Heat transfer to liquid metals flowing turbulently and longitudinally through closely spaced rod bundles

NUCLEAR ENGINEERING AND DESIGN 23 (1972) 273-294 NORTH-HOLLAND PUBLISHING COMPANY

HEAT TRANSFER

TO LIQUID METALS

AND LONGITUDINALLY

THROUGH

FLOWING

CLOSELY

TURBULENTLY

SPACED ROD BUNDLES

PART I O E DWYER and H C BERRY Brookhaven Nattonal Laboratory, Upton, New York 11973, USA

Reeetved 10 July 1972 Tins paper presents results of a theoretical study of heat transfer to llqmd metals m fully developed turbulent, mhne flow through unbaffled, spacer-free rod bundles The bundles have eqmlateral mangular arrangement, and the rod spacings, rod design, and ranges of independent variables covered were chosen with reference to hqmd-metal-cooled nuclear reactor apphcatlons Three different sets of thermal boundary condataons are considered (A) umform heat flux m the axml darecuon with uniform temperature m the circumferential dlrect~on, on the outer surface of the cladding, (B) uniform heat flux m both dtrectlon~ on the outer surface of the cladding, and (C) umform heat flux m both dtrectlons on the tuner surface of the cladding The results of the thtrd set are presented m Part II In the present paper, rod-average heat-transfer coefficients and c~rcumferenttal varmtlons of temperature and heat flux on the rod surface are presented as functions of P/D and ~Pe Five different methods of esttmanng the eddy dfffuslvlty of momentum m the elrcumferenttal direction are employed, and the results compared The reference method, which appears to be quite satisfactory, assumes the equahty of the eddy dfffuslvatles of momentum m both the clrcumferenUal and radtal directions The results, presented m both tabular and graphical forms, are reported m terms of convement dimensionless groups, correlated by seml-empmcal equatmns, and compared with previous results.

1. Introduction This paper is the sLxth in a s e n e s [ 1--5] on the general subject o f heat transfer to hqmd metals flowing longitudinally through u n b d f l e d , spacer-free rod bundles. Here, we are deahng with the case o f fully developed heat transfer, where both molecular and eddy conductwmes are taken into account The prewous paper [5] dealt with the same problem, with the exception that molecular conducuon only was considered Thus, the relatave tmportance of eddy conductlwty m tins problem can be readily seen by companng the remits m that paper w~th those here m part I, and with those m part II [6]. The scope o f the present study, summarized m table 1, was earned out for the foUowmg three thermal boundary condmons (A) uniform wall heat flux m the axial direction, and uniform wall temperature m * This work was performed under the ausptces of the Umted States Atomtc Energy Commission

the clrcumferenUal direction, on the outside surface o f the cladding, (B) umform wall heat flux m all directions on the outside surface of the eladchng, and (C) umform wall heat flux m all directions on the tn~de surface of the cladding Boundary c o n d m o n (A) represents the lmaltmg sltuaUon for boundary c o n d m o n (C), as the thickness and/or the thermal conduetwlty of the cladding increase, and boundary c o n d m o n (B) represents the hrmtmg situation for boundary cond~taon (C), as the thickness and/or the conductwlty o f the cladding decrease Moreover, all three boundary condmons tend toward coexistence as the P/ D ratio ~s increased The present paper covers the vanous assumptions that were made m the analys~s, the computational procedures, and the results for the ltmmng thermal condmons (A) and (B). The results for boundary c o n d m o n (C), where claddmg thickness and thermal conductlwty are taken into account, are presented m part II [6], and compared with expertmental results recently obtained at Brookhaven with mercury.

0 E Dwyer, H C Berry, Heat transfer to hqutd metals, part I

274

Table 1 Scope of present study (parts I and I1) 1. Rod arrangement, equtlateral triangular

2. PID range, 1.001 tcL1.30 3. ~Pe range, 20 to 5000 4. Prandtl range, 0,0048 to 0 022 5 Range of cladding tluckness, 2 5 to 30% of rod dtameter 6. Range of cladding conductwlty, 10 to 400% of coolant conductwlty

/

2 Computational procedure For heat transfer considerations, a typical reactor fuel rod may be considered to consist of a (ceramic) fuel core contained m an alloy tube (cladding). The small space between the fuel and the cladding may be f'dled with an inert gas or with sodium. In the present study, we are considering only a two-region heat transfer model 0.e., the cladding and the coolant), assuming a uniform heat flux on the umer wall o f the cladding. This is valid for typical LMFBR Pael rods and P/D down to about 1.04 [3, 7]. For a given P/D, the parameter having the 8reatest influence on the circumferentaal vanataon of the heat flux on the tuner wall o f the cladchng is k/kf The less this is, the less will be the flux vanatton, for LMFBR fuels the ratio falls m the low range 0.03 to 0.04. Fig. 1 is a schematic cross sectmn of any three-rod portion o f an equdaterally spaced, triangularly arrayed rod bundle. Because of symmetry, the 3 0 0 - 6 0 0 - 9 0 ° triangle shown at rod c, and m f ' ~ 2 is the elemental area that must be considered. The governing energy equalaon for the elemental coolant flow area (efdc in fig 2) is . . 1 at . a2t (kf + ICe,r)r~-r + (kf + ke, r)ar2 --

+

ake,r at (kf + ke, 0 ) a2t 0--7- a-; + \ao2

1 at ake, o ~ dt + r 2 aO a--O-- - t.pop ~-~. "

(1)

It is based on the conditions (a) the flow is turbulent, (b) the velocity and temperature fields are fully developed, and (c) slgmficant axial heat conduction does not exist. Furthermore, it is based on the assumptions


) ' \\

I)

U/:'

I

v

Fig. 1. Cross sectton of a three-rod portion of a rod bundle

with equilateral triangular spacing The cross-hatched area represents an elemental coolant-flow area. f

AREA oob REPRESENTS CERAMIC CORE ~'~r=P/2

AREA cdbo REPRESENTS CLADDING

'l /~"

AREA e-fd-c REPRESENTS COOLANT FLOW AREA

I

"~d

Ae I

I

---_--~

~ ~ . . . .

_L L__] C

O

t

-" ~ "~ ~

. ~-,,

CENTER O

Fig. 2. Schematic diagram to illustrate the nature of the two-

regmn heat transfer problem and particularly the type of grid used m settmg up the f'tmte~tfference equaUons for the coolant regmn (a) the effects of the transverse temperature vanaUon on the physical propertms of the coolant are negligible, and (b) secondary flow effects on the temperature field are negligible. The vahditms of both of these assumptions denve from the fact that, due to the high thermal conductivltes of hquid metals, transverse temperature variations are, m most pracacal situatmns, relatively small The boundary con0mons for eq. (1) are

at/ao=o,

at0=O

°,30°,

(2)

0 E Dwyer, H C Berry, Heat transfer to hquM metals, part I

aT 30

where 0, as shown In fig 2, IS the clrcumferentaal angle measured from the straight hne that connects the center points of two adjacent rods (cos 0) at _ sin 0 at P ar ~ - - 3 0 ' atr-2cos0, (3)

aT _ sin 0 aT

and

-kf(at/ar)r 2 = -kw(at/ar)r 2 = qw2(0)

(4)

Boundary conditions (2) and (3) take cognizance of the fact that the isotherms in the elemental coolant flow sectaon are perpendicular to the boundary flow hnes (fig. 2). Boundary condition (4) says that the heat flux at r 2 is some function of 0 This is not known a priori but is calculated (by an iteratlve process) from the differenttal equation govermng the heat flow in the cladding This equation is a2t

--+-

1 at

+

1 a2t

-0

(5)

ar 2 r ~- r 2 302 ' for wluch the boundary conditaons are

(6)

t = f(O),

(7)

at r = r 2 .

Boundary condition (7) IS also not known a priori but must be determined from eq (1). It Is thus clear that eqs. (1) and (5) are coupled through boundary cond,tlons (4) and (7). The simultaneous solution of eqs. (1) and (5) was obtamed by an ,teratlve process [3]. Eq. (1) was solved numerically, using a system of finite difference equations For ease in handling, it was put rata the dimensionless form

R aT ~ ~- + (I+K)

" aR"

aKr aT

l+Koa2 T

+ -- -- + -aR aR R 2 302

aR

qw2

'

m the coolant a t R = 1, or

aT kf %2(0 aR

kw qw2

In the cladding at R = 1.

(4a)

Eq. (5) can be readily solved [3] to yield the temperature in the cladding as a functaon of r and 0 This relation can then be dlfferentmted to give rlqwl

-w\~-]r2- 72

6kw r2

1-(rl/r2)12n ] cos (6n0), X n~-_lnAn [ t+(ra/r2)12n

(12)

where ~_/6 tw, 0 cos (,6nO)dO (13) 0 Eq. (12) gwes the clrcumferentlally local flux for the heat leaving the cladding at r 2. This is obviously the same flux for the same heat entermg the coolant stream at r2, which is therefore boundary (4) for eq.

A n = (12/rr) j

2 1. Velocity distribution {X/~(p/D)2_~lr

(8)

where

g = r/r 2 ,

(9)

kft

T= - , qw2r2 and

Kr=ke, r/k f ,

%2 (0)

(3a)

COS 0 '

(1).

