Chapter 10 LOW PRANDTL NUMBER THERMAL-HYDRAULICS*

Chapter LOW PRANDTL NUMBER THERMAL-HYDRAULICS*. Introdction This chapter is an introdction into the field of momentm and heat transfer in lo Prandtl nmber flids. In order to read this chapter a basic knoledge in flid dynamics and thermodynamics is reqired. To limit the content not all formlas for technical configrations are listed; therefore mch better literatre sorces are available in engineering libraries or other specific sorces. The first section illminates the specific characteristics of the liqid metals, hich represent the largest class of lo Prandtl nmber flids. Rather specific lo Prandtl nmber flids like ferromagnetic flids or metal/oil sspensions are not considered in this contet. It is clearly dedicated to the single phase thermal-hydralics of liqid metals obeying the Netonian la. The terminology sed in this report as ell as the conservation eqations are part of Chapter. In the sections three and for are restricted to the laminar momentm and heat transfer althogh they are not of major importance in technical applications they act in many sitations as the pper or loer limit appearing in a technical set-p. Here, first the momentm transfer is described, becase the convective heat transport is alays copled ith the simltaneosly appearing momentm transport. Analytic soltions like for tbe or plate flo as ell as self-similar soltions of the bondary layer eqations and copled approimate soltions are sketched. Especially the latter can be sed to perform technically relevant problem soltions. For some technical configrations formlas or literatre sorces are given. Both paragraphs deal solely ith the technically most relevant case, namely the forced convective flo, in hich the flo is controlled by a shaping of the dct or by a spplied pressre gradient. The sections five and si deal ith the trblent momentm and energy transport throgh channels and dcts. De to the importance of this field most of the space is dedicated to it. Natral or boyant convection are in many applications sperposed to the forced convective thermal energy transport phenomena. These types of flos are called mied convective flos and are analytically not accessible and even a nmerical treatment reqires a lot of effort. Hence, ithin the section si the athor tries to indicate for hich dimensionless parameter set a forced trblent thermal energy transport eists and here the transition region starts. Boyant or natral convection both for the laminar and the trblent energy echange are not considered in this contet. Althogh to-phase flos as ell as free srface flos gain more and more importance in technical applications this rather specific topic is also not treated in the frameork of this collection. * Chapter lead: Robert Stieglitz (FZK, Germany). Thanks are de to the many researchers ithin the FZK (in epressis Dr. G. Grötzbach, Dr. U. Müller, Dr. H. Hoffmann, Prof. K. Rehme) and from otside (Prof. V. Sobolev, Dr. C.B. Reed) ho helped by providing literatre or alloing access to rare papers, and to Dr. Takizka for the helpfl discssion concerning this chapter. 399

. Specific featres of liqid metals Liqid metals are considered in many nclear and non-nclear processes associated ith thermal-hydralic aspects. In the nclear field liqid metals are both sed in fission and fsion concept stdies. While in the nclear fsion lithim or lithim alloys allo to merge the fel generation problem ith the heat removal from the fsion reaction, see e.g. [Malang, et al., 99] in the nclear fission the se of liqid metals is mch broader. In the fast breeder concept for instance sodim is sed as coolant hile in neer breeder concepts often lead or lead alloys are sed, since they ensre compared to sodim in any case that no positive void coefficient appears. A comprehensive stdy on the fel and coolant aspects as ell as their advantages and disadvantages may be taken from [David, 5]. Also in the non-nclear energy generation liqid metals are sed as heat transfer medim in solar plants, here the snlight is reflected by nmeros mirrors onto a heat echanger operated ith liqid metals, see [Benemann, 996]. In refinery and casting processes of metals like steel, copper, tin, alminim etc., as ell as glasses, hich behave in the molten condition similar like liqid metals, the same heat transfer problems appear as in the poer generation field. The main difference beteen the metals and the other media is that they have a significantly higher thermal condctivity O(W/mK) and loer specific heat capacity c p [J/(kgK)]. In case of the heavy liqid metals like lead, lead-alloys or mercry often the kinematic viscosity Q (m /s) is considerably smaller than that of e.g. air or ater. The thermal condctivity combined ith the specific heat capacity can be compressed in a characteristic nmber the so-called Prandtl nmber Pr. The Prandtl nmber is an essential non-dimensional parameter in convective heat transfer problems. The physical sense of the Prandtl nmber is that it eights the transport coefficients of momentm to that of thermal energy. Ths, the Prandtl nmber describes the ratio of momentm diffsion to the thermal transport in the flid and it is defined as: Pr U Q c O p Q N ith N O U c p (.) here c p is the specific heat capacity, U the density (kg/m 3 ) and O the thermal condctivity. N is often called in tetbooks the temperatre condctivity or thermal diffsivity. In contrast to gases or light liqid the heat capacity hardly depends on the pressre. While engine oils have sally Prandtl nmber of the order O( - 6 ) and conventional media like air or ater reveal an order of O() Prandtl nmber the liqid metals ehibits significantly smaller Prandtl nmbers in the range Pr = 3 -. Table.. shos the Prandtl nmber of e.g. mercry compared to air ater or engine oil for different temperatres. There eists no Netonian flid in the range beteen the liqid metals and the gases for e.g..5 < Pr <.5. Table... Typical Prandtl nmber for different flids from [Beitz & Küttner, 986] Temperatre Mercry Air Water Engine oil qc.88.7 3.6 4.7. 4 qc.49.7 7... 4 qc.6.7.74.6. Let s consider a flo of a Netonian flid ith the temperatre T over a semi-infinite plate ith the constant temperatre T. In Figre.. sch a configration is illstrated for flids ith Pr <<, Pr ~ and Pr >>. Case (a) refers to liqid metals, hich have good condctivity and a lo viscosity, it is Pr <<. Here the thickness of the viscos bondary layer G v is negligibly small and to 4

Figre... Illstration of the inflence of the Prandtl nmber on the magnitde of the viscos and thermal bondary layers in a to-dimensional flo over plate ith the constant all temperatre T y T y T y G T G G G th G v G th G v G v G th T T T (a) (b) (c) G v G th G v ag th Pr<< Pr~ G v!!g th Pr>> estimate the temperatre bondary layer the velocity profile can be replaced by G (). In gas or ater flos the thickness of the thermal and the viscos bondary layer are of the same order of magnitde, it is Pr ~. Liqid like cool ater (<qc) and especially oils have poor thermal condctivity bt their viscosity is relatively large, as shon in case (c). Here the viscos bondary determines almost the hole flo field. In forced convective systems nder laminar flo conditions, moleclar condction of heat epressed by the Prandtl nmber controls the thermal energy transfer process, irrespective of hether the coolant is a liqid metal or any other Netonian flid. Hence, there is no fndamental difference beteen the thermal behavior of the three types of flids described above nder these conditions. And accordingly non-dimensional correlations developed to describe the heat transfer performance can be generally applied eqally ell to liqid metals in spite of their lo Prandtl nmber. Under trblent flo conditions, hoever, eddy condction of heat becomes important and the process of heat transfer is determined by both moleclar and eddy condction over the varios flo regions in the flid stream. While in ordinary flids like air and ater moleclar condction is only of importance near the all (in the viscos sblayer), in a liqid metal the magnitde of the moleclar condctivity is of the same order as that of the eddy condctivity. Ths, the moleclar condction is felt by the flo not only in the bondary layer bt also to a significant etend in the trblent core of the flid stream. Therefore, the fndamental details of the heat transfer mechanism in liqid metals differ significantly from those observed for instance in air and as a reslt relationships (or correlations) developed to determine the heat transfer coefficients for trblent flos obtained in those flids cannot be sed. A frther conseqence of the even greater importance of moleclar condction of thermal energy in trblent liqid metal flo is that the concept of the hydralic diameter cannot be sed so freely to correlate heat transfer data from systems hich differ in configration bt retain a similar basic flo pattern. As an eample in Pr ~ flids basic heat transfer data for flo throgh circlar pipes can be sed to predict Nsselt nmbers (N) for flo parallel to a rod bndle by evalating the hydralic diameter for the latter and sing this in the non-dimensional correlations for the circlar tbe. Sch methods are fond to be invalid for liqid metal systems, and accordingly theoretical, nmerical or eperimental heat transfer relationships mst be developed to deal ith each specific configration, see [Reed, 987] or [Dyer, 976]. Also a lot of effort has been dedicated to evalate heat transfer coefficients in standard geometries in the recent years the available liqid metal heat transfer data sho qite a bit of scatter. Several phenomena have been proposed to eplain the scatter and the corresponding lack of correlation ith 4

nmerical predictions or theoretical approimations. They inclde: non-etting or partial etting of the flid-solid interface, gas entrainment, the possibility of oide formation or other srface contaminants, and mied convection effects. Especially, the latter three inflences the heat transfer considerably and by tracking nmeros literatre sorces pblished in the past decades it trned ot that they are mostly responsible for the eistence of the large scatter. In this contet it shold be mentioned that for many liqid metal eperiments a detailed description of the thermal and viscos bondary conditions is incomplete, hich makes it difficlt to jdge abot the applicability of the obtained data to dra estimation to more general geometries. And, even if great care is take on the bondary conditions in a pre- and post-test analyses of the eperiment, in the heated state asymmetries in the set-p may appear leading to a co-eistence beteen mied and forced convection sggesting different heat transfer coefficients than nder clean conditions, see [Lefhalm, et al., 4]. Folloing an etensive amont of eperimental investigations on the effects of etting on the liqid metal heat transfer a general consenss has been reached on the sbject. This is that etting or lack of etting, in or of itself, does not significantly affect liqid metal heat transfer. Hoever, non-etting combinations of liqid metals and solid srfaces can sffer more readily from gas entrainment problems and at elevated temperatres of oidation at the solid liqid interface; imprities and particles can more easily become trapped at a non-etting solid-liqid interface, ths redcing heat transfer. Hence care shold be taken to avoid these problems in system designs. Finally, in liqid metal systems, niform all temperatre bondary conditions (althogh difficlt to obtain in the eperiment) yield loer Nsselt nmbers than constant all heat fl bondary conditions for the same Peclet nmber. This is in contrast to Pr ~ -flids, in hich the to bondary conditions make only a little difference in the Nsselt nmber. Related to the heavy liqid metals for many thermal-hydralic configrations often no Nsselt nmber correlation eists. Hoever, a lot of eperiments in rather generic geometries ere condcted for the sodim cooled fast breeder and in the contet of the fsion engineering commnity sing alkali metals as operation flid. In many cases the Prandtl nmber of the individal flids is close to the heavy liqid metals as shon in Figre... Nevertheless, great care has to be taken on the validity of the chosen heat transfer correlation even if the Prandtl nmber matches the heavy liqid metal considered, becase the viscosity of the heavy liqid metals is considerably smaller than that of the alkali metals. Figre... Moleclar Prandtl nmber as a fnction of temperatre in [qc] for different flids The thermophysical data for lead and lead-bismth are taken from this book, hile the data for sodim, the etectic sodim-potassim alloy (Na K 78 ) are from [Fost, 97], mercry (Hg) from [Lyon, 95], lithim from [Addison, 984], lead-lithim (Pb 83 Li 7 ) from [Schlz, 986] and [Smith, et al., 984] and gallim-indim-tin from [Barleon, et al., 996].6.4 Akali metals Li Na Na K 78 heavy liqid metals Pb Pb 45 Bi 55 Pb 83 Li 7 Hg Ga 68 In Sn Pr. 3 4 5 6 7 8 9 T [ C] 4

By mltiplying the hydralic Reynolds nmber Re ith the Prandtl nmber Pr the Peclet nmber Pe is obtained hich can be conceived as the ratio of the convective heat transport verss the moleclar condction. Both the Reynolds nmber and the Peclet nmber are defined as: Re d Q and Pe Re Pr d N (.) here is a characteristic velocity of the flo configration considered and d a characteristic length scale of the problem. The Nsselt nmber correlations eperimentally obtained can only then sed for a heat transfer assessment or a transfer from an alkali metal to the heavy liqid metals if both Prandtl nmber and the valid Peclet nmber regime coincide..3 The conservation eqations Within the contet of this chapter e restrict orselves to the consideration of flos in channels or closed cavities ithot the entrainment of an additional mass sorce or a mass transfer beteen different species. Consider the flo of a single phase and single component flid in a fied control volme, then hole mass flo entering the volme mst leave the volme. This is epressed by the continity eqation, hich describes the conservation of mass. Becase liqid metals belo their boiling point are almost incompressible (U = const.) the continity eqation for them can be epressed by or vectori al v y z (.3) here & is the velocity vector composed of the velocity components (,v,) in, y, and z-direction respectively and ( ) is the divergence operator. The dynamic behavior of flid motion is governed by a set of eqations called the momentm eqations or the eqation of motion. The derivation considering a defined control volme is similar to that of the continity eqations. The momentm eqations can be ritten in the folloing form: U v t y z v v v v U v t y z U v t y z U f U f y U f V W y z W V yy y z W y y W yz y W z z W zy z V z zz (.4) here f = (f, f y, f z ) is a body force being of gravitational, electrical or magnetic origin, V is the stress normal to the srface and W the shear stress tangential to the sides of the control volme. It has been fond eperimentally that, to a high degree of accracy, stresses in many flids are related linearly to the rates of strain (derivatives of the velocity components). It can be shon, see e.g. [Lamb, 945] or [Schlichting, 979] that for Netonian flids the epressions are: 43

44 P W W P W W P W W P P V P P V P P V z v y ; z ; y v ; z p ; y v p ; p zy yz z z y y zz yy 3 3 3 (.5) here P = U Q is the dynamic viscosity of the flid and p the pressre. Sbstitting the Eq. (.5) into (.4) yields the fll momentm eqations, hich are called the Navier-Stokes eqations. Nearly all analytical investigations involving viscos flids are based on them. They are general in the sense that they are valid for compressible Netonian flids ith varying viscosity. When the density and the viscosity are constant that is, hen the flid is incompressible and the temperatre variations are small the Navier-Stokes eqations simplify to: p f dt d z y z p f z y v t ; z v y v v y p f z v y v v v t v ; z y p f z y v t z y notation vectorial in the or Q U P U U P U U P U U (.6) here =. is the Laplacian operator given by = (/ + /y + /z ). Any flid flo problem hich involves the determination of the velocity components and the pressre distribtion as a fnction of the spatial co-ordinates and the time reqires simltaneosly the soltion of the continity eqation and the Navier-Stokes eqations nder the specific bondary and initial conditions. Althogh these set of eqations are in most cases too comple to be solved analytically they may be solved by nmerical means. Nevertheless, there eist some cases here the natre of the flo is sch that they can be simplified considerably for an analytical soltion. A similar approach as for momentm and continity eqation derivation can be applied to dedce the energy eqations. This derivation implies the first la of thermodynamics, hich coples the thermal energy ith the ork done by the system and the total internal energy of the control volme. The complete derivation of the energy eqation is described in the tetbooks by [Keenan, 94], [Van Vylen and Sonntag, 979] or [Jischa, 98]. The thermal energy eqation finally reads to: W W W V V V O U z z v y y v z y v T dt d z yz y zz yy (.7)

in hich O is the thermal condctivity of the flid, the internal energy the and T the temperatre. The internal energy is liked to the flid enthalpy i by I = + p/u. The operator d/dt denotes the total derivative. If the relations for stress and strain acting pon a flid element in a Netonian flid [Eq. (.5)] are sbstitted into Eq. (.7), it redces to: ) ª ««v y d U O T p P) ith (.8) dt º v v» z y y z» z 3 ¼ here is the internal energy of the flid per nit mass and ) is called the dissipation fnction. The first term on the right hand side represents the net rate of heat condction to the flid particle per control volme, the second term is the rate of reversible ork done on the control volme and the last term is the rate at hich viscos forces do irreversible ork in form of e.g. viscos dissipation or viscos heating per nit volme. If one considers an incompressible flid ith d = c p. dt the energy eqation takes the form: ) ª ««v y dt (.9) U c p O T P) ith dt º v v» z» y y z z ¼ Additionally, if the thermal condctivity O is constant it redces to: dt O T dt U c P U p c p ) (.) The continity, Navier-Stokes and energy eqations provide a comprehensive description of the thermal energy transfer in a flo field. These eqations, hoever, present insrmontable mathematical difficlties de to the nmber of eqations to be simltaneosly satisfied and the presence of non-linear terms sch as /. Becase of these non-linearities, the sperposition principle is not applicable and comple flos may not be componded from simple flo as e.g. possible for potential flos, see [Schlichting and Trckenbrodt, 96]. Eact soltions ere obtained for some simple cases, here the nonlinear terms are either small (approimate soltions) or identically zero. This class of soltion appears in slo motion or creeping flos and is important for the theory of lbrication. In most practical heavy liqid metal applications the nonlinear terms are most often of greater magnitde than the other terms in the Navier-Stokes eqations. The hydralic Reynolds nmber defined in Eq. (.) is a dimensionless qantity hich measres the ratio of the inertia effects to the viscos effects in a flid. Creeping flos are therefore characterised by small Reynolds nmbers, hereas in most practical flos the Reynolds nmber is far above nity. Finally, to important observations are orth mentioning. First, the velocity and temperatre fields ill be copled if the flid has a temperatre dependent density and/or viscosity. Secondly the temperatre field can become similar to the field nder certain conditions. If p =, ) = and f = and if Pr = then the soltions for the velocity and temperatre fields are similar, provided that the bondary conditions are also similar. Bt as shon in Section. for any liqid metal flo the moleclar Prandtl nmber is far belo nity so that a similarity of both fields does practically not appear. 45

.4 Laminar momentm echange This sbsection is dedicated to describe the basic concepts to treat the laminar momentm echange. The main aim is to obtain the local friction coefficient c() of a flo in a geometry and the main ideas for the simplification of the Navier-Stokes eqations in terms of an asymptotic approach. The reslts obtained in this chapter are necessary to elaborate the difference beteen the laminar and the trblent flo and they form the basis for the nderstanding of the heat transfer phenomena appearing in both types of flos. The laminar momentm echange of a steady to-dimensional flo of an incompressible flid ith a constant kinematic viscosity Q is governed by the folloing eqations: v y, (.) v y v v v y p Q U y p v v P U y y ; This represents a system of partial non-linear differential eqations of the elliptic type. A general soltion does not eist and there are only a fe eact soltions nder rather restrictive bondary conditions like the channel or dct flo. Fortnately, most of the laminar momentm echange processes can be simplified considerably. For large Reynolds nmbers nearly all echange processes take place in a thin layer, the so-called bondary layer. The bondary layer approimation is not restricted to flos over solid alls, also free jets or ake flos ehibit bondary layer character..4. Channel or tbe flo Consider a steady planar flly developed flo in a dct as shon in Figre.4.. Here flly developed means that the velocity component in -direction does not change. Becase / = also v/y =, hich means that the v-component of the velocity is constant. Eclding sction or bloing immediately yields that v =. For Eqs. (.)b, c the convective terms on the left side diminish and from the force balance (.)c only the epression p/y = remains. Ths, the pressre p depends only from the flo direction and the force balance in flo direction reads to: dp d d dw (.) P dy dy and since is only a fnction of y the partial derivative can be replaced by the simple derivative d. From the stress tensor only W = P/y remains. An integration immediately yields that the shear stress is linear and de to symmetry, hich reqires W(y = H) = one obtains: W dp d H (.3) 46

This correlates the all shear stress ith the pressre gradient. In dimensionless form it reads to: y W W y H (.4) With W = P/y and a frther integration the velocity distribtion is obtained. Setting (y = H) = ma yields: ma W H P dp d H ' ph (.5) P PL here 'p/l=-dp/d is the pressre drop along the channel length L. Finally the dimensionless velocity distribtion reads to: ma y H y H (.6) hich is the classical parabolic Hagen-Poiseille profile. Figre.4.. Shear stress and velocity distribtion of a flly developed channel flo for a planar dct and a circlar pipe Wy (y) r Wr H y R W W The same reslt is obtained for the flly developed laminar circlar dct flo sing the same procedre. Withot describing the fll details the folloing set is obtained for the configration shon in Figre.4.: r W W ma r R ; ' pr 4PL ' p 4PL r R r ; ma r R ; (.7) Here, R is the radis of the pipe and r the radial co-ordinate. The factor ½ is eplained by the fact that in a tbe flo compared to a channel flo a ratio the cross-section on hich the pressre acts is tice as large as the srface on hich the viscos forces act to compensate the pressre. The flo rate V in a circlar tbe can be calclated to: 47

V r R S' pr ³ S r r dr 8PL r 4 (.8) It is important to note the dependencies V ~ 'p/l and V ~ R 4. For the same applied pressre gradient an increase of the diameter by % yields an increase of the flo rate of 46%. If one defines a mean velocity by applying = V/A one obtains = ½ ma. For engineering prposes the pressre loss is one of the most interesting parameters. Taking V = SR and sing the diameter D = R, one gets for the pressre drop over a dct length of L: ' p U L D 64 Re (.9) Defining a friction coefficient c L in the ay 'p = U/ L/D one obtains the friction la for a tbe flo epressed by: 64 (.) c L Re Herein Re is the hydralic Reynolds nmber defined as Re = D/Q, hich can be conceived as a force ratio of the inertial forces verss the viscos ones. This shos simply, ho the pressre drop and the friction coefficient depend on the Reynolds nmber. The Reynolds nmber is in all momentm echange processes the only appearing variable. Ths, for all laminar flos the Reynolds nmber correlations of any other liqid can be sed and transferred to heavy liqid metals..4. Bondary layer eqations The basis for the derivation of the bondary layer eqations is the eqation set (.)a-c. Figre.4. shos the sed local co-ordinate system and the observed velocity distribtion. The order of thickness of the bondary layer is a priori not knon; the folloing assessment is aimed to give an estimate on this magnitde. Figre.4.. Co-ordinate system and observed velocity distribtion y () oter flo G (,y) bondary layer flo First e introdce the dimensionless variables in the folloing ay: L ; y G y ; p U ; p (.) 48

hich are chosen in sch a ay that they are of order one in magnitde O(). In a net step the length L and the mean velocity of the oter flo are taken as reference qantities. Frther on there is: v v ; U U U (becase U const.) (.) Introdcing these variables into the continity Eq. (.)a the folloing declaration can be made: v ~ G L (.3) becase c, c and yc are of O(). Introdcing the variables into the Navier-Stokes Eq. (.)b in -direction leads to the reslt that as determining parameter the Reynolds nmber appears, hich characterises viscos flos. Retrning to the assessment and postlating that the viscos terms are of the same order of magnitde as the inertial terms and the pressre force immediately yields that the thickness of the bondary layer scales as: G (.4) ~ L Re A similar insertion of the non-dimensional scaling into Eq. (.)c (in y-direction) ehibits the assertion that for Re >> the derivative of the pressre in y-direction diminishes. This means that the pressre normal to the all does not change ithin the bondary layer; it is rather given by the vale from the inviscid oter potential flo. The pressre p in bondary layer theory is not an nknon qantity it is rather given as a bondary condition. All these order of magnitde estimates lead to the bondary layer eqations, hich ere initially ritten by Ldig Prandtl in 94. v, (.5) y dp( ) v Q y U d y These are to eqations to determine the nknons, v = f(,y). Becase of Bernolli s eqation p() + U/. () = const. the pressre and the velocity gradient of the inviscid oter flo are copled. The bondary conditions of the eqation system are: y y : G : v ( ), (.6) The system of the bondary layer eqations is parabolic and represents an initial bondary problem. De to the simplification of the viscos terms diffsion processes are only active in the y-direction. There is no pstream fl of information and all can be calclated straight forards donstream. This is an enormos simplification compared to the elliptic Navier-Stokes eqations, for hich a simltaneos soltion for all domains is reqired. There are some eact and approimate soltions of the bondary layer eqations hich can be fond in [Schlichting, 979] or [Jischa, 98]. 49

