Complete nlytil model of loop het pipe with flt evportor B. Siedel, V. Srtre, F. Lefèvre To ite this version: B. Siedel, V. Srtre, F. Lefèvre. Complete nlytil model of loop het pipe with flt evportor. Interntionl Journl of Therml Sienes, Elsevier, 205, 89, pp.372-386. <0.06/j.ijthermlsi.204..04>. <hl-0286580> HAL Id: hl-0286580 https://hl.rhives-ouvertes.fr/hl-0286580 Sumitted on Mr 206 HAL is multi-disiplinry open ess rhive for the deposit nd dissemintion of sientifi reserh douments, whether they re pulished or not. The douments my ome from tehing nd reserh institutions in Frne or rod, or from puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseignement et de reherhe frnçis ou étrngers, des lortoires pulis ou privés.
Complete nlytil model of loop het pipe with flt evportor Benjmin Siedel, Vlérie Srtre, Frédéri Lefèvre Université de Lyon, CNRS INSA-Lyon, CETHIL UMR5008, F-6962, Villeurnne, Frne Université Lyon, F-69622, Villeurnne, Frne Astrt A stedy-stte nlytil model hs een developed to determine the thermohydruli ehviour of loop het pipe with flt evportor. Its min originlity lies in the omintion of energy lne equtions for eh omponent of the system with 2D nlytil solutions for the temperture field in the evportor. Bsed on Fourier series expnsion, het trnsfer in the wik s well s in the evportor sing re urtely modelled, enling thorough onsidertion of the prsiti het fluxes. The model is sed on the therml ontt resistne etween the wik nd the sing, the therml ondutivity of the wik nd the ommodtion oeffiient. This nlytil method offers simple solution tht n e implemented in LHP design nlysis without the need of lrge omputtionl resoures. A sensitivity nlysis hs een rried out to evlute the influene of severl prmeters on the LHP ehviour. The results show tht the min prmeters of the model re independent. Therefore, they ould e experimentlly determined using n pproprite test enh with only few temperture mesurements. The model hs een vlidted with set of experimentl dt from the literture. A good greement is found etween the theoretil nd the experimentl results. Keywords: Loop Het Pipe, Anlytil, Modeling, Sensitivity nlysis. Introdution Loop Het Pipes LHP re two-phse ooling systems le to pssively trnsport high mounts of het over distnes up to severl meters. Developed in the 970 s, these devies hve proven their reliility in mny sptil pplitions nd re now ndidtes for terrestril ooling solutions. Indeed, their speifi design offers roustness nd flexiility for wide vriety of prtil pplitions [, 2]. As onsequene, mny efforts hve een dedited to understnd their opertion in order to optimize their design. A LHP is omplex therml system inluding n evportor onneted to the het soure, ondenser to dissipte the het lod nd vpour nd liquid lines to trnsport the working fluid etween oth omponents. Coupled thermo-hydruli phenomen govern the LHP ehviour nd need to e understood to enle orret system designing. Prmeters suh s evportion effiieny, het losses to the mient nd prsiti het flux in the evportor s well s ondenstion het trnsfer n e of gret influene on the loop opertion. A lot of ppers onerning LHP omplete modelling n e found in the literture [36]. However, these numeril models often imply omplex lgorithms nd lrge omputtionl resoures whih re not lwys ville for upstrem pre-design pplitions. An nlytil model offers Corresponding uthor Emil ddress: vlerie.srtre@ins-lyon.fr Vlérie Srtre the dvntge to provide solution without n exessive omputing time nd tht n e esily implemented. Some nlytil models of LHP re found in reserh works. Mydnik et l.[7] ited y Luny et l.[8] suggested losed form solution of LHP nlytil model estlishing the energy lne in the reservoir nd the pressure lne in the whole system. Furukw[9] developed omplete nlytil model of LHP le to enhne the sizing of the system nd to study the influene of mny geometril prmeters on the loop opertion. However, the model requires the operting temperture s input prmeter. Yet in most ses, the evportor temperture is the min expeted output of LHP model. Luny et l.[8] proposed losed-form expressions of the operting temperture of the LHP, for oth the vrile nd the fixed ondutne mode. Their model is sed on energy lne equtions on eh system omponent nd on thermodynmi equtions. The therml links in the reservoir re defined s equivlent therml resistnes. Their solution is useful tool in the LHP design. However, the het trnsfer in the evportor is not urtely determined nd hs to e djusted in ordne with experimentl dt. The purpose of the present study is to present omplete nlytil model of Loop Het Pipe urtely tking into ount het nd mss trnsfer in the evportor struture. The model is developed for flt disk-shped evportor geometry. This model is sed on the nlytil study of Luny et l.[8]. However, the present study rests upon two 2-D nlytil solutions desriing the tem- Preprint sumitted to Interntionl Journl of Therml Sienes Jnury 0, 204
perture field in the wik nd the evportor sing. Both solutions enle to determine prsiti het losses through the wik nd the evportor ody, the sensile het given to the liquid flowing through the porous struture s well s the het dissipted y evportion t the wik-groove interfe. A similr pproh ws implemented in the se of onventionl het pipes y Lefèvre nd Lllemnd [0] nd lter extended y Lips nd Lefèvre []. These fetures, oupled with energy lnes nd thermodynmi reltionships in the rest of the LHP, give simple solution for the operting temperture. An itertive proedure is implemented to lulte the two-phse length in the ondenser. T ext Q ext,e Body Evportor/Reservoir Q in Q Qw Qev T e T we Wik T v Condenser two-phse T sink 2. Model desription T ext Q ext,r T r Q su P 2.. Anlytil Model of the LHP The therml stte of the omplete LHP n e determined using energy lne equtions nd thermodynmi reltionships. Figure presents the operting priniple of the LHP nd the links etween its omponents. The totl het lod to e dissipted y the evportor Q in is onduted through the wik or through the evportor ody so tht: Q in = Q w +Q T ext T r,in Liquid line P Thermodynmi reltionship Fluid therml