1 aT ago +R 2 30 a T

aT

(2a)

at R - P/D

P/D 3 0 '

k ( a t '~

-kw(at/ar)r~ = qwl = a constant,

(l+Kr)

at 0 = 0 °, 30 ° ,

-0,

aR

275

(10)

In order to solve eq. (I), for a gwen set of condl. tlons, the velocity field In the elemental flow area must first be determined In the present study, this was done by using the Van Karman law-of-the-wall equataons [8] u+=y+,

(y+~<5)

u + = - 3 0 5 + 5 . 0 1 n y +,

K 0 =ke,o/k f •

(11, Xla)

Boundary conditions (2), (3), and (4) now become

u+=55+251ny

÷,

(14) (530)

(15) (16)

0 b, Dwver H C Berm, tleat trans]et to ltqutd metals part I

276

winch were originally obtained from circular-tube data The justification for this is based on the experimental results of Elfler and Nijslng [9] and of Levchenko [10] These investigators measured velocity profiles for flow through rod bundles having P/D varymg from 1 00 to 1 15 and found that the standard lawof-the-wall relationship for pipe flow held In eqs (14), (15), and {16), u + and) ,+ refer to condmons along a line f normal to the rod surface, for a given circumferential angle 0 (fig 3) From flmd-dynamm and heat transfer standpoints, there are close slmdantms between flow longitudinally through rod bundles and flow m the inner p o m o n s of annuh For example, the hexagon m fig 1 can be approximated by a circle of the same area The relation between rod radius (r2) and the radius of this circle (r m) is analogous to that between the inner radms and the radms-of-maxlmum-veloclty of an annulus This is the basis of the so-called equwalent-annulus model [11 ] for analyzing heat transfer to llqmds flowing uihne through rod bundles This model relates P/D and r o/r, through the equation

(ro/r)-~(P/D) x/evr3/rr(P/D) - 1

;/0_) 0343 =

'

(17)

which is based on eq (23) in ref [12] Eq (17) says that for P/D 1 30, the upper limit in the present study, the equwalent ro/r ~ is 1 813 For ro/r ~ up to about this value, Brighton and Jones [13] working with air, and Hlavac, Nlmmo, and Dwyer [14] working with mercury, found that the u + vs y+ plots were linear and

close t~) those lor pipe flow Consequently, by mw/kmg the analogy between flow through lod bundles and that through the mner portions of annuh, there is justlhcatlon for using the standald law-ot-the-wall equations for calculating velocity distributions m rodbundle flow tor P/D up to about I 3 Tins P/D the upper limit in the present study ~ somewhat above 1 15 the upper hmlt nr the Elfler Nllslng e\pernneJ> tal study [¢)] Belore it is possible to use eqs {14) to (lo) to ~.,tlculate tire local time-average hnear velocity (u} a l gwen values of)' and 0, one must know r w as a fun~tmn 0 Tins, m turn, depends on tire pressure drop (~p/AL), which, m turn, depends on the friction ta~.toi ( / ) The frlctaon factor was explessed as the latlo J/Jct, where J'ct represents the ~lrcular-tube trictlon factor This was calculated from J~, ~ 0 0 0 5 8 + 0 4 3 2 R e

0 312

t,S}

which represents the Moody fnctlon-tactor curve for smooth pipes m the range of Reynolds numbeis covered in tins study When calculating fct by means o f e q (18), Re is the rod-bundle Reynolds number In the present study, f/-/ct was determined by the method of successive approxlmauon It was systematlcally varied until the average linear velocity as determined from the calculated local linear velocity equaled the average hnear velocity corresponding to the Reynolds number on winch the local linear velocities were based In other words, f/tot was varied until the equation

(l/Ac) f

VdAc=/~Re/DeP

{19}

was satisfied It was found that//Jct increased slightly with Reynolds number, and a balance was struck, t e a final value off/fct was chosen such that the ratio

( 1/A c)

I

?

vdA c / I~ Re/DeP

(20)

h

,I

e

I ~/ / ~ -~~"

^

. r2

"-.

C

Fig 3 Schematic drawing to illustrate the geometry of the problem The are,, abdc represents the cladding

0

was very slightly below unity at the lower Reynolds numbers, and very shghtly above umty at the Ingher Reynolds numbers This procedure was quite satisfactory, because the local velocities, as calculated in each case, were dwlded by the ratio (20) before being used m e q (1)

0 E Dwyer, H C Berry, Heat transfer to hqmd metals, part I Table 2 Dependence off/let on P/D

,'°' c[, ox ( x ° 8 av )3

I/At P/D

Present study

Ibragunov et al [5, 15]

1.001 1 01 1 02 1 03 1 04 1.05 1.06

0.640 0 775 0.855 0 910 0 950 0 980 1 000

0 653 0 706 0 795 0 819 0.943 0 990 1 023

1 07 1 10

1 015

-

1.15 1 20

1 040 1 065 1.080

1 30

1 100

1 40 1 60 1 75 1 80 2.00

1 115 1 130 1 140 1 145 1 150

proposed by Ibragmaov et al. [16], where x =.4 c

(24)

av

1 6f/6[ c 7r 0

1 - exp

(

7.7fi(0)) ] --d0. X0 8fiav

(25)

Spot checks showed good agreement between velocity distributions, both radml and carcumferentaal, calculated in the present study with those measured by Elfler and Nijsmg [9].

1 087 1 115 1 129 1 149 1 166 1 199 1 224 1 233 1 268

Ap _ fo2aP/2gcDe

(21)

-¢w (6A cApl A L )/Itr2

(22)

AL and the average, wall shear stress, r-w, IS given by

(23)

and

A tabulauon of~fret used m the present study is gwen m table 2, compared wlthflfet calculated from the correlaUon of Ibragimov et al. [5, 15] The latter gave raUo (20) values that departed appreciably more from umty than did those calculated from f/fct recommended by the present authors However, it turned out that the heat transfer results obtemed with both sets of/fret values were negligibly different. For ex,: ample, the greatest chfference occurred at the tughest Peclet number (5000) and the largest P/D (1 30), where the rod-average Nusselt number for limiting thermal boundary condmon (13) when using the present f/tier was just 1% lower than that obtmned when usang the Ibraglmov value The reason is that, in channelflow, forced-convection, hquld-metal heat transfer, the temperature field in the coolant is not senmwe to slgmficant vanaUons m the velocaty field, The pressure drop through the rod bundle is glvon by the standard, channel-flow equation

=

277

The local wall shear stress, rw(0), was calculated by

2 2.1 Numencal solution of eq (8) As menUoned earher, eq (8) was solved numerically, using a system of fimte difference equaUons, and employing a refinement of the Gauss ehmmatlon method. The type of gnd used Is tUustrated m fig. 2. The radial distance betWeen r 2 and ½P was dlvaded mto M equal increments, and the 30 ° angle rote N equal angles. For radial distances beyond r = ~P, the radial increments of the grid were deterrmned by the mtersecUons of the angle grid lines with the vemcal flow-boundary line. There are two advantages to flus (a) gnd points exterior to the elemental flow regmn are ehrmnated, obwatang the need for addmonal equaUons, and (b) a fine gnd is obtmned for radial distances beyond Xp by stmply increasing the number (Le., decreasing the size) of the angular increments. This latter advantage Is particularly important for low P/D. M and N were therefore vaned, wlule kept as hagh as possible (conmtent with the storage capacity of the computer), until the best combmaUons were found. These comblnattons are listed m table 3. Table 3

P/D range

M

N

1.001 to 1.05 1.06 to 1 20 1 30

20 40 80

20 15 10

278

0 t~ Dwver, H C Berry tteat transfer to hquM metals, part I

2 3 Handhng the eddy thermal conducttvzty problem The eddy thermal conductlwtles and their first derwatwes are needed, in both the radml and circumferential directions The usual approach in estimating the eddy thermal conductivity is to calculate the eddy dilfusivity of m o m e n t u m transfer, eM, convert that to ett by some means, and then calculate ke(= Cvpe H I from

sing eH=J(e M) that was employed by the analyst 3 Finally. the approach taken m the present stud\ allows the design engineer to leadIly find the uppe~ hmlt of a heat transfer coefhclent by simply taking equal to umty (The lower hnnt is obtained when = 0, e g . when molecular conduction only [5 ] la consadered )

~[t At each point In the elemental coolant flows area o1 the flowing stream, one can theoretically estimate ke m both the radial (l e , ke, r) and circumferential (I e , ke,o) darectlons, by the above means, and employ them (and their corresponding derivatives) as local rallies In the numerical solutmn o f e q (1) or (8) However, there is no good way of converting e M to elt, particularly m the circumterentlal direction so the above approach was not used Rather, e H was assumed to be equal to e M , or, In other words, eH/e M --= ff = 1 where ~ IS the overall, average, effectwe value of eH/e M, in both the radial and circumferential directions There are three justifications for taking this particular approach 1 In fully developed heat transfer to hquld metals flowing turbulently in pipes [17] and annuh [11], Nusselt numbers can be satisfactorily correlated in terms of f P e , as the independent variable In the present study, this was also shown to be true for inline flow through rod bundles, as shown in table 4 It was also shown to be true for the dependent variable of circumferential surface temperature variation In calculating the results in table 4, all local values ot eM, r and eM, 0 were multlphed by the same t}- to give local values of ell, r and ell, o , from which local values ofke, r and ke, o were obtained 2 Since ~7 was taken as u m t y in the present study, f P e = Pe, and the Nusselt values, although calculated as functions of Pe, can be reported as functions of ~Pe This means that in a practical situation f must first be estimated, before Nusselt values can be determined from the correlations given herein This is considered to be a more feasible approach than that of estimating local values of ff and then developing correlatmns giving the Nusselt value as a function of Peclet number only, because expressing Nu=f(~bPe) allows the design engineer to readily employ improved methods of estimating ~ as they come along The other approach restricts the design engineer to the m e t h o d of expres-

2 3 1 Estimating As stated above, m order to make practical use ~1 the results of this study, one must first have some means of evaluating ~ Also, in order to make com. parlsons between the present theoretical predictions and experimental results, one must also be able to evaluate ~). for each situation For this the semi-empirical relationship [18]