.4.3 Smmary and comments The main aim to treat the laminar momentm eqations is to determine the local c L () and/or the global friction coefficient c W. Many tetbooks sho that these vales can be given in a form c L () = f(re m ) or c W = f(re n L). Herein, Re is the local Reynolds nmber and Re L the Reynolds nmber bilt ith the characteristic dimension of the body. For a dct or a tbe flo n = - is obtained. The friction coefficient does not change in flo direction for a flly developed flo. This is different to a bondary layer flo along a plate, becase there the bondary layer thickness gros along the flo direction hile the local friction coefficient decreases ith. In this case the local friction coefficient yields an eponent m = -/ and the total reslting friction coefficient ehibits a coefficient n = -/. Crcial in this contet is that the only appearing dimensionless qantity is the Reynolds nmber, hich can be conceived as the ratio of the inertial forces verss the viscos ones. For a flly developed flo the inertial force is replaced by the driving pressre gradient. One can distingish beteen different approaches to describe laminar flos: Analytic soltions of the Navier-Stokes eqations can be only obtained for a fe rather specific eceptions like the plane dct or the circlar pipe flo. Nmeric soltions of the Navier-Stokes eqations are etremely time consming. Here, a system of non-linear partial differential eqations of the elliptic type mst be solved. This is a bondary vale problem and no consective methods like for parabolic partial differential eqations can be applied. For some technically interesting cases like the developing flo into a tbe or the flo near the leading edge of a plate, calclations ere performed. In the engineering practice, hoever, these soltions are of minor importance. In the limiting case Reof the Navier-Stokes eqations mtate into the bondary layer eqations. In many practically important cases the Reynolds nmber is large enogh to apply the bondary layer eqation. An eception in this contet is the leading edge problem for hich the prereqisite dg/d << and / << /y is not flfilled. Eact soltions of the bondary layer eqation eist only for specific oter flos acting as bondary condition. The terminology eact means that this is not an analytic bt an arbitrarily eact nmeric soltion of an ordinary differential eqation. In case of the self-similar soltions the individal velocity profile does not alter its shape in flo direction and it can by an appropriate co-ordinate transformation redced to the similar profile. For an arbitrary oter flo the velocity profile changes its shape in flo direction; there eist no self-similar soltions and the bondary layer eqation can only be solved nmerically. This can be performed in to ays. Nmeric soltions of the bondary layer eqations eist for nmeros cases. By means of difference methods the parabolic bondary layers eqation can be nmerically integrated as a initial bondary problem sing a consective approach. Becase of stability reasons implicit methods althogh they reqire more effort are preferred compared to simpler eplicit schemes. The problems arising sing the individal techniqes are in detailed discssed in the book of Cebeci and Bradsha (977). Integral schemes to solve the bondary layer eqations are simple to treat and practically most often sed. Often sed are the schemes by Pohlhasen nd v. Karman. The effort reqired for compting time as ell as the memory consmption lo. Also the soltion of to copled 4

non-linear ordinary differential eqations is not problematic. These integral methods are still in se to calclate especially trblent bondary layers. An ehastive discssion on the different types of nmeric approaches may be taken in the monograph by Walz [966]. Despite the ndeniable advantages of the integral methods one crcial disadvantage eists; the qality of the achieved reslts depends on the assmption of the chosen profile. The ser mst kno a lot of properties of the soltion in order to formlate an appropriate profile. Empirical soltions mst be sed in all cases here comple geometries do not allo a direct nmerical soltion. Finally, for completeness to problem circles shold be shortly mentioned. This is the higher order bondary layer theory. This is a pertrbation theory hich is called the matched asymptotic epansions. It is a frther development of Prandtl s bondary layer concept, here the bondary layer eqation is the first order eqation in the hierarchy. Effects of higher order like inclinations, etc., can be treated in this kind of concept. An overvie on the different model ideas is given in the books by [Van Dyke, 975] or [Cole, 968]. The not treated rather comple isse are three-dimensional bondary layers, hich appear in many technical applications. Here, the gracios reader shold refer to the tet books, e.g. [Schlichting, 979] or [White, 974]..5 Laminar energy echange In a moving flid energy and momentm are echanged if: a) thermal energy is spplied to or removed from the flid; b) the kinetic energy of the flo is of the order of the inner energy so that by dissipation the temperatre of the flid is increased. In Case (a) one distingishes beteen forced and free convective flos. In forced convective flos an eternal forcing bean means of a spplied pressre gradient or a given oter flo is present, hile in free or natral convection flos boyancy forces cased by density differences (hich are originating from temperatre differences) are the driving sorce of the flo. Restricting orselves to steady flos of single component Netonian flids one obtains the energy eqation in the form described in Eq. (.8) (see Section.3). The set of eqations to be solved for the laminar energy echange reads to: v p ; U v P Ug ; y y y v v p v v U v P Ug y; y y y T T T T p p U c p v O v ) y y y (.7) here the heat condctivity O is assmed to be constant, g is the gravity vector in the form g = (g, g y ) and ) is the dissipation fnction defined in Eq. (.9)b. This section deals ith the laminar flo and forced convection heat transfer characteristics of a variety of dcts interesting to heating and cooling devices in nclear engineering. The reslts presented here are applicable to straight dcts ith aially nchanging cross-sections. Also the dct 4

alls are considered smooth, non-poros, rigid, stationary and etted. Frthermore, the dct alls are assmed to be niformly thin, so that the temperatre distribtion ithin the solid alls has negligible inflence on the convective heat transfer in the floing flid. Ths this section covers the steady, incompressible laminar flo of a constant property Netonian flid. All forms of body forces are neglected; moreover, the effects of natral convection, phase change, mass transfer or chemical reactions are omitted. A complete representation of the laminar heat transfer correlations and a detailed description of the methods can be fond in the monographs by [Shah and London, 978]..5. Types of laminar dct flo For types of laminar dct flos eist; namely, flly developed, hydrodynamically developing, thermally developing, and simltaneosly developing. The latter means that the flo is at the same time hydrodynamically developing and thermally developing. A brief description of these flos is given here ith an aid in Figre.5., hich depicts a flid ith the niform velocity and temperatre T entering a dct of an arbitrary cross-section at =. Figre.5.. Types of laminar dct flo for constant all temperatre bondary condition (a) Hydrodynamically developing flo folloed by thermally developing (and hydrodynamically developed) flo (b) Simltaneosly developing flo in liqid metals (Pr << ) Solid lines denote the velocity distribtion hile the dotted grey lines represent the temperatre profiles = T=T =(,y,z) T=T T=T (,y,z) =(y,z) =(y,z) T=T (y,z) y y z G G th T =T T <T T <T = hydrodynamically developing flo =l hy thermally developing flo =l th flly developed flo (a) = T=T T=T (,y,z) =(,y,z) =(y,z) T=T (y,z) y y z G th G = T <T simltaneosly developing flo for Pr< l th <l hy (b) =l d T <T flly developed flo Referring to Figre.5., sppose that the temperatre of the dct all is kept at the entering flid temperatre T and there is no generation or dissipation of heat ithin the flid. In this case the flid eperiences no gain or loss of thermal energy. In sch a case of an isothermal flo, the effect of viscosity spreads across the dcts cross-section commencing at =. The hydrodynamic bondary layer develops according to Prandtl s bondary layer theory (see Sbsection.4.) and its thickness G 4

gros proportional to Re /. The bondary layer separates the flo field in to domains; a viscos region near the all and an essentially inviscid region arond the dcts centreline. At = l hy the viscos effects have completely spread across the holes dcts cross-section. The domain d d l hy is called the hydrodynamic entrance region and the flo in this domain is called the hydrodynamically developing flo. In the entrance region the velocity depends on all three spatial co-ordinates, hile for > l hy the velocity profile is independent of the aial co-ordinate. After the flo becomes hydrodynamically developed consider that the dct all temperatre is dropped compared to the entering flid T < T. In this case the thermal effects diffse gradally from the dct all commencing at =l hy. The etent of the thermal diffsion is denoted by the thickness of the thermal bondary layer G th, hich gros along the aial co-ordinate. A similarity analysis based on the bondary layer approimation approach immediately reveals that the thermal bondary layer gros as G th () ~ (Re. Pr) /. Moreover, applying the bondary layer approach the thermal bondary layer also divides the flo field in to regions; a heat affected region close to the all and an naffected domain in the dcts centre. A reslt of the dimensional analysis is also that the ration of the thermal bondary compared to the viscos one scales as: G th (.8) ~ G Pr At = l th the thermal effects have spread throghot the hole dcts cross-section and beyond this point the flo is called thermally developed. The region l hy d d l th is termed the thermal entrance region. Here the temperatre varies ith all three spatial co-ordinates. The simltaneosly developing flo is displayed in the loer graph of Figre.5. for a liqid metal. In this case the viscos and the thermal effects diffse simltaneosly from the dct all toards the dcts centre commencing at =. The essential parameter here is the Prandtl nmber, hich denotes the ratio of the kinematic viscosity to thermal diffsivity. The kinematic viscosity is the diffsion rate for momentm (velocity) in the same sense that the thermal diffsivity is the diffsion rate for heat (temperatre). If Pr =, the viscos and thermal diffse throgh the flid at the same rate. This eqality of diffsion rates does not garantee that the viscos and thermal bondary layers in close dct flos ill be of the same thickness at defined aial position. The reason for this parado lies in the fact that ith Pr =, the applicable momentm and energy differential eqations do not become analogos. As depicted in the Figre.5.(b), ithin the region d d l d viscos and thermal effects spread simltaneosly toards the dct centre. Accordingly, this region is referred to as the combined entrance region. It is obvios that the length l d depends on the Prandtl nmber. In this region both velocity and temperatre depend on all three space co-ordinates. Only for > l d hen the flo is flly developed the temperatre and velocity become aially invariant and depend only on y and z, i.e. = (,y) and T = T(,y)..5. Flid flo and heat transfer parameters The flid flo characteristics of all dct flos is epressed in terms of certain hydralic parameters. For the hydrodynamically developing flo, the dimensional aial distance + is defined as: Re d h (.9) here d h is the hydralic diameter defined by d h = 4A/P ith A the dcts cross-section and P the etted perimeter. The hydrodynamic entrance length l hy is defined as the aial distance reqired to attain 99% of the ltimate flly developed maimm velocity hen the entering flo is niform. The dimensionless hydrodynamic entrance length is epressed by l hy + = l hy /(d h. Re). 43

The flid blk mean temperatre, also referred to as the miing cp or flo average temperatre T m is defined as: T m (.3) T da A ³ A The circmferentially averaged bt aially local heat transfer coefficient D is defined by: q,,m m D T T (.3) here T,m is the all mean temperatre and T m is the flid blk mean temperatre given by Eq. (.3). The heat fl q and the temperatre difference (T,m T m ) are by natre vector qantities. In the notation (.3) the direction of heat transfer is from the all to the flid, and consistently the temperatre drop is from the all to the flid. In contrast, if q represents the heat fl from the flid to the all the temperatre difference entering Eq. (.3) ill be (T m T,m ). The flo length averaged heat transfer coefficient D m is the integrated vale from = to in the ay: (.3) m d ³ D D The ratio of the convective condctance D to the pre moleclar condctance O/d h is defined as a Nsselt nmber N. The circmferentially averaged bt aially local Nsselt nmber N is defined as: N D d q d (.33) O h O T,,m h T m Ths the Nsselt nmber is nothing else than the dimensionless temperatre gradient at the all. The Nsselt nmber can also be conceived as a ratio of to different lengths, namely the ratio of the characteristic length to the local thickness of the thermal bondary layer. The mean Nsselt nmber N m based on D m in the thermal entrance region reads to: N m ³ D m d h q d (.34) N d O O,m h ' T m The epression for ('T) m cold become complicated and depend pon the thermal bondary conditions. The dimensionless aial distance * is defined as: * (.35) Pe d RePr d h h The thermal entrance length l th is defined as the aial distance reqired to achieve a vale of the local Nsselt nmber N, hich is.5 times the flly developed Nsselt nmber vale. The dimensionless thermal entrance length is epressed as l th * = l th /(d h. Pe). Also often the Stanton nmber St is sed. It describes the ratio of the heat fl transferred from the all to the enthalpy difference of the oter flo and is defined as: St D U c p q U c p ' T N RePr (.36) 44

.5.3 Thermal bondary conditions In order accrately interpret the highly sophisticated heat transfer reslts in the ensing sections, a clear nderstanding of the thermal bondary conditions imposed on the dct alls is absoltely essential. A systematic eposition of the bondary conditions is provided by [Shah and London, 978]. Here e focs on the technically most important ones. a) Uniform all temperatre T ith circmferentially and aially constant all temperatre, hich is epressed by Eq. (.37). It appears mostly in condensers or evaporators. T const. (.37) b) Convective ith aially constant all temperatre and finite thermal resistance normal to the all. It is in principle the same as a) ecept that the all thermal resistance is finite in these applications, hich can yield to an pstream heating of the flid. Especially in liqid metal heat transfer eperiments this condition mostly appears, becase they can not be directly heated de to their good electric condctivity. The formlation of this bondary condition reads to: T T y,z;t T, y,z ; (.38) T O n t D O D D e T T here T is the temperatre at the otside of the heater/cooler, T the flid/all interface temperatre, D the heat transfer coefficient at the flid/all interface, O the heat condctivity of the all, t its thickness, n the all normal nity vector and D e the heat transfer coefficient at the entrance, here the heat transfer starts. c) Constant all heat fl ith circmferentially constant all temperatre and aially constant all heat fl. This condition applies for electric resistance heating, nclear heating and heat echangers having nearly identical flid capacity rates. Hoever, this condition only applies if the all materials are thermally highly condcting, i.e. O >> O or the all is considerably thinner than the characteristic dct dimensions. This condition is formlated by: y,z ; T T ; q q (.39) d) Uniform aial and circmferential heat fl, hich is nearly the same condition as for case c), bt it refers to bondary conditions here the all material has a lo thermal condctivity and the all thickness is niform. It reads to: q const. (.4) e) Convective ith aially constant all heat fl and a finite thermal resistance normal to the all. Also this condition is nearly the same as case c) ecept for the finite thermal resistance normal to the all. Moreover, there is assmed that there is negligible heat condction along the dcts circmference. This condition is epressed by: q q T y,z; D T T O n (.4) 45

f) Finally, the condctive bondary condition ith aially constant all heat fl and finite heat condction along the alls circmference, hich is an etension of the case e). This bondary condition is described by: q q q O T n here s is the circmferential co-ordinate. O t O T s y,z; (.4) This list of bondary conditions applies both for the laminar and the trblent heat transfer phenomena and it demonstrates the necessity for a detailed description of the eperimental set-p. Only a complete description of the thermo-physical properties of the materials sed, the heater type applied and the eact geometric dimensions allos to jdge on the thermal bondary condition to be applied for the analysis of the eperiment. Apart from the choice of the correct bondary conditions it says nothing abot the qality of the performance of the eperiment. Even a laminar eperiment may be sperimposed by boyancy effects hich considerably changes the eperimental reslts..5.4 Laminar heat transfer in circlar dcts The circlar dct is the most idely sed geometry in flid flo and heat transfer devices. Accordingly, it has been analysed in detail for varios bondary conditions. Also available in literatre is a lot of information on the effects of viscos dissipation, flid aial condction, thermal energy sorces and aial momentm diffsion. Some of the reslts ill be shon ithin this sbsection..5.4. Flly developed flo The velocity profile and the leading correlations for a flly developed laminar flo in a circlar dct are elaborated already in Sbsection.4.. In this contet e refer to these eqations and sho the reslts for the different thermal bondary conditions. a) Uniform all temperatre (condition a) The temperatre distribtion in circlar dcts for non-dissipative flos in the absence of flo ork, thermal energy sorces and flid aial condction is given eactly. The epression for the flid blk mean temperatre is given by the folloing asymptotic formla applicable for * >.335, see [Bhatti, 985]: T Tm *. T T e. 8948ep( O ) ith O 743644 (.43) It shold be noted that althogh the local temperatre T is both a fnction of the radial and the aial co-ordinates the flid mean blk temperatre T m depends on the aial co-ordinate only (the dimensionless temperatre (T T)/(T T m ) is only a fnction of the radis. The Nsselt nmber of the flly developed flo can be calclated to: O (.44) N * 3. 6568 46

For flly developed flos and the thermal condition a) is the reslt is independent of the Prandtl nmber of the flid. Hoever, if the Peclet nmber Pe is smaller than, the inflence of the aial flid condction is not negligible anymore. In this parameter regime the asymptotic epressions presented by [Michelsen and Villadsen, 974] are recommended: N 4. 8654. 8346 Pe for Pe.5 3. 656794 4. 487 / Pe for Pe! 5. (.45) b) Uniform all heat fl (q = const. condition d) The temperatre distribtion in circlar dcts for non-dissipative flos in the absence of flo ork, thermal energy sorces and flid aial condction can be ritten analytically in the folloing form: T T T ª r º ª 3 r / R º T Tm O ; O Tm R q d (.46) 6 ¼ ¼ h 48 N D d h The Nsselt nmber obtained for constant all heat fl is for a flly developed laminar flo abot % larger than that for constant all heat fl. The qalitative shape of the flid blk mean temperatre distribtion the all heat fl and the all temperatre profile as a fnction of the aial co-ordinate is qalitatively depicted in Figre.5.. Figre.5.. Qalitative shape of the flid blk mean temperatre, the all temperatre and the all heat fl for the case of a constant all heat fl q = const. (condition d) left graph and a constant all temperatre T = const (condition a) right graph T T q const. T m T q T =const. q =const. T m q c) Convectively heated or cooled dct all (condition b) For this bondary condition the temperatre is assmed to be constant aially bt the dct has a finite thermal resistance normal to the all. The thermal resistance can be embedded in an eternal convective heat transfer coefficient D e, hich is inclded in the dimensionless Biot nmber Bi defined as Bi = D e d h /O. The Biot nmber can also inclde the effect of the all thermal resistance. In this case Bi = /R, here R = (O t )/(O d h ) + O/(D e d h ). The limiting cases are the constant all temperatre condition corresponding to Biof and the constant heat fl condition implying Bi =. Hence the Biot nmber can be conceived as a ratio of the thermal resistance of the all compared to that of the flid. For a constant D e [Hickman, 974] developed an asymptotic soltion for the Nsselt nmber in the form (.47). The overall mean Nsselt nmber N o,m then reads to: 47

N 48 (.47) Bi D d h qd h ; ith No,m 59 No,m N Bi O OT Tm Bi N o,m is qite insensitive to circmferential variation of Bi according to [Sparro, et al.,978]..5.4. Hydrodynamically developing flo The problem of hydrodynamic flo development has been theoretically investigated by nmeros scientists. Depending on the Reynolds nmbers varios soltions can be categorised as: a) soltions involving the bondary layer eqations valid for Reynolds nmber Re > 4; b) soltions involving the Navier-Stokes eqations for Re < 4; c) creeping flo soltion for Reo, hich hardly eist in liqid metal flos. a) Soltions involving the bondary layer simplification The varios soltions sing this approach are revieed and classified in detail by the book of [Shah and London, 978]. Among them the nmerical soltion of [Hornbeck, 964] is technically most sed. According to these reslts the dimensionless hydralic developing length l hy + and the dimensionless pressre gradient 'p * can be calclated sing:. 5 64 3. 74. 565 ; ' p 3. 74 4. * l hy (.48) b) Soltions involving the Navier-Stokes eqations Close to the dct inlet the aial momentm diffsion and the radial pressre variation are of importance. The proper acconting of these effects introdces the Reynolds nmber as a parameter in the soltion and also reqires a carefl specification of the inlet flo velocity profile. Using modern comptational flid dynamic code packages this problem can be arbitrarily eact solved for different inlet velocity distribtions. Assming a constant velocity far of the inlet the analysis by [Chen, 973] can be sed as most accrate for the hydralic developing length. It reads to: l. 6. 56 hy Re. 35 Re (.49).5.4.3 Thermally developing flo Within this sbsection a hydrodynamically flly developed flo is assmed for the velocity distribtion in the circlar dct. The temperatre profile is alloed to develop nder the individal thermal bondary conditions. a) Constant all temperatre (T = const.) Neglecting viscos dissipation, flid aial condction and thermal energy sorces the soltion of the problem is called in the literatre the Graetz problem [Graetz, 885], ho solved this configration 48

analytically. Based on the Graetz and etended Leveqe soltions, the local Nsselt nmber N can be obtained from the folloing formlas depending on the aial co-ordinate * : N *. 77 3. 7 3. 657 6. 874 3 *. 488 * ep 57. for for * * d. t. (.5) [Hasen, 943] presented the folloing correlation for the mean Nsselt nmber N m valid in the entire range of * : N m 3. 66. 668 * / 3. 4 * / 3 (.5) The error is in the range of % compared to a fll nmerical soltion. The effect of flid aial condction is negligible for Pe > 5, according to [Hennecke, 968]. Hoever, the thermal developing length l * th increases considerably from.33 for Pe = f to.5 for Pe =. The dimensionless temperatre distribtion T * as a fnction of r/r and * is depicted in Figre.5.3(a) and the mean local Nsselt nmber N and the mean Nsselt nmber N m is shon in Figre.5.4(a). b) Constant all heat fl (q = const.) Similar to the Graetz problem ith T = constant soltion, the thermal entrance length for the thermally developing flo ith q = const. is of large practical interest. The soltion obtained for this problem can be analytically epressed in terms of a series as demonstrated by [Siegel, et al., 958] or [Hs, 965]. Based on the Graetz type and Leveqe soltions also the mean local Nsselt nmber N and the mean Nsselt nmber N m can be epressed by approimation formlas, hich here derived by [Bird, et al., 96]. N * / 3. 3. * / 3. 3. 5 4. 364 8. 68 3 *. 56 * ep 4 for 5-5 for d for * * * d 5 d. 5 t. 5-5 -3-3 (.5) N m * / 3. 953 4. 364. 7 / * for for * * d.3 t.3 (.53) The error made by the approimation is abot r% compared to the eact soltion. From the analysis of [Hennecke, 968] the effect of flid aial condction is negligible for Pe > if * >.5, hich is less than for constant all temperatre. The thermal entrance or developing length l th * is fond to be.4357, see [Grigll and Tratz, 965]. c) Convectively heated or cooled dct all (condition b in Section.5.) The convective bondary condition can also be solved by means of a series epansion incorporating the Biot nmber, see [Hs, 968]. The local Nsselt nmbers as a fnction of the dimensionless distance * for different Biot nmbers are shon in Figre.5.4(b). The crves corresponding to Bi = (q = const.) and Biov (T = const.) are the same as in Figre.5.4(a). 49

Figre.5.3. Dimensionless temperatre distribtion in the thermal entrance region of a circlar dct and a thermally developing laminar flo as a fnction of the dimensionless length * for the thermal condition T = constant (a) and the condition q = constant (b) from [Grigll and Tratz, 965] T= (T -T ) (T -T e )..8.6.4. T= 8O (T -T) 3 (q d h )..8.6.4. 3 3 * 5 r/r 3 * r/r 5 (a) (b) Figre.5.4. Nsselt nmber as a fnction of the dimensionless entrance length * for a hydralically developed and thermally developing laminar flo in a circlar tbe a) Local and mean Nsselt nmbers N and N m for T = constant and q = constant b) Local Nsselt nmber N for mied convective thermal bondary conditions epressed by the Biot nmber 8 8 5 N 8 N m (q =const.) N (q =const.) N m (T =const.) N (T =const.) R N 4 6 8 Bi = Bi = Bi 8 Bi = Bi = R 5 4.364 3.657 4-5 5-4 5-3 5-5 - 5-4 5-3 5-5 - * * (a) (b).5.4.4 Simltaneosly developing flo All reslts presented so far have been based on the rather idealised assmption of an already hydralically flly developed flo. This assmption is in most cases far aay from the practical se in liqid metals, becase in flos ith the Prandtl nmber being far belo nity, the hydralic entrance length is considerably longer than the thermal one (see discssion in Section.5.). The idealisation of a flly developed velocity profile at the dct inlet may lead to a significant error in the predicted performance. In sch cases, the reslts pertaining to simltaneosly developing velocity and temperatre profiles shold be applied. An inherent reslt of the both flo types developing is that the soltion depends immediately on the Prandtl nmber. Sbseqent e discss the reslts for a circlar dct for different thermal bondary conditions. 4

a) Uniform all temperatre (T = const.) For Prov the flo is hydralically already developed before the temperatre profile starts developing. This represents the limiting case discssed in Sbsection.5.3.3. According the Nssel nmbers crves coincide. In case of a good condcting flid, assme idealised that Pr = the temperatre profile develops mch faster than the velocity profile. So in this limiting case, hile the temperatre profile develops the velocity distribtion remains nearly nchanged. The non-modified velocity profile ith its small bondary layer has a large flo rate close to the all, hich immediately yields considerably larger heat transfer from the all epressed by the Nsselt nmber. The soltion for the limiting cases ere already obtained by [Graetz, 883] and rediscovered by [Leveqe, 98]. For the ideal case of Pr = and Prov the soltions can be epressed by analytic means, hile for the other Prandtl nmber series epansions incorporating Bessel fnctions and nmerical approimations are necessary. The mean and the local Nsselt nmbers N and N m as a fnction of the dimensionless entrance length * for different Prandtl nmbers are displayed in Figres.5.5(a) and (b). Figre.5.5. Local and mean Nsselt nmbers N (a) and N m (b) as a fnction of the dimensionless entrance length * for a simltaneosly developing flo in a circlar tbe ith the thermal bondary condition T = constant and for different flids epressed by the Prandtl nmber N (T =const.) 8 4 6 8 4 Pr =5. Pr 8 Pr = Pr =.7 Pr =. 5-3 5-5 - 5 * R 5.783 3.6568 N m (T =const.) 44 4 36 3 8 4 6 8 4 5-3 5-5 - 5 (a) (b) * Pr= Pr =.7 Pr =. Pr =5. Pr Note that for Pr = the local and the mean Nsselt nmber approach the vale of 5.783, hereas the remainder of the crves approach the vale of 3.6568. The thermal entrance lengths for the simltaneosly developing are according to [Bhatti, 985] fond to be: 8 R 5.783 3.6568 l * th. 8 for Pr (.54). 37 for Pr. 7. 33 for Pr o f b) Uniform all heat fl (q = const.) A similar approach like for the previos thermal bondary condition can be sed for the case ith a niform all heat fl. It is interesting to note, that the crve for the local Nsselt nmber in case of Pr = tends for large * to the asymptotic vale of 8, hich is abot 4% more than for the niform all temperatre condition. The dependence of the local Nsselt nmber as a fnction of the entrance length is shon in Figre.5.6. Not only the heat transfer is larger in for the niform all heat fl 4

