Four Parameter Heat Transfer Turbulence Models for Heavy Liquid Metals

Transcription

1 J. Energy Power Sources Vol., No., 015, pp Receved: January 9, 015, Publshed: February 8, 015 Journal of Energy and Power Sources Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals Sandro Manservs and Flppo Menghn DIN, Lab. d Montecuccolno, Unversty of Bologna, Va de Coll, 16, Bologna, Italy Correspondng author: Sandro Manservs (sandro.manservs@unbo.t) Abstract: In advanced Gen IV nuclear reactors heavy lqud metals are consdered as coolant for ther hgh conductvty and specfc neutronc propertes. These fluds have a very low Prandtl number and show a pecular heat transfer where conducton can be the domnant mechansm at very hgh Reynolds numbers. In ordnary fluds varous turbulence models are avalable to match the expermental data: Smlarty between velocty and thermal turbulent felds s assumed n almost all commercal Computatonal Flud Dynamcs codes and the smple eddy dffusvty model wth constant turbulent Prandtl number s mplemented. In low Prandtl number fluds ths model fals to reproduce standard correlatons buld from expermental data. Therefore t s mportant to develop new heat transfer turbulence models that are able to reproduce numercally the physcal behavor. In ths wor we present dfferent turbulence models to study the heat transfer n heavy lqud metal turbulent flows. Results obtaned wth the smple eddy dffusvty model are reported. More complex four parameter turbulence models are also presented and numercal results n smple geometres are reported. For a large range of forced flows wth no smlarty between velocty and thermal felds a four parameter turbulence model s a powerful tool for predctng the heat transfer. Keywords: Turbulence models, heat transfer, heavy lqud metals. 1. Introducton For the next GEN IV nuclear reactors, heavy lqud metals are often consdered n the engneerng communty as coolant fluds because of ther physcal propertes ncludng hgh conductvty, low vscosty and excellent neutronc propertes for fast-spectrum nuclear reactors. In the desgn of these reactors CFD computatons are usually performed wth smple eddy dffusvty models by mposng a constant turbulent Prandtl number. Ths model, mplemented n standard commercal codes as the only heat transfer turbulence model, s nown to be napproprate for no untary Prandtl fluds because there s a strong dssmlarty between thermal and velocty felds [1]. Usng these smple eddy dffusvty models an expermental nvestgaton to assgn the correct turbulent Prandtl number s needed for any dfferent geometry. Therefore t could be very useful to be able to predct the space dstrbuton of the turbulent Prandtl number by usng a turbulence model. On the last several years four parameter turbulence models have been developed rapdly startng from the Algebrac Flux Model (AFM) based on mplct or explct formulaton [-9]. The mplct methods rely on an algebrac soluton of the correspondng transport equaton whle the explct ones approxmate the mplct term by ntroducng the velocty and temperature tme scales (see [-6] and references theren). Heat transfer two-equaton models have been tested aganst DNS smulatons for low Reynolds numbers n very smple Cartesan geometres but a full test aganst expermental correlatons and data has not been yet nvestgated n a satsfactory way [, 4, 9-10]. Expermental data on heavy lqud metals over a large range of Prandtl and Reynolds numbers are dffcult to fnd n lterature and often these data do not agree among

2 64 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals dfferent authors because these experments are very dffcult to carry on and there are many error sources, see [1]. For smple cylndrcal geometry, one can consder the two man correlatons whch have been studed n lterature for lqud metals flowng n crcular ppes wth a unform heat flux at the wall: The theoretcal modfed Lyon equaton and the Krllov-Ushaov expermental correlaton. These correlatons are usually consdered for the evaluaton of the heat transfer coeffcents n cylndrcal geometry. The heat transfer coeffcent h = q/dt s usually defned n order to characterze the heat transfer between a sold and a flud, where convecton and conducton processes are nvolved. The quantty q s the heat flux through the sold surface and DT the