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AECL-8365

ATOMI C ENERG Y ££^ S L ENERG1 E ATOMIQU E O F CANAD A LIMITE D ^fi 9 D U CANADA , LIMITE E HEA T TRANSFE R COEFFICIENT S FO R LEA D MATRIXIN G I N DISPOSA L CONTAINER S FO R USE D REACTO R FUE L

COEFFICIENT

S D E TRANSFER T D E CHALEU R DAN S LE S MATRICE S D E PLOM B DE S CONTENEUR S D'EVACUATIO N D U COMBUSTIBL E D E REACTEUR S P . M . Mathew , M . Taylor , P . A . Kruege r

Whiteshel

l Nuclea r Researc h Etablissemen t d e recherche s

Establishmen

t nucleaire s d e Whiteshel l

Pinawa

, Manitob a RO E 1L O

Februar

y 198
5 fevrie r

ATOMIC ENERGY OF CANADA LIMITED

HEA T TRANSFE R COEFFICIENT S FO R LEA D MATRIXIN G I N DISPOSA L CONTAINER S FO R USE D REACTO R FUE L b y P.M . Mathew , M . Taylo r an d P.A . Kruege r

Whlteshel

l Nuclea r Researc h Establishmen t

Pinawa

, Manitob a RO E 1L O 198
5 Februar y

AECL-836

5 COEFFICIENTS DE TRANSFERT DE CHALEUR DANS LES MATRICES DE PLOMB DE S CONTENEUR S D'ÉVACUATIO N D U COMBUSTIBL E D E RÉACTEUR S pa r P.M . Mathew , M . Taylo r e t P.A . Kruege r

RÉSUM

É O n évalue , dan s l e cadr e d u Programm e canadie n d e gestio n de s déchet s d e combustibl e nucléaire , de s matrice s métallique s ayan t u n ba s poin t d e fusio n pou r détermine r leu r capacit é d e suppor t d e l'envelopp e de s conteneur s d'évacuatio n d u combustibl e irradi é e t d e protectio n supplémen - tair e contr e le s radionuclide s libérés . L a matric e métalliqu e serai t incorporé e a u conteneu r pa r coulée . Afi n d'étudie r le s processu s d e transfer t d e chaleu r pendan t l a solidifica - tion , o n a employ é un e techniqu e stationnair e dan s laquell e l e méta l coul é

étai

t l e plom b pou r détermine r l e coefficien t d e transfer t d e chaleu r globa l entr e l e plom b e t certain s de s matériau x d e conteneur s retenus . L'existenc e d'u n espac e d'ai r entr e l e plom b coul é e t l e matéria u d u conteneu r sembl e contrôle r l e coefficien t d e transfer t d e chaleu r global . Le s observation s expérimentale s indiquen t qu e l a topographi e d e l a surfac e d u matéria u d e conteneu r influ e su r l e transfer t d e chaleu r e t qu'un e surfac e plu s liss e entraîn e u n transfer t d e chaleu r plu s élev é qu'un e surfac e rugueuse . E n outre , le s résultat s expérimentau x on t montr é u n coefficien t d e transfer t d e chaleu r croissan t e n fonctio n d e l'augmentatio n d e l a différenc e d e tempéra - tur e à traver s l e socl e d e conteneur . U n modèl e d e flexio n d e socl e mi s a u poin t peu t explique r le s résultat s observés .

L'Énergi

e Atomiqu e d u Canada , Limité e

Établissemen

t d e recherche s nucléaire s d e Whiteshel l

Pinawa

, Manitob a RO E 1L 0 198
5 févrie r

AECL-836

5

HEAT TRANSFER COEFFICIENTS FOR LEAD MATRIXING

I N DISPOSA L CONTAINER S FO R USE D REACTO R FUE L b y P.M . Mathew , M . Taylo r an d P.A . Kruege r

ABSTRAC

T I n th e Canadia n Nuclea r Fue l Wast e Managemen t Program , meta l matrice s wit h lo w meltin g point s ar e bein g evaluate d fo r thei r potentia l t o provid e suppor t fo r th e shel l o f disposa l container s fo r use d fuel , an d t o ac t a s a n additiona l barrie r t o th e releas e o f radionuclides . Th e meta l matri x woul d b e incorporate d int o th e containe r b y cast - ing . T o stud y th e hea t transfe r processe s durin g solidification , a steady - stat e techniqu e wa s used , involvin g lea d a s th e cas t metal , t o determin e th e overal l hea t transfe r coefficien t betwee n th e lea d an d som e o f th e candidat e containe r materials . Th e existenc e o f a n ai r ga p betwee n th e cas t lea d an d th e containe r materia l appeare d t o contro l th e overal l hea t transfe r coeffi - cient . Th e experimenta l observation s indicate d tha t th e surfac e topograph y o f th e containe r materia l influence s th e hea t transfe r an d tha t a smoothe r surfac e result s i n greate r hea t transfe r tha n a roug h surface . Th e experi - menta l result s als o showe d a n increasin g hea t transfe r coefficien t wit h increasin g temperatur e differenc e acros s th e containe r bas e plates ; a mode l develope d fo r base-plat e bendin g ca n explai n th e observe d results . Atomi c Energ y o f Canad a Limite d

Whiteshel

l Nuclea r Researc h Establishmen t

Pinawa

, Manitob a RO E 1L 0 198
5 Februar y

AECL-836

5

CONTENTS

Pag e LIS T O F SYMBOL S i 1 . INTRODUCTIO N 1 2 . REVIE W O F DEFINITION S 2 2. 1 THERMA L CONDUCTIO N I N A HOMOGENEOU S MEDIU M 2 2. 2 BOUNDAR Y CONDITION S 2 2. 3 OVERAL L HEA T TRANSFE R RAT E I N A COMPOSIT E SYSTE M 4 3 . EXPERIMENTA L 6 3. 1 GENERA L DESCRIPTIO N 6 3.1. 1 Material s 6 3.1. 2 Containe r an d Hea t Sourc e 7 3.1. 3 Coolin g Syste m 7 3.1. 4 Castin g Procedur e 8 3. 2 HEA T TRANSFE R EXPERIMENT S 8 3.2. 1 Determinatio n o f Overal l Hea t

Transfe

r Coefficien t 8 3.2. 2 Determinatio n o f Convectiv e Hea t

Transfe

r Coefficien t an d Hea t Los s 9 4 . RESULT S 1 0 4. 1 OVERAL L HEA T TRANSFE R COEFFICIEN T 1 0 4. 2 CONVECTIV E HEA T TRANSFE R COEFFICIEN T FO R WATER - FLO W AN D HEAT-LOS S ESTIMATE S 1 0 5 . DISCUSSIO N 1 1 6 . CONCLUSION S 1 3

ACKNOWLEDGEMENT

S 1 4

REFERENCE

S 1 4 TABLE S 1 6 continued.. .