resistne Condution The wik is ssumed to e fully sturted with liquid. The therml het flux Q w is trnsverslly onduted through the evportor wll, the wll-wik interfe nd then dividesup: prtq ev isevportedtthewik-grooveinterfe wheres the rest is dissipted y ondution nd onvetion with the liquid flowing through the porous struture nd with the liquid in the reservoir. Q is onduted longitudinlly through the evportor wll to the reservoir nd prt of it, Q ext,e, is given y onvetion to the mient. Both the het flux through the wik Q w nd the het flux onduted through the evportor sing Q re funtions of the reservoir, the groove, the wik nd the evportor tempertures T r, T v, T we nd T e. The sme dependene pplies for Q ev nd Q ext,e : Q w = ft r,t v,t e 2 Q ev = ṁ l h lv = ft r,t v,t we 3 Q = ft r,t e 4 Q ext,e = ft r,t e 5 Q ext,e is lso funtion of T ext, whih is given dt of our model. As result, the het lod Q in n lso e expressed s funtion of these four tempertures: Q in = ft r,t v,t we,t e 6 An nlytil expression of Q in will e derived in susetions 2.2 nd 2.3. The prt of Q w tht is not dissipted y evportion is the trnsversl prsiti het flux. Prt of this flux is onduted through the wik nd relesed to the 2 T sink T,o T v Condenser suooling Convetion Phse-hnge Contt Figure : LHP shemti nodl network reservoir wheres the rest is dissipted y onvetion due to the liquid flow inside the porousstruture. At the interfe etween the wik nd the evportor envelope, there is temperture gp T e -T we due to ontt resistne R defined s: R = S T e T we Q w 7 where S is the ontt surfe etween the wik nd the evportor ody nd T e nd T we re the tempertures on the envelope side nd on the wik side, respetively. A glol het lne on the evportor/reservoir gives the following eqution: Q in = Q ev +Q sen +Q su +Q ext,e +Q ext,r 8 where Q sen is the sensile het given to the liquid, Q su is the suooling due to the liquid entering the reservoir
nd Q ext,r is the het flux dissipted to the mient y the reservoir. The determintion of Q sen nd Q su leds to: Q sen = ṁ l p,l T v T r 9 Q su = ṁ l p,l T r T r,in 0 where T r,in is the temperture of the liquid oming from the ondenser nd flowing k to the reservoir. To evlute the het trnsfer given y the reservoir to the mient, it is ssumed tht its surfe is t uniform temperture equl to T r. Het trnsfer with the mient Q ext,r is then pproximted y: Q ext,r = h ext S r T r T ext where S r is the totl externl surfe of the reservoir. A single expressionof Q in n e derived y omining equtions 3, 5 nd 7 to : Q in = ft r,t v,t we,t e,ṁ l,t r,in 2 In some ses, the reservoirn e full of liquid. This phenomenon hs een extensively studied y Adoni et l.[2]. In the present model, the existene of two-phse equilirium in the reservoir is ssumed. As shown y Luny et l. [8], thermodynmi reltionship exists etween the sturtion temperture inside the grooves nd the one t the liquid-vpour interfe in the reservoir: T = T v T r = T P P v + P l ρ l g H 3 where ρ l is the liquid density nd H is the elevtion of the ondenser ompred to the evportor. The slope of the pressure-temperture sturtion urve is given y the Clusius-Clpeyron eqution: T P = T /ρ v /ρ l h lv 4 The model n lso ope with non-ondensle gses NCG, whih n e generted for vrious resons in the LHP nd umulte in the reservoir, s explined y Singh et l. [3]. We ssume tht in operting onditions, these NCG re drined to the reservoir. In order to tke into ount the overpressure generted y the NCG P NCG, eqution 3 is modified: T = T v T r = T P P v + P l ρ l g H +P NCG 5 The vpour line is onsidered diti so tht the vpour enters the ondenser with temperture equl to T v. Furthermore, the ondenstion temperture T nd the vpour temperture T v re linked with the thermodynmi reltionship 4. Sine the pressure drops in the vpour line re low, it is ssumed tht ondenstion ours t temperture T v T T v. In the prt of the ondenser where liquid suooling ours, het trnsferwiththe hetsinkndwiththe mient 3 re lulted onsidering onvetive het trnsfer h sink nd h ext, respetively: T,o = T sink +T v T sink 6 πd,i L L 2φ exp ṁ l p,l /h l +D,i /h sink D,o wheret,o isthetempertureoftheliquidttheondenser outlet, L nd L 2φ re the lengths of the ondenser nd of the two-phse region respetively. D,i nd D,o re the inner nd outer ondenser dimeters. In similr wy, the liquid line het lne is: T r,in = T ext +T,o T ext 7 πd l,i L l exp ṁ l p,l /h l +D l,i /h ext D l,o where D l,i nd D l,o re the inner nd outer dimeters of the liquid line nd L l its length. Additionlly, in the ondenser, the het exhnge with the het sink in the twophse zone is equl to the ltent het of the ondensing vpour: ṁ l h lv = πl 2φ T v T sink 8 h ond D,i + h sink D,o Equtions 3, 7, 8, 2, 5 to 7 nd 8 form set of 8 independent equtions with 8 unknowns: T we, T e, T r, T r,in, T,o, T v, ṁ l nd L 2φ. Their solution leds to the determintion of the omplete therml stte of the LHP. The vlues of Q, Q w nd Q ev equtions 2-4 n e lulted using severl methods. In the lrge mjority of LHP models from the literture, het trnsfer in the evportor is simply desried using equivlent therml resistnes sed on the geometril hrteristis nd the thermophysil properties of the evportor. However, suh method does not tke into ount dequtely the het flux in the wik nd the evportor ody sine the determintion of the therml resistnes requires 2D or 3D pproh. Thus, in the present study, n urte therml nlysis of the evportor hs een onduted, sed on n nlytil pproh. 2.2. Anlytil Therml Model of the Wik The first prt of the nlytil model of the evportor dels with het nd mss trnsfer inside the porous struture. As shown in Figure 2, prt of the porous wik is modelled, ordered on one side y the liquid ulk of the reservoir nd y hlf of fin nd hlf of groove on the other side. The 2-D sttionry het eqution in the wik is expressed s: 2 T w x 2 + 2 T w y 2 = 0 9 where T w is the temperture of the porous struture nd x nd y re the xis oordintesfig. 3. A non-dimensionl temperture is defined s: T w = λ efft w T r φ 0 20
