Model Discussion of Transpiration Cooling with Boiling

Transcription

1 Transp Porous Med DOI /s Model Dscusson of Transpraton Coolng wth Bolng Kuan We Janhua Wang Mao Mao Receved: 25 July 2011 / Accepted: 2 Aprl 2012 Sprnger Scence+Busness Meda B.V Abstract The two-phase mxture model has been wdely used to descrbe the performances of flud flow and heat transfer wthn porous meda wth lqud phase change. However, ths model was based on two mportant assumptons: the temperature s constant T f = const n two phase regon, whle flud temperature s locally equal to the sold matrx temperature T s = T f. These assumptons result n an nveracous numercal phenomenon,.e.,: a thermal nsulatng layer wthn the porous matrx n numercal smulatons. Ths numercal phenomenon s not real, because the sold matrx s made of thermal conductve materal. To modfy the mathematcal model of the transpraton coolng problem wth bolng, ths paper presents an mproved model, whch s based on that the Gbbs free energy of lqud phase and vapor phase are equal n two-phase regon. Temperature varaton n two-phase regon s consdered, and flud temperature s locally dfferent from the sold matrx temperature T s = T f, therefore the local heat transfer through the convecton between sold and flud s consdered as well. Numercal calculatons of the transpraton coolng problem wth bolng are carred out wth the mproved model. The numercal results such as the varatons of temperatures of flud and sold, the saturaton and pressure of flud wthn porous meda, are reasonable, and the nveracous ssue of the thermal nsulatng layer s successfully resolved. Keywords Transpraton coolng Porous meda Lqud coolant Phase change Lst of Symbols z Coordnate L Length T Temperature h Specfc enthalpy k Thermal conductvty K. We B J. Wang M. Mao Department of Thermal Scence and Energy Engneerng, Unversty of Scence and Technology of Chna, Jnzha Road No. 96, Hefe , Anhu, People s Republc of Chna e-mal: wkuannjlgdx@gmal.com

2 K.Weetal. H c p s t u j ṅ K k r p g d p p c q h sl h sv R M g B ṁ Q D N Specfc revsed enthalpy Specfc heat capacty Saturaton Tme Velocty Dffusve mass flux Volumetrc mass rate of evaporaton Absolute permeablty Relatve permeablty Pressure Gravtatonal constant Averaged partcle dameter of porous meda Capllary pressure Volumetrc heat generaton rate nduced by convecton Convectve heat transfer coeffcent between sold and lqud Convectve heat transfer coeffcent between sold and vapor Mole gas constant Mole mass Specfc Gbbs free energy Saturaton equaton assocated wth the phase state Mass flow rate of coolant Heat flux on the hot sde surface Dscrete system of the equatons Nodes number Greek Symbols Ɣ Effectve dffuson coeffcent ρ Densty ε Porosty λ Relatve moblty μ Knetc vscosty σ Surface tenson coeffcent α Specfc surface η Entropy ξ Intermedate varable ζ Intermedate varable χ Phase state Subscrpts s Sold l Lqud v Vapor f Flud sat Saturated state eff Effectve

