SAND80-1057 Unlimited Release

STEADY NATURAL-CONVECTION HEAT-TRANSFER EXPERIMENTS IN A HORIZONTAL ANNULUS FOR THE UNITED STATES SPENT-FUEL SHIPPING-CASK-TECHNOLOGY PROGRAM

Ronald D. Boyd

( j f l ) Sandia National Laboratories

«MM«msNQN£intauM«

SAMD80-10 5 7 Distribution Unlimited Pelease Category IJC-79C Printed April 1981

DY MATURAL--CONVECTI0N IIF.AT-TRANSFER EXPERIMENTS ••: A HORIZONTAL ANNULUS EOR Tt!E UNITED STATES fU'ENT- FI'P.L S!ITPPT\TG-CASR-TECin'OLOGY PROGRAM

R. n. Boyd i '1 h Thermal Sciences Pcp.'.r Lrr.ent. ^andi a Ma t i ona 1 bahora (.or if s A1 buquerquc. Mow Mexico °7]Rc;

•,o natural ennvective heat transfer arror,:i, an ar.nulus with nr boundaries was s'judiei usinq a Ma..h-7.chr. !or inter-or. The annubjs was Formed, by .an inner hexaoor.a 1 cylinder <•;;: l. •••] ~or.ee nt r ic ci rcc lnr cy * ir.-icr. Th i s ennf i cur a t1 on in !.*r,' d i irons inns, a liquid nctal fast- ! roc ier reactor

\io] s.i*-asse;:ib ;y inside a shipping cunLniner . Huririq the V.i- -ciii'i!.:s was filled with a sinn'e aas, nit'ncr neon, air, kryniui., or xi-*non, at a pressure of -IIM.-.I 0.r MPn.

,"!i> i I'liipi-Tri! .irt* measurement s, both local and meat; N'ussel t [!-.'-.] 1 a'. Hi'.' surface of the inner cylinder were evaluated

:-,.-( ic5r, rjf the Rayleiqh number (Ra^J '. is the :-.axi.num r::) . The da ta correlator: for the mean Mussel t and h n..ir.ber.s is oiven by Ku.= 0.701 Ra,, " ,

[]+C* ( r 0 / r L ) !

l -4

i 1 r | i 'i the mini mum rad ius of the inner cy I i rider , Th is cor re-it inn is appl i eabl e to the present annu lus (C* = 1.263) as we 11 to isothermal concentric circular cylinders (C* - 1.0),

W T » « i n * l ¥ TUtS inCKMEII IS UNUMOEO

TABLE OF CONTENTS

INTRODUCTION

EXPERIMENTAL STUDY

2.! APPARATUS

2.2 PROCEDURE

!-:FS'"I.TS

•7">:C!.;'S rov

LIST OF FIGURES

The Macb-7.chnder Interferometer (45°) Used to Experimentally Record the Two-Dimersioiial Temperature Distribution.

The Test Cell With the Annulus Formed by an Inner :iexnqonal Cy1inder Inside an Outer Concentric "irculnr Cy 1 inder .

An I nt 'Tf eroqrarn Showing the Two-Dimenslonal I sntherrr.s in the Annulus Formed by nn Inner '!ex and an Outer Concentric Circular Cylinder. Cas-Arqon: Pa R - 3.61 x 10 ; l.enqth o f Hex Si des = l . M ? <-•".

An ! ntcrfcroqrain Showing the Two-Dimensional iso!.heri"R in the Annulus Formed by an Inner "ex and a'i r»u''-T Onncr-ntric Circular Cylinder. "•TS-y«n^n; P"ip - \.&4 x ! 0 q ; benqth of Hex Side- - 7. i$? r-n.

•\r. ! r *-̂ r f'T'inran; Showing the Two-!) imens i ona 1 i toM-[ n-n<; in thf1 ^.nnnlu? Formed by an Inner M»x and m Outer Concentric Circular Cylinder. "••is-'ip;,;!; Pn p = «. 1 r, x 10'-; T.enqth of Ilex

"in,..'ris i or. 1 ess Kad i a 1 Temperature His Lr i but ions .!' " dc.'l ̂ ̂ /'.inuiKii" Iinr.it ions . Cas-Arqon; ^'^R

3 . {• 1 X i n '.

*.r.-iu 1-i r "̂ i l-' r i bu I i <>n of t ho Nnsselt Number ."it 1 1 *. ~,i.-rac .»f !)>•• !'px (R-P T.TbU- 1). r'vj..-r I ;-"nt al R.-suHs •">(" the Mrun Nusselt Number i V'i ) i= '. I'HTT'Mir.n of the Ray! e tq.h Number (Ra R) f -r *• h r> 'innnliis '•'' IPPP'1 by an I nn^r Hex Inside a , •orw..- t r i '• C i rnii-ir Cy 1 inder .

blS'" OF TABU'S

I.-"-..I '.'ussidl Number Distribution a I. the Surface • f !).•• ';••:<. lqonal Cylinder

S'i:;i: ..I i / >f V. xpe r linen ta 1 Results

NOMENCLATURE

= g e o m e t r y f a c t o r f o r t h e a n n u l u s ( s e o c q u a t i o n 1 an

r e f e r e n c e 19)

= g r a v i t a t i o n a l a c c e l e r a t i o n

= l o c a l > J u s s e l t nun'oe r, •T/ -r ' / (T • - 7.^}

= mean va lue of Ku , o b t a i n e d over '.he ,-ippl K-ai le in

I or

= p r e s s u r e of the qas in the annu" -.<=

- r a t i o o f t h e a n n u l u s c a s p r e ^ s u r ^ r n f h e n r f s s • : r .-•

s t-in-"Jaril c o m ! i. t i o n s

- ; a . " ! i a ! c o o r d i n a t e .measure ; ! f r o r I h o o - n i . 1 : ; •:' ';>.•

P a y W i a h :i . i :nber, [ o f"'2 T* S J ] / - (" '^ • • h l e r s i '•

ex i i l i<:i t l y f o r u ^ s ^ s o n l y )

" h a r-'ict o r i s t i c lun- . : ln f o r t h e P ."i y 1 •* i .in nin"i>*r.

