Chapter 10 LOW PRANDTL NUMBER THERMAL-HYDRAULICS*

Transcription

1 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

2 . 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 qc qc 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

3 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

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 T [ C] 4

5 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

6 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 (.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)

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

8 .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

9 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

10 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

11 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

12 .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

13 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

14 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

15 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

16 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

17 .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

18 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 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 (.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 *

19 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 Pe for Pe / 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

20 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] 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: ; ' p * 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 hy Re. 35 Re (.49) 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