TWO-DIMENSIONAL TRANSIENT RADIATIVE HEAT TRANSFER USING DISCRETE ORDINATES METHOD

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1 TWO-DIMENSIONAL TRANSIENT RADIATIVE HEAT TRANSFER USING DISCRETE ORDINATES METHOD Zhxong Guo and Sunl Kumar Department of Mechancal, Aerospace and Manufacturng Engneerng Polytechnc Unversty, 6 Metrotech Center, Brooklyn, NY 1121, USA ABSTRACT. The S-N dscrete ordnates (DO) method s developed for the frst tme to solve transent radatve heat transfer n a two-dmensonal rectangular enclosure wth absorbng, emttng, and ansotropcally scatterng medum subject to dffuse and/or collmated laser rradaton. The transent DO method s used to solve several example problems and compared wth the exstng results and the Monte Carlo predctons. Good agreement between the transent DO solutons and other predctons s found. Fnally, the transent DO method s appled to nvestgate the characterstcs of short-pulsed laser radaton nteracton and transport wthn bologcal tssues. INTRODUCTION Wth the advent of the short-pulsed laser wth the duraton of the order of femtoseconds, transent laser radaton transport through turbd meda has attracted a great deal of attenton n recent years [1], partcularly for applcatons n bo-medcal treatment and dagnostcs. One mathematcal model for descrbng short-pulsed laser transport s tme-dependent radatve transfer equaton. The soluton of the hyperbolc transent radatve heat transfer equaton s then of great nterest. Sgnfcant progress has been made n the development of soluton method of radatve heat transfer n partcpatng meda n recent decades. However, the analyss of radatve heat transfer n most engneerng problems tradtonally neglects the effect of lght propagaton speed. In the applcatons of short-pulsed lasers, such a neglectng may nduce sgnfcant errors [1-5]. Most prevous studes on transent laser transport are based on the parabolc dffuson approxmaton [2,3] or have utlzed the stochastc Monte Carlo (MC) method [4,5]. However, the dffuson approxmaton s hardly applcable to thn tssues or tssues havng varyng dstrbutons of optcal propertes and complex geometres. The MC method s tme-consumng and the results are subject to statstcal error due to practcal fnte samplngs. Few studes have addressed the soluton of the entre hyperbolc transent radatve transfer equaton. The addng-doublng method [6] was proposed to solve the transent response of a slab medum wth constant external source. Kumar et al. [7] consdered the soluton of the hyperbolc transent radatve equaton by usng the P 1 models n 1D planar medum. More recently, Mtra and Kumar [8] examned several numercal methods for 1D transent radatve transport n absorbng-scatterng medum, n whch dscrete ordnates method, P -N model, dffuse approxmaton, and two-flux method have been dscussed. Tan and Hsu [9] developed an ntegral equaton formulaton for transent radatve transfer. Guo and Kumar [1] extended the radaton element method to consder the transent radatve transfer. Mtra et al. [11] appled the hyperbolc P 1 model to transent radatve transfer n a 2D rectangular medum. Wu and Wu [12] solved the transent ntegral equaton usng quadrature method n 2D cylndrcal lnearly ansotropcally scatterng meda. However, the P 1 model underestmates apparently the lght propagaton speed [8], and the ntegral formulaton s dffcult to be appled to complex geometres wth Me ansotropcally scatterng meda. In the soluton of mult-dmensonal steady state radatve transfer n partcpatng meda, the dscrete ordnates (DO) method has been one of the most wdely appled methods [13-15]. The DO method requres a sngle formulaton to nvoke hgher order approxmatons, ntegrates easly nto