bt also the thermal entrance length increases. In case of ell moleclar condcting flids it is by 5% larger than for the niform all temperatre. The analysis performed by [Bhatti, 985b] yields the folloing vales: l * th. 4 for Pr (.55). 53 for Pr. 7. 43 for Pr o f Figre.5.6. Local Nsselt nmbers N as a fnction of the dimensionless entrance length * for a simltaneosly developing flo in a circlar tbe ith the thermal bondary condition q = constant and for different flids N (q =const.) 8 4 6 8 Pr =5. Pr 8 Pr = Pr =.7 Pr =. R 8 4 4.3636 5-3 5-5 - 5 *.5.5 Smmary on the laminar heat transfer Within this section the laminar heat transfer has been considered for different thermal bondary conditions. The description has been focsed to describe the main effects of developing length, heat local and global transfer performance nder different thermal bondary conditions. The effect of the flid properties in the non-dimensional sense ere fond to play a crcial role for developing flos. This section as intended to define the terminology of the individal transport properties, hich ill be kept the same throghot this chapter on thermal-hydralics. The central task of this laminar heat transfer section as to determine the all heat fl transferred at the flid all interface in dependence on the specific geometries (here mainly the circlar dct) and on the flid properties. More details on the heat transfer in other geometries may be taken from [Shah and London, 985]. The all heat fl is given by q = -O(T/n) and it depends only on the moleclar heat condctivity of the individal flid and the temperatre gradient at the all. Althogh the laminar heat transfer is a pre moleclar transport process the flo field plays an essential role. For the convective transport of internal energy or enthalpy the flo is responsible and inflences the temperatre gradient in a significant manner. It is obvios that the heat transfer can be enhanced by increasing the flo velocity. The faster energy is convectively transported the more energy can be removed from the all. For incompressible flids the flo field is determined by only one characteristic nmber. In forced convective problems this is the Reynolds nmber,hile in boyant convection the Grashof nmber appears. In both characteristic nmbers the moleclar viscosity Q appear, hich is responsible for the momentm echange. For the heat echange in a 4

floing liqid the ratio of the moleclar viscosity Q to the moleclar heat condctance N = O(U c p ), the so-called Prandtl nmber, is of crcial importance. In contrast to the Reynolds or the Grashof nmber the Prandtl nmber is a pre material constant and it depends only on the flid properties. One normalises the all heat fl ith an appropriate temperatre difference and this is named the heat transfer coefficient D = q /'T. The heat transfer coefficient can be represented non-dimensional in different manners. The Stanton nmber is the ratio of the all heat fl to the enthalpy difference density and does not contain the heat condctivity. More often sed is the Nsselt nmber, in hich the heat transfer coefficient mltiplied ith a characteristic dimension is related to the moleclar heat condctivity. In case of pre forced convection the reslts can be compressed in the ay: N (.56) n f Pr ith n Re The epression N / Re increases ith rising Prandtl nmber. This does not mean that the heat transfer or the heat transfer coefficient also increases ith increasing Prandtl nmber, becase the Nsselt nmber itself contains the moleclar heat condctivity. For a flly developed flo (both thermally and hydrodynamically) the heat transfer is given only by a constant Nsselt nmber. The constants depend niqely on the chosen thermal bondary condition. The Prandtl nmber inflences only the heat transfer in the thermal entrance (or developing) region. In order to determine the heat transfer correlation there eist in principle the folloing possibilities: Analytic soltions inclding the hole set of conservation eqations. Mostly, this is only applicable for drastic simplifications like the flly developed dct flo or the Coette-flo. Nmeric soltion of the conservation eqations. This reqires a lot of effort in compting time and memory capacity, since the eqations are of elliptic type. Hoever, the progress of the compters in calclation poer makes many problems even no tractable. Eact soltions of the bondary layer eqations. The similar soltions are of major importance for the determination of heat transfer correlations, since they are available in tables, diagrams or in form of series approimations. Nmeric soltions of the bondary layer eqations reqire mathematical effort, bt are ith commercially available comptational flid dynamics codes easier accessible than in the past. Approimate soltions sing integral methods. They are less accrate than nmeric soltions bt they are still in practical se. The nmeric effort is lo and comple and epensive codes are not necessary. Analogies beteen heat and momentm transfer are only applicable for Pr = flids and are reslts of eact soltions. Empirical correlations. They mst be sed if comple geometries do not allo another soltion..6 Trblent momentm echange Trblent flos are of immense technological importance for all liqid metal applications as they occr normally nder sal operation conditions. The main advantage of trblent flos over laminar flos is that they are capable of providing vastly enhanced heat and mass transfer rates. Hoever this is at the epense of increased friction factors accompanying the trblent flos. 43

The flid flo and heat transfer characteristics of all dcts are described in terms of certain hydrodynamic and thermal parameters. Many of them have been eplained in the sections of the laminar momentm and energy echange. In order to describe trblent flos additional parameters that are pecliar to trblent dct flos ill be defined..6. Description of trblence In the year 883 Osborne Reynolds shoed ith his famos color stripe ater eperiment that a laminar tbe flo above a certain flo rate trns into a trblent flo. This critical threshold or border for the transition is given by a critical Reynolds nmber. He fond in the eperiments that the transition occrs at Re crit.3 and spposed that the laminar-trblent transition is a stability problem. For the development of trblence mainly to qestions are of importance: Ho develops trblence from a laminar flo? This is essentially the stability problem of the laminar trblent transition, hich has been initially solved by Tollmien 99 by means of a pertrbation analysis. A detailed description of the methodology, the assmptions sed and the interpretation of the obtained reslts for this type of analysis is far beyond this chapter. Here, a etensive amont of literatre eists. As eamples the books of [Schlichting, 979], [Lin, 96], [Betchov and Criminale, 967] and [White, 974] are recommended for the interested reader. Ho emerges trblence in an already trblent flo? Why remains a trblent flo trblent? This is mainly a qestion of the prodction of trblence. Trblent flos are alays nsteady, three-dimensional and stochastic. From eperiments it is knon that the laminar trblent transition is accompanied by: an increase of the pressre drop; an increase of the bondary layer thickness. This is cased by trblent diffsion. De to the intense macroscopic oscillation motion the trblent velocity profile becomes fller and the trblent bondary layer gets thicker than in the laminar case. All variables in the trblent flo depend on the spatial co-ordinates, y and z and the time t. Restricting the considerations to incompressible single component flos the nknons of the isothermal flo are, v, and p hich are all fnctions of the three space co-ordinates and the time. Consider a in the mean steady flo, hich means that it is statistically steady, then the flo variables can be decomposed in a time averaged mean vale pls the flctations arond the mean vale. For the velocity this so-called Reynolds decomposition reads to: i,t t, (.57) i i i here e have sed the tensorial notation in the form i = (, v, ). The time averaged mean is given by: i,t i i i t t (.58) ³' i i dt ' t t here the time interval 't, over hich is integrated, mst be sfficiently large bt not too large to retain the capability to stdy long term nsteady changes of the flo. By definition the time average of the flctations diminishes, bt the mean vale of their sqares is far from being zero. The trblence intensity or RMS vale is given by: 44

I i v (.59) The relative intensity of the velocity flctations is defined: T (.6) i i is called the trblence level. According to [Jischa, 98] the trblence level is abot % donstream a sieve, approimately % close to a all and more than % in a free jet. Generally, the trblent flo types can be divided in three classes: Isotropic trblence: The statistic properties in the hole flo field are the same and are independent of the direction ( v ). They are invariant against any translation and rotation of the co-ordinate system. Homogeneos trblence: All statistic properties depend only on the direction and not on the location. They are translation invariant. Anisotropic or shear trblence: This is of practical interest in technical applications, since it appears in bondary layer flos, free jets, dct flos, etc..6. Reynolds eqations for trblent flos and derivation of transport eqations The nsteady Navier-Stokes eqations are generally considered to be valid for trblent flos in the continm regime. Hoever, the compleity and random natre of the trblent motion prevents an eact soltion of these eqations. Most of the present day prediction methods for trblent flos are based on the time averaged Navier-Stokes eqations. These eqations are referred to as the Reynolds eqations. Time averaging the eqation of motion gives rise to ne terms hich can be interpreted as apparent stress gradients or heat fl qantities associated ith the trblent motion. These ne qantities mst be related to the mean flo variables throgh trblence models. This transfer process introdces frther assmptions and approimations. Ths, this attack on the trblent flo problem throgh solving the Reynolds eqations of motion does not follo entirely from fist principles, since additional assmptions mst be made to close the system of eqations. In the sbseqent listing of the Reynolds eqations, one in the mean steady flo of an incompressible Netonian flid is considered. Reynolds form of the continity eqation: i i v y z Reynolds form of the momentm eqation: i p U j P j i j i j U i j (.6) (.6) Here appears the apparent or virtal Reynolds stress tensor W tr = -U i j hich differs from the moleclar one hich is given by W ij = P j / i i / j. It is important to note that the additional Reynolds stresses are only registered in relation to the temporal mean motion. The eqation of motion 45

for the instantaneos qantities does not contain any additional trblence term. The transition from the instantaneos eqations of motion to the Reynolds eqation of motion makes the trblence problem obvios. The price of the temporal averaging is the introdction of the nknon Reynolds stress tensor, hich originates from the non-linear convective terms in the Navier-Stokes eqations. The major task in solving the trblence problem is to correlate the Reynolds stress tensor ith the temporal mean velocity field by means of appropriate trblence models. The methodology to solve the nknon Reynolds stress tensor is performed sing balancing eqations. Like any other transport qantity it mst obey a common form. These types of balancing eqations often called transport eqations are the basis for nearly all ne closre assmptions. The starting point for the formlation of the transport eqations is the derivation of the Navier-Stokes eqation for the instantaneos qantities. Folloing a strict mathematical derivation process according to [Rotta, 97] the transport eqations for the Reynolds stress tensor is obtained: C v j vi v v i j vk viv j viv k v j v k Q k k k k k º p v iv j G kjvi G ki v j» p ª v v i j «viv j vk Q U j i k «k U» ¼ PSC P D DS (.63) Herein C is the convective change and P the trblence prodction, hich is the prodct of the Reynolds stress tensor ith the velocity gradients of the time averaged mean velocity. DS represents the trblent dissipation, hich is essentially a negative prodction, PSC is pressre shear correlation, hich contribtes similar like the diffsion D to the reorganisation of the flo. The prodction term is the most essential one, since it describes that only in flos ith a velocity gradient (shear flos) trblence can be maintained. The velocity gradient of the time averaged velocity is the generator of trblence. The necessary energy is taken from the mean flo. The tensor eqation eists of nine components. An important case of the transport eqation is obtained by the formal contraction i = j, for hich the transport eqation of the trblent kinetic energy k is obtained. If one defines k as k = / v j / v the transport eqation for the trblent energy reads to: k v j v k v j v k H k k C P º v j vk» ª p k «k v k Q Q k «U k j» ¼ D (.64) in hich H is the dissipation. The trblence energy is not averaged and the trblent dissipation H is given by: H v Q v v j j k k k j (.65) The transport eqations play an essential role in the development of modern trblent flo calclation methods. Althogh the Reynolds stress tensor can be described by the transport eqations, these eqations does not solve the closre problem, since they contain nknon correlation fnctions. For the correlation fnctions ne transport eqations can be formlated, hich themselves contain ne nknon correlations. The order of the obtained tensors increases ith each step by the factor one. 46

To smmarise, the closre problem can not be solved directly. There eist alays more nknons the available eqations. The only closre possibility is the se of semi-empirical closre assmptions, hich in the simplest form can be applied directly to the Reynolds eqations like the Prandtl miing length model..6.3 A flashlight on trblence modelling Several attempts have been made to solve the trblence closre problem by the introdction of a sccession of trblence models. The simplest one of them is Prandtl s miing length model, see [Prandtl, 98]. This model and its variations on the hand of [Taylor, 95], [v. Karman, 93] and [Van Driest, 956] have proven reasonably adeqate for plain to-dimensional flos. For the general case of three-dimensional flos higher order trblence models are reqired. They tilise one or more partial differential eqations derived from the modified Navier-Stokes eqations for qantities like the trblent kinetic energy k, the kinetic energy dissipation H, and the components of the trblent stress tensor W ij. The trblence model employing the single partial differential eqation for the trblent kinetic energy in conjnction ith the algebraic epression for the trblence length scale (e.g. Prandtl miing length l) is referred to as the one eqation model or k-l-model. Another model, employing the partial differential eqations for the kinetic energy and its dissipation is called the to-eqation k-h model, hich has bee initially proposed by [Harlo and Nakayama, 968] and modified by [Jones and Lander, 97] and [Lander and Spalding, 974]. It is technically most often sed and spread nearly throghot all commercially available codes. Many applications of the k-h model have made se of all fnctions to treat the near all region. Alternatively, additional terms have been added to the k and H to etend their applicability to the viscos sblayer by [Jones and Lander, 97] or [Chien, 98]. In this connection the viscos sblayer is often referred to as the region of lo trblence Reynolds nmber (k / l/q). The inner model is crcial for comple trblent flos as for eample those containing flo separation or severe property variations. The ncertainty of sch inner-region modelling for comple flos appears to limit the range of the applicability of the k-h model. The k-h models that have been modified so that they are applicable in the viscos sblayer [Jones and Lander, 97 or Rodi, 98] are knon as lo Reynolds nmber k-h models. Details of sch models are beyond the aim of this article. Modifications to the k-h model to inclde the effect of boyancy and streamline crvatre on the trblence strctre have also been proposed. Nmeros other to-eqation models have been sggested, the most freqently sed being the Ng-Spalding [Ng-Spalding, 97] model, and the [Wilco and Traci, 976] model. All of these models employ a modelled form of the trblence kinetic energy eqation, bt the modelling for the gradient diffsion term is different. The most striking difference, hoever, is in the choice of dependent variable for the second model transport eqation from hich the length scale is determined. Often to-eqation models coeist; a famos eample is the so-called shear stress trblence model (SST). The SST model combines the advantages of the k-h model ith the ones of the k-z model, here Z is the vorticity. It takes advantage of the fact that for sing in the near all region the k-z model an analytical soltion for the viscos sblayer is knon for small y + vales, see e.g. [Wilco, 986]. The matching of the k-z model close to all alls to the k-h model in the rest of the flid domain is performed by means of blending fnctions. The complicated mlti-eqation models involve soltions of the partial differential eqations for all components of the trblent stress tensor and are referred to as stress eqation models. A brief description of the trblence models together ith pertinent references is amongst the tremendos 47

literatre available in this field given in the articles and books by [Reynolds, 976] and [Arpaci and Larsen, 984]. Here only the main ideas of these Reynolds stress models (RSM) are otlined. RSM models are those models that do not assme that the trblent shearing stress is proportional to the rate of strain. That is for a -D incompressible flo: U v v z P t y (.66) in hich P t is the trblent viscosity being P t ~k /H. The most idely sed Reynolds stress models are those of [Hanjalic and Lander, 97], [Lander et al., 975] and [Donaldson, 97]. The RSM reqire at least three additional partial differential eqations (PDE) compared to the k-h model and the RSM mst still tilise approimations and assmptions in modelling terms hich presently are difficlt to measre eperimentally. A simplification of the Reynolds stress modelling knon as an algebraic stress or fl model has a large poplarity in isothermal flos. This approach is discssed in detail in [Rodi, 98]. In the algebraic stress modelling it is generally assmed that the transport of the Reynolds stresses is proportional to the transport of the trblent kinetic energy. For bondary layer flos ithot boyancy effects, the algebraic stress model reslts in: v C P k (.67) H y hich is identical to the reslts obtained from the k-h model. Hoever, in the algebraic stress model, C P becomes a fnction of the ratio of the prodction to dissipation of the trblent kinetic energy rather than a constant..6.4 Bondary layer approimations Many technical flos have bondary layer character. The laminar bondary layer eqations as derived in Section.4. can not be transferred to the trblent case becase the oscillation velocities vc k are rapid in space and time and so their derivatives are not negligible compared to the other terms, althogh their absolte magnitde is an order of magnitde smaller than the mean vales. Ths the bondary layer approimation mst be directly derived ithin the Reynolds eqations and the transport eqations. The procedre is similar to that for the laminar flo and the details may be taken from tetbooks like [Brmeister, 983] or [Anderson, et al., 984]. For a to-dimensional flo the bondary layer eqations read to: v y, U v y dp d P U v y y G (.68) in hich p G is the pressre at the border of bondary layer to the blk flo. A similar procedre can be performed for the transport eqations and one obtains according to [Jischa, 98] the transport eqation for the shear stress and the trblent kinetic energy eqation in the form (.69) for a -D flo. 48

ª º v v ª v v v p v v v Q «v Q v y C y P «y y z z» ¼ U y ª k k º p k v v v H «k v Q Q» y y y «U y y» ¼ C DS P PSC D p º» y «y U» ¼ D (.69) The most terms in both transport eqations can be measred eperimentally, ecept for the local pressre oscillations. Fortnately the pressre oscillations do not appear in the eqation for the trblent kinetic energy. Most flos are confined by alls and by approaching the all the velocity oscillations hence the Reynolds shear stress approaches zero. lim yop y W (.7) If e assme a flly developed -D trblent flo near a all all statistic properties depend only on the co-ordinate y. Since the derivate of the velocity in aial direction vanishes, the continity eqation immediately yields that the mean velocity in y-direction is also zero. In this specific case convective terms and the pressre are zero in the Reynolds eqation and it remains: P U v y y (.7) With the bondary condition at the all one gets: W Q v (.7) U y Obviosly only the parameters Q, W /U appear and one defines shear stress velocity W as: W W U f (.73) here f is the friction coefficient f = W /(U ). Defining the dimensionless co-ordinates + and y + in the ay: W and y y W Q (.74) the velocity distribtion close to the all can be given in an niversal manner + = f(y + ). The soltion of f in the immediate vicinity of the all is immediately obtained becase the velocity flctations close to the all are zero and at the all the non-slip condition holds, so that + = y +. This is the so-called viscos sb-layer, in hich the moleclar shear stress is significantly larger than the Reynolds stress. With increasing distance from the all the Reynolds stress gros considerably and simltaneosly the moleclar part becomes negligible. This yields to a logarithmic epression for the dimensionless velocity in the form: 49

lny C C (.75) hich as initially determined by Prandtl 93. Beteen the viscos sblayer and the flly trblent bondary layer a transition layer appears, for hich different bt mostly logarithmic fnctions can be sed. An overvie on the different definitions beteen viscos sblayers and flly developed bondary layer may be taken from [Shah and London, 978]. For the transition of the anisotropic trblent bondary layer toards the isotropic core flo the logarithmic las loose their applicability. Within this region the so-called ake las are introdced. There eist nmeros articles for the velocity profile description in pipe flos, in hich many differ only by the constants chosen for the ake profile. A comprehensive pictre on the dimensionless velocity distribtion in liqid metal flos can be fond in the PhD thesis by [Fchs, 973]. Neer reslts can be taken from [Zagarola, 996, 997], [Barenblatt and Chorin, 998] [Go and Jlien, 3]..6.5 Smmary The nmerical prediction of trblent bondary layer flos is mainly possible on the basis of semi-empirical closre assmptions. They are applied directly in the Reynolds eqations or in the transport eqations. The applicability of closre methods of the first order is rather limited and ths mainly closre assmptions of higher order are applied, hich nmerically can be applied ith rather small effort. Fortnately for the formlation of semi-empirical assmptions a lot of eperimental data eist and so the trblent momentm transfer can be satisfactory described for the forced convective liqid metal flo in simple geometries ith a high degree of accracy. Regarding the trblent momentm transport in flly developed forced convective flos the only inflencing parameter is the Reynolds nmber and ths pressre drop correlations, friction factors as ell as prodction and dissipation rates can be taken from standard engineering handbooks for the individal geometry considered in the application. This is completely different to the trblent energy echange, since there net to the velocity data temperatres have to be determined. One shold be carefl to define a heavy liqid metal eperiment ith a small heat transfer rate to be a solely forced convective flo if only Reynolds nmber is qite large. This holds especially for horizontal arrangements of the eperiments. Mercry eperiments performed by [Gardner, 969] or [Lefhalm, et al., 4] have shon that even at Reynolds nmbers Re > 5 boyancy forces can become important and modifies the local flo qantities..7 Trblent energy echange Regarding the trblent energy the similar procedres as for the trblent momentm echange can be applied. Unfortnately less eperimental data are available for the trblent energy echange in liqid metals than for the trblent momentm echange here also data from other flids can be applied. Ths, a lot of correlations appear hich ere derived only from one eperiment sometimes ith speclations on their bondary conditions. So often no ncertainty estimates can be given. The energy eqation of a steady flo of an incompressible flid in the temperatre notation rites to: U c p v k T k T O k or U c p T T y T z T O T y T z (.76) 43

.7. Reynolds eqations for the trblent energy echange When the trblent stream involves heat transfer its instantaneos temperatre T i can be decomposed according to Reynolds in a flctating component and a time averaged vale. The time averaged component mst not be confsed ith the blk mean temperatre T m. The former is a local qantity at a discrete point, hereas the latter is a flo average qantity across the hole dcts cross-section. The Reynolds approach leads to the folloing eqation: U c p vk T k T (.77) O Uc p v k T k k RH in hich RH is the apparent or virtal heat fl often referred to as the Reynolds heat fl. The Reynolds heat fl eists only de the temporal averaging of the eqations. It describes a macroscopic echange originating from the oscillating components. In contrast to the Reynolds shear stress tensor the Reynolds heat fles are vectorial qantities, becase the transported property contains a scalar in comparison to the vectorial momentm. Also for the heat fl transport eqations can be derived. In contrary to the momentm transport the trblent energy echange is referred to as a passive transport process. Similar to the momentm echange also bondary layer approimations can be applied to the energy eqation. For a -D in the mean steady flo of an incompressible flid they read to: T T c v y T O Uc y y v T U p p (.78).7. Analogies beteen flid flo and heat transfer parameters In analogy ith Neton s la of friction for the laminar shear stress W a la of the friction for the apparent trblent stress W tr as introdced by [Bossinesq, 877]: W P y and W tr P t y (.79) here P t is an apparent viscosity also referred to as a virtal or eddy viscosity. Note that the eddy viscosity is not a property of the flid like the dynamic viscosity P, bt depends on the time average flo velocity. In analogy ith the moleclar momentm diffsivity Q, eddy diffsivity for momentm H M is often reqired to characterise trblent flo. Q and H M are defined as: Q P U and H M P t U (.8) In analogy ith Forier s la of heat condction for laminar flo, a la of heat condction for trblent flo is introdced. These las are: q O T y and q tr O tr T y (.8) 43

here O tr is an apparent (virtal or eddy) condctivity, hich is also not a flid property, bt depends on the time average flo velocity as ell as on the time-average blk mean temperatre. In analogy ith the thermal diffsivity N, an eddy diffsivity for heat transfer, H H is defined. N O U c p and H H O U c tr p (.8) Frthermore, in analogy ith the moleclar Prandtl nmber Pr, a trblent Prandtl Pr t nmber is introdced: Pr Q N and Pr t H H M H (.83) Eperiments on the trblent Prandtl nmber in circlar tbes ehibit that the trblent Prandtl nmber depends on the Reynolds nmber Re, the moleclar Prandtl nmber Pr and the distance from the all epressed by (y/r). Sch a relation in the form: Pr t f Re, Pr, y R v v T T y y (.84) is epected. [Eckert and Drake, 97] collected measrements of the trblent Prandtl nmber for varios flids inclding heavy liqid metals. They rote, The eperimental difficlties are obviosly so large, that no satisfying coherence beteen the individal data sets can be established for the liqid metals. [Fchs, 973] describes the problems in more detail. The main error sorces are: determination of the temperatre and velocity gradients from a measred temperatre and velocity distribtion; determination of the correlation v T, as ell as hich is in liqid metals hardly possible de to the lack of adeqate measrement techniqes. Consider a flly developed trblent flo in a circlar dct. Within sch a problem the reslting all shear stress W and the reslting heat fl q can be described sing the trblent echange qantities in the folloing ay: W q d dy P U H or Q H M dt q dt c H or N H dy O U p H dy W U U c p M d dy and H (.85) The Reynolds analogy In flly trblent flos Q << H M and N << H H ecept for the immediate vicinity of the all. If one assmes that the trblent energy transport has a constant ratio as the trblent momentm transport, in the etreme case H H = H M this yields immediately Pr t =. This major assmption is called the Reynolds analogy and yields immediately for Pr t = : 43

dt c p q d W (.86) This can be easily integrated, assming at the all non-slip for the mean velocity and the all T y and for the oter flo the position is sed at hich the mean velocity and temperatre T for the mean temperatre is attained, i.e. = and T = T m. Both qantities are defined: c p Tm ³ T dt q d W ³ or c p q W T T m (.87) Assming the Reynolds analogy in the tbe this immediately yields for the friction factor c O and for the Stanton nmber St the folloing reslt: c O W N c (.88) O 8 and St U Re Pr 8 The Reynolds analogy yields good reslts for moleclar Prandtl nmbers larger than Pr >.7. It is also embedded in a lot of commercial code systems sing the k-h model. It definitely fails describing the heat transport in liqid metals becase the initial bondary condition of H H ~H M and constant is not flfilled for this type of flids. Ths, any approach of a constant Prandtl nmber assmption that means sing the Reynolds analogy ill lead to miscellaneos reslts. This problem is knon since [Martinelli, 947] and has dring the last to decades been revieed critically by several athors like [Grötzbach and Wörner, 99], [Kays, 994], [Carteciano and Grötzbach, 3] or [Arien, 4]. The Prandtl analogy Within this approach the flo field is separated into to domains a viscos sblayer and a flly trblent blk flo. In the viscos sblayer the moleclar viscosity is larger than the eddy viscosity and the moleclar condctivity dominates over the eddy diffsivity, i.e. Q!! H M and N!! H H. This yields: W U Q d dy, q U c p N dt dy and c p dt q -Pr d W (.89) The relations can be integrated from the all to the border of the laminar sblayer, for hich = and T = T is assmed. For the flly trblent blk the model takes se of the Reynolds analogy ith the assmptions Q << H M and N <<H H and sing them the Eqs. (.85) can be integrated from the border of the sblayer toards the blk, hich yields the reslt: c p q ª º T T»¼ m W «Pr (.9) The velocity at the border of the viscos sblayer in direction of the blk is knon since there + = y + holds. Setting y + = 5 the relation (.9) knon as Prandtl analogy is obtained, hich reads for the Stanton nmber to: St N Re Pr 5 c c O O / 8 / 8 Pr (.9a) 433