CONTENTS (concluded)

Pag e

FIGURE

S 1 8

APPENDI

X 2 8 A.I . A MODE L FO R BASE-PLAT E BENDIN G AN D IT S EFFEC T O N HEA T TRANSFE R 2 8 A.2 . DEFINITION S O F SURFAC E ROUGHNES S PARAMETER S 3 0 A.3 . ERRO R ESTIMATIO N 3 1 - i - LIS T O F SYMBOL S a thicknes s o f bas e plat e h hea t transfe r coefficien t o f th e ai r ga p h . hea t transfe r coefficien t o f th e insulatio n h hea t transfe r coefficien t o f th e bas e plat e h overal l hea t transfe r coefficien t h convectiv e hea t transfe r coefficien t fo r th e coolin g water/containe r interfac e k therma l conductivit y (als o proportionalit y constant ) k . "herma l conductivit y o f lea d k ? therma l conductivit y o f containe r materia l k therma l conductivit y o f ai r a t T k therma l conductivit y o f th e insulatio n i containe r diamete r q hea t flu x q hea t flu x a t th e air-gap/lea d interfac e q . hea t flu x throug h th e containe r sid e wall s q hea t flu x throug h th e lea d q hea t flu x a t th e base-plate/air-ga p interfac e q hea t flu x throug h th e composit e boundar y q hea t flu x a t th e base-plate/coolin g wate r interfac e u hea t flu x i n th e x-directio nHx r ^ outsid e radiu s o f containe r r £ radia l distanc e fro m th e containe r centre-lin e t o th e thermocoupl e positio n i n th e insulatio n R ar c radiu s o f ben t bas e plat e T temperatur e T . surfac e temperatur e o f th e lea d a t th e lead/air-ga p interfac e T. . surfac e temperatur e o f th e containe r materia l a t th e containe r base-plate/air-ga p interfac e T - surfac e temperatur e o f th e containe r materia l a t th e base-plate/coolin g wate r interfac e T roo m temperatur e (28 3 K ) - ii - T . temperatur e o f th e insulatio nins r T lea d temperatur e a t sam e heigh t a s thermocoupl e i n insulatio n T temperatur e o f th e coolin g wate r T H heatin g elemen t temperatur e x distanc e withi n lea d X widt h o f ai r ga p a t T X c fina l air-ga p widt h afte r base-plat e deflectio n A X deflectio n a t th e centr e o f bas e plat e a linea r expansio n coefficien t o f containe r materia l cf > deflectio n angl e o f bas e plat e

Subscrip

t o denote s paramete r a t T

1. INTRODUCTION

I n th e Canadia n Nuclea r Fue l Wast e Managemen t Program , lo w melting-poin t materials , suc h a s lead , a lea d 1 wt. % antimon y alloy , zinc , an d a n aluminu m 7 wt. % silico n alloy , ar e bein g evaluate d fo r th e immobili - zatio n an d disposa l o f use d fuel . Thes e material s hav e th e potentia l t o p-T'Vid e interna l suppor t fo r th e shel l o f th e disposa l container , t o with - st t ' th e hydrostati c pressur e i n a floode d disposa l vaul t [1-3] . More - over , i.h e meta l matri x coul d ac t a s a n additiona l barrie r t o th e releas e o f radionuclides . T o perfor m th e aboir e functions , a high-integrit y meta l matri x withou t void s i s required . Th e meta l matri x woul d b e cas t int o th e containe r t o envelo p th e used-fuel . Solidificatio n o f th e liqui d meta l involve s hea t transfe r acros s th e matri x metal/containe r interface . I f th e hea t transfe r durin g solidificatio n i s no t controlled , suc h tha t contractio n durin g th e liquid-to-soli d transformatio n i s no t compensate d for , the n void s ca n for m i n th e matri x [4] . Further , th e hea t transfe r condition s determin e th e contac t tim e betwee n th e liqui d meta l an d th e container , an d th e fue l clad - din g material , Zircaloy-4 . Sinc e interactio n ca n occu r betwee n thes e component s durin g casting , e.g. , dissolutio n o f containe r an d claddin g material s i n th e liqui d meta l matrix , a rapi d solidificatio n proces s i s desirabl e [2] . T o stud y th e parameter s controllin g th e hea t transfe r processe s i n metal-raatrixe d containers , compute r simulatio n o f th e solidificatio n proces s ha s bee n initiate d usin g finite-elemen t method s [4] . T o compar e th e modellin g result s wit h solidificatio n experiments , hea t transfe r coefficient s a t th e containe r boundarie s ar e required . Th e objectiv e o f thi s stud y wa s t o develi p a techniqu e fo r deter - minin g th e hea t transfe r coefficient s betwee n candidat e matrixin g material s an d candidat e containe r materials . I n th e wor k reporte d here , lea d wa s use d a s th e matrixin g materia l an d AIS I Typ e 316
L o r Typ e 30
4 austeniti c - 2 - stainles s stee l (S S 316
L an d S S 304
, respectively) , Incone l 62
5 an d AST M grade-1 2 titaniu m (Gr-1 2 Ti ) wer e chose n a s containe r tes t materials . 2 . REVIE W O F DEFINITION S 2. 1 THERMA L CONDUCTIO N I N A HOMOGENEOU S MEDIU M A temperatur e gradien t V T withi n a homogeneou s mediu m result s i n a hea t flux , q , withi n th e medium , whic h i s expresse d b y Fourier' s la w o f hea t conductio n a s [5 ] (1 ) wher e th e constan t o f proportionality , k , i s th e therma l conductivit y o f th e mediu m an d ma y depen d upo n temperatur e an d pressure . I f th e temperatur e distributio n withi n th e mediu m i s linear , suc h a s i n a homogeneou s mediu m o f fixe d k durin g steady-stat e hea t transfe r i n on e dimension , Equatio n (1 ) ca n b e writte n a s d xdx x~ - x. wher e 1^ an d T ^ ar e th e temperature s a t position s x ~ an d x, , respectively .