2 (a) (b) Fgure 1. (a) System geometry; (b) a control volume. control volume transport codes, and s applcable to complete Me ansotropc scatterng phase functon and nhomogeneous meda. Based on these characterstcs, the DO method has been selected n the present study for mplementaton nto mult-dmensonal transent radaton transport n absorbng, emttng, and ansotropcally scatterng meda. The transent DO soluton s verfed by comparson aganst exstng steady state DO soluton and transent Monte Carlo predcton n several exemplfed problems. The equvalent sotropc scatterng results are compared wth the ansotropc scatterng modelng wth truncated Legendre polynomals phase functon n the transent doman. Fnally, the transent DO method s appled to nvestgate the short-pulsed laser nteracton and transport n lvng tssues. MATHEMATICAL MODEL For 2D Cartesan coordnates as shown n Fg. 1 (a), the hyperbolc transent radatve transfer equaton of dffuse ntensty I n the dscrete ordnate drecton ŝ s formulated as 1 I I I + ξ + η + βi = βs, = 1,2,K n (1) c t x y where the extncton coeffcent β s the sum of absorpton coeffcent κ and scatterng coeffcent σ s, c s the speed of lght n medum, and S s the radatve source term: n ω S = ( 1 ω ) Ib + w j Φ j I j + Sc, = 1, 2, K n (2) 4π j = 1 where scatterng albedo ω = σ s / β, Φ j represents scatterng phase functon, and S c s the source contrbuton of collmated rradaton. A quadrature of order n wth the approprate angular weght w j s used n the S-N dscrete ordnates method. The scatterng phase functon may be approxmated by a fnte seres of Legendre polynomals as j Φ C P (cosϕ) (3) = M k = k k here, cos ϕ = ξ ξ j + ηη j + µ µ j. The C k s are the expanson coeffcents of the correspondng Legendre functon. ξ, η, and µ are the three drecton cosnes of the dscrete drecton ŝ. The enclosure walls are dffusely reflectng. The dffuse ntensty at wall 1 s 1 ε n / 2 w Iw = ε w Ibw + w j I j ξ j (4) π ξ j < Smlarly, we can set up relatons for the rest three walls.

3 The collmated laser sheet s normally ncdent upon the center of wall 1 wth spatal wdth d c and ts ntensty s Ic ( x, ξ c, t) = I exp( β x)[ H ( t x / c) H ( t t p x / c)] δ ( ξc 1), y ( d c / 2, dc / 2) (5) where H(t) and δ are the Heavsde and the Drac delta functons, respectvely, t p and I are the pulse wdth and ncdent ntensty of the on-off square laser pulse. The collmated component S c n Eq. (2) s then wrtten as ω Sc = I c Φ ( ξcξ ) (6) 4π In the regon where no collmated laser rradaton s passng through, I S =. c = c Once the ntensty feld s obtaned, the ncdent radaton G and the net radatve heat fluxes Q x and Q y can be obtaned as n G = w j I j + I j=1 n c, Q x = j w j I j + Ic j=1 n ξ, Qy = η j w j I j (7) j= 1 NUMERICAL SOLUTION To solve the dscrete ordnate equaton (1), the fnte volume approach s employed. The enclosure s dvded nto small control volumes by MX MY meshes. In each control volume as shown n Fg. 1 (b), the spatally and temporally dscretzed equaton can be expressed as V ( I P I P ) ( E E W W ) ( N N S S ) ( P P ) + ξ A I A I + η A I A I = βv I + S c t (8) where, AE = AW = y, AS = AN = x, V = x y, and I P s the control volume average ntensty at prevous tme step. In order to solve Eq. (8), the weghted damond dfferencng scheme s ntroduced [13-16]: I P = γ y I N + ( 1 γ y ) IS = γ x I E + (1 γ x ) IW (9) Concernng the determnaton of the values of γ x and γ y, many types of spatal dfferencng scheme have been dscussed. In the present study, the postve scheme, whch guarantees postve radatve ntensty n terms of the spatal and angular grds [15], s appled. The fnal dscretzaton equaton for the cell ntensty n a generalzed form becomes 1 ξ η I P S I I c t + P + x y x + β γ y x β γ y β I = (1) P 1 ξ η β c t γ β x γ β y x y where I x s the x-drecton face ntensty where the beam enters ( = IW for ξ >, and = I E for ξ < ), and I y s the correspondng y-drecton face ntensty. A tme dscretzed term was added n Eq. (1) for the consderaton of transent radatve transfer. If the tme step t s nfntely large, Eq. (1) s consstent wth the steady state form. It should be noted that an mplct scheme was used n Eq. (1) for the tme-dependent term. An ntal feld of ntensty must be gven based on the physcal realty. In the present study, the ntal values of ntensty at all dscrete ordnates everywhere n the feld are set equal to zero. Actually, the transent solvng procedure s very smlar to the teratve soluton for steady state radatve transfer. Hence, we are not gong to descrbe the soluton procedures n detal. However, t should be notced that the ntroducton of transent soluton does not smply mean an addton of soluton method for steady

