Criticality Safety

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Chapter V

CRITICALITY SAFETY IN

PROCESSING PLANTS

A. Plant Features with Criticality Potential

P r ocessin g pla n ts con ta in a m ult iplicit y of w or k st a t ion s, a n d a r ea s for bot h lon g-t er m a n d

s hor t -t er m s tor a ge. C r it ica lit y s a fet y con sid er a t ion s g o bey on d t he a n a ly sis of ea ch of t h es e

in t er m s of s ub cr it ica l in div id ua l u nit s or s t or a g e a r r a y s. Th e pr og res sion of fis sile m a t er ia l

t hr ou gh a pla n t in volves t ra n sfer s a n d specia l h a ndlin g dur in g w h ich u nusua l con dit ion s

m a y be en cou nt er ed . I t is im por ta n t t ha t t hese oper a tion s be g over ned by pr oced ur es a n d

be ca r r ied ou t b y w e ll-t r a in ed per son n el.

C on sider a pla nt for processin g high ly enr ich ed ura nium a s solids, such a s fa brica t ion of

w ea pon com pon en ts or fuel elem en ts for r ea ct or s.

I t is essent ia l t o a void t he effect of

m a ss ive fis sile u nit s fa llin g t og et h er or en cou nt er in g ot h er u nit s a s t h e r es ult of a n a ccid en t

w it h t ra n sfer eq uipm en t. Min im um spa cin g bet w een un it s ca n be m a in ta in ed by t he use of

b ir d ca g es , pr ov id ed t h er e a r e a p pr opr ia t e pr oced ur es f or loa d in g a n d u nloa d in g t h em .

I n a pla n t for scr a p r ecover y or pr ocess in g ir ra d ia t ed fu el, t he oper a tion s in volvin g fissile

solut ions must be ca refully pla nned. I t is not ew or t hy t ha t a ll cr it ica lit y a ccident s t ha t

h a ve occu rr ed in pr oces sin g pla n t s h a ve in volv ed s olu t ion s. Mis ha ps t h a t h a ve led t o t h es e

a ccid en t s in clu de s olu t ion lea k a g e, pr ecipit a t i on , d is solu t ion of s olid s, in st r um en t f a ilu re,

a n d t r a ns fer a m on g v es sels . Avoid a nce of t h es e m is ha ps ca lls for con t in ued cooper a t ion of

cr it ica l it y s a f et y a n d op er a t i n g p er s on n el.

I n g en er a l, bot h ph ysica l a n d a dm in ist ra t ive cr it ica lit y sa fet y pr a ct ices m ust be t a ilor ed

t o specific pla n t con dit ion s. Th is r eq uir em en t in evit a bly w ill r eq uir e ju dg men t. S pecia l

eva lua t ion a lso m ay be req uired beca use t her e is no “st a nda rd” pla nt for w hich un iversa l

cr it ica l it y s a fet y r ecipes ca n b e d ef in ed .

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B.

Administration

P r ov is ion s of S t a n d a r d AN S I /AN S -8. 19,

A dm i ni st r at i ve Pr act i ces f or Nucl ea r C r i ti ca l i ty

Safeiy,

a re of m ajor sign ifica nce in pr ocessin g pla n ts.

Th is S ta n da rd r ecogn izes t ha t

cr it ica lit y s a fet y r eq uir em en t s m ust con t ribu t e t o t h e ph ysica l a n d econ om ic fu nct ion s of a

pla nt in a ba la nced ma nn er . Accordin gly , it pla ces no req uiremen t on t h e form of pla nt

or ga n iz a t ion . I n st ea d , r eq uir em en t s of t h e S t a n da r d a r e expr es sed in t er ms of m a n a gem en t ,

oper a t ion a l su per visor s, a n d a cr it ica lit y s a fet y s ta f f pr ovid ed by m a n a gem en t .

The S ta n da r d em ph a sizes t ha t effect ive cr it ica lit y con tr ol, like ot her bra n ch es of sa fet y,

req uires t he posit ive support of ma na gemen t a nd implemen t a tion by supemisors w it h

a ssist a nce of t he crit ica lit y sa fet y st a ff.