Steady-stat

e hea t transfe r occur s whe n th e temperatur e a t ever y poin t withi n th e body , includin g th e surfaces , i s independen t o f time . 2. 2 BOUNDAR Y CONDITION S Th e hea t flu x int o o r ou t o f a mediu m wil l b e determine d b y condition s a t th e boundarie s o f th e medium . Conside r th e boundar y condi - tion s o f a n experiment , show n schematicall y i n Figur e 1 , i n whic h axia l - 3 - hea t flo w occur s throug h a cylindrica l lea d castin g withi n a metalli c con - tainer . Assum e th e sid e wall s o f th e containe r ar e insulated . A tempera - tur e gradien t ca n b e establishe d withi n th e lea d i f a constan t hea t flu x q i s provide d t o th e fre e lea d surfac e a t th e ope n en d o f th e container , an d i f th e containe r bas e plat e i s coole d b y movin g water . Unde r steady-stat e conditions , a linea r temperatur e distributio n i s attaine d withi n th e lead . I f a n ai r ga p exist s betwee n th e containe r base-plat e materia l an d th e lead , an d i f hea t transfe r throug h th e ga p i ? b y conductio n only , a linea r temperatur e distributio n i s als o obtaine d withi n th e gap . Le t T . an d T . b e th e surfac e temperature s o f th e lead/air-ga p interfac e an d th e base-plate / air-ga p interface , respectively . A linea r temperatur e distributio n wil l als o exis t withi n th e base-plat e material , wit h T ~ bein g th e surfac e temperatur e a t th e base-plate/coolin g wate r interface , an d T bein g th e wate r temperature . I n th e steady-stat e hea t transfe r situatio n describe d above , th e followin g boundar y condition s apply : (I ) Th e hea t flu x q throug h th e lea d i s q - k . ^ O )it 1 dx, wher e k ^ i s th e therma l conductivit y o f lead . Th e temperatur e gradien t dT/d x ca n b e determine d experimentall y fro m th e temperature-distanc e relationshi p withi n th e lead . (2 ) Th e surfac e temperatur e T o f th e lea d a t th e lead/air-ga p interfac e ca n b e determine d b y extrapolatio n o f th e temperatur e distributio n withi n th e lead . (3 ) A t th e base-plate/wate r interface , th e hea t flu x q enterin g th e coolin g wate r an d th e temperatur e differenc e betwee n th e base-plat e surfac e an d th e coolin g wate r ar e relate d b y

Newton'

s la w o f cooling , h (T , - T ) (4 ) w 2 w v ' where h is the convective heat transfer coefficient. Tow £. an d T ca n b e determine d experimentally . Thes e boundar y condition s determin e th e hea t flo w throug h th e syste m describe d above . 2. 3 OVERAL L HEA T TRANSFE R RAT E I N A COMPOSIT E SYSTE M Th e hea t transfe r rat e i n a composit e system , suc h a s tha t show n i n Figur e 1 , i s governe d b y th e therma l conductivitie s o f lead , air , an d th e base-plat e materia l an d th e hea t transfe r processe s a t th e syste m interface s o r boundaries . Th e quantitie s dx/k , i n Equatio n (3 ) an d 1/ h i n

Equatio

n (4 ) ma y b e regarde d a s therma l resistance s fo r th e conductio n an d convectio n processes , respectively . Th e overal l resistanc e t o hea t flo w (Figur e 1 ) fro m th e lea d t o th e coolin g wate r (wit h a temperatur e T ) ha s thre e components : (1 ) th e interfac e resistanc e 1/ h , du e t o formatio n o f a n ai r ga p betwee n th e lea d an d th e bas e plat e ( h i s th e hea t transfe r coefficien t fo r th e gap) ; (2 ) th e base-plat e resistanc e 1/ h ( h = k^/a , an d i s th e hea t transfe r coefficien t fo r th e bas e plate ; k . an d a ar e th e therma l conductivit y an d th e thicknes s o f th e bas e plate , respectively) ; (3 ) th e convectiv e resistanc e a t th e coolin g water/containe r interface , 1/ h ( h i s th e convectiv e hea t transfe r coefficien t fo r th e water/containe r interface) . Sinc e th e thre e resistance s ar e connecte d i n series , th e overal l resistanc e 1/hj . ca n b e writte n a s h h h ht c p w o r t i =t = 111~ h~+ h~c p w Th e hea t flu x throug h eac h o f th e thre e resistance s ca n b e writte n separatel y a s - 5 - q c = h c (T i - T t ) (ai r gap ) (6 ) q p = h p (T 1 - T 2 ) (bas e plate ) (7 ) % = h w ( T 2 " V (coolin g water ) (8 ) Th e hea t flux , q , throug h th e composit e boundar y show n i n Figur e 1 , i s give n b y Th e steady-stat e hea t transfe r rat e throug h th e lead , q , i s J o give n b y Equatio n (3) . Fo r unidirectiona l hea t flo w throug h th e lead , ai r ga p an d bas e plat e t o th e coolin g water , wit h n o hea t los s t o th e sid e walls , w e ca n writ e th e followin g equation : (10 ) Fro m Equation s (9) , (3 ) an d (10) , th e overal l hea t transfe r coefficien t i s give n b y k £- \ Th e convectiv e hea t transfe r cv.afficien t fo r wate r ca n b e evaluate d similarl y fro m Equation s (8) , (3 ) an d (10) , t o yiel d h w ' "w ' (T, - T Th e temperatur e differenc e acros s th e bas e plate , AT , ca n b e calculate d b y combinin g Equation s (7 ) an d (3) , t o give * - 6 - 3 . EXPERIMENTA L 3. 1 GENERA L DESCRIPTIO N Th e experimenta l apparatu s consiste d o f a n insulate d cylindrica l containe r i n whic h th e lea d wa s cas t wit h a heatin g elemen t locate d o n th e fre e surfac e o f th e lea d an d a wate r spra y coolin g syste m t o extrac t hea t fro m th e containe r bas e plat e (se e Figur e 2) . Thi s arrangemen t allowe d unidirectiona l steady-stat e hea t transfe r condition s t o b e established . A n insulate d plat e prevente d wate r seepag e throug h th e insulatin g materia l t o th e sid e wal l an d an y los s o f hea t associate d wit h it . 3.1. 1 Material s