4 Surface Heat Flux κ =.1 Exact Soluton [17] t* = 1 t* = 2 t* = 5 t* = X κ = X κ = 1 L = W = X (a) (b) (c) Fgure 2. Non-dmensonal surface heat flux for a square enclosure wth cold black walls and hot absorbng medum: (a) κ =.1; (b) κ = 1.; (c) κ = 1. state radatve transfer. Its sgnfcance s emboded by the ncorporaton of lght propagaton effect n mcroscale short tme radaton transport. Some short tme radaton phenomena, such as the broadenng of short pulse through scatterng medum, can only be observed n transent soluton [1]. Fveland [13] has ntroduced lmtaton on the spatal dfferental step. For transent radatve transfer, a lmtaton on tme step should also be mposed. Snce a lght beam always travels wth a velocty c, the travelng dstance c t between two neghborng tme steps should not exceed the control volume spatal step,.e., c t < Mn{ x, y}. Thus, f we ntroduce non-dmensonal varables t* = β ct, x* = x/l, and y* = y/w, we have ξ η t* < Mn, (11) 1-γ x 1-γ y The choce of quadrature scheme n the DO method s arbtrary. In the present calculatons, the S-12 approxmaton (n = 84, whch computes 84 fluxes over the hemsphere) s generally used. The values of dscrete ordnate quadrature sets and weghts can be found n Table 2 of Fveland [14]. RESULTS AND DISCUSSION At frst, the transent DO method s appled to a square enclosure wth cold, black walls, and a purely absorbng medum that s suddenly rased to and mantaned at an emssve power of unty. The predcted surface heat fluxes at dfferent tme nstants for three dfferent absorpton coeffcents are plotted n Fgs. 2 (a), (b) and (c), respectvely. It s seen that the heat flux ncreases as the tme proceeds. After t* = 5., the change versus tme s nvsble and the results reach to a steady state solutons. The results at long tme stages are compared wth exact soluton [17] n steady state. Excellent agreement was found. Then a boundary ncdent problem n a square enclosure s studed, where wall 1 s suddenly heated and mantaned at hot wth unty emssve power, but the rest walls and the medum are kept cold. The medum s ansotropcally scatterng wth Me phase functon F2, whch was lsted n Table 1 of Km and Lee [15] n detal. The asymmetrc factor g for the strong forward phase functon F2 s g = The non-dmensonal ncdent radaton and net heat fluxes along the centerlne (y* = ) are dsplayed n Fgs. 3 (a) and (b), respectvely, for dfferent tme nstants. The sold crcle marks are the steady state values predcted usng S-14 DO method [15]. As tme advances, t s seen that the radaton s propagatng to the larger x end. The transent results gradually match to the steady state solutons. The mnor dfference between the steady state soluton and the transent soluton at

5 Incdent Radaton G S-14 Soluton at Steady State [15] ω = 1 βl = βw = x* (a) t* =.5 t* = 1. t* = 2. t* = 4. t* = 8. Net Radatve Heat Flux Q y ω = 1 βl = β W = x* (b) t* =.5 t* = 1. t* = 2. t* = 4. t* = 8. Fgure 3. Incdent radaton (a) and net radatve heat flux n the y-drecton (b) along the centerlne for a square enclosure wth one hot wall and cold ansotropc scatterng medum..8 One hot wall: dffuse rradaton Three cold walls at y* =. Reflectance.6.4 κ =.1 mm -1 σ si = 1. mm -1 L = W = 1 mm at y* =.25 DO Method MC Method.2 at y* =.48 Cold medum: sotropc scatterng t* Fgure 4. Comparson of DO method wth MC method for temporal profles of reflectance. t* = 8. may be attrbuted to the dfferent order approxmatons used n the two solutons. The DO method s examned n transent doman by comparson aganst the Monte Carlo predcton for sotropcally scatterng medum wth black walls n Fg. 4, where temporal dstrbutons of reflectance are shown. Wall 1 s assumed to be hot and rradated dffusely, other walls and the