I t id en t ifies a s socia t e d r es pcm sib ilit ies , ca l ls f or

effect iv e t r a in in g of per son n el a n d con cis e oper a t i ng pr oced ur es , a n d h a s s ect ion s on pr oces s

ev a lu a t ion , m a t er ia l con t r ol, a n d pla n n ed r es pon se t o cr it ica l it y a ccid en t s.

c Training

Th e t r a in in g pr og ra m for per son s in volved in oper a t ion s w it h fissile m a t er ia l sh ou ld m a lie

s a fet y con sid er a t ion s, in clu din g cr it ica l it y s a fet y , a n in t eg ra l pa r t of a pr og ra m t h a t pr ov id es

n eces sa r y job s kil ls . S t a n d a r d AN S 1\ AN S -8.

20, NucZea r Cr i t i ca l i t y Saf et y T ra in i ng,

applies

t o per son n el a s socia t e d w i t h oper a t i on s w h er e t h er e is t h e pot en t ia l for a cr it ica l it y a ccid en t .

P rovision s of t he S ta nda rd a re con sist en t w it h t he pr ecept t ha t sa fet y educa t ion w ill be

m ost mea nin gful a n d r ea dily a ssimila t ed if it is clea rly releva n t t o oper a tion s. I t follow s

t ha t loca l supervision sh ould pa rt icipa t e in crit ica lit y sa fet y t ra in in g, or con duct it w it h

t h e s uppor t of cr it ica l it y s a fet y s pecia l is t s. Appr opr ia t e t r a in in g of s uper vis or y per son n el is

implied.

The S ta nda rd ca lls for t ra in in g in t he recogn it ion of crit ica lit y a la rms a nd t h e proper

respon se t o t hem. Tra in in g sh ould be support ed by discussion of select ed crit ica lit y

a ccid en t s. S t r a t t on )s h is t or y of n uclea r a ccid en t s30

d es cr ibes ea ch in s ufficien t d et a il t o be

h elpfu l for t h is pu rpos e.

Accoun ts of r ea l a cciden t exper ien ce in t ra in in g t a lks ca n h elp

k eep t h e a u dien ce a w a k e.

D. Criticality Alarms and Response

C rit ica lit y a la r ms ha ve t w ice in it ia t ed lifesa vin g eva cua t ion of a r ea s in w hich a ccid en ts

occurred.30

Th e va lue of su ch sy st em s is t h er efor e clea r in a r ea s for pr oces sin g sig nifica n t

a moun ts of fissile ma t er ia l. G uida nce for t he design , in st a lla t ion , a nd ma in t en a nce of

such sy st em s m a y be obt a in ed fr om S t a nd a rd ANS I \ANS -8.3,

Cr i ti ca l i ty A cci den t A l ar m

System.

Th is docum en t dir ect s t ha t a n a cciden t a la rm sy st em must be con sider ed for a ny

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a r ea con ta in in g m or e t ha n a t hr esh old q ua n tit y of fissile m a ter ia l. The S ta n da rd ca lls for

a n ea sily r ecogn ized sign al for im media t e eva cua t ion in ca se of a n a la r m. I t r ecom men ds

t ha t t he respon se of t he a la rm syst em t o ra dia t ion be t est ed a t lea st mon th ly, ea ch sign a l

gen er a t or be t est ed a t lea s t on ce ever y t hr ee m on t hs, a n d a n eva cu a tion d rill be per for med

a t lea s t a n nua lly .

Th e exist en ce of a n a la r m sy st em ca r ries w it h it cer ta in r espon sibilit ies. Th e sy st em m ust

be ma in t a in ed t o provide con fiden ce t ha t it w ill fun ct ion if n eeded, a nd t o min imize t he

fr eq uen cy of fa lse a la rm s. F alse a la rm s ca n h ave a n ega t ive im pa ct on sa fet y by cr ea t in g a

pot en t ia l for in jur y a s a r esu lt of pr ecipit ous r espon se.