' i.'^per,-! L u r e

voforencp t e m p o r a l a r c , ( T . - T , ) 1+C* *

t .hormnl l i f f u s i v i t y at. s l a n < i n r - l m m ! i l i "•;-..•

t ht?r;rta 1 r o o f f i i ' i o r .L of v o l ;i;'.:< •t v • .• .'yj-.,!-.-; ;.

:-ax i iiiUJ" a n n u l . i s o a p wi.!f h , (i : • ) : ;

f.-'i:!'!*1 s h i f t o r o r . l o r n u n i l ^ r

mi Mil M ' t_-oor.lir.at-o i n n i s n v •• I f r •"! t h n !_">p .

. i i i r i n '. u:-i

'. i ;.iM-:.it i <: v i ;• • -s i t y a t s L a i . l a i- "i . - . M M i ! i o i , :

( r . . . ' . - , ) - ' •

Subscripts i - evaluated at the inner cylinder 0 = evaluated at the outer cylinder n = denotes normal vector

I. INTRODUCTION

Natural convection heat transfer is an important mechanise

of energy transfer in rrany practical systems. Numerous appl ica­

tions are found in energy storage, reactor safety, passive solar

systems, and reactor waste transport and storage.

This experimental study deals with the measurement of the

heat transfer across a horizontal annulus which is formed by an

inner hexagonal cylinder and an outer concentrir r-ireular cylin­

der, ^ e geometry simulates, in two dimensions, a lion id metal

fast breeder reactor radioactive fuel subassembly inside a ahi n-

ninn container- This geometry is also similar t'-> .1 radioactive

fuel nin inside a horizontal reactor subassembly.

The objective of the experiments is to measure the local

and ;nean heat transfer at the surface of the inner hexagonal

cy1inder-

Very little work has been done on natural convection heat

transfer in a horizontal annulus former? by an inner hexagonal

cyl inder anil an outer circular cyl inder. K] irna (!_} measured the

overa 11 heat transfer coefficients due to natural convection

using argon, helium, and 1iquid sodium. Pope, ct a 1 - (2) used

the correlation technique of Itoh, et a 1. ( 3 ), to correlate the

data produced by Klimc. Schimmel (4) made a demonstrat ion

inter ferogram which displayed conduction-dominated heat transfer,

using a holographic interferometer. Gartling (_5 ) made two-

dimensional numerical finite element predictions for the case

of a hexagonal cylinder resting on the bottom of an outer

circular cylinder. The natural convective flow regime studied

1

by Gartling was near the transition between thermal conduction

anil laminar transport (or the psuedo-conduction flow regime).

The natural convective flow in a horizontal annulus formed

by an inner hexagonal cylinder and an cute., concentric circular

cylinder is similar to the natural convection flow in the annulus

formed by concentric circular cylinders. However, the presence

of corners and flat surfaces in the former annulus maKes the two

cases different enouqh to warrant additional study.

Numerous authors, e.q. (6-15), have studied the natural

convective flow between horizontal concentric cylinders. Kuehn

((i) , n̂.' :ehn and Holds tein ( Z - ^ ' have reviewed the literature

thorough -f . Us i nq a Mach-Zehnder inter ferometer, th<?y examined

experimentally and numerically the local and mean heat transfer

coefficients for both concentric and eccentric cylinders. Their

experimental results covered a range of the Rayleiqh number

(based on the gap width, ) of 2.2 x 10 2 <_ Ra < 7.7 x 1 0 ' (in

tneir definition for Pa , T* = T .-'!') . Origull an-1 H;r.:f (Q) ,

also using a Mach-?ehnder interferometer, presented experimental

results of the Nusselt number (Mn.) as a function of the Hrashof

number (Gr . , for 10 < fir, < 1 0 ) and the rat i o of l-he gap width

to the diameter of th<? inner cylinder- They also presented,

using smoke dispersion, tvo-dimensionn\ photographs of the

streamlines of the flow fields. Lis (J-0̂ ) pressurized the annulus

fluid to achieve turbulent heat transfer, with the Reyleiqh number

being as hiqh as 1 0 . Many other studies and correlational

analyses, e.g. (11-15), have been conducted on the natural convec­

tive transport between concentric cylinders.

2

II. EXPERIMENTAL STUDY

A 19.7 cm aperture Mach-Zehnder interferometer (see Figure

1) was used to record the steady two-dimensional temperature

d istribution in the horizontal annul us formed by an inner hexa­

gonal cylinder (hex) and an outer concentric circular cylinder.

The light source for the interferometer was a SO.Q inW fle-Ne "W

laser. The data records (or interferograms) we re analysed to

yield the !• cal temperature distribution. The local and mean

heat flux were obtained from the temperature distribui inns.