For Pr = the Prandtl analogy is similar to the Reynolds analogy. The impact of the moleclar Prandtl nmber is reflected by taking it into accont in the viscos sblayer. The factor 5 is arbitrary. Bt, de to the increase of the trblent Prandtl nmber ith decreasing moleclar Prandtl nmber, as observed in nmeros eperiments, the Prandtl analogy is only very restricted applicable to liqid metal flos. The largest hint is the assmption that ecept for the viscos sblayer the trblent Prandtl nmber is set to nity, althogh the thermal bondary layer is considerably larger than the viscos one. V. Karman analogy Folloing the ideas of Prandtl, v. Karman divided the flo field instead of to into three layers, the viscos sblayer (y + < 5) a transition layer (5 < y + < 3) and the trblent ake (y + > 3). In the intermediate transition layer he assmed an identity of the moleclar and the trblent qantities, i.e. Q H M and N H H, and obtained for the Stanton nmber the folloing relation. The eact derivation may be taken from the book of [Gebhardt, 97]: St N Re Pr 5 c O c O / 8 / 8 Pr ln > 5Pr / 6@ (.9b) Again for Pr = the Reynolds analogy is obtained. Frther analogies [Martinelli, 947] etended the three layer model of v. Karman in the ay that he considered both the moleclar heat condctivity and the eddy diffsivity in the fll trblent range. Using this approach his analogy is applicable also to liqid metals. The considerations of Martinelli [Martinelli, 947] and especially its restrictions are etensively discssed in the books of [Rohseno and Choi, 96] as ell as in [Kndsen and Katz, 958]. The reslts obtained consist of a nmber of tables. [Rohseno and Choi, 96] even etended Martinelli s ideas by implementing a relation of Pr t = f(pr). Hoever, a comparison of the reslts ith the eperiment sho only satisfactory reslts for Peclet nmber larger than Pe > 7. Moreover, the trblent Prandt nmber remains space independent, hich does not hold for liqid metals as the folloing paragraph denotes..7.3 Eperimental observations of the trblent heat transport The eqation (.84) in particlar makes clear that to evalate the trblent Prandtl nmber at any point it is necessary to measre for qantities: the trblent shear stress, the trblent heat fl, the velocity gradient and the temperatre gradient. This is the reason hy direct measrements of trblent Prandtl nmbers in dct flos are relatively scare and hardly available in bondary layers. Despite the large eperimental difficlties it is clear that the trblent Prandtl nmber is larger than nity in case of the liqid metals, becase liqid metals possess a large moleclar condctivity compared to their moleclar viscosity. A trblence parcel ill transfer in liqid metals easier energy than momentm to the adjacent flid. Ths, the trblent Prandtl nmber shold increase ith decreasing moleclar Prandtl nmber. There is an inflence of the all distance (y/r) on the trblent Prandtl nmber in flly developed flos. In the majority of the eperimental investigations Pr t seems to decrease ith increasing distance from the all, see [Azer and Chao, 96] or [Davies, 969]. For the flly developed trblent flo in circlar tbes the local trblent Prandtl nmber Pr t as evalated in detail by [Fchs, 973] in sodim ith a moleclar Prandt nmber Pr =.7 as a fnction of y/r. The eperimental reslts are depicted in Figre.7.(a). This general tendency is spported by the measrements of [Sleicher, et al., 973] or [Bron, et al., 957]. 434

Besides the dependence of the trblent Prandtl nmber on the radial co-ordinate the dependence of the measred mean trblent Prandtl nmber Pr tm on the moleclar Prandtl nmber seems to be even larger, as depicted in Figre.7.(b). The eperimental data on the mean trblent Prandtl nmber define sho a qite large scatter for the liqid metals, hich can be partially be eplained by the reasons mentioned in [Fchs, 973]. Another aspect may be also the fact of the determination of the mean trblent Prandtl nmber Pr tm. For an eact formlation it shold be gained from the Lyon-Integral [Lyon, 95] in the form: ³ r / R r / R r dr r dr ³ ³ dr H M Pr H Pr ³ r M r Q Prt Q Prt dr (.9) and assming a constant moleclar Prandtl nmber in the cross-section of interest one gets: H M Pr Q Pr t N N min (.93) here N min is the Nsselt nmber in case of pre heat condction. Some athors tried based on different assmptions to derive formlas for the mean trblent Prandtl nmber. This aspect ill be discssed in the contet of the different closre models in the net section. Figre.7.. (a) Measred trblent Prandtl nmber Pr t as a fnction of the dimensionless distance (y/r) in a flly developed trblent circlar dct flo ith a constant all heat fl (q = const.) from [Fchs, 973]. b.) Measred mean trblent Prandtl nmber Pr tm for different moleclar Prandtl nmbers Pr for a flly developed trblent circlar dct flo ith a constant all heat fl (q = const.) for to different Reynolds nmbers Re =. 4 (open symbols) and Re = 5. The eperimental data are from [Fchs, 973] for Na, [Bhr, et al., 968] for O NaK and ' Hg, [Sbbotin, et al. 96] Na, PbBi, [Sleicher, 958] Õair, [Allen and Eckert, 964] +ater 4 4 centerline Re=5. 3 3 ncertainty region for Re=. 4 Prt 3 Re=. 4 all Pr tm Re=6. 4 +.. Re=8. 4 ncertainty region for Re= 5....4.6.8. Re=. 5 y/r Pr (a) (b) 435

From the eperimental observations the folloing statements on the mean trblent Prandtl nmber Pr tm can be made: For Pr t (air ater, oil, etc.) the trblent Prandtl nmber seems to be constant and independent of the Reynolds nmber and moleclar Prandtl nmber. The inflence of the Prandtl nmber and the Reynolds nmber increases both ith decreasing Prandtl nmber and decreasing Reynolds nmber. The trblent Prandtl nmber increases ith decreasing Prandtl nmber and decreasing Reynolds nmber..7.4 Closre methods for the trblent heat fl The knoledge of the trblent Prandtl nmber is a central problem of all theoretical considerations concerning the trblent heat transfer in to-dimensional bondary layers or in dct flos. A large nmber of models have been pblished in the past hich address the prediction of the trblent Prandtl nmber Pr t for sch sitations. There is a very strong inflence of the moleclar Prandtl nmber on the vale of Pr, for flids ith very lo Prandtl nmbers (liqid metals). Additionally, here is an inflence of the all distance on the vale of the trblent Prandtl nmber hich tends to increase Pr, close to the all. This increase in Pr, near the all is especially important for high Prandtl nmber flids becase of the very thin thermal bondary layer. Otside this thin layer near the all the trblent Prandtl nmber seems to be constant for Pr >. Becase of the strong dependence of Pr, on moleclar Prandtl nmber for lo Prandtl nmber flids (liqid metals) effort has been taken in the past to develop prediction models for Pr t, hich are able to take the dependence of Pr t, on the moleclar Prandtl nmber as ell as the dependence of Pr t, on the all distance into accont. In principle all the proposals can be divided into mainly for classes (of corse containing nmeros amont of sbclasses): a) semi-empirical or empirical closre methods of zero and first order; b) trblent Prandtl nmber from analytic soltions; c) modelling of the trblent heat fl by means of transport eqations at different levels; d) direct nmerical simlation of the hole set of the time-dependent Navier-Stokes eqations. The length of this chapter etends a little compared to the previos ones. Bt the crrently ecessive se of nmerical simlation tools to design critical components of liqid metal cooled devices in many technical fields reqires a principle nderstanding of hat is modelled in the individal closre assmptions for the trblent heat fles. The type of this modelling plays a decisive role in the analysis of the reslts. It is clear, that not all the details of the modelling can be described ithin this contet. Hoever, the main ideas, the assmptions and the approaches yielding to the specific reslts for the different types of closre methods are flash-lighted. Moreover, the limiting parameter range for their validity is given as far as knon to the athor. Hopeflly, a list of the most relevant literatre is cited in this contet. At the end of each section a sometimes rather personal assessment is given on the individal class of closre type. a) Semi-empirical closre methods of zero and first order Revies of the eisting ork of these types of closre methods can be fond in the overvie article by [Reynods, 975] or in [Kays, 994]. Within these types of models a hole branch of assmptions is hided. 436

The simplest ones are prely empirical correlations, in hich the trblent Prandtl nmber is mainly fitted to eperimental data from varios eperiments, as e.g. by [Notter, 969]. Some orkers, thogh aare of attempts to predict the dependence of the eddy diffsivity ratio an moleclar diffsivities and position ithin the flo, have rejected these analytical reslts, in vie of the contradictions among the several models, beteen models and eperiments, and even beteen different eperiments. In this vie [Knz and Yeraznis, 969] adopted for the for the trblent Prandtl nmber of trblent liqid metal flos in a circlar dct: Pr ª H M ep«9 3 «Q t.. 64 º»» ¼ (.94) With this they accept deviations of r.5 from the eperiments they consider and frther, it does not contain a dependence of the radial co-ordinate. Again, in the vie of the scatter of their measrements, [Qarmby and Qirk, 97, 974] felt that it as impossible to isolate the dependence of the trblent Prandtl nmber on the moleclar Prandtl nmber and the dct Reynolds nmber. Nevertheless, they fond an important variation across the pipe and formlated for their individal eperiment the relation: 4 y / R Pr (.95) t This yields a trblent Prandtl nmber near the all of Pr t =.5, hich is considerably to lo for liqid metal flos [see Figre.7.(a)], hile close to the dcts centre a vale of Pr t is obtained, hich is acceptable for Re > 5. 4. These prely empirical reslts offer neither practical nor fndamental advantages. And as matter of fact the effort spent on these methods ended arond the late seventies. A more theoretical approach offered the individal miing length models already mentioned in the contet of the analogies beteen heat and mass transfer. These analyses seek to accont for the echange of momentm and heat transfer beteen the instantaneos srrondings and a lmp of flid hich moves across gradients of velocities and temperatre. At the end of its jorney the lmp (mostly considered to be a sphere) is assmed to mi instantaneosly ith the srronding flid; the transfer rate is calclated sing the terminal properties of the moving element. Clearly, these types of models are inconsistent in that no accont is taken of that fraction of the transferred (heat or momentm) fl hich is conveyed only part ay along the miing length. This implies more or less a proportional relation beteen trblent heat and trblent momentm fl knon from the Reynolds analogy. Several types of miing length models ere proposed ranging from the beginning fifties p to the end of the sities. The most famos are the Jenkins type miing models after [Jenkins, 95] ith several modifications, hich are listed in Table.7. and basically yield the trblent Prandtl nmber as a fnction of the moleclar Prandtl nmber and H M /Q. They do only marginally contain a local dependence of Pr t on y/r.. Table.7.. Different Jenkins type miing length models and their range of validity for the trblent Prandtl nmber Pr t Athor Fnctional form Empirical constants Validity Jenkins, 95 f(pr, H M /Q) All Pr Sleicher and Tribs, 957 f(pr, H M /Q) f(h M /Q) All Pr Rosheno and Cohen, 96 f(pr) Pr << Tien, 96 f(pr, H M /Q) All Pr Senechal, 968 f(y Qc/(l N)) Pr < 437

The other class of miing length models is formed by the Deissler type models after [Deissler, 95], hich is focssed on mainly small Prandtl nmber flids as the liqid metals. They take into accont only the heat transfer from the moving element and relate the length scale of an imaginary flid sphere in the temperatre gradient field to the miing length. On this scale only moleclar diffsion is assmed. These types of models are listed in Table.7. ith their range of validity. A problem here is similar as in the Jenkins models that the velocity field is copled directly to the temperatre field, althogh knoing from the eperiments that the statistical featres of both are different. Table.7.. Different Deissler type miing length models and their range of validity for the determination of the trblent Prandtl nmber Pr t Athor Fnctional form Empirical constants Validity Deissler, 95 f(pe) Pr << Lykodis and Tolokian, 958 f(pr) Pr << Aoki, 963 f(pr Re.5 ) Pr << Mizshina and Sasano, 963 f(pr, H M /Q) 3 Pr < Mizshina, et al., 969 f(pr, H M /Qa/l) 3 Pr < Wassel and Catton, 973 f(pr, H M /Q) 3 All Pr The more advanced miing length models adopt a more sophisticated pictre of the trblent activity. Ths the defining featre of this category is not the se of an immtable miing length bt the introdction of a discrete element hich gains or looses heat and momentm as it moves throgh the blk of the flid. Nevertheless, also this analysis is constrained by the assmption of a basic analogy beteen momentm and heat transfer, at least in its statistical behavior. In the article by [Reynolds, 975] the individal aspects of these type of models are discssed as ell as their advantages and disadvantages in se. They seem to fit qite ell for the trblent Prandtl nmber in dct and plate flo bt do not generally accont for the temporal behavior of the temperatre and flo field. These more advanced models are listed in Table.7.3. Table.7.3. Different varied miing length models and their range of validity for the determination of the trblent Prandtl nmber Pr t Athor Fnctional form Empirical constants Validity Azer and Chao, 96 f(pe, Pr, y/r) + Pr << Bleev, 96 f(pr, Re, geometry) 5 All Pr Dyer, 963 f(pr, H M /Q) Pr << Tyldesley and Silver, 968 f(pr) All Pr Tyldesley, 969 f(pr,?) All Pr Ramm and Johannsen, 973 f(pr, Re, geometry) 5+ All Pr These semi-empirical types of closre methods contains also another grop of models hich is based on the analysis of the Reynolds eqations ithot the direct formlation of transport eqations. Additionally, [Kays and Craford, 993] developed a prediction model for Pr, hich can be sed for all moleclar Prandtl nmbers. The model contains to empirical constants hich mst be determined from available eperimental data. [Cebeci, 973] adopted van Driest s idea of near-all damping of the miing length to propose a trblent Prandtl nmber concept. This model as etended by [Chen and Chio, 98] for liqid metal flo in pipes by incorporating the enthalpy thickness G T, in Pr t. Althogh the predictions of Chen and Chio agree ell ith measred Nsselt nmbers for liqid metal flos, their model has the disadvantage that an additional nonlinearity is introdced into the 438

energy eqation by sing the enthalpy thickness G T, in Pr t. This restricts the possibility of analytical soltions of the energy eqation for hydrodynamically flly-developed flos. Frthermore, the von Kármán constant is changed in the miing length epression for their calclation. This might lead to misleading reslts for larger moleclar Prandtl nmbers. Smmary on semi-empirical or empirical closre methods of zero and first order All semi-empirical closres for the trblent heat fl independent of their order do not give detailed insight in the trblent echange mechanisms beteen the velocity flctations and the temperatre oscillations. For engineering prposes some of them can be applied to assess the heat echange, bt one shold be carefl on the model, its validity and the empirical constants chosen, becase they often apply only for one flid in a rather limited parameter field. By natre the Reynolds analogy can give only reliable reslts for large Reynolds nmbers, in hich the thermal bondary layer remains small and the trblent Prandtl nmber tends to nity. As soon as one arrives to loer Reynolds nmbers, this analogy leads to inaccrate reslts. This shold be taken into accont also for nmerical simlations sing the k-h model ithot an additional trblent heat fl modelling. As benchmark stdies have shon, see [Arien, 4] this can lead to severe discrepancies. b) Trblent Prandtl nmber from analytic soltions [Yakhot, et al., 987] present an analytic soltion based on hat they call the renormalisation grop analysis method and this is reslt is apparently devoid of any empirical inpt. The relative simple eqation reads to: ª Pr ««Pr - eff -. 793 º». 793» ¼.65 ª Pr ««- eff. 793 º»» ¼.35 - Pr. 793 H / Q M ith Pr eff H / Q H M / Q Pr Pr t M (.96) It is not clear hether this reslt is epected to be valid over the entire bondary layer or only in the region here the logarithmic la holds. It is remarkable that the reslts obtained ith that relationship matches the nearly all eperimental observations qite ell. And it seems even to ork for higher moleclar Prandtl nmber. [Kays, 994] fitted the relation (.96) by an asymptotic crve in the form: Pr t. 7 (.97). 85 ith Pet H M QPr Pe t here Pe t is the trblent Peclet nmber. It looks similar like a sggestion performed by [Reynolds, 975] bt in that case Pr t is an average throghot the hole bondary layer. [Lin, et al., ] modified the renormalisation grop analysis performed by [Yakhot, et al., 987]. Within the model a qasi-normal approimation (see [Cho and Chin, 94]) is assmed for the statistical correlation beteen the velocity and temperatre fields. De to the statistical approach the different time scales of the temperatre flctations and the velocity flctations remain in the model and are copled together by the ave nmber k. A differential argment leads to derivation of the trblent Prandtl nmber Pr t as a fnction of the trblent Peclet Pe t nmber, hich in trn depends on the trblent eddy viscosity H M. The fnctional relationship beteen Pr t and Pe t is comparable to that of [Yakhot, et al., 987] bt it contains in contrast to the Yakhot, et al. soltion the spectral properties of both oscillating fields. Bt, the reslting formla for the effective trblent Prandtl nmber Pr eff is very close to Eq. (.96). It reads to: 439

ª Pr «Pr - eff - /3 - /3 º ª Pr º (.98) eff HM / Q» ith Pr - eff ¼ Pr ¼ HM / Q HM / Q Prt Pr The apparent resemblance beteen the formlas (.96) and (.98) is qite remarkable not only becase of the similarity in mathematical strctre, bt becase the reslts are basically derived entirely from different approaches to renormalisation grop analysis of trblence. Pr t can be represented in terms of Pe t for any given moleclar Prandtl nmber Pr. A comparison ith the direct nmerical simlation reslts (in hich the hole set of time dependent Navier-Stokes eqation is solved) of [Kasagi, et al., 99] and [Kim and Moin, 987] shos ecellent agreement for a Prandtl nmber of Pr =.. Also a comparison of the reslts ith the eperimental data taken from [Bremhost and Krebs, Sheriff and O Kane, and Fchs, 973] all measred in liqid sodim at the centreline of thermal trblent pipe flo shos ecellent agreement, see Figre.7.(a). Net, they noticed that the Peclet nmber Pe t is a thermal analogy of the Reynolds nmber. The reason hy Pr t alays tends to as Pe t becomes sfficiently large old be that at this time the thermal flo trblence is completely convection dominated and the temperatre field T responds immediately to any change of the velocity field v. Figre.7.. a.) The trblent Prandtl nmber Pr t verss the trblent Peclet nmber Pe t plotted based on Eq. (.98) for liqid sodim (Pr =.7) in comparison ith eperimental data measred by [Bremhost and Krebs, 99, ], [Sheriff and O Kane, 98, '] and [Fchs, 973, O]. b.) The scaling la of the thermal energy spectrm E Htr (k) verss the normalised ave nmber k/k for Pr << after [Lin, et al., ] in comparison ith eperiments by [Boston and Brling, 97]. 3 4 3 slope (-5/3) E Htr - - comptation slope (-7/3) - -3-3. Pe t (a) -4-3 - - k/k (b) [Batchelor, 958a, b] as the first to propose scaling las for the thermal energy spectrm for Pr << and Pr >>, respectively, leaving a proportional constant C B ndetermined. Sbseqently, many athors have tried to estimate or measre the vale of C B nder varios flo conditions. In theory, [Kraichnan, 97] had the estimate C B =.8, and [Gibson and Scharz, 963a] had the estimate C B =.9, hile [Yakhot, et al., 987] obtained C B =.6 according to their H-based renormalisation grop analysis. [Kerr, 99] obtained nmerically the vale C B =.6. In eperiment, [Gibson and Scharz, 963b] had C B =.35, [Grant, et al., 968] obtained C B =.3, and [Boston and Brling, 97] got C B =.8. 44

De to the retaining of the spectral information in the model by [Lin, et al., ] the scaling las for the thermal energy spectrm (E Htr ) verss the ave-nmber k can be obtained. Sch a thermal energy spectrm as a fnction of the normalised ave-nmber is depicted in Figre.7.(b). It is visible that in the inertial-convective range the thermal energy spectrm follos a poer la of -5/3 and in the inertial-condctive range of a slope of -7/3 is obtained. Especially, the latter is different to large Prandtl nmber flids, here in the inertial condctive domain a slope of - is observed, see eperiments by [Grant, et al., 968]. Ths, an essential reslt of the renormalisation grop analysis is that the Batchelor constant is not a niversal constant, bt depends on the shape of the actal thermal energy spectrm. The sal Smagorinsky model is sed for the large-eddy simlation of the Navier-Stokes eqation. One can constrct a parallel model for trblent transport of the passive scalar, hich yields that constant C P depends on Pr t -/ and on the ave nmber k. In order to model the trblent thermal transport process at a all bondary the srface reneal concept initially introdced by [Danckerts, 95] can also be sed. This as recently modified by [Tricoli, 999]. The basic assmption considers the srface to be covered by a mosaic of laminar floing patches of flid, here transport occrs only by moleclar diffsion These flid patches are spposed to be periodically replaced by ne ones srface reneal model or to form viscos layers that periodically gro and collapse groth-breakdon model. These are nsteady state -D models. It has also been proposed that the flid patches are arranged in a reglar repetitive pattern of steady state bondary layers developing for a given length. These simple models are able to eplain a nmber of qalitative featres observed in trblent transport. The model derived by [Tricoli, 999] appeal to postlated mechanistic pictres of trblence and the concepts and the qantities involved have no fndamental relationship ith the correlated trblent flctations, the sole qantities that actally determine trblent transport. He constrcted a theory able to predict trblent heat transport from fndamental information, namely the thermal diffsivity and the normal trblence intensity in the flid blk. This theory applies to heat transfer in trblent incompressible flo for Pr <<. His approach is based on the primary ingredients of trblent transport, i.e. the trblence flctation and the moleclar momentm or heat diffsivity and does not contain empirical epressions for sch qantities eddy diffsivity models or postlates a physical pictre of the transport mechanism at the all srface srface reneal. The predictions sho an ecellent agreement ith available eperimental data and empirical correlations for the trblent heat transport in liqid metal flos in pipes. The most remarkable reslts of this approach refer to a relationship of the Nsselt nmbers in a flly developed trblent circlar dct flo ith different thermal bondary conditions. This relation valid for Pe > 3 reads to: N T const S N const q (.99) Smmary on trblent Prandtl nmber derivations from analytic soltions The analytic class of soltions for the trblent Prandtl nmber may be difficlt to handle in the contet of a nmerical treatment of a commercial code package. Bt it gives important reslts to the near all behavior of the heat fl and ths contribtes considerably to the treatment of the thermal all fnctions. Another aspect is that, especially ithin the renormalisation grop analysis, the temporal behavior of the temperatre field is still a part of the final reslt. This allos verifying the validity of direct nmerical simlations (DNS), hich are in principle eact bt may se a too coarse grid. Only the analytic soltions are able to provide near all data of the kinetic and thermal energy spectrm, becase crrent measrement technologies are still in a stage here they hardly can resolve the viscos and thermal bondary layers simltaneosly and the analytic soltions are able to provide assessments 44

in comple geometries, hich are crrently not accessible ith a direct nmerical simlation. Unfortnately, for engineering applications to assess a mean trblent heat transport form a all this type of analysis is time consming and reqires a lot of effort. c) Modelling of the trblent heat fl by means of transport eqations at different levels The modelling of the trblent heat fles by means of the transport eqations contains in some sense also to the semi-empirical methods, becase the development of each transport eqation generates a set of constants a priori not knon. These constants can be either eperimentally determined or calclated by eact soltions (analytic ones or a direct nmerical simlation). The modelling of the trblent heat fl transport eqations follos the same systematic scheme as shon in the section of the trblent momentm echange. A first proposal to model the trblent heat fles sing transport eqations as made by [Jischa and Rieke, 979]. The final form of their transport eqation for bondary layer type flos reads for the heat fl in all normal direction to: y v T v v T Pr v T p T... > v T @ T v d p Q v v H... Convection y Ucp d cp Pr j j Pr odction Dissipation U y y Redistribtion Diffsion (.) Generally, all types of trblent transport eqations ehibit the same strctre consisting of convective terms, trblence prodction, dissipation, redistribtion and finally diffsion. Every of the nely modelled transport eqations contain correlation or time averaged prodcts not knon. The challenging task after formlating these fl transports, dissipation or destrction terms is to model the nknon correlations or gradients in an appropriate manner. This can be condcted in form of a gradient approach like the Bossinesq approach for the shear stress, here v ~ / y as set, or even an additional transport eqation if a simple gradient approach can not match all the transport processes taking place in the individal problem. In the ork by [Jischa and Rieke, 979 and 98] the trblent flctations v are assmed to be proportional to the trblent kinetic energy k and for the dissipation of the trblent kinetic energy the Prandtl miing length approach has been sed. The latter is problematic ith respect to liqid metals since the moleclar heat condction is not negligible. Finally, the dissipation of the gradients of the velocity and temperatre flctations ere set to be proportional to the trblent heat fles in the form Q / Prv T C Q D / Prv T / L, here C D is a constant and L the sqare of a characteristic length. All these assmptions yield finally to a simplified transport eqation, in hich the trblent heat fl is a fnction of the temperatre gradient. Althogh this model is rather crde, since the trblent heat fl model is only taking into accont a forced convective transport, it ehibits the dependence of the trblent Prandtl on the distance from the all and on the Reynolds nmber. As a disadvantage it shold be clearly noted that the thermal eddy diffsivity H H is still linked to the momentm diffsivity H M assming a similar statistical behavior of the trblence and that it is valid only for high Peclet nmbers in prely forced convective flos. Ths, any kind of mied or even boyant flo can not be treated. In the past decade a considerable effort has been spend on the development of a more detailed modelling of the trblent heat fles, the trblent temperatre variancet and its dissipation or destrction H T. A first order for-eqation type model has been developed by [Nagano, et al., 988, 994 and 995], hich as able to represent the physics in more detail, bt reqired of corse data for 44