Pr(e../v) l 4 M

"

mlL-,~

appears to be as good as any tor our present pur poses The constants in eq (26) were obtained from heat transfer results obtained on wide P/D spacings, and (em/V)max values were obtained by employing the equivalent-annulus model, (e M/V)max was presented graplcally [18, 19] forP/D down to 1 38 laor P/D below this, (eM/V)max in the radial direction begins to vary more and more with peripheral angle O, so that one is obliged to think in terms of [(eM,r/V)max]av and rewrite eq ( 2 6 ) a s = 1

1 82

(26a)

Pr[(eM,r/V)max ]lay4 Values of [(eM,r/V)max]av, calculated in the present study, are shown in fig 4 It is obvious that there is very little effect of P/D on [(eM,r/V)max]a v In the 1 O0 ~ P/D < 1 40 range It is also noteworthy that, for P/D = 1 40, the curve in fig. 4 agrees perfectly with the earlier calculations [ 18] The present authors employed the same constants in eq (26a) for close spacings as were developed for eq (26) for wide spacings, because there are as y e t insufficient reliable experimental data on closely spaced rod bundles to re-evaluate the constants Moreover. although eq (26a) only employs e M In the radial direction, ~ IS presumed (in the present study) to take care of both radial and circumferential heat transport m the elemental coolant flow area

O E. Dwyer, H C Berry, Heat transfer to hquld metals, p a t t i

2 4. Radtal c o m p o n e n t o f eddy thermal c o n d u c t w t t y The radial eddy thermal conductwaty, ke, rlS cal. culated from the radial eddy dlffuslvlty of m o m e n t u m transfer by assurmng that the radial eddy dlffuswltles of heat and momentum transfer are equal, or that

Table 4 Heat transfer results obtamed m the present study that show that the product ~Pe is a prune mdependent variable. In the calculatmns, it was assumed that eM,0 = eM,r, eH,r = ~eM,r, and ell,0 = ~eM, 0 For each row ~-Pe = 500

P/D

Pe

~

[N--ut]q

tw,0 tw,30 t-w,0 - tb

1.01

500 700 1000 2000 5000

1.000 0.714 0.500 0.250 0 100

0 514 0.516 0.514 0.513 0.512

3 41 3.42 3.42 3 42 3.43

1.10

500 700 1000 2000 5000

1.000 0.714 0 500 0.250 0 100

8.89 8.87 8 93 8.89 8.82

2 28 2.28 2 30 2.30 2.30

1.30

500 700 1000 2000 5000

1 000 0 714 0 500 0.250 0.100

15 51 15 47 15.54 15.58 15.46

0.264 0.263 0.264 0.264 0.262

279

eH,r/eM, r = ~ = 1 The radial momentum dlffuslvlty xs defined by the equatmn Trgc/p = (v + eM,r ) d v / d r ,

(27)

where do/dr lS the slope of the velocity profile along a gwen radial hne 33 (see fig 3) and can be obtained from either eq (15) or (16), depending on the value o f y +. A force balance over the circumferential distance r ( d 0 ) in fig. 5 gives

(

rr = -

7r

27r

27r

rdO

(28)

and a force balance over the clrcumferenUal distance r2d# gwes a similar equation, with r r replaced by rr2 and r replaced by r 2 Dividing one equation by the

10 4

I0: g E

0

W

I0: I I

d

I0 I0 4

2

3 4

e elO 5

2

3 4

e elO 6

34

e~O r

REYNOLDS NUMBER, Deva p

P

Fig.4. Values of [(eM,r/v)max]av for fully developed, longitudinal, turbulent flow through rod bundles hawng equilateral trtangular spacing.

280

0 E D~ver H C Berm' lteat transfer to hqutd metals, part I

I ? CO 5 ~

--.>. I

rd8

"~- . / /

-

-

r

r 2

--

P2

Fig 5 Drawing used to write the radial-shear-stress equation (eq 28) other, gives rr

r~ (½P)2 _ r 2 cos2 0

7r '

r

-

I 2 (,7P)

{29)

r 29 cos 2 0

Finally, combining eqs (27) and (29), and evaluating d v/dr from the law-of-the-wall equations, eM, r = 0 ,

(30)

f o r y + ~< 5, and

eM, r

=u* . r

. . . . . 5 0 L(½P) 2 - r 2 cos 2 0

v,

(31)

for 5 < y + ~< 30, and r 2 r - r 2 [ (}P)2 - r2 cos 20 ] -

eM,r

-

u* 2 5

u,

(,32)

L({-P) 2 - r 2cOs 20

f o r y + > 30 Since ell, r (= ke, r/OCp) IS assumed equal to CM,r, we have ke, r = eM,rP 9

(33)

This was the method of evaluating ke, r for use in eqs (8) and (11) In eq (8), we have the quantity 3 K r / O R wluch is basically the rate of change of ke, r with radial distance In setting up the fimte-dlfference equations for the z2~ vs --A0 grid, the values of b K r / 3 R at the wall points were taken as one-half those at the first mesh point away from the wall This was done to give proper weight to thus quanUty in the first A r Increment next to the wall when the first mesh point away from the wall was at a radial distance greater than that corresponding to y+ = 5

2 5 C~rcum]erenttal compo,zen;,~? eddy thermal~ o~ du~ ttvtlv The eddy dlffuslvlty of heat transfer m the ClrCUlnferentxal direction can be estimated m a manner s,m~lar to that m the radial direction, 1 e . by estimating eM,~ and by assuming that ell, o - ~M o But :he pltJb]em is there is no good way ol predicting eM, O t~r m line flow through rod bundles Consequently, the approach taken here is that eM, o must be evaluated by some seml-elnplrtcal method Vanou~ method~ ,.,m b~ reed, until one is lound that gwes heat transfer plc &~.tIons that agree with dependable expenmenta[ wsult~ Unlortunately In t h l s P / D range the numbe~ and dependablhty ot experimental results are ye~ ~ adequate In an earlier study [20], evaluating CM,o, and from it, eIi 0, was circumvented by firsl estabhshmg the velocity field and then calculating values of cM along the flow tmles (1 e . curves perpendicular to the constant-velo~lty lines) Since the velocity-flow line~ were not too different from the heat-flow lines, valuer,. eH were directly obtained for the directions of heat flow ~Ihe temperature field was then determined by trial-and-error heat-transfer calculations tot curved heat-flow paths This method oi determlng Nusselt numbers is, however, very time consuming, and only a few cases were run The simplest approach is to assume that, a~ all points in the elemental flow area (Ac), eM, 0 -- c M , and this was done m the present study However other methods of estimating eM, o were also employed, in certain cases, and the Nusselt results are reported later, tor comparison These other methods will no~ ~.be briefly discussed For flow in a pipe, it is generally agreed that in the turbulent core eM,0 is practically independent of radial distance and roughly equal to the maximum value of eM, r Comparable measurements for m-hne flow through rod bundles are not available On the basts ot pipe-flow behavior, Rapier [21] postulated that the circumferential eddy dlffuslvlty ot m o m e n t u m transfer for rod-bundle flow could be approximated by eM o = 0 1 ~X/rw gc/p ,

(34)

which says that eM, 0 varies from a minimum at 0 = 0 -~ to a maximum at 0 = 30 ° However, oa the basis ot lnterchannel mixing experiments conducted by Rowe and Angle [22], which indicated that circumferential

0 E Dwyer, H C Berry, Heat transfer to hqutd metals, partl

velocity fluctuations in the 0 ° region were significantly greater than those m the 30 ° region, Oberlohn [23] modified eq. (34) to read O. l(Y30 ) eM, 0 =

ogc

y

(35)

but he found this change resulted in mordlnarely large Nusselt numbers at the low P/D ratios Moreover, he found that when eM,0 was taken equal to eM,r, the resultang Nusselt numbers turned out to be reasonable, according to lus method of analysis On the basis of dlspemon experiments in open rectangular channels, Elder [24] found that eddy diffusion m the darection perpendicular to the direction of flow and parallel to the channel base was roughly three tlmes that in the dlrectaon perpendicular to both the direction of flow and the channel base. From has resuits, we get the relataon 7

eM,L = 0 0136v Re r

(36)

whach N1jslng, Gargantml, and Etfler [25] * modified to 7

eM,0 = 0 0136v Re0r

(37)

for rod-bundle flow, where Re 0 is the penpherally local Reynolds number. Elfier and Nijstng [9] measured velocity profdes for water flowing m rod bundles. At two P/D ratios, they made compansons between their experimental results and velocity fields calculated from the basic momentumbalance dffferenUal equation. At P/D = 1 05, they found that the velocity field calculated from the momentum equataon when using eM,0 values by eq. (36) agreed better with the expenmental remits than that calculated using eM,0 by eq. (37). AtP/D = 1 10, the satuatton was reversed. However, m both cases, the differences between the calculated, and experimental resuits were not large As stated earher, in the present study, the standard method of evaluating eM,0 was to assume it equal to * Ref [25] actually used a coefficient of 0 0115 instead of 0.0136. The difference would have only a very minor effect on the final heat transfer results The discrepancy results from the fact that, in denying eq. (36), ref. [25] employed the fnctmn factor as represented by the common relation 0 079 Re-~, whale the present authors employed relations gwen dtrectly m ref [24]

281

eM, r at all points m the elemental flow area Nusselt values based upon this method are compared later with values based on four alternate methods of evaluating eM,0, Le.,

1 Eq. (34), 2. Eq. (37), 3. Taking eM,0 equal to (eM,r)max, as calculated (for a gwen value of 0) from eq. (32) 4. Taking eM,0 equal to three times eM,r at all points. Alternate method no. 3 Is a close approximation of eq (34), while no. 4 is a rough approxtmation of eq. (37).