Reservoir vpour liquid wik groove Thus eqution 23 eomes: T wx,y = A m YosmπX 25 m=0 Comining equtions 2 nd 25 yields: mπb 2 A m Y+ 2 A m Y Y 2 = 0 26 Solving the previous differentil eqution leds to: A 0 Y = A 0 Y +A 02 if m = 0 27 wll A m Y = A m e mπby +A m2 e mπby otherwise 28 nd leds to: with X = x, Figure 2: Evportor ross-setion 2 T w X 2 + B 2 2 T w Y 2 = 0 2 Y = y nd B = 22 where nd re the lengths of the modelled region in the x nd y diretions respetively, λ eff is the wik effetive therml ondutivity nd φ 0 is n ritrry het flux. y T r 0 T we 0 T v x Figure 3: Shemti of the wik model A generl expression of the non-dimensionl temperture field expnded in 2-D Fourier series is given y: Tw X,Y = A m YosmπX+B m YsinmπX m=0 23 For X = 0 nd X =, n diti oundry ondition is onsidered sed on symmetry hypothesis: T w X = T w X=0 X = 0 24 X= 4 The oundry ondition for Y = 0 is set temperture profile orresponding to the evportor temperture T we for one side x 0 nd to the groove temperture T v for the other side x. At the juntion etween the wll nd the groove, the temperture singulrity is treted y onsidering liner vrition of the temperture etween two fitive points 0 nd. The slope of the temperture grdient nd thus the position of these points depend on the mximum evportion rte t the liquid-vpour interfe tht n e lulted using the kineti gs theory, whih gives het trnsfer oeffiient h ev [4]: h ev = ev 2 ev ρ v h 2 lv T st 2πRTst M 0.5 P st 2ρ v h lv 29 where ev is the ommodtion oeffiient. Suh n pproh ws lredy used in [5] to ope numerilly with the temperture singulrity t the triple line. In the se of the evportion of thin liquid film, the ommodtion oeffiient is defined s the rtio of the tul evportion rte to theoretil mximl phse hnge rte. A oeffiient equl to unity desries perfet evportion while lower vlue represents inomplete evportion. In the se of wter, vlues vrying from 0.0 to re suggested in the literture[6]. We ssume the following reltionship etween h ev nd the position of 0 nd : h ev = λ eff 0 30 The oundry ondition for Y = 0 is then defined s: λ eff T we T r if 0 < X 0 φ 0 λ eff Tw X,0 = T v T r +T we T v X φ 0 0 if 0 < X < λ eff T v T r φ 0 For Y =, the reservoir temperture is set: if X < 3 T wx, = 0 32
The non-dimensionl temperture field is then: Tw X,Y = A 0Y +A 02 33 + Am e mπby +A m2 e mπby osmπx with A 0 = λ eff T v T r +T we T v 0 + φ 0 A 02 = λ eff T v T r +T we T v 0 + φ 0 A m = 2 λ eff T we T v φ 0 m 2 π 2 A m2 = 2 λ eff φ 0 T we T v m 2 π 2 os 0 34 35 mπ 0 os mπ e 2mπB 36 os mπ 0 os mπ 0 e 2mπB 37 In eqution 33, the liquid flow inside the porous struture is not tken into ount. This flow is twodimensionl. However, in the present study, we ssume D flow inside the wik, onsidering homogeneous volumetri soure inside the wik q defined s: q = 4ṁ l p,l T v T r πd 2 w 38 where ṁ l is the totl liquid mss flow rte in the wik nd p,l is the speifi het of the liquid. This ssumption respets the energy lne nd enles to derive n nlytil expression for the influene of the liquid flow inside the wik. The superposition priniple enles to dd simple model of het trnsfer in the wik with homogeneous soure to the previous developed nlysis. It hs to e noted tht 2D pproh of het nd mss trnsfer in porous wik ws presented y Co nd Fghri [7], ut the model is not entirely nlytil. The het eqution in the wik is thus: 2 T y 2 = q 39 λ eff nd the oundry onditions re: T y = 0 T y = = 2 q λ eff 40 T y = = T r 4 This leds to the non-dimensionl temperture field implied y the liquid flow inside the wik: T = q 2φ 0 Y 2 3Y +2 42 This solution is dded to the previous one to give generl expression of the non-dimensionl temperture field in the wik, with onsidertion of -D onvetion: T t = T w +T 43 5 The previously desried model enles to lulte the het flux through the wik Q w nd the het dissipted y evportion Q ev : Q w = S 0 + 2 w T t λ eff 0 y dx 44 y=0 [ 0 + = S w φ 0 A 0 3q 2φ 0 + BA m A m2 sin mπ ] 0 + Q ev = S w T t λ eff 0 + y dx 45 2 y=0 [ = S w φ 0 0 + A 0 3q 2φ 0 BA m A m2 sin mπ ] 0 + 2.3. Anlytil Therml Model of the Evportor Body The model of the wik is suffiient if the het losses through the evportor ody re negligile. Nevertheless, the ody is usully mde of high ondutive mteril. Therefore, it is generlly neessry to estimte the het trnsferred y ondution from the evportor to the reservoir nd to the mient. A seond nlytil model is developed to desrie the evportor sing. As shown in Figure 4, the evportor wll is unwrpped nd modelled s retngulr domin. At x = 0 nd x =, n diti ondition is ssumed due to the symmetry. At y = 0, onvetive het trnsfer with the mient is tken into ount on the whole externl surfe, inluding the heting setion 0 x 0, where het flux φ in is lso pplied. The inner prt of the ody is highly influened y the reservoir temperture. The retngulr shpe hosen to represents the evportor ody does not onsider the effet of the grooves on the distortion of the temperture field etween the se plte nd the wik. We ssume, therefore, set temperture profile, with the evportor temperture T e on one side 0 x 0 nd the reservoir temperture T r on the otherside x with liner profile in-etween. The sum of two solutions is neessry to tke into ount the omplete set of oundry onditions. The first one orresponds to set temperture profile in the inner prt of the wll, in ontt with the wik nd the liquidvpour ulk in the reservoir. Convetive het losses to the mient re ssumed for the whole externl surfe of the evportor/reservoir. This hypothesis onsiders tht the eletroni omponent to ool down dissiptes het with the LHP on one side nd with the mient on the other. The seond solution dds the het input to the evportor.