temperature dfference between the wall T w and the surroundng flowng coolant T b. The coeffcent h s usually evaluated wth the ntroducton of the Nusselt number Nu = hd h /λ, where D h s the hydraulc dameter and λ s the thermal conductvty of the lqud. Comparng these two expressons, t s easy to note that the theoretcal equaton and the expermental correlatons cannot match for large range of Reynolds numbers, f a constant turbulent Prandtl number s consdered. Other correlatons have been proposed for dfferent geometres, le trangular rod bundle geometry or annular geometry. These correlatons are very mportant snce they are commonly used by engneers to predct heat transfer n nuclear reactor cores. For trangular rod bundle geometry the most common employed correlaton s the one by Krllov and Ushaov, whle for annular geometry Dwyer, Hartnett and Irvne have proposed correlatons based on expermental data. For a revew on these correlatons see [6-11] and references theren. The present wor addresses an effort to mprove the predcton of turbulent heat flux n flows under complete dssmlarty between velocty and thermal felds as t happens for a LBE flud. We consder the standard SED turbulence model wth a constant value of the turbulent Prandtl number whch can be found n commercal codes and three more complex four equaton turbulence models. In the next secton we derve the transport equatons of the turbulence models just mentoned and we report them. In the followng secton we report some numercal results obtaned wth these models n order to show the features of each of them. Fnally we draw our concluson.. Turbulence Models for Lqud Metals.1 Reynolds Averaged State Equatons The moton of ncompressble fluds s descrbed by the equatons of conservaton of mass, momentum and energy. In order to descrbe turbulent flows, one can apply the Reynolds averagng procedure to the fundamental dynamc equatons and obtan a set of transport equatons for averaged felds. Throughout the rest of the paper, the usual Ensten notaton for summaton over repeated ndces s assumed. In order to solve for the flow varables we consder the system of conservaton equatons u u = = 0 (1) x u ρ + ρu t u x = x j σ j x ρu ' u ' + ρg () T T T ρcp +u = λ ρcpu' T' +Q t x x x (3) x where u and T are the averaged velocty and temperature felds. Moreover, the Reynolds stress tensor u'u' s the averaged product of velocty fluctuatons and the turbulent heat flux densty u't' s the averaged product of velocty and temperature fluctuatons. We also defne the stress tensor and the velocty deformaton tensor as u u j σj : = pδj + μsj, Sj : = + (4) x j x where μ s the molecular vscosty. In Eqs. (1)-(3) the unnown Reynolds stress u'u' and the turbulent heat flux u't' can be seen as solutons of two transport evoluton equatons. The Reynolds stress transport equaton for each stress component of the tensor u'u' can be wrtten as [7]

3 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals 65 u' u j' + u t x u ' u ' = L j j + D T j + j + P E (5) where P and L are the stress producton term and the lamnar dffuson term defned by u j u Pj = u ' ul' + u j' ul' xl xl (6) L u ' u j' D j = ν x l xl The remanng terms D T, Φ, E descrbng the turbulent dffuson, the pressure stran and the dsspaton respectvely, must be modelled n order to obtan the soluton of Eq. (5). The turbulent heat flux transport equaton for each component u't' can be wrtten as [7] L T u ' T' +u u ' T' = Dt + Dt + Πt + Pt Et (7) t x where P t s the producton term gven by T u P t = u ' ul' + ul' T' (8) xl xl Unle the prevous case both the molecular and turbulent energy dffuson terms D T must be modelled, along wth the pressure correlaton Π t and the destructon rate vector E t.. Approxmaton of the Momentum Turbulent Transport Equaton We can approxmate the Eq. (5) wth a two-equaton turbulence model. A great varety of models s avalable n lterature. For the momentum flux, we consder the standard sotropc eddy-vscosty assumpton and defne the turbulent netc energy and ts dsspaton rate by 1 u' κ = u' u' ε = ν :, :, : (9) x j Cm wth turbulent vscosty ν t = C μ /ε and C μ = By tang Eq. (5) for = j = 1,, 3 and addng the three components we can wrte an equaton for the trace κ as follows: κ κ +u = t x x j ν + σκ κ + P ε (10) x j j j wth u u u j u P = u ' u j' = ν : t + (11) x j x j x x j In a smlar way one can wrte an equaton for the turbulent energy dsspaton rate ε as +u = ν+ +C1 P C t x x j σ (1) x j κ but sometmes t s convenent to solve the equaton for the specfc dsspaton rate defned as ω = ε/(c μ κ), namely ω ω ω ω +u = ν+ +C1 P Cω t x x j σ (13) ω x j κ wth P defned by Eq. (11). Many turbulence models can be used for the defnton of the eddy vscosty ν t and of the coeffcents C 1ε and C ε or C 1ω and C ω..3 Approxmaton of the Energy Turbulent Transport Equaton We can approxmate the Eq. (7) wth a two-equaton turbulence model n a smlar way as we have done for the momentum turbulence model. For the heat flux, we neglect the D t and E t terms and approxmate the total dervatve term as [4-6] Cm u ' T' u T +ul' T' = u ' ul' + Et (14) R τu xl xl wth τ t = Rτ u, τ u = κ/ε. In analogy wth the defntons n Eq. (9) we ntroduce the square averaged temperature fluctuatons and ts dsspaton rate gven by 1 ν T' t = T' εt = :, :, Pr x t j t κ : (15) Cmt and defne the characterstc tme τ t = κ t /ε t and the rato R = τ t /τ u = (κ t /ε t )/(κ/ε). From the algebrac Eq. (14), called Algebrac Flux Model (AFM), dfferent approxmatons can be consdered. By neglectng E t together wth the second term on the left nvolvng mean velocty gradent and assumng R = 0.5 we can recover the smple Eddy Dffusvty Model (SED). In the SED model we replace each term of the tensor u'u' by ts trace κ and wrte

4 66 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals T u ' T' = (16) Prt0 x where Pr t0 = s the turbulent Prandtl number whch s assumed to be a constant. Agan from the algebrac Eq. (14) by approxmatng the velocty gradent wth a constant term proportonal to τ u = κ/ε and neglectng the E t term we can wrte T T u ' T' = R = (17) R +Cm Cμ x Prt x where R +Cγ Pr t = Prt0Cα (18) R wth C constants and R = (κ t /ε t )/(κ/ε). Clearly ths expresson neglects the dsspaton term E t n Eq. (14). Ths contrbuton s often taen nto account as T Et = Bν,,Pr,R (19) CαPrt0 x The functon B(ν, ν t, Pr, R) can tae several forms and the nterested reader can consult [-6]. The fnal form of the turbulent Prandtl number n the thermal turbulence model can be wrtten as [, 4] 1 R + Cγ Prt = Prt0 Cα (0) Bν,,Pr,R+ 1 R The thermal flux Eq. (17) and the turbulent Prandtl number Eq. (0) can be used to solve the energy equaton for the flud T T T Q +u = α + αt + (1) t x x x ρc p wth the turbulence thermal conductvty defned by α t = ν t /Pr t. In ths model the turbulent Prandtl number s not constant and depends on R and therefore on κ t and ε t or ω t. The equaton for the averaged temperature squared fluctuatons κ t s defned by the followng transport equaton [6] κt κt α t κt +u = α+ + Pt εt () t x x σ κ x t where T T P t : = u' T' = (3) x Prt x In a smlar way an equaton for ε t can be wrtten as [6] ε t εt α t ε t +u = α+ t x x j σt x j εt εt C p1pt Cd1εt + C pp C ε κ κ + d t (4) where P s defned by Eq. (11). The coeffcents can be assumed constant or as model functons. For detals on the κ t -ε t model see [6-8]. If one solves the system Eqs. ()-(4) then the functon R can be computed as R = (κ t /ε t )/(κ/ε)..4 Boundary Condtons The boundary condtons have to be defned for every formulaton of the four parameter turbulence model. The wall functon approach s not consdered here for heavy lqud metal. The boundary condtons on the sold surfaces are mposed on a cylndrcal surface at a dstance δ from the physcal wall, nsde the flud. The dstance δ depends on the mesh sze of the boundary layer regon. We refne the mesh untl the surface defned by the dstance δ les n the vscous lamnar regon, then we enforce the near-wall boundary condtons. For κ-ε and κ-ω models the near wall boundary condtons are κ = 0, ϵ = 6ν κ, ω = 6ν 1 (5) δ βδ where we have assumed full sotropy for the dsspaton on the wall. We remar that usng these boundary condtons and κ-ε model the two varables are coupled on the wall because of the presence of κ n ε boundary condton. On the contrary, usng a κ-ω formulaton the two varables do not depend on each other on the wall, so the equatons can be solved n a segregated way. For the thermal felds the boundary condtons to be mposed depend on the thermal problem. If a constant temperature s assgned then the square averaged temperature fluctuatons κ t have to be set to zero on the wall. If a constant heat flux s mposed on the wall then κ t can be dfferent from zero and a dfferent boundary condton has to be set. A dscusson over the approprate boundary condtons to be set and over the dfferences arsng n the results obtaned can be found n [6]. In the followng we wll use mxed heat flux