3 Model Dscusson of Transpraton Coolng wth Bolng sf Sold flud 0 Reference state n Inlet of porous meda c Coolant tank 1 Introducton In recent years, transpraton coolng has been wdely used n aerospace and hypersonc vehcles as an effcent actve coolng technque. Prevous numercal nvestgatons have ndcated that compared to regeneratve coolng and flm coolng, transpraton coolng can enhance the coolng effcency 35 and 13 %, respectvely Lands and Bowman 1996; Leontev In some applcatons, coolant tself can also be a part of payload of vehcles. So t s an ssue worthy of consderaton how to protect the heated part wth mnmum coolant consumpton. van Foreest et al expermentally nvestgated transpraton coolng usng lqud water. Ther result shows that compared to the transpraton coolng usng gaseous ntrogen, transpraton coolng usng lqud water can reduce the temperature to the same level wth only one ffth of coolant consumpton. It ndcates that the transpraton coolng usng lqud coolant can be consdered as a potental actve coolng approach. Bau and Torrance 1982 expermentally proved that there would be three regons, wthn porous meda when flud bolng occurs: subcooled lqud regon, two-phase regon and superheated vapor regon. Rahl et al expermentally studed convectve bolng n porous meda usng low bolng pont pentane. Ther result ndcated that the saturaton temperature of flud would decrease n two-phase regon along the flow drecton. The reason s that flud pressure decreases along the flow drecton and ths pressure corresponds to saturaton temperature. Although many experments on the transpraton coolng wth bolng were carred out, theoretcal research on ths problem has yet to be deeply looked at. The mathematcal models to descrbe macroscopc two-phase flow wth phase change n porous meda can be dvded nto three categores: sngle-equaton model, two-equaton model, three-equaton model Duval et al The sngle-equaton model assumes that the volume averaged temperatures of three phases are equal Wang and Beckermann 1993; Benard et al. 2005; Brdge and Wetton Two-equaton model deems that the volume averaged temperatures of lqud and vapor are equal, but the volume averaged temperatures of flud and sold matrx are dfferent Vafa and Sozen Three-equaton model s based on that the volume averaged temperatures of three phases sold, lqud, and vapor are all dfferent Lkhansk et al. 1997; Duval et al In the energy equaton of all models, there s a latent heat term of phase change ṅ h lv. Ths would cause two problems. On one hand, to obtan a closed system of equatons, an expresson for ṅ s needed Duval et al On the other hand, h lv s always set to be constant for convenence Vafa and Sozen 1990 or for that there s no expresson of h l,sat or h v,sat n the form of the ndependent varables Fchot et al In the case of transpraton coolng, the permeablty of porous meda can vary n the order of magntude from to Soller et al There s a very hgh level of pressure gradent, or a very sharp change of h lv n porous meda. So the change of h l,sat, h v,sat and h lv n the two-phase regon should be consdered. To solve the frst problem, Wang and Beckermann 1993 ncorporated the latent heat term of phase change nto mxture enthalpy by changng the ndependent varable for the energy equaton from temperature to mxture enthalpy n the sngle-equaton

4 K.Weetal. Fg. 1 Effectve dffuson coeffcent versus saturaton usng the TPMM L et al two-phase mxture model TPMM model. The latent heat term of phase change dsappears from ths modfcaton. The governng equatons of the models mentoned above are mostly nonlnear. After ntroducng mxture parameters and the two assumptons mentoned above, the governng equatons of TPMM become lnear Wang and Beckermann Furthermore, the model s based on mxture enthalpy, makng t easer to trace the phase change nterface by enthalpy method. Therefore the TPMM has been wdely appled Peterson and Chang 1997; Yuk et al However, L et al found that nterpolaton of the effectve thermal dffusvty needs specal treatment when TPMM s used to smulate two-phase flow wth phase change n porousmeda. Fgure1 dsplays the curve of the effectve thermal dffusvty wth saturaton n two-phase regon. It can be seen that the effectve thermal dffusvty vares severely, even ncreasng 10 orders of magntude. In two-phase regon ths varaton s very low, but near subcooled lqud regon and superheated vapor regon when saturaton s close to ether 0 or 1, t rses rapdly. In other words, compared to the subcooled lqud regon and superheated vapor regon, the two-phase regon s a thermal nsulatng layer, due to ts extremely low thermal effuson coeffcent. When TPMM s used n smulatng transpraton coolng wth bolng, heat flux wll be dffcult to pass through the two-phase regon due to the thermal nsulatng layer. In fact, n the structure of transpraton coolng, sold matrx s usually domnant, and the conductve effect of the sold matrx plays an mportant role n the heat transfer process. Therefore ths numercal result s not a real physcal phenomenon. The reason for the thermal nsulatng layer are the two nveracous assumptons, T s = T f and n two-phase regon T f = const. Dscusson regardng the two assumptons wll be dscussed at length n subsequent chapter. In order to solve the problem wth TPMM and the second problem of the latent heat term of phase change, ths artcle puts forward a new model, whch s based on the two-equaton model T s = T f and the assumpton that the Gbbs free energy of lqud phase and vapor phase are equal n the two-phase regon. It s worth mentonng that the latter assumpton can

