Numerical simulation ofturbulent macrow and heat

transfer in multi-channelnarrow-gap fuel element

Huang Jun Wang QW and Tao WQSchool of Energy and Power Engineering Xirsquoan Jiaotong University

Xirsquoan Shaanxi PRC

Keywords Turbulent macrow Heat transfer

Abstract A computational study of convective heat transfer for turbulent macrows in multi-channelnarrow-gap fuel element has been carried out using a general marching procedure The macruiddistribution adjustment among seven annular-sector channels is based on the assumption of thesame pressure drop in these passages It was found that the inlet velocities of the bilateral channelsare lower than those of the middles and the axial local heat transfer coefregcients for the sevenchannels do not approach the fully developed constant value At each cross section the peripherytemperature distribution is not uniform while the local temperature distribution along axialcoordinate is of sinuous type with the peak at x = 07plusmn08 m At the same Reynolds number theaveraged Nusselt numbers of water in Channel 1 and Channel 7 are higher than those in themiddles The maximum surface temperature increases almost linearly with the inlet watertemperature whereas it decreases almost asymptotically with the inlet average velocity

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httpwwwemeraldinsightcom0264-4401htm

NotationD = channel equivalent diameter mF = mass macrow rate kgsh = heat transfer coefregcient Wm2 8 Cl = mixing lengthL = total length of the cooled channel

mNu = Nusselt numberp = pressure PapAuml = small pressure variation PapAring = space-average pressure over the

cross section PaPr = Prandtl numberq = heat macrux Wm2

r = radial distance radial coordinatem (Figure 1)

Re = Reynolds numberriic = inner radial distance of the inner

casing m (Figure 1)

rooc = outer radial distance of the outercasing m (Figure 1)

Su Sv Sw = volumetric source term in x r umomentum equation

t = temperature 8 Cu = axial velocity component msv = radial velocity component msw = angular velocity component

msx = axial distance axial coordinateu = angular distance angular

coordinate radm = macruid dynamics viscosity

kgmsmt = turbulent viscosity kgmsmeff = equivalent macruid viscosity

m + mt

sT = turbulent Prandtl number

This research work was supported by the National Natural Science Foundation of China(No 59806011 59995460-2)

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Received June 2001Accepted January 2002

Engineering ComputationsVol 19 No 3 2002 pp 327-345

q MCB UP Limited 0264-4401DOI 10110802644400210423972

1 IntroductionIn a high macrux engineering test reactor (HFETR) the multi-channel narrow-gapfuel element with enriched uranium is employed to meet the demand for high-performance High-velocity high-pressure water macrows through seven narrow-gap annular channels between concentric fuel layers each of which is ofsandwich type consisted of two aluminum layers and a uranium one (Figure 1)Uranium layers generate enormous heat macrux uniform peripherally but sinuousaxially and are cooled by the high-pressure and high-velocity water Each

Dp = pressure drop for a channelf = the dependent variable f

Superscriptsn = temporal solution to the

difference equation at nth step

Subscriptsi = innerin = channel inlet locationsmax = maximum

Figure 1Schematic diagram of thephysical domain

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water channel is separated by three regns along the peripheral forming threeannular-sector channels To keep the reactor operating safely the maximumsurface temperature for the channels in the fuel element should not surpass thedesigned value It is therefore of great practical importance to investigate thetemperature distribution in this kind of fuel element

Some researches have been conducted to investigate turbulent macrow and heattransfer in the fuel element experimentally or theoretically Due to the extremelyhigh price and more importantly the difregculties to arrange thermocouples inthe narrow channels experimental data are very limited (Chen and Jiang 1984)The numerical method is a useful alternative in this situation (Yu 1981 Wang1981) In the works by Yu (1981) and Wang (1981) the fuel element was dividedinto several districts and their temperature distributions were solved separatelyinstead of solving the entire region Many simplifying assumptions andempirical equations had to be employed to regll the coupling of the separateddistricts These include

(1) one-dimensional or two-dimensional conduction model was used for thefuel region

(2) along the fuel layer element surface the convective heat transfercoefregcients were determined from empirical equation

(3) thermophysical properties were temperature-independent

(4) the inlet velocities for the seven channels were assumed to be the sameas each other

All these assumptions made the accuracy of the numerical results quitequestionable and far from satisfactory to give enough information for guidingthe operation

As mentioned earlier annular-sector ducts are involved in this conreggurationAlthough convective macrow and heat transfer in annular-sector ducts has beenperformed by several authors (Lin et al 1995 Nida 1980 Soliman 1987Sparrow et al 1964) no results are provided in the literature for multi-passageannular-sector ducts subjected to a non-uniform heat macrux The key issuesincluded in the numerical simulation of this problem are as follows

(1) the macrow distribution in different subchannels is unknown prior rather itshould be determined during the computational procedure with a giventotal macrow macrux

(2) the conduction in the fuel layer and the convection of the water channelssurface are coupled making the problem of conjugated type

(3) the variation of the macruid thermophysical properties should be taken intoaccount because of high heat macrux released by the fuel layer

The objective of this article is to develop a three-dimensional marchingcalculation procedure mathematical modeling for the fuel element which includes

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seven narrow-gap passages six uranium layers and fourteen aluminum layersas schematically shown in Figure 1 The macruid distribution among passages isadjusted during the simulation procedure so as to guarantee the same pressuredrop between the inlets and outlets of each channels Some thermophysicalproperties of water such as viscosity and Prandtl number are considered to betemperature-dependent In such kind of the peculiar structure of the channelsno recirculating macrow in the axial direction may be expected and the macrow ismainly of parabolic type Therefore a three-dimensional parabolic model isselected to simulate the macrow procedure Although various sophisticatedturbulence models have been developed (Hanjalic 1994 Ramdhyani 1997) themixing length theory is considered as the right choice for the presentsimulation A parametrical study is also conducted where the effects of theinlet water temperature and velocity are examined

2 Mathematical modelAs sketched in Figure 1 six ring-like fuel layers and inner and outer casingsare assembled concentrically with regns used as spacings to regx their positionEach fuel layer consists of three layers The middle one is uranium thegenerator of heat macrux while the bilateral layers are aluminum The inner casingthe outer casing and regns are also made of aluminum High-velocity high-pressurewater macrows through the seven narrow-gap annular gaps keeping the system inthe desired temperature It is assumed that the macrow is steady turbulent andincompressible with temperature-dependent thermoproperties The physicalproperties of uranium and aluminum are supposed to be constant because theyalmost have no change in the range of temperature studied Heat macrux generatedby uranium layers is uniform peripherally (u) but sinuous axially (x ) (Figure 1)which could be expressed by

q(x) = qmax acute sinp(100x + 9)