5 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals 67 boundary condtons κ t = 0, ϵ t = 6α κ t, ω δ t = 6α 1 (6) βδ by mposng null temperature fluctuatons on the wall and ε t or ω t as n Eq. (6) n order to obtan the correct value of R = Pr on the wall [6]. 3. Numercal Results In ths secton we report the numercal results obtaned wth all the turbulence models just presented n smple geometres, namely plane, cylnder and annulus wth unform heat flux on the wall. We have smulated fully developed turbulent flows of a lead-bsmuth eutectc alloy wth physcal propertes gven n Table 1 and a Pr A constant temperature T = K and a flat velocty profle are assumed on the nlet secton wth the Reynolds number of approxmately 5, ,000. We assume a unform heat flux of q = 3.6e + 5 W m - on the heated walls and standard outflow condtons on the outlet. In the next two paragraphs we report the results obtaned wth the standard SED model and we compare these ones wth those obtaned wth a κ-ω-κ t -ε t model. In the last two paragraphs we present the results obtaned wth the κ-ε-κ t -ε t and wth the κ-ω-κ t -ω t four parameter turbulence models. 3.1 κ-ω-sed Model In ths paragraph we report some numercal results obtaned wth the standard SED model n cylndrcal geometry. All the commercal codes have a zero-order turbulence energy model based on a standard momentum turbulence model. Ths turbulence energy model, called SED (smple eddy dffusvty) model, s smply defned by T T u ' T' = R = (7) R +Cm Cμ x Prt x where the turbulent Prandtl number Pr t s assumed to be nown and constant. The turbulence model s needed for the computaton of the eddy vscosty whle the eddy thermal dffusvty s drectly proportonal to the eddy vscosty by mean of the turbulent Prandtl number. For Table 1 LBE propertes at the reference temperature T = K. Propertes Values Unts Densty ρ 10,340 Kg m -3 Dynamc vscosty μ Pa s Thermal conductvty λ 10.7 W (m K) -1 Specfc Heat capacty Cp J (Kg K) -1 cylndrcal geometres some theoretcal and expermental correlatons are avalable for the computaton of the Nusselt number. As a reference correlaton, we can use the one that has been developed theoretcally by Lyon startng from the Naver-Stoes equaton and the expermental one obtaned by Krllov. The Lon correlaton s well nown n the followng form Pe Nu = (8) Pr t whle the expermental Krllov-Ushaov correlaton can be wrtten as Nu = 4.5+ Pe (9) The computatonal doman for the cylndrcal geometry, as shown n Fg. 1a, conssts of a smple cylnder wth a total length L = 3 m and dameter D = 55.8 mm. We assume a unform heat flux of q = 360,000 W m - on the heated wall and compare the results wth the above mentoned correlatons (8) and (9) for the predcton of the Nusselt number. In ths geometry we consder the smple SED approach, where only the sngle κ-ω turbulent model s consdered. Ths may be used to test how the CFD approach can deal wth ths type of problems wthout any turbulence energy transport model. In Fg. 1b one can see the asymptotc Nusselt number Nu as a functon of the Peclet number Pe for three cases: Emprcal Krllov-Ushaov correlaton (K), modfed Lyon correlaton wth Pr t = 0.85 (L) and computed values wth the SED model mplemented n Fluent (F) and n the FEM code, an n-house fnte element code (E). From the computatonal pont of vew we see that, n the explored range of veloctes, the smple standard SED turbulent heat transfer model reproduces the correlaton n Eq. (8) wth a constant Prandtl number