Commercia

l purit y lea d (AST M B29-79 ) wa s use d fo r th e casting . Bas e plate s wer e mad e fro m eac h o f th e followin g containe r tes t materials : grade-1 2 titanium , Incone l 625
, an d Typ e 316
L an d Typ e 30
4 stainles s steel , al l o f 6.35-m m nomina l thickness . T o characteriz e th e lead/containe r base-plat e interfac e throug h whic h hea t flo w occurs , a s describe d late r i n th e experimenta l procedure , th e surfac e topograph y o f th e base-plat e material s wa s determine d usin g a profilomete r alon g a profil e lengt h o f th e plat e ( L = 1 0 mm) , a t tw o ran - doml y selecte d locations . Evaluatio n o f th e surfac e roughnes s a t th e tw o location s showe d agreemen t t o withi n ±5% . A s a n example . Figur e 3 show s a sectio n o f th e tota l profil e lengt h L fo r Typ e 30
4 stainles s steel . Th e centre-lin e o f th e surfac e profil e i n Figur e 3 wa s define d s o tha t th e tota l are a abov e th e lin e wa s equa l t o th e tota l are a belo w th e lin e ove r th e tota l profil e lengt h L . Tabl e 1 list s th e surfac e roughnes s parameter s (roo t mea n squar e o f th e surfac e height s (o) , mea n slop e o f surfac e height s (m) , an d maximu m surfac e heigh t (y, . )) > an d th e measure d plat e thicknesse s (a ) [6] . Th e abov e quantitie s ar e define d i n th e Appendix . - 7 - 3.1. 2 Containe r an d Hea t Sourc e Th e experimenta l ri g (se e Figur e 4 ) consiste d o f a carbo n stee l pip e wit h a n insid e diamete r o f abou t 15 2 m m an d a lengt h o f abou t 17 8 mm , t o whic h th e bas e plat e wa s attache d usin g eigh t equall y space d screws . A n alumina-base d cerami c cemen t wa s applie d alon g th e join t betwee n th e bas e plat e an d th e pip e t o preven t escap e o f molte n lea d durin g th e castin g pro - cess . Si x insulate d chrome1-alume l thermocouple s wer e positione d insid e th e cylinde r a t distance s o f 5 , 15 , 30
, 45
, 6 0 an d 7 5 m m fro m th e bas e plat e alon g th e centra l axis . Th e thermocouple s wer e supporte d b y tw o thi n Typ e 3u 4 stainles s stee l bracket s positione d wel l awa y fro m th e temperatur e measurin g points . Severa l Typ e 30
4 stainles s stee l hook s wer e spot-welde d t o th e containe r sid e wall s t o preven t displacemen t o f th e lea d relativ e t o th e containe r bas e plat e durin g handling . Th e hea t sourc e consiste d o f a 1500-
W spira l elemen t attache d t o a machine d coppe r plat e o f hig h therma l conductivity . Th e elemen t wa s insulate d wit h a n alumin a cerami c past e an d alumin a powder . T o furthe r reduc e hea t loss , a structura l insulatio n materia l (Marinit e I* ) wa s use d t o cove r th e spira l element . Th e remainin g spac e abov e th e heatin g elemen t wa s fille d wit h Marinit e I containin g tw o holes , throug h whic h th e thermo - coupl e an d electrica l suppl y wire s wer e passed . Th e hoie e wer e the n fille d wit h alumin a powder . 3.1. 3 Coolin g Syste m Th e coolin g syste m consiste d o f tw o Typ e 30
4 stainles s stee l cylinders , 15 0 an d 30
0 m m i n diameter , an d 7 5 an d 10 0 m m i n height , respec - tively . Th e smalle r cylinde r wa s supporte d insid e th e large r on e b y fou r Typ e 30
4 stainles s stee l leg s (se e Figur e 2) . Coolin g wate r entere d throug h fou r hole s i n th e lowe r par t o f th e insid e cylinde r an d wa s directe d t o th e containe r bas e plat e b y vanes . Wate r flowe d ou t t o th e large r outsid e cylinde r throug h a ).18-ra m ga p betwee n th e bas H plat e an d * Trad e name : Manvill e Corp. , Keii-Cary l Ranch , Denver , Colorad o - 8 - th e to p o f th e insid e cylinder . Th e wate r flo w rat e wa s monitore d wit h a calibrate d flo w meter . 3.1. 4 Castin g Procedur e Afte r assembly , th e containe r wa s place d o n th e wate r coolin g apparatu s an d mor e insulatio n wa s applie d t o th e sid e wall s (se e Figur e 2) . Befor e lea d casting , th e containe r wa s preheate d b y force d air . Lea d wa s melte d i n a resistance-heate d furnace . Th e coolin g wate r flo w rat e wa s se t a t 1 8 L/mi n an d th e lea d wa s cas t i n les s tha n 3 0 s . Th e prehea t an d cast - in g temperature s fo r experimenta l container s wit h differen t base-plat e material s ar e give n i n Tabl e 2 .

Toward

s th e en d o f solidification , a propan e ga s torc h wa s applie d t o th e to p o f th e lea d surfac e t o reduc e shrinkage . Afte r cooling , th e to p surfac e o f th e lea d wa s machine d fla t s o tha t th e distanc e t o th e bas e plat e wa s 101.
6 m m everywhere . On e containe r wa s sectione d longitudinally , machine d an d etche d t o revea l th e microstructure . Figur e 5 show s th e grai n siz e t o b e larg e an d primaril y oriente d i n th e directio n o f hea t flow , i.e. , toward s th e containe r bas e plate . 3. 2 HEA T TRANSFE R EXPERIMENT S 3.2. 1 Determinatio n o f Overal l Hea t Transfe r Coefficien t T o determin e th e overal l hea t transfe r coefficient , h , betwee n th e solidifie d lea d an d coolin g water , steady-stat e hea t transfe r experi - ment s wer e conducted . I n thes e tests , th e heatin g elemen t wa s controlle d a t 423
, 47
3 o r 55
3 K . Th e wate r flo w rat e wa s se t a t 11.0 , 18.0 , o r 25.
2 L/mi n unti l th e temperatur e gradien t i n th e lea d remaine d constan t fo r a t leas t 180
0 s . Th e overal l hea t transfe r coefficien t wa s calculate d usin g

Equatio

n (11) . - 9 - 3.2. 2 Determinatio n o f Convectiv e Hea t Transfe r

Coefficien

t an d Hea t Los s T o establis h th e rate-determinin g ste p i n th e overal l hea t trans - fe r . iroug h th e bas e plate , th e convectiv e hea t transfe r coefficien t fo r th e base-plate/wate r interfac e wa s determine d usin g Typ e 30
4 stainles s stee l a s th e base-plat e materia l an d lea d a s th e cas t metal . Th e hea t los s throug h th e insulatio n o f th e containe r sid e wall s wa s determine d i n th e sam e experiment . Th e convectiv e hea t transfe r coefficient , h , ca n b e calculate d usin g Equatio n (12 ) i f th e hea t flu x throug h th e lea d an d th e temperatur e differenc e betwee n th e externa l surfac e o f th e bas e plate , T ^ > an d th e bul k wate r temperature , T , ar e known . Sinc e thi s require s accurat e temperatur e measurements , tw o calibrate d chromel-alume l thermocouple s wer e use d t o measur e th e base-plat e an d wate r temperatures . On e o f th e thermocouple s wa s spot-welde d t o th e centr e o f th e bas e plate . Th e temperatur e reading s o f th e calibrate d thermocouple s wer e determine d t o withi n ± 0. 1 K usin g a precisio n potentiometer . T o obtai n a conservativ e estimat e o f th e hea t los s throug h th e insulate d sid e wall s o f th e container , a hig h heating-elemen t temperatur e o f 57
3 K wa s selected . Thermocouple s wer e place d a t distance s o f 5 , 15 , 25
, 35
, 4 5 an d 5 5 m m verticall y abov e th e bas e plate . A n additiona l thermocoupl e wa s place d insid e th e insulation , 4 0 m m fro m th e containe r wal l an d 5 5 m m fro m th e containe r bas e plate . Th e wate r flo w rat e wa s se t a t 11. 0 L/mi n an d th e syste m wa s brough t t o stead y state . Th e temperature s insid e th e lea d an d insulatio n wer e monitore d o n a Digistri p recorder . - 10 - 4 . RESULT S 4. 1 OVERAL L HEA T TRANSFE R COEFFICIEN T Th e variatio n o f temperatur e wit h distanc e fro m th e bas e plat e i s presente d i n Figure s 6 t o 9 fo r differen t base-plat e materials - I n al l cases , th e temperatur e showe d a linea r relationshi p wit h distance , wit h a correlatio n coefficien t greate r tha n 0.99 , indicatin g tha t steady-stat e hea t transfe r existe d durin g th e measurements . Th e surfac e temperatur e o f th e lea d a t th e lead/air-ga p interface , T. , an d th e slop e dT/d x wer e determine d b y linea r regressio n analysis . Th e averag e wate r temperature , T , wa s 27
5 K fo r al l experiments . Usin g a valu e o f k , = 34.
6 W/(m.K ) fo rw 1 th e therma l conductivit y o f lea d [7] , Equatio n (11 ) wa s use d t o calculat e th e overal l hea t transfe r coefficient , h . I n Figur e 10 , th e overal l hea t transfe r coefficient s fo r Typ e 30
4 an d Typ e 316
L stainles s steel , grade-1 2 titanium , an d Incone l 62
5 ar e plot - te d agains t th e lead/air-ga p interfac e temperature , T. . Th e erro r bar s wer e estimate d usin g th e metho d describe d i n th e Appendix . 4. 2 CONVECTIV E HEA T TRANSFE R COEFFICIEN T FO R WATER-FLO W AN D

