 j

9 Q det, W Forward Monte Caro Backward Monte Caro Number of photon bundes, N opening ange θ max, deg Figure 3: Detector fuxes and required number of photon bundes for one-dimensiona sab with normay incident coimated irradiation. some segments may ie totay inside this coumn, some partiay, and some not at a). Starting points distributed across the detector are chosen as in forward Monte Caro, and a direction for the backward trace is picked from Eq. (3). Again, a scattering distance is found from Eq. (22), after which the bunde is scattered into a new direction. However, rather than having fixed energy, the backward-traveing bundes accumuate energy according to Eq. (2) as they trave through regions with a radiative source. The tota fux hitting the detector is cacuated by adding up bunde energies according to Eq. (2). Resuts are incuded in Fig. 3, and are seen to coincide with forward Monte Caro resuts to about one variance or better (discrepancy being arger at arge max, since the absoute variance increases). However, the number of required bundes remains essentiay independent of opening ange at about 2, (and, simiary independent of detector area). Since the tracing of a photon bunde requires essentiay the same cpu time for forward and backward tracing, for the probem given here the backward Monte Caro scheme is up to 25, times more efficient than forward Monte Caro. Expanding on the previous exampe, for an acceptance ange of max=, we wi now assume that the medium absorbs as we as scatters radiation, using absorption coefficients of m and 5 m. Forward as we as backward Monte Caro wi be used, and aso both standard ray tracing as we as energy partitioning. Forward Monte Caro standard ray tracing: The soution proceeds as in the previous exampe, except that aso an absorption ength F is chosen simiar to Eq. (22). If the sum of a scattering paths exceeds F, the bunde is terminated. Forward Monte Caro energy partitioning: The soution proceeds as in the previous exampe, except the energy of each bunde hitting the detector is attenuated by a factor ofopq, when is the tota (scattered) path that the bunde traves through the ayer before hitting the detector. Backward Monte Caro standard ray tracing: The soution proceeds as in the previous exampe, except for two changes. First, the oca scattering source must be attenuated by absorption of the 9

10 3 Tabe : Comparison between four different Monte Caro impementations to cacuate irradiation onto a detector from a coimated source. Forward MC Forward MC Backward MC Backward MC Standard Energy partitioning Standard Energy partitioning (m Nƒ ) Q det Nƒ Q det Nƒ Q det Nƒ Q det 9 22ƒ ƒ ƒ.2 9 7ƒ.2 2 ƒ ƒ ƒ ƒ ƒ, ƒ ƒ ƒ.2 Variance of 5% (a other data have variance of 2%) direct beam, and Eq. (2) becomes n 3 sq I r i ŝ i 2 R 2 z 2j e F ˆŠE s z dz s zj Œ Q 2 R 2 3 e Ž z j e Ž z 2j s zj (25) j z j j and are scattering abedo and extinction coefficient, respectivey. And again, an absorp- whereœ F F tion ength is chosen, and the addition in Eq. (25) is stopped as soon as the tota path reaches or the bunde eaves the ayer (which ever comes first). Backward Monte Caro energy partitioning: Again, the scattering source must be attenuated as in Eq. (25), but the exponentia attenuation term in Eq. () must aso be retained. Thus, n 2 e Ž z < F < HR r d sq I r i ŝ i (2) 2 R 2 e R where the integrand contributes ony where the source is active r, but attenuation of the bunde takes pace everywhere ( r tota distance aong path from r i to ). The rest of the simuation remains as in the previous case. Resuts are summarized in Tabe. As expected, if standard ray tracing is empoyed, the number of required bundes grows astronomicay if the absorption coefficient becomes arge, both for forward and backward Monte Caro. Whie backward m Monte Caro retains its advantage (indeed, the forward Monte Caro simuation for 5 coud ony be carried out to a variance of 5%), the reative growth of required bundes appears to be worse for backward Monte Caro. If energy partitioning is empoyed, the number of bundes remains unaffected by the absorption coefficient for both, forward and backward Monte Caro. + In a fina exampe a point source of strength Q W, ocated at x y z 5L wi be considered for a purey scattering medium. Again, fux hitting the detector wi be compared using forward and backward Monte Caro methods. The forward Monte Caro simuation is amost identica to that of the first exampe, except that a photon bundes are now emitted from a singe point, but into random directions. In the backward Monte Caro simuation, the detector fux again consists of a direct and a scattered component and, again, the direct component is zero, this time because a direct radiation hits the detector at an ange arger than the acceptance ange. The I n are then found from Eqs. (2) and () as sq I nr i ŝ i 2 j C j e E s r r x r rx 2 d (27)