6 68 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals Lyon correlaton provdes results whch are closer to the expermental Krllov-Ushaov correlaton. Unfortunately an expresson for the varable turbulent Prandtl number, such as Eq. (30), s nown only emprcally and studed for partcular geometres and boundary condtons. For dfferent geometres and boundary condtons new expressons have to be found based on expermental results. 3. κ-ω-κ t -ε t Model Fg. 1 Cylndrcal geometry, and asymptotc Nusselt number Nu, as a functon of the Peclet number Pe, for the cylnder ppe heated wth constant heat flux: Emprcal Krllov correlaton (K), Lyon correlaton (L) and the computed values wth the smple dffusvty model (Pr t = 0.85) mplemented n Fluent (F). (Pr t = 0.85). Ths correlaton may be vald for water and ar but t does not seem to be approprate for lqud metals. In fact the predctons of ths model for lqud metals are qute dfferent from the values of the expermental correlaton proposed by Krllov-Ushaov. In partcular the values of the Nusselt number calculated by CFD smulatons are greater than data from the expermental correlaton. In order to match the expermental data a number of formulas have been suggested for Pr t whch are vald for dfferent ranges of the Reynolds numbers [1]. For example, n order to reproduce the expermental data of the Krllov-Ushaov correlaton n ths range of Pe, one can use Lyon correlaton and defne varable turbulent Prandtl number Pr t as [1] Pe Pr t = (30) Pe Wth these values of the turbulent Prandtl number, In ths paragraph we report the numercal results obtaned wth the κ-ω-κ t -ε t model and compare them wth the standard SED model ones. Snce n commercal codes the use of the κ-ω model s very popular we construct a κ-ω-κ t -ε t four parameter model and devote ths secton to the computaton of the heat transfer coeffcent for cylndrcal and annular geometry n forced flows wth no gravty by usng the four parameter κ-ω-κ t -ε t model. As ntroduced n the prevous secton, ths model conssts of a κ-ω system for the transport of turbulent momentum and a κ t -ε t two-equaton model for the transport of turbulent energy as defned n Eqs. (10)-(13) and Eqs. ()-(4). In order to valdate the four parameter model we would le to compare the computatonal results wth the Krllov-Ushaov expermental data represented through the correlaton (9) for the case of a cylndrcal geometry wth unform heat flux on the wall. For these computatons we have used the coeffcents defned n [4-5], namely C d1 = 0.9, C p = The dsspaton term n Eq. (19) has been modelled as [4] R 10 B (31) 3/ 4 ν,,pr,r = exp R / 5 1/ 4 et Pr 1 + Pr Rt wth R t = κ²/νε, R et = (1 + (Pr)½)R d and R d = δ (εν)½/(ν) where δ s the dstance from the wall. The coeffcent C p1 has been chosen to be 0.98 to match the Krllov expermental correlaton and C d = 1.9(1-0.3(exp(-Rt /45)-1))(1-exp(-Rd )). The coeffcent C γ s set to 0.5/(Pr)¼ and C α s defned as [4] C α = 1/ 4 7 / 4 1 exp Ret Rd (3)

7 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals 69 In order to match the lmt to the wall, the ntroducton of the κ-ω-κ t -ε t model allows us to have a turbulent Prandtl number whch vares n space throughout the computatonal doman. Fg. a shows the temperature dfference DT between the wall temperature T w and bul temperature T b for dfferent nlet veloctes v = 0.5 (A), (B), 0.5 (C), 0.77 (D) and 1.0 (E) obtaned by usng the κ-ω-κ t -ε t model. We remar that the fnal drop n DT near the outlet s due to the boundary condtons and all the temperature measurements should be taen n the thermal fully developed regon. In Fg. b, the asymptotc Nusselt number Nu, as a functon of Peclet number Pe, for the cylnder ppe heated wth constant heat flux s shown. One can see the emprcal Krllov-Ushaov correlaton (K), Lyon correlaton (L) and the κ-ω-κ t -ε t model (P). As the graph shows the four parameters model reproduces qute well the expermental data by Krllov-Ushaov. In Fg. 3 the tme rato R = βωκ t /ε t (Fg. 3a) and the temperature T (Fg. 3b) profles along the dameter at z =.5 m for dfferent nlet veloctes v = 0.5 (A), 0.77 (B) and 1 (C) are shown for the cylnder ppe smulatons wth the κ-ω-κ t -ε t model. The basc computatonal doman for the annular geometry s shown n Fg. 4a and t conssts of an annulus of total length L = 3 m wth an outer and nner Fg. the axal profles of DT = T w - T b for dfferent nlet axal velocty v = 0.5 (A), (B), 0.5 (C), 0.77 (D) and 1.0 (E) m/s by the κ-ω-κ t -ε t model; the asymptotc Nusselt number Nu computed by usng the emprcal Krllov-Ushaov correlaton (K), the Lyon correlaton (L) and the κ-ω-κ t -ε t model (P). Fg. 3 Tme rato R = βωκ t /ε t, and temperature T profles along the dameter at z =.5 m for dfferent nlet veloctes v = 0.5 (A), 0.77 (B) and 1 (C) m/s for the κ-ω-κ t -ε t model.