118

sup3 acute(1)

where x is the axial distance The thermal conductivities of aluminum anduranium are 2007 Wm8 C and 1689 Wm 8 C while their densities are2700 kgm3 and 5060 kgm3 respectively The thickness of both the uraniumand the aluminum of each fuel layer are 05 mm The total length of the cooledchannels is one meter The outer diameter Do and the inner diameter Di of eachfuel layer and the inner and outer casings are listed in Table I It should benoted that the number system in Table I was also shown in Figure 1

Considering the symmetry of the present problem only one-third of the fuelelement needs to be taken into account (that is u = 2p=3 Figure 1) In this macrowproblem there exists a predominant direction of macrow (the axial direction x) inwhich upwind convective macrow greatly inmacruences the downwind thus it can betreated as parabolic in the streamwise direction therefore the streamwise

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diffusion of momentum and energy can be omitted (Patankar and Spalding1972)

The control volume based fully implicit regnite difference method wasapplied to solve the macrow of the channels in the present work (Patankar andSpalding 1972) The governing equations for the macrow and thermal regelds maybe written as

Continuityshy u

shy x+

1

r

shy w

shy u+

shy v

shy r+

v

r= 0 (2)

Momentum

shyshy x

(ru 2) +1

r

shyshy r

(rruv) +1

r

shyshy u

(ruw) =1

r

shyshy r

meffrshy u

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy u

shy u

sup3 acute+ Su

(3a)

shyshy x

(ruv) +1

r

shyshy r

(rrv 2) +1

r

shyshy u

(rvw) =1

r

shyshy r

meffrshy v

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy v

shy u

sup3 acute+ Sv

(3b)

shyshy x

(ruw) +1

r

shyshy r

(rrvw) +1

r

shyshy u

(rw 2) =1

r

shyshy r

meffrshy w

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy w

shy u

sup3 acute+ Sw

(3c)where Su Sv Sw are the volumetric source terms deregned as

Su = 2shy p

shy x+

1

r

shyshy r

rmeffshy v

shy x

sup3 acute+

1

r

shyshy u

meffshy w

shy x

sup3 acute(4a)

Sv = 2shy p

shy r+

shyshy x

meff

shy u

shy r

sup3 acute+

1

r

shyshy r

rmeff

shy v

shy r

sup3 acute+

1

r

shyshy u

meff

shy (w=r)shy u

sup3 acute

22meff

r

1

r

shy w

shy u+

v

r

sup3 acute+

rw 2

r(4b)

Sw = 21

r

shy p

shy u+

shyshy x

meff

shy u

r shy u

sup3 acute+

1

r

shyshy r

rmeff

1

r

shy v

shy ushy

w

r

sup3 acutesup3 acute(4c)

Fuel layersInner casing No 1 No 2 No 3 No 4 No 5 No 6 Outer casing

Do (mm) 14 21 28 35 42 49 56 63Di (mm) 12 18 25 32 39 46 53 60

Table IValue of Do and Di

for six fuel layersand the inner and

outer casings

Numericalsimulation of

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Energy

shyshy x

(rut) +1

r

shyshy r

(rrvt) +1

r

shyshy u

(rwt) =1

r

shyshy r

r(m=Pr + mt=sT )shy t

shy r

sup3 acute

+1

r

shyshy u

m=Pr + mt=sT

r

shy t

shy u

sup3 acute(5)

where meff = m + mt sT = 09 mt = rl 2 shy u=shy yshyshy shyshy

The algebraic mixing length model proposed by Patankar (1979) is adoptedin the present study The model takes account of the proximity of both the regnsurfaces and tube wall as well as of the gradients in the radial andcircumferential directions Since the available wall functions of the standardk-e model account for the inmacruence of only single wall they are not suitable forthe present problem where the inmacruences of both the tube wall and the regnsurface are important near the regn region (Figure 1) The resultant mixinglength l is calculated by

1

l=

1

lp+

1

lc(6)

where lp is the mixing length considering a pipe macrow without regns lc is themixing length if the inter-regn surface is likened to a parallel plate channel(Patankar 1979) Equation (6) was employed to evaluate the mixing length atall points in the inter-regn space

Equations (2plusmn5) are completed by the following set of boundary conditions

for u = 0 or u =2

3p u = 0 v = 0 w = 0

shy t

shy u= 0 (7a)

for r = riic or r = rooc u = 0 v = 0 w = 0shy t

shy r= 0 (Figure 1) (7b)

for x = 0 t = tin v = 0 w = 0 u = uin( j) j = 1 7 (Figure 1) (7c)

Since there is no reverse macrow in the main macrow direction and the diffusion ofmomentum heat is negligible in that direction the downstream pressure regeldhas little inmacruence on the upstream macrow conditions It is this convenientbehavior of the boundary-layer macrows that enable us to employ a marchingintegration from an upstream station to a downstream one This procedure canbe regarded as a boundary-layer method

Note that in equations (4a) p can be thought of as a form of space-averagedpressure over a cross section and p in Equations (4b) and (4c) is the smallpressure variation governing the macrow distribution in the cross-section Thegradient shy p= shy x is supposed to be known (or calculated) before we proceed to

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get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

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(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

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where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

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335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

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Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

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turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

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43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

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339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

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different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

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341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

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variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

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5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

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water channel is separated by three regns along the peripheral forming threeannular-sector channels To keep the reactor operating safely the maximumsurface temperature for the channels in the fuel element should not surpass thedesigned value It is therefore of great practical importance to investigate thetemperature distribution in this kind of fuel element

Some researches have been conducted to investigate turbulent macrow and heattransfer in the fuel element experimentally or theoretically Due to the extremelyhigh price and more importantly the difregculties to arrange thermocouples inthe narrow channels experimental data are very limited (Chen and Jiang 1984)The numerical method is a useful alternative in this situation (Yu 1981 Wang1981) In the works by Yu (1981) and Wang (1981) the fuel element was dividedinto several districts and their temperature distributions were solved separatelyinstead of solving the entire region Many simplifying assumptions andempirical equations had to be employed to regll the coupling of the separateddistricts These include

(1) one-dimensional or two-dimensional conduction model was used for thefuel region

(2) along the fuel layer element surface the convective heat transfercoefregcients were determined from empirical equation

(3) thermophysical properties were temperature-independent

(4) the inlet velocities for the seven channels were assumed to be the sameas each other

All these assumptions made the accuracy of the numerical results quitequestionable and far from satisfactory to give enough information for guidingthe operation