5 Model Dscusson of Transpraton Coolng wth Bolng smultaneously solve the problem wth TPMM as assumng that T f = const n two-phase regon and the second problem of the latent heat term of phase change. Numercal smulatons of one-dmensonal, steady state, constant propertes transpraton coolng problem wth bolng are carred out wth the mproved model. 2 Descrpton of the Improved Model 2.1 Analyss of the Reasons Causng Inveracous Thermal Insulatng Layer In the TPMM Wang 1997, the effectve thermal dffusvty s calculated by Ɣ h = ρ v /ρ l h v,sat /h lv D + k eff dt dh 1 and dt dh = 1 ρ l c pl, H ρ l 2h v,sat h l,sat 0, ρ l 2h v,sat h l,sat <H ρ v h v,sat 2 1 ρ v c pv, H > ρ v h v,sat The frst term on the rght handsde of Eq. 1 vares smoothly n the whole range, but accordngtoeq.2, dt /dh has a large jump on the phase change nterface. Ths jump results n very large varaton of the effectve thermal dffusvty, as shown n Fg. 1. By analyzng the governng equatons of the TPMM, one can fnd that the reason of ths problem s the assumpton of constant temperature n two-phase regon T f = const. Its worthy mentonng though, that TPMM cannot be closed wthout ths assumpton. Although n the TPMM there s the governng equaton of saturaton, ε ρ ls t + [ρ uλ ] = j + ṅ, 3 one cannot gve the expresson of the second term on the rght hand sde of Eq. 3. For ths reason, TPMM needs a governng equaton for saturaton. To solve ths problem, the assumpton of constant temperature n two-phase regon was ntroduced. Because of ths assumpton, the energy equaton of the TPMM smplfes nto the governng equaton of saturaton n two-phase regon. At the same tme, t remans to be the equaton of temperature n subcooled lqud regon and superheated vapor regon. TPMM became a closure model, thus. It s ths assumpton that causes the thermal nsulatng layer n the two-phase regon. If one can establsh an ndependent energy equaton for the sold matrx wthout the assumptons of T s = T f and T f = const n two-phase regon, heat flux can pass through the two-phase regon by the conducton of the sold matrx, and the nveracous thermal nsulatng layer wll not exst, ether. 2.2 Governng Equatons In the model presented, the assumpton of constant temperature n two-phase regon s replaced wth that the Gbbs free energy of lqud and vapor phase are equal n two-phase regon. The sngle-equaton model s replaced wth the two-equaton model.

6 K.Weetal Classcal Equatons The classcal equatons of mass conservaton of lqud and vapor n the two-phase flow wth phase change n porous meda used n ths paper can be expressed as: εsρ l + ρ l u l = ṅ, 4 t ε1 s ρ v + ρ v u v = ṅ. 5 t In Eqs. 4 and5, ε s porosty, whch s supposed to be constant; s s the volume fracton of the lqud phase n entre flud.e., saturaton; ρ l s the densty of lqud; ρ v s the densty of vapor; and ṅ s the volumetrc mass rate of evaporaton. The classcal Darcy law as the momentum equaton of lqud and vapor s: u l = K k rl p l ρ l g, 6 μ l u v = K k rv p v ρ v g, 7 μ v In Eqs. 6 and7, K s the absolute permeablty that can be obtaned by Carman Kozeny equaton: ε 3 dp 2 K = ε 2, 8 where d p s the characterstc sze of porous meda; k rl and k rv are the relatve permeablty of lqud and vapor, respectvely. The correlaton k rl = s 3 and k rv = 1 s 3 are used to calculate the relatve permeablty of lqud and vapor; p l and p v are the pressure of the lqud and vapor; μ l and μ v are the knetc vscosty of the lqud and vapor. The relatonshp between p l, p v, and capllary pressure, s calculated by the followng correlaton: σ [ p c s = p v p l = s s 2 K/ε 1/ s 3] 9 In Eq. 9, σ s the surface tenson coeffcent on the lqud vapor nterface. Add Eq. 4 wth Eq. 5, one obtan Eq. 10: εsρ l + ε 1 s ρ v + ρ l u l + ρ v u v = t Add Eqs. 6and7 nto Eq. 10, one obtan Eq. 11: εsρ l + ε 1 s ρ v t + [ ρ l K k rl p l + ρ l g + ρ v K k ] rv p v + ρ v g μ l μ v In the mproved model, the heat transfer between the flud and sold n the pores s consdered T s = T f, therefore ndependent energy equatons of flud and sold matrx are needed. Consderng that the energy equaton of the flud s a partal dfferental equaton of second order, two boundary condtons are requred. However, for transpraton coolng problem wth bolng, the hot sde surface boundary condton for the energy equaton of the flud s dffcult to determne. Heat can be brought out of the porous meda wth exteror flud flow; at the same tme, heat can be brought nto the porous meda by the flowng flud. In consderaton of the conductvty of the sntered alloy s much larger than that of the flud n = 0. 11