Tn ter f eromet ric techniques ar:; ideal for both time-averaaed

and i nstantaneous heat transfer measuremeni s in t ransparent media

because: (1) the test medium is not perturbed bv m external

nri/ne, {?.) the i nter feromet er has essen! ial b no i.ner' i,i, and

( 1) Tier mane nt whole- field quant i t at i. ve data records a re nbta i Tied .

The di sadvantaqes ;irp; (1) lirop optical systems are usually

pxnensive, and { 2 ) the inter fern meter runs I tip H! -iblo and v i bra t i

isola ted.

As noted previously for concentric cylinders, the Nusselt

number can be di rect]y corre 1 a ted to the Ray1^ i ah number and the

ratio of the rail i i . However, since a large number of expeiiments

p\r^ required to produce such results for the hex inside the cir-

circular cylinder, it was decided to conduct all experiments

using prototypic cross-sectinna! dimens ions.

The present study is Phase 1 (pressure 0,5 MPa) of a

broader heat transfer study in which the annulus pressure, gas

molecular weiqht (helium, neon, air, argor, 1jypton, or xenon),

and the temperature difference are experimental parameters

3

COLLIMATIMG LENS

BEAM SPLITTER 3SH1 UNEXPANDED He-!\l(

USER BEAM

TEST LEG

POLARIZER

MAGING LENS

OBSERVATION PLANE

F i g u r e 1. The Mach-^ehnder I n t e r f e rometer M r> c) Used t o E x p e r i m e n t a l l y Record the Two-Dimensional Tempera ture Di.sLriLuLi.on.

4

w h i c h a r e u s e d t o a c h i e v e l a r g e d i f f e r e n c e s , From t e s t t o t e s t ,

i n t h e R a y 1 e 1 g h n u m b e r .

2.1 A p p a r a t u s

T h e t e s t c e l l , w h i c h i s p o s i t i o n e d in t h e t e ^ t 1 en o f t h e

i n t e r f e r o m e t e r , i s shown i n F i g u r e 2. The Leal re 1 1 c o n s i s t s o f

a n i n n p r h e x wh i ch i s s e a t e d on -i sma 11 s t r ind -i n d ''• mcenf • r i ca 1 I y

1 o f a t e d i n s i d e a h o r i z o n t a l I ' i r c u U r c y l i n d e r . l i e r . m s ^ ^T t h.p

s i z e ^F t h e t p ^ r c e l l , o n l y . i hou t h a l f o f i t s e r o s s - s ^ e t [on W t-, p

i l l u m i n a t e d . T h i s i s s u f f i c i e n t : 5 inco t h e a n n u l u s f I '-w i s

*rymr"o t r ; ca 1 a b o u t a v e r t i c a l p l a n e p a s s i n g H i m u a h I h - I ' l ' i i t-^-

o f t a p c e l l .

T h e i n n e r h e x i s o r i e n t a t e d w i t h l.wn F l a t s i d e s 1.. i'i.'."';.-.,-.' .

T h e l i e : ' , w h i c h '. 'as f a b r i c a t e d F r o n s o l i d n l u r i : - . : i ! : ; ? 0 7 % ".'(: ~. *

s I: >cl: , i s 14 . 0 cm 1 onu an 1 h a s s i x s i d e s wh i r)\ a r e 7 . 3 3 c:- i p.

w ; d ;. l'j. Th. i s cy 1 i ndet ; was h e a t e d 1>y a c i r o i l - i r ev 1 i rid r i ca i v t

j s i s -

t ' v e h e a ' p r [2.r>4 cm in d i a m e t e r ) w h i c h was l o e a t e d in i i u J • . •en te r

• > F i h e iifx . F o u r t ' n e r m o c o i i p ! e s w e r e p o s i L i o::ed : : ist l e i ow ( 3

inr?) t r i e B i j r f . i c e of f o u r o f I l ie s i x s i d e s .»F I In- h e " ' - " , ' 3 ,

2 . ' 3 , a n d r a d ; a r i s ) . : , 'o te Lhat = 0 i s , i ' ! h e ' o n o F ; h o h e x .

On a n i v e n l ie/ , s i d e , t h e t h e r m o c o u p l e s w e r e l o c a ' e d .it a x i a '

p o s i t i o n s o f 2 . 0 , 4 . 8 , 6 . 4 , a n d 7 . 0 L - m F r o n o n e e n : c f t h e

r y l i r d e r s o t h a t a x i a l t e m p e r a t u r e v a r i a t i o r . s c.^-ild K e a s s e s s e d .

A l l t h e r m o c o u p l e a n d h e a t e r e l e c t r i c a l l e a d s w e r e p a u s e d t h r n u a h

a s l o t l o c a t e d n e a r t h e r e a r o f t h e t e s t e e l ': .

T h e o u t e r c i r c u l a r c y l i n d e r , a l s o a 1 mninur" , i s 2 0 . 3 em l o n q

a n d h a s .in i n s i d e d i a m e t e r o f 1 9 . 0 cm. T h r e e t h e r m o c o u p l e s w e r e

l o c a t e d i n a v e r t i c a l p l a n e p a s s i n g t h r o u g h t h e c e n t e r o f t h e

COOLV.T Port' (5i

uinowiu PQCIS

Figure 2. The Test Cell With the A.nnulus Formed by an Inner Hexagonal Cylinder Inside an Outer Concentric Circular Cylinder

6

length of the test cell and along the surface of the outer cylin-

der at = 0 , /3, and 2 / 3 . Inside the cross-section of this

cylinder were located circumferential channels (see Fig-jre 2b)

through which flowed a coolant, ethylene glycol. The coolant

was maintained at an isothermal tenperature by a reelrculat i ng

constant temperature bath. The flow lines for the coolant were

coiled to reduce vibration transmission to the optical table.