3. Results Most of the results obtained m the present study were calculated for a Prandtl number of 0.004 834, the Prandfl number of sodmm at 800°F Thas was done because of the tmportance of sodium as a nuclearreactor coolant. The heat transfer coefficients are presented as Nusselt numbers, as functmns of ~Pe, the effect of Prandfl number Is fortunately not large m the hqmd-metal range A few results were obtamed for mercury at a Prandtl number of 0.022. The thicker the cladding, and the greater ItS thermal conductavity relatave to that of the coolant, the more the heat transfer behawor approaches that for uniform wall heat flux in the axaal direction and urnform outer-wall temperature in the clrcumferenUal &rectlon (boundary condmon A). And when the mua. tion is reversed, the heat transfer behavior approaches that for uniform outer wall heat flux m all directions (boundary condmon B) These cases are of practical interest, and are also used in the correlataons for the more general case.

3 1 Ltmmng thermal boundary condltton (,4) 3.1.1. Rod-average heat transfer coefficients These are presented as rod-average Nusselt numbers, [Nut] r. The dependence of ~--ut]r on bothP/D and ~Pe is dlustrated by the remits given m table 5 The greater the value of ~Pe, the less the effect of P/D ratio on the__rod-average Nusselt numbers. The shght increase m [Nut] r over the range 20 < ~Pe < 500 (for a given P/D ) is due primarily to the change m shape of the velocity profile, whale the large increase over the range 500 < ~Pe < 5000 Is due primarily to the increase m

O E Dw~ er H C Berry, tteat Transl6r to hqutd metals part 1

282

fabk 5 Rod-average Nusselt numbers for tully developed m-hne flow ot liquid metals through unbaffled rod bundks, tor ltmltmg thermal boundar~ condition (AI, Pr = 0 0048

P/D

.

~Pe=20

ffPe=50

1001 101

222 273

25[ 3 11

1 02 1 03

3 35 400

1 04 1 05 1 06

+Pe=100

~Pe=200

tpPe=1000 bPe-2000

fPe=5001J

278

719

410

528

394 4 81 5 67

503 6 07 70v

645 7 72 893

71~ 8 -0

1[18~ 13~2

3 84 457

345 4 25 504

10 37 1l 94

16 24 18.5

4 63 5 24 5 81

5 28 5 94 657

5 79 6 49 7 15

6 47 7 22 793

8 01 8 85 964

10 03 11 02 1192

13 35 14 60 1561

20 99 22 95 23 98

107

632

7 11

772

8q3

10 ~1

1269

1698

2550

1 10 1 12

7 55 8 19

844 9 10

9 10 9 79

996 10 68

11 88

14 47

18 76

28 88

12 66

15 32

19 79

30 45

1059 1154 1223 1281

11 5t 12 48 1319 1379

1 15

893

988

1 20

982

1081

125 130

1046 1100

1148 1205

.

.

.

.

.

.

.

.

,oo 901 7o

6o

.

.

.

.

.

.

.

,\

.

.

!

\

kPr:o00481

8o

\ 2ooo

\

!

22F

301 ____S~ 2o

! \i

IO

I0 -3

SPe-500

10.2

6 810_~

I0 0

[P-DIID

Fig 6 Variation of the rod-average heat transfer coefficient with P/D, for hmltmg thermal boundary condition (A) The relation Is actually plotted as h tD/kf versus (P D)/D for greater unlversahty and precision

.

.

1354

1629

2095

32 18

14 56

17 40

22 24

34 05

1529 1593

1818 1888

2312 2388

3528 3608

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

eddy conductivity resulting lrom the Increase in flow rate Whereas the Nusselt number always increases with an increase in P/D, other parameters remaining constant, the rod-average heat transfer coefficient does not The family of curves in fig 6 shows the manner in which the rod-average heat transfer coefficient depends on P/D and f P e , for a fixed rod diameter There, for convenience, the coefficient is given m the dimensionless form htD/kf, and P/D is represented by (P-D)/D Since htDe/kf xs a function only of P/D and ~Pe, so is htD/kf, as the definition o f D e 111 the Nomenclature will reveal It IS seen that, for each value ol f P e , the heat transfer coeffic]ent passes through a maximum as P/D is varied, the sharpness depending on the value of ~Pe The P/D at which the maximum occurs, also dependent on f P e , varies from 1 025 at ~Pe = 5000 to 1 06 at f P e = 20 AtP/D below that at which (htD/kf)ma x occurs, circumferential heat flow dominates in the elemental coolant flow area A C ' whale at P/D ratios above that at which (htD/kf)max occurs, radial heat flow dominates 3 1 2 Circumferential variation of local wall heat flux The results of the present study, showing the circumferential varlataon o f the local wall heat flux, are summarized in table 6 As one would expect, the variation Ofqw2/qw 2 decreases with both P/D ratio and ~Pe~

0 E Dwyer, H C Berry, Heat transfer to hqutd metals, part I

283

Table 6 Orcumferentlal vanaUon of the local wall heat flux for m-hne turbulent flow through unbaffled rod bundles, for fully developed flow and heat transfer, for Itmltmg thermal boundary condition (A) at Pr = 0 0048 qw2/q'w2

P/D

~Pe

0°

2°

5°

10°

15°

20 °

25 °

28 °

30 °

1 001

20 50 100 1000 5000

0 0000 0 0000 0.0001 0 0005 00009

0 0000 0 0000 0 0001 0 0015 00023

0.0025 0 0052 0 0084 0 0179 00178

0.0980 0 1289 0 1443 0 1604 01520

0 567 0 598 0 607 0 613 0608

1 484 1 474 1 474 1 455 1469

2 430 2 395 2 375 2 356 2349

2 768 2 729 2 710 2 712 2714

2 840 2 803 2 787 2 806 2808

101

20 50 100 1000 5000

0 0172 00291 00384 0 0540 00535

0 0205 00337 0.0439 0 0598 00590

0 0439 00646 00787 0 0952 00929

0 2060 02439 0 2578 0 2716 0.2673

0 686 0707 0.713 0 717 0718

1 487 1472 1 461 1 445 1458

2 268 2224 2206 2 181 2179

2 544 2492 2476 2 465 2452

2 601 2550 2 537 2 539 2527

1 02

20 50 100 1000 5000

00791 0 1079 0 1239 0.1387 0 1394

00855 0.1155 0.1314 0 1461 0 1465

0 0 0 0 0

1239 1588 1741 1874 1879

0 0 0 0 0

3190 3549 3671 3710 3769

1 776 1 790 0 793 0 770 0 799

2461 1 440 1 429 1 421 1 423

2099 2.049 2 030 2026 2 006

2 319 2 262 2 246 2257 2 226

2 364 2 308 2 294 2 316 2 285

1 04

20 50 100 1000 5000

0 0 0 0 0

0 0 0 0 0

0 0 9 0 0

2918 3290 3418 3570 3553

0 4959 0.5224 0 5311 0 5397 0 5410

0 882 0887 0 887 0 883 0 890

1 387 1.366 1 357 I 339 1.346

1 826 1 783 1 769 1 756 1 749

1.973 1 925 1 912 1 912 1 900

2 003 1 955 1.944 1 951 1 940

1 07

20 50 100 1000 5000

0.4262 04567 0 4675 0 4671 0.4788

0.4352 0.4652 0 4758 0 4747 04791

0 4829 05103 0 5199 0 5157 05222

0 660 0677 0 684 0.667 0683

0 950 0952 0 952 0 953 0948

1 289 1273 1 267 1 267 1 261

1.568 1538 1 528 1 538 1 527

1 658 1625 1 6156 1 637 1 625

1 676 1643 1 635 1 662 1.651

1 10

20 50 100 1000 5000

0 0 0 0 0

562 585 593 601 597

0 570 0 595 0 600 0 608 0 605

0.612 0 632 0 638 0 644 0.642

0 757 0 769 0 774 0.773 0 773

0.978 0 978 0 978 0 971 0 973

1 220 1.207 1 203 1.195 1 197

1 411 1 391 1 384 1 383 1 383

1 472 1 452 1 443 1 451 1 452

1 484 1 464 1 455 1 468 1 470

1 20

20 50 I00 I000 5000

0 801 0811 0 814 0 813 0 814

0 805 0815 0.818 0 817 0 818

0 827 0836 0 838 0 836 0 837

0 899 0904 0 905 0 901 0 901

0 998 0998 0 997 0 993 0 993

1 099 1.094 1 092 1 091 1 090

1 175 I 166 I 164 I 170 I 170

1 198 I 189 I 187 I 198 I 199

1 203 I 194 I 192 1 205 1 206

1.30

20 50 100 1000 5000

0898 0 903 0 904 0 898 0.897

0900 0 905 0 906 0 900 0 899

0912 0 916 0 916 0 911 0 910

0949 0 951 0 951 0 947 0 945

I000 1 000 0 999 0 997 0 996

1051 1 049 1 048 1 050 1 050

1089 1 085 1 085 1 092 1.094

I I00 1 096 1 096 1 106 1 110

I 102 I 098 1 098 1 110 1 115

2335 2729 2846 3035 3022

2424 2815 2940 3117 3103

~ n c e u a was the only actual v a n a b l e m Pe, increase m ~bPe m e a n s a s h g h t l y f l a t t e r v e l o c i t y p r o f d e and greater values o f k e m the t u r b u l e n t core, b o t h o f w h i c h t e n d

to decrease the clrcumferentaal variation o f the local h e a t transfer behavior. F r o m the m f o r m a t l o n given m tables 5 and 6, val-

0 E Dwver, tt C Bern, lteat transter to hqutd metal~, part I

284

fable 7 Rod-average Nusselt numbers tor tully developed m-hne flo,s of llqmd metals through unbattled rod bundles tot hmmng thermalboundary eond]tlon (B), Pr = 0 0048