vpour liquid wik T e φ in Q 0 h ext T ext y d T e T x,y T r 0 h T ext ext = + T r First solution 0 x y d T 2 x,y Seond solution T = 0 0 0 φ = 0 x φ in Figure 4: Evportor sing modelling Using the sme mthemtil proedure s previously, the het eqution eomes: where: 2 T X 2 + D 2 2 T Y 2 = 0 46 X = x ; Y = y d ; D = d ; T = λ T T r φ 0 d 47 The oundry onditions for the first solution T re: T X T Y = T X=0 X = 0 symmetry 48 X= = Bi T + h ext T r T ext Y=0 φ 0 T X, = with λ T e T r φ 0 d Bi = dh ext λ if 0 < X 0 49 λ φ 0 d T e T r X if 0 0 X < 0 if < X < 50 wheres for the seond solution T2 : T2 X = T 2 X=0 X = 0 5 X= T2 φ in if 0 < X 0 Y = φ 0 Y=0 0 if X > 52 0 X, = 0 53 T 2 6 The omintion of equtions46 to53 nd the ddition of oth solutions led to the non-dimensionl temperture field in the evportor ody: TX,Y = C 0 Y +C 02 54 + Cm e mπdy +C m2 e mπdy osmπx with λ C 0 = Bi Bi +φ 0 d φ in T e T r 0 + +T r T ext 2 0 55 φ 0 λ C 02 = T e T r 0 + Bi T r T ext Bi +φ 0 d 2 + 0 φ in 56 φ 0 os mπ 0 os mπ 0 e mπd + mπd+bi mπd Bi e mπd mπ 0 +e 2mπD 57 os mπ 0 os mπ 0 e mπd + mπd Bi mπd+bi e mπd mπ 0 +e 2mπD 58 C m = 2 λ T e T r φ 0 d m 2 π 2 2 φ in φ 0 m 2 π 2 D sin C m2 = 2 λ T e T r φ 0 d m 2 π 2 +2 φ in φ 0 m 2 π 2 D sin The het dissipted to the mient in the evportor setion of the ody is: Q ext,e = 0 0 h ext πxt x,0 T ext dx 59 + h ext π 0 T x,0 T ext dx 0 = h ext π 0 0 2 T r T ext + φ 0d λ C 02
+h ext π φ 0d λ mπ C m +C m2 mπ mπ0 +os 2 + 0 mπ sin mπ The het trnsferred y therml ondution through the evportor ody to the reservoir setion is lulted y integrting the Fourier s lw t y = Fig. 4. Thus, the totl het losses through the ody is: d T Q = Q ext,e + π 0 λ 0 x dy 60 x= = Q ext,e +d 0 φ 0 πsin mπ C m e mπd C m2 e mπd 2.4. Solving proedure The solving proedure is presented in Figure 5. The set of equtions is not liner. Thus n itertive proedure is used to solve it. After initilistion of the prmeters of the model, the two-phse length in the ondenser L 2φ is set, ording to the energy lne for given het input Q in 8. L 2φ hs mjor influene on the determintion of the temperture of the liquid entering in the reservoir T r,in equtions 6-7. Then, the thermophysil properties re lulted, s well s the pressure drops in the trnsport lines. K-oeffiients n e defined to reformt the expression of Q w, Q ev, Q nd T r,in s funtions of T r, T v, T e nd T we : no Prmeter initiliztion, Q in Setting of the two-phse length in the ondenser L 2φ ording to Eq. 8 Clultion of the thermophysil properties nd the pressure drops Clultion of the K- oeffiients See Appendix A Determintion of the vpour temperture T v Eq. B. Is the energy lne in the ondenser Eq. 8 stisfied? yes Q in i + = Q in i + Q in Q in i + > Q mx yes Plot of the operting urve Figure 5: Solving lgorithm flowhrt no Q w = K T r +K 2 T v +K 3 T we using 34,36,37,44 6 Q ev = K 4 T r +K 5 T v +K 6 T we using 34,36,37,45 62 Q ext,e = K 0 T r +K T e +K 2 using 56,57,58,59 63 Q = K 7 T r +K 8 T e +K 9 using 57,58,60,63 64 T r,in = K 3 T v +K 4 using 6,7 65 The detiled expression of these oeffiients is presented in Appendix A. Equtions,7-, 5-7,6-65 re solved nd give seond-order expression whih enles to lulte the vpour temperture T v see Appendix B. This proedure is iterted until the energy lne is stisfied in the ondensereqution 8. The sme method is omputed for eh het input Q in i. Sine the het trnsfer oeffiient with the het sink is generlly muh lower thn the ondenstion het trnsfer oeffiient, the ondenstion therml resistne is negleted to simplify eqution 8. Eqution 5 is funtion of the pressure drops P v nd P l. These prmeters, due to the frition fores in 7 the vpour nd the liquid lines, depend on the fluid flow regime. They re lulted s follows: P = f 2ρD ṁ 2 L 66 whereais the ross-setionreofthe tue. Forsmooth tue wll of dimeter D nd length L, the frition ftor f is expressed y: 64/Re if Re 2000 f = 0.032 if 2000 < Re < 950 0.36Re 0.25 if Re 950 A 67 Eqution 5 lso inludes the pressure of nonondensle gses P NCG. In order to tke into ount the NCG, it is neessry to lulte the liquid level in the reservoir tht depends on the het lod. We ssume tht the void frtion of the two-phse flow in the ondenser is 0.5 nd tht the vpour density is negligile ompred to the liquid density. The liquid level e l in the reservoir is
therefore expressed s: e l = S w m f εs w e w L lπdl,i 2 L 2 L 2φπD,i 2 ρ l 4 4 68 where m f is the totl fluid hrge in the system nd e w is the wik thikness. The totl volume of NCG nd vpour in the reservoir V v is equl to: V v = S w e r e l 69 with e r the thiknessofthe reservoir. Consideringthenonondensle gses s idel gses, their prtil pressure is lulted y: P NCG = m NCGRT r M NCG V v 70 Extensive studies hve een undertken to develop models le to predit urtely the effetive therml ondutivity of porous struture. Thus, mny different orreltions n e found in the literture to determine the effetive therml ondutivity of porous mteril [224] Tle 2 presents the results otined for nikel wik of 75% porosity sturted with wter, using vrious orreltions. This thermophysil property depends not only on the ondutivity of the mterils onstituting the wik ut lso on geometril prmeters suh s the porosity, the men pore dimeter nd the pore size distriution. The results re very different ording to the hosen orreltion. This shows tht this prmeter is diffiult to evlute urtely. For the stndrd se, n effetive therml ondutivityequlto5w m K ishosen, whihorresponds to the wik properties of Singh et l.[9]. where R is the idel gs onstnt nd m NCG nd M NCG re the NCG totl mss nd the molr mss, respetively. To solve equtions 6-8, the het sink het trnsfer oeffiient h sink is determined in ordne with the ondenser design, wheres the het trnsfer oeffiient of the liquid h l is lulted ssuming lminr fully-developed flow nd onstnt Nusselt numer Nu D = 4.36 [8]. Equtions nd 7 depend on the het trnsfer oeffiient with the mient h ext. It is given y the orreltion of Churhill nd Chu [8] for free onvetion on the surfe of n isotherml ylinder: h ext = λ ir D 0.60 + 0.387R 6 D +0.559/Pr 9 6 8 27 2 7 This orreltion is vlid for Ryleigh numers R D lower thn 0 2. The determintion of the effetive therml ondutivity of the porous struture will e disussed in the next setion. 3. Results nd disussion This setion presents sensitivity nlysis to show the influene of different prmeters on the LHP performne. The LHP geometry onsidered for this nlysis is similr to stndrd systems used for eletroni ooling pplitions. Its geometril hrteristis re sed on the experiments of Singh et l.[9] nd Choi et l.[20]. A vlidtion of the model is presented in the lst prgrph. 