Chapter 2. Monte Carlo Ray-Trace Formulation

Transcription

1 Chpter 2. Monte Crlo Ry-Trce Formulton Mny het trnsfer problems requre the defnton of the rdtve exchnge mong surfces n n enclosure. Even though the formulton of rdton exchnge problems s not prtculrly dffcult, the result s usully set of tedous ntegrl equtons or ther lgebrc equvlent, lrge system of coupled lner lgebrc equtons. Some smplfyng ssumptons nd pproxmtons usully hve to be mde n order to resolve the nherent mthemtcl dffcultes. For exmple, the worklod my be reduced tremendously by ssumng tht the medum fllng the cvty does not prtcpte n the rdtve exchnge nd tht the surfces behve only s dffuse nd gry emtters, bsorbers, nd reflectors of rdton. These ssumptons led to the defnton of confgurton fctors tht ccount for the geometrcl effects. However, snce no surfce s purely dffuse, the vldty of usng confgurton fctors s lmted. In 1962 Sebn, n wrtten dscusson of pper by Sprrow et l. suggested tht the reflectvty of mny surfces of prctcl engneerng nterest could be expressed s the sum of dffuse nd speculr component. Usng ths pproxmton the reflectvty cn be wrtten d s ρ = ρ + ρ (2.1) Bsed on ths de, Mhn nd Eskn [1981, 1984] ntroduced the concept of the rdton dstrbuton fctor. In hs book, Mhn [2002] defnes the totl rdton dstrbuton fctor D j s the frcton of the totl rdton emtted from surfce element tht s bsorbed by surfce element j, due both to drect rdton nd to ll possble reflectons wthn the enclosure. Ths defnton ncludes drectonl emsson nd bsorpton nd bdrectonl reflecton. The model developed n ths thess ssumes only dffuse emsson nd bsorpton nd dffuse-speculr reflecton nd so the totl, dffuse- 12

2 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton speculr dstrbuton fctor D j s used. Ths smplfyng ssumpton s often justfed n the desgn nd modelng of rdometrc nstruments by the judcous choce of cotngs to be used n the fbrcton of the nstrument. In other words, the nstruments re desgned to be menble to ccurte therml rdtve nlyss usng the totl, dffuse-speculr rdton dstrbuton fctor. 2.1 The Totl, Dffuse-Speculr Rdton Dstrbuton Fctor The totl, dffuse-speculr rdton dstrbuton fctor D j s defned s the frcton of totl therml rdton emtted dffusely from surfce element tht s bsorbed by surfce element j, due to drect rdton nd to ll possble dffuse nd speculr reflectons wthn the enclosure. Usng ths dstrbuton fctor, the rdtve energy emtted by surfce tht s bsorbed by surfce j my be wrtten s Q j 4 = ε A σt D (W) (2.2) j where ε s the hemsphercl, totl emssvty of surfce, A s the re of surfce ( m 2 ), σ s the Stefn-Boltzmn constnt ( Wm -2 K -4 ), nd T (K) s the temperture of surfce. Equton 2.2 my be tken s the defnton of dstrbuton fctor hs three useful propertes [Mhn, 2002]: D j. The 1. Conservton of energy n j=1 j 0 D = 1., 1 n (2.3) 2. Recprocty ε A D = ε A D, 1 n, 1 j n (2.4) j j j j 3. Combnton of recprocty nd conservton of energy 13

3 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton n =1 j ε A D = ε A 1 j n. (2.5) j j In Equtons 2.3, 2.4 nd 2.5, n s the number of surfce elements nto whch the enclosure hs been dvded, ε s the hemsphercl, totl emssvty of gven surfce element, nd A s ts surfce re. The conservton of energy relton nd the recprocty relton my be used to detect nd elmnte errors mde durng the estmton of dstrbuton fctors or to fnd unknown dstrbuton fctors from known dstrbuton fctors (but not both). It s noted tht t s possble to defne other dstrbuton fctors; for exmple, f the enclosure hs n openng through whch rdton my pss, the openng cn be treted s blck membrne tht bsorbs ll rdton ncdent from the nteror nd through whch power Q o psses nto the enclosure wth specfed ngulr dstrbuton. Then the dstrbuton fctor D = 1 j n (2.6) oj Q oj/q o cn be defned where Q oj s the power (W) enterng the enclosure through openng o tht s bsorbed by surfce element j. The vlue of the dstrbuton fctor does not depend solely on geometry but lso on the rdtve chrcterstcs of the surfces mkng up the cvty. Ths mens tht t s not possble to use the trdtonl methods for fndng confgurton fctors to estmte dstrbuton fctors. Dstrbuton fctors re estmted usng the Monte Crlo ry-trce (MCRT) method. 2.2 The Monte Crlo Ry-Trce (MCRT) Method The dscusson n ths secton s lberlly dpted from Mhn [2002]. The Monte Crlo ry-trce (MCRT) method solves rdton het trnsfer problems by performng sttstcl smplng experments on computer usng rndom numbers. The method s nmed fter the cptl of the prncplty of Monco, fmous for 14