7 Model Dscusson of Transpraton Coolng wth Bolng general, ths artcle adopts the assumpton used by Wang and Wang 2006 whch s that the nfluence of flud conducton can be neglected. At the same tme, the nfluence of thermal dsperson and volume heat source s gnored Sozen and Vafa Fnally, the energy equatons based on the enthalpy balance are: t [εsρ lh l + ε 1 s ρ v h v ] + ρ l u l h l + ρ v u v h v = q sf, 12 t [1 ε ρ sh s ] = k s,eff T s qsf. 13 Here, ρ s s the densty of the sold phase porous matrx; h s, h l and h v are the specfc enthalpy of the sold, lqud, and vapor, respectvely; k s,eff s the effectve conductvty of sold phase whch can be calculated by k s,eff = 1 ε k s ; q sf s the heat flux between the sold matrx and flowng flud wthn pores, and can be calculated by: q sf = h sl α sf T s T f n lqud regon, 14a q sf = h sv α sf T s T f n vapor regon, 14b q sf = sh sl α sf T s T f + 1 s h sv α sf T s T f n two-phase regon 14c where the convectve coeffcents wthn pores n dfferent phase regons be calculated by: h sα = k α /d p Pr 1/3 α Re0.6 p,α, α = vapor,lqud. α sf s the specfc surface of porous matrx, and can be calculated by: State Parameters and Propertes α sf = 6 1 ε d p. 15 In the mproved model, all the state parameters and propertes have to be expressed as the functons of the ndependent varables, namely p l, T f, T s and s. Just lke Benard et al dd, the followng assumptons are ntroduced: h 1 α T α p α, T α only depend on T α,thats, h α T α p α, T α = c p,α T α ; h 2 α p α p α, T α only depend on p α,thats, h α p α p α, T α = ξ α p α. At the reference pressure and temperature, the reference enthalpy can be calculated by h α p 0, T 0 = h α,0.byntegratngdh α p α, T α = h α T α p α, T α dt α + h α p α p α, T α p α wth the reference enthalpy, one can obtan T α p α h α p α, T α = h α,0 + c p,α τ dτ + T 0 p 0 ξ α π dπ. 16 From thermodynamcs equaton, dh α p α,η α = T α dη α + ρ 1 α dp α,whereη s entropy, one can get: dη α p α, T α = c p,α dt α 1 1 ξ α p α dp α. 17 T α T α ρ α Ths mples usng Maxwell relatons: T α 1 ρα ξ α p α T α = 0. 18

8 K.Weetal. We thus defne that: 1 ρ α ξ α p α = ζ α p α. 19 T α As a result, one can deduce the expresson of the densty of all phases: ρ α p α, T α = Add Eq. 19 nto Eq. 17, one can get: 1 T α ζ α p α + ξ α p α. 20 dη α p α, T α = c p,α T α dt α ζ α p α dp α. 21 T α By ntegratng Eq. 21 wth the reference entropy η α,0, one can obtan: η α p α, T α = η α,0 + T α T 0 c p,α τ α dτ τ ζ α πdπ. 22 p 0 Assumng the densty of lqud phase to be constant ncompressble, one can deduce that: ζ l p l = 0andξ l p l = 1/ρ l. Assumng deal gas for vapor phase, one can deduce ζ v p v = R/Mp v and ξ v p v = 0, where R s molar gas constant, M s the molar mass of vapor phase. For sold matrx, one can obtan the sold enthalpy h s = h s,0 + c p,s T s,where h s,0 = h s p 0, T 0. The specfc heat capactes of the lqud, vapor, and sold are assumed to be constant. The fve constants h s,0, h l,0, h v,0,η l,0 and η v,0 cannot be chosen arbtrarly. In ths artcle, water s used as coolant, so the reference state s chosen to be p 0 = Pa and T 0 = K. Consderng that the reference state s the phase equlbrum state, one can deem that the Gbbs free energy of lqud and vapor s the same,.e., g v p 0, T 0 = g l p 0, T 0 and η v p 0, T 0 η l p 0, T 0 = h lv,0 /T 0. At ths pont, one can choose h s,0 = 0, h l,0 = 0,η l,0 = 0, h v,0 = h lv,0 and η v,0 = h lv,0 /T 0. The state parameters and propertes appear n Eqs are all expressed as the functons of p l, T f, T s and s: h s = h s,0 + c p,s T s, 23 h l = h l,0 + c p,l T f T ρ l p l p 0, 24 h v = h lv,0 + c p,v T f T 0, 25 ρ v = Mp v RT f = M p l + p c s RT f. 26 Add Eqs nto Eqs , one can get Eqs : [ εsρ l + ε 1 s M p ] l + p c s t RT f ρ l K k rls μ l p l + ρ l g + + Mp l+p c s RT f K k rvs pl + p c s = 0 27 μ v + Mp l+p c s RT f g [ ] εsρl c p,l T f T 0 + ρ 1 l p l p 0 t +ε 1 s Mp l+p c s RT hlv,0 f + c p,v T f T 0