F alse a la rms a lso t en d t o d est roy

con fid en ce in t h e s ys t em . U n a n n ou nced d rills a r e n ot en dor sed .

Th e respon se t o a n a la rm is t o be govern ed by a n emergen cy pla n w it h elemen ts given in

t he a d min ist ra t ive S t a nd a rd ANS 1\ ANS -8.19. F ur th er fea t ur es of a n em er gen cy pla n a r e

being considered.

E lem en ts of t he em er gen cy pla n in clud e pr oced ur es for eva cu a tion t o specified a ssem bly

s t a t ion s, a ct ion s a f t er a s sem bly , a n d t r ea t m en t of in ju red a n d expos ed per son s in a ccor d a n ce

w i t h a d va n ce a r r a n g em en t s.

P erson n el must be t ra in ed in t h eir proper respon se t o t h e a la rm in cludin g t h e use of.

eva cu a t ion r ou t es a n d d esig na t ed a s sem bly poin t s. E m er gen cy pla n s m ust be k ept cu rr en t ;

evolut ion of a pla n t ca n in fluen ce t he pr ocedur es t o be follow ed in t he even t of a n a la r m.

E Material Control

On e cr it ica lit y a ccid en t occu rr ed beca u se a con cen t ra t ed fissile solu t ion in a poly et h ylen e

cylin der w a s m ist a ken for a dilut e solut ion .30

Th is occur ren ce em ph a sizes t he va lue of

la b elin g or ot h er pos it iv e id en t ifica t i on of fis sil-em a t er ia l in h elpin g t o a v oid r ou t in g er r or s

w it h in a pla nt .

Also of va lue a re post ed limit s a t w ork st a tion s a nd st ora ge a rea s. I f

obser ved , for exa m ple, in t he t ra n sfer of m a ter ia l a lon g a glove-box lin e, post ed lim it s ca n

pr ev en t in a d ver t en t ov er loa d in g of a b ox.

La belin g a nd post ed lim it s ca n not t a ke t he pla ce of up-t o-da t e pr ocedur es used by w ell-

t r a i n ed per s on n el , b ut s h ou ld m a k e er r or s l es s li kely . C om pu t er iz ed a c cou n t in g p roced u r es ,

such a s pr oposed for sa fegua rds, sh ould con tr ibut e fur t her t o t he reduct ion of t ra nsfer

errors.

P r ovision s for h a ndlin g fissile m a ter ia l dur in g in ven t or ies m ust be a s ca r efully pla n ned a s

for r egu la r pla n t oper a t ion . Th is n eed is em ph a sized by t he t hr ee cr it ica lit y a ccid en ts t h a t

r es ul t ed f rom m is d ir ect i on of s ol ut i on s d ur in g i nv en t or y .30

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An occa sion a l r eq uir emen t t ha t sh ould be a n ticipa t ed is t he emer gen cy st or a ge of fissile

m a t er ia l t ha t ca n a ccum ula t e a s t he r esu lt of in ter rupt ion s of n or ma l oper a t ion s. Mish a ps

such a s fa ult y processin g or eq uipmen t fa ilure ma y in t errupt t he flow of solut ion s, a nd

a ccid en t s or ot h er d isr upt ion s m a y pr even t m a t er ia l fr om lea v in g t h e pla n t .

I n a pla n t la y ou t , t h e con ven ien ce of pr oper oper a t ion s sh ou ld be con sid er ed . To be a v oid ed ,

for exa m ple, is t r a n sfer of m a t er ia l t h rou gh a w or kin g a r ea w h en a n ot h er con ven ien t r ou te is

a v a ila b le, a n d u nn eces sa r y pr oces sin g of d iffer en t fis sion a b le m a t er ia l s in t h e s a m e a r ea . To

illu st ra t e, use of t he sa m e fur na ce for ca st in g en rich ed a n d n a t ur a l ur a nium , except d ur in g

in depen den t ca mpa ign s, could con tr ibut e t o t he con fusion of feed it em s. An exa mple of

m a kin g m ish a ps in con ven ien t is t o t r a n sfer fissile m a t er ia l on a sin gle pla n e, a s w it h specia l

ca r t s . Tr a n sfer b y cr a n e ov er ot h er fis sile m a t er ia l w ou ld be ob ject ion a b le.