the nknon constants appearing in the ne transport eqations and information on the statistical behavior of the flctations near the bondaries especially the alls. Especially, close to the all or in mied and even more in boyant convective flos the trblence field is strongly anisotropic, hich does not allo to match the trblent shear stresses v ith the trblent kinetic energy k, see [Drbin, 993a]. The details of the more advanced shear stress modelling close to the alls may be taken from [Drbin, 993a and b], [Mancea, et al., ], [Oi, et al., ] or [Speziale and So, 998]. Also the modelling of the trblent heat fl evoltion eqations is still problematic and the reported reslts are only slightly better than obtained ith loer order closres hich based on the eddy diffsivity formlation, see [Nagano and Kim, 988]. Ths in many forced convective problems the to-eqation approaches acconting for the temperatre variance T and its destrction H T are crrent state of the art, see [Grötzbach, et al. 4] and [Karcz and Badr, 5]. A lot of attention mst be paid to the destrction rate of temperatre variance transport eqation and this especially close to the all. Different approaches are generally employed to predict the generation of H T. The details on the different approaches may be taken from [Sommer, et al., 99], [Abe, et al. 995], [Shikazono and Kasagi, 996], [Deng, et al., ] or in a comprehensive stdy by [Nagano, ]. Net to all near flos also any boyancy inflenced flo ehibits a strong anisotropy. In sch types of flo even a second order description of the trblent transport of heat shold be applied [see Grötzbach, et al., 4], hich means the se of independent transport eqations for the three components of the trblent heat fl vector. These kinds of trblence models called Trblent model for boyant flo miing (TMBF) proposed by [Carteciano, et al., 997], [Carteciano and Grötzbach, 3] belong to this class of models. Smmary on modelling of the trblent heat fl by means of transport eqations at different levels Any comptational flid dynamic calclation, hich is condcted to assess the liqid metal heat transfer in trblent flos mst be performed by a skilled ser, hich is aare of the specific featres appearing in liqid metal flos. In prely forced convective flos, hich may even not appear for hydralic Reynolds nmber beyond Re > 5 (see [Gardner, 968] or [Lefhalm, 4]) a k-h model may ehibit sensible reslts if the thermal all fnction formlation chosen is correct and applies for the problem considered. In strongly anisotropic flos liqid metal flos associated ith heat transfer the k-h model sing a constant trblent Prandtl nmber of Pr t =.9 (Reynolds analogy) yields nphysical reslts. In this contet higher order closre approaches are necessary in hich the trblent heat fles are modelled in sch a sense that the transport eqations contain the statistical featres of the thermal field. By performing sch an approach nknon triple correlations and constants are generated. The spatial distribtion of the correlations may be directly evalated in benchmark eperiments, hich is de to the limited capability of the crrently available measrement techniqes difficlt. More often the correlations and constant are determined by the comparison of the modelling data ith the eact soltion obtained by means of direct nmerical simlation. d) Direct nmerical simlation (DNS) The direct nmerical soltion of the time dependent Navier-Stokes eqations represents in principle an eact soltion of the problem. Althogh the progress in memory storage capacity and compting speed increased considerably in the past decade the application of a DNS is still limited to rather heat transfer problems in simple geometries. Nevertheless, the progress made in the compter technology allos no to perform DNS-calclations hich are not only slightly trblent as the first ones by [Kim and Moin 989], here it as difficlt de to the lo Reynolds nmber to detect the logarithmic region of the velocity profile. The DNS performed in the recent years here focsed to 443

stdy the effect of the Prandtl nmber both in case of isotropic trblence [Kasagi, et al., 993], in near all flos ([Kaamra, et al., 998, 999] or [Piller, et al. ]) and in boyancy governed flos [Otic, et al., 5] (here only a fe pblications of eemplary natre are cited). An immediate reslt of these comptations is the spatial and temporal dependence of the statistical properties of the thermal and the momentm field on the different bondary conditions, hich is especially eperimentally not accessible. Ths, the DNS is crrently the only available tool to qalify and verify the completeness and qality of compter codes based on a transport eqation modelling. It reveals their modelling deficits and offers the possibility to model more adapted higher order closres. In this sense DNS can be conceived as a complementary tool for a more advanced trblence modelling for technical and engineering applications, from hich the DNS is still far aay..7.5 Heat transfer correlations for engineering applications All the heat transfer correlations referred to in this sbsection are valid for the single-phase heat transfer of a trblent liqid metal flo in a specified geometry. They are gained by eperiments performed since the late forties p to the present. For many of the described and shon heat transfer correlations the hole set of bondary conditions ere not available, so that a final jdgement on the qality of the obtained data can not be dran. Especially, data on the homogeneity of the prodced all heat fl, the temperatre ncertainty, the inlet-conditions for the flo field and the methodology to determine if the flo or the temperatre field is flly developed or not. Another aspect in this contet is the qestion, ho the Nsselt nmber is determined. In many pblications it is nclear if this has been performed sing a global or a local method. The srvey of the heat transfer correlations does not cover the ncertainties of the trblent velocity profiles, since for most of the liqid metal eperiments condcted in the past, ecept for a fe rather specific ones, a fll set is not available. Taking into accont the nmeros heat transfer correlations being on the market this sbsection tries to otline, ho large the ncertainty margin for the individal correlation is and here the deviation to the other arise from. Bt, before trning into the technical description of the forced convective heat transfer processes some critical remarks to the terminology forced convection shold be mentioned. Despite the considerable effort made in the comptational flid dynamics, for engineering prposes, heat transfer correlations play still a significant role. It may be to assess, ho mch heat can be transferred in a particlar set-p or they may be sed in the contet of a system analysis. Many technical configrations consist of several devices and each of them acts as a heat sink (heat echanger) or a heat sorce (by viscos dissipation as a valve or a mechanical pmp, by direct electric heating like an electro-magnetic pmp, or the flid flo arond a the reactor thermal sorce). Crrent comptational simlations are by far not capable to represent the hole set three-dimensionally. Hence, the component interaction is mainly performed by one-dimensional models taking into accont the heat transfer correlations and the friction factor in the flo components in order to predict the temporal system performance in dependence of normal and abnormal operation cases..7.5. Free convection distortion in liqid metal heat transfer Free convection becomes important at lo Reynolds nmbers in liqid metals and sally inflences the heat transfer by distorting the according temperatre and velocity profiles from the epected symmetrical behavior. This effect is even more emphasised for heavy liqid metals. Free 444

convection distortion have been already observed by [Schrock, 964] in sodim-potassim (NaK) for Reynolds nmber p to.6. 4, by [Lefhalm, et al., 3 and 4] in lead-bismth p to Re =.. 5 and [Gardner, 969] in mercry for Re = 3. 5 in horizontal tbes, hile in vertical circlar tbes [Koalski, 974] detected free convection distortion p to Reynolds nmbers of Re = 9. 4. All the eperimental data ehibit that the velocity profile rapidly distorts as the heat fl is increased. At high heat fl a limiting profile shape ith the centreline velocity ell belo the mean velocity. As a matter of this fact the eperimentally determined Nsselt nmber in even in a flly developed pipe flo depends on several parameters, sch as hether the pipe flo is pards or donards, the pipe is horizontal or vertical, the direction of the heat fl (cooled or heated) and the vale of the Rayleigh nmber. Figre.7.3(a) displays the measred Nsselt nmber variations in dependence of the angle of the perimeter for a horizontal trblent mercry pipe flo having free convection distortion. In an order of magnitde estimation performed by [Bhr, 967] he introdced a parameter, say Z, hich can be conceived as a ratio of the boyancy forces verss the inertia forces. This parameter is defined as: Z Ra d h Re L ith Ra Gr Pr, Gr 3 d h Eg' T dt (.) and ' T d h Q d Herein, Re is the hydralic Reynolds nmber, d h the hydralic diameter, Ra the Rayleigh nmber, Gr the Grashof nmber, E the epansion coefficient, g the gravity constant, Q the kinematic viscosity and 'T the aial temperatre difference. He analysed a series of vertical and horizontal heat transfer eperiments in pipe flo and fond that if Z >. 4 free convection affects the forced convective measred Nsselt nmber N. Also he is mentioning that the flid properties especially for larger temperatre differences shold be evalated at the blk mean temperatre T m = (T in + T ot )/. Figre.7.3. a.) Measred local Nsselt nmber at several angles of the perimeter in a trblent mercry in a horizontal pipe ith convection distortion effects from [Gardner, 969]. b.) Dependence of the Nsselt nmber on the Z-parameter, in a vertically pard directed trblent mercry pipe flo ehibiting free convection effects. N 3 6 9 average for large Pe 48/ N 6 Pe.4. 3.4. 3 7.5. 4.5. ^ ^ [Koalski, 974] [Horsten, 97] [Jacoby, 97] [Koalski, 974] [Horsten, 97] [Jacoby, 97] [Horsten, 97] 8 3 4 5 Pe (a)..3.5.7 Z (b) When the flo is dominated by forced convection, the folloing qalitative pictre of the effects of free convection on the forced convection heat transfer is observed. Boyancy forces enhance the trblent heat transfer to liqid metals for donards oriented flo hile it retards it for pards directed flo. Boyancy forces inflence the convective heat transfer indirectly throgh their effect on shear stress distribtion in the liqid metal. In vertically donard flo, the boyancy forces operate 445

against the mean flo direction and increase the isothermal shear stress near the heated srface. In contrast to the pards flo the shear stress is redced by the boyancy. The increase in shear stress in the donards floing case leads to an increase of the trblence prodction and hence ill enhance the heat transfer. The opposite holds for the pard flo. If the temperatre gradients are large the boyancy effects become more and more important and the enhancement of heat transfer occrs in both pard and donard flo as observed by [Jackson, 983]. The effects of boyancy on heat transfer to liqid metals are for the relatively light alkali metals less than for ordinary flids becase of the redced importance of trblent eddy condction at lo Peclet nmbers. For heavy liqid metals, hoever, de to their lo viscosity they are of crcial importance. This qalitative pictre is spported by Figre.7.3(b), hich ehibits that the Nsslet nmber is a fnction of the parameter Z initially decreasing, then groing as Z increases. Z is essentially proportional to the applied heat fl. As the boyancy forces become comparable to the inertia forces of the flo the Nsselt nmber reaches a minimm, begins to increase, and finally appears to reach a limiting vale. It can be seen that the effects of free convection distortion can affect the measred vale of the Nsselt nmber and cold, therefore, eplain some of the ell-knon scatter in the Nsselt nmber data for the liqid metal heat transfer. Needless to say, the effects of free convection distortion mst be considered hen dealing ith liqid metal heat transfer systems, especially for heavy liqid metal systems..7.5. Trblent heat transfer in circlar dcts The trblent heat transfer in circlar dcts is one of the most often investigated configration of the liqid metal heat transfer. Similar as in the laminar energy echange one can distingish beteen the three cases: a) flly developed flo (either hydralically and thermally); b) hydralically developing; c) thermally developing. a) Flly developed trblent flo For liqids ith Prandtl nmber arond nity both the temperatre and the velocity distribtion can be calclated approimately in terms of a series epansion. Bt, for liqids ith Pr << (as ell as for Pr >> ) no closed form formlas for the temperatre distribtion are available either for the condition of a constant all temperatre (T = const.) or a constant all heat fl (q = const.). A ridiclosly large nmber of correlations, both theoretical and empirical have been developed for the heat transfer in a smooth circlar dct. [Shah and Johnson, 98] pt many of them together in tablar form, hich is etended by the correlations knon to the athors in the contet of this handbook. The correlations for constant all temperatre (T = const.) are displayed in Table.7.4 hile for the condition of a constant all heat fl (q = const.) they are shon in Table.7.5. Case: T = constant Referring to the least error in the scatter of all eperimental data the correlation proposed by [Notter and Sleicher, 97] seems to fit best to all data for the case of a constant all temperatre. The comptational ork and dimensional analysis in the late seventies performed by [Leslie, 977] and nmerical calclations by [El-Hadidy, et al., 98] yield that there are no different eponents regarding the Reynolds and the Prandtl nmber. Their main reslt is that for T = const. the Nsselt 446

nmber is proportional to N~Pe.8, hich holds for Pe >. and pre convective flo. For a long time it as difficlt to calclate the heat transfer in a circlar tbe by means of a direct nmerical simlation, becase of the singlarity at the dcts centreline. Hoever, a transformation of the problem made it accessible for a DNS and [Y, et al., ] shoed a niversal soltion in principle valid for all Prandtl nmbers. They tried to match their findings ith the different all heat fles and to derive a niqe correlation. The folloing set of eqations is recommended as the best choice for Pr t > Pr, bt is sbject to possible significant improvement, becase still some simplifications had to be applied to the DNS. N N NPr Pr Pr NPr t ith 7 5 3. - r r N Pr N Pr 8 Prt Pr t Pr 8. 54 / 3.436 ln > r @ ; Pr Prt,y o f NPr Prt NPr Prt 3 NPr Prt NPr NPr Prt, y o f Re f / and N Pr Pr t 45 5 / (.) (a-d) here r + is the dimensionless tbe co-ordinate (r + = rq(w. U) ½ ), and + the dimensionless velocity normalised by ( + = Q(W. U) ½ ), Pr the moleclar and Pr t the trblent Prandtl nmber, Re the Reynolds nmber and f the (Fanning-) friction factor (f = W /(U )). According to their estimates on the error compared to the [Notter and Sleicher, 97] correlation is redced by more than 5%. Althogh some simplifications and assmptions ere introdced (for details see e.g. [Chrchill, or Chrchill, et al., ]) they fond a good representation for all of the compted vales of N ithot the introdction of any eplicit empiricism compared to other DNS calclations. In either event, a correlation for trblent Prandtl nmber Pr t like the sbseqent named ones (.3)(a,b) is necessary to obtain nmerical vales of N sing a pocket compter. Also empirical epressions (from the DNS) sch as the to eqations (.)(c,d) are needed for N = N(Pr = ) and N = N(Pr = Pr t ) for vales of r + intermediate to or above those for hich the comptations ere carried ot by [Y, et al., ]. Pr t ith ª v 35 ««v º»» ¼ ª 3/ v ) 9Pr ««v. 5 Prt. 85 Pr ) ª v. 5Pr««v º»» ¼ and º» )» ¼ U v v W or (.3) (a,b) The later epression already appeared in the article by [Jischa and Rieke, 979]. 447

It is interesting to note in this contet that the ratio of the all Nsselt nmbers for the constant all temperatre and the case of constant all heat fl fond in the DNS stdies is nearly eact the same as the analytical reslt derived by [Tricoli, 999], see Eq. (.99). That means that the Nsselt nmbers for a constant all temperatre are in any case loer than for a constant all heat fl at least by this ratio for a flly thermally and hydrodynamically developed trblent pipe flo. Table.7.4. Flly developed trblent flo Nsselt nmbers in a smooth circlar dct for Pr <. ith a constant all temperatre (T = const.) from eperiments Investigator Correlation Remark [Gilliland, et al., 95] N. 8 3. 3. Pe d Pr d. and 4 d Re d 5. 6, rather speclative description of the bondary conditions. [Sleicher and Tribs, 957] N. 9. 4. 8. 5 Re Pr d Pr d. and 4 d Re d 5. 6, the predictions are ithin a bandidth of -33% and +9.5% of [Notter & Sleicher, 97]. [Hartnett and Irvine, 957]. 8 N Nslg. 5 Pe d Pr d. and 4 d Re d 5. 6, the predictions are 4% belo the [Notter & ith Nslg 5.78 Sleicher, 97] correlation assming a slg velocity profile [Azer and Chao, 96] N. 77. 5. 5 Re Pr d Pr d. and 4 d Re d 5. 5, the predictions are ithin +4.% and -8.6% of the [Notter & Sleicher, 97] correlation. [Notter and Sleicher, 97] N. 85. 93 4. 8. 56 Re Pr.4 d Pr d. and 4 d Re d 6, the predictions are based on a nmerical analysis and the minimm error of the clod of eperimental data. [Chen and Chio, 98] N. 85. 86 4. 5. 56 Re Pr d Pr d. and 4 d Re d 5. 6, the predictions are ithin a bandidth of +36% and -% of [Notter & Sleicher, 97]. Case: q = constant The flly developed Nsselt nmbers compted from the [Notter and Sleicher, 97] correlations covers a ide range of the Reynolds nmbers and moleclar Prandtl nmbers ith the smallest error to all eperimental data crrently available. [Heng, et al., 998] condcted a nmerical stdy for the constant heated flly developed circlar tbe case based on the models proposed by [Chrchill, 997a and b]. The integrals to be solved are rather comple and can not be sed in the contet of an engineering approach. Hoever, the reslts obtained by [Heng, et al., 998] almost coincide ith the DNS of [Y, et al., ]. The maimm deviation fond occrs for small r +, here [Heng, et al., 998] nderpredict the Nsselt nmber by.8%. Since the eperimental data basis for the niform all heating is considerably larger than for the niform all temperatre, the empirical correlation derived by [Notter and Sleicher, 97] is recommended to fit best all eperimental data available. The deviation to the DNS data and other nmerical reslts is for its valid Reynolds nmber range ithin the range of +5% over-prediction of the mean N. All preceding heat transfer reslts pertain to niform heat fl or niform all temperatre arond the tbes perimeter. The technically important problem of circmferentially varying bt aially constant all heat fl as solved by [Reynolds, 963], [Sparro and Lin, 963] and [Gartner, et al., 974]. To the best knoledge of the athor the only available eperimental data are available by [Black and Sparro, 967], nfortnately not for liqid metals. Nevertheless, the analysis by [Gartner, 448

et al., 974] is performed also lo Prandtl nmber flids. Assming a cosine heat fl variation at a given cross-section of the dct, the local Nsselt nmber arond the perimeter can be calclated by: q M q b cos M yields NM bcosm / N G / b cos M (.4) here N is the vale of the niform all heat fl, q the mean heat fl, b a specified constant and G the circmferential heat fl fnction calclated by [Gartner,974]. G is nity for laminar flos for all Prandtl nmbers. The tablated vales can be taken from Table.7.6. Table.7.5. Flly developed trblent flo Nsselt nmbers in a smooth circlar dct for Pr <. ith a constant all heat fl (q = const.) from eperiments Investigator Correlation Remark [Lyon, 949, 95].8 N 5.5Pe d Pr d. and 4 d Re d 5. 6, the [Sbbotin, et al., 96] predictions are ithin +33% and -6.5% of [Notter & Sleicher, 97]. [Lbarski and Kafman, 955]. 4 N.65Pe d Pr d. and 4 d Re d 5, significant nderprediction of [Notter& Sleicher, 97] by -43%. [Sleicher and Tribs, 957].9. N 6.3.6Re Pr d Pr d. and 4 d Re d 5. 6, the predictions are ithin a bandidth of -3% and +6% of [Notter & Sleicher, 97]. [Hartnett and Irvine, 957].8 N Nslg.5Pe d Pr d. and 4 d Re d 5. 6, the predictions are 44% belo the ith Nslg 8 [Notter & Sleicher, 97] correlation assming a slg velocity profile [Dyer, 963].8 ª.8Re º d Pr d. and 4 d Re d 5. 6, N 7. 5«RePr -. 4» the predictions are ithin +3% and «H M / Qma» ¼ -6.5% of the [Notter & Sleicher, 97] correlation. here H M / Qma. 37Re f [Skpinski, et al., 965]. 87 N 4.8.85Pe d Pr d. and 4 d Re d 5. 6, the predictions are ithin +% and - 8% of the [Notter & Sleicher, 97] correlation. [Notter and Sleicher, 97].85. 93 N 6.3.67Re Pr.4 d Pr d. and 4 d Re d 6, the predictions are based on a nmerical analysis and the minimm error of the clod of eperimental data [Chen and Chio, 98].85. 86 N 5.6.65Re Pr d Pr d. and 4 d Re d 5. 6, the predictions are ithin a bandidth of +34% and -7% of [Notter & Sleicher, 97] [Lee,983]. 833 N 3.Re. d Pr d. and 5. 3 d Re d 5, here its predictions are ithin +5% and -44% of [Notter & Sleicher, 97]. 449

Table.7.6. Calclated circmferential heat fl fnction G for the se in conjnction ith Eq. (.4) for a flly developed trblent pipe flo ith varying thermal bondary conditions as a fnction of the Reynolds and Prandtl nmber from [Gartner, et al., 974] G Pr Re = 4 Re = 3. 4 Re = 5 Re = 3. 5 Re = 6..9989.9937.956.859.4853.3.999.963.85.54.3..9499.795.4567.85.867.3.7794.475.55.888.336 b) Hydrodynamically developing trblent flo Several attempts have been made to solve the problem of trblent flo development in a smooth circlar dct starting ith a niform velocity profile at the dcts inlet. The closed form analytical soltion of [Zhi-qing, 98] is particlarly sitable for engineering applications. According to this analysis the velocity distribtions in the hydrodynamic entrance region is given by: ma / 7 y/ G for d y d G for G d y d R ; ma 4 G R 5 G R (.5) here the hydrodynamic bondary layer thickness G, hich varies ith the aial co-ordinate in accordance ith the relation: /d Re h /4 G. 439 R 5 4 ª G G «. 577. 793 «R R G. 68 R 3 4 G º. 64» R» ¼ (.6) (a) Althogh ne DNS stdies calclate the entrance length problem nearly eact for engineering prposes the folloing relation by [Zhi-qing, 98] matches for a smooth circlar dct the hydralic developing length l hy /d h qite ell. The accracy is arond r%. According to their proposal the developing length, defined as the aial distance at hich the hydrodynamic bondary layer groing from the dct all reaches the dcts centreline, reads to: l d hy h. 359 Re /4 (.6) (b) c) Thermally developing trblent flo The problem of a hydralically developed and a thermally developing flo has been analytically, nmerically and eperimentally considered by many athors. Case: T = constant For the case of a constant all temperatre this problem is often referred to as the trblent Graetz problem. The first, ho treated this problem considering liqid metals analytically here [Sleicher and Tribs, 956]. [Notter and Sleicher, 97] provided analytical soltions for nmeros 45