8 70 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals dameter of D = 64 mm and d = 8 mm, respectvely. We assume that the heat source, located outsde the annulus, generates the same unform heat flux q as dscussed n the prevous secton and study some confguratons where the unform heat flux s appled to the nner or the outer wall. In partcular the annular geometry wth a constant heat flux on the nner wall and the outer adabatc wall s partcularly nterestng because t resembles the rod bundle geometry. For the correct computaton of the Nusselt number one must be n the regon of fully developed flow. Fg. 4b shows the temperature dfference DT between the wall and the bul temperature for dfferent nlet veloctes. The lnes become nearly horzontal after ~.3 m so that the asymptotc Nusselt number can be evaluated n ths regon. In Fg. 5, by usng the κ-ω-κ t -ε t model for the annular geometry wth heat flux on nner wall, we show the profles of the tme rato R = βωκ t /ε t (Fg. 5a) and temperature square fluctuatons κ t (Fg. 5b) along the radus at z =.5m for dfferent nlet veloctes v = (A), 0.5 (B), 0.77 (C) and 1 (D) m/s. Fg. 6a shows a comparson between the correlatons proposed by Dwyer (D), Hartnett/Irvne (H), Krllov (K)and Lyon (Pr t = 0.85) (L). In the same fgure the κ-ω-κ t -ε t model (E) and the SED model mplemented n Fluent (F) are shown for the annulus heated from the nner wall. The two models predct a dfferent Nu for ths range of Pe. The SED model predcts a very hgh Nu compared wth all the expermental correlatons, whle the four parameter model reproduces qute well the predctons of Hartnett and Irvne correlaton. In Fg. 6b, the Nusselt Fg. 4 Annular geometry, and axal profles of temperature dfference DT = T w - T b for the nlet veloctes v = (A), 0.5 (B), 0.77 (C) and 1.0 (D) m/s for unform heat flux on nner wall. Fg. 5 Tme rato R = βωκ t /ε t, and temperature square fluctuatons κ t along the radus at z =.5 m for dfferent nlet veloctes v = (A), 0.5 (B), 0.77 (C) and 1 (D) m/s wth the κ-ω-κ t -ε t model for the annular geometry and unform heat flux on nner wall.