As mentioned earlier annular-sector ducts are involved in this conreggurationAlthough convective macrow and heat transfer in annular-sector ducts has beenperformed by several authors (Lin et al 1995 Nida 1980 Soliman 1987Sparrow et al 1964) no results are provided in the literature for multi-passageannular-sector ducts subjected to a non-uniform heat macrux The key issuesincluded in the numerical simulation of this problem are as follows

(1) the macrow distribution in different subchannels is unknown prior rather itshould be determined during the computational procedure with a giventotal macrow macrux

(2) the conduction in the fuel layer and the convection of the water channelssurface are coupled making the problem of conjugated type

(3) the variation of the macruid thermophysical properties should be taken intoaccount because of high heat macrux released by the fuel layer

The objective of this article is to develop a three-dimensional marchingcalculation procedure mathematical modeling for the fuel element which includes

Numericalsimulation of

turbulent macrow

329

seven narrow-gap passages six uranium layers and fourteen aluminum layersas schematically shown in Figure 1 The macruid distribution among passages isadjusted during the simulation procedure so as to guarantee the same pressuredrop between the inlets and outlets of each channels Some thermophysicalproperties of water such as viscosity and Prandtl number are considered to betemperature-dependent In such kind of the peculiar structure of the channelsno recirculating macrow in the axial direction may be expected and the macrow ismainly of parabolic type Therefore a three-dimensional parabolic model isselected to simulate the macrow procedure Although various sophisticatedturbulence models have been developed (Hanjalic 1994 Ramdhyani 1997) themixing length theory is considered as the right choice for the presentsimulation A parametrical study is also conducted where the effects of theinlet water temperature and velocity are examined

2 Mathematical modelAs sketched in Figure 1 six ring-like fuel layers and inner and outer casingsare assembled concentrically with regns used as spacings to regx their positionEach fuel layer consists of three layers The middle one is uranium thegenerator of heat macrux while the bilateral layers are aluminum The inner casingthe outer casing and regns are also made of aluminum High-velocity high-pressurewater macrows through the seven narrow-gap annular gaps keeping the system inthe desired temperature It is assumed that the macrow is steady turbulent andincompressible with temperature-dependent thermoproperties The physicalproperties of uranium and aluminum are supposed to be constant because theyalmost have no change in the range of temperature studied Heat macrux generatedby uranium layers is uniform peripherally (u) but sinuous axially (x ) (Figure 1)which could be expressed by

q(x) = qmax acute sinp(100x + 9)

118

sup3 acute(1)

where x is the axial distance The thermal conductivities of aluminum anduranium are 2007 Wm8 C and 1689 Wm 8 C while their densities are2700 kgm3 and 5060 kgm3 respectively The thickness of both the uraniumand the aluminum of each fuel layer are 05 mm The total length of the cooledchannels is one meter The outer diameter Do and the inner diameter Di of eachfuel layer and the inner and outer casings are listed in Table I It should benoted that the number system in Table I was also shown in Figure 1

Considering the symmetry of the present problem only one-third of the fuelelement needs to be taken into account (that is u = 2p=3 Figure 1) In this macrowproblem there exists a predominant direction of macrow (the axial direction x) inwhich upwind convective macrow greatly inmacruences the downwind thus it can betreated as parabolic in the streamwise direction therefore the streamwise

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diffusion of momentum and energy can be omitted (Patankar and Spalding1972)

The control volume based fully implicit regnite difference method wasapplied to solve the macrow of the channels in the present work (Patankar andSpalding 1972) The governing equations for the macrow and thermal regelds maybe written as

Continuityshy u

shy x+

1

r

shy w

shy u+

shy v

shy r+

v

r= 0 (2)

Momentum

shyshy x

(ru 2) +1

r

shyshy r

(rruv) +1

r

shyshy u

(ruw) =1

r

shyshy r

meffrshy u

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy u

shy u

sup3 acute+ Su

(3a)

shyshy x

(ruv) +1

r

shyshy r

(rrv 2) +1

r

shyshy u

(rvw) =1

r

shyshy r

meffrshy v

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy v

shy u

sup3 acute+ Sv

(3b)

shyshy x

(ruw) +1

r

shyshy r

(rrvw) +1

r

shyshy u

(rw 2) =1

r

shyshy r

meffrshy w

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy w

shy u

sup3 acute+ Sw

(3c)where Su Sv Sw are the volumetric source terms deregned as

Su = 2shy p

shy x+

1

r

shyshy r

rmeffshy v

shy x

sup3 acute+

1

r

shyshy u

meffshy w

shy x

sup3 acute(4a)

Sv = 2shy p

shy r+

shyshy x

meff

shy u

shy r

sup3 acute+

1

r

shyshy r

rmeff

shy v

shy r

sup3 acute+

1

r

shyshy u

meff

shy (w=r)shy u

sup3 acute

22meff

r

1

r

shy w

shy u+

v

r

sup3 acute+

rw 2

r(4b)

Sw = 21

r

shy p

shy u+

shyshy x

meff

shy u

r shy u

sup3 acute+

1

r

shyshy r

rmeff

1

r

shy v

shy ushy

w

r

sup3 acutesup3 acute(4c)

Fuel layersInner casing No 1 No 2 No 3 No 4 No 5 No 6 Outer casing

Do (mm) 14 21 28 35 42 49 56 63Di (mm) 12 18 25 32 39 46 53 60

Table IValue of Do and Di

for six fuel layersand the inner and

outer casings

Numericalsimulation of

turbulent macrow

331

Energy

shyshy x

(rut) +1

r

shyshy r

(rrvt) +1

r

shyshy u

(rwt) =1

r

shyshy r

r(m=Pr + mt=sT )shy t

shy r

sup3 acute

+1

r

shyshy u

m=Pr + mt=sT

r

shy t

shy u

sup3 acute(5)

where meff = m + mt sT = 09 mt = rl 2 shy u=shy yshyshy shyshy

The algebraic mixing length model proposed by Patankar (1979) is adoptedin the present study The model takes account of the proximity of both the regnsurfaces and tube wall as well as of the gradients in the radial andcircumferential directions Since the available wall functions of the standardk-e model account for the inmacruence of only single wall they are not suitable forthe present problem where the inmacruences of both the tube wall and the regnsurface are important near the regn region (Figure 1) The resultant mixinglength l is calculated by

1

l=

1

lp+

1

lc(6)

where lp is the mixing length considering a pipe macrow without regns lc is themixing length if the inter-regn surface is likened to a parallel plate channel(Patankar 1979) Equation (6) was employed to evaluate the mixing length atall points in the inter-regn space