Prandtl nmbers in the range from < Pr < 4. All soltions presented are of infinite series type like in the laminar case. Bt, in contrast to the laminar flo the individal eigenvales and constants are fnctions of the Reynolds and the Prandtl nmber. Related to the analytical soltions the thermal entrance length l th can be calclated according to the soltion of [Notter and Sleicher, 97]. The deviation to neer nmerical calclations performed by [Weigand, 996] is of marginally natre (<%) ecept for Reynolds nmbers less than 4, here aial condction becomes important. In this contet the thermal entrance length is defined as the aial distance at hich the local Nsselt nmber N reaches.5 times the Nsselt nmber of the flly developed flo problem, i.e. N =.5. N. The calclated thermal entrance length as a fnction of the Reynolds nmber is displayed in Figre.7.4.a for varios Prandtl nmbers according to [Notter and Sleicher, 97]. Certain correlations for the Nsselt nmbers in the thermal entrance region of a smooth circlar dct ere developed by several athors. Hoever, many of them ere applicable to only to a limited range of the Reynolds nmber and the Prandtl nmber. A comprehensive stdy relevant for liqid metals as only performed by [Chen and Chio, 98]. They propose the folloing correlations for the local and the mean Nsselt nmbers valid for /d h and Pe > 5. : N. 4 N 7. 8 / d Nf / d h m ª h º and ln / d Nf / d h / d h ¼ h (.7) here N v is the Nsselt nmber of the flly developed flo. They recommend in this contet the Notter-Sleicher-correlation ith: Case: q = constant. 85 N 4. 5. 56Re Pr.86 f (.8) A similar approach as performed by [Becker, 956] and as modified by [Genin, et al., 978]. While the latter spported their analytical stdies by several eperiments they fond that the local Nsselt nmber in the thermally developing flo can be calclated according to: N N f -. / d h. 85. 86. 6 ith Nf 5. 6. 65Re Pr Pe (.9) (a) In a parametric stdy they sho that their correlation is valid for.9. < Pe <.8. 3 ith an accracy of r9%. The thermal entrance length corresponding to N =.5 N v as fond to be best represented by their eperimental data ith the correlation: l d th h Pe. Pe (.9) (b) [Lee, 983] confirmed their stdies and, moreover, he stdied the effect of aial condction on the soltion for constant all heat fl. Similar to the previos athors he fond, that the thermal developing length for a constant all heat fl are larger than those for a constant all temperatre. The normalised thermal developing length l th /d h as a fnction of the Reynolds nmber for a constant all heat fl is depicted in Figre.7.4(b). 45

l th /d h Figre.7.4. Normalised thermal developing length l th /d h as a fnction of the Reynolds nmber in a hydralically flly developed flo of a smooth circlar tbe for different Prandtl nmbers from [Notter and Sleicher, 97] 4 35 3 5 5 5 Pr..6.4.3.. r.4 3 5 8 4 3 5 8 5 3 5 8 6 Re d) Simltaneosly developing trblent flo a) T = constant, b) q = constant Pr..4..3.4.6. Pr.6.3.. r 3 5 8 4 3 5 8 5 3 5 8 6 Re (a) (b) The simltaneosly trblent flo in a smooth dct is affected appreciably by the type of dct entrance configration. [Boelter, et al., 948] and [Mills, 96] condcted an etensive eperimental stdy for varios inlet configrations into a smooth circlar dct. While the first employed a constant all temperatre the latter performed a series for a constant all heat fl. As a reslt of all these measrements it trned ot that the measred local Nsselt nmbers are sbstantially larger than that of thermally developing flo (at hydralically developed flo), nderscoring the decisive role played by the dct entrance configration. Especially, the sqare inlet region for /d h < ehibits a strange Nsselt nmber distribtion ith a steep increase and then a rapid drop for /d h >, hich is apparently cased by the flo contraction folloed by the re-epansion in the vicinity of the dct inlet. While for liqids ith Pr t.7 Sparro and co-orkers ([Sparro, et al., 98], [Sparro and O Brien, 98], [Sparro and Grdal, 98], [Sparro, et al., 98], [Sparro and Chaboki, 984], [La, et al., 98], [Sparro and Bosmans, 983], [Wesley and Sparro, 976] and [Molki and Sparro, 983]) performed etensive semi-analytical and eperimental stdies the data basis and the eperimental verification for liqid metals is rather sparse. [Chen and Chio, 98] presented correlations for the simltaneosly developing flo of liqid metals in a smooth circlar dct ith niform velocity profile at the inlet. The correlations for the local Nsselt nmber N valid for d /d h d 35, Pe > 5. and Pr d.3 are given by:...3.6 Pr N N N N f f. 4. 88 / d 5 / d h. 86 / d. 5 h / d h h A / d ln h, B (.) here for a constant all temperatre (T = const.) the constants A and B read to: A 4 / d h (.), B. 9 9 hile for constant all heat fl (q = const.) A and B vanish to zero. 45

.7.5.3 Trblent heat transfer in a flat dct The flat dct heat transfer problem has been stdied qite etensively by many athors in the past as this type of dct constittes the limiting geometry for the family of rectanglar and concentric annlar dcts. In this contet a dct is considered to be flat if its height (a) to idth (b) ratio is smaller than., i.e. a/b d.. a) Flly developed flo [Kays and Leng, 963] presented comprehensive trblent heat transfer reslts for arbitrarily prescribed heat fles at the to dct alls. Based on their analysis, the flly developed Nsselt nmber for the constant all heat fl case can be represented from: N q const N JM (.) here J is the ratio of the prescribed heat fles at the to dct alls, i.e. for J = one all is heated and the other adiabatic, for J = both are heated ith the same heat fl and J = - one is heated and the other cooled ith the same rate. The Nsselt nmber N and the inflence coefficient M entering Eq. (.) can be taken from Table.7.7 for the case of J =. Table.7.7. Nsselt nmbers and inflence coefficients for flly developed trblent flo in a smooth flat dct ith niform heat fl at one all and the other adiabatic (J = ) from [Kays and Leng, 963] Pr Re = 4 Re = 3. 4 Re = 5 Re = 3. 5 Re = 6 N M N M N M N M N M. 5.7.48 5.78.445 5.8.456 5.8.46 5.8.468. 5.7.48 5.78.445 5.8.456 5.88.46 6.3.46.3 5.7.48 5.8.445 5.9.45 6.3.45 8.6.4. 5.8.48 5.9.445 6.7.44 9.8.47.5.333.3 6..48 6.9.48..39 3..33 6..55 In order to verify hether the N predictions can be derived from the circlar dct correlations, the predictions of the Eq. (.) are compared to the Notter-Sleicher-correlation in Table.7.5. Comparing the calclated data and bilding the ratio N flat dct /N pipe this ratio varies from +.57 to -.55 in the Reynolds nmber range 4 < Re < 6. Similar to the reslts for the circlar dct, it is fond that the Nsselt nmbers in a flat dct for a constant all temperatre are loer than for a constant all heat fl. Certain empirical correlations are developed to calclate the flly developed heat transfer in a smooth flat dct for different bondary conditions. For J = (one all is heated and the other adiabatic) three correlations are available, one presented by [Bleev, 959], [Dyer, 965] and [Dchatelle and Vatrey, 964]. They read to: N N N q q q const const const 5.. 8. Pe, (.3) 5. 6. 95Pe. 775 5. 85. 34Pe. 9, 453

All these correlations are claimed to be valid for d Pr d.4 and 4 d Re d 5. The latter epression derived by [Dchatelle and Vatrey, 965] matches the analytic reslt obtained by [Kays and Leng, 963] ith an accracy of +4% to -% best for J = and is ths recommend for se. Many neer DNS calclations (see [Kasagi and Otsbo, 993] or [Kaamra, 995]) sho that the Nsselt nmber varies only marginally ith increasing Peclet nmber p to Pe ~ 3.. The only eperiment matching this finding ithin an accracy of r6% is the Dchatelle and Vatrey eperiment. [Dyer, 965] presented the folloing empirical correlation for both alls heated (J = ):. 688 N 9 49 596 q const.. Pe (.4) DNS reslts from [Lyons, et al., 993] and [Kim and Moin, 989] considering a similar problem, hoever, sho that the Dyer correlation yields to too optimistic Nsselt nmbers (abot % larger than the DNS vales). A detailed analysis performed by [Kaamra, et al., 998] shos that the correlation derived by [Sleicher and Rose, 975] matches the eperimental reslts for J = best. For the case of a constant all temperatre ith J = (one all is heated and the other adiabatic) only one relevant eperiment is available, hich as performed by [Seban, 95]. It yields to the folloing correlation: (a). 8 N 5 8 T const.. Pe (.4) A critical revie on the qality of this relation can not be given since DNS data for this specific case cold not be fond in literatre. Bt the loer Nsselt nmber for the case of constant all temperatres compared to constant all heat fl looks qite sensible. b) Thermally developing flo The thermally developing trblent flo in a flat dct ith niform and eqal all temperatres has been solved analytically by [Sakakibara and Endo, 976] and by [Shibani and Özisik, 977]. The soltion can be epressed by a series epansion similar like for the laminar flo. [Hatton and Qarmby, 963] stdied the problem ith one part of the plates being at a constant temperatre and the opposite being inslated. The type of soltion corresponds to the ones mentioned above and is of series epansion type. [Hatton, et al., 964] and [Sakakibara, 98] presented semi-analytical soltions for the case ith constant all heat fl. [Faggiani and Gori, 98] stdied analytically the effect of aial flid condction on thermally developing flos in a Reynolds nmber range of 7.6. 3 d Re d 7.4. 4 for Prandtl nmbers. d Pr d.. They conclded that the inflence of aial flid condction is to significantly redce the Nsselt nmbers in the thermal entrance region. This effect hoever occrs mainly at eakly trblent flo (Re d 4 ) and for small /d h < 5. Unfortnately, in none of the analytical and semi-analytical soltions the order of magnitde for the thermal developing length l th is mentioned or a Nsselt nmber correlation for the thermal entrance region is developed in dependence on the Reynolds nmber and the Prandtl nmber. Moreover, to the knoledge of the athors no eperimental verification or a direct nmerical simlation calclation eists, hich cold give an estimate on the qality of the derived soltions. c) Simltaneosly developing flo Also the information on the simltaneosly developing trblent flo beteen parallel plates is relatively sparse. Here, hoever, an eperiment condcted by [Dchatelle and Vatrey, 964], in hich 454 (b)

the local Nsselt nmbers have been measred eists. In their set-p one all as inslated and to the opposite one a constant all heat fl as spplied. Figre.7.5 shos that the ratio of the local Nssel nmber N to the flly developed flo Nsselt nmber N attains nity at Pe/(/d h ) = 5. Figre.7.5. Normalised local Nsselt nmber for simltaneosly developing trblent flo in a smooth flat dct for Pr =. ith one all heated and the other inslated from [Dchatelle and Vatrey,964] 8 5 a N /N 3 5 8 3 Pe/(/d h ) Ths the thermal entrance length l th for the simltaneosly developing flo can be estimated to l th /d h = Pe/5. For the thermally developing bt hydrodynamically developed flo these athors estimate a significantly smaller thermal entrance length, i.e. l th /d h = Pe/8..7.5.4 Trblent heat transfer in a rectanglar dct It is generally difficlt to give engineering correlations for the heat transfer in rectanglar dcts, since the attainable Nsselt nmbers depend not only on the heat fl transferred to the flid bt also to the thermal bondary conditions for the all, the symmetry, the shape of the corners ith the associated secondary flo, the pre-conditioning of the flo and many other aspects. Ths, the correlations presented in this contet can only be sed as an assessment of the potential heat transfer capability in a rectanglar dct. The correlations shon are far aay to perform a detailed design of a heat transfer nit becase of a lack of detailed eperimental data or verification by a direct nmerical simlation. The database for both of the latter named classes is rather sparse for liqid metals. For an estimate on the heat transfer capability of a specific geometrical set-p a CFD simlation is recommended. It can give a more detailed insight in the heat transfer mechanisms than the se of the sbseqent correlations, even if rather rogh models like the simple k-h model (ith all the deficits mentioned in Sbsection.7.4) are sed in the simlation. A simple correlation is available for estimating flly developed Nsselt nmbers in trblent flo of liqid metals in rectanglar dcts either for a constant all heat fl and a constant all temperatre. The correlation as derived by [Hartnett and Irvine, 957] for a niform velocity distribtion (slg flo) and a pre moleclar condction heat transfer mechanism. This correlation is given by: N. slg. 5Pe 8 N 3 N ith N slg slg 5. 78 8. for T for q const. const. (.5) 455

In case of thermally developing or simltaneosly developing flos no analytical and hardly eperimental data are available. There are some correlations on the market, hich look qite inconsistent to the reslts obtained for the circlar tbe and the sbseqently described concentric annls problem..7.5.5 Trblent heat transfer in a concentric annls The flo in a concentric annls like a rod in a circlar tbe is one of the most stdied trblent thermal heat transfer problems. Especially in contet ith nclear systems, here typically circlar fel rods are placed ith a specific arrangement, the concentric annls is often considered as a soltion of the single problem. There eist nmeros possibilities of thermal bondary conditions for this problem and not all of them are covered by a correlation. Belo a consistent representation to the circlar dct flo is tried and e concentrate or focs only to concentric annli. a) Flly developed flo [Rensen, 98] measred the flly developed Nsselt nmbers in a concentric annls, ith the inner all sbjected to a niform heat fl and the oter to be thermally inslated. The radis ratio r *, hich is defined as r * = r i /r o, here r i is the radis of the inner annls and r o that of the oter one as in his eperiment r * =.549. From the eperiment condcted for sodim (Pr.5) in a Reynolds nmber regime from 6. 3 < Re < 6. 4 he obtained a Nsselt nmber correlation in the form:. 8 N 5. 75. Pe (.6) [Hartnett and Irvine, 957] presented an analytic soltion for liqid metals ith small Prandtl nmbers [see Eq. (.5)], hich reqires the knoledge of the slg Nsselt nmbers for the conditions of constant all heat fl and constant all temperatre. The slg Nsselt nmbers N slg for these to cases are depicted in Figre.7.6(a). For the case here one all is sbjected to a niform heat fl and the other is inslated, hich is in literatre often referred to as the fndamental soltion of the second kind, the slg Nsselt nmbers are shon in Figre.7.6(b). Figre.7.6. Slg flo Nsselt nmbers for smooth concentric annlar dcts ith (a) constant all temperatre and constant all heat fl and (b) inner or oter all heated and the other all being adiabatic from [Hartnett and Irvine, 957] q =const. T =const 9.86 r i r o N slg 9 8. 8 r i 7 r o 6 5.78..5. r * (a) 9 8 7 inner srface heated oter srface heated 6..5. r * (b) 456

The prediction of Eq. (.5) for r * =.549 in conjnction ith Figre.7.6(b) for the case of the inner all heated and the oter all inslated are 3% loer than those of Eq. (.6). Related to this specific eperiment Eq. (.6) is limited to be valid for 8 d Pe d 354. [Dyer, 966] developed semi-empirical correlations for liqid metal flos ith Pr <.3 in concentric annli ith one all sbjected to a niform heat fl and the other all inslated. For the case of the oter all heated the eqations read to: ma n N A B E Pe ith.5. 354. 333 A 5.6, B. 848 - * * * r r r. 333. 833 n. 78, * * r r. 8 E ith. 4 PrH M / Qma H M H M Q Q ma,core, (.7) An epression for (H M /Q) ma,core applicable to a circlar dct (r * = ) as developed by [Bhatti and Shah, 987] hich is (H M /Q) ma,core =.37 Re f.5, ith f the Fanning friction factor. The agreement ith the eperimental data from [Petrovichev, 959] is ecellent. The data of [Baker and Sesonske, 96] ehibit abot 5% higher Nsselt nmbers, hoever, an analysis of the data immediately ehibits that their flo as not flly developed. In the eperiment by [Nimmo and Dyer, 966] the validity of the eqations as etended to Pe ~ 3. 3. For the case of the inner all heated the [Dyer, 966] correlation read to: N A A B E Pe.686 4.63 * r n, B ith. 43. 54 * r,n. 657. 883. 75 * * r r (.8) The eqations in Eq. (.8) demonstrated their validity in the Prandtl nmber range.5 d Pr d.3 for Peclet nmbers 3. d Pe d 5. The accracy in this range is abot -5%. The predictions of Eq. (.7) for r * = are ithin +% and -3% of the prediction of Eq. (.3c) derived by [Dchatelle and Vatrey, 965]. For Peclet nmbers belo Pe =.. 3 the ncertainty level falls belo 6%. b) Thermally developing flo [Qarmby and Anand, 97] provided an eigenvale soltion for the concentric annli problem for different r * and Prandtl nmbers covering the Reynolds nmber range. 4 d Re d.4. 5. This soltion looks qite ell for large Prandtl nmbers bt nderestimated the data of [Rensen, 98] A detailed measrement of the thermal entrance length l th /d h as only performed in the eperimental stdies by [Rensen, 98]. In his set-p ith the inner all heated (q = const.) and the oter adiabatic he fond for sodim (Pr =.54) a strong dependence of the thermal entrance length on the Reynolds nmber, hich is tablated in Table.7.8. Here, a flly developed flo is established if the criterion N <.5N is matched. 457

Table.7.8. Thermal entrance length l th /d h for thermally developing flo in a smooth concentric annls ith r * =.549 for sodim (Pr =.54) ith the inner all heated and the oter adiabatic after [Rensen, 98] c) Simltaneosly developing flo Re l th /d h. 3 (laminar) 8. 6. 3 (laminar-trb. transition). 8. 3.7. 4 3.. 4 6. 5. 4 3. 6. 4 3.5 There appears to be little information available on simltaneosly developing flo in concentric annli. An integral analysis presented by [Roberts and Barro, 967] ho also condcted an etensive eperimental series fond that the reslts are not very mch different to those obtained for thermally developing flos ith a hydrodynamically developed velocity profile..7.5.6 Trblent heat transfer over rod bndles Since the sities the adeqate cooling performance of the fel elements sing liqid metals as a major isse for the feasibility of fast breeder reactors. Many eperimental investigations ere performed to stdy the heat transfer in trblent flo throgh rod bndles. They can be classified locally into three regions, the United States, the Eropean Union and the former USSR. Withot complaining completeness, nmeros eperimental heat transfer correlations have been developed, hich are valid in a ide range of geometrical, thermal and flo bondary conditions in the folloing papers: [Sbbotin, et al., 96,964,967,97], [Friedland, et al., 96a, b], [Borishanskii and Firsova, 963 and 964], [Maresca and Dyer, 964], [Nimmo and Dyer, 965], [Kalish and Dyer, 967], [Borishanskii, et al., 967, 969], [Pashek, 967], [Hlavac, et al., 969], [Zhkov, et al. 969a, b, 978, 98], [Marchese, 97], [Gräber and Rieger, 973], [Weinberg, 975], [Möller and Tschöke, 978]. The comprehensive overvie by [Weinberg, 975] on the eperimental stdies ehibits that the correlations obtained are etremely sensitive to depositions of imprities on the srface, the qestion of the etting especially to heavy liqid metals and the srface roghness. As one of his conclsions he mentioned that reliable correlations are only available for very fe cases. One of the problems associated ith the heat transfer over rod bndles is according to [Rehme, 987] that only in a fe eperiments the temperatres are measred locally at different rods and that the temperatre differences are often de to the high thermal condctivities of the liqid metals so small that reliable Nsselt nmber correlations are difficlt to derive from these data. Most of the eperiments ere performed in trianglar or heagonal arrays or bndles, hich arise from the design intended to se or sed in fast breeder reactors. The pitch (P) to pin (D) ratio, here P is the distance beteen the center of to neighboring fel rods and D is its diameter, ranges from. d P/D d.95 and the Peclet nmber from d Pe d 4.5. 3. In this contet rod bndles from 7 to 37 pins ere investigated. Only one eperiment is available in sqare arrays ith 9 heated rods in a clster of 5 by [Ushakov, et al., 963]. The problem of the trblent Prandtl nmber for the calclation of the Nsselt nmber has been already discssed in Sbsection.7.4 in the contet of the closre methods for the trblent heat fl. 458

For the liqid metal heat transfer in rod bndles and the case of vanishing Prandtl nmber [Zhkov, et al., 973] and [Ushakov, et al., 974] derived a general correlation. This correlation as first pblished by [Sbbotin, et al., 975] and etensions ere reported by [Ushakov, et al., 976, 977 and 979]. This general correlation as checked against nmeros reslts, both eperimental data and theoretical predictions and can be recommended for rod bndles arranged in a trianglar or heagonal arrangement and reads to: N N N ith m lam lam 3. 67 9 P 7. 55 D 3 > @ P / D «P / D. 56. 9 ª ««6 P D. 6. 3 P D 8 and. 4H º»» Pe. 5» ¼ 3. 6P / D 7 P D P D. 8. 86 P / D P / D. 5H 3. K K m (.9) The laminar Nsselt nmber N lam as originally proposed by [Sbbotin, 974]. The thermal modelling parameter H K takes into accont the effects of heat condction in the fel (O ), the cladding (O ) and the flid (O 3 ) and can be estimated by: H K O O 3 / / r / r O O ith / r / r O O (.) here r and r denotes the distances of the fel rod ros from the centre of the bndle. The limiting case of H K are. for peripherally constant heat fl and infinity for peripherally constant temperatre. Eq. (.9) is valid in the ranges from. d P/D d. and the Peclet nmber from d Pe d 4. 3 and. dh K < v. In Figre.7.7 the Nsselt nmber calclated after Eq. (.9) is shon as a fnction of the Peclet nmber for different P/D. Figre.7.7. Nsselt nmbers in flly developed flo in rod bndles arranged in a trianglar array as a fnction of the Peclet nmber Pe and P/D for constant all heat fl N 5 P/D=.5 P/D=.3 P/D=. P/D=. P/D=.5 5 P/D=. 3 5 8 3 5 8 Pe 459

In case P/D gros beyond.3 the effect of H K diminishes and Eq. (.9) redces to: N 3 P P 3. 67. 9. 7. 55 Pe D D 9 P D P D 56 (.) It is valid for. d P/D d. and from d Pe d 4. 3. The thermal hydralic design is sally performed by sbchannel codes, hich compte flid and srface temperatres averaged over each sbchannel. Therefore, it is important to kno the peripheral variations of the rod srface temperatres sperimposed on the average temperatre. The dimensionless peripheral variation of the srface temperatre is correlated by [Sbbotin, 974 and 975] and [Ushakov, 979] and reads to Eq. (.). Eq. (.) is valid in the ranges from. d P/D d.5, d Pe d. 3 and H K >.. 'T ma,lam is the maimm temperatre variation of the flly developed laminar flo in a bndle. [Rehme, 987] fond that the agreement beteen Eq. (.) and most of the eperimental data is better than %. Correlations of the Nsselt nmbers for heat transfer to liqid metals for design prposes hich are only of approimate natre can be taken from [Todreas and Kazimi, 993], [Kazimi and Carelli, 976] or [Tang, et al., 978]. In some ranges they are very pessimistic and according to [Rehme, 987] they shold not be sed for the design. ' T J ' T ' T ma ' T JPe. 8. 3H ma ma,lam O ma,lam 3 T E ma P D T q D 3( K ith and. P / D min ). 4 E and 5 log P. 65 P D ª ««. ep 6. 4( P D ) lnh «tanh. 99 «P D. 6 «. 84.. 6 D, K º»»»»» ¼ (.) In case of thermally developing flos along rod bndles the Nsselt nmber vales may significantly deviate from the circlar dct becase of the strong geometric non-niformity of the sbchannels. If the coolant area per rod is taken to define an eqivalent annls ith zero shear at its oter bondary, it is fond that the Nsselt nmber predictions are accrate ithin r% for P/D t., see e.g. [Nijsing, 97]. Especially, the entrance region is problematic since the entrance length is sensitive to Re, Pr and the geometry. For a hydralically flly developed flo only one eperiment is available by [Sbbotin, 975], here he detected that flly developed thermal conditions in sodim cold not be reached for l th /d h belo (!) for Pe >. Their eperimental data ere obtained for a P/D =.5. For loer P/D ratios the thermal developing length is even larger, becase the heat transfer beteen the sbchannels is redced. This has been confirmed by the measrement of [Möller and Tschöke, 98] obtained in a 9-rod bndle ith P/D =.3. For a corner sbchannel and Pe = 5 a thermal entrance length of l th /d h = 7 as measred. The se of fins redces the thermal entrance length considerably, since it leads to a better lateral miing. In the tetbook by [Stasilevivis and Skrinska, 988] different options for fins on the rods are discssed and optimisation gidelines are given. 46

.8 Some final remarks This report mainly focses on the forced convective single phase heat transfer of liqid metals in smooth dcts. Althogh the athor is aare that in many technical applications this rather idealised case ill hardly appear. Ths, this chapter reqires in the ftre an etension to the heat transfer appearing in mied and prely boyancy governed flos. A fe aspects also need a closer vie in contet ith all kinds of heat transfer in liqid metals. In recent years, many papers addressed the qestion of etting and the inflence of srface roghness, hich is sometimes taken as an eplanation for a mismatch of data. Althogh many eperimenters spend a lot of effort to avoid this isse, it obviosly leads to severe effects in thermal-hydralic components and as a conseqence to loer Nsselt nmbers. Bt, the data basis for a closed discssion on these to topics seemed to me too sparse either theoretically and eperimentally to dedicate an individal chapter to it. The natre of this chapter based only on the eperimental reslts old be rather of speclative natre. A topic also not considered p to no is the flo and the heat transfer arond obstacles, hich occrs in many reactor and metal refinement processes. Althogh here a mch broader data basis is available it is difficlt to strctre this rather comple contet in a short paragraph. Bt since it is important for the eperimental investigation of heat transport phenomena it shold be integrated in a net step of the liqid metal handbook. Many advanced reactor concepts make se of the gas lift principle by means of a convection enhancement injecting gas into the heavy liqid metal. The reslting to-phase flo is theoretically difficlt to describe de to the large density differences and crrently only limited accessible to the modelling. Nevertheless, also here a large variety of eperiments eist, hich ere performed in the contet of the specific technical configration sed for the individal technical application. Associated ith the to-phase flo problem is the treatment of free liqid metal srfaces, as they may appear in free srface targets of accelerator driven systems or other metallrgical processes. Here, mainly trblent flos appear and crrently a lot of effort is spent on the modelling of these free srfaces. The net isse of the liqid metal shold contain a chapter dedicated only to this problem. 46

