Effects of nonlinear gradient index on radiative heat transfer in a one-dimensional medium by the DRESOR method

Transcription

1 Joural of Phyc: Coferece Sere Effect of olear gradet dex o radatve heat trafer a oe-dmeoal medum by the DRESOR method To cte th artcle: Z C Wag et al 0 J. Phy.: Cof. Ser Related cotet - Fte volume method for radatve heat trafer a utructured flo olver for emttg, aborbg ad catterg meda Mocef Gazdallah, Véroque Feldhem, Kla Claramut et al. - Radatve heat trafer hoeycomb tructure-ne mple aalytcal ad umercal approache D Ball, R Coquard ad J Radraaloa - The Smplfed Sphercal Harmoc Method For Radatve Heat Trafer Mchael F Modet ad Sheghu Le Ve the artcle ole for update ad ehacemet. Th cotet a doloaded from IP addre o 7/06/08 at 4:8

2 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 Effect of olear gradet dex o radatve heat trafer a oe-dmeoal medum by the DRESOR method Z C Wag, Q Cheg, G H Wag, Z F Huag ad H C Zhou,,3 State Key Laboratory of Coal Combuto, Huazhog Uverty of Scece ad Techology, Wuha , Hube, P. R. Cha State Key Laboratory of Cotrol & Smulato of Poer Sytem & Geerato Equpmet, Departmet of Thermal Egeerg, Tghua Uverty, Bejg 0084, P. R. Ch Emal: hczh@mal.tghua.edu.c (H C Zhou) Abtract. The DRESOR (Dtrbuto of Rato of Eergy Scattered by the medum Or Reflected by the boudary urface) method appled for radatve heat trafer a oedmeoal medum th a olear gradet dex ad gray boudary urface. I th propoed method, the DRESOR value calculated by the Mote Carlo method expre quattatvely the mpact of catterg o radatve trafer ad the radatve tety th hgh drectoal reoluto of hgh preco ca be ealy obtaed. Wth gve meda charactertc ad boudary codto, the temperature ad radatve flux dtrbuto de the medum are calculated uder the codto of radatve equlbrum. It ho, the cae tuded, that the DRESOR method ha a good accuracy. The temperature dtrbuto have a ode th dfferet kd of e chaged gradet dex dtrbuto uder the ame boudary emvty. The mpact of the gradet dex o the radatve heat trafer coderable, ad the ame a that of the rato of t ampltude ad average dex. Bede, the effect of optcal thcke, boudary emvty ad catterg phae fucto o radatve trafer alo hould be pad adequate atteto.. Itroducto A oe bac eergy ad formato trafer mode, radatve heat trafer ha bee greatly developed ad dely ued hgh-tech feld, epecally uder vacuum ad hgh temperature codto. Normally, the refractve dex of the partcpatg medum alay aumed to be cotat. Th dealg mplfe the calculato, hle brg ucertaty of reult ad uavodable error. The fact that the refractve dex of the medum affected by t compoet, temperature ad ome other factor. Wth the developmet of cetfc reearch, the gradet dex materal have mportat applcato backgroud ad broad propect for developmet, hch make u try to obta t tramo charactertc ad mecham, ad grap quck ad coveet calculatg method. Hoever, a the ray propagate curved path, hch are determed by the Fermat prcple [], the oluto of radatve heat trafer a medum th gradet dex much more dffcult tha that a medum th uform dex. 3 To hom ay correpodece hould be addreed. Publhed uder lcece by IOP Publhg Ltd