Equations (2plusmn5) are completed by the following set of boundary conditions

for u = 0 or u =2

3p u = 0 v = 0 w = 0

shy t

shy u= 0 (7a)

for r = riic or r = rooc u = 0 v = 0 w = 0shy t

shy r= 0 (Figure 1) (7b)

for x = 0 t = tin v = 0 w = 0 u = uin( j) j = 1 7 (Figure 1) (7c)

Since there is no reverse macrow in the main macrow direction and the diffusion ofmomentum heat is negligible in that direction the downstream pressure regeldhas little inmacruence on the upstream macrow conditions It is this convenientbehavior of the boundary-layer macrows that enable us to employ a marchingintegration from an upstream station to a downstream one This procedure canbe regarded as a boundary-layer method

Note that in equations (4a) p can be thought of as a form of space-averagedpressure over a cross section and p in Equations (4b) and (4c) is the smallpressure variation governing the macrow distribution in the cross-section Thegradient shy p= shy x is supposed to be known (or calculated) before we proceed to

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get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

Numericalsimulation of

turbulent macrow

333

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

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where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

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Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

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43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

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different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

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variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

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seven narrow-gap passages six uranium layers and fourteen aluminum layersas schematically shown in Figure 1 The macruid distribution among passages isadjusted during the simulation procedure so as to guarantee the same pressuredrop between the inlets and outlets of each channels Some thermophysicalproperties of water such as viscosity and Prandtl number are considered to betemperature-dependent In such kind of the peculiar structure of the channelsno recirculating macrow in the axial direction may be expected and the macrow ismainly of parabolic type Therefore a three-dimensional parabolic model isselected to simulate the macrow procedure Although various sophisticatedturbulence models have been developed (Hanjalic 1994 Ramdhyani 1997) themixing length theory is considered as the right choice for the presentsimulation A parametrical study is also conducted where the effects of theinlet water temperature and velocity are examined

2 Mathematical modelAs sketched in Figure 1 six ring-like fuel layers and inner and outer casingsare assembled concentrically with regns used as spacings to regx their positionEach fuel layer consists of three layers The middle one is uranium thegenerator of heat macrux while the bilateral layers are aluminum The inner casingthe outer casing and regns are also made of aluminum High-velocity high-pressurewater macrows through the seven narrow-gap annular gaps keeping the system inthe desired temperature It is assumed that the macrow is steady turbulent andincompressible with temperature-dependent thermoproperties The physicalproperties of uranium and aluminum are supposed to be constant because theyalmost have no change in the range of temperature studied Heat macrux generatedby uranium layers is uniform peripherally (u) but sinuous axially (x ) (Figure 1)which could be expressed by

q(x) = qmax acute sinp(100x + 9)

118

sup3 acute(1)

where x is the axial distance The thermal conductivities of aluminum anduranium are 2007 Wm8 C and 1689 Wm 8 C while their densities are2700 kgm3 and 5060 kgm3 respectively The thickness of both the uraniumand the aluminum of each fuel layer are 05 mm The total length of the cooledchannels is one meter The outer diameter Do and the inner diameter Di of eachfuel layer and the inner and outer casings are listed in Table I It should benoted that the number system in Table I was also shown in Figure 1

Considering the symmetry of the present problem only one-third of the fuelelement needs to be taken into account (that is u = 2p=3 Figure 1) In this macrowproblem there exists a predominant direction of macrow (the axial direction x) inwhich upwind convective macrow greatly inmacruences the downwind thus it can betreated as parabolic in the streamwise direction therefore the streamwise

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diffusion of momentum and energy can be omitted (Patankar and Spalding1972)

The control volume based fully implicit regnite difference method wasapplied to solve the macrow of the channels in the present work (Patankar andSpalding 1972) The governing equations for the macrow and thermal regelds maybe written as

Continuityshy u

shy x+

1

r

shy w

shy u+

shy v

shy r+

v

r= 0 (2)

Momentum

shyshy x

(ru 2) +1

r

shyshy r

(rruv) +1

r

shyshy u

(ruw) =1

r

shyshy r

meffrshy u

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy u

shy u

sup3 acute+ Su

(3a)

shyshy x

(ruv) +1

r

shyshy r

(rrv 2) +1

r

shyshy u

(rvw) =1

r

shyshy r

meffrshy v

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy v

shy u

sup3 acute+ Sv

(3b)

shyshy x

(ruw) +1

r

shyshy r

(rrvw) +1

r

shyshy u

(rw 2) =1

r

shyshy r

meffrshy w

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy w

shy u

sup3 acute+ Sw

(3c)where Su Sv Sw are the volumetric source terms deregned as

Su = 2shy p

shy x+

1

r

shyshy r

rmeffshy v

shy x

sup3 acute+

1

r

shyshy u

meffshy w

shy x

sup3 acute(4a)

Sv = 2shy p

shy r+

shyshy x

meff

shy u

shy r

sup3 acute+

1

r

shyshy r

rmeff

shy v

shy r

sup3 acute+

1

r

shyshy u

meff

shy (w=r)shy u

sup3 acute

22meff

r

1

r

shy w

shy u+

v

r

sup3 acute+

rw 2

r(4b)

Sw = 21

r

shy p

shy u+

shyshy x

meff

shy u

r shy u

sup3 acute+

1

r

shyshy r

rmeff

1

r

shy v

shy ushy

w

r

sup3 acutesup3 acute(4c)

Fuel layersInner casing No 1 No 2 No 3 No 4 No 5 No 6 Outer casing

Do (mm) 14 21 28 35 42 49 56 63Di (mm) 12 18 25 32 39 46 53 60

Table IValue of Do and Di

for six fuel layersand the inner and

outer casings

Numericalsimulation of

turbulent macrow

331

Energy

shyshy x

(rut) +1

r

shyshy r

(rrvt) +1

r

shyshy u

(rwt) =1

r

shyshy r

r(m=Pr + mt=sT )shy t

shy r

sup3 acute

+1

r

shyshy u

m=Pr + mt=sT

r

shy t

shy u

sup3 acute(5)

where meff = m + mt sT = 09 mt = rl 2 shy u=shy yshyshy shyshy

The algebraic mixing length model proposed by Patankar (1979) is adoptedin the present study The model takes account of the proximity of both the regnsurfaces and tube wall as well as of the gradients in the radial andcircumferential directions Since the available wall functions of the standardk-e model account for the inmacruence of only single wall they are not suitable forthe present problem where the inmacruences of both the tube wall and the regnsurface are important near the regn region (Figure 1) The resultant mixinglength l is calculated by