HEAT-LOS

S ESTIMATE S A plo t o f lea d temperatur e agains t distanc e fro m th e bas e plat e i s show n i n Figur e 1 1 fo r th e experimen t use d t o determin e th e convectiv e hea t transfe r coefficient . A linea r relationshi p wa s obtained , indicatin g steady-stat e hea t transfer . Th e lead/air-ga p interfac e temperature , T. , wa s 42
1 K an d th e slop e dT/d x wa s 72
5 K/m . Th e base-plate/coolin g wate r interfac e temperature , T_ , an d th e wate r temperature , T , wer e 295.
6 K an d 295.
0 K , respectively . Usin g Equatio n (12) , th e convuctiv e hea t transfe r 4 ?coefficient, h , was calculated to be 4.2 x 10 W/(m~.K) for a water flow W 4 2 rat e o f 11. 0 L/min . Th e tota l hea t flux , q , wa s 2. 5 x 1 0 W/ m . - 11 - Th e thermocoupl e locate d i n th e insulatio n indicate d a temper - ature , T , , o f 34
4 K . Th e therma l conductivit y o f th e insulatin g HI S ry material , k . , i s 5.4 8 x 10 ~ W/(m.K) . Th e lea d temperature , T , a t th e in s s sam e heigh t (5 5 m m fro m th e containe r bottom ) wa s 46
2 K (Figur e 11) . I f th e containe r wal l surfac e temperatur e i s assume d t o b e 46
2 K , a con - servativ e estimate , th e hea t flux , q , throug h th e sid e wall s ca n b e calculate d fro m [5 ] h , ( T - T . ) (14 ) in s s in s k . tnS r~ (15 )ins r In wher e h . = hea t transfe r coefficien t o f th e insulation ,ins r ~ = radia l distanc e fro m th e containe r centre-lin e t o th e thermocoupl e position , an d r . = outsid e radiu s o f th e container . _