3.. Stndrd se The properties of this stndrd LHP, hving flt disk-shped evportor, is defined in Tle. The system is supposed to e in horizontl orienttion. The wik is mde of nikel nd the working fluid is wter. The min prmeters for the wik re the effetive therml ondutivity, the ommodtion nd the ontt therml resistne etween the wik nd the sing. 8 Correltion λ eff W m K Alexnder 5.82 Chudhry-Bndhri 4.06 Mxwell 6.96 Prllel sheme 23 Zehner-Shlunder.75 Tle 2: Effetive therml ondutivity lultion with wtersturted nikel wik, 75% porosity Informtion from the literture onerning the therml ontt etween porous struture nd solid se plte is very sre. This prmeter depends on mny geometril nd mnufturing hrteristis. Choi et l.[20] investigted new tehniques to enhne the therml ontt ondutne of evportors in LHPs. Severl hnnel designs hve een tested nd ompred. An estimtion of the ontt therml resistne inluding in prtie therml ondution in the se plte, in the wik nd n evportion resistne is given nd vlues rnging from 7 0 5 to 3 0 4 m 2 K W hveeen otined. Bsedon this work, onstnt ontt resistne of 0 4 m 2 K W is hosen for the stndrd se. Although the ommodtion oeffiient is key prmeter in the determintion of the LHP temperture, its vlue is very diffiult to predit. In the literture, severl studies re dedited to the determintion of this prmeter for vrious fluids[6]. However, the sttering of the results onfirms the diffiulty of desriing the evportion urtely. For wter, vlues rnging from 0.0 to hve een found. Therefore, n ommodtion oeffiient equl to 0. is hosen for the stndrd se. The tempertures t different lotions in the LHP re plotted in Figure 6 for het inputs rnging etween 0W nd 0 W. A temperture differene of severl degrees etween T e nd T we shows the impt of the therml ontt resistne etween the wik nd the sing, prtiulrly t high het inputs. Sine the mss flow rtes in the trnsport lines nd the ondenser re moderte due to the high
Evportor design Flt disk-shped Wik dimeter 40 mm Wik thikness 3 mm Wik effetive therml ondutivity 5W m K Vpour groove width mm Wik-wll ontt resistne 0 4 K m 2 W Aommodtion oeffiient 0. Reservoir depth 0 mm Evportor dimeter 4 mm Evportor wll mteril Stinless steel Working fluid Wter Fluid hrge 7 g Condenser nd trnsport lines i/o dimeters 2/2.4 mm Trnsport lines length 200 mm Condenser length 00 mm Het sink temperture 22 C Het trnsfer oeffiient with the het sink 2000W m 2 K Amient temperture 22 C Het trnsfer oeffiient with the mient 5W m 2 K Tle : Definition of the stndrd LHP for the sensitivity nlysis ltent het of vporiztion of wter, pressure drops in the loop re not signifint. Therefore, the sturtion tempertures in the reservoir nd in the grooves, T r nd T v, re lmost equl. The vlues of T,o nd T r,in re lso very similr euse the het trnsfer etween the liquid line nd the mient is limited. T C 60 40 20 00 80 60 40 20 T e T we T v T r T,o T r,in 20 40 60 80 00 Q in W Figure 6: Temperture of the LHP in the stndrd se A ler trnsition etween the vrile ondutne mode nd the fixed ondutne mode is oserved round 60W. For lower het inputs, the temperture of the liquid exiting the ondenser is onstnt nd equl to the het 9 sink temperture. Aove this point, T,o strts to inrese nd the shpe of ll the temperture urves in the LHP is qusi-liner. Figure 7 shows the temperture in the porous struture in the stndrd se, for totl het lod Q in of 50W. For y = 0mm, the temperture profile shows shrp disontinuity t the trnsition etween the groove nd the fin. The intense evportion ourring t x = 0.5mm uses the onvergene of the hetlines towrd this point nd shrp distortion of the lines in its neighourhood. As y inreses, the temperture profile flttens until the temperture is uniform in the x-diretion for y > 2mm. Then, the lterntion etween the fin nd the groove does not hve ny influene nd the temperture field in the wik in ontt with the liquid ulk of the reservoir is uniform. As defined in Eqution 3, the differene etween the groove temperture nd the reservoir temperture is set only y the pressure losses in the trnsport lines, the hydrostti pressure differene nd the non-ondensle gses prtil pressure. Smooth tues with reltively lrge dimeter led to redued pressure losses. Therefore, the LHP opertes with groove temperture nd reservoir temperture lmost equl. As onsequene, the onvetive ooling due to the liquid flow in the wik is extremely low nd het trnsfer in the porous struture is minly ontrolled y het ondution. Figure 8 shows the effet of non-ondensle gses on the evportor temperture. Vrious quntities of ir rngingetween ñgnd200 ñgresimultedin thereservoir. For NCG mss elow 0 ñg, orresponding to prtil pressure P NCG equl to out 400P, NCG do not hve determinnt influene on the evportor temperture. However, when the mss of NCG in the LHP is lrger, its prtil pressure is more importnt thn the sum
y mm 3 2.5 2.5 0.5 8.2 C 82.2 C 83.2 C 79.7 C 80.2 C 80.7 C 8.7 C 82.7 C T r 79.7 C 80.2 C 80.7 C 80.7 C 80.2 C 0 79.7 C 0 T we 0.5 T v x mm Figure 7: 2-D temperture field in the wik of the pressure drops in the whole loop. The influene of NCG eomes then importnt t low het input, leding to signifint inrese of the evportor temperture. As onsequene, the shpe of the hrteristi urve of the LHP is flttened. T e C 40 30 20 0 00 90 80 70 60 No NCG µg NCG 0 µg NCG 00 µg NCG 200 µg NCG 400 µg NCG 20 40 60 80 00 Q in W Figure 8: Influene of the NCGs on the evportor temperture These results re in ordne with Singh et l. s experimentl study [3]. Their work shows tht the net effet of the generted NCG in the LHP ws to produe n overll 0 rise in the stedy-stte operting temperture. Besides, it ws oserved tht the performne degrding effet of the NCG ws more pronouned t low het lods. 3.2. Sensitivity nlysis The LHP opertion depends on mny distint prmeters: the geometril design size, shpe of the system, the thermophysil properties of the working fluid nd of the mterils of the loop elements, the het trnsfer hrteristis inside nd outside the LHP nd the pressure losses in the system. Most of these prmeters re esy to determine if the LHP geometry nd the working fluid properties re known. However, some prmeters re very diffiult to determine