The outer surface channels were insulated with 1..' cm of

min-K insulation. The thermocouple data wore recorded to

within + 0 . 0 5 K.

The test cell windows are made of fused silici (qrade A)

and have a wedge ancle of less than 3-4.2 n d . Tho widows ave

25.4 cm in tliarcter antf 4.5 cm thick .

The principles of adjustment an:l operaticn of the Mach-Zehnder

interferometer have been described by many authors, e.n. ( 16-18).

2.2 Procedure

Prior to all experimental runs, the interferometer and test

cell alignment were checked and corrected. The a ] I .-mncnt of

the auxili ary opt i ca 1 equ ipment ( i.e. , every tiiiivi downs ' r e.iiti

of the second beam splitter of the interferometer) was also

checked. Appropriate adjustments were made to the imaoina syst•MI

so that the midplane, along the axis, of the test cell was imaged

nni-n the observation screen. The test cell was then filled {to

0.07 MPa) with research-grade gas and subsequently evacuated.

This fill-evacuate sequence was repeated at least four times.

The annulus was then pressurized to a predetermined 'evel.

7

Finally, the entire system was left overnight so that a state

of thermal equilibrium could be attained.

An isothermal infinite fringe adjustment was made to the

i nterferometer . The alignment of the test cell was again checked.

Photographs of the infinite fringe setting were taken so that

the uniformity of the interference field and the optical adjust­

ments could be evaluated. The predetermined recirculating bath

temperature was set. Temperatures in the interferometer's

reference and test cell leas were recorded. The power for the

resistive heater was set to a predetermined level. The apparatus

was left and a 1 lowed to reach a s: eady state, whi eh -:sua 1 ly

occurred from six to eleven hours after the experimental runs

began.

After steady state was a11ained, the initial data aegui-

sitic."- was begun. The data acquisition included recording the

1 oc.-'l harometr ic pressure, the reference and test cell tempera­

tures, the circulating bath's temperature, the test coll cage

pressure, and the power veneration level of the resistive header.

Soon after these measurements were recorded, a number of inter-

ferograms were recorded. The experimental parameters were

controlled to match the characteristics of Polaroid 105 p/n film.

The camera used was a Kolleiflex (model SI.-66) with a 150 mm lens.

~ypica1 exposures and f-stop set t: ings were 0.25 s and f-2.8,

respective!y.

",ocal temperature distributions were obtained from each

. - *.<:.- ferogr-im along seven (•• = /6, i/3,-/2, 2 / 3 , 5 / 6 , /IB,

-: - .--:;) radial lines across the annulus. The measurement

of the fringe location from the negatives was made using a tool-

maker's microscope- Such measurements were made starting at

the surface of the hex. The local temperature was ohtained by

relating the fringe order number to the ^nrrespnri-iino index

of refraction charge fietermine-'i from t'n& snrf^rF- r e^pera' ur.° o <~

the hex, the 1 oca 1 harom.et rif pressu ro, the tppt r-e": 1 .!,=,.•: ̂ nres-

s uro, the reference 1 en tempera t i:r e and proper: IPS rv~ * h*» 'PR'

HIT: reference }ases. The data pnin,fS p.̂ a r t-h" S!--.T~O ,>f r h p

hoy *ior? fi t ted to i st r a i ah t line. Th i s nr^vi-i"-: i " w : -~>s- ! ~a

*• i on of the t emperat ur" --jrad i en t {or NIIRSP] 1 rvirV'f r ' •..,-»-. ̂co -^^

'̂ ound.-iry layer temperature profile is linear near -.r. .•*--.••>-.--: => ;

surface. "'he Payleioh number and f-he +• p"ppr-^ uro • : F-,.; ,.-•-.-.

across the annn1 us w e r ° «va!uat ed for ea^h annn '• a r p^= • * : -̂ r..

^ach radial poo i +• j on , all nas proper*- i es *'pr° e vn ' ua * **d ••* * •-*-•.=•

-i r i ' hmet i c mean for hu ! k ) t empera t- i;r « , T V H O a • '.;u = «=•-•: * .-> n-!

''ay 1 o i gh numbers wor <=» r>bt a i ned by i nt en rat- ing t he respp'"' i ve

guar*- i ty over the total interval of the angular 1 i s t r i l'«u* i <'.n .

\̂s will be seen later, ext ra neons !'" second a ry ) "rinses

were superimposed on the i n t or f erogram, 'f'hpse frinnes were hie

to the interference of the front and hack surface reflections

from the beam splitter upstream of the basic inter ferometer com­

ponents. A polarizer was used to reduce the contrast of these

secondary fringes in the annul us. Spatial filtering in the

Fourier transform plane was attempted but the spatial separation

of the secondary fr inges from the zeroth order 1 ight was not

great enough to make this ccmpLetely effective. The combination

of the polarizer, spatial filtering, and the interference of the

tesf an<i reference l eqs of the i n t e r f e r o r - e t o r r e i u c e l the c o n t r a s t

-;f the seron^l^ry f r i n a e s t o n lo-'e'. wV.*>r" "hoy were not •-> n .nsanr t i

10

The r e s u • t of a pa r t i c u l a r e xper ; ncnt. a 1 r ;:n i s ^r. : ::* er f er ~ -

qram which i s a p a t t e r n of l i a h t ar.d '! -"* r "k f : i ;v"es ' ~-r I ; r.ep ' . The-

f r i n g e s a r e a s s o c i a t e d wi th chrir.aes i p. ' ho ' "a • . r. iex f ;••:-fr.i.--

t i on which i s p r o p o r t i ona ' * '; t he Ir.r.'i! t e rp i '-*.; .;!-••• : r. ?]•.••• nr:vs.\.;s.