REFERENCES Abe, K., T. Kondoh, Y. Nagano (995), A Ne Trblence Model for Predicting Flid Flo and Heat Transfer in Separating and Reattaching Flos. II. Thermal Field Calclations, Int. J. Heat Mass Transfer, Vol. 38, pp. 467-48. Addison, C.C. (984), The Chemistry of the Liqid Alkali Metals, J. Whiley & Sons, Ltd., ISBN 479589. Anderson, D.A., J.C. Tannehill, R.H. Pletcher (984), Comptational Flid Mechanics and Heat Transfer, Hemisphere, Ne York. Aoki, S. (963), A Consideration on the Heat Transfer in Liqid Metals, Bll. Tokyo Inst. Tech., Vol. 54, pp. 63-73. Arien, B. (4), Assessment of Comptational Flid Dynamic Codes for Heavy Liqid Metals ASCHLIM, EC-Contract FIKW-CT--8-Final Report. Arpaci, V.S., P.S. Larsen, Convective Heat Transfer, Prentice-Hall, Engleood-Cliffs, N.J. Azer, N.Z., B.T. Chao (96), A Mechanism of Trblent Heat Transfer in Liqid Metals, Int. J. Heat and Mass Transfer, Vol., p. ff. Azer, N.T., B.T. Chao (96), Trblent Heat Transfer in Liqid Metals Flly Developed Pipe Flo ith Constant Wall Temperatre, Int. J. Heat and Mass Transfer, Vol. 3, pp.77-83. Baker, R.A., A. Sesonske (96), Trblent Heat Transfer in Concentric Annli of a NaK Heat Echanger, Ncl. Science Eng., Vol. 3, p. 83ff. Barenblatt, G.I., A.J. Chorin (998), Trblence: An Old Challenge and Ne Perspectives, Meccanica, Vol. 33, pp. 445-468. Barleon, L., K-J. Mack, R. Stieglitz (996), The MEKKA Facility A Fleible Tool to Investigate MHD-flo Phenomena, FZKA-Report FZKA-58. Batchelor, G.K. (959a), Small-scale Variation of Convected Qantities like Temperatre in Trblent Flid Flos. Part. General Discssion and the Case of Small Condctivity, J. Flid Mech., Vol. 5, pp. 3-33. Batchelor, G.K., I.D. Hoells, A.A. Tonsend (959b), Small-scale Variation of Convected Qantities like Temperatre in Trblent Flid. Part. The Case of Large Condctivity, Jornal Flid Mech., Vol. 5, pp. 34-39. Becker, H.L. (956), Heat Transfer in Trblent Tbe Flo, Appl. Sci. Res., Ser. A, Vol. 6, pp. 47-9. 46

Beitz, W., K-H. Küttner (986), Dbbel-Taschenbch des Maschinenbas, Springer-Verlag, 5 th Edition, ISBN 3-54-48-7, p. 358 ff (in German). Benemann, J. (996), Stats Report on Solar Throgh Poer Plants, Pilkington Press. Betchov, R., O.W. Criminale (967), Stability of Parallel Flos, Academic Press, Ne York. Bird, R.B., W.E. Steart, E.N. Lightfoot (96), Transport Phenomena, Wiley, Ne York. Bhatti, M.S. (985), Flly Developed Temperatre Distribtion in a Circlar Dct Tbe ith Uniform Wall Temperatre, Oens-Corning Fiberglas Corporation, Granville, Ohio. Bhatti, M.S. (985b) Limiting Laminar Heat Transfer in Circlar and Flat Dcts by Analogy ith Transient Heat Transfer Problems, Oens Corning Fiberglas Corporation, Granville, Ohio. Bhatti, M.S., R.K. Shah (987), Trblent and Transition Flo, Convective Heat Transfer in Dcts, Chapter 4, in Handbook of Single-phase Convective Heat Transfer, (Kakac, Shah and Ang eds.), John Wiley and Sons, Ne York, ISBN -47-87-3. Black, A.W., Sparro, E.M. (967) Eperiments on Trblent Heat Transfer in a Tbe ith Circmferentially Varying Bondary Conditions, J. Heat Transfer, Vol. 89, pp. 58-68. Boelter, L.M.K., G. Yong, H.W. Iverson (948), An Investigation of Heaters XXVII Distribtion of Heat Transfer Rate in the Entrance Section of a Circlar Tbe, NACA TN45. Borishanskii, M.V., E.V. Firsova (963), Heat Echange in the Longitdinal Flo of Metallic Sodim Past a Tbe Bank, At. Energiya, Vol. 4, pp. 584-585. Borishanskii, M.V., E.V. Firsova (964), Heat Echange in Separated Bndles of Rods ith Metallic Sodim Floing Longitdinally, At. Energiya, Vol. 6, pp. 457-458. Borishanskii, M.V., M.A. Gotovskiy, E.V. Firsova (967), Heat Echange hen a Liqid Metal Flos Longitdinally Throgh a Bndle of Rods Arranged in a Trianglar Lattice, At. Energiya, Vol., pp. 38-3. Borishanskii, M.V., M.A. Gotovskiy, E.V. Firsova (969), Heat Transfer in Liqid Metals Floing Longitdinally in Wetted Rod Bndles, At. Energiya, Vol. 7, pp. 549-55. Boston, N.E.J., R.W. Brling (97), An Investigation of High-ave Nmber Temperatre and Velocity Spectra in Air, J. Flid Mech., Vol. 55, pp. 473-49. Bossinesq, J. (877), Theorie d écolement torbillant, Mem. Pres. Acad. Sci, Paris, Vol. 3, p. 46ff. Bremhost, K., L. Krebs (99), Eperimentally Determined Trblent Prandtl Nmbers in Liqid Sodim at Lo Reynolds Nmbers, Int. J. Heat Mass Transf., Vol. 35, pp. 35-359. Bron, H.E., B.H. Amstead, B.E. Short (957), Temperatre and Velocity Distribtion and Transfer of Heat in a Liqid Metal, Trans. Am. Soc. Mech. Engrs., Vol. 79, pp. 79-85. 463

Bhr, H.O. (967), Heat Transfer to Liqid Metals, ith Observations of the Effect of Sperimposed Free Convection in Trblent Flo, PhD Thesis, Univ. Cape Ton, Cape Ton, Soth Africa. Bhr, H.O., A.D. Carr, R.E. Balzhiser (968) Temperatre Profiles in Liqid Metals and the Effect of Sperimposed Free Convection in Trblent Flos, Int. J. Heat and Mass Transfer, Vol., pp. 64-654. Bleev, N.I. (959), Problems of Heat Transfer, Pblishing hose of the USSR Acad. Sci., Mosco. Bleev, N.I. (96), Theoretical Model of the Mechanism of Trblent Echange in Flid Flo, Coll. Heat Trans. USSR Acad. Sci., pp. 64-98. Brmeister, L.C. (983), Convective Heat Transfer, Wiley-Interscience, Ne York. Carteciano, L.N., D. Weinberg, U. Müller (997), Development and Analysis of a Trblence Model for Boyant Flos, 4 th World Conf. on Eperimental Heat Transfer, Flid Mechanics and Thermodynamics, -6, Brssels, Vol. 3, pp. 339-347. Carteciano, L.N., G. Grötzbach (3), Validation of Trblence Models in the Compter Code FLUTAN for a Free Hot Sodim Jet in Different Boyancy Flo Regimes, FZKA-Report, FZKA-66. Carteciano, L.N., G. Grötzbach (3), Validation of Trblence Models for a Free Hot Sodim Jet ith Different Boyancy Regimes Using the Compter Code FLUTAN, Forschngszentrm Karlsrhe FZKA-66. Cebeci. T. (973), A Model for Eddv Condctivitv and Trblent Prandtl Nmber, ASME Jornal of Heat Transfer, Vol. 95, pp. 7-34. Cebeci, T., P. Bradsha (977), Momentm Transfer in Bondary Layers, Hemisphere Pblishers, Washington. Chen, R.Y. (973), Flo in the Entrance Region at Lo Reynolds Nmber, J. Flids Engn., Vol. 95, pp. 53-58. Chen, C.J., J.S. Chio (98), Laminar and Trblent Heat Transfer in the Pipe Entrance Region for Liqid Metals, International Jornal of Heat and Mass Transfer, Vol. 4, pp. 79-89. Chien, K.Y. (98), Prediction of Channel of Bondary Layer Flos ith a Lo Reynolds Nmber Trblence Model, AIAA J., Vol. (), pp. 33-38. Cho, P.Y., J. Chin (94), Wave Nmber Analysis in Trblent Flo, J. Phys., Vol. 4, p. ff. Chrchill, S.W. (997a), Ne Simplified Models and Formlations for Trblent Flo and Convection, AIChE Jornal, Vol. 43, pp. 5-4. Chrchill, S.W. (997b), The Prediction of Trblent Convection ith Minimal Eplicit Empiricism, Thermal Science and Engineering, Vol. 5, pp. 3-3. Chrchill, S.W. (), The Prediction of Trblent Flo and Convection in a Rond Tbe, Advances in Heat Transfer, Vol. 34, pp. 55-36. 464

Chrchill, S.W., M. Shinoda, N. Arai (), A Ne Concept of Correlation for Trblent Convection, Thermal Science and Engineering, Vol. 3, pp. 49-65. Cole, J.D. (968), Pertrbation Methods in Applied Mathematics, Blaisdell Pblishers, Waltham. Danckerts, P.V. (95), Significance of Liqid-film Coeffcients in Gas Absorption, Ind. Engng. Chem., Vol. 43(6), pp. 46-467. David, S. (5), Ftre Scenarios for Fission Based Reactors, Nclear Physics A, Vol. 75, pp. 49-44. Davies, F. (969), Eddy Viscosity and Eddy Condctivity: A Statistical Approach and Eperimental Verification, PhD Thesis ETH-Zürich, No. 47 (969). Deissler, R.G. (95), Analysis of Flly Developed Trblent Heat Transfer in at Lo Peclet Nmbers in Smooth Tbes ith Application to Liqid Metals, Res. Memo E5F5 US Nat. Adv. Comm. Aero. Deng, B., W. W, S. Xi (), A Near-all To-eqation Heat Transfer Model for Wall Trblent Flos, Int. J. Heat Mass Transfer, Vol. 44, pp. 69-698. Drbin, P.A. (993a), A Reynolds Stress Model for Near-all Trblence, J. Flid Mech., Vol. 49, pp. 465-498. Donaldson, C. (97), Calclation of Trblent Shear Flos for Atmospheric and Vorte Motions, AIAA J., Vol., pp. 4-. Dchatelle, L., L. Vatrey (964), Determination des coefficients de convection dún alliage NaK en ecolement trblent emtre plasqes plates paralleles, Int. J. Heat and Mass Transfer, Vol. 7, pp. 7-3. Drbin, P.A. (993b), Application of Near-all Trblence Model to Bondary Layers and Heat Transfer, Int. J. Heat Flid Flo, Vol. 4, pp. 36-33. Dyer, O.E. (963), Eddy Transport in Liqid Metal Teat Transfer, A. I. Ch. E. J., Vol. 9, pp. 6-68. Dyer, O.E. (965), Heat Transfer Throgh Liqid Metals Floing Trblently Beteen Parallel Plates, Nc. Sci. Eng., Vol., pp. 79-89. Dyer, O.E. (966), Recent Developments in Liqid Metal Heat Transfer, Atomic Energy Revie, Wien, Vol. 4(), pp. 3-94. Dyer, O.E. (976), Liqid Metal Heat Transfer, Sodim-NaK Engineering Handbook, Vol., pp. 73-9. Eckert, E.R.G., R.M. Drake (97), Analysis of Heat and Mass Transfer, McGra-Hill Book Corporation, Ne York. El Hadidy, M.A., F. Gori, D.B. Spalding (98), Frther Reslts on the Heat Transfer to Lo Prandtl Nmber Flids in Pipes, Nm. Heat Transfer, Vol. 5, pp. 7-7. 465

Faggiani, S., Gori, F. (98), Inflence of Streamise Moleclar Heat Condction of the Heat Transfer Coefficient for Liqid Metals in Trblent Flo Beteen Parallel Plates, J. Heat Transfer, Vol., pp. 9-96 Fost, O.J (97), Sodim-NaK Engineering Handbook, Vol., Sodim Chemistry and Physical Properties, Gordon and Breach Science Pblishers, ISBN67734. Friedland, A.J., O.E. Dyer, W.M. Maresca, C.F. Bonilla (96a), Heat Transfer to Mercry in a Parallel Flo Throgh Rod Bndles of Circlar Rods, Int. Dev. in Heat Transfer, Am. Soc. Mech. Eng., Ne York, Part III, pp. 56-534. Friedland, A.J., C.F. Bonilla (96b), Analytical Stdy of Heat Transfer Rates for Parallel Flo of Liqid Metals Throgh Tbe Bndles. Part II, AIChE J., Vol. 7, pp. 7-. Fchs, H. (973), Wärmeübergang an strömendes Natrim, Eidg. Institt für Reaktorforschng, Würelingen, Bericht Nr. 4, Scheiz. Gardner, R.A. (969), Magneto Flid Mechanic Pipe Flo in a Transverse Magnetic Field ith and ithot Heat Transfer, PhD Thesis Prde University, West Lafayette, Ind. Gartner, D., K. Johannsen, H. Ramm (974), Trblent Heat Transfer in a Circlar Tbe ith Circmferrentially Varying Thermal Bondary Conditions, Int. J. Heat and Mass Transfer, Vol. 7, pp. 3-8. Gebhardt, B. (97), Heat Transfer, MacGra Hill Book Corp., Ne York. Genin, L.G., E.V. Kdryavtseva, Y.A. Pakhotin, V.G. Sviridov (978), Temperatre Fields and Heat Transfer for a Trblent Flo of Liqid Metal on an Initial Thermal Section, Teplofiz. Vysokikh. Temp., Vol. 6 (6), pp. 43-49. Gibson, C.H., W.H. Scharz (963a), The Universal Eqilibrim Spectra of Trblent Velocity and Scalar Fields, J. Flid Mech., Vol. 6, pp. 365-384. Gibson, C.H., W.H. Scharz (963b), Detection of Condctivity Flctations in a Trblent Flo Field, J. Flid Mech., Vol. 6, pp. 357-364. Gilliland, E.R., R.J. Msser, W.R. Page (95), Heat Transfer to Mercry, Gen. Disc. on Heat Transfer, Inst. Mech. Eng. and ASME, London, pp. 4-44. Gräber, H., M. Rieger (973), Eperimentelle Unterschng des Wärmeübergangs an Flüssigmetalle (NaK) in parallel drchströmten Rohrbündeln bei konstanter nd eponentieller Wärmeflssdichteverteilng, Atomkernenergie, Vol. 9, pp. 3-4. Graetz, L (883), Über die Wärmeleitngsfähigkeiten von Flüssigkeiten. Part, Ann. Phys. Chem., Vol. 8, pp. 79-94. Graetz, L. (885), Über die Wärmeleitngsfähigkeiten von Flüssigkeiten, Part, Ann. Phys. Chem., Vol. 5, pp. 337-357. Grant, H.L., B.A. Hghes, W.M. Vogel, A. Moilliet (968), The Spectrm of Temperatre Flctations in Trblent Flo, J. Flid Mech., Vol. 34, pp. 43-44. 466

Grigll, U., Tratz, H. (965), Thermischer Einlaf in asgebildeter laminarer Rohrströmng, Int. J. Heat and Mass Transfer, Vol. 8, pp. 669-678. Grötzbach, G., M. Wörner (99), Analysis of Second Order Transport Eqations by Nmerical Simlation of Trblent Convection in Liqid Metals, Proc. 5 th Int. Topical Meeting on Nclear Reactor Thermal-hydralics (NUETH-5), Am. Nc. Soc., Vol., pp. 358-365. Grötzbach, G., A. Batta, C-H. Lefhalm, I. Otic (4), Challenges in Thermal and Hydralic Analyses of ADS Target Systems, 6 th Int. Conf. on Nclear Thermal Hydralics, Operations and Safety (NUTHOS6), Nara, Japan, 4-8 Oct. 4, paper ID N6P5. Go, J., P.Y. Jlien (3), Modified Log-ake La for Trblent Flo in Smooth Pipes, Jornal of Hydralic Research, Vol. 4, No. 5, pp. 493-5. Hanjalic, K., B.E. Lander (97), A Reynolds Stress Model of Trblence and its Application to Asymmetric Shear Flos, J. Flid Mech., Vol. 5, pp. 69-638. Harlo, F.H., P.I. Nakayama (968), Transport of Trblent Energy Decay Rate, Los Alamos Scientific Laboratory Report LA 3584. Hartnett, J.P., T.F. Irvine (957), Nsselt Vales for Estimating Liqid Metal Heat Transfer in Non-circlar Dcts, AIchE J., Vol. 3, pp. 33-37. Hatton, A.P., A. Qarmby (963), The Effect of Aially Varying and Unsymmetrical Bondary Conditions on Heat Transfer ith Trblent Flo Beteen Parallel Plates, Int. J. Heat and Mass Transfer, Vol. 6, pp. 93-94. Hatton, A.P., A. Qarmby, I. Grndy (964), Frther Calclations on the Heat Transfer ith Trblent Flo Beteen Parallel Plates, Int. J. Heat and Mass Transfer, Vol. 7, pp. 83-83. Hasen, H. (943), Darstellng des Wärmeübergangs in Rohren drch verallgemeinerte Potenzbeziehngen, VDI-Zeitng, Sppl. Verfahrenstechnik, No. 4, pp. 9-98. Heng,L., C. Chan, S.W. Chrchill (998), Essentially Eact Characteristics of Trblent Convection in a Rond Tbe, Chemical Engineering Jornal, Vol. 7, pp. 63-73. Hennecke, D.K. (968), Heat Transfer by Hagen-Poiselle Flo in the Thermal Development Region ith Aial Condction, Wärme-nd Stoffübertragng, Vol., pp. 77-84. Hickman, H.J (974), An Asymptotic Stdy of the Nsselt-Graetz Problem, Part : Large Behavior, Int. J. Heat and Mass Transfer, Vol. 96, pp. 354-358. Hlavac, P.J., O.E. Dyer, M.A. Helfant (969), Heat Transfer to Mercry Floing In-line Throgh an Unbaffled Rod Bndle: Eperimental Stdy on the Effect of Rod Displacement on Rod-averaged Heat Transfer Coefficients, J. Heat Transfer, Vol. 9, pp. 568-58. Hornbeck, R.W. (964), Laminar Flo in the Entrance Region of a Pipe, Appl. Sci. Res., Vol. A3, pp. 4-3. Horsten, E.A. (97), Combined Free and Forced Convection on Trblent Flo of Mercry, PhD Thesis, University of Cape Ton, Cape Ton, Soth Africa. 467

Hs, C.J. (965), Heat Transfer in a Rond Tbe ith Sinsoidal Wall Heat Fl Distribtion, AIchE J., Vol., pp. 69-695. Hs, C.J. (968), Eact Soltion to Entry Region of Laminar Heat Transfer ith Aial Condction and the Bondary Conditions of the Third Kind, Chem. Eng. Sci., Vol. 3, pp. 457-468. Jackson, J.D. (983), Trblent Mied Convection Heat Transfer to Liqid Sodim, Int. J. Heat Flid Flo, Vol. 4, pp. 7-. Jacoby, J.K. (97), Free Convection Distortion and Eddy Diffsivity Effects in Trblent Mercry Heat Transfer, MS Thesis, Prde University, West Lafayette, Indiana. Jenkins, R. (95), Variation of the Eddy Condctivity ith Prandtl Modls and its Use in Prediction of Trblent Heat Transfer Coefficients, Proc. of the Heat Transfer and Flid Mechanics Institte, Stanford University Press, Stanford, CA. Jischa, M., H.B. Rieke (979), Abot the Prediction of Trblent Prandtl Nmbers and Schmidt Nmbers from Modelled Transport Eqations, Int. J. Heat and Mass Transfer, Vol.. pp. 547-555. Jischa, M. (98), Konvektiver Impls-, Wärme- nd Stoffastasch, Vieeg Verlag, Branscheig, ISBN 3-58-844-9 (in German). Jones, W.P., B.E. Lander (97), The Prediction of Laminarization ith a To-eqation Model of Trblence, Int. J. Heat and Mass Transfer, Vol. 5, pp. 3-34. Kalish, S., O.E. Dyer (967), Heat Transfer to NaK Floing Throgh Unbaffled Rod Bndles, Int. J. Heat and Mass Transfer, Vol., pp. 533-558. Karcz, M., J. Badr (5), An Alternative To-eqation Trblent Heat Diffsivity Closre, Int. J. of Heat and Mass Transfer, Vol. 48, pp. 3-. Kasagi, N., Y. Tomita, A. Kroda (99), Direct Nmerical Simlation of Passive Scalar Field in a Trblent Channel Flo, ASME J. Heat Transfer, Vol. 44, pp. 598-66. Kasagi, N., Y. Ohtsbo (993), Direct Nmerical Simlation of Lo Prandtl Nmber Thermal Field in a Trblent Channel Flo, in Trblent Shear Flos, Vol. 8, Springer, Berlin, pp. 97-9. Kaamra, H. (995) Direct Nmerical Simlation of Trblence by Finite Difference Scheme, in The Recent Developments in Trblence Research, International Academic Pblishers, pp. 54-6. Kaamra, H., K. Ohsaka, H. Abe, K. Yamamoto (998), DNS of Trblent Heat Transfer in Channel Flo ith Lo to Medim-high Prandtl Nmber Flid, Int. J. of Heat and Flid Flo, Vol. 9, pp. 48-49. Kaamra, H., H. Abe, Y. Matso (998), DNS of Trblent Heat Transfer in Channel Flo ith Respect to Reynolds and Prandtl Nmber Effects, Int. J. of Heat and Flid Flo, Vol., pp. 96-7. Kays, W.M., E.Y. Leng (963), Heat Transfer in Annlar Passages: Hydrodynamically Developed Trblent Flo ith Arbitrarily Prescribed Wall Heat Fl, Int. J. Heat and Mass Transfer, Vol. 6, pp. 537-557. 468

Kays, W.M., M.E. Craford (993), Convective Heat and Mass Transfer, 3 rd Ed. McGra-Hill, Ne York. Kays, W.M. (994), Trblent Prandtl Nmber Where Are We?, Trans. of the ASME, Vol. 6, pp. 85-95. Kazimi, M.S., M.D. Carelli (976), Heat Transfer Correlations for Analysis of CRBRP Assemblies, CRBRP-ARD-34, Westinghose Electric Corp., Madison, Penn. Keenan, J. (94), Thermodynamics, Wiley, Ne York. Kerr, R.M. (99), Velocity, Scalar and Transfer Spectra in Nmerical Trblence, J. Flid Mech., Vol., pp. 39-33. Kim, J., P. Moin (989), Transport of Passive Scalars in a Trblent Channel Flo, Proc. 6 th Int. Symp. on Trblent Shear Flos, Tolose, 7-9 Sept. 989, Springer Verlag Berlin-Heidelberg, pp. 85-96. Kim, J., P. Moin (989), Transport of Passive Scalars in a Trblent Channel Flo, in Trblent Shear Flos, Vol. 6, Springer, Berlin, pp. 85-96. Kndsen, J.G., D.L. Katz (958), Flid Dynamics and Heat Transfer, MacGra Hill Book Corp., Ne York. Koalski, D.J. (974), Free Convection Distortion in Trblent Mercry Pipe Flo, M.S. Thesis, Prde Univ., West Lafayette. Kraichnan, R.H. (97), Inertial-range Transfer in To- and Three-dimensional Trblence, J. Flid Mech., Vol. 47, pp. 55-535. Knz, H.R., S. Yeraznis (969), An Analysis of Film Condensation, Film Evaporation and Single Phase Heat Transfer for Liqid Prandtl Nmbers from 3 to 4, J. Heat Transfer, Vol. 9C, pp. 43-4. Lamb, H. (945), Hydrodynamics, Dover, Ne York. La, S.C., E.M. Sparro, J.W. Ramsey (98), Effect of Plenm Length and Diameter on Trblent Heat Transfer in a Donstream Tbe and on Plenm Related Pressre Losses, J. Heat Transfer, Vol. 3, pp. 45-4. Lander, B.E., D.B. Spalding (974), The Nmerical Comptation of Trblent Flos, Comp. Meth. Appl. Mech. Eng., Vol. 3, pp. 69-89. Lander, B.E., G.J. Reece, W. Rodi (975), Progress in the Development of a Reynolds Stress Trblence Closre, J. Flid Mech., Vol. 68, pp. 537-566. Lee, S.L. (983), Liqid Metal Heat Transfer in Trblent Pipe Flo ith Uniform Heat Fl, Int. J. Heat and Mass Transfer, Vol. 6, pp. 349-356. 469