9 Model Dscusson of Transpraton Coolng wth Bolng t + ρ l K k rls μ l + Mp l+p c s RT f p l + ρ l g c p,l T f T 0 + ρ 1 l p l p 0 pl + p c s K k rvs μ v + Mp l+p c s RT f g hlv,0 + c p,v T f T 0 =q sf [ 1 ε ρs c p,s T s ] = ks,eff T s qsf Saturaton Equaton In order to close the governng equatons, a new saturaton equaton s derved. In the mproved model, the assumpton of constant temperature n two-phase regon s replaced wth the assumpton that the Gbbs free energy of lqud phase and vapor phase are equal n two-phase regon. Defne χ as the phase state parameter, where χ = 1 denotes subcooled lqud regon, χ = 2 denotes two-phase regon, χ = 3 denotes superheated vapor regon. One can wrte the saturaton equaton as B p l, s, T f,χ = 0, where 28 B p l, s, T f, 1 = s 1 = 0 n lqud regon, B p l, s, T f, 2 = g l p l, T f g v p v, T f = 0 n two-phase regon, B p l, s, T f, 3 = s = 0 n vapor regon. 30a 30b 30c Snce, g = h T η, one can get: B p l, s, T f, 2 = h l p l, T f T f η l p l, T f h v p v, T f + T f η v p v, T f = c p,l T f T p l p 0 T f c p,l ln T f ρ l T 0 h lv,0 + c p,v T f T 0 + T f η v,0 +c p,v ln T f T 0 R M ln p l+ p c s By now, Eqs andB p l, s, T f,χ = 0 can form a system based on p l, T f, T s, s and χ. Note that h l,sat, h v,sat and h lv are not constant. They can change wth p l. Obvously, ths system s not closed due to lack of governng equaton for χ. For ths reason, the governng equaton of χ s needed. Note that χ s a dscrete varable, whle p l, T f, T s and s are contnuous varables. It can be expected that the governng equaton of χ s n the form of logcal judgment statement. Consderng that ths knd of equaton can only be used n the process of numercal calculaton, we wll ntroduce the governng equaton of χ n the secton about the numercal soluton method. p Physcal Model and Boundary Condtons The physcal model consdered n ths artcle s a porous matrx wth a thckness of L = 0.1m, as sketched n Fg. 2. The upper surface of the matrx s exposed to a heat flux of Q, lqud coolant s njected from reservor at a temperature of T c wth a mass flow rate of ṁ nto the porous matrx, as shown n Fg. 2. If the lqud coolant evaporates fully n the porous meda, there wll be a large temperature gradent near the hot surface, namely a hgh thermal stress wll do harm to the porous matrx. If the hot surface was n the two-phase regon, ths large temperature gradent would be declned. Therefore an deal coolng state whch conssts only