1

l=

1

lp+

1

lc(6)

where lp is the mixing length considering a pipe macrow without regns lc is themixing length if the inter-regn surface is likened to a parallel plate channel(Patankar 1979) Equation (6) was employed to evaluate the mixing length atall points in the inter-regn space

Equations (2plusmn5) are completed by the following set of boundary conditions

for u = 0 or u =2

3p u = 0 v = 0 w = 0

shy t

shy u= 0 (7a)

for r = riic or r = rooc u = 0 v = 0 w = 0shy t

shy r= 0 (Figure 1) (7b)

for x = 0 t = tin v = 0 w = 0 u = uin( j) j = 1 7 (Figure 1) (7c)

Since there is no reverse macrow in the main macrow direction and the diffusion ofmomentum heat is negligible in that direction the downstream pressure regeldhas little inmacruence on the upstream macrow conditions It is this convenientbehavior of the boundary-layer macrows that enable us to employ a marchingintegration from an upstream station to a downstream one This procedure canbe regarded as a boundary-layer method

Note that in equations (4a) p can be thought of as a form of space-averagedpressure over a cross section and p in Equations (4b) and (4c) is the smallpressure variation governing the macrow distribution in the cross-section Thegradient shy p= shy x is supposed to be known (or calculated) before we proceed to

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get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

Numericalsimulation of

turbulent macrow

333

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

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334

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

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336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

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338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

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340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

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342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

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344

diffusion of momentum and energy can be omitted (Patankar and Spalding1972)

The control volume based fully implicit regnite difference method wasapplied to solve the macrow of the channels in the present work (Patankar andSpalding 1972) The governing equations for the macrow and thermal regelds maybe written as

Continuityshy u

shy x+

1

r

shy w

shy u+

shy v

shy r+

v

r= 0 (2)

Momentum

shyshy x

(ru 2) +1

r

shyshy r

(rruv) +1

r

shyshy u

(ruw) =1

r

shyshy r

meffrshy u

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy u

shy u

sup3 acute+ Su

(3a)

shyshy x

(ruv) +1

r

shyshy r

(rrv 2) +1

r

shyshy u

(rvw) =1

r

shyshy r

meffrshy v

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy v

shy u

sup3 acute+ Sv

(3b)

shyshy x

(ruw) +1

r

shyshy r

(rrvw) +1

r

shyshy u

(rw 2) =1

r

shyshy r

meffrshy w

shy r

sup3 acute+

1

r

shyshy u

meff

r

shy w

shy u

sup3 acute+ Sw

(3c)where Su Sv Sw are the volumetric source terms deregned as

Su = 2shy p

shy x+

1

r

shyshy r

rmeffshy v

shy x

sup3 acute+

1

r

shyshy u

meffshy w

shy x

sup3 acute(4a)

Sv = 2shy p

shy r+

shyshy x

meff

shy u

shy r

sup3 acute+

1

r

shyshy r

rmeff

shy v

shy r

sup3 acute+

1

r

shyshy u

meff

shy (w=r)shy u

sup3 acute

22meff

r

1

r

shy w

shy u+

v

r

sup3 acute+

rw 2

r(4b)

Sw = 21

r

shy p

shy u+

shyshy x

meff

shy u

r shy u

sup3 acute+

1

r

shyshy r

rmeff

1

r

shy v

shy ushy

w

r

sup3 acutesup3 acute(4c)

Fuel layersInner casing No 1 No 2 No 3 No 4 No 5 No 6 Outer casing

Do (mm) 14 21 28 35 42 49 56 63Di (mm) 12 18 25 32 39 46 53 60

Table IValue of Do and Di

for six fuel layersand the inner and

outer casings

Numericalsimulation of

turbulent macrow

331

Energy

shyshy x

(rut) +1

r

shyshy r

(rrvt) +1

r

shyshy u

(rwt) =1

r

shyshy r

r(m=Pr + mt=sT )shy t

shy r

sup3 acute

+1

r

shyshy u

m=Pr + mt=sT

r

shy t

shy u

sup3 acute(5)

where meff = m + mt sT = 09 mt = rl 2 shy u=shy yshyshy shyshy

The algebraic mixing length model proposed by Patankar (1979) is adoptedin the present study The model takes account of the proximity of both the regnsurfaces and tube wall as well as of the gradients in the radial andcircumferential directions Since the available wall functions of the standardk-e model account for the inmacruence of only single wall they are not suitable forthe present problem where the inmacruences of both the tube wall and the regnsurface are important near the regn region (Figure 1) The resultant mixinglength l is calculated by

1

l=

1

lp+

1

lc(6)

where lp is the mixing length considering a pipe macrow without regns lc is themixing length if the inter-regn surface is likened to a parallel plate channel(Patankar 1979) Equation (6) was employed to evaluate the mixing length atall points in the inter-regn space

Equations (2plusmn5) are completed by the following set of boundary conditions

for u = 0 or u =2

3p u = 0 v = 0 w = 0

shy t

shy u= 0 (7a)

for r = riic or r = rooc u = 0 v = 0 w = 0shy t

shy r= 0 (Figure 1) (7b)

for x = 0 t = tin v = 0 w = 0 u = uin( j) j = 1 7 (Figure 1) (7c)

Since there is no reverse macrow in the main macrow direction and the diffusion ofmomentum heat is negligible in that direction the downstream pressure regeldhas little inmacruence on the upstream macrow conditions It is this convenientbehavior of the boundary-layer macrows that enable us to employ a marchingintegration from an upstream station to a downstream one This procedure canbe regarded as a boundary-layer method

Note that in equations (4a) p can be thought of as a form of space-averagedpressure over a cross section and p in Equations (4b) and (4c) is the smallpressure variation governing the macrow distribution in the cross-section Thegradient shy p= shy x is supposed to be known (or calculated) before we proceed to

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332

get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

Numericalsimulation of

turbulent macrow

333

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

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334

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

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Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

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338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

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different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

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342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

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344

Energy

shyshy x

(rut) +1

r

shyshy r

(rrvt) +1

r

shyshy u

(rwt) =1

r

shyshy r

r(m=Pr + mt=sT )shy t

shy r

sup3 acute

+1

r

shyshy u

m=Pr + mt=sT

r

shy t

shy u

sup3 acute(5)

where meff = m + mt sT = 09 mt = rl 2 shy u=shy yshyshy shyshy

The algebraic mixing length model proposed by Patankar (1979) is adoptedin the present study The model takes account of the proximity of both the regnsurfaces and tube wall as well as of the gradients in the radial andcircumferential directions Since the available wall functions of the standardk-e model account for the inmacruence of only single wall they are not suitable forthe present problem where the inmacruences of both the tube wall and the regnsurface are important near the regn region (Figure 1) The resultant mixinglength l is calculated by