P/D 1 001 1 01 1 02 1 03

qJPe=20

CPe=50

bPe=100

~,Pe=200

~Pe=500

0 183 0371 0656 1 056

0 191 0402 0 731 1 194

0 198 0426 0 785 1 283

0 206 0456 0 846 l 396

0 223 0514 0977 1 64l

0 31t,

0 268 0708 1 45~ 2 51~

t 012 ,2 181 3 87~ 6 0(,

1 58

t 79

1 93

2 11

249

3 01

3 89

220

250

271

296

~46

425

5 51

860

106 107

291 365

330 4 12

356 445

390 486

460 578

554 692

7 "5 9 I~

1151 1435

1 10 1 12

5 75 6 87

6 42 7 63

6 89 8 16

7 53 890

l 15

8 13 9 44

1 25 t30

1025 1084

8 97 10 33 11 16 1179

9 55

1 20

10 37 11 84 12 72 1337

8 89 1046 12 12 13 73 14 69 1551

10 85 12 74 14 "72 16 60 1766 1828

t4 00 /6 34 18 7 ~ 20 94 22 15 234~

2l )24 93 28 2" 31 2o 327349z

l I Ol

l l 88 1250

3 2 Ltmttmg thermal boundary condttton fB)

F Ii

b

50

FPr:O0048] 4~

,~Pe=50O0

t 04

70[

~

0 242 0590 1 142 l 966

~Pe=2000

105

ues o f the local h e a t t r a n s f e r c o e f f i m e n t , ht, 0 , c a n be readily c a l c u l a t e d

601

+Pe=1000

EO,,Z,

L

2~3C' i

zyl

i4 2

~l 5

~

4

6

g

2

~

4 ~-

I0 I2

I

le

I

IP b ~ / [ )

Fig 7 Variation of the rod-average heat transfer coefficient with P/D ratio and ~Pe, for llmmng thermal boundary condltaon (B) Th~ results are plotted as -htD/kf versus (P-D)D for greater usefulness and precision

t

3 2 1 Rod-average h e a t t r a n s f e r coefficient These, m the f o r m o f Nusselt n u m b e r s , arc summ a n z e d m table 7, w h e r e the e f t e c t s of P/D and ~ P e are clearly evxdent In c o m p a r m g the results m table v wxth those for l l m l t m g b o u n d a r y c o n d i t i o n ( A ) m table 5, it is seen t h a t t h o s e in table 7 are conslderabl~ l o w e r at the closer spacmgs and s h o w a m u c h greater d e p e n d e n c y o n P/D Fig 7, a n a l o g o u s to fig 6, is a p l o t o f htD/k f versus (P -D)/D and, m effect, shows the d e p e n d e n c y o f the rod-average h e a t t r a n s f e r coefficxent o n P/D a n d ~bPe, o t h e r t h m g s b e i n g e q u a l It Is seen t h a t the shapes of the curves m the two figures are quite d i f f e r e n t T h o s e m fig 7 have relatwely sharp maxama, since for h m l t m g t h e r m a l b o u n d a r y c o n d i t i o n (B) t h e r e Is considerably more c i r c u m f e r e n t i a l h e a t flow m the e l e m e n t a l c o o l a n t flow area, t o r the l o w e r P/D ratxos, t h a n t h e r e Is for h m l t l n g b o u n d a r y c o n d t n o n ( A ) As P/D mcreases, c i r c u m f e r e n t i a l flow o f h e a t m the e l e m e n t a l c o o l a n t flow area d ] m m t s h e s and h m l t m g t h e r m a l b o u n d a r y c o n d m o n s ( A ) a n d (B) t e n d t o w a r d coexistence It wall b e n o t t c e d t h a t the rnaxxma o f the curves all o c c u r at the sarneP/D, l e , 1 12 S l m d a r curves l o t slug-flow [1] a n d for t u r b u l e n t flow w i t h m o l e c u l a r c o n d u c n o n only [5] also s h o w n r n a x t m a t h a t o c c u r at this same P/D T h e p r a c t i c a l slgmficance o f th]s particular P/D Is o b v i o u s

0 E Dwyer, H C Berry, Heat transfer to hqutd metals, part I

285

Table 8 Ctreumferentml variation of the wall temperature for m-hne, turbulent flow through unbaffled rod bundles for hmltlng boundary condltaon (B), for fully developed flow and heat transfer, and for Pr = 0 0048

kf(tw,O-tb)/qw 2Oe P/D

~Pe

0°

5°

1 001

20 50 100 1000 5000

20 41 20 O0 19 71 18 05 16 15

12 12 11 10 7

1 01

20 50 100 1000 5000

1 02

10°

15°

20 °

25 °

26 °

28 °

47 09 80 06 87

6 71 6 38 6.11 4 53 2 75

3.21 2 96 2 76 1 69 0 791

1.023 0 866 0 754 0 305 0 077

-0 -0 -0 -0 -0

7 57 721 6 95 5 58 3 67

6 30 595 5 71 4 40 2 70

3 99 370 3 50 2 42 1 26

2 O0 179 1 64 0 948 0 400

0 588 0456 0 374 0 102 0 000

20 50 100 1000 5000

4 12 3 83 3 65 2 77 1 60

3 58 3 31 3 14 2 31 1 27

2 41 2 19 2.05 1 39 0 674

1 23 1 07 0 978 0.561 0 227

0 0 0 0 -0

198 276 312 294 176

- 0 340 -0.407 - 0 432 - 0 355 - 0 201

-0 -0 -0 -0 -0

528 580 590 431 232

-0 -0 -0 -0 -0

591 637 642 456 242

- 0 238 -0308 - 0 339 - 0 298 - 0 164

- 0 335 -0398 - 0 421 - 0 340 - 0 180

- 0 466 -0516 - 0 530 - 0 393 - 0 201

-0 -0 -0 -0 -0

509 556 566 411 207

301 344 356 281 139

- 0 390 - 0 425 -0.430 - 0 319 - 0 154

-0 -0 -0 -0 -0

420 452 455 332 158

1.03

20 50 100 1000 5000

2 53 2 32 2 21 1 58 0873

2 24 2 04 1 94 1 36 0720

1 55 1 40 1 31 0 850 0411

0 799 0 693 0 633 0 355 0146

0 190 0 127 0 095 0 006 -0015

- 0 195 - 0 226 - 0 235 - 0 185 -0093

- 0 242 - 0 269 - 0 275 - 0 206 -0102

-0.305 - 0 326 - 0 328 -0.235 -0112

-0 -0 -0 -0 -0

326 345 346 244 116

1.05

20 50 I00 I000 5000

1 17 1 07 I Ol 0698 0 364

1 05 0 958 0 904 0613 0 312

0 759 0 682 0 637 0410 0 197

0 0 0 0 0

408 355 324 187 081

0 101 0 071 0 056 0011 0 001

- 0 102 -0 115 -0 I19 -0094 -0 045

-0 -0 -0 -0 -0

127 138 141 106 050

-0 -0 -0 -0 -0

161 169 170 122 057

-0 -0 -0 -0 -0

173 180 179 128 059

1.07

20 50 I00 I000 5000

0 0 0 0 0

660 601 567 396 194

0 0 0 0 0

600 545 514 354 171

0 448 0.403 0 377 0 249 0 161

0 0 0 0 0

256 225 207 124 056

0 0 0 0 0

083 065 055 020 009

-0 035 -0.044 -0 047 -0 044 -0 018

-0 -0 -0 -0 -0

050 057 060 052 022

-0 -0 -0 -0 -0

070 076 077 063 026

-0 -0 -0 -0 -0

077 082 083 066 027

I I0

20 50 I00 I000 5000

0 0 0 0 0

356 325 307 205 104

0 0 0 0 0

329 300 283 188 094

0 260 0 235 0.220 0 143 0 070

0 0 0 0 0

169 151 140 086 042

0 0 0 0 0

084 072 065 036 018

0 024 0 016 0 012 0 002 0 003

0.006 0 000 -0 004 -0 008 -0 002

0 -0 -0 -0 -0

002 0O4 006 010 003

1.20

20 50 I00 I000 5000

0 140 0 128 0.120 0 0819 0 0434

0 134 0 123 0 116 0.0789 0 0418

0 0 0 0 0

122 112 105 0709 375

0 106 0097 0 091 0.0601 0 0319

0 089 0081 0 076 0 0495 0 0264

0 078 0070 0 065 0 0417 0 0224

0 076 0069 0.064 0 407 0 0219

0 074 0067 0 062 0 0393 0 0212

0 073 0 066 0 062 0.0388 0 0209

1.30

20 50 I00 I000 5000

0 1025 0 0945 0.0895 0.0623 0 0327

0 I011 0 0932 0 0882 0 0613 0.0322

0.0973 0 0896 0 0847 0.0585 0 0307

0 0921 0.0848 0 0800 0 0548 0 0287

0.0869 0 0799 0 0753 0 0509 0 0266

0 0832 0 0764 0.0718 0 0481 0 0250

0 0827 0 0759 0.0714 0 0477 0 0248

0.0820 0 0753 0 0707 0 0472 0 0245

0 0818 0 0751 0.0705 0 0470 0 0244

320 226 176 023 018

-0 -0 -0 -0 -0

234 283 299 251 128

-0 -0 -0 (--0 -0

0 016 0 009 0.006 -0 002 0 000

30 °

O E Dwver H C Berry lteat transler to hqutd metals part 1

286

i able 9 Maximum ctrcumferentlal surtace temperature varmtmna tor ln-hne flow through unbatfled rod bundles, t(,~ tulb developed tiow and heat transfer, for hmltmg thermal boundary c o n d m o n (B), and tor Pr = 0 0048 The results are expressed m terms of ttl, dlmenMonless ratio (tw, 0 - t w , 3 0 ) / ( ~ , 0 tb)