F. Process Startup

B efore in it ia l opera t ion of a pla nt , or of a module t h a t is n ew or revised, con firma tion

of t he proper con dit ion of it s compon en t s is ma nda t ory .

C on fir m a t ion in clu des t es t in g

of in st rum en ta t ion , va lves, sea ls, t ra n sfer devices, a nd ven tila t ion a nd fir e-pr ot ect ion

eq u ipm en t . At t h is poin t , a d eq ua cy of t r a in in g sh ou ld be est a b lis hed .26

I t is a lso im por ta n t t o r ea s sess cr it ica lit y sa fet y befor e st a r tup. Th e in it ia l a s sessm en t ca n

be in flu en ced by ev olu t ion a r y ch a n ges d ur in g con st r u ct ion .

E ven t hough t he effect .of ea ch

ch a n ge h a s been con sid er ed , t h e a s -con st r uct ed con fig ur a t ion s hou ld be exa m in ed .

At t h is st age, it is a ppropr ia te t o recon sider ma t t ers of judgmen t a bout t h e a deq ua cy

of t he exper imen ta l ba sis for eva lua t in g t he crit ica lit y sa fet y of opera t ion . J u dgmen t is

in volved in d ecis ion s con cer nin g t h e a ppr opr ia t en ess of d ir ect ly a pplica b le exper im en t s,

of exper imen ts used for va ~ d a tin g ca lcula t ion s, or of a ddit ion a l sa fet y m ar gin s-a pplied

w hen va lida t ion is q uest ion a ble. An y doubt usua lly ca n be r esolved by mea n s of n eut ron -

m u lt i pl ica t i on m ea s u r em en t s a s ou t lin ed in S t a n d a r d AN S 1\ AN i9-8.6~ S a f et y in C o’’~ d u ct i ng

,,,

Subcr i t i ca l Neu t r on -M u l t i p l i ca ti on M easu remen t s i n S i tu .

Th ese meas uremen ts , con ducted

d ur in g s t epw i se in t r od uct ion of fis sile m a t er ia l , w ou ld id en t ify s a fely s ubcr it ica l con d it ion s.

In gen era l, t hey w ould simply pr ovide r ea ssur an ce t ha t n orm al oper a tion is a ccept a ble.

Th ey m ust n ot ca u se per son n el t o r ela x con cer nin g a ccid en t pot en t ia l.

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G. Maintaining Safety Provisions

D u rin g pla n t oper a t ion , con t in uou s ob ser va t i on a n d per iod ic s ur vey s a r e m ea n s of g ua r d in g

a g a in st a d ver se effect s of evolu tion a ry ch a nge in con dit ion s or pr a ct ices. H a s a vess el t h a t

cou ld con t a in m or e t h a n a cr it ica l volu me been br ou gh t in t o a pr oces s..a r ea ? H a s eq uipm en t

for fissile ma t er ia l been used for ot her ma t er ia l?

S hould fea t ur es of fir e pr ot ect ion be

r eview ed beca u se of ch a n ged pla n t con t en t ? H a ve pr eca u t ion s a g a in st t h e con seq uen ces of

n a tur a l d isa st er s such a s ea r th qua ke, flood , or t or na d o been r ela xed over t im e? Th e list of

q uest ion s d oes n ot st op h er e. I n fa ct , it d epen ds on d et a iled pla n t fea t u res , r egu la t ion s, a n d

t he policy of pla n t m a na gem en t. Th us, t he w ish for a un iver sa l ch eck list w ould be fu tile.

H. Examples of Plant Application

1.