1

l=

1

lp+

1

lc(6)

where lp is the mixing length considering a pipe macrow without regns lc is themixing length if the inter-regn surface is likened to a parallel plate channel(Patankar 1979) Equation (6) was employed to evaluate the mixing length atall points in the inter-regn space

Equations (2plusmn5) are completed by the following set of boundary conditions

for u = 0 or u =2

3p u = 0 v = 0 w = 0

shy t

shy u= 0 (7a)

for r = riic or r = rooc u = 0 v = 0 w = 0shy t

shy r= 0 (Figure 1) (7b)

for x = 0 t = tin v = 0 w = 0 u = uin( j) j = 1 7 (Figure 1) (7c)

Since there is no reverse macrow in the main macrow direction and the diffusion ofmomentum heat is negligible in that direction the downstream pressure regeldhas little inmacruence on the upstream macrow conditions It is this convenientbehavior of the boundary-layer macrows that enable us to employ a marchingintegration from an upstream station to a downstream one This procedure canbe regarded as a boundary-layer method

Note that in equations (4a) p can be thought of as a form of space-averagedpressure over a cross section and p in Equations (4b) and (4c) is the smallpressure variation governing the macrow distribution in the cross-section Thegradient shy p= shy x is supposed to be known (or calculated) before we proceed to

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get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

Numericalsimulation of

turbulent macrow

333

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

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334

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

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336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

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338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

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different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

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342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

get the lateral pressure gradient shy p= shy u and shy p= shy r In the conregned macrow weregard shy p= shy x as uniform over a cross-section and then obtain it from theintegral mass-conservation eqs(Patankar 1972) Here an assumption is made asfollows

p = p + p (8a)

shy p

shy x=

shy p

shy x

shy p

shy u=

shy p

shy u

shy p

shy r=

shy p

shy r(8b)

Such an assumption de-couples the longitudinal and the lateral pressuregradients which may be calculated in a different way In this paper thepressure regeld is determined by regrst calculating an intermediate velocity regeldbased on an estimated pressure and then obtaining appropriate correction soas to satisfy the continuity equation This kind of treatment to pressure isnecessary to make the equations parabolic resulting in the freedom to solve athree-dimensional problem with a two-dimensional computer storage eventhough the macrow is three-dimensional and the full equations are elliptic Thedetails of solution procedures can be found in the method proposed byPatankar (1972) Lin et al (1995) presented an example of usage of the marchingprocedure which solved the developing macrow and heat transfer in annular-sector ducts

3 Numerical procedureThe above governing equations were discretized by the regnite volume approachThe convection term in the cross-section is approximated by the power lawscheme The velocity regeld of the cross-section were solved by an elliptic solverusing SIMPLE algorithm to deal with the coupling between velocity andpressure regeld While the governing eqsof the axial velocity was discretized byone-sided scheme with an assumed axial pressure drop The computation ofone-step forward was considered converged if the axial mass macrow rate balancemeets the required accuracy The details of this computation procedure may befound in Patankar (1972) Here we focus our presentation on the detailedprocedure for obtaining an appropriate macrow distribution at the channel inletand revealing the conjugated character between the solid fuel layer and thecooling water The solution procedure for the velocity and temperature regelds inthe subchannels and solid regions is as follows

(1) Assuming the inlet velocity distribution for the seven ducts according tothe given total macrow rate

(2) Solving the velocity distribution regeld for the annular-sector ductsseparately The macruid thermophysical properties are determined by thelocally averaged temperature determined from energy balance equation

Numericalsimulation of

turbulent macrow

333

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

EC193

334

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

EC193

336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

(3) Updating the inlet velocities for the seven channels according to thecomputed pressure drops in each channel

(4) Going back to step 2 until the following criterion for the pressure dropsis satisreged

j Dp( j)j= 17 2 Dpmean j max

Dpmean 3 (9)

where Dpmean is the average value of pressure of the seven channels

(5) Calculating the temperature regeld for the entire region including annular-sector channels uranium layers aluminum layers regns and inner andouter casings

(6) The above procedures from step 2 to step 5 is repeated so that the macruidphysical properties are updated by the introduction of the newly solvedtemperature distribution regeld The iteration stops when the requiredconvergence criteria are satisreged

A careful grid-independence study was carried out to ensure that results arebasically independent of grid system For this purpose three grid systems20(r) pound 98(u) pound 30(x) 30(r) pound 136(u) pound 40(x) 40(r) pound 156(u) pound 50(x) are testedIt is found that for most cases examined the maximum relative error in thevelocity and temperature solutions between grid systems 30 pound 136 pound 40 and40 pound 156 pound 50 is within 15 per cent Therefore the accuracy of the solutionfrom the 30 pound 136 pound 40 grid system is thought to be satisfactory and it isemployed in all computations

For the longitudinal pressure gradient the following convergence criterionis enforced

shy p

shy x

n+ 1

2shy p

shy x

n

shy p

shy x

n

shyshyshyshyshyshyshyshy

shyshyshyshyshyshyshyshy 1 pound 102 5 (10)

At each marching step the convergence criteria for stopping iteration in ellipticcomputation is as follows

Fn+ 1 2 Fn

Fn

shyshyshyshyshy

shyshyshyshyshy 1 pound 10 2 5 (11)

where f denotes v wThe local heat transfer coefregcients h(x) and the Nusselt numbers Nu(x) for

the seven channels are obtained by

hj(x) =qj(x)

tw j(x) 2 tb j(x)Nu(x) =

hj(x) cent Dj

kj = 1 2 7 (12)

EC193

334

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

EC193

336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

where qj(x ) is the local heat macrux tb j(x) and tw j(x) are the local bulktemperature and wall temperature respectively tb j(x) is calculated from

tbj(x) =

Z 23p

0

Z r ij+ 1

roj

utrdrdu

Z 23p

0

Z rij+ 1

roj

urdrdu

(13)

where ro j and ri j+ 1 are the outer and inner radius of each water channelThe channel averaged Nusselt number is deregned as

Numj =

Z L

0

Nuj(x)dx

L(14)

4 Results and discussionComputations were performed regrst for a basic case with tin = 508 C F =161 kg=s and qmax = 1188 pound 1010 W=m3 Then parametric studies wereconducted by changing water inlet temperature and inlet velocity from 208 Cto 80 8 C and 6 ms to 15 ms respectively The presentation will start fromthe macrow and heat transfer characteristics of the basic case followed by theeffects of the inlet temperature and velocity on the maximum surfacetemperature