9 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals 71 Fg. 7 Plane channel geometry, and cylnder ppe geometry. Fg. 6 Asymptotc Nusselt number Nu as a functon of Peclet number Pe for the annular geometry heated wth constant heat flux over the nner wall and for dfferent dameter ratos D e /D wth constant heat flux over the nner or external wall: Hartnett and Irvne correlaton (H), Lyon correlaton (L), κ-ω-κ t -ε t model (E) and the SED model mplemented on Fluent (Pr t = 0.85) (F). number s compared for dfferent rato between nner (D ) and external dameter (D e ) wth constant heat flux over the nner or external wall. The κ-ω-κ correlaton (H). The Fgure shows that the Nusselt number changes rapdly wth low ratos between the nner and outer dameter when the nner wall heatng s consdered. Moreover a good agreement wth the expermental correlaton s obtaned wth the use of the four parameter turbulence model for both nner and outer heated annulus and dfferent dameter ratos. 3.3 κ-ε-κ t -ε t t -ε t results (E) are compared wth the Hartnett-Irvne Model On the last several years four parameter turbulence models have been developed rapdly startng from expermental databases wth several dfferent Prandtl number fluds. These models are all κ-ε-κ t -ε t models based on the orgnal wors n [4-5]. In these wors the momentum turbulent κ-ε model s a second order approxmaton whle the energy turbulent κ t -ε t model s based on mxed energy and momentum tme. In ths paragraph we present the heat transfer as predcted by a second order κ-ε-κ t -εε t model n two smple geometres, a channel plane and a cylnder both wth unform heat flux on the wall. For mplementaton detals one can see [6]. The computatonal doman for the two test cases s shown n Fg. 7. On the left the plane geometry s represented wth a channel half wdth of 30.5 mm, on the rght the cylnder wth dameter of 60.5 mm. We assume a unform heat flux of q = 360,000 W m - on the heated wall and compare the results wth DNS geometry and Krllov correlaton for the predcton of the Nusselt number n the cylndrcal test case. In Fg. 8 the asymptotc Nusselt number as computed wth the four parameter κ-ε-κκ t -ε t turbulence model s compared wth DNS and expermental correlaton. In Fg. 8a, the plane channel resultss are reported, Fg. 8b the cylnder results. As one can see from ths Fgure the match wth expermental correlaton s almost perfect, whle for the DNS data there s a lttle dfference. The turbulence model can be just a compromse between expermental and DNS data where these two data sets dsagree. For heat transfer results n dfferent geometres the nterested reader can consult [6, 11-1]. data for the plane

10 7 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals have obtaned the equaton for ω drectly from the ε equaton, so a new cross-correlaton term appears, ω ω ν t ω +u = ν+ t x x j σω x j (33) ω + ν + C1 P Cω σ ω x j x j κ If one uses a κ-ω turbulence model for the dynamcal turbulence then t s obvous to ntroduce ω t = ε t /(C μ κ t ) so that R can be easly computed as ω/ω t. In ths case an Fg. 8 Asymptotc Nusselt number Nu as a functon of the Peclet number Pe for the plane channel geometry and for the cylnder ppe heated wth constant heat flux: DNS data by Kawamura (DNS) [10], emprcal Krllov correlaton (Krllov) and values obtaned wth the κ-ε-κ t -ε t model. 3.4 κ-ω-κ t -ω t Model In ths last paragraph we report some results n plane and cylndrcal geometry obtaned wth the κ-ω-κ t -ω t turbulence model. The most mportant feature of ths model s the robustness and stablty of ts mplementaton whch s due to the possblty of uncouple the varables on the wall and use an mposed Drchlet condton for both κ and ω. Many turbulence models can be used for the defnton of the eddy vscosty ν t and of the coeffcents C 1ε and C ε or C 1ω and C ω. For the κ-ε formulaton we have used the model by Nagano et al. reported n [4, 6], whle for the κ-ω formulaton we approprate transport equaton has to be found for ω t and ths can be accomplshed by substtutng the defnton of ω t n the transport equaton of ε t. By dong so the equaton for ω t becomes t +u t t = x x j α α + σ t t t x j t t t + ν+ C (34) p Pt Cd t 1 1 t σ ωt x j x j κt t + C pp Cd κ and n order to solve ths equaton a model obtaned from the κ t -ε t model has to be used. The test case geometres are the same as reported n Fg. 7 and n ths case we compare the results only wth Krllov correlaton. In Fg. 9 the asymptotc Nusselt number as computed wth the κ-ω-κ t -ω t turbulence model s reported for the plane channel case (Fg. 9a) and for the cylnder case (Fg. 9b).Ths model over predcts the heat transfer for the cylnder test case but the overall result s qute satsfactory. 