2From Equations (14) and (15), with r. = 7.62 x 10 m and

- 2 ? Tj - 11-6 2 x 1 0 m , q . wa s calculate d t o b e 13 2 W/m~ . B y takin g th e containe r side-wal l surfac e are a an d th e containe r base-plat e surfac e are a int o account , th e hea t flo w rate s throug h th e sid e wal l an d t o th e coolin g wate r wer e calculate d t o b e 3. 8 W an d 423.
2 W , respectively . Therefore , th e hea t los t throug h th e sid e wal l wa s les s tha n 1 % o f th e hea t flo w t o th e coolin g water , confirmin g tha t th e hea t flo w wa s predominantl y uni - directional . Thi s i s consisten t wit h th e linea r relationshi p observe d betwee n temperatur e an d distance - 5 . DISCUSSIO N Figur e 1 0 show s tha t th e overal l hea t transfe r coefficient , h , increase d wit h increasin g surfac e temperatur e o f th e lea d a t th e - 12 - lead/air-ga p interface , T. , fo r al l material s studied . Th e rat e o f increas e wa s highes t ro r Incone l 62
5 an d lowes t fo r Typ e 30
4 stainles s steel . A mode l ha s bee n develope d (se e th e Appendix ) t o explai n th e observe d dependenc e o f h o n T. . Thi s mode l consider s th e base-plat e bendin g cause d b y a temperatur e difference , AT , acros s th e plate . Th e widt h o f th e ai r ga p betwee n th e bas e plat e an d th e cas t lea d a t roo m temperatur e i s decrease d b y thi s process . Thi s increase s th e air-ga p conductanc e and , hence , th e overal l hea t transfe r coefficien t wit h increasin g AT , whic h i s proportiona l t o T. . T o tes t th e model , th e temperatur e differenc e A T acros s th e bas e plat e wa s calculate d usin g Equatio n (13 ) an d plotte d agains t h (se e Figur e 12) . Equatio n (A.12 ) wa s use d t o predic t th e chang e i n h wit h AT . Th e calculate d curve s ar e show n i n Figur e 1 2 usin g dat a fro m Tabl e 3 . Th e mode l satisfactoril y predict s th e dependenc e o f h o n A T fo r temperatur e difference s < 6°C . A t highe r temperatur e differences , th e calculate d hea t transfe r coefficient s wer e greate r tha n thos e obtaine d experimentally . A s suggeste d i n th e model , i t i s probabl e that , a t hig h temperatur e differ - ences , th e restrain t provide d b y th e attachmen t o f th e containe r circum - ferenc e t o th e bas e plat e restricte d fre e bendin g o f th e bas e plate . A close r examinatio n o f th e dat a i n Table s 1 an d 3 indicate s tha t th e hea t transfe r coefficient s a t roo m temperature , h , depen d o n th e paramete r m/a , whic h describe s th e surfac e roughnes s o f th e base-plat e materia l [8] . A highe r m/ a valu e indicate s a smoothe r surface . Incone l 62
5 an d Typ e 30
4 stainles s stee l ha d th e smoothes t an d roughes t surfaces , respectively . Th e tes t result s indicat e tha t a smoothe r base-plat e surfac e provide s greate r hea t transfer , sinc e th e averag e ai r ga p betwee n th e cas t lea d an d bas e plat e i s smaller . Thi s i s t o b e expecte d since , durin g th e castin g process , liqui d lea d doe s no t full y penetrat e int o th e micro-pit s o f a roug h surface , becaus e o f surfac e tensio n effects . Th e effectiv e averag e ga p widt h forme d durin g th e subsequen t coolin g will , therefore , b e greate r fo r a roug h surfac e tha n fo r a smoothe r surface . - 13 - T o determin e th e rate-determinin g coefficien t i n th e overal l hea t transfe r process , w e conside r th e hea t transfe r coefficient s i n Equatio n (5 ) a t roo m temperatur e T ( T = 28
8 K) , denote d b y th e subscrip t "o" . h h hco po wo Th e convectiv e hea t transfe r coefficient , h , i s no t temperatur e dependent , an d th e hea t transfe r coefficien t fo r th e bas e plate , h , i s p o onl y weakl y temperatur e dependent . Th e valu e o f h wa s determine d t o b e4 24.2 x 10 W/(m .K) in Section 4.2. Table 3 shows that h ranges from 1469 t o 280
6 W/(m 2 .K) , an d h fro m 12 0 t o 30
0 W/(m 2 .K) , dependin g o n th e therma l conductivit y an d th e surfac e roughnes s o f th e bas e plate . Sinc e h i s muc h smalle r tha n h an d h , th e overal l hea t transfe r coefficien t i s primaril y determine d b y th e hea t transfe r coefficien t o f th e ai r gap , h .co T o reduc e th e interactio n tim e betwee n th e liqui d meta l an d th e container , an d th e fue l claddin g material s durin g th e matrixin g process , th e tota l solidificatio n tim e ha s t o b e reduced . Thi s wil l requir e hig h hea t transfe r throug h th e cas t metal/containe r Interface . A goo d metallurgica l bon d betwee n th e solidifie d materia l an d th e containe r wil l provid e considerabl y bette r hea t transfe r tha n on e wit h a n ai r gap . 6 . CONCLUSION S Th e followin g conclusion s ca n b e draw n fro m thi s work : (1 ) Steady-stat e hea t transfe r experimenta l technique s ca n b e use d t o determin e th e overal l hea t transfe r coefficien t betwee n lea d cas t o n a candidat e containe r materia l an d coolin g water . - 14 - (2 ) Th e developmen t o f a n ai r ga p betwee n th e cas t lea d an d th e containe r materia l appeare d t o contro l th e overal l hea t transfe r o f th e syste m and , hence , th e solidificatio n rat e o f th e matri x material . (3 ) Th e surfac e roughnes s o f th e containe r materia l influence s th e overal l hea t transfe r coefficient , i.e. , a smoothe r surfac e provide s highe r hea t transfe r tha n a roug h surface . (4 ) li. e experiment s showe d a n increas e i n th e hea t transfe r co - efficien t wit h increasin g temperatur e a t th e cas t lead/air - ga p interface . Thi s i s attribute d t o th e bendin g o f th e base-plat e material , an d a theoretica l mode l ha s bee n develope d t o explai n thi s behaviour . (5 ) T o decreas e th e tota l solidificatio n tim e o f a meta l matri x i n used-fue l wast e disposa l containers , goo d chemica l bondin g betwee n th e containe r materia l an d th e cas t meta l i s required .

ACKNOWLEDGEMENT

S Th e author s wis h t o than k F . Hughes , Mechanic s an d Material s

Branch

, Whiteshel l Nuclea r Researc h Establishment , fo r th e measurement s o f base-plat e surfac e characteristic s an d V . Krishnan , M . Schankula , K .

Nuttall

, an d J.L . Crosthwait e fo r reviewin g th e manuscrip t an d fo r valuabl e comments .

REFERENCE

S 1 . D.J . Cameron , "Fue l Isolatio n Researc h fo r th e Canadia n Nuclea r Fue l Wast e Managemen t Program, " Atomi c Energ y o f Canad a Limite d

Report

, AECL-683 4 (1982) . - 15 - 2 . P.M . Mathew , J.S . Nadeau , F . Weinberg , A.CD . Chaklade r an d K .

Nuttall

, "Investmen t o f Irradiate d Reacto r Fue l i n a Meta l

Matrix,

" Canadia n Metallurgica l Quarterl y 22_
, 10 7 (1983) . 3 . P.M . Mathew , F . Weinberg , J.S . Nadea u an d A.CD . Chaklader , "Fille r Metal s fo r Container s Holdin g Irradiate d Fue l Bundles, " Metal s Technolog y 9_ , 37
5 (1982) . 4 . P.M . Mathe w an d P.A . Krueger , "Meta l Matri x Integrit y an d Relate d

Technolog

y Developmen t i n th e Canadia n Nuclea r Fue l Wast e

Managemen

t Program, " _i n Proceedings , 198
3 Annua l Meeting ,

Material

s Researc h Society , Symposiu m o n Nuclea r Fue l Wast e

Management

, Boston , Mass. , 198
3 Novembe r 13-18 , pp . 583-589
. 5 . W.H . Adams , Hea t Transmission , 3r d Edition , McGraw-Hill , Ne w York , 1954
. 6 . J.W . DeVaal , "Definin g Surfac e Parameter s fo r Modellin g Therma l

Contac

t Resistance, " unrestricted , unpublishe d Whiteshel l Nuclea r

Researc

h Establishmen t Report , WNRE-6 4 (1983) . 7 . Internationa l Nicke l Compan y Inc . Ne w York , Propertie s o f Som e Metal s an d Alloys , 196
8 November . 8 . M.M . Yovanovich , "Ne w Contac t an d Ga p Conductanc e Correlation s fo r

Conformin

g Roug h Surfaces, " jlr i Proceedings , 16t h Thermophysic s

Conference

, America n Institut e o f Aeronautic s an d Astronautics , Pal o Alto , California , 198
1 Jun e 23-25
, p . 1164
. - 16 - Typ e Typ e AST

MTHICKNESS a, ROOT MEAN

SLOP E O F SURFAC E HEIGHT S m,

Base-Plat

e

Materia

l 316
L stainles s stee l 30
4 stainles s stee l grade-1 2 titaniu m

Incone

l 62
5OF a (mm ) 6.79 3 6.60 9 6.62 9 6.67

2TABLE

SQUAR E1 O F SURFAC E HEIGHT S AN D TH E MAXIMU M BAS E 1 < 5 1 5 7 2 .3LATES 0 40
2 81
0 13 3 39
70.
0 . 0 . 0 .SURFACE m 28
0 39
7 36
0 19

2a, MEAN

HEIGH T y___. , y.max(Mm) 8.54 3 38.04
9 19.71 0 /.32 2m/a (1/pm ) 0.05 2 0.02 5 0.05 1 0.08 0