P/D

~Pe=20

~9Pe=100

~ Pe=500

~,Pe = 1000

~,Pe=20(lO

t. l ' e : S n l ) t ,

l 001 10l l 02 1 03 104

3 835 2998 2 978 3 017 3022

4 019 3 203 3 223 3 282 3291

4 284 '413 ~ 430 3 485 3486

.t 474 ',53t 544 ? 594 3 592

4 7,'I 3689 3 667 3 702 368~

;, 183 3925 q 840 3 834 3 ,75

1 05 l 06 107 1 10 1 12

2 968 2 859 2688 2 031 1 612

3 220 3 091 2892 2 160 1 700

3 432 ~ 296 3071 2 278 1 792

~ q08 3 ~95 3 198 2 334 1 838

3 581 3 43l 3 190 2 341 1 83 -7

3 636 ~ 452 3 J82 2 290 1 784

1 15 1 20 1 30

1 118 0620 0 225

1 177 0 648 0 237

i 239 0 692 U 264

] 2"71

1 z/O

o 715

0 722

U 280

u 291

1 227 0 706 0 292

3.2 2 Circumferential variation of local wall temperature Local values of the surface temperature of the rod, tw,o, are given in table 8 m terms ot the dimensionless ratio (t~, o - t b)kf/qw2D e, the reciprocal of the local Nusselt number At low P/D, the surface temperature In the victmty of the 30 ° regaon is less than the bulk temperature of the coolant (at the same axial position), which accounts for the negative values in the table Table 9 gives values of the dlmensmnless ratio = 12~

~=l

0 004~ I

I;~

~~-,

D LAMINAR LINE [ 2 ]

/ I

i01

o-

182

~

~ ~(~1

FLOW

.

~

MOLECULAR-

)

CONDUC%ON CURVE [ 5 ]

SLUG- ~LL'I~ LINE [,]

.~

" SINGLE, THEORETICAL VALUE~ PUBLISHED IN I966 [ 2 0 ] ~ .i0 2

~ "iO 3

"10 4

Pe

Ftg 8 Nusselt-vs-Peclet relatmnshlps for heat transfer to hquld metals flowing m-hne through free, unbaffled rod bundles Curve A - A is for fully developed turbulent flow with f = 1 Curve C - C is for fully developed turbulent flow, and molecular conduction only Curve B - B is for tully developed turbulent flow with ~- evaluated by means of eq (26a) The predicted curve for turbulent flow, m a real sltuatmn, ts therefore C B-B

(tw, 0 tv~,30)/(tw,o-tb) , which is the rattu ot the maximum circumferential variation In wall tempmature (over the arc 0 to 30 °) to the average temperature difference between the rod surface and the tiowlng coolant, required in evaluating the thermal stress in the cladding It is also useful in correlating circumferential temperature profiles for the more general case of uniform wall heat flux on the mner wall of the cladding [6[

3 3 Comparison oj present results with those of previous theorettcal and experimental stu&es Unfortunately, there are very few theoretical or experimental results with which the present results can be compared Curve A A m fig 8 represents the present rod-average Nusselt results forP/D = 1 20 and h m m n g the> mal boundary c o n d m o n (B) Curve C- C is the so-called "molecular-conducUon" curve [5], hne D xs the "lain mar-flow" hne [2], and hne E is the "slug-flow" line [11 To make rigorous comparisons with previous theoretical results, the latter must of course be based on the same boundary conditions and on equahty of the eddy dlffUSwlty of heat and m o m e n t u m transfer The only previous theorencal results are by Dwyer [20] He calculated five cases, two of which are the black dots in fig 8 Their agreement with curve A - A Is stir

O E Dwyer, H.C. Berry, Heat transfer to hquld metals, part I

287

Table 10 Comparisons of rod-average Nusselt numbers and ctteumferentml surface temperature varmlaons obtained m two different theoretical analyses They were calculated for fully developed turbulent flow and for lUnltmg thermal boundary condition (B) tw,0-tw,30 tw,o-t b

[NUtlq

P/D

~Pe

Present study

Dwyer [ 20 ]

Present study

Dwyer [20]

1 10 1 I0 1 20 1 20 1 375

206 1015 98 885 998

76 11 2 10.9 15 8 19 0

61 96 10.5 15 3 18 1

2 21 2 33 0 65 0 71 0.16

1.99 2 14 0 48 0 56 0 16

pnslngly good, consxdenng the fact that a wholly different, less accurate computaUonal method was employed Table 10 presents his rod-average Nusselt remits [20], along with those of m the present study, which run somewhat lugher, particularly at P/D = 1 10 The same ~s true for the c~rcumferentaal surface temperature vanatmns, wluch are also shown m table 10. The Prandtl numbers m the present and earher studies were 0.0048 and 0.0074, respectively, but the effect of flus difference on the heat transfer behavaor is neghglble. When curve A - A m fig. 8 is corrected for ~ 4= 1, we ge~ curve B - B , the lower end of wluch connects with the molecular-conductaon curve C - C . Thus, the predicted experimental curve is C - B - B In the lowPeclet range, It is obwously theoretically Lrnposslble for expenmental data on "clean", gas-free, particulate-free systems m turbulent flow to gtve Nusselt numbers below the molecular-conductaon curve. Obeqohn [23] pubhshed the results of an analytical study with objectwes stmdar to those of the present

study but with different methods of computatxon. Table 11 presents a comparison of rod-average Nusselt results obtained in the two studies, where the equahty of eM, 0 = e M r was assumed These Nusselt numbers allow for eH]e M :/: 1 0 b e r j o h n employed Delssler's [26] correlalaon = BPe [ 1 - exp (-1/BPe)]

(38)

for converting local e M to ell, wluch then gave the local eddy conductivity In the present paper, voth e H = eM, the calculated Nusselt number was corrected by employing ~ from eq. (26) or (26a). The present results m table 11 are generally slgnlfxcantly lugher than those by Obeqohn, partacularly at the lugher Peclet numbers. It Is beheved that lus results are low, pnnclpally because he employed experimental pipe-flow Nusselt numbers [27] for evaluating B in eq. (38) which were themselves low. The Oberjohn results m table 11 were taken off small plots and therefore may not be correct to the last daglt shown. His calculations were for Pr = 0.007 35, rather than 0.004 834,

Table 11 Predicted values of [Nut]q for fully developed, turbulent, m-hne flow of hqmd metals through spacer-free, unbaffled rod bundles

P/D=I.02

P/D=1 05

P/D=I.IO

p/D=I.20

Pe

Present study

Ober~hn [23]

Present study

Oberlohn [23]

Present study

Oberjohn [23]

Present study

Oberjohn [23]

100 200 500 1000 2000 5000

0.73 0.75 0.85 1.05 1.40 2.17

0.70 0.72 0 80 0 91 1.25 2.1

2 48 2.56 3.00 3 92 5 31 8.60

2.3 25 2.7 31 43 75

6 37 6.54 7 55 9.70 13 0 21 0

6.0 6.2 7.0 8.1 11.0 -

10 32 10 52 11.1 14.6 20 0 31.1

99 10 0 11 0 13 9 17.0 27.5

288

O E Dwyer, H C Berry, Heat transfer to hqutd metals, part I ~0

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Fig 10 Comparison of present theoretical predictions and expertmental results of Graber and Rleger [31] for fully developed m-line flow of NaK on the shell rode of a shell-and-tube heat exchanger, under thermal boundazy condmon closely approximating that of uniform wall heat flux tn all directions

o, Pr =0 0 0 7

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9. C o m p a r i s o n

per[mental

between

theoreUcal

results for fully developed

predictions

and ex-

m-line turbulent

flow of

hqmd metals through rod bundles The differences irl Prandtl number, shown m each part of the figure, do not stgmficantly affect the Nusselt-Peclet relationship

but the effect of this difference on the Nusselt number, at the same Peclet number, ~s negligible. Strictly speaking, it Is not posmble to compare the present theoretical results and pubhshed experimental data, because no data have been obtained rather under hrmtmg thermal boundary condmon (A) or (B) All experimental condmons were between these two hmtts, the extents to which the conditions more nearly approached either (A) or (B) dependang pnmarfly on the heater-rod cladding thickness and conductwity However, as P/D increases, both of these lnnltmg boundary condmons and all actual experimental conditions tend to approach each other Reference to tables 5 and 7 shows that, at the lughestP/D m the present study (1.e. 1.30), the rod-average Nusselt numbers for boundary (A) and (B) are no more than 3% apart. On thin barns, we are justified m comparing these predicted Nusselt numbers w~th those experimentally determined by Bonshansky et al.-[28] for P/D 1.30, as in figs. 9a and 9b, where the dog-leg shaped theoretical curves correspond to C - B - B in fig. 8. The difference m

Prandtl number is not slgmficant. The agreement m fig 9a between theory and expertment IS not bad conmdenng the difficulty of hquld-metal heat transfer expernnents In fig. 9b, expertmental results for Pr = 0 007 (presumably sodium) fall very low, even below the "laminar-flow" line Bonshansky et al did not suggest posmble reasons. However, channel-flow, hqmdmetal heat transfer results often fall below theoretical pred~cUons, particularly at the lower Prandtl numbers [19,29]. The dashed curves m fig 9a and 9b are actually the same curve, predmted by Bonshansky et al. [30] in an earlier paper, and satd to represent both Pr = 0.007 and 0.03. It allows for e H d: eM, but the method of calculation is not g~ven m detail. Presumably, eddy conducUon m the circumferential dlrecUon was taken equal to that m the radial d~rect~on, at all points in the elemental coolant area It so happens that agreement between the two theoretical curves in fig. 9b ~s excellent Fig. 10 shows experimental results recently obtained by Graber and Rleger [31 ] when flowing NaK countercurrently m a shell-and-tube heat exchanger. Data were taken on the central tube of a 13-tube bundle under conditions closely approaclung umform wall heat flux, and for fully developed flow and heat transfer The data points, clearly demonstratmg the typical dog-leg curve, fall about 8% below the presently predicted curve, wluch is actually not bad agreement.