3 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 A e proce, Dtrbuto of Rato of Eergy Scattered by the medum Or Reflected by the boudary urface method, ko a the DRESOR method, a propoed by Zhou [-5] to addre the radatve trafer problem. Hoever, the refractve dex codered aumed to be uform all of the above tude. Jut recetly, Wag [6] uccefully expaded the DRESOR method to a oedmeoal lear gradet dex medum th black boudare. A the developmet of multlayer flm materal ad meda, cyclcal chaged gradet dex gradually evoked people atteto ad the e chaged gradet dex oe of the mot ormally codered. Hoever, thg tur out to be mafetly dffcult he coderg complcated gradet dex dtrbuto ad gray boudare. The preet ork therefore amed at dcug the effect of olear gradet dex o radatve heat trafer ug the DRESOR method a oe-dmeoal medum th e chaged gradet dex. The bac prcple of the DRESOR method, the calculatg of radatve tety ad the relevat quatte ll be ho Secto. Radatve trafer th to e chaged gradet dex codered three cae. The proper dcrete grd umber ad drecto umber combato determed frt. The the frt cae take a a example to exame the accuracy of the propoed method. The the follog to cae are take to dcu the mpact of the gradet dex, the rato of t ampltude ad average dex o the temperature dtrbuto. Bede, the effect of optcal thcke, boudary emvty, catterg phae fucto ad catterg albedo o radatve equlbrum are vetgated, too. Thee ll be decrbed detal Secto 3. Fally, ome cocluo ll be dra Secto 4. Nomeclature N dcrete elemet of medum τ optcal thcke R d DRESOR value, /m 3, or /m θ polar agle I radatve tety, W/(m².r) Greek letter Subcrpt ŝ drectoal vector, (rad) κ aborpto coeffcet, /m b blackbody q radatve flux, W/(m ) δ catterg coeffcet, /m urface or all T temperature, K Ф catterg phae fucto 0 the left all refractve dex β extcto coeffcet, /m N+ the rght all L phycal dtace, m σ Stefa-Boltzma cotat the grd M dcrete umber of ρ/ε urface reflectvty/emvty agle ω catterg albedo. Prcple.. Bac prcple of the DRESOR method The tegral form of the radatve trafer equato a aborbg, emttg-catterg medum th gradet dex ca be expreed a [7, 8] : Ir (,) Ir ( r, ) = exp( d ) S( r, )exp( d ) d, β + β β () r Where Sr (, ) the ource term, hch : ω Sr (, ) = ( ω) Ib( r) + (, ) Φ(, ) Ω, 4 4π Ir π d () r The correpodg boudary codto of the o-traparet urface th arbtrary charactertc [7] : I( r, ) = ε( r) I( r) + ρ( r,, ) Ir (, ) d Ω. (3) r b π < 0

4 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 Accordg to the DRESOR method, after rertg equato () ad (3) the ubmttg them back to equato () e ca get the follog expreo. I order to avod duplcato, detal ca be foud the prevou ork doe by Wag et al [6]. Ir (,) r r = (,, )[ πε ( ) ( )] + (,, )[4 πβ ( ω) ( )] + [ πε ( ) ( )] R d r r r Ib r da R d r r Ib r dv r Ib r V r r r + + r ( )exp[ β ] 4 πβ ( ω) ( ) (,, )[ πε ( ) 0 π d Ib r Rd r r r 0 r r 4πβ b r I ( r )] da + R (,, )[4 πβ ( ω) ( )] exp[ d r r r Ib r dv β d] β d. (4) V r Where Rd( r, r, ), Rd( r, r, ), Rd( r, r, ) ad Rd ( r, r, ) are all DRESOR value a defed the DRESOR method [, 3]. The DRESOR value th the ytem are eetal for calculatg the radatve tety the DRESOR method. The DRESOR value Rd ( r, r, ) deote the rato of the eergy cattered by the ut volume aroud the pot r to a ut old agle aroud the drecto to that emtted from a ut volume aroud the pot r. Smlarly, e could get the defto of Rd( r, r, ), Rd( r, r, ) ad Rd( r, r, ). The detaled calculato method for the DRESOR value ca be foud the lterature [9, 0]. Calculatg quattatvely the mpact of catterg o radatve trafer, a combed th the DRESOR value, oe of the ma feature of the DRESOR method,... Dcreted calculato of tety ad relevat quatte Take the e chaged gradet dex to coderato, the gradet dex ca be decrbed a: x ( ) = + ( )( π x/ L) (0 x L ). (5) 0 ε0 T0 4 3 θ θ N+ εn+ TN+ 0 ε0 T0 5 7 θ 6 8 N+ εn+ TN+ θ θ x x 3 4 θ θ θ θ θ 3 N0/ N0-3+ N0 3 N0/ N0-3+ N0 (a) e model th > (b) e model th < Fgure. Phycal model ad ray tracg ytem. A ho fgure, a oe-dmeoal, em-traparet, gray, aborbg, emttg ad catterg lab, th thcke x L ad gray boudare, codered. The ytem dcretzed to N grd ad M drecto each grd. The phycal legth of each grd x= x L / N. The mdpot of each grd 3