PREHEA

T AN D

Base-Plat

e

Materia

l Typ e 316
L stainles s stee l Typ e 30
4 stainles s stee l AST M grade-1 2 titaniu m

Incone

l 62

5TABLE 2

CASTIN

G TEMPERATURE S

Containe

r Sid e Wal l

Temperatur

e (K ) 50
4 44
5 48

6•

45

8Casting

Temperatur

e (K ) 64
8 62
8 62
4 65
9 - 17 - TABL E 3

THERMA

L CONDUCTIVITY , k.., , HEA T TRANSFE R COEFFICIENT , h , AN D LINEA R

EXPANSIO

N COEFFICIENT , g , O F BASE-PLAT E MATERIALS , AN D OVERAL L HEA T TRANSFE R COEFFICIENT , h _j _ A T ROO M TEMPERATUR E

Materia

l Typ e 30
4 stainles s stee l AST M grade-1 2 titaniu m Typ e 316
L stainles s stee l

Incone

l 62
5k2 (W/(m.K) ) 16. 3 18. 6 16. 3 9. 8h P (W/(m 2 .K) ) 246
6 280
6 240
0 146
9a (1/K )

1.73xlO~

5

9.8xl0"

6

1.6xlO~

5

1.28xlO"

5hto (W/(m 2 .K) ) 15 0 26
0 29
0 30
0 - 18 - HEA T FLU X q

TEMPERATUR

EBASE PLATE

BASE-PLAT

E RESISTANC E

AIR-GA

P RESISTANCE S \CONVFXTIVE ''RESISTANC E FIGIX E 1 : Schemati c Representatio n o f Ch e Steady-Stat e Temperatur e

Distributio

n an d th e Resistance s t o Hea t Transfe r i n a Composit e Syste m o f Cas t Lead/Air-Gap/Water-Coole d Bas e Plat e

TO RECORDER

STEE L PIPE •

THERMOCOUPLE

S BAS E PLATE -

SEPARATIN

G

INSULATE

D PLAT E WATE R I

NINSULATION

,152.4mm .

HEATIN

G

ELEMEN

TTO POWER

/

THERMOCOUPL

Ei76.2 mm;

I

STAINLES

S STEE L

CYLINDER

S WATE R I N "WATE R OU T \SUPPORTING LEGS FIGUR E 2 : Schemati c Diagra m o f th e Experimenta l Apparatu s - 20 - 4 0 - 2 0E3- H O i °w s-20 - -4 0 -1 X fl? xi+l1 1 1 1 / W ^

CENTRE-LIN

E 11(

1500 1000 1500 2000 2500PLOTTED PROFILE LENGTH (ym)

FIGUR E 3 : Surfac e Roughnes s Profil e o f Typ e 30
4 Stainles s Stee l Bas e Plat e - 21 -

THERMOCOUPL

E WIRE

STO POWER INSULATING HARDBOARD

RETAININ

G RIN G

INSULATIN

G HARDBO A

INSULATIN

G BACKFIL L

ALUMIN

A CERAM I

HEATIN

G ELEMEN T COPPE R CASIN G STEE L TUBIN G

THERMOCOUPL

E

SUPPORT

Soooooooooooooootooooooooooooooooooooooooooooooooooao0O0O00000OOOD0OOOO•1O00OOOOO0OO Ou ooooooooooo 0 c oooooooooo 0

OOOOOOOOO

O 0 ^ 2. ° - £.°_ £ iLP_2. ° °» P ° ° 9 ° a °

CERAMI

C CEMEN T EA D BAS E PLAT E 25
0 m m FIGUR E 4 : Th e Containe r an d Hea t Sourc e - 22 - BAS E PLAT E I I M I I I i | J jDIRECTION OF HEAT FLOW FIGUR E 5 : Cros s Sectio n o f a Lea d Castin g Unidirectionall y

Solidifie

d an d Etche d - 23 - >6 I FIGUR E380 36
0 - 34
0 - 32
0 - 30
0 2 0 4 0 6 0 8 0

DISTANC

E FRO M BAS E PLAT E (mm ) 6 : Lea d Temperatur e Versu s Distanc e Fro m Typ e 30
4

Stainles

s Stee l Bas e Plat e fo r Variou s Heatin g

Elemen

t Temperatures , T H , an d Wate r Flo w Rates , V w 1w H

O<•18.0 L/min

11. 0 L/mi n 36
0 34
0 32
0 - 30
0 28

0.25.2 L/min!

• 18. 0 L/min j * 11. 0 L/min ! 2 0 4 0 6 0

DISTANC

E FRO M BAS E PLAT E (mm )80 FIGUR E 7 : Lea d Temperatur e Versu s Distanc e Fro m Typ e 316
L Stainles s Stee l Bas e Plat e fo r Variou s Heatin g Elemen t Temperatures , H 'and Water Flow Rates, V - 24 - IH w380 36
0 34
0 32
0 30

0TH=553K

25.
2 L/mi n 18. 0 L/mi n 11. 0 L/mi n 2 0 4 0 6 0

DISTANC

E FRO M BAS E PLAT E (mm )80 FIGUR E 8 : Lea d Temperatur e Versu s Distanc e Fro m Grade-1 2 Titaniu m Bas e Plat e Fo r Variou s Heatin g Elemen t Temperatures , T., , an d Wate r Flo w Rates , V 36
0 34
0

1S 320

H 30
0 T =423

KH J v• 25.2 L/min

o 18. 0 L/mi n a 11. 0 L/mi n 28
0 0 2 0 4 0 6 0

DISTANC

E FRO M BAS E PLAT E (mm )80 FIGUR E 9 : Lea d Temperatur e Versu s Distanc e Fro m Incone l 62
5 Bas e Plat e fo r Variou s Heatin g Elemen t Temperatures , T u , an d Wate r Flo w Rates , Vn - 25 - S 5 MO P SWPi, w o800 60
0 40
0 20

0INCONEL 625

TYP E 30
4 STAINLES S STEE

LTYPE 316L STAINLESS STEEL

GRADE-1

2

TITANIU

M 28
8 30

0320340350

LEAD/AIR-GA

P INTERFAC E TEMPERATUR E T . (K ) FIGUR E 10 : Overal l Hea t Transfe r Coefficien t fo r

Variou

s Base-Plat e Material s - 26 - 48
0 46
0 - 144
0 42
0 - 40
0

020 40DISTANCE FROM BASE PLATE (nun)60

FIGUR E 11 : Steady-Stat e Temperatur e Profil e Use d t o Determin e th e

Convectiv

e Hea t Transfe r Coefficien t - 27 - 80
0 g 60

0MOMPM

W o400 20 0

3MODEL

TYP E 316
L STAINLES S STEE

LINCONEL 625

GRAD E 1 2

TITANIU

M TYP E 30
4 STAINLES S STEE L w o2 4 6 8 10 12

TEMPERATUR

E DIFFERENC E ACROS S BAS E PLAT E A T (K ) FIGUR E 12 : Compariso n Betwee n Measure d an d Calculate d