Lefhalm, C.H., N.I. Tak, G. Groetzbach, H. Piecha, R. Stieglitz (3), Trblent Heat Transfer Along a Heated Rod in Heavy Liqid Metal Flo, Proceedings of the th International Topical Meeting on Nclear Reactor Thermal Hydralics (NURETH-), Seol, Korea, 5-9 October 3. Lefhalm, C-H., N-I. Tak, H. Piecha, R. Stieglitz (4), Trblent Heavy Liqid Metal Heat Transfer Along a Heated Rod in an Annlar Cavity, Jornal of Nclear Materials, Vol. 335, pp. 8-85. Leslie, D.C. (977), A Recalclation of Trblent Heat Transfer to Liqid Metals, Lett. Heat and Mass Transfer, Vol. 4, pp. 5-33. Leveqe, M.A. (98), Le lois de la transmission de chaler par convection, Ann. Mines, Mem., Ser., Vol. 3, pp. -99, pp. 35-36, pp. 38-45. Lin, C.C. (96), The Theory of Hydrodynamic Stability, Cambridge University Press. Lin, B.S., C.C. Chang, C.T. Wang (), Renormalization Grop Analysis for Thermal Trblent Transport, Physical Revie E, Vol. 63, pp. 634-63. Lbarski, B., S.J. Kafman (955), Revie of Eperimental Investigations of Liqid Metal Heat Transfer, NACA Technical Note, TN-3336. Lykodis, P.S., Y.S. Tolokian (958), Heat Transfer in Liqid Metals, Trans. Am. Soc. Mech. Engnrs., Vol. 8, pp. 653-666. Lyon, R.N. (949), Forced Convection Heat Transfer Theory and Eperiments ith Liqid Metals, USAEC Report, ORNL-36, Oak Ridge National Laboratory. Lyon, R.N. (95), Liqid Metal Heat Transfer Coefficients, Chem. Engng. Progr., Vol. 47 (), pp. 75-79. Lyon, R.N. (95), Liqid Metals Handbook, Naveos P-733, nd ed. Lyons, S.L., T.J. Hanratty, J.B. McLaghlin (99), Direct Nmerical Simlation of Passive Heat Transfer in a Trblent Channel Flo, Int. J. Heat Mass Transfer, Vol. 34, pp. 49-6. Malang, S., L. Barleon, L. Bühler, H. Deckers, S. Molokov, U. Müller, P. Norajitra, J. Reimann, H. Reiser, R. Stieglitz (99), MHD Work on Self-cooled Liqid-metal Blankets Under Development at the Nclear Research Center Karlsrhe, Perspectives in Energy, Vol., pp. 33-3. Mancea, R., S. Parnei, D. Larence (), Trblent Heat Transfer Predictions Using the v f Model on Unstrctred Grids, Int. J. Heat Flid Flo, Vol., pp. 3-38. Marchese, A.R. (97), Eperimental Stdy of Heat Transfer to NaK Floing In-line Throgh a Tightly Packed Rod Bndle, 3 th Nat. Heat Transfer Conf., Denver, Col., AIChE Paper No.36, Am. Inst. Chem. Eng. Maresca, M.W., O.E. Dyer (964), Heat Transfer to Mercry Floing In-line Throgh a Bndle of Circlar Tbes, J. Heat Transfer, Vol. 86, pp. 8-96. Martinelli, R.C. (947), Heat Transfer to Molten Metals, Trans. of the ASME, Vol. 69, p. 947ff. 47

Michelson, M.L, J. Villadsen (974), The Graetz Problem ith Aial Heat Condction, Int. J. Heat and Mass Transfer, Vol. 7, pp.39-4. Mills, A.F. (96), Eperimental Investigation of Heat Transfer in the Entrance Region of a Circlar Condit, J. Mech. Eng. Sci., Vol. 4, pp. 63-77. Mizshina, T., T. Sasano (963), The Ratio of the Eddy Diffsivities of Heat and Momentm and its Effect on the Liqid Metal Heat Transfer, Int. Dev. Heat Trans., pp. 66-668. Mizshina, T., R. Ito, F. Ogina, H. Mramoto (969), Eddy Diffsivities for Heat in the Region Near the Wall, Mem. Fac. Engng. Kyoto Univ., Vol. 3, pp. 69-8. Möller, R., H. Tschöke (978), Local Temperatre Distribtions in the Critical Dct Wall Zones of LMFBR Core Elements. A Stats of Knoledge, Unsettled Problems and Possible Soltions, Proc. ANS/ASME Int. Top. Meet. Nc. Reac. Thermal Hydralics, Saratoga, NY, Vol. 3, pp. 87-88. Molki, M., E.M. Sparro (983), In-tbe Heat Transfer for Skeed Inlet Flo Cased by Competition Among Tbes Fed by the Same Plenm, J. Heat Transfer, Vol. 5, pp. 87-877. Nagano, Y., C. Kim (988), A To-eqation Model for Heat Transport in Wall Trblent Shear Flos, Trans. ASME, J. Heat Transfer, Vol., p. 583ff. Nagano, Y., M. Shimada, M.S. Yossef (994), Progress in the Development of a To-eqation Heat Transfer Model Based on DNS Data Bases, Proc. Int. Symp. on Trblence, Heat and Mass Transfer, Lisbon, 9- Ag. 994, pp. 3..-3..6 Nagano,Y., M. Shimada (995), Rigoros Modeling of Dissipation-rate Eqation Using Direct Simlations, Jpn. Soc. Mech. Eng. Int. J. Ser. B, 38, p. 5. Nagano, Y. (), Modelling Heat Transfer in Near-all Flos, in Closre Strategies for Modelling Trblent and Transitional Flos, B.E. Lander, N.D. Sandham (Eds.), Cambridge University Press, Cambridge. Ng, K.H., D.B. Spalding (97), Trblence Model for Bondary Layers Near Walls, Phys. Flids, Vol. 5, pp. -3. Nimmo, B., O.E. Dyer (965), Heat Transfer to Mercry Floing In-line Throgh a Rod Bndle, J. Heat Transfer, Vol. 87, pp. 3-33. Nimmo, B., O.E. Dyer (966), Eperimental Stdy on Trblent Flo of Mercry in Annli, Nc. Sci. Eng., Vol. 9, p. 48ff. Nijsing, R. (97), Heat Echange and Heat Echangers ith Liqid Metals, AGRD-LS-57-, AGRD Lectre Series No. 57, on Heat Echangers by J.J. Gino. Notter, R.H. (969), To Problems in Trblence, PhD Thesis, University of Washington, Seattle. Notter, R.H., C.H. Sleicher (97), A Soltion to the Trblent Graetz-Problem III Flly Developed and Entrance Region Heat Transfer Rates, Chem. Eng. Sci., Vol. 7, 73-93. 47

Ooi, A., G. Iaccarino, P.A. Drbin, M. Behnia (), Reynolds Averaged Simlation of Flo and Heat Transfer in Ribbed Dcts, Int. J. Heat Flid Flo, Vol. 3, pp. 75-757. Otic, I., G. Grötzbach, M. Wörner (5), Analysis and Modelling of the Temperatre Variance Eqation in Trblent Natral Convection for Lo-Prandtl-nmber Flids, J. Flid Mech., Vol. 55, pp. 37-6. Pashek, M. (967), Stdy of Local Heat Transfer and Thermal Fields in a Seven Rod Bndle ith Aial Flo of Sodim, UJV-85, Nc. Res. Inst., CSSR. Petrovichev, V.I. (959), Heat Transfer in Concentric Annli, Atomn. Energ., Vol. 7, pp. 366-369. Piller, M., E. Nobile, T.J. Hanratty, (), DNS Stdy of Trblent Transport at Lo Prandtl Nmbers in a Channel Flo, J. Flid Mech., Vol. 458, pp. 49-44. Prandtl, L. (98), Bemerkngen über den Wärmeübergang in einem Rohr, Phys. Z., Vol. 9, pp. 487-489. Qarmby, A., R.K. Anand (97), Trblent Heat Transfer in the Thermal Entrance Region of Concentric Annli ith Uniform Wall Heat Fl, Int. J. Heat and Mass Transfer, Vol. 3, pp. 395-4. Qarmby, A., R. Qirk (97), Measrements of the Radial and Tangential Eddy Diffsivities of Heat and Mass in Trblent Flo in a Plain Tbe, Int. J. Heat and Mass Transfer, Vol. 5, pp. 39-37. Qarmby, A., R. Qirk (974), Aisymmetric and Non-aisymmetric Trblent Diffsion in a Plain Circlar Tbe at High Schmidt Nmber, Int. J. Heat and Mass Transfer, Vol. 7, pp. 43-48. Ramm, H., K. Johannsen (973), Radial and Tangential Trblent Diffsivities of Heat and Momentm in Liqid Metals, Prog. Heat and Mass Transfer, Vol. 7, pp. 45-58. Reed, C.B. (987), Convective Heat Transfer in Liqid Metals, Chapter 9 from Handbook of Singlephase Convective Heat Transfer, S. Kakac, R. Shah, W. Ang (Eds.), John Wiley & Sons, ISBN 47-87--3. Rehme, K. (987), Convectvie Heat Transfer Over Rod Bndles, Kakac, Shah, Ang (Eds.), John Wiley & Sons, Ne York, ISBN -47-87-3. Rensen, Q. (98), Eperimental Investigation of the Trblent Heat Transfer to Liqid Sodim in the Thermal Entrance Region of an Annls, Nc. Eng. Design, Vol. 68, pp. 397-44. Reynolds, W.C. (963), Trblent Heat Transfer in a Circlar Tbe ith Variable Circmferential Heat Fl, Int. J. Heat and Mass Transfer, Vol. 6, pp.445-454. Reynolds, W.C. (976), Comptation of Trblent Flos, Ann. Rev. Flid Mech., Vol. 8, pp. 83-8. Roberts, A., H. Barro (967), Trblent Heat Transfer in the Vicinity of the Entry of an Internally Heated Annls, Proc. Inst. Mech. Engnrs., Vol. 8, Pt3H, pp. 69-76. Rodi, W. (98), Trblence Models and Their Application in Hydralics, Monograph Int. Association of Hydralic Research, Delft, The Netherlands. 47

Rodi, W. (98), Progress in Trblence Modelling for Incompressible Flos, AIAA, Paper 8-45. Rosheno, W.M., L.S. Cohen (96), Trblent Transfer of Heat, MIT Heat Transfer Lab Report. Rohseno, W.M., H. Choi (96), Heat, Mass and Momentm Transfer, Prentice Hall, Engleood- Cliffs, Ne Jersey. Rotta, J.C. (97), Trblente Strömngen, B.G. Tebner Stttgart. Sakakibara, M., K. Endo (976), Analysis of Heat Transfer for Trblent Flo Beteen Parallel Plates, Int. Chem. Eng., Vol. 8, pp. 78-733. Sakakibara, M. (98), Analysis of Heat Transfer in the Entrance Region ith Flly Developed Trblent Flo Beteen Parallel Plates The Case of Uniform Wall Heat Fl, Mem. Fac. of Eng. Fki Univ., Vol.3 (), pp. 7-. Schlichting, H., Trckenbrodt (96), Aerodynamik des Flgzegs, Springer Verlag, Berlin. Schlichting, H., (979), Bondary Layer Theory, 7 th ed., McGra-Hill, Ne York. Schrock, S.L. (964), Eddy Diffsivity Ratio in Liqid Metals, PhD Thesis, Prde Univ., West Lafayette, Ind. Schlz, B. (986), Thermophysical Properties in the System LiPb, KfK-444. Shikazono, N., N. Kasagi (996), Second-moment Closre for Trblent Scalar Transport at Varios Prandtl Nmbers, Int. J. Heat Mass Transfer, Vol. 39, pp. 977-987. Seban, R.A. (95), Heat Transfer Throgh a Flid Floing Trblently Beteen Parallel Walls ith Asymmetric Wall Temperatres, Trans. ASME, Vol. 7, pp. 789-795. Seban, R.A., T.T. Shimazaki (95), Heat Transfer to a Flid Floing Trblently in a Smooth Pipe ith Walls at Constant Temperatre, Trans. ASME, Vol. 73, pp. 83-89. Senechal, M. (968), Contribtion to Convection Theory for Liqid Metals, Thèse 3e cycle, Paris. Shah, R.K., A.L. London (978), Laminar Flo Forced Convection in Dcts, Spplement to Advances in Heat Transfer, Academic, Ne York. Shah, R.K., R.S. Johnson (98), Correlations for Flly Developed Trblent Flo Throgh Circlar and Non-circlar Channels, Proc. 6 th Int. Heat and Mass Transfer Conf., Indian Inst. of Techn., Madras, India, D75-D95. Sheriff, N., D.J. O Kane (98), Sodim Eddy Diffsivity and Heat Measrements in a Circlar Dct, Int. J. Heat Mass Transfer, Vol. 4, pp. 5-. Shibani, A.A., M.N. Özisik (977), A Soltion to Heat Transfer in Trblent Flo Beteen Parallel Plates, Int. J. Heat and Mass Transfer, Vol., pp. 565-573. Siegel, R., E.M. Sparro, T.M. Hallman (958), Steady Laminar Heat Transfer in a Circlar Tbe ith Prescribed Wall Heat Fl, Appl. Sci. Res., Vol. A7, pp. 386-39. 473

Skpinski, E., J. Tortel, L. Vatrey (965), Determination des coefficients de convection d n allage sodim-potassim dans n tbe circlaire, Int. J. Heat and Mass Transfer, Vol. 8, pp.937-95. Sleicher, C.A., M. Tribs (956), Heat Transfer in a Pipe ith Trblent Flo and Arbitrary Wall Temperatre Distribtions, 956 Heat Transfer and Flid Mechanics Institte, Stanford Univ. Press, Stanford, CA, pp. 59-78. Sleicher, C.A., M. Tribs, Jr. (957), Heat Transfer in a Pipe ith Trblent Flo and Arbitrary Wall Temperatre Distribtion, Trans. ASME, Vol. 79, pp. 789-797. Sleicher, C.A. (958), Eperimental Velocity and Temperatre Profiles for Air in a Trblent Pipe Flo, Trans. Am. Soc. Mech. Engns., Vol. 8, pp. 693-73. Sleicher, C.A., A.S. Aad, R.H. Notter (973), Temperatre and Eddy Diffsivity Profiles in NaK, Int. J. Heat and Mass Transfer, Vol. 6, pp. 565-575. Sleicher, C.J., M.W. Rose (975), A Convenient Correlation for Heat Transfer to Constant and Variable Property Flids in Trblent Pipe Flo, Int. J. Heat Mass Trans., Vol. 8, pp. 677-683. Smith, D.L., et al. (984), Blanket Comparison and Selection Stdy, Final report; ANL/FPP-84-, Vol., Chapter 6. Sommer, T.P., R.M. So, Y.G. Lai (99), A Near-all To-eqation Model for Trblent Heat Fles, Int. J. Heat Mass Transfer, Vol. 35, pp. 3375-3387. Sparro, E.M., S.H. Lin (963), Trblent Heat Transfer in a Tbe ith Circmferentially Varying Temperatre and Wall Heat Fl, Int. J. Heat and Mass Transfer, Vol. 6, pp. 866-867. Sparro, E.M., S.V. Patankar, H. Shahrestani (978), Laminar Heat Transfer in a Pipe Sbjected to Circmferentially Varying Eternal Heat Transfer Coefficients, Nm. Heat Transfer, Vol., pp. 7-7. Sparro, E.M., K.K. Moram, M. Charmchi (98), Heat Transfer and Pressre Drop Characteristics Indced by a Slat Blockage in a Circlar Tbe, J. Heat Transfer, Vol., pp. 64-7. Sparro, EM., J.E.O. O Brien (98), Heat Transfer Coefficients on the Donstream Face of an Abrpt Enlargement or Inlet Constriction in a Pipe, J. Heat Transfer, Vol., pp. 48-44. Sparro, E.M., M. Molki, S.R. Chastain (98), Trblent Heat Transfer Coefficients and Flid Flo Patterns on the Srface of a Centrally Positioned Blockage in a Dct, Int. J. Heat and Mass Transfer, Vol. 4, pp. 57-59. Sparro, E.M., L.D. Bosmans (983), Heat Transfer and Flid Flo Eperiments ith a Tbe Fed by a Plenm Having Non-aligned Inlet and Eit, J. Heat Transfer, Vol. 5, pp. 56-63. Sparro, E.M., U. Grdal (983), Heat Transfer at an Upstream Facing Srface Washed by Flid En Rote to an Apertre in the Srface, Int. J. Heat and Mass Transfer, Vol. 4, pp. 85-857. Sparro, E.M., A. Chaboki (984), Sirl Affected Trblent Flid Flo and Heat Transfer in a Circlar Tbe, J. Heat Transfer, Vol. 6, pp. 766-773. 474

Speziale, C.G., R.M.C. So (998), Trblence Modelling and Simlation, in The Handbook of Flid Dynamics, R.W. Johnson (Ed.), CRC Press LLC, Boca Raton, Florida, pp. 4. 4.. Stasilevivis, J., A. Skrinska (988), Heat Transfer of Finned Tbe Bndles in Crossflo, Hemisphere Pblishing Corporation, ISBN--896-36-3. Sbbotin, V.I., P.A. Ushakov, B.N. Gabrianovich, A.V. Zhkov (96), Heat Echange Throgh the Flo of Mercry and Water in a Tightly Packed Rod Pile, Sov. At. Energy, Vol. 9, pp. -9. Sbbotin, V.I., A.K. Papovyants, P.L. Kirillov, N.N. Ivanovskii (96), A Stdy of Heat Transfer to Molten Sodim in Tbes, At. Energiya (USSR), Vol. 3, pp. 38-38. Sbbotin, V.I., P.A. Ushakov (964), Heat Removal from Reactor Fel Elements Cooled by Liqid Metals, Proc. 3 rd Int. Conf. of Peacefl Uses of Atomic Energy, Geneva, Vol. 8, United Nations, Ne York, pp. 9-3. Sbbotin, V.I., P.A. Ushakov, A.V. Zhkov, V.D. Talanov (967), The Temperatre Distribtion in Liqid Metal Cooled Fel Elements, At. Energiya, Vol., pp. 37-378. Sbbotin, V.I., P.A. Ushakov, A.V. Zhkov, E.Y. Sviridenko (97), Temperatre Fields of Fel Elements in the BOR Reactor Core, Sov. At. Energy, Vol. 8, pp. 6-6. Sbbotin, V.I. (974), Heat Echange and Hydrodynamics in Channels of Comple Geometry, Proc. 5 th Int. Heat Transfer Conf., Tokyo, Vol. 6, pp. 89-4. Sbbotin, V.I., P.A. Ushakov, A.V. Zhkov, N.M. Matykhin, Y.S. Jr év, L.K. Kdryatseva (975) Heat Transfer in Cores and Blankets of Fast Breeder Reactors A Collection of Reports, Symp. of CMEA Contries: Present and Ftre Work on Creating AES ith Fast Reactors, Obninsk, Vol.. Sbbotin, V.I., M.K. Ibragimov, M.N. Ivanovskii, M.N. Arnolbov, E.V. Nomofilov (99), Heat Transfer from the Trblent Flo of Heavy Liqid Metals in Tbes, The Soviet Jornal of Atomic Energy, Vol. (), pp. 769-775. Tang, Y.S., R.D. Coeffield, R.A. Markley (978), Thermal Analysis of Liqid Metal Fast Breeder Reactors, Am. Nc. Soc. Taylor, G.I. (95), Transport of Vorticity and Heat Throgh Flids in Trblent Motion, Phil. Trans. Roy. Soc. London, A5, pp. -6. Tien, C.L. (96), On Jenkins Model of Eddy Diffsivities for Momentm and Heat, J. Heat Transfer, Vol. 83C, pp. 389-39. Todreas, N.E., M.S. Kazimi (993), Thermalhydralic Fndamentals Nclear Systems I, Francis and Taylor, ISBN -896-935-. Tricoli, V. (999), Heat Transfer in Trblent Pipe Flo Revisited Similarity La for Heat and Momentm Transport in Lo Prandtl Nmber Flids, International Jornal of Heat and Mass Transfer, Vol. 3, pp. 535-54 Tyldesley, J.R., R.S. Silver (968), The Prediction of Transport Properties of a Trblent Flid, Int. J. Heat and Mass Transfer, Vol., pp. 35-34. 475

Tyldesley, J.R. (969), Trasnport Phenomena in Free Trblent Flos, Int. J. Heat and Mass Transfer, Vol., pp. 489-496. Ushakov, P.A., V.I. Sbbotin, B.N. Gabrianovich, V.D. Talanov, E.Y. Sviridenko (963), Heat Transfer and Thermalhydralic Resistance in Tightly Packed Corridor Bndle of Rods, Sov. At. Energy, Vol. 3, pp. 76-768. Ushakov, P.A. (974), Calclation of the Temperatre Fields of Bndles of Fel Elements in Aial Trblent Flos of Heat Transfer Media ith Vanishing Small Prandtl Nmber, High Temp., Vol., pp. 677-685. Ushakov, P.A, A.V. Zhkov, N.M. Matykhin (976), Thermal Fields of Rod-type Fel Elements Sitated in Rectanglar Lattices Streamlined by a Heat Carrier, High Temp., Vol. 4, pp. 48-488. Ushakov, P.A, A.V. Zhkov, N.M. Matykhin (977), Heat Transfer to Liqid Metals in Reglar Arrays of Fel Elements, High Temp., Vol. 5, pp. 868-873. Ushakov, P.A. (978), Problems of Flid Dynamic and Heat Transfer in Cores of Fast Reactors, COMECON Symp., Teplofizika I gidrodynamika aktivnoi zony I parogeneratov dlya bystrykh reaktorov, Marianske Lazne, CSSR, Paper ML78/, pp. 4-35. Ushakov, P.A. (979), Problems of Heat Transfer in Cores of Fast Breeder Reactors, Heat Transfer and Hydrodynamics of Single-phase Flo in Rod Bndles, Izd. Naka, Leningrad (in Rssian). Van Driest, E.R. (956), On Trblent Flo Near a Wall, J. Aerosp. Sci., Vol. 3, pp. 7-. Van Dyke, M. (977), Pertrbation Methods in Flid Mechanics, Parabolic, Stanford. Van Vylen, G.J., R.E. Sonntag, Fndamentals of Classical Thermodynamics, nd edition, Wiley, Ne York. Von Karman (93), Mechanische Ähnlichkeit nd Trblenz, Nachr. Ges. Wiss. Göttingen, Math. Phy. Klasse, No. 5, pp. 58-76. Walz, A. (966), Strömngs- nd Temperatrgrenzschichten, G. Bran Verlag Karlsrhe (in German). Wassel, A.T., I. Catton (973), Calclation of Trblent Bondary Layers Over Flat Plates ith Different Phenomenological Theories of Trblence and Variable Trblent Prandtl Nmber, Int. J. Heat and Mass Transfer, Vol. 6, pp. 547-563. Weigand, B. (996), An Etract Analytical Soltion for the Etended Trblent Graetz Problem ith Dirichlet Wall Bondary Conditions for Pipe and Channel Flos, Int. J. Heat Mass Transfer, Vol. 39 (8), pp. 65-637. Weinberg, D. (975), Temperatrfelder in Bündeln mit Na-Kühlng- Thermo- nd Fliddynamische Unterkanalanalyse der Schnellbrüter-Brennelemente nd ihre Relation zr Brennstabmechanik, Kernforschngszentrm Karlsrhe, KfK-3. Wesley, D.A., E.M. Sparro (976), Circmferentially Local and Average Heat Transfer Coefficients in a Tbe Donstream of a Tee, Int. J. Heat and Mass Transfer, Vol. 9, pp. 5-4. 476

White, F.M. (974), Viscos Flid Flo, McGra-Hill Comp., Ne York. Wilco, D.C., R.M. Traci (976), A Complete Model of Trblence, AIAA, Paper 76-35, San Diego. CA. Wilco, D.C. (986), Mltiscale Model for Trblent Flos, 4 th Aerospace Science Meetings, American Institte of Aeronatics and Astronatics. Yakhot, V., S.A. Orszag, A. Yakhot (987), Heat Transfer in Trblent Flids Pipe Flo, Int. J. Heat Mass Transfer, Vol. 3, pp. 5-. Y, B., H. Ozoe, S.W. Chrchill (), The Characteristics of Flly Developed Trblent Convection in a Rond Tbe, Chemical Engineering Science, Vol. 56, pp. 78-8. Zagarola, M.V. (996), Mean-flo Scaling of Trblent Pipe Flo, PhD Dissertation, Princeton University, N.J. Zagarola, M.V., A.J. Smits (997), Scaling of the Mean Velocity Profile for Trblent Pipe Flo, Physical Revie Letters, Vol. 78, No., pp. 39-4. Zhi-qing, W. (98), Stdy of Correction Coefficients of Laminar and Trblent Entrance Region Effect in Rond Pipe, Appl. Math. Mech., Vol. 3 (3), pp. 433-446. Zhkov, A.V., V.I. Sbbotin, P.A. Ushakov (969a), Heat Transfer from Loosely Spaced Rod Clsters to Liqid Metal Floing in the Aial Direction, Liqid Metals, Engl. Translation, NASA Rep. NASA-TT-F5, pp. 49-69. Zhkov, A.V., L.K. Kdryatseva, E.Y. Sviridenko, V.I. Sbbotin, D.V. Talanov, P.A. Ushakov (969b), Eperimental Stdies of Temperatre Fields Using Models, Liqid Metals, Engl. Translation, NASA Rep. NASA-TT-F5, pp. 7-89. Zhkov, A.V., M.N. Matykhin, A.B. Mzhanov, Y.Y. Sviridenko, P.A. Ushakov (973), Methods of Calclating Heat Echange of Liqid Metals in Reglar Lattices of Fel Elements and Some Ne Eperimental Data, FEI-44, Inst. of Physics and Energy, Obninsk, AEC-TR-7549. Zhkov, A.V., M.N. Matykhin, E.V. Nomofilov, A.P. Sorokin, V.S. Yrev (978), Temperatre Distribtion in Non-standard and Deformed Fel Pin Grids of Fast Reactors, COMECON Symp. Teplofizika I gidrodynamika aktivnoi zony I parogeneratov dlya bystrykh reaktorov, Marianske Lazne, CSSR, Paper ML78/. Zhkov, A.V., M.N. Matykhin, Y.Y. Sviridenko (98), Einflss von Geometriestörngen af die Temperatrfelder nd den Wärmeübergang in den Unterkanälen von Brennelementbündeln Schneller Reaktoren, FEI-979, Fiziko-energeticheskii Institt Obninsk. German Translation: KfK-Tech. Rep. 675. 477