11 3 3 Q det, W Forward Monte Caro Backward Monte Caro Number of photon bundes, N Scattering coefficient σ s, m - Figure : Detector fuxes and required number of photon bundes for one-dimensiona sab with interna point source. where the E Q j are the straight paths the bunde traves between scattering events. Equation (27) must be integrated numericay, and this can be done using a simpe Newton-Cotes scheme; here no optimization of the quadrature was attempted, except that away from the source the number of integration points was minimized for sma E (arge s). Aternativey, the integra can be obtained statisticay from 2 sq I nr i ŝ i 2 j E Q j N n e E s r r n x r r n x 2 (28) where the r n are N random ocations chosen uniformy aong path E Q j+ Resuts for detector fux as function of scattering coefficient are shown in Fig.. For sma vaues of s the number of photon bundes required to achieve a reative variance of 2% is much smaer for the backward Monte Caro method, as expected, since the voume with secondary scattering (i.e., the Source S ) is reativey arge, and the detector is sma. However, as s increases, the size of the secondary scattering voume decreases, and backward Monte Caro becomes ess and ess efficient. For both methods arge s mean smaer E Q j, eading to increased tracing effort for each individua bunde. Numerica integration via Eq. (28) was generay much more efficient than Newton-Cotes quadrature, with N usuay being sufficient (since the integra is evauated so many times). However, for arge s this method became inefficient, requiring many photon bundes to achieve a 2% reative variance. In addition, a methods became inefficient for s * m. Summary A comprehensive formuation for backward Monte Caro simuations, capabe of treating emitting, absorbing and anisotropicay scattering media, media with diffuse or coimated irradiation (with arge or sma footprints), media with point or ine sources, etc., has been given. The basic

12 backward Monte Caro simuation of Waters and Buckius [3] was reviewed, and was extended to aow for coimated irradiation, point sources, and other sources of sma voume/area and/or sma soid ange range. In addition, the method was extended to aow standard ray tracing (bundes of fixed energy) as we as energy partitioning (bundes attenuated by absorption). Sampe resuts for radiation hitting a sma detector show that forward Monte Caro methods degrade rapidy with shrinking detector size and acceptance ange. Backward Monte Caro, on the other hand, is unaffected by detector size, but requires a reativey arge radiation source, which in the case of coimated irradiation or point sources needs to be created artificiay by separating direct and scattered radiation. Even for reativey arge detectors/opening anges, using backward Monte Caro can resut in severa orders of magnitude esser computer effort, and becomes the ony feasibe method for very sma detectors. Simiary, using energy partitioning in strongy absorbing media aso reduces numerica effort by orders of magnitude for, both, forward and backward Monte Caro simuations. References [] Modest, M. F., 993, Radiative Heat Transfer, McGraw-Hi, New York. [2] Siege, R. and Howe, J. R., 992, Therma Radiation Heat Transfer, Hemisphere, New York, 3rd ed. [3] Waters, D. V. and Buckius, R. O., 992, Monte Caro Methods for Radiative Heat Transfer in Scattering Media, In Annua Review of Heat Transfer, 5, Hemisphere, New York, pp [] Gordon, H. R., 985, Ship perturbation of irradiance measurements at sea. : Monte Caro simuations, Appied Optics, 2, pp [5] Coins, D. G., Bättner, W. G., Wes, M. B., and Horak, H. G., 972, Backward Monte Caro cacuations of the poarization characteristics of the radiation emerging from spherica-she atmospheres, Appied Optics,, pp [] Adams, C. N. and Kattawar, G. W., 978, Radiative transfer in spherica she atmospheres I. Rayeigh scattering, Icarus, 35, pp [7] Nishita, T., Miyawaki, Y., and Nakamae, E., 987, A shading mode for atmospheric scattering considering uminous intensity distribution of ight sources, Comput. Graphics, 2, pp [8] Sabea, P, 988, A rendering agorithm for visuaizing 3D scaar fieds, Comput. Graphics, 22, pp [9] Edwards, D. K., 983, Numerica methods in radiation heat transfer, In Shih, T. M., ed., Proc. Second Nationa Symposium on Numerica Properties and Methodoogies in Heat Transfer, Hemisphere, pp [] Waters, D. V. and Buckius, R. O., 992, Rigorous Deveopment For Radiation Heat Transfer In Nonhomogeneous Absorbing, Emitting And Scattering Media, Internationa Journa of Heat and Mass Transfer, 35, pp [] Case, K. M., 957, Transfer Probems and the Reciprocity Principe, Review of Modern Physics, 29, pp