6 12 3 L = W = 1 mm κ =.1 mm -1 Normalzed G (x 1 5 ) 8 4 σ s = 3. mm -1 g = Transmttance (x 1 3 ) Isotropc modelng Ansotropc modelng t* = β ct Ι Fgure 5. Comparson between equvalent sotropc scatterng results and drect ansotropc smulatons for temporal profles of transmttance and normalzed ncdent radaton n a square subject to an mpulse laser rradaton. medum are cold. Other parameters are, L = W = 1 mm, κ =.1 mm -1, and σ si = 1. mm -1. The MC results n the current paper are calculated based on the algorthm developed by Guo et al. [12]. It s seen that the transent DO results are n excellent agreement wth those predcted by the Monte Carlo method. For sotropcally scatterng medum, we found that even a lower order DO approxmaton (S-8) can predct accurate results. The equvalent sotropc scatterng results are then compared wth the correspondng drect ansotropcally scatterng smulatons wth phase functon F2 n Fg. 5 for temporal profles of transmttance at the center of the output wall and of normalzed ncdent radaton at the center of the enclosure. The square medum s dvded by meshes and s subject to an mpulse laser rradaton at the center of wall 1. It s seen that the equvalent sotropc scatterng results approach closely the Me phase functon ansotropcally scatterng predctons except at the early tme stages. Ths fndng s consstent wth our prevous fndng [18] for an optcally thck and forward scatterng medum wth Henyey-Greensten phase functon. Although the nput laser pulse s mpulse,.e., wth nfntely small pulse wdth, the transmtted pulse and the temporal dstrbuton of ncdent radaton are clearly broadened wth fnte pulse wdth as shown n Fg. 5. Such a phenomenon s a salent feature of short pulse laser transport n scatterng medum and s only observable by performng transent smulaton wth the ncorporaton of lght propagaton effect. Fnally, short-pulsed laser transport n lvng tssues s nvestgated usng the DO method. Parameters are: L = 1 mm, W = 29.9 mm, κ =.1 mm -1, and σ si = 1. mm -1. The refractve ndex of the tssue s 1.4. The spatal wdth of the ncdent mpulse laser s d c =.1 mm (to smulate a laser mposed through a 1 µm optcal fber). The control volume sze s.1 mm.1 mm n am to smulate precsely the transent laser transport n mcroscale area and to smulate detectors usng optcal fber. Tme resoluton s t =.2 ps. Internal reflecton s not consdered because boundary s

7 Incdent Radaton G κ =.1 mm -1 σ si = 1. mm -1 L = 1 mm W = 29.9 mm t = 4 ps t = 2 ps t = 4 ps t = 8 ps t = 16 ps t = 24 ps t = 32 ps t = 4 ps t = 48 ps Transmttance (x 1 6 ) y = y = 1 y = 2 y = 3 y = 4 y = 5 y = 6 y = 7 y = x* (a) tme (ps) (b) Fgure 6. Incdent radaton along the centerlne (a) and temporal dstrbutons of transmttance at varous locatons (b) n a rectangular tssue subject to an mpulse laser rradaton. matched when the optcal fbers are nserted nto lvng tssue. Fg. 6 (a) shows the non-dmensonal ncdent radaton along the centerlne (y* = ) at varous tme nstants. It s clearly seen that the sudden peak, whch represents the ballstc component of the laser, s propagated from small x to large x untl t passes through the medum wth the speed of lght, and the peak value s substantally reduced n the process of propagaton. The dffuse component due to multple scatterng events also forms a second maxmum pont along the x-drecton and the dffuse apex s also propagated gradually from the small x to the center of the x-axs. At long tme stages, the profle of the ncdent radaton s symmetrc along the center poston x* =.5. As tme proceeds, the value of the ncdent radaton becomes smaller and smaller. The temporal transmttance profles at dfferent locatons are shown n Fg. 6 (b). It s seen that, wth the ncrease of dstance between the detector and the laser ncdent axs, the peak poston of the transmtted pulse moves to large tme nstant and the transmtted pulse wdth ncreases. However, the magntude of the transmtted pulse decreases. CONCLUSIONS The dscrete ordnates method s formulated to study two-dmensonal transent radatve heat transfer n ansotropcally scatterng, absorbng and emttng medum subject to dffuse and/or collmated short-pulsed laser rradaton. The transent DO soluton s verfed by comparson wth the exstng publshed results and/or wth the Monte Carlo smulaton for a varety of exemplfed problems. It s found that the present method s accurate and can be used to predct all transent radatve quanttes. The temporal dstrbutons of transmttance and dvergence n equvalent sotropc scatterng modelng are found to approach closely the predctons of drect modelng of strong forward ansotropc scatterng wth truncated Legendre polynomals phase functon n most of the transent doman except at early tme nstants. The transent DO method s appled to study the characterstcs of short-pulsed laser nteracton and propagaton wthn lvng tssues. It s found