preisely theoretilly s it hs een seen in the previous setion: the effetive therml ondutivity of the wik λ eff, the ontt resistne etween the wik nd the evportor envelope R nd the ommodtion oeffiient ev. Furthermore, the het sink het trnsfer oeffiient h sink nd the het trnsfer oeffiient with the mient h ext re lso lulted using orreltion eing inherently inurte. Thus, in the present setion, sensitivity nlysis is onduted on these prmeters to see their influene on the model. The sensitivity oeffiient of the funtion T = fx i,x j,x k,... in reltion to the prmeter x i is defined s: S i = T 72 x i xj,x k,... In order to ompre severl prmeter sensitivities, it is onvenient to define reltive sensitivities: Si T = x i S i = x i 73 x i xj,x k,... This oeffiient enles to quntify the vrition T used y reltive vrition x i /x i of the prmeter x i. The greter the solute vlue of the oeffiient, the more the funtion is sensitive to the prmeter. Unless otherwise mentioned, the sensitivity nlysis is onduted with the stndrd LHP defined in Tle. A slight vrition 5% of eh studied prmeter is pplied, the vlue of the other prmeters eing onstnt. This effet on the tempertures t different lotions of the LHP is predited y the model. Figure 9 presents the reltive sensitivity of T e, T v, T r,in nd T,o to the het trnsfer oeffiient with the mient. Sensitivities oft e nd T v resignifintforlowhet inputs nd derese with the het lod. Het trnsfer with the mient ools the loop down nd leds to lower opertionl temperture. As expeted, h ext does not hve ny influene on T r,i nd T,o in vrile ondutne mode euse their vlue is only set y the het sink. For higher het lods, the inrese of the het losses to the mient led to lower opertionl temperture. As onsequene, lrger prt of the ondenser is used to ondenste the vpour, the suooling length in the ondenser is smller nd T,o inreses. The sme effet is oserved on
T r,in, ut prtilly offset y the het losses from the liquid line to the mient. 0 0 dt/dh ext *h ext K 2 0-2 -4-6 -8-0 -2-4 T e T v T r,in T,o dt v /dx i *x i K -0-20 -30-40 -50-60 λ eff R ev h sink 20 40 60 80 00 Q in W 20 40 60 80 00 Q in W Figure 0: Reltive sensitivity of T v to vrious prmeters Figure 9: Reltive sensitivity of the LHP tempertures to h ext In mny experimentl onfigurtions, n dequte therml insultion of the entire LHP enles to redue onsiderly the het losses to the mient. In the following, this prmeter is set to zero in order to etter highlight the effet of the other prmeters on the LHP. The reltive sensitivity of the vpour temperture to the other prmeters is shown in Figure 0. It is ler tht het trnsfer inside the evportor funtion of the prmeters λ eff, R nd ev governsthe opertion in vrile ondutne mode wheres t high het lods, h sink eomes the dominnt prmeter. Indeed, t low het lods, T,o equls T sink so the het trnsfer oeffiient with the het sink hs no influene on the suooling of the liquid nd on the LHP opertion in generl. In fixed ondutne mode, het trnsfer in the ondenser sets the LHP opertionl temperture nd the sensitivity of T v to h sink is liner. The prmeters ev, λ eff nd R hve moderte influene on the vpour temperture nd this influene dereses s the het lod inreses. The reltive sensitivity of T e to the sme prmeters is shown in Figure. The influene of the ommodtion oeffiient nd of the het trnsfer with the het sink is lmost the sme s for T v. However, the sensitivities to R nd λ eff hve different ehviour. While the ontt resistne hs limited effet on T v, its influene on T e is lrge nd inreses with the het input. Indeed, the differene etween T e nd T we is proportionl to the het trnsfer rte Q w nd to R. As the het input Q in inreses, Q w inreses lmost linerly nd leds to higher sensitivity of T e to the ontt resistne. The prtiulr shpe of the reltive sensitivity of T e to λ eff isdisussedinthe following; it requiresmoredetiled nlysisontheeffetofλ eff onthelhpthermlehviour. dt e /dx i *x i K 20 0 0-0 -20-30 -40-50 -60 λ eff R ev h sink 20 40 60 80 00 Q in W Figure : Reltive sensitivity of T e to vrious prmeters Figure 2 shows the vrition of the vpour temperture with the effetive therml ondutivity of the wik for severl het lods. An optiml vlue of λ eff is found etween nd 2W m K whtever the het input. ThisminimumvlueofT v isonsequeneoftheevolution of the het trnsferred from the sing to the evportion zone. As the therml ondutivity of the wik inreses,
the het entering the wik eomes lrger t the expense of the longitudinl prsiti het flux. Figure 3 shows the distriution of het trnsfer in the evportor. The lrgest prt of the het lod Q in enters the wik Q w, wheres the rest is thermlly onduted through the evportor ody Q. When the vlue of λ eff is very low, inresing the ondutivity enhnes oth the evportion Q ev nd the trnsversl prsiti het flux Q w Q ev. When λ eff exeeds W m K, the inrese of the het flux entering the wik Q w is smller, leding to n inrese of the trnsversl prsiti het flux t the expense of Q ev, tht deresess λ eff eomes lrger. The mximum evportion het flux leds to minimum vpour temperture Figure 2. The sme onlusion hs een drwn in the numeril study of Siedel et l.[5]. Q/Q in % 00 95 90 85 5 0 5 Q w Q ev Q w -Q ev Q 50 40 30 20 Q=0W Q=30W Q=50W Q=70W Q=90W Q=0W 0 0.5 2 3 5 0 20 λ W.m -.K - eff Figure 3: Distriution of het trnsfer in the evportor Q in = 50W T v C 0 00 90 80 5W m K in the stndrd se, the sign of the slope of the temperture urve hnges s the het flux inreses. As onsequene, the sign of the sensitivity of T e to λ eff hnges when inresing the het lod, s it is shown in Figure. This is not the se for T v, sine the minimum of the urves T v λ eff does not depend on the het flux Figure 2. 70 0.5 2 3 5 0 20 λ W.m -.K - eff Figure 2: Influene of λ eff on T v The evolution of the temperture differene T e T v with λ eff for severl het inputs is given in Figure 4. As expeted, this differene dereses when the therml ondutivity of the wik inreses. Indeed, higher vlue of λ eff leds to lower therml resistnes in the evportor Figure. Figure 5 presents the evolution of T e with λ eff, whih is onsequene of the results otined in figures 2 nd 4. An optiml effetive therml ondutivity exists, for whih T e is miniml. However, its vlue is lso dependent on the het input Q in, ontrry to the temperture of the vpour Figure 2. The optiml vlue of λ eff otined for the minimum vlue of T e inreses from 2W m K to 0W m K with the inrese of the het input from 0W to 0W. Therefore, there is not n optiml vlue of the therml ondutivity of the wik ut rnge of optiml vlues depending on the het lod. Figure 5 shows tht t given λ eff for exmple 2 T e -T v K 80 70 60 50 40 30 20 0 Q=0W Q=30W Q=50W Q=70W Q=90W Q=0W 0 0.5 2 3 5 0 20 λ W.m -.K - eff Figure 4: Influene of λ eff on T e T v This sensitivity nlysis shows tht the min prmeters of the model re independent. Therefore, their in-