5 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 the calculato pot, hle the to boudare are amed a 0 ad N+ grd, repectvely. The emvty of the boudary all are ε 0 ad ε N +, hle the temperature of the boudary all are kept at T 0 ad T N +, repectvely. The aborpto ad catterg coeffcet of the medum, κ ad σ, are kept cotat ad the extcto coeffcet β = κ + σ. After dcretzato, the refractve dex of the medum a arbtrary grd, (), cotat, ad vare amog dfferet grd. The tety at grd drecto θ, I (, θ ), ca be got by ug the dcretzg equato (4). The tegrato take alog curved path le for dfferet drecto θ, a ho fgure. For the path le ad 33 fgure, there ll be full-reflecto at grd 3 ad, becaue of ymmetry, aother full-reflecto ll appear at grd N-3+. The le ll propagate betee thee to grd tll the eergy they brg totally aborbed ad cattered by the medum. So the dcretzed formula of tety ca be decrbed a follo: N 3+ I (, θ ) = 4 πβ ( ω) Ib( ) + πε I 0 b(0) Rd(0,, θ) + πε N+ Ib( N+ ) Rd( N+,, θ) 4πβ = 3 (6) N + 4 ( ) 4 ( 4) x πβ ω Ib Rd ( 4,, θ ) [ exp( τ ) exp( τ ) ], 4= Hoever, a for the other path le fgure, o matter hether there ll be full-reflecto, the path le ll ed at the boudary urface. So the correpodg formula of tety ca be ho a: I (, θ ) N = ( ) πε Ib( ) + πε N+ Ib( N+ ) Rd( N+,, θ) + x 4 πβ ( ω) I( 3 b 3) Rd( 3,, θ) π 3 = alog the path N + πε I 0 b(0) Rd(0,, θ) exp( τ ) + x 4 ( ) I( πβ ω 4 b 4) Rd( 4,, θ ) 4πβ 4 = (7) + 4 πβ ( ω ) Ib( ) + N Ib( N ) Rd( N,, ) I 0 b(0) Rd(0,, ) πε θ πε θ [ exp( τ ) exp( τ ) ]. It orth otg that alog the tegrato path, refer to the boudary of the ytem ad the ed of the tegrato path, ad vare th dfferet tegrato le. Varyg th θ, the tegrato le uch a le, 44, 55, 66, 77 ad 88, ho fgure, demotrate dfferet path accordg to the dtrbuto of refractve dex de the medum, ad ed at dfferet urface. The optcal thcke ca be calculated alog the path. The tegrato path are curved betee the adjacet grd ad j accordg to Decarte-Sell la. Th the other ma feature of the DRESOR method, obtag the radatve tety th hgh drectoal reoluto of hgh preco ealy, hch play a vtal role radatve mage proceg ome dutral applcato. After the tety of each grd every drecto got, t of o dffculty to calculate the radatve flux q () of each grd ug the follog formula [] : π q ( ) = πi (, θ)( θ)co( θ) θ. (8) θ = 0 Whe the radato atta equlbrum de the medum, e ca get the temperature dtrbuto of the medum [] : π 0.5 T ( ) = [ I (, )( ) ]. π θ θ θ (9) 4 σ θ = 0 4