The f r i i .Tes , t h e r e f o r e , r e p r e s e n t i so the rms '-.'"• ". v~ ?: :••**.".s i .•.,-. .s ! ,

ar. example of which can he seen in F i n u r e 1 ' • —sr.v. *.-.:,, '•--.._

3 . r. 1 x n 3 > .

T r. spec t i ion of F i qure 3 e v e a l s some i r. t er >- s t : r. : ••'"•.•» '•' ! - ; ' « ' r t s -

'_ics shout LIID two- 'Umensional t e m p e r a t u r e d i s i r i ! • : • • ". : :. v ,

• innu I us forced ! • / a hexaqona 1 cy1 inde r (hex ) i t.s i.k- : •• : -<>::; r i .•

. r i r r i i l . i r r y l i r i i l e r . P . o u n d a i y l a y e r f l o w s e x i s t : . • ' : ) • ' • ' • -; ;;• f d . ' e s ,

!.*••>., -i 'l j a c n r . t . tr> b o t h t h e h e x a u o n a l a n d c i r c u l a r .-y " : : . : • - ' : s , . r

i.:,<> ? i P . n n ] i ] H . Tn a d d i t i o n , a n i n s t a b i l i t y a p p e a r s ! : . ' ' . ' f *->w a n d

i s ' - h a r a c t c r i z e ' l Ivy t h e i . p n e r h a l f o f t h e a n n u l u s : . . M H I . H I i n s t f v . l y

l . i i n i n a r f l o w a n d t h e l o w e r h a l f h e i n q a s t e a d y 1 <-,!••;;,a r r 1 • >w . ""his

ft . 1 S x \c\m. !t should h" noted tha* a r i m i l a r nor* i^ I 'y uris teadv

flow ovjsf , s at a hi oner mean Ray I. in number { ! n ' , > -. i M r - = T on the

nap t h i c k n e s s ; t h i s va 1 n*» won 1,1 t ..» in"* IMSI-M • TI ic 1 •' i t ht? a n n u l u s

Formed hy rorifpn t r i e cy I i ruler s ( 7 } ,

The i n n e r thermal boundary l aye r ad jac^ : ' t n t he hex i n c r e a s e s ,

hut not m n n o t o n i r a l l y , in t h i c k n e s s ( thermal d i f f u s i o n ! as i t p r o ­

ceeds upward. The spac i no of the i so therms d e c r e a s e s as t he hex

i s approached from the norma 1 d i r ^ c H on , Th i s impli es tha I t he

c o r r e s p o n d i n g t e m p e r a t u r e gradient , or h e a t f lux i n c r e a s e s as t he

hex i s app roached . F u r t h e r , as t he gas flows around the c o r n e r

1 1

F i quro 3 . An ! n t n r f ore irarr. Showi nq r.ho Two-U imonsi onnl I so the rms i r. the Annul us Formed by an Inner liex and an n u u j r Confontr i c C i r cu1a r C y l i n d e r . Gas-Argon; Ra^ = i . f > 1 x 10*; l.onqth of He* Sidos = 7 .3 32 cm.

12

at = 90° (horizontal mid-plane of the interferogram), the

isotherms def lect towards the hex surface. As ^he nas continues

to fl<w fllnna the upper inc l ine fnear = rr>n} n r the hex, the

t her r, ^ 1 hound a ry 1 n-pr di f fuses and anpea r s m become unstable.

It is in terest inn to note th i t the therma 1 hm:r: d--> ry i ayer rer?. i r.s

^t *" T~hed to the hex. ( for 3'* ) - i r te r the -̂ -.-lop,*- .^r '-^unda ry

! awr F'ipnr--j'(?fi n* the upper cr.rner ' •-- 3.'1"1. ^f •->.;• ht>:. 'TTh e

scp i r ^ i i ' v : ~>f t !'.f ™or.-=r>*- :™ !- ' - :n : try 'ay--' : :• ».-.r:- • r. : ; •- r e a l -

* • ~n ' i i -sc-rv i t I oris hy -> r.\:r~\-r?r o i *_"- or ""^ ! - . - • • • - ; ( ? T.- • ri : - he *.:prer

'"y n n l o r . . r i

struct ••rn i n !

bound. -! ry ! aynr

thn i n t e n s i t y

->i; '^r i so the rm '--f t h : •- hr.nn-?nry }-->.y?r. As t f r -

nr in ' i c r . t o d i f f u s e nlr-nn the o u t e r c y l i n d e r ,

f hc unsteady f l ow if, -.! * enuat cd . ' ' c low t he

^hi F mda horizontal midplane of the annulus f - ^(' ),

1 ayer appears to lie s teady and ] ami na r .