10 K.Weetal. Fg. 2 Physcal model of transpraton coolng wth bolng of subcooled lqud regon and two-phase regon wthn the porous matrx s consdered n ths paper. A numercal nvestgaton of one-dmensonal, steady state, constant propertes transpraton coolng problem wth bolng s carred out wth the mproved model. To solve the problem, the followng boundary condtons are appled: s z=0 = 1, 32a K ṁ z=0 = ṁ n = ρ l p l + ρ l g p l + p c s z=l = p atm, 32b μ l z = 0 T f = T c T s = T c, 32c z = L k s,eff T s = Q. 32d 4 Numercal Soluton TosolveEqs.27 29andB p l, s, T f,χ = 0 usng fnte volume method, these equatons are represented n one form,.e. Eq. 33: D p l, s, T f, T s,χ = 0 33 In the dscretzaton of Eq. 33, the fnte volume method wth staggered grd s used. Frst-order upwnd scheme s used for the convecton term, and central dfference scheme s used for the dffuson term. The effectve conductvty on the nterface of numercal cell s obtaned by harmonc mean. For steady state problems, the ntal teraton parameters of each node,p 0 l,, T 0 f,, T 0 s,, s 0 and χ 0, are nput values, and s the ndex of the th node. The dscretzaton of Eqs and B p l, s, T f,χ = 0 can be expressed wth Eq. 34: D p 0 l,, s0, T 0 f,, T 0 s,,χ 0 = Equaton 33 represents a seres of coupled nonlnear equatons, so Eq. 34 s solved usng Newton s method. Detals of Newton s method can be found n Benard et al

11 Model Dscusson of Transpraton Coolng wth Bolng After each teraton, one can obtan new values p 1 l,, s1, T 1 f,, T 1 s,. The new values can be used to update χ 0 accordng to Eq. 35: If χ 0 = 1 and B If χ 0 = 2 and B If χ 0 = 2 and B If χ 0 = 3 and B Else χ 1 = χ 0. p 1 l,, s1, T 1 f,, 2 > 0, then χ 1 = 2; p 1 l,, s1, T 1 f,, 1 > 0, then χ 1 = 1; p 1 l,, s1, T 1 f,, 3 < 0, then χ 1 = 3; p 1 l,, s1, T 1 f,, 2 < 0, then χ 1 = 2; The above teraton s performed step by step, untl the followng convergence crteron s satsfed: N 1 N N N δp l, + δt l, + δt s, + δs < N =1 =1 =1 = Grd-Independent Valdaton Consderng that the temperature gradent near the hot sde s much hgher than that near the cold sde, an exponental dstrbuton grd s adopted n numercal smulaton. Three grds wth dfferent node numbers 148, 160, and 170 are used to valdate grd-ndependency. The maxmum cell length of the three grds are all m, however, the ratos of reducton of the three grds are dfferent, whch are , , and , respectvely. Under the condtons of Q = W/m 2,ε = 0.35, d p = m, T c = K, k s = 30 W/m K and ṁ = 1.5 kg/m 2 s, numercal smulatons were carred out, and the sold matrx temperature dstrbuton wthn the porous meda was obtaned usng the three grds. The propertes used n numercal smulaton are that of water at the reference state lsted n Table 1. As llustrated n Fg. 3, the varaton of the sold matrx temperature n the rectangle frame s very large; therefore the temperature varaton n ths regon s magnfed n the central fgure. From the magnfed fgure, t can be observed that the numercal results of the three grds are very consstent wth each other. So t can be regarded that the results s grd-ndependent. In the followng calculatons, all the results are obtaned wth the frst grd. Table 1 Propertes of saturated water under normal atmosphere Property Lqud Vapor Unt Densty Ideal gas kg/m 3 Specfc heat J/kg K Knetc vscosty e e 05 m 2 /s Conductvty W/m K Surface tenson coeffcent N/m Latent heat of evaporaton e+06 J/kg

12 K.Weetal. Fg. 3 Sold matrx temperature dstrbuton wth dfferent grd 6 Results and Dscusson 6.1 Dstrbuton of Temperature and Saturaton Fgures 4 and 5 llustrate the dstrbutons of saturaton, sold matrx temperature, and flud temperature wthn porous meda at dfferent coolant mass flow rate: 0.5, 1.5, and 6.0 kg/m 2 s. In the centre of Fg. 4, the varaton of saturaton n the rectangle frame s magnfed. Fg. 4 Effect of changng coolant mass flow rate on saturaton dstrbuton