6 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369// Reult ad dcuo 3.. Objectve tuded, cae ad codto of calculato Coder the radatve trafer proce de the oe-dmeoal medum th to boudare, a ho fgure. The boudare are dffue gray all th dfferet, gve temperature. The catterg albedo, the optcal thcke ad the olear gradet dex dtrbuto de the medum are et before calculato. The temperature ad radatve flux dtrbuto de the medum are to be aalyzed quattatvely. I thee cae, the temperature of boudare are T (0) = 000K ad T( N + ) = 500K. All the calculato are performed ug a Itel Petum computer. 3.. Cae : the purely emttg ad aborbg medum Frt, the fluece o the accuracy of the reult ad the computg tme are examed to determe the proper dcrete grd umber N ad drecto umber M combato. The catterg albedo ω = 0.0 ad the gradet dex of the medum x ( ) =.8 0.6( x/ L. ) The boudare are all black ad the optcal thcke τ =.0. A ho fgure ad table, t obvou to ee that the reult of the DRESOR method are table uder all thee codto. Whe N 00 ad M 8, the fluece of N ad M qute eak. Whle the computg tme of M = 80 doe t creae too much compared th that of M = 8 uder the ame N = 00 ; hoever, the computg tme of N = 000 uacceptable compared th that of N = 00 th the ame M = 80. So N = 00 ad M = 80 are ued the calculato belo (00,8) (00,80) 00 (00,800) (0,80) (000,80) x/l Fgure. The reult uder dfferet value of N ad M combato. Table. The computg tme uder dfferet value of N ad M combato. (N, M) Computg tme (N, M) Computg tme (0, 80).5685 (00, 8) (00, 80) (00, 800) (000, 80)

7 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 I th cae, to dfferet gradet dex dtrbuto ad four dfferet kd of boudary emvty are codered DRESOR 00 DCRT 7 CRTP Mehle method FEM Fgure 3. Temperature feld uder radatve equlbrum the purely emttg ad aborbg medum. ε 0 =0., ε N+ =.0; curve : (x)=.+0.6(πx/l), curve : (x)=.8-0.6(πx/l); ε 0 =0.7, ε N+ =0.7; curve 3: (x)=.+0.6(πx/l), curve 4: (x)=.8-0.6(πx/l); ε 0 =.0, ε N+ =.0; curve 5: (x)=.+0.6(πx/l), curve 6: (x)=.8-0.6(πx/l); ε 0 =.0, ε N+ =0.; curve 7: (x)=.+0.6(πx/l), curve 8: (x)=.8-0.6(πx/l). The reult of temperature dtrbuto are ho fgure 3. Thee are compared th thoe got by Huag [] ug the dcrete curved ray-tracg method (DCRT), Ta [3] th combed curved raytracg ad peudo-ource addg method (CRTP), Lu [4] ug the mehle method ad Lu [5] through a fte elemet method (FEM). The reult predcted by the DRESOR method ho excellet agreemet th thee method uder dfferet gradet dexe ad boudary emvty. The relatve error are belo 0.35%. Ulke that of lear gradet dex dtrbuto [6], the temperature dtrbuto have a ode of thee e chaged gradet dex dtrbuto uder the ame codto of boudary emvty. The reao that the e chaged gradet dex caue the pecal ray propagato path: complete ymmetry th to de of the ceter, hch make the ode affected jut by the character of to urface rather tha the gradet dex. Jut a ho fgure 3, f ε 0 = ε N +, the ode ll be the mdpot of the medum ad t radatve equlbrum temperature tay cotat. Hoever, th ode ha a hftg tred toard the bgger boudary emvty de f ε0 ε N +. The effect of optcal thcke o radatve trafer uder olear gradet dex ho fgure 4. A e ca ee, th the creae of optcal thcke, the temperature dfferece betee t to de creae hle the temperature dfferece betee a e chaged gradet dex ad a cotat gradet dex frt creae ad the decreae. I order to carate the etvty of the optcal thcke, τ =.0 ued the follog cae Cae : the aborbg, emttg ad otropc catterg medum I th cae, the boudary emvty aumed to be ε0 = ε N + = 0.7, the catterg albedo ω = 0.3. The effect of the gradet dex dtrbuto vetgated. 6