4 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton ts csno, becuse the rndom numbers could be obtned by droppng bll n spnnng roulette wheel [Anon, 2002]. The MCRT method s sometmes descrbed s sttstcl pproch where gme, or model, wth the sme behvor of the physcl process beng modeled replces the specfc problem, producng the sme outcome s the physcl process but wth the dvntge tht the gme s eser to ply. In ths pproch rdton energy emtted from gven surfce s dvded nto lrge number N of dscrete nd unform energy bundles. Thus, solvng therml rdton problem by the MCRT method mples trcng the hstory of these bundles from ther emsson by surfce element, through seres of reflectons on other surfce elements, to ther bsorpton by one of the surfce elements j of the enclosure. Usng the propertes of the enclosure nd the lws of probblty t s possble to determne the number of energy bundles N j emtted by surfce nd bsorbed by surfce j. Then the dstrbuton fctors my be estmted s N D j j (2.7) N As N becomes ncresngly lrge, the ccurcy of the estmte ncreses. In prctce the dstrbuton fctor but lrge vlue of D j cn be estmted to n rbtrry degree of ccurcy usng fnte N. Once ll the propertes of the surfces re defned, the MCRT method cn be used to determne the vlue of the dstrbuton fctor. The logc flow chrt for pplyng the MCRT method to obtn totl, dffuse-speculr rdton dstrbuton fctors s shown n Fgure 2.1. The stndrd technque s to relese lrge number of energy bundles from rndomly selected loctons on gven surfce element nd then to trce ther pths through seres of reflectons untl they re bsorbed by nother (or the sme) surfce element. The number of energy bundles used depends on the requred ccurcy nd on the vlblty of computer resources. Normlly, 100,000 bundles s consdered mnmum number to relese from ech surfce of the enclosure for sttstclly sgnfcnt 15

5 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton results, wth one mllon beng common. The norml steps n numercl mplementton of the MCRT method re descrbed below. The step numbers refer to the block numbers n the logc flow chrt, Fgure 2.1 Determne the Locton of Emsson of n Energy Bundle 1 Determne the Drecton of Emsson of the Energy Bundle 2 3 Determne the Intersecton pont of the Energy Bundle wth the Enclosure Wlls Determne the Surfce n whch the Energy s Absorbed Absorbed Determne f the Energy Bundle s Reflected or Absorbed 4 Reflected Yes N <N mx? 6(d) Dffuse Determne f the Reflecton s Speculr or Dffuse Speculr 5 6(s) No Determne the Drecton of the Dffuse Reflecton Determne the Drecton of the Speculr Reflecton Stop Fgure 2.1 Logc flow chrt for mplementng the Monte Crlo ry-trce method for gven source surfce n dffuse-speculr, gry enclosure 1. Determne the locton of emsson of n energy bundle In the typcl pplcton to rdometrc nstruments the energy bundle s often not relly emtted from surfce element but rther enters the cvty through n openng, the perture. A pseudo-rndom number genertor s used to generte very long sequence, usully tens of mllons, of hghly uncorrelted rndom numbers unformly dstrbuted 16

6 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton between zero nd unty. For further nformton bout pseudo-rndom number genertors the reder s referred to Appendx D n Mhn [2002]. Three rndom numbers, R x, R y, R z, re drwn sequentlly from populton of prevously generted rndom numbers. Then for exmple n the prtculr cse of plnr rectngulr surfce, the locton of emsson s n prncple defned by x = x y = y z = z mn mn mn + (x x ) R (2.8) mx mx mn mn y x + (y y ) R (2.9) + (z z ) R (2.10) mx mn z In Equtons 2.8 through 2.10, the subscrpts mn nd mx denote, respectvely, the mnmum nd mxmum vlue of the pproprte coordnte on the surfce element. Note tht n prctce only two of these equtons re ctully needed to determne the coordntes (x, y, z) of the pont of emsson of the energy bundle, becuse the thrd coordnte cn be clculted usng the equton of the plne, n x 0 y 0 z 0 = (x - x ) + n (y - y ) + n (z - z ) 0 (2.11) In Equton 2.11, n x, n y, nd n z re the components of the unt vector norml to the plne, nd x 0, y 0, nd z 0 re the coordntes of ny pont on the surfce. 2. Determne the drecton of emsson of the energy bundle. Sprrow nd Cess [1966] gve expressons for the drecton of dffuse emsson s functon of two ndependent coordntes: the crcumferentl ngle φ n the plne of the surfce wth respect to locl unt surfce tngent t, nd the zmuth ngle θ wth respect to the locl unt surfce norml n. These two ngles re gven by φ = 2πR (2.12) φ 17