13 Model Dscusson of Transpraton Coolng wth Bolng Fg. 5 Effect of coolant mass flow rate on temperature dstrbuton As llustrated n Fg. 4, the nterface between subcooled lqud regon and two-phase regon gradually closes to the hot surface as the mass flow rate of coolant decreases. When the flud on the hot surface s stll n two-phase regon, the saturaton on the hot surface ncreases wth coolant mass flow rate, and ths trend s dentcal wth the work of Sh and Wang As shown n Fg. 5, the temperature dfference between flud and sold matrx can almost be neglected n the subcooled lqud regon and n part of the two-phase regon near the subcooled lqud regon. The sold temperature rses all along, whereas the coolant temperature decreases at some pont. If coolant mass flow rate s equal to 0.5kg/m 2 s, the flud temperature falls at pont z = ; f coolant mass flow rate s equal to 1.5kg/m 2 s, the flud temperature falls at pont z = ; f coolant mass flow rate s equal to 6.0kg/m 2 s, the flud temperature falls at pont z = Ths local thermal non-equlbrum phenomenon s dentcal wth Yuk et al. 2008, and ths flud temperature fallng phenomenon s dentcal wth the expermental result of Rahl et al Fgure 5 llustrates that the temperature dfference between flud and sold on the hot sde surface rses monotonously when coolant mass flow rate decreases. Ths non-equlbrum trend s also consstent wth Yuk et al These numercal results ndcate that the mproved model can successfully avod the nveracous thermal nsulatng layer and demonstrate the real temperature varaton: sold temperature rses contnuously due to heat conducton, but for flud temperature varaton t s a complcated process, because accordng to Eq. 31, the flud temperature s related to both the lqud pressure and the vapor pressure n the two-phase regon. Ths varaton process cannot be demonstrated by TPMM due to ts key assumptons T s = T f and T f = const n two-phase regon. 6.2 Dstrbuton of Pressure and Velocty Fgure 6a, b dsplay the dstrbutons of lqud/vapor pressure and capllary pressure. Fgure 7 llustrates coolant mass flow rate at ṁ = 1.5kg/m 2 s, and the portons of lqud and vapor

14 K.Weetal. Fg. 6 a Dstrbuton of lqud and vapor pressure and b dstrbuton of capllary pressure flow. In these fgures, the varatons n the rectangle frame are magnfed n the centre of the fgures. AsshownnFg.7, there s a negatve vapor mass flow rate n the two-phase regon, at the same pont, the lqud mass flow rate rses to keep the mass conservaton. Ths s a countercurrent flow phenomenon. Ths phenomenon can be explaned by Fg. 6a, b. In Fg. 6b, the capllary pressure ncreases along z-axs postve drecton, and n Fg. 6a, the lqud pressure decreases along the same drecton. When the fallng speed of the lqud pressure s lower than the rsng speed of the capllary pressure, the counter-current flow phenomenon occurs. However, near the dotted lne n Fg. 7, the flow drecton of vapor phase reverses to z-axs

15 Model Dscusson of Transpraton Coolng wth Bolng Fg. 7 Dstrbuton of mass flow rate n porous meda postve drecton. Ths s a strange phenomenon, whch mght be because the fallng speed of the lqud pressure has surpassed the rsng speed of the capllary pressure. Accordng to Eq. 6, we can fnd the mmedate cause of ths phenomenon, that s on one hand the relatve permeablty of lqud s a cubc functon of saturaton, and on the other hand, as llustrated n Fg. 7, the mass flow rate of lqud mantans the same order of magntude along the z-axs, so when the saturaton falls to a specal level on the hot sde surface, the relatve permeablty decreases severely, that makes the absolute value of the gradent of lqud pressure rsng sharply to mantan the mass flow rate of lqud n the same order of magntude along the z-axs. As shown n Fg. 5, n the two-phase regon, the flud temperature rses at frst, then decreases. Accordng to Eq. 31, the flud temperature s related to both the lqud pressure and the vapor pressure n the two-phase regon. However, n comparson wth Fgs. 5 and 6a, one can fnd that the varaton trend of the flud temperature s dentcal wth that of the vapor pressure. In other words, the relatonshp between the flud temperature and the vapor pressure s much closer than that between the flud temperature and the lqud pressure. 7Concluson Aware of the nveracous thermal nsulatng layer problem n TPMM, ths artcle developed an mproved model that s based on enthalpy balance. A numercal nvestgaton of onedmensonal, steady state, constant propertes transpraton coolng problem wth bolng s carred out wth the mproved model. The followng conclusons can be drawn: The mproved model s able to avod the nveracous thermal nsulatng layer problem. In the two-phase regon, the flud temperature rses at frst then falls, and the flud temperature depends manly on the vapor pressure.