T e C 200 80 60 40 20 00 80 Q=0W Q=30W Q=50W Q=70W Q=90W Q=0W hnge of slope 0.5 2 3 5 0 20 λ W.m -.K - eff Figure 5: Influene of λ eff on T e etween the experimentl results nd the lulted tempertures of the evportor wll nd of the vpour in the grooves. A good greement is found etween the experimentl dt nd the model. Severl unknown prmeters were identified y omprison of the predited vlues with the experimentl dt of Singh et l.: the het trnsfer oeffiient etween the ondenser wll nd the het sink, the evportor wll thikness, the ontt resistne nd the ommodtion oeffiient. As result, stright-tue equivlent ondenser is simulted with het trnsfer oeffiient h sink fixed to 3.2kW m 2 K, onsidering n outside dimeter of 2.4 mm for the tues. The ommodtion oeffiient is equl to 0.4. The vlue of R is set to 0 5 K m 2 W. The vlue of R tht enles to fit t est theresultsisverysmll, ut it hstoe notedtht T v is not relly the experimentl mesurement of the vpour temperture, ut the temperture of the tue t the exit of the evportor. Therefore, these experimentl results re not suffiient to estimte urtely the prmeters. Nevertheless, the order of mgnitude of the prmeters re lose to the one defined for the stndrd se. fluene n e differentited from eh other. Thus, the vilility of preise experimentl dt of severl representtive tempertures of the LHP for vrious het inputs my theoretilly led to preise determintion of these prmeters nd provide the model with dequte input prmeters. This sensitivity nlysis lso shows the lrge influene of the tested prmeters on the LHP opertion. Their inurte determintion n led to mjor error on the LHP opertion predition. 3.3. Model vlidtion The present nlytil model is ompred to n experimentl dt set from Singh et l. [9] for vlidtion purpose. These uthors studied the opertionl hrteristis of flt disk-shped evportor LHP, 30 mm in dimeter, using wter s working fluid. The 3 mm thik porous wik is mde of sintered nikel nd its therml effetive ondutivity is onsidered equl to out 5.87W m K using Alexnder s formul nd sed on Singh et l. s study[22]: T C 00 95 90 85 80 75 70 65 60 55 T e experimentl T v experimentl T e model T v model 20 30 40 50 60 70 Q in W ε 0.59 λl λ eff = λ l λ wm 74 Figure 6: Comprison etween the model nd dt from Singh et l. [9] where λ l nd λ wm re the therml ondutivities of the liquid nd of the wik mteril respetively nd ε is the porosity, equl to 75%. The wik is emedded in opper evportor. The vpour nd liquid lines, of internl dimeter 2mm, re50mm nd290mm longrespetively. Afinnd-tue ondenser, 50 mm long, dissiptes het y fored onvetion of ir t mient temperture i.e. 22 C. No prsiti het trnsfer through the evportor ody is tken into ount, sine n O-ring sel prevents het ondution to the reservoir. Figure 6 shows the omprison 3 4. Conlusion In this pper, omplete nlytil model of LHP hs een developed. Its originlity lies in the omintion of energy lne equtions for eh omponent of the system with nlytil solutions for the temperture field in the evportor. Bsed on Fourier series expnsions, het trnsfer in the wik s well s in the evportor sing re
urtely modelled. This nlytil method offers simple solution tht n e implemented in LHP design nlysis without the need of lrge omputtionl resoures. A sensitivity nlysis hs een undertken to ssess the influene of five prmeters on the loop opertion. This nlysis enles etter omprehension of the operting mehnisms of the LHP s well s omprtive study of the prmeters ffeting its tempertures. It ppers tht these prmeters n e experimentlly determined using n pproprite test enh with only few temperture mesurements. This model hs een vlidted with set of experimentl dt from the literture. A good greement hs een met etween the simultion nd the experimentl results. The model results show tht onvetion inside the wik does not ply mjor role nd n e negleted. Moreover, the temperture field in the wik is lmost uniform fr from the grooves. In ordne with experimentl dt from the literture, the presene of NCGs in the reservoir leds to n inrese of the evportor temperture. This degrding effet is more pronouned t low het lods. Nomenlture A Fourier series oeffiient ross-setionl re [m 2 ], 0, length [m] ev ommodtion oeffiient B length rtio Fourier series oeffiient length [m] C Fourier series oeffiient, 0, length [m] p speifi het [J.kg.K ] D dimeter [m] length rtio d length [m] e thikness [m] f frition ftor g grvittionl elertion [m.s 2 ] H height [m] h het trnsfer oeffiient [W.m 2.K ] h lv enthlpy of vporiztion [J.kg ] K oeffiient L length [m] M molr mss [kg.mol ] m, n Fourier series inrement m f totl fluid hrge [kg] ṁ mss flow rte [kg.s ] P pressure [P ] Q het trnsfer rte [W] q volumetri het soure [W.m 3 ] R ontt resistne [K.m 2.W ] S surfe re [m 2 ] S i solute sensitivity Si reltive sensitivity T temperture [K] T non-dimensionl temperture V volume [m 3 ] X, Y non-dimensionl oordintes x i,j,k sensitivity prmeter x, y xis oordintes [m] Greek Symols differene ε porosity λ therml ondutivity [W.m.K ] φ,φ 0 het flux [W.m 2 ] ρ density [kg.m 3 ] Susripts 2φ two-phse ir ir evportor ody ondenser, onvetive, ontt e evportor eff effetive ev evportion ext externl, mient i inner in input, inlet l liquid NCG non ondensle gs o outlet, outer r reservoir sen sensile sink het sink su suooling t totl inluding onvetion v vpour w wik we wik side of the wik-envelope interfe wm wik mteril Non Dimensionl Numers Bi Biot numer P r Prndtl numer R Ryleigh numer Re Reynolds numer Referenes [] Y. F. Mydnik, Miniture loop het pipes, in: 3 th Interntionl Het Pipe Conferene, 2004, pp. 2335. [2] Y. Mydnik, Loop het pipes, Applied Therml Engineering 25 2005 635657. 4