0 E Dwyer, H C Berry, Heat transfer to hqutd metals, partl

289

Table 12 Comparison of rod-average Nusselt numbers, [N-Ut]q'calculated with eM 0 values that were evaluated by five different methods, as described m section 2-5 The calculations ~ere made with Pr = 0 0048 Method of evaluating eM0 Alternate no 1

Alternate no 2

Alternate no 3

Alternate no 4

P/D

~Pe

1 01

100 500 1000 2000 5000

0 43 0 51 0 59 0 71 1 01

0 47 0 66 0 82 1 04 1 55

0 50 0 75 0 96 1 25 1 88

0 46 0 64 0 79 1 00 1 51

0 48 0 64 0 78 1 00 1 46

1 05

100 500 1000 2000 5000

2 71 3 46 4 25 5 51 8 60

2 89 4 11 5 29 7 23 11 98

3 04 4 63 6 22 8 55 13 98

2 86 4 06 5 22 7 16 11 72

3 04 4 40 5 83 8 08 13 15

1 10

100 500 1000 2000 5000

6 89 8.89 10 85 14 00 21 57

7 14 9 72 12 23 16 23 25 65

7 35 10 35 13.16 17 45 27 35

7 11 9 63 12 14 16 06 25 36

7 36 10 02 12 58 16 51 25 33

1 20

100 500 1000 2000 5000

11 01 13.73 16 60 20 94 31 20

11 14 14 11 17 21 21 84 32 70

11 23 14 36 17 55 22 26 33 20

11 12 14 08 17 15 21 77 32.61

11 22 14 10 17 13 21 66 32 22

1 30

100 500 1000 2000 5000

12 50 15 51 18 28 23 45 34 95

12 57 15 74 18 63 24 01 35 91

12 63 15 89 18 83 24 24 36 18

12 56 15 71 18 61 23 96 35 84

12 65 15 71 18 51 23 72 35 23

ell, 0 = EH,r

To theoretically check experimental results for (penphencally) local Nusselt numbers, even forP/D as high as 1.30, one must take cladding thickness and thermal conductavlty into account This wall be done in part II

[6]. Moreover, to theoretlcaUy check experimental results for rod-average Nusselt numbers forP/D from 1 05 to ~ 1.20, one must also take cladding thickness and thermal conductwlty into account [3, 4]. For P/D below "- 1 04, one must, m addition, take the fuel core into account [7]

3 4 Effect o f method ofevaluatmg eM,o As explmned m sectaon 2 5, the standard method o f evaluating eM,0 m the present study was to assume ~t equal to eM,r at all points m the elemental coolant

flow area This assumpUon was also used by Bonshansky et al [30], Bender and Magee [32], and Oberjohn [23] Two arguments m support o f this approach are (1) the results, if they are m error, err on the conservatwe stde, and (2) the predicted Nusselt numbers are already higher than nearly all expertmental values thus far obtmned. The four alternate methods described of evaluating CM,0 were all tested m the present study In table 12, It Is seen that the four alternate methods give Nusselt numbers fmrly close together and generally higher than those by the standard m e t h o d m the present study, except at P/D > 1.2. As one would expect, the greatest difference shows up at low P/D and lugh Peclet numbers. However, these conditions are more academic than practical, because of the associated high pressure

0 E Dwt er tt C Berry, tleat transJer to lzqutd metala, part /

290

[able 13 Circumferential variation el surtace temperature ot cladding as lntlueneed by method ot estimating eM, 0 I tie ~akulatlons ~ t re earned out out with Pr = 0 0048 The results are expressed In terms el the dzmensionless ratio

Ix\ ~

gb

Jq Method el evaluating eM,0

P/D 1 01

1 05

~Pe

Alternate

Allernate

eH,O - qt,*

|ltl

llt~

I

100

3 20

500 1000 2000 5000

3 4l 3 54 3 69 3 92

3 2l 3 44 3 59 3 79 a09

100

500 1000 20OO 5000

3 22 3 43 ~ 5l 3 58 3 63

3 1t, 32" 29 3 29 2~

2

klte~ nat, *10

2

M~ernate 1~*k' ~"

3 22 3 48 3 65 3 86 4 II

3 22 ~ 45 3 60 _~81 409

'~21 49 3 64 _~ ~0 'gq~

s ll) 3 04 300 2 92

3 3 3 3 3

129 3l 31) 28

;06 ~cJ5 2 9< ~ 83 2 '~3

206 2 01

I 96 1 ~6

208 2 04

t ~)3 ; 0

l 96 l 85 1 67

i 63 1 50 I 3q

t99 1 90 u 72

1 )t, 1 59 I 22

3 11

100

2 16

500 1000 2000 5000

228 2 33 2 34 2 29

1 20

100 500 1000 2000 50O0

0 0 0 0 0

65 69 72 72 70

0 60 0 56 0 54 0 50 044

t) 55 0 45 0 40 0 36 033

0 6l o 58 0 56 0 52 046

~ ~4 ~ 45 ,.J 40 0 35 t~30

1 30

100 500 1000 2000 5000

0 0 0 0 0

24 26 28 29 29

022 0 21 020 0 19 0 16

020 0 16 0 14 0 12 0 11

022 0 22 021 0 20 0 18

t, 19 t.J lfi ~115 0 13 i) 12

l 10

d r o p s resulting f r o m the high velocities F o r e x a m p l e ,

Whereas the m e t h o d o f evaluating e M,0 ha~ tile

for D = 0 250 in, P / D = 1 01, and a P e c l e t n u m b e r as

greatest influence on [Nu t]q at the l o w e r P / D ratios,

low as 100, the average h n e a r velocity o f s o d i u m at

It has the greatest i n f l u e n c e on the relative circum-

8 0 0 ° F is a p p r o x i m a t e l y 27 ft/sec Table 12 was calculated for the h m l t l n g t h e r m a l b o u n d a r y c o n d i t i o n o f u n i f o r m h e a t flux in all direc-

ferential t e m p e r a t u r e variation

tions f r o m the o u t e r surface o f the cladding, w h e r e

[(tw, 0 tw,30)/(~w,o - t b ) l q at the f u g h e r P / D ratios This is illustrated In table 13 The d i f f e r e n c e s b e t w e e n the n u m e r i c a l values increase w i t h in ~ P e , as e x p e c t e d

the e f f e c t o f the m e t h o d o f evaluating eM, 0 is the greatest F o r limiting t h e r m a l b o u n d a r y c o n d i t i o n (A),

Since the results in table 13 are for limiting thermal b o u n d a r y c o n d i t i o n (B), they r e p r e s e n t u p p e r h m l t s

the d i f f e r e n c e s are m u c h smaller, as one w o u l d e x p e c t , because o f the c o n s i d e r a b l e r e d u c t i o n in c i r c u m f e r e n t ial h e a t t r a n s f e r In e a c h e l e m e n t a l cross-sectional cool-

for real s i t u a t i o n s The results In tables 12 and 13 are n o t only based

a n t - f l o w area (see table 18 in p a r t II [6])

on the e q u a l i t y o f ell, r a n d eM, r b u t also o n the e q u a l i t y o f ell, 0 and eM, 0 The latter a s s u m p t i o n t e n d s

0 E Dwyer, H C Berry, Heat transfer to hqmd metals, part I

291

kw

= [({P/D } cos 0) 2 - 1] D = penpheraUy local equivalent diameter for m-hne flow through rod bundles [ft] = 2geDe(AP/AL)/o2p = friction factor for fully developed ln-hne, turbulent flow through unbaffled rod bundles [dimensionless] = same as f, except for ctrcular tube = factor for converting from lbf to lbm, 32 1711b m ft)/(lbf sec2)] = qw2/(-i-w,o - tb) = rod-average turbulentflow heat transfer coefficient for ltmmng thermal boundary condition of uniform heat flux in all directions on the outer surface of the cladding [Btu/(h ft 2 °F)] = qw2/(Tw,o - t b ) = rod-average turbulentflow heat transfer coefficient for limiting thermal boundary condition of uniform heat flux in the axial direction and umform temperature in the circumferential direction, on the outer surface of the cladding [Btu/(h ft 2 .°F)]. = thermal c o n d u c t m t y of ceramic fuel core [Btu/(h ft °F)] = eddy thermal conductlvaty for heat flow in any given direction [Btu/(h ft °F)] = eddy thermal conductavlty for heat flow in the radlaldlrectlon [Btu/(h ft °F)] = same as ke, r, except in the circumferential direction. = molecular thermal conductivity of coolant [Btu/(h. ft" °F)I. = thermal conducttvaty of cladding

K

= k e / k f [dlmensxonless].

also to magnify the differences between the results obtamed when assuming eM, 0 = eM, r at all points, and those obtained wath the four alternate methods of evaluatang eM, 0 There are strong experimental indications that the ratao eH,o/eM, o (= ~00) is much less than eH,r/eM, r (= ~r)" The greater mixing lengths and the lower velocmes of the circumferential eddies m the zero-degree region (Le, between adjacent rods) apparently greatly reduce the capacity of liquid-metal coolants to transport heat, compared to their capacity to conduct it In the 30 ° regton, the question of the relataon between ell, 0 and eM, 0 is irrelevant, because there IS an insignificant amount of circumferential heat flow m that regaon anyway. The clrcumferenUal variation of the wall shear stress over the 30 ° arc increases as the P/D ratio decreases, until at some low P/D ratio, it gwes rise to a significant amount of secondary flow. Little is known of the Importance of this secondary flow in llqmd metal heat transfer However, indications are that Its contribution is small, because of the relatively low secondary-flow velocities [33], and high thermal conductwmes Further progress m understanding rod-bundle hquldmetal heat transfer depends heavily on the production of dependable experimental data under carefully controlled conditions, particularly for low P/D