An i ntermedi ate f 1 ov reqi on exists between Hie two t hernia I

boundary layers. The temperature in this rcqion is stratified

in the anqular direction. There are inflections (or a reversal

in the temperatu-e gradient) of the temperature distributions

a long the radial direction at most angular locations. However,

the temperature variation in the radial direction is usually

smal1.

13

report. **'1 in -l^ail h'jre w;is ru-i usinn helium at a lower Rayleinh

:i'i"ibpr than th^r shown in !•'i n*i r** r>. When a steady state w-is

•ir'iipve'l, a larger thermal cell 'ippeared in approximately the

sainn location as that seen in Finure "5 • This cellular configu-

ra t ion i s a consequence of the interact ion between the two

14

i gure 4 . An Interferon ram Showi nq the T\%\ -Dimonsi on.i J Isotherms in the Annu ] us Forme-1 'ry an Surer V. and an Cuter Concentric Circular CvJ:nik~r. c Xenon; R a R = 1.64 >: 10 4; Lennth of Hex Sides 7.3 32 cm.

15

Figure 5. An Interferogram Showing the Two-Dimensional Isotherms in the Annulus Formed by an Inner Hex and an Ou'ier Concentric Circular Cylinder. Gas-Neon; Ra R = 8.15 x 10 2; Length of Hex Sides = 7.332 cm.

thermal boundary layer flows on either side of the annulus. The

interaction occurs as the two boundary laye-s diffuse thermal

energy at smaller values of the mean R' ,.. number. The

appearance of the thermal cell is a^so dependent upon the ratio

of the mean gap width to the mean radius of th'.1 inner boundary

of the annulus. As this ratio increases, the thermal cell will

disappear. More eelIs wi11 like ly appear when th is rstic is

decreased. The Ijcation of the corner at = 90° is another

factor which probably a f fects the appearance and locat ion of the

thermal cell(s).

The dimons ion 1 ess radial temperature profiles for 3::^::

(Ha,, = 3.61 x 10 ) are presented in Fiquro 6. These prof i I or,

are typical of rr.any of those observed at other Rayleiqh numbers.

There ire some similarities between the profiles for the annul':.1?

formed by concentric circular cy 1 inders (8 } ar.d i_hat forme * by a

hex placed inside a concentric circular cylinder. The boundary

layer reqions are well iefined and the tempera t;: 1 -„ iiivor.ii.nris in

the intermediate reqior. are showr.

Detailed inspection of the local Vusselt number [base^ on

the maximum annulus qap width) distribution around the hex reveals

some interestinc differences from that found around the inner

cylinder for the case of concentric circular cylinders, Piaure

7 shows the anqular distribution of the Musselt number around

the hex for the five qases (also see Table 1). The obvious dif­

ference is that the local Nusselt number at the hex surface does

not decrease monotonica1ly from the bottom ( = 180°) to the top

( = 0) of the hex. The effect of the corners of the hex is also

17

DIMENSIONLESS RADIUS, r - r ,

Figure 6. Dimonsionless Radial Locations. Gas-Ar .'o:

"emuerature Distributions at .Selected ;\no,uK ; Ra n = 3.61 >: 10 J

GRAVITY 5 t VECTOR

57T 6

R a R X 1 C T 3

x 0.815 0 2.477 o 3.610 • 6.070 A 16.35o

7T 3

2

f

Anquldr Distribution of the Nussc 1 * N-.nr.ber Surface or" the Hex (See Table 1)

TABLE 1 LOCAL NUSSELT NUMBER DISTRIBUTION AT THE SURFACE OF

THE HEXAGONAL CYLINDER (SEE FIGURE 7) Rayleigh Number Ra„xlO~3

Angular Local Location Nusselt (Degrees) Number, :

-10 5.13 30 0.85 60 3.45 90 7.50

120 5.65 150 7.15 180 5.57

'-10 7.41 30 3.08 60 4.57 90 7.38 120 6.53 150 7.92 180 7.61

'VlO 14.19 30 2.36 60 6.05 90 7.93

120 8.09 150 9.98 1B0 14.96

-10 5.31 30 4.65 60 6.59 90 9.30

120 7.86 150 11.40 180 9.84

-••10 14.40 30 11.35 60 9.56 90 10.64 120 9.39 150 12.78 1B0 9.56

Argon

Krypton

0.B15

16.353

20

obvious. As noted earlier, the inner thermal boundary layer does

not separate (as does the velocity boundary layer) at = 30°.

Separation of the inner thermal boundary layer would have caused

the local Nusselt number, for 10° - <_ 30°, to decease; however,

s ince separation did not occur the local 'vussel t number increase.-.,

in some cases drastically. This increase in the \~usselt number

is also due to recirculation of the fluid from the vicin"tv or the

outer boundary layer towards the inner boundary layer. T t should

be noted that, unl ike the present case, the loca 1 \*u^selt r.umber

around the inner circular cylinder, for the concentric cylinder

case, decreases monotonically as the fluid flows towards the top

of the inner cylinder (£5) .

Since the field of view from the interferograms was limited,

the location of the data points at 1B0° and 10° are

approximate. The supporting stand for the hex prevented nhser v-

ations from being made at = 180°. Rather, the observation was

made near the corner at = 150° and normal to the bottom nor i-

;•orlta 1 surface of the hex. A similar approach was used for =

10°. In future experiments, the test eel 1. wi 1 I be mo /ed sn that

the plume near • = 0° and 10° can he observed. Future test

cell modifications wi 11 include replacing the s tai.d for the hex

with Lmall metal pins, and lengthening the hex.