[3] T. Ky, J. Ku, T. T. Hong, M. K. Cheung, Mthemtil modeling of loop het pipes, in: 37 th AIAA Aerospe Sienes Meeting nd Exhiit, 999. [4] A. Delil, V. Bturkin, G. Gorenko, P. Gkl, V. Ruzykin, Modelling of miniture loop het pipe with flt evportor, in: Proeedings of the 32 nd Interntionl Conferene on Environmentl Systems, 2002. [5] P. Chung, An improved stedy-stte model of loop het pipes sed on experimentl nd theoretil nlyses, Ph.D. thesis, Pennsylvni Stte University 2003. [6] L. Bi, G. Lin, H. Zhng, D. Wen, Mthemtil modeling of stedy-stte opertion of loop het pipe, Applied Therml Engineering 29 2009 26432654. [7] Y. F. Mydnik, Y. G. Fershtter, N. N. Solodovnik, Loop het pipes: Design, investigtion, prospets of use in erospe tehnis, Teh. Rep. 9485, SAE Interntionl, Wrrendle, PA Apr. 994. [8] S. Luny, V. Srtre, J. Bonjour, Anlytil model for hrteriztion of loop het pipes, Journl of Thermophysis nd Het Trnsfer 22 4 2008 62363. [9] M. Furukw, Model-sed method of theoretil design nlysis of loop het pipe, Journl of Thermophysis nd Het Trnsfer 20 2006 2. [0] F. Lefèvre, M. Lllemnd, Coupled therml nd hydrodynmi models of flt miro het pipes for the ooling of multiple eletroni omponents, Interntionl Journl of Het nd Mss Trnsfer 49 7-8 2006 375383. [] S. Lips, F. Lefèvre, A generl nlytil model for the design of onventionl het pipes, to e pulished, Interntionl Journl of Het nd Mss Trnsfer. [2] A. Adoni, A. Amirjn, V. Jsvnth, D. Kumr, P. Dutt, Theoretil studies of hrd filling in loop het pipes, Journl of Thermophysis nd Het Trnsfer 24 200 7383. [3] R. Singh, A. Akrzdeh, M. Mohizuki, Opertionl hrteristis of the miniture loop het pipe with non-ondensle gses, Interntionl Journl of Het nd Mss Trnsfer 53 7 200 3473482. [4] V. P. Crey, Liquid-vpor phse-hnge phenomen, Hemisphere, New York, 992. [5] B. Siedel, V. Srtre, F. Lefèvre, Numeril investigtion of the thermohydruli ehviour of omplete loop het pipe, Applied Therml Engineering 6 2 203 54553. [6] I. Emes, N. Mrr, H. Sir, The evportion oeffiient of wter: review, Interntionl Journl of Het nd Mss Trnsfer 40 2 997 29632973. [7] Y. Co, A. Fghri, Anlytil solutions of flow nd het trnsfer in porous struture with prtil heting nd evportion on the upper surfe, Interntionl Journl of Het nd Mss Trnsfer 37 0 994 525533. [8] F. P. Inroper, D. P. DeWitt, Fundmentls of het nd mss trnsfer, John Wiley & Sons, 996. [9] R. Singh, A. Akrzdeh, M. Mohizuki, Opertionl hrteristis of miniture loop het pipe with flt evportor, Interntionl Journl of Therml Sienes 47 2008 50455. [20] J. Choi, B. Sung, C. Kim, D.-A. Bor-Tsiu, Interfe engineering to enhne therml ontt ondutne of evportors in miniture loop het pipe systems, Applied Therml Engineering 60 2 203 37378. [2] S. Mo, P. Hu, J. Co, Z. Chen, H. Fn, F. Yu, Effetive therml ondutivity of moist porous sintered nikel mteril, Interntionl Journl of Thermophysis 27 2006 30433. [22] R. Singh, A. Akrzdeh, M. Mohizuki, Effet of wik hrteristis on the therml performne of the miniture loop het pipe, Journl of Het Trnsfer 3 8 2009 08260. [23] K. Boomsm, D. Poulikkos, On the effetive therml ondutivity of three-dimensionlly strutured fluid-sturted metl fom, Interntionl Journl of Het nd Mss Trnsfer 44 4 200 827836. [24] C. Li, G. P. Peterson, The effetive therml ondutivity of wire sreen, Interntionl Journl of Het nd Mss Trnsfer 49 2-22 2006 4095405. 5
Appendix A. Detiled expression of the K-oeffiients K = S w 0 + [ K 2 = S w 2 λ eff 0 m 2 π 2 sin + 0 + λ eff 6 +ṁ l p,l πdw 2 mπ 0 + os λeff 0 + [ K 3 = S w 2 λ eff 0 m 2 π 2 sin λ ] 2 eff 0 + K 4 = S w 0 + λeff [ K 5 = S w 2 λ eff 0 0 + K 6 = S w [ + λ eff 0 + K 7 = h ext π 0 m 2 π 2 sin λeff 2 λ eff 0 m 2 π 2 sin 0 + 6 ṁ l p,l mπ 0 + ṁ l p,l 6 πd 2 w mπ 0 + 0 + mπ 0 + ] 0 0+ 2 + 2 Bi + mπ0 2λ 0 m 2 π 2 os [ mπ e 0 πsin mπd + πh ext πd 2 w A. mπ0 mπ e 2mπB os e 2mπB ] A.2 mπ0 os mπ0 os ] 6 ṁ l p,l os πd 2 w mπ0 os mπ mπ mπ λ e mπd +e mπdmπd+bi mπd Bi mπ e + 0 πsin mπd + πh ext λ e mπd +e +mπdmπd Bi ] mπd +Bi K 8 = h ext π 0 0 0 + 2 2 Bi + mπ0 + 2λ 0 m 2 π 2 os [ mπ e 0 πsin mπd + πh ext λ e mπd +e mπdmπd +Bi os os os os +os +os mπ mπ mπ mπ mπ +os mπ e 2mπB e 2mπB A.3 A.4 mπ e 2mπB e 2mπB A.5 mπ e 2mπB e 2mπB mπ0 mπ + 0 sin mπ0 mπ + 0 sin mπ0 mπ + 0 sin mπd Bi mπ e + 0 πsin mπd + πh ext mπ0 mπ +os + 0 sin λ mπ mπ 6 A.6 A.7 A.8
e mπd +e +mπdmπd Bi ] mπd +Bi K 9 = h ext π 0 0 φ 0 d 0 φ in 2 λ φ 0 + 2φ in m 2 π 2 sin mπ0 [ mπ 0 πsin +e 2mπD mπ + 0 πsin +e 2mπD ] Bi + T ext e mπd πh ext λ e mπd + πh ext λ +os mπ mπ +os mπ mπ mπ0 mπ + 0 sin mπ0 mπ + 0 sin A.9 K 0 = h ext π 0 0 0+ 2 + 2 Bi + mπ0 mπ 2λ 0 m 2 π 2 os os [ e mπd +e mπdmπd +Bi mπd Bi K = h ext π 0 0 0 + 2 2 Bi + mπ0 mπ + 2λ 0 m 2 π 2 os os [ e mπd +e mπdmπd +Bi mπd Bi K 2 = h ext π 0 0 φ 0 d 0 φ in 2 λ φ 0 Bi + T ext 2φ in m 2 π 2 sin mπ0 πhext λ mπ mπ [ +e 2mπD +e 2mπD ] πl L 2φ K 3 = exp exp K 4 = exp + exp ṁ l p,l /h l D,i +/h sink D,o πl l ṁ l p,l /h l D l,i +/h ext D l,o πl l ṁ l p,l /h l D l,i +/h ext D l,o πhext λ +os mπ mπ + e mπd +e +mπdmπd Bi mπd +Bi πhext λ +os mπ mπ + e mπd +e +mπdmπd Bi mπd +Bi exp +os T ext mπ0 mπ + 0 sin πl l ṁ l p,l /h l D l,i +/h ext D l,o πl L 2φ ṁ l p,l /h l D,i +/h sink D,o mπ0 mπ + 0 sin ] mπ0 mπ + 0 sin ] T sink A.0 A. A.2 A.3 A.4 Appendix B. Seond order eqution for the determintion of T v 0 =Tv 2 [ p,l K 3 K 6 K +K 2 +K 7 K 4 +K 5 K 3 +K 8 +K 8 R /S K 6 K +K 2 K 3 K 4 +K 5 ] +T v [ p,l K 3 K 3 +K 8 K 4 T +K 3 K 4 K K 6 K 8 TR /S +K 6 K T K 7 T +K 9 Q in +h lv p,l K 4 K 6 K +K 2 +K 7 +K 8 K +K 2 K 4 +K 5 K 3 +K 8 +K 3 K 8 R /S +h lv K K +K 2 +K 7 +K 3 K 7 R /S K 3 +K 8 +K 3 K 8 R /S K 0 +h ext S r ] + p,l K 4 K 6 Q in K 9 + T K 6 K +K 7 +K K 8 R /S K 4 K 3 +K 8 +K 3 K 8 R /S 7 B.
+h lv T K 3 +K 8 K 4 +K 0 K 6 +K K +K 7 +K 3 K 8 K 4 +K 0 R /S K 3 K 7 K +K K 6 K 8 R /S +h lv K 3 +K 8 +K 3 K 8 R /S Q in K 2 +h ext S r T +T ext K 6 +K +K 3 K R /S Q in K 9 8