7 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton nd -1 θ = sn R θ (2.13) where R φ nd R θ re two consecutve rndom numbers unformly dstrbuted between zero nd unty. The ngles φ nd θ nd the norml nd tngent vectors re used together to determne the drecton cosnes (l, m, n) of the pth long whch the emtted energy bundle trvels n the globl coordnte systems. The drecton cosnes re defned s l = cosα (2.14) m = cosβ (2.15) n = cosγ (2.16) where α, β, nd γ re the ngles tht drected lne mkes wth respect to the coordnte xes x, y, nd z, respectvely, s shown n the Fgure 2.2. Angles φ nd θ re defned wth respect to the locl coordntes (t, n), whch must be trnsformed to globl coordntes (x, y, z). The relton between the locl nd globl coordntes s shown n Fgure 2.3. Vector mnpulton nd/or coordnte trnsformton re requred to relte (t, n) nd (x, y, z), nd thus the ngles φ nd θ detls of ths process re gven n Mhn [2002]. to the drecton cosnes (l, m, n). The z Pth of emtted energy bundle γ β α y x Fgure 2.2 Illustrton of the drecton cosnes 18

8 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton n Pth of emtted energy bundle θ φ t z x y Fgure 2.3 Reltonshp between locl nd globl coordntes 3. Determne the pont of ntersecton of the emtted energy bundle wth the enclosure wlls Usng the equtons of the surfces tht mke up the enclosure nd the equtons of the lne for the pth long whch the energy bundle trvels (the ry ), ll the possble ponts of ntersecton of the energy bundle wth the wlls cn be found. Becuse the equtons of lne descrbe n nfntely long lne nd, n the cse of enclosures consstng of plnr surfces, the equtons of the surfces descrbe plnes nfnte n extent, mny cnddte ponts of ntercton wll generlly be found. Therefore t s necessry to elmnte ll of the ponts of ntersecton except the one correspondng to the ctul ntersecton of the pth of the energy bundle nd the enclosure wll. Ths problem s llustrted n Fgure

9 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton Equtons 2.17 d b S(x, y, z) = 0 c Fgure 2.4 Intersectons of strght lne wth the surfces of n rbtrry enclosure In the stuton depcted n Fgure 2.4, n energy bundle s emtted from Pont n the drecton of Pont b. The equtons of the lne re gven by x x y y z z T = = = (2.17) l m n where T s the length of the lne connectng x, y, z wth x, y, z. Equtons 2.17 nd the equton of ech surfce of the enclosure, S(x,y,z) = 0, consttute sets of four equtons n the four unknowns x, y, z, nd T. In the exmple of Fgure 2.4 when Equtons 2.17 re solved wth the equtons of the surfces, four ponts re obtned:, b, c, d. From the fgure, t s cler tht Pont b s the desred pont. However, ensurng tht the computer selects the correct pont s tsk ccomplshed through the followng three-step process: 1. Frst, elmnte the pont of orgn (Pont ). Ths pont wll lwys emerge s cnddte. One wy to do ths s to compre the coordntes of cnddte ponts of ntersecton between the lne nd the surfce wth the coordntes of the emsson 20

10 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton pont. If these coordntes re the sme, wthn some specfed tolernce, ths pont s rejected. 2. Next, elmnte the bck cnddte (Pont c). Ths my be ccomplshed by formng the dot product of the outwrd-drected surfce norml S(x, y, z) wth the vector (x c x ) + (yc y ) j + (zc z )k drected from the source pont towrd the cnddte pont. A postve vlue of ths dot product ndctes tht the cnddte les n front of the source, whle negtve vlue ndctes tht the cnddte les behnd the source nd therefore must be rejected. 3. Fnlly, choose the nerest cnddte from mong ll the cnddtes tht le n front of the source. In order to do ths, t s necessry to compre the dstnces from the source to ll the cnddtes n front of t, nd then to choose the shortest one. Equton 2.17 gves the dstnce T between ny two ponts. Equtons 2.17 my be wrtten n the form x = x + lt (2.18) y = y + mt (2.19) z = z + nt (2.20) The form of Equtons 2.18 through 2.20 vods the possblty of dvson by zero tht could occur f one or two of the drecton cosnes hve vlues of zero (whch occurs whenever the pth of n energy bundle s norml to gven coordnte xs). 4. Determne whether the energy bundle s bsorbed or reflected Usng the ssumpton tht ll surfces re opque dffuse-speculr, gry reflectors, t s possble to wrte tht d s α = 1 ρ = 1 ρ ρ (2.21) 21