After the angular Nusselt number distribution around the

hex was obtained for all gases, the mean Russell number was com­

puted for each experimental run. The results for this mean

Nusselt number ( N u ) were plotted, in Figure B, as a function of

the Rayleigh number (Ra R) (see Table 2 for more detail). Also

21

included in Fin/ur<- ^ are prototype data generated by Klima (!).

'isino a linear regression analysis it is found that the data in pirrire " can he described by

NT; = 0.794 Ra R"

Ra R = q T*P 2P 3/ ,, t

P 3 = " r ^ 1 [l.n + c* * ' 1 ] - 4 ,

T* = (T;-T o)[1.0 + C * * " 1 ] " 1 (1)

1.20?, for lex inside a circular cylinder C* = (0.66 < Pr < 0.72)

1.0, for concentric circular cylinders (0.6 < Pr < 3100)

I f - -

The characteristic radius .md reference tenperritures user] for the

Ray ! eiah rripber have been demons t rat ed , by Royd (JQ ) t to be annl i -

cable to annuli of arbitrary cross-sections. It is also important,

for comparison purposes, to note that the Rayleigh number based on

? ani T* is approximately two orders of magnitude loss than a

Rayleigh number based on '. and ^ i ' ^ o ' * M a n Y investigators mis­

conceive that concept by indicating, for example, the turbulent

flow regime occurs at a given value of Rayleigh number with no

regard to defining the reference length or temperature.

t P enters the definition for Ra explicitly for y-ases only.

I'ABLK 2

SUMMARY OF iiy.PKRI MENTAL RKSUI-TS*

Mean Gas Test Cell Temperature

(Molecular Pressure, Di fference 'T (K) W e i q h t ) P (MPa)

Neon ( 2 0 . 2 )

0 . 5 0 7

A i r ( 2 9 . 0 )

0 . 5 3 2

A r a o n o . 5 1 ? * - '. J ,

K r y p t o n ( 8 3 . 7 )

0 . 5 2 5

Xenon ( 1 3 1 . 3 )

0 . 5 2 1

R a y l e i q h M u s s e l t B a r o m e t r i c D i f f e r e n c e (K Nmnbe r , N u m b e r , P r e s s u r e f o r = 1 and

R a p x 1 0 " 4 Nu • (MPa) P = 0 . 5 0 7 MPa

0 . 0 8 1 5 4 . 9 9 0 . 0 8 5 7 3 . 1 6

0 . 2 4 7 7 6 . 17 0 . 0 8 3 6 0 . 6 7 3

0 . 3 M O 8 . 17 0 . 0 8 3 9 0 . 7 4 1

0 . 6 0 7 C 7 . 9 0 0 . 0 8 5 4 0 . 4 9 3

1 . 6 3 5 0 1 0 . 9 5 0 . 0 8 3 4 0 . 300

*The reference qns in the reference leq of the interferometer w.is .iir.

frinqe shift or or-lcr nuinher res nl t i nq from a spec i f led teinpemturo d i f ference .

•0.5000 0.661

Figure 8. Experimental Results of the Mean N U S H Number { R.iR) for t he Annu I us For met l I Cy1inder

Finally, there were no r empe ra r M r« variations npas-ire'l a*

the surface of and alono the lennth of the h^x . ~h i ̂ i~>plipd

that the annjlus flow, over most of the length -f >"•-,•-• h^v, -"is

t wo-d imens ion a \ . In addition, the t ennpra turf •liff-*r*ir,ce a --r->s <s

the annu ]us was "ker-t to a mini ivim s- > that re f ract j on er,'p:^.s

were minimised. Howeve r, s i noe the 1 enoth of rhe 'r.^x w.is sh'~>r t e

than the inside lenath of the test cell, end pffpot? WP?-H r.o*

minimi zed . In future e xne r i rnent s , the 1 Himt h ~> r t f,p V> x wi ! T

pxt.pii'l alona the entire inside lennth of thf 'PSI --PI!.

IV. CiNCt.l'STONS

The results of an e xper i ment a 1 st uiiy of r h e st eaily natural

!->->n«/pfi ion in a horizontal a n n u l n s formed by an inner h e x a n o n a 1

'•••/ '. i n'ler ann 1 an outer concept- r i <" c i r c u l a r cy! in ;.»r h a v e b e e n

nr^spri! e<*. T h e quant it i es neasur e-1 inrl'iip temper at r.re ^ i s t r i hu —

*IP, ari'i local an-: npan M u s s e ! t n u m b e r s . r virinO =>. qiven e x p e r i -

•"oi *• -i ] run , the ar.nu 1 u-? wa s r ' 11 eH wi *- h ei -.her n ^ r •?'>, H -"TOP ,

•-T-ypt^n, or xenon n a s .

"he results of the experimental runs w e r e in?erferoqrans

wh i i*h w e r e ana 1 y ze^l t n nive the t enpera fire ills', ri bu t ions a c r o s s

• I: • anriu 1 us .