8 that the ballstc component of the laser propagates wth the speed of lght at the tssue and ts value s substantally reduced wth the advance of propagaton. The dffuse component due to multple scatterng also forms a second maxmum ncdent radaton nsde the tssue, but fnally the profle s symmetrc along the center of the square. The ncdent radaton s strongly affected by ts mcroscale space poston and tme nstants. The temporal shape of the transmtted pulse s strongly nfluenced by the poston of the detector. Wth the ncrease of dstance between the detector and the laser ncdent axs, the peak poston of the transmtted pulse moves to large tme nstant and the transmtted pulse wdth ncreases. ACKNOWLEDGMENTS The authors acknowledge partal support from the Natonal Scence Foundaton grant AW 9963 (CTS-97321) admnstrated by Sanda Natonal Laboratores, Shawn Burns, Project Manager. REFERENCE 1. Kumar, S., Mtra, K., Mcroscale Aspects of Thermal Radaton Transport and Laser Applcatons, Advances n Heat Transfer, Vol. 33, pp , Yoo, K. M., Lu, F., Alfano, R. R., When Does the Dffuson Approxmaton Fal to Descrbe Photon Transport n Random Meda, Physcal Revew Letter, Vol. 64, pp , Brewster, M. Q., Yamada, Y., Optcal Propertes of Thck, Turbd Meda from Pcosecond Tme-Resolved Lght Scatterng Measurements, Int. J. Heat Mass Transfer, Vol. 38, pp , Wlson, B. C., Adam, G., A Monte Carlo Model for the Absorpton and Flux Dstrbutons of Lght Tssue, Medcal Physcs, Vol. 1, pp , Guo, Z., Kumar, S., San, K.-C., Mult-dmensonal Monte Carlo Smulaton of Short Pulse Laser Radaton Transport n Scatterng Meda, J. Thermophy. Heat Transfer, Vol. 14, pp , Rackml, C. I., Buckus, R. O., Numercal Soluton Technque for the Transent Equaton of Transfer, Numercal Heat Transfer, Vol. 6, pp , Kumar, S., Mtra, K., Yamada, Y., Hyperbolc Damped-Wave Models for Transent Lght-Pulse Propagaton n Scatterng Meda, Appled Optcs, Vol. 35, pp , Mtra, K., Kumar, S., Development and Comparson of Models for Lght-Pulse Transport Through Scatterng-Absorbng Meda, Appled Optcs, Vol. 38, pp , Tan, Z. M., Hsu, P.-F., An ntegral Formulaton of Transent Radatve Transfer Part 1 Theoretcal Investgaton, J. Heat Transfer, (submtted). 1. Guo, Z., Kumar, S., Radaton Element Method for Transent Radatve Transfer n Plane-Parallel Inhomogeneous Meda, Numercal Heat Transfer, Part A, (n press). 11. Mtra, K., La, M.-S., Kumar, S., Transent Radaton Transport n Partcpatng Meda wthn a Rectangular Enclosure, J. Thermophy. Heat Transfer, Vol. 11, pp , Wu, C.-Y., Wu, S.-H., Integral Equaton Formulaton for Transent Radatve Transfer n an Ansotropcally Scatterng Medum, Int. J. Heat Mass Transfer, Vol. 43, pp , Fveland, W. A., Three-Dmensonal Radatve Heat Transfer Solutons by the Dscrete-Ordnates Method, J. Thermophy. Heat Transfer, Vol. 2, pp , Fveland, W. A., The Selecton of Dscrete Ordnate Quadrature Sets for Ansotropc Scatterng, ASME HTD-Vol. 72, pp , Km, T. K., Lee, H., Effect of Ansotropc Scatterng on Radatve Heat Transfer n Two-Dmensonal Rectangular Enclosures, Int. J. Heat Mass Transfer, Vol. 31, pp , Modest, M. F., Radatve Heat Transfer, McGraw-Hll, Inc., New York, Shah, N., New Method of Computaton of Radaton Heat Transfer n Combuston Chambers, Ph. D. dssertaton, Dept. of Mech. Eng., Imperal College of Scence and Technology, London, Guo, Z., Kumar, S., Equvalent Isotropc Scatterng Formulaton for Transent Short-Pulse Radatve Transfer n Ansotropc Scatterng Planar Meda, Appled Optcs, Vol. 39, pp , 2.