8 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 A ho fgure 5, A/P deote the rato of ampltude ad average dex of the gradet dex. A e ca ee, dog olated reearch o the mpact of the ampltude or the average dex correct ad the reult eem chaotc. So the effect caued by the ampltude coupled the average dex vetgated. Frt, oce the A/P decded, uch a he A/P equal 0.5 fgure 5, o matter hat the ampltude ad the average dex are, the temperature dtrbuto are completely overlapped th each other. The, th the decreae of the A/P, the propagate path of the radato ho fgure (a) creae, hch make the mpact of the boudare o t earby medum greater. I coequece, the temperature dfferece betee the to de creae ad o doe the temperature gradet τ=0., (x)=.+0.6(πx/l) τ=0., (x)=.0 τ=.0, (x)=.+0.6(πx/l) 00 τ=.0, (x)=.0 τ=0.0, (x)=.+0.6(πx/l) τ=0.0, (x)= x/l Fgure 4. Temperature feld uder radatve equlbrum th dfferet kd of optcal thcke (x)=0.+0.6(πx/l), A/P=3 (x)=.+(πx/l), A/P=0.83 (x)=.+0.6(πx/l), A/P=0.5 (x)=.4+.(πx/l), A/P=0.5 (x)=.+0.6(πx/l), A/P=0.7 (x)=.+0.(πx/l), A/P= Fgure 5. Temperature feld uder radatve equlbrum the otropc catterg medum Cae 3: the aborbg, emttg ad aotropc catterg medum 7

9 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 I th cae, the gradet dex dtrbuto x ( ) =.8 0.6( x/ L. ) To lear catterg phae fucto; Φ= + bµµ, ( b =, ), are vetgated epl=epl=.0,b= 50 epl=epl=.0,b= - 00 epl=epl=0.7,b= epl=epl=0.7,b= epl=epl=0.,b= epl=epl=0.,b= Fgure 6. Temperature feld aotropc catterg medum. Radatve flux, kw/m 300.0k 50.0k 00.0k 50.0k 00.0k 50.0k epl=epl=.0,b= epl=epl=.0,b= - epl=epl=0.7,b= epl=epl=0.7,b= - epl=epl=0.,b= epl=epl=0.,b= Fgure 7. The correpodg radatve flux dtrbuto of fgure 6. Fgure 6 ad 7 ho the effect of boudary emvty o the radatve equlbrum. Here, the catterg albedo ω = 0.7. Three combato of boudary emvty are calculated. A e ca ee, th a decreae the boudary emvty, the eergy emtted from the to de reduce, hch tur make the eergy aborbed ad cattered by the medum reduce. So the radatve equlbrum flux the decreae, a ho fgure 7. I other ord, the heat effect o the medum of the to boudare decreag gradually. Of coure, th make the temperature dfferece of t to de loer ad the temperature curve flatter. At the ame tme, t alo orth otg that; eve though the effect of aotropc catterg phae fucto o temperature dtrbuto, a ho fgure 6, eaker tha that caued by the boudary 8

10 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 emvty, t caot be gored. Compared th forard catterg b =, the backard catterg b = ehace the radatve tety the egatve drecto, hch, accordg to formula. (8), reduce the radatve flux uder radatve equlbrum. Hoever, the backard catterg b = creae the tramttg legth to ome degree, hch mea that the heat effect o the medum of the to boudare greater tha that of b =. That hy uder the ame codto, the flux of the backard catterg b = le tha that of forard catterg b =, hle the temperature dfferece of the backard catterg b = hgher tha that of forard catterg b = ω=0.7,b= ω=0.7,b= ω=0.5,b= ω=0.5,b= ω=0.,b= ω=0.,b= Fgure 8. Temperature feld uder radatve equlbrum aotropc catterg medum th ε 0 =ε N+ = k Radatve flux, kw/m 00.0k 80.0k 60.0k ω=0.7,b= ω=0.7,b= - ω=0.5,b= 40.0k ω=0.5,b= - ω=0.,b= ω=0.,b= - 0.0k Fgure 9. The correpodg radatve flux dtrbuto of fgure 8. Fgure 8 ad 9 ho the effect of catterg albedo ω o radatve equlbrum. Here, the boudary emvte are ε = ε N + = 0.7. Three dfferet catterg albedo; ω = 0.7, ω = 0.5 ad ω = 0., are calculated. Whe the catterg albedo decreae, the aborbg capacty of the medum creae 9