4. Conclusons The present wor nvestgates heat transfer models based on varable turbulent Prandtl number for lqud metal flows by usng both momentum and energy transport turbulence equatons. Smple geometrcal cases have been nvestgated by usng four turbulence models, namely the smple eddy dffusvty (SED), κ-ε-κ t -ε t, κ-ω-κ t -ε t and κ-ω-κ t -ω t turbulence models developed for turbulent flow feld rght up to the wall wthout the use of wall functons. In the LFR desgn wth

11 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals 73 defne a specfc turbulent Prandtl number for any dfferent geometry, but the model tself s capable of computng a correct one. For a large range of forced flows a four parameter turbulence model s a powerful tool for predctng heat transfer n flows wth large velocty/thermal feld dssmlartes. The use of a κ-ω formulaton allows mplementng a more robust and stable solver and smulatons wth ths model n more complex geometres wll be performed and reported n future wors. Fg. 9 Asymptotc Nusselt number Nu as a functon of the Peclet number Pe for the plane channel geometry and for the cylnder ppe heated wth constant heat flux: Emprcal Krllov correlaton (Ky) and values obtaned wth the κ-ω-κ t -ω t model. lqud metal coolant CFD computatons are usually performed wth smple eddy dffusvty models by mposng a constant turbulent Prandtl number. Ths model s nown to be napproprate n many geometrcal confguratons for non-unty Prandtl fluds, as we have shown n ths wor. By usng these smple eddy dffusvty models all the CFD smulatons need an expermental verfcaton to defne the approprate turbulent Prandtl number. The results we have obtaned wth the four parameter turbulence models show how a better match wth experments can be attaned usng a more complex turbulence model le the ones we have presented. Usng these models there s no need to References [1] X. Cheng, N. Ta, Investgaton on turbulent heat transfer to lead-bsmuth eutectc flows n crcular tubes for nuclear applcatons, Nuclear Engneerng and Desgn 36 (4) (006) [] B. Deng. W. Wu, S X, A near-wall two-equaton heat transfer model for wall turbulent flow, Int. J. of Heat and Mass Transfer 44 (001) [3] R.Z. Mehran, B.T. Farzad, Implct algebrac model for predctng turbulent heatflux n flm coolng flow, Int. J. Numer. Meth. Fluds 64 (010) [4] Y. Nagano, M. Shmada, Development of a two equaton heat transfer model based on drect smulatons of turbulent flows wth dfferent Prandtl numbers, Phys. Fluds 8 (1996) [5] K. Abe, T. Kondoh, Y. Nagano, A new turbulence model for predctng flud flow and heat transfer n separatng and reattachng flows II. Thermal feld calculatons, Int. J. Heat Mass Transfer 38 (8) (1995) [6] S. Manservs, F. Menghn, A CFD four parameter heat transfer turbulence model for engneerng applcatons n heavy lqud metals, Internatonal Journal of Heat and Mass Transfer 69 (014) [7] H. Hattor, Y. Nagano, M. Tagawa, Analyss of turbulent heat transfer under varous thermal condtons wth two-equaton models, n: W. Rod, F. Martell (Eds.), Engneerng Turbulence Modellng and Experments, 1993, pp [8] B.E. Launder, G.J. Reece, W. Rod, Progress n the development of a Reynolds-stress turbulence closure, Journal of Flud Mechancs 37 (3) (1975) [9] C.B. Hwang, C.A Ln, A low Reynolds number two-equaton κ t -ε t model to predct thermal feld, Int. J. of Heat and Mass Transfer 4 (1999) [10] H. Kawamura, H. Abe, Y. Matsuo, DNS of turbulent heat transfer n channel flow wth respect to Reynolds and Prandtl number effects, Int. J. of Heat and Flud Flow 0 (1999)

12 74 Four Parameter Heat Transfer Turbulence Models for Heavy Lqud Metals [11] S. Manservs, F. Menghn, Trangular rod bundle smulatons of a CFD κ-ε-κ t -ε t turbulence model for heavy lqud metals, Nuclear Engneerng and Desgn 73 (014) [1] J. Paco, K. Ltfn, A. Batta, M. Velleber, A. Class, H. Doolaard, F. Roelofs, S. Manservs, F. Menghn, M. Böttcher, Heat transfer to lqud metals n a hexagonal rod bundle wth grd spacers: Expermental and smulaton results, Nuclear Engneerng and Desgn, 014, dx.do.org/ /j.nucengdes (n Press)