Overal

l Hea t Transfe r Coefficient s - 28 -

APPENDI

X A.I . A MODE L FO R BASE-PLAT E BENDIN G AN D IT S EFFEC T O N HEA T TRANSFE R Assum e tha t th e containe r bas e plat e ha s thicknes s a an d diamete r { , a t roo m temperatur e T , an d neglec t edg e restrain t du e t o attachmen t t o th e stee l pipe . I f th e to p surfac e temperatur e o f th e plat e i s raise d t o T . (T , > T ) , i t wil l expan d t o diamete r s... Similarly , i f th e botto m sur - fac e temperatur e o f th e plat e i s lowere d t o T , (T , < T ) , i t wil l contrac t t o diamete r j,_ . Therefore , th e bas e plat e wil l ten d t o ben d upward s (se e Figur e A-l) . Le t R represen t th e ar c radiu s o f th e ben t bas e plate . Th e distanc e AX , whic h i s th e base-plat e deflectio n correspondin g t o a tempera - tur e differenc e A T = T^ - T^, fo r a bendin g angl e * , ca n b e calculate d fro m geometrica l considerations : * 2 h " " a( V wher e a i s th e linea r therma l expansio n coefficien t o f th e base-plat e material . Fro m Equation s (A.I ) t o (A.3) , w e obtai n a( l - "( T " T ) ) R = gA T ° 2 (A.4 ) A X = R( l - co s $ ) (A.5 ) Fo r smal l value s o f th e bendin g angl e $ , 2 CO S A a 1 -7 (A.6) - 29 - Fro m Equation s (A.4 ) an d (A.5) , i t wa s estimate d that , fo r th e experimenta l conditions , th e bendin g angl e • wa s les s tha n 0. 1 degrees . Th e widt h o f th e ai r ga p can , therefore , b e take n a s unifor m ove r th e base-plat e area . Fro m Equation s (A.I) , (A.3) , (A.4) , (A.5 ) an d (A.6) , th e maximu m deflectio n a t th e centr e o f th e bas e plate , &X, i s i 2 a ( 1 - a( T - T )) A T . X = J ! 8 a ° 2 (A.7 ) I f th e therma l conductivit y o f ai r a t T i s k , an d i f th e radiativ e hea t transfe r throug h th e ai r ga p i s neglecte d du e t o th e lo w temperature s durin g th e experiment , the n th e widt h o f th e ai r ga p betwee n th e lea d an d th e bas e plate , X , a t T i s give n b y o o (A.8 ) wher e h = hea t transfe r coefficien t fo r th e ai r ga p a t T .co o Sinc e th e widt h o f th e ai r ga p decrease s accordin g t o Equatio n (A.7) , th e fina l air-ga p width , X , an d th e correspondin g hea t transfe r coefficient , h , afte r th e base-plat e bendin g ca n b e calculate d a s follows : X c = X Q - ^ X (A.9 ) k h c = T' c - 30 - Fro m Equation s (A.7 ) t o (A.10) , th e hea t transfe r coefficien t fo r th e ai r ga p i s give n b y (A.11 )h =cka hCOoa(1ka - "< v8aV) AT Sinc e th e conductanc e o f th e bas e plat e an d th e convectiv e hea t transfe r coefficien t remai n unchanged , th e overal l hea t transfe r coeffi - cient , takin g int o accoun t th e effec t o f base-plat e bending , ca n b e writte n a s h = = i (A.12 ) t;"AT( l -»(T o -T 2 ) ) h _ 8 a kto a wher e h i s th e overal l hea t transfe r coefficien t a t temperatur e T •to o Th e valu e o f h wa s determine d b y linea r extrapolatio n o f h . i n Figur e 1 0 t o a lead/air-ga p interfac e temperatur e T ( T = 28
8 K) . A . 2 . DEFINITIO N O F SURFAC E ROUGHNES S PARAMETER S Th e surfac e topograph y o f th e base-plat e materia l i s character - ize d b y th e followin g parameters : (1 ) Mea n slope , m : m i s th e averag e slop e o f th e surfac e profile , expresse d b y N zyi " X i(A.13) wher e y i s th e profil e heigh t fro m th e centre-lin e a t a distanc e x . - 31 - an d N i s th e numbe r o f segment s alon g th e tota l profifl e lengt h L . (2 ) Roo t mea n squar e o f surfac e height , o : o i s define d a s th e squar e roo t o f th e arithmeti c mea n squar e o f th e vertica l distanc e fro m th e centre-line . N (3 ) Maximu m profil e height , y y i s th e maximu m profil e heigh t alon g L . A. 3 ERRO R ESTIMATIO

N(A.14)

T o estimat e th e erro r i n th e overal l hea t transfe r coefficien t h , conside r Equatio n (11) : k , B h t - - (T^ T )i w wher e B = dT/dx . B y partia l differentiatio n o f Equatio n (11 ) an d b y considerin g onl y th e maximu m positiv e error , w e obtai n 6h t a B "< V T ) I T = I + (T.- T ) < A - l t l w wher e * stand s fo r th e partia l derivatives . T o estimat e th e erro r i n h , conside r th e dat a point s belongin g t o th e sam e heatin g elemen t temperature , T^ , i n Figure s 6 t o 9 . Th e maximu m erro r i n B , 5B , fo r th e sam e T, j valu e wa s estimate d fro m th e differenc e betwee n th e maximu m an d minimu m value s o f th e slope s B in th e abov e figures . T . an d B wer e determine d fro m regressio n analysi s o f th e - 32 - dat a points . Th e erro r i n th e temperatur e determinatio n o f T . an d T wa s ± 1 K . Wit h thes e values , th e maximu m erro r fo r al l heatin g elemen t temperature s wa s calculate d usin g Equatio n (A.15 ) an d th e error s ar e give n i n Tabl e A-l . TABL E A- l ERRO R ESTIMATE S FO R OVERAL L HEA T TRANSFE R COEFFICIENT S

Materia

l Typ e 30
4 stainles s stee l Typ e 316
L stainles s stee l AST M grade-1 2 titaniu m

Incone

l 62

5Heating

Elemen

t

Temperatur

e (K ) 42
3 47
3 55
3 42
3 47
3 55
3 42
3 47
3 55
3 42
3 47
3 55

3Max. Error

( ± % ) 4. 6 2. 8 7. 0 6. 6 4. 5 6. 6 4. 8 3. 9 4. 0 8. 0 9. 2 4.

1Absolute Error

( ± W/(nT.K) ) 7 412
2 5 1 8 2 9 1 6 1 3 1 7 3 5 4 4 2 7 - 33 - I T 4 V V 2 FIGUR E A-l : Schemati c Representatio n o f Base-Plat e Bendin g

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