41 Flow velocity distributionAs mentioned above at a given total macrow rate macrow velocity distributionamong different channels is adjusted according to the same pressuredrop principle Table II presents the results of the inlet average velocitydistribution

Some important observations can be made from the table First the inletvelocities of the two bilateral channels are apparently lower than those of themiddle Secondly among the regve middle channels the average inlet velocityincreases slightly from inner to outer The observations can be understoodfrom the effects of viscosity velocity and equivalent diameter on pressure dropIt is well known that pressure drop for a channel will increase with the increaseof the viscosity and velocity of the water while it will decrease with the

Channel number 1 2 3 4 5 6 7

Averaged inlet velocity (ms) 9558 10108 10173 10191 10201 10207 9533

Table IIChannel-averagedinlet velocities for

the channels underidentical pressure

drop

Numericalsimulation of

turbulent macrow

335

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

EC193

336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

increase of the channel equivalent diameter Water in the two bilateral channelsis heated by one-side wall with the other wall insulated This fact results in alower water temperature and hence a higher viscosity than the water in themiddle channels Therefore to keep the same pressure drop as the others thebilateral channels should have smaller inlet velocities On the other handthe regve middle channels were heated by two-side walls they should havehigher inlet velocities comparing to the two bilateral channels mainly due totheir high water temperature The small difference in the inlet velocity amongthe regve channels came from their different equivalent diameters

42 Temperature DistributionFigures 2plusmn4 show the variations of the local bulk temperature of water in theseven channels uranium layers and aluminum layers (including the inner andouter casings) respectively with the axial distance Apart from seven watercooled channels there are six uranium layers and fourteen aluminum layersWe group the aluminum layers into two classes outer aluminum layers andinner aluminum layers according to their relative position to the channels towhich they are close For a better understanding of the different positions ofthe numbered objects the four numbering systems are shown in Figure 1 Itcan be seen from Figure 2 that the water temperature in the two bilateralchannels is much lower than that of the others Figure 3 shows that theuranium temperature of the regrst layer is much lower than those of the othersbecause of its better cooling condition while the temperature of the sixthuranium layer although still lower than the values of the other four layers is abit higher than that of the regrst one This is basically because of the lowerconvective heat transfer coefregcient of the Channel 7 which will be presented inlater discussion The aluminum layer temperature distributions presented in

Figure 2Local bulk temperatureof water along the axialdirection

EC193

336

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

Figure 4 indicates a similar variation pattern as that of uranium that is theregrst inner aluminum layer and the seventh outer one have the lowesttemperature at each cross-section while the regfth outer aluminum layer and thesixth inner layer possess the highest temperatures To sum up the temperatureof the regfth uranium layer and the related aluminum layers are the highestThus for the safe operation of the reactor more attention should be paid to thetemperature situation of the regfth uranium layer

It is interesting to note that from Equation (1) the highest volume heatsource occurs at the axial location of x = 05 m where sine function equals 1and q(x) = qmax However all the temperature distribution patterns shown inFigures 3 and 4 exhibit their summit at the location around x = 07plusmn08 mrather than 05 m showing some delay between the distributions of volumetricheat source and the temperature in uranium layer This is also a noticeablefeature for the design and operation of such kind of reactor

Figure 5 presents the local maximum surface temperature for fourteenaluminumlayers along the axial direction which shows the similar characteristicsto the curves shown in Figure 4

Attention is now turned to the temperature distribution at the cross-sectionsin different axial location Take channel 2 as a representative for the middlechannels Its temperature contours are shown in Figure 6 for four axiallocations Two features may be noted from the reggure First in each of thetemperature contour of the four cross-sections there is a core region in whichthe water temperature is more or less uniform The radian span of this coreregion decreases with the increase of axial location indicating the developmentof the thermal boundary layer along the macrow direction Second the temperaturegradients normal to the inner and outer circular surfaces in the center region ofthe peripheral direction are much greater than those in the two end regions near

Figure 3Local bulk temperature

of uranium layers alongthe axial direction

Numericalsimulation of

turbulent macrow

337

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

the regns in each cross-section This implies that the local convective heat transfercoefregcient of the center region is much higher than those of the two ends Thiswill lead to non-uniform peripheral surface temperature distribution of eachchannel To have a better understanding for this non-uniformity Figure 7provides the peripheral temperature for channel 2 at x = 0795 m It can be seenclearly that the end temperatures are signiregcantly higher than their counterpart inthe center region the difference ranging from 108 C to 15 8 C for the parametersstudied in this paper This is also a noticeable character for the design andoperation of such kind of reactor

Figure 4Local bulk temperatureof aluminum layersalong the axial direction

EC193

338

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

43 Heat transferHeat transfer coefregcient distribution is of great importance in the presentapplication Figure 8 represents the axial variation of the local heat transfercoefregcient for the seven channels As seen from the reggure all the curves do notapproach to a constant value with the increase of axial location This is resultedfrom the non-uniform axial heat macrux distribution (Equation (1)) For the middlechannels which are heated by both the inner and outer walls the local heattransfer coefregcient increases to a maximum value at certain locations and thendecreases monotonically While for the one-wall-heated bilateral channelsthe local heat transfer coefregcient exhibits the entrance problem character at the

Figure 5Local maximum surface

temperature foraluminum layers along

the axial direction

Numericalsimulation of

turbulent macrow

339

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

very beginning of the channel and then takes the variation pattern of themiddle channel heat transfer coefregcient with about 30plusmn40 per cent percentagereduction in the absolute value

The effect of the Reynolds number on the averaged Nusselt number ispresented in Figure 9 It can be noted that the bilateral walls also exhibit a

Figure 7Temperaturedistribution along theperipheral directionfor Channel 2 atx = 0795 m

Figure 6Isothermos at differentchannel cross-sectionsfor Channel 2

EC193

340

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

different characteristic from the middles At the same Reynolds number theaveraged Nusselt numbers of Channel 1 and Channel 7 are higher than those ofthe others which seems to be contradictory to Figure 8 It should be noted thatin Figure 8 the water viscosities of Channel 1 and Channel 7 are much higherthan those of other channels due to their lower temperatures and their velocitiesare lower (Table II) Therefore the Reynolds numbers in the bilateral channels1 and 7 are lower than that of the middles (Channels 2 to 6) In the meantime atthe same Reynolds number the Nusselt numbers of the relatively outerchannels are a little smaller than that of the inner ones