11 Eurotherm Coferece No. 95: Computatoal Thermal Radato Partcpatg Meda IV IOP Publhg Joural of Phyc: Coferece Sere 369 (0) 0004 do:0.088/ /369//0004 correpodgly. Wth forard catterg b =, he the catterg albedo ω chage from 0.7, 0.5 to 0., the radatve tety decreae, o doe the radatve flux. Hoever, the temperature dfferece betee the to de creae becaue of the creae the aborbg capacty of the medum. Hoever, th backard catterg b =, the decreae the catterg albedo ω reduce the radatve tety the egatve drecto, hch make the radatve flux creae. At the ame tme, the ehacemet effect of backard catterg a e dcued above lkely to decle, hch make the temperature dfferece betee the to de decreae. Bede, he the catterg albedo ω reduce, o doe the effect of aotropc catterg. A ho fgure 8, the dfferece betee b = ad b = become maller ad maller. 4. Cocluo I th paper, the DRESOR method ha bee appled to radatve trafer olear gradet dex meda, th a DRESOR value troduced. The temperature ad radatve flux dtrbuto de the medum ere calculated three cae. The mpact of the gradet dex dtrbuto, the varato of the ampltude ad the average dex, the optcal thcke, the catterg albedo, the boudare emvty ad the catterg phae fucto ere all vetgated. The ma cocluo follo belo. Th method could deal ell th the radatve trafer problem a oe-dmeoal emtraparet lab th olear gradet dex ad complex boudary charactertc. The temperature dtrbuto have a ode th dfferet e chaged gradet dex dtrbuto uder the ame boudary emvty. The mpact of the gradet dex o the radatve heat trafer the ame a that of the rato of t ampltude ad average dex. The boudary emvty ha a gfcat mpact o radatve heat trafer, hle the effect of optcal thcke, the catterg phae fucto much eaker. Hoever, they caot be gored ad hould be pad adequate atteto. Referece [] Bor M ad Wolf E 997 Prcple of optc 6d (Cambrdge: Cambrdge Uverty Pre) [] Zhou H C 005 Vual detecto prcple ad techology of the furace flame (Bejg: Scece Pre) [3] Zhou H C, Che D L ad Cheg Q 004 J. Quat. Spectroc. Radat. Trafer [4] Zhou H C, Cheg Q, Huag Z F ad He C 007 J. Quat. Spectroc. Radat. Trafer [5] Cheg Q ad Zhou H C 007 J. Heat Tra-Traacto of the Ame [6] Wag Z C, Cheg Q, Wag G H ad Zhou H C 0 J. Quat. Spectroc. Radat. Trafer [7] Mhara D ad Mhal B W 984 Foudato of radato hydrodyamc (Ne York: Oxford Uverty Pre) [8] Fumero S ad Allaaj F 009 J. Quat. Spectroc. Radat. Trafer [9] Huag Z F, Cheg Q, He C ad Zhou H C 008 Heat Trafer-Aa Reearch [0] Cheg Q, Luo Z X ad Zhou H C 00 It. J. Nol Sc Numer [] Modet M F 003 Radatve heat trafer d (Sa Dego: Academc Pre) [] Huag Y 00 A tudy o the thermal radatve trafer gredet dex emtraparet medum Ph.D the (Harb: Harb Ittute of Techology) [3] Ta H P, Huag Y ad Xa X L 003 It. J. Heat Ma Trafer [4] Lu L H 006 It. J. Heat Ma Tra [5] Lu L H 005 It. J Heat Ma Tra Ackoledgemet Th tudy a upported by the Natoal Natural Scece Foudato of Cha (No , 5056, 50065) ad the Fudametal Reearch Fud for the Cetral Uverte (No. 0TS43). 0