De, o

Acknowledgment

ke,/,

The authors gratefully acknowledge the valuable assistance of Robert J Claghom, a student appointee at Brookhaven during the summer of 1971, with the computer programming and acqumtlon of results

ke, o

f

fct gc [ht] q

[ht]r

k ke

kf

[Btu/(h. ft °F)] Nomenclature h c

An c

c. D De

= elemental coolant flow area, defined by area fdce in fig. 2 [ft2]. = expansion coefficients in Founer cosine series ofeq. (12) [°F]. = quantity defined in eq (25) for use m eq. (23) [dlmensmnless] = specific heat of coolant [Btu/(lb m • °F)] = 2r 2 = daameter of rod [ft]. = [2X/~(P/D)2/Tr-1] D = equivalent diameter or rod bundle for m-hne flow [It]

= ke, r/k f [dimensionless]. = ke,o/k f [dimensionless]. Ko M = number of radial increments between r = r 2 a n d r = ~-P N = number of angular increments between 0 = 0 ° and 30 °. [~t]q = [ht]qDe/kf = rod-average, turbulent-flow Nusselt number for ltmltmg thermal boundary condition of uniform heat flux on outer surface of cladding [dimensionless]. [Nut]~- = [ht]TDe/k f = rod-average, turbulent-flow Nusselt number for h m m n g thermal bound-

292

ap/aL P Pe

O/: Dwver, 1t C Berry, Heat transCer to hquld metals, part I ary c o n d m o n s of u n i f o r m heat flux m the axial direction and u n i f o r m temperature m the circumferential d l r e c n o n on the outm surtace of cladding [dimensionless] = pressure drop for fully developed turbulent flow through rod bundles [ I b t/ft 31 = D t c h or distance between r o d c e n t m ~ lilt = D e V a P C p / k f = Peclet n u m b e r [ d n n e n s n m -

less] Dr

= Cpla/k t = Prandtl n u l n b e r [dnnensxonlebs]

qx~ 1

= u n i f o r m heat flux on nmer surlace ol daddlng [Btu/(h ft 21] = peripherally local heat flux on outer surtace of cladding [Btu/(h ft2)] = (6/re),! n / 6 q w 2 d O = average value ol qw2 [Btu/(h t't2)] = radial distance [ft] = inner radius of cladding [ft] = outer radms of cladding If t] = r/r 2 [dimensionless] = DeVaP/la, rod-bundle R e y n o l d s n u m b e l [&mensmnless] = D e oUo 0/11 = peripherally local Reynolds n u m b e r [dnnensmnless] = mner radius of a n n u l u s [ft] = radial &stance defined m fig 1 [ft] = outer radius of a n n u l u s [ft] = temperature at radial distance t and angular &stance O [°F] = bulk temperature ol coolant [°F] = peripherally local value of temperature at outer surfa~.e of cladding [°F] = (6/rr) o f ~/6 t w 0 dO = circumferential average value o f t ~ u [ °F] = value often,0 at 0 ° [°F] = value of t~ 0 at 0 = 30 ° [°F] I ' ( r w g c / p F = l n c t t o n velomty at angle 0 [ft/hr] = u / u * = dimensionless velocity parameter at radml distance r and angle 0 = average value o f u at angle 0 [ft/hr] = linear velocity at radial distance r and angle 0 [ft/hr] = # Re/D e p = kf Pe/DeC p P = average linear velocity m elemental coolant area A c [ft/hr] = q u a n U t y defined b y eq (24) [dimensionless] = distance measured n o r m a l to rod surface at angle 0 [ft] = v u * / u = d~mens~onless d~stance measured

qv, 2 qw2 r

rI r2 R Re Re 0

rm

r l

tb tw,O lw,o tW,O

tw,30 lt* it +

vo t)

0d

X l' y+

1;~0 1'0 v"a~

n Ol n]al to rod su rl ace at angle O = nornla] dlstant.c lrom ~od surlace to htle Ol m a x i m u m velo~.lty at ,ingle 0 lit] = Fma \ = m a x i m u m value ot ~ lit] -- ~;mm = m l n m i u m value ,,t ( I lq - (6/rrl jr r/6 i>d0 = cn~,l,nlerentlal avelage

_-

value ol v lit] -- axial distance lit]

; t *x

( ; r e e k letters

eli

- a~,/Cpp = eddy dllhtslvlty ol heat u a n s t e t

ell p

: h e r / C ~ p = eddy dlffuslvlt}, ot heat n a n >

61t,0

=/~e o / C p p - eddy dlffllSaVltV of heat tran~-

m any given d l r e c n o n [ft2/h] !er In the radial dnectJorl [lt2/h]

cM CM,L

eM, r

eM 0

ler in the clrcutnfercntlal dlrectlou [ft2/h] = lae ;p - eddy dlftuavlty of nlomen tuin tran> ter in any given direction [ft2/hl = eddv dflfUSlWty ol m o m e n t u m transler m latelal dlrecuon, with ~efereuce to flow m open, h o n T o n t a l recl angulm channels [fl2/h] = P e p / P = eddy dltlUswltv ot m o l n e n t u m transter in the radial dlrecUon [lt2/h] = #e,o/P = eddy dtffuslvUv ot m o m e n t u m I ranster m the circumferential d l r e c u o n

[I t2/h] 0

~e ~e,r #e,0 t, P r, % rw

~r ~0

= Ln~.unrferentml angle ~fig 2) mesurecl flora qralght line connecting the centei point,, o( two adjacent rods [degrees or 1admnsl = absolute molecular wscoslty [ l b m / l f l hi I = eddy viscosity for shear stress m any given darecuon [ l b m / ( f t h)l = edd) viscosity lor ahear stress m ~adlal &tectlon [ l b m / ' ( t t h)l = eddy vtscoslty for shear stress m ctrcumterentlal direction l i b m/(ft h)] =/.z/p = kinematic molecular viscosity It t2/h = density of coolant [1 b m / f t 31 = ladtal shear stress at radius r and angle 0 [lbt/It2] = shear stress at r 2 arid angle 0 [ l b f / t t 2] = (6/rr)fn/6 rwdO = circumferential average o value of [ l b t / f t 2 ] = eH/e M {dlmensmnless] = overall, average, effective value ol = ett,r/eM, r [dunensmnless] = ell, o/eM, 0 [dmmnstonless]

0 E Dwyer, H C Berry, Heat transfer to hquzd metals, part I

References [1] O E Dwyer and H C. Berry, Slug-flow Nusselt Numbers for In-Line Flow Unbaffled Rod Bundles, Nucl Sct Eng 39 (1970) 143 [2] O E Dwyer and H C Berry, Lammar-Flow Heat Trans-fer for In-Line Flow Trough Unbaffled Rod Bundles, Nucl Scl Eng 42 (1970)81 [ 3] O E Dwyer and H C Berry, Effects of Cladding Thickness and Thermal Conducttvlty on Heat Transfer to Llqmd Metals Flowing In-Lme Through Bundles of Closely Spaced Reactor Fuel Rods, Nucl Sct Eng 40, (1970) 317 [4] O E Dwyer and H C Berry, Effects of Cladding Thickness and Thermal Conductwlty on Heat Transfer for Lammar In-Line Flow Through Rod Bundles, Nucl Scl Eng 42, (1970) 69 [5] O E Dwyer and H C Berry, Turbulent-Flow Heat transfer for In-Line Flow Through Unbaffled Rod Bundles Molecular Conduction Only, Nucl Sct Eng [6] O E Dwyer, H C Berry and P J Hlavac, Heat transfer to Llqmd Metals Flowing Turbulently and Longltudmally Through Closely Spaced Rod Bundles, Part II, Nucl Eng DesJgn 23 (1972) 295 [7] R A Axford, Two-Dn'nensaonal, Multlregton Analysts of Temperature Fteids m Reactor Tube Bundles, Nucl Eng Design 6, (1967) 25 [8] T Von Karma.n, Turbulence and Skm Friction, J Aeronaut Sct 1, (1934) 1 [9] W Etfler and R Ntjsmg, Expertmental Investtgatlon of Velocity Dtstnbutton ans Flow Resistance m a Triangular Array of Parallel Rods, Nucl. Eng Design 5, (1967) 22 10 Y D Levchenko, V I Subbotm and P A Ushakov, The Dtstrlbutlon of Coolant Velocities and Wall Stresses in closely Packed Rods (transl) Atomnaya Energlya 22, (196~) 218 [11] O E Dwyer and P S. Tu, Analytical Study of Heat Transfer Rates for Parallel Flow of Ltqutd Metals Through Tube Bundles Part I, Chem Eng Progr Symposmm Series, 56, No 30, A I Ch E (1960) 183 [12] W M Kays and E Y Leung, Heat Transfer m Annular Passages Hydrodynamtcally Developed Turbulent Flow wtth Axbitrartly Prescribed Heat Flux, Int J Heat Mass Transfer 6 (1963)537 [13] J A Brighton and J B Jones, Fully Developed Turbulent Flow m Annuh, Trans ASME, J. Basle Eng. 86 (1964) 835 [14] P J Hlavac, B G Ntmmo and O E. Dwyer, Flutd-Dynamlc Study of Fully Developed Turbulent Flow of Mercury m Annuh, Trans ASME., J Heat Transfer, accepted for pubhcatton [15] M K lbraglmov, I A Isupov, L L Kobzar and V I Subbotm, Calculation of Hydrauhc Resistivity Coefficients for Turbulent Flow in Channels of Non-circular Cross Sectton (transl), Atomnaya Energwa 23 (1967) 300

293

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