11 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton where d ρ s the dffuse component of reflectvty nd ρ s s the speculr component. The bsorptvty my be nterpreted s the probblty tht n ncdent energy bundle wll be bsorbed. Then ech tme tht n energy bundle ntercepts surfce, the next vlble rndom number R α, unformly dstrbuted between zero nd unty, s drwn nd compred wth the bsorptvty. If the rndom number s less thn or equl to the bsorptvty, the energy bundle s bsorbed; otherwse, t s reflected. If the energy bundle s bsorbed, the vlue of N j s ncremented nd control returns to Step 1 wth the emsson of new energy bundle. If the energy bundle s reflected, the logc contnues to the next step. 5. Determne f the reflecton s dffuse or speculr The speculrty rto component of reflectvty to the reflectvty, rs s surfce property, defned s the rto of the speculr r s s s ρ ρ = (2.22) s d ρ ρ + ρ Ths rto cn be nterpreted s the probblty tht reflecton from the surfce n queston s speculr. A rndom number R s, s drwn nd compred wth r s. If unformly dstrbuted between zero nd unty, R s s less thn or equl to r s, the reflecton s speculr; otherwse t s dffuse. If the reflecton s speculr, the logc brnches to Step 6(s), wheres f the reflecton s dffuse, the logc brnches to Step 6(d). 6(s) Determne the drecton of the speculr reflecton A reflecton s sd to be speculr when the pths of the ncdent nd reflected energy bundles nd the surfce norml ll le n the sme plne, nd the ngle between the reflected energy bundle nd the surfce norml s equl to the ngle between the 22

12 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton ncdent energy bundle nd the surfce norml. These reltonshps re llustrted n Fgure 2.5. The rules llustrted n ths fgure my be combned to yeld V r = V 2(V n)n = V + 2 V cosθ n (2.23) where V = (x x ) + (y y ) j + (z z )k (2.24) b b b In Equton 2.24, (x b, yb, z b ) re the coordntes of the pont where the energy bundle ntersects the recevng surfce nd (x, y, z ) re the coordntes of the source pont. n V θ θ V r - (V n) n b V - (V n)n Fgure 2.5 Reltonshp between ncdent nd speculrly reflected rys The result gven by Equton 2.23 my be nterpreted n terms of the drecton cosnes. Knowng tht the mgntude of the reflected vector s gven by T = ( V ) + ( V ) + ( V ) r r,x r,y r,z (2.25) then Vr,x l r = (2.26) T r 23

13 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton Vr,y m r = (2.27) T r nd Vr,z n r = (2.25) T r 6(d) Determne the drecton of the dffuse reflecton When n energy bundle s reflected dffusely, t retns no knowledge of ts hstory before sufferng the reflecton. Tht s, the energy bundle hs been reborn s newly emtted energy bundle. Ths mens tht dffuse reflecton s exctly lke dffuse emsson. Then the current logc step would be dentcl to Step 2, s the dshed lne n the block dgrm of Fgure 2.1 ndctes. The steps descrbed bove re repeted for lrge number of energy bundles, ledng to s fnl result the dstrbuton fctor estmted by N D j j (2.29) N where N j s the number of energy bundles emtted by surfce nd bsorbed by surfce j, nd N s the totl number of energy bundles emtted by surfce. 2.3 MCRT n FELIX (Functonl Envronment for Longwve Infrred exchnge) The mjorty of the numercl experments done n ths reserch were crred out usng the progrm FELIX developed by Félx Nevárez-Ayl [2002]. The progrm receves the geometres from CAD-bsed softwre such s AutoCd nd dvdes every surfce nto fcets. Ech fcet s trngle locted n spce whose poston nd orentton re defned by ts three vertces nd the norml to the fcet. Usng ths pproch the ntersecton pont between n energy bundle nd fcet cn be found by solvng smultneously the equtons of the pth followed by the energy bundle nd the 24

14 Mrí Crstn Sánchez Monte Crlo Ry-Trce Formulton plne tht contns the fcet. The reder s referred to Nevárez-Ayl [2002] for more detled dscusson of FELIX. In ths chpter n overvew of the Monte Crlo ry-trce method ws presented. The theoretcl foundtons of ths method long wth the methodology for pplyng the MCRT method to rdton het trnsfer problems were dscussed. In the next chpter sttstcl protocol for determnng the uncertnty ssocted wth estmtes of the dstrbuton fctors usng from the MCRT method s presented. 25