Figure 8Local heat transfer

coefregcients along theaxial direction

Figure 9Variation of the averaged

Nusselt number withReynolds number

Numericalsimulation of

turbulent macrow

341

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

44 Effects of the inlet water temperature and velocitySince temperature distribution is the major concern of the present studythe computational results for the extended parameters of the inlet watertemperature and inlet velocity are provided for the effects on the maximumchannel surface temperature (so-called hot spot temperature) and the maximumuranium temperature Figure 10 presents the numerical results It should benoted that when the effect of one parameter is examined all other parametersremain the same as for the basic case From Figure 10(a) an almost linear

Figure 10Variation of maximumtemperature with inletwater temperature andvelocity

EC193

342

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

variation of the maximum surface temperature with the inlet temperaturemay be observed This is an expected outcome since the variation of thewall temperature is not large enough to make an appreciable change of theconvective heat transfer coefregcient Thus with the same heat macrux the localtemperature difference between wall and water remains the same resulting thelinear variation of the wall temperature with water Figure 10(b) shows that themaximum temperature of the aluminum surface and the uranium decrease withthe increase of inlet velocity The curves shown in the reggure exhibit somewhatasymptotic character when inlet velocity goes quite high (from 6 to 15 ms)This is because that the heat transfer resistance from the uranium to the watermainly lies in the solid conduction when the surface convective heat transfercoefregcient exceeds some speciregc values Therefore the increase of water inletvelocity will decrease the wall temperature efregciently only when the water-sideheat transfer resistance is predominated

It should be noted that all the temperature all the computations presented inthe present paper are for the ideal case This means that the factors which leadto the additional heat transfer resistances are not taken into account Thesefactors include the thin water reglm between the aluminum surface and the space(regn) the oxidization of the aluminum surface and the distortion of the uraniumlayers The computational results are compared with the experimental datagiven by Chen and Jiang (1984) in the case of tin = 3498 8 C F = 157 kg=s andqmax = 1001 pound 1010 W=m3 in the presence of the above effects The water reglmand the oxidized aluminum coating had the thickness of 05 mm and 005 mmrespectively while the uranium distortion was 0125 mm thick Table III showsthe results It can be found that after considering the effects of water reglmoxidized aluminum coating and the uranium distortion a good agreementbetween numerical and experimental results is achieved with the maximumdeviation about 14 per cent This agreement indicates that the presentnumerical approach can be used to predict the macruid macrow and temperature regledof the multi-channel fuel elements

Point Channel PositionExperimental

data ( 8 C)Computational

results ( 8 C)Deviation(per cent)

1 2 x = 070 m 12338 11258 872 3 x = 070 m 11478 10860 543 4 x = 070 m 11405 11357 044 5 x = 075 m 13172 11934 945 6 x = 070 m 11011 12401 1266 4 x = 065 m 13058 11225 1407 4 x = 075 m 11399 11435 038 5 x = 070 m 12648 11918 589 5 x = 070 m 12705 11917 62

Table IIIComparison

betweenexperimental and

computationalresults

Numericalsimulation of

turbulent macrow

343

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

5 ConclusionsThree-dimensional parabolic turbulent forced convection heat transfer andmacruid macrow characteristics in multi-channel narrow-gap fuel element with non-uniform heat macrux are studied numerically The boundary-layer method with amixing length turbulent model was proved to be able to simulate the macrow andpressure characteristics in the multi-channels The main conclusions are

(1) The inlet velocities of the bilateral channels are apparently lower thanthose of the middle and among the regve middle channels the averageinlet velocity is almost the same with very slight increase from inner toouter

(2) Because of the non-uniform heat macrux axial local heat transfer coefregcientsfor the seven channels donrsquot approach fully developed constant value

(3) For both aluminum and uranium layers the local temperature distributionalong axial coordinate is of sinuous type with the peak at x = 07plusmn08 mTheir maximum temperature occurs in the regfth uranium layer

(4) At each cross-section the periphery temperature distribution is notuniform rather the temperature of the two ends are appreciably higherthan those of the center region

(5) At the same Reynolds number the averaged Nusselt numbers of waterin Channel 1 and Channel 7 are higher than those of the middles Andamong the regve middle channels the Nusselt number for the relativelyouter channel is a little smaller than the inner one at the same Reynoldsnumber

(6) The maximum surface temperature increases almost linearly with theinlet water temperature while decrease almost asymptotically with theinlet average velocity

ReferencesChen DL and Jiang F (1994) ordfFlow and temperature measurement for fuel element in HFETRordm

Nuclear Science and Engineering Vol 4 No 2 pp 178-82 (in Chinese)

Hanjalic K (1994) ordfAdvanced turbulence closure models a review of current status and futureprospectsordm Int J Heat Mass Transfer Vol 15 No 3 pp 178-203

Lin MJ Tao WQ and Lue SS (1995) ordfStudy on friction factor of developing and developedlaminar macrow in annular-sector ductsordm J Thermal Science Vol 4 No 3 pp 180-84

Lin MJ Wang QW and Tao WQ (2000) ordfDeveloping laminar macrow and heat transfer inannular-sector ductordm Heat Transfer Engineering Vol 21 pp 53-61

Nida T (1980) ordfAnalytical solution for the velocity distribution laminar macrow in an annular-sector ductordm International Journal of Chemical Engineering Vol 20 pp 258-65

Patankar SV and Spalding DB (1972) ordfA calculation procedure for heat mass and momentumtransfer in 3-D parabolic macrowsordm Int J Heat Mass Transfer Vol 15 pp 1787-1806

Patankar SV Ivanovic M and Sparrow EM (1979) ordfAnalysis of turbulent macrow and heat transferin Internally Finned Tubes and Annuliordm ASME J of Heat Transfer Vol 101 pp 29-37

EC193

344

Ramdhyani S (1997) ordfTwo-equations and second-moment turbulence models for convectiveheat transferordm in Advances in Numerical Heat Transfer Minkowycz WJ Sparrow EM(Eds) Vol 11 pp 171-99

Soliman HM (1987) ordfLaminar heat transfer in annular sector ductsordm ASME J Heat TransferVol 109 pp 247-9

Sparrow EM Chen TS and Jonson VK (1964) ordfLaminar macrow and pressure drop in internallyregnned annular ductsordm Int J Heat Mass Transfer Vol 7 pp 583-5

Wang JF (1981) ordfThermal performances of multi-casing fuel element with regns in HFETRordmNuclear Power Engineering Vol 2 No 3 pp 50-5 (in Chinese)

Yu EJ (1981) ordfComputational prediction of the hot point temperature in multi-casing fuelelementordm in Computations and experiments on thermal engineering in nuclear reactorNuclear Power Press Beijing pp 35-41

Numericalsimulation of

turbulent macrow

345