16 K.Weetal. Closer to the hot sde surface, larger the temperature dfference between lqud and sold matrx, namely, a hgher degree of local thermal non-equlbrum; as the coolant mass flow rate decreases, the degree of local thermal non-equlbrum on the hot sde surface ncreases contnuously. There s a vapor counter-current flow phenomenon n part of the two-phase regon near the lqud regon due to capllary pressure, and the vapor flow drecton wll return to postve z-axs drecton near the hot surface. Acknowledgments The project s supported by the Natural Scence Foundaton of Chna No References Bau, H.H., Torrance, K.E.: Bolng n low-permeablty porous materals. Int. J. Heat Mass Transf. 25, Benard, J., Eymard, R., Ncolas, X. et al.: Bolng n porous meda: model and smulatons. Transp. Porous Meda 60, Brdge, L.J., Wetton, B.R.: A mxture formulaton for numercal capturng of a two-phase/vapor nterface n a porous medum. J. Comput. Phys. 225, Duval, F., Fchot, F., Quntard, M.: A local thermal non-equlbrum model for two-phase flows wth phasechange n porous meda. Int. J. Heat Mass Transf. 47, Fchot, F., Duval, F., Tregoure, N.: The mpact of thermal non-equlbrum and large-scale 2D/3D effects on debrs bed refloodng and coolablty. Nucl. Eng. Des. 236, Lands, J.A., Bowman, W.J.: Numercal study of a transpraton cooled rocket nozzle. In: 32nd AIAA/ASME/SAE/ASEE Jont Propulson Conference and Exhbt, July 1 3, Lake Buena Vsta, FL, pp Leontev, A.I.: Heat and mass transfer problems for flm coolng. ASME J. Heat Transf. 121, L, H.Y., Leong, K.C., Jn, L.W. et al.: Transent two-phase flow and heat transfer wth localzed heatng n porous meda. Int. J. Therm. Sc. 49, Lkhansk, V.V., Loboko, A.I., Khoruzh, O.V.: Two phase convecton n a porous medum wth nonunform volume heat release. Atomc Energy 82, Peterson, G.P., Chang, C.S.: Heat transfer analyss and evaluaton for two-phase flow n porous-channel heat snks. Numer. Heat Transf. A 31, Rahl, F., Topn, F., Tadrst, L. et al.: Analyss of heat transfer wth lqud vapor phase change n a forced-flow flud movng through porous meda. Int. J. Heat Mass Transf. 39, Sh, J.X., Wang, J.H.: A numercal nvestgaton of transpraton coolng wth lqud coolant phase change. Transp. Porous Meda 87, Soller, S., Krchberger, C., Kuhn, M.: Expermental nvestgaton of coolng technques and materals for hghspeed flght propulson systems. In: 16th AIAA/DLR/DGLR Internatonal Space Planes and Hypersonc Systems and Technologes Conference, Oct , Bremen, Germany, AIAA Sozen, M., Vafa, K.: Analyss of the non-thermal equlbrum condensng flow of a gas through a porous bed. Int. J. Heat Mass Transf. 33, Vafa, K., Sozen, M.: Analyss of energy and momentum transport for flud flow through a porous bed. ASME J. Heat Transf. 112, van Foreest, A., Sppel, M., Gulhan, A. et al.: Transpraton coolng usng lqud water. J. Thermophys. Heat Transf. 23, Wang, C.Y.: A fxed-grd numercal algorthm for two-phase flow and heat transfer n porous meda. Numer. Heat Transf. B 32, Wang, C.Y., Beckermann, C.: A two-phase mxture model of lqud gas flow and heat transfer n capllary porous meda I. Formulaton. Int. J. Heat Mass Transf. 36, Wang, J.H., Wang, H.N.: A dscusson of transpraton coolng problems through an analytcal soluton of local thermal nonequlbrum model. J. Heat Transf. Trans. ASME 128, Yuk, K., Abe, J., Hashzume, H. et al.: Numercal nvestgaton of thermoflud flow characterstcs wth phase change aganst hgh heat flux n porous meda. ASME J. Heat Transf. 130,