'•v-r iel.it ively small Hayleiqh n u m b e r s ( Pa.. R . 1 ri x 1 0 ^ } ,

fin- i.-mn^ary layer flow .ilonq the upper li.t 1 f of the inner h e x a -

•niii-ii .-ylinler b e c o m e s unstable- T h i s instability is t remspor te.l

in :-!*•} "utt'i' c y l i n d e r w h e r e it is lampel as the outer b o u n d a r y

I.iyo: diffuses. T h i s J e s u i t s in the upper h a l f of the a n n u l u s

h n v i n i an u n s t e a d y laminar flow an-.1, the lower h a l f havinq an

apparent steady laminar flow. T h e e x i s t e n c e of the c o r n & r s of

the inner hexaqor.al c y l i n d e r , w h e n c o m p a r e ^ with ari inner c i r c u l a r

c y l i n d e r , e n h a n c e s the m e a n h e a t t r a n s f e r .

V. A C K N O W L E D G M E N T

T h e a u t h o r w o u l d like to thank the very c a p a b l e Paul C.

Montoya who provided plumbinq and electrical assistance in the

laboratory.

26

V. REFERENCES

Viraa, B. B., "LMFBR Spent Fuel Transport: Single Assembly Heat Transport Test," ORKL-TM-4936, 1975, Oak Ridge National Laboratory, Tennessee.

Pope, R. B-, "Evaluation of Heat Transfer in Conceptual Designs of Spent Fuel Shippinn Casks for the United States Breeder Reactor Technology Program," SAND77-1781, May 1978, Sandia Laboratories, Albuquerque, New Mexico.

Itoh, M., Nishiwaki, N. , and Hi rata, M., "A Kew Method for Correlating Heat-Transfer Coefficients for Natural Convection in Horizontal Cylindrical. Annul i," Internat ional Journal of Heat and Mass Transfer, Vol. 13, 1970, pp. 1364-1368.

Schimmel, W. P., Jr., "An Optical Measurements Lai ratory for Determining Heat Transfer Coefficients," SAND76- 0162, 1976, Sandia Laboratories, A1buquerquo, New Mexico.

Gartljng, D. K., "Convective Heat Transfer Analysis by thp Finite Element Method," Computer Methods in Applied Mechanics and Engineeri nn, Vol. 12, No. 3, December 1977, pp. 365-382.

Kuehn, T. !!., "Natural Convection Heat Transfer from a Horizonta1 Circular Cylinder to a Surround inn Cylindrical '•'n^lnsur"," PhD Pi ssert at ion, University of Minnesota, \ ^ 7 ^ .

Kuehn, T. H . , an 1 C-ol dstei n, R . J . , "An Kxper inenta 1 Study of Natural Convection Heat Transfer in Concentric and Keren-trie ! ! T i zonta 1 -"y ! i ndr i ca 1 Annul i , " Journal of lleat Transfer, Vol. 100, 1^78, pp. 6 3 5-640.

Kuehn, T. K., and Goldstein, ^. •T . , "An K xper i "lent ~ 1 an-theoretical Study of Natural Convection in the Annnlus Between "ori-'ontal Concentric Cylinders," Journal of Fluid Mechanics, Vol. 74, Part 4, 1976, pp. 695-7 l 9~." "~" " ~ " "

C-riqull, "., and liauf, W., "Natural Convection in Horizontal Cy1indrica1 Annuli, " Third International Heat Trans fer Con­ference, Chicaqo, 1966, pp. IR2-19 5.

I is, J., "Experimental Investigation of Natural Convection Heat Transfer in Simple and Obstructed Horizonta1 Annu L i, " Third International Heat Transfer Conference, Chicago, 1966, pp. 196-204.

Kuehn, T. H., and Gotdstein, R. J., "Correlating Equations for Natural Convect ion Heat Transfer Between Horizontal Circular Cylinders," International Journal of Heat and Mass Transfer, Vol. 19, 1976, pp. 1127-1134,

27

12. Raithby, G. D. , and Hollands, K. G. T., "A General Met'nn.i of obtaining Approximate Solutions to Laminar and Tur'i'ii' cnt F'ree Convection Problems, " Advances in_ Heat Transfer, \ol. 1 ! , 197S, pp. 265-315.

H . Charr ier-Mojtabi , M , C. , Mojtahi, A . , and C^l^aninne, J . P. , "\'umer ica I Solution of a Flow Due to a Matural Cnnvec*' i ̂ n in Horizontal Cylindrical Annulus, " Journal of Heat Transfer, Vol. 101, 1979, pp. 171-173.

14. Patanker, S. V. , Ivanovic, M., and Sparrow, P* . M., "Analysis of Turbulent Flow and Heat Transfer in Internally Finned Tuhfs and Annu 1 i, " Journal of Meat Transfer, Vol . 101, 197Q, r,p. 29-37.

15. Van de Sande, F.., and Hamer, R- J. G., "Steady and Transient-natural Convection in Enclosures Between Horizontal Circular Cylinders (Constant Heat Flux)," International Journal of Heat and Mass Transfer, Vol. 22, 1979, pp. 361-370.

16. Haul;, W., and Grigull, U., "Optical Methods in "eat Transfer," Advances in Heat Transfer, Vol. 6, Academic Press, Vew York, NY, 1970, pp. 133-366.

17. Kckert, E. R. G., and Goldstein, R. J., Measurements in Heat. Trans fer , Second Edition, ^cG raw-Hi 11, New York , MY , I Q 7 6 . pp. 241-293.

18. MerzKirch, W. , Flow Visualization, Academic Press, Mew York, MY, 1974, pp. 62-184.

19. Boyd, R. D., "A Mew Correlation Theory for Steady Natural Convective Heat Transport Data For Horizontal Annuli," 20th National Heat Transfer Conference (August, 1981), (To bo presented).

28