Dissolver for Water-Reactor Fuel

Th e sa fe geomet ry of a 100-lit er dissolver for ch opped U (3.2)OZ fuel elemen ts is t o be

explor ed . Th e sh a pe of t he d issolver sh ould be sim ple a n d it is t o be sur roun ded by a st ea m

ja cket . Full w a t er reflect ion sh ould be a ssumed t o a llow for w a t er in t he st ea m ja cket a nd

for incidenta l reflect ion.

Ta ble 9 sh ow s a limit in g va lue of 26.4 cm for t he subcr it ica l dia met er of a lon g cy lin der of

heterogeneous oxide.86

Th is v a lu e i s es sen t ia l ly t h e in sid e d ia m e t er of 10-in ch S ch ed ule-5S

pipe. The dia met er limit for solut ion is significa nt ly gr ea t er . B eca use a cylin der of t his

dia met er h as a ca pa cit y of 55 lit er s per met er of len gt h, t he heigh t of a 100-lit er dissolver

w ould be a bout 1.8 m. A design st udy w ill sh ow w het h er t h is h eight meet s fun ct ion a l

requirements.

S hould t his lon g, sma ll dia met er prove t o be un desira ble, a n a lt ern a t ive w ould be a n

a n n ula r t a n k su rr ou nd in g a n eu t ron -a b sor bin g m a t er ia l t o r ed uce n eu t ron exch a n ge w it h in

t he con figura t ion . I f t h e a bsorbing ma ter ia l is w a t er a nd t h e in side dia met er is a t lea st

30 cm , t he a n nula r t hickn ess ca n be a ppr oxim a ted by a r eflect ed in fin it e sla b specified in

Ta ble 9 t o be 12.6-cm t hick. I f a ddit ion a l con serva t ism is desired, a t hickn ess of 10 cm

a nd a n in side dia met er of 40 cm ma y be a ssumed for t h e design st udy, t he ca pa cit y of

w h ich is a b ou t 157 lit er s per m et er . Accor din gly , a ves sel of 100-lit er ca pa cit y w ou ld h a ve

n ea r -eq u ila t e ra l ext er n a l d im en sion s. B e for e a d opt ion , t h e a ccept a b ilit y of t h e fin a l d es ig n

sh ou ld be con fir med eit h er by a va lid a t ed ca lcula t ion or by in s it u n eu tr on -m ult iplica t ion

meaimrements.2g

Of cou rs e, t h is d iss olver en com pa s ses m or e t h a n t h e sim ple con t a in er . I n t h e fir st pla ce, t o

a ccom mod a t e ir ra d ia t ed fu el, it m ust be on e com pon en t of a sh ield ed fu el-h a nd lin g sy st em .

The con ta in er must be modified for in troduct ion of t he ch opped fuel, dra in ing solut ion ,

a n d w it h dr a w a l of r esid ua l solid s. S pa r gin g t o fa cilit a t e u nifor m d issolu tion a lso m a y pr ove

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d es ir a ble. Th e u lt im a t e cr it ica lit y s a fet y eva lu a t ion m us t t a ke in t o a ccou nt a u xilia r ies a n d

in t er a ct ion w i t h ot h er com pon en t s.

F ur th er , t her e m a y be specia l r eq uir em en ts for ca m pa ign in g fuels fr om d iffer en t sou rces,

for in st a nce, t h e fuel up t o 4 w t%

ZS 5Uin t ile follow in g exa m ple of pla n t a pplica t ion . I f

t he possibilit y of h a ndlin g fuel a t somew ha t more t ha n 3.2 w t% 235U ca n be foreseen , it

sh ould be more effect ive t o pla n for it a t t his st age t ha n t o a da pt t o it la t er . Act ua lly , t h e

“con ser va t ive” a n nula r t hickn ess of 10 cm m a y pr ove t o be suit a ble for fuel en rich men ts of

n ea r ly 5 w t l o235U .

..-

2. Storage of Low-Enriched Uranium Solution

C on sid er v es sels for s t or in g a v a r iet y of u ra n iu m s olu t ion s in w h ich t h e 235Uen r ich m en t w i ll

n ot exceed 4 w t ?lo a nd t he ura nium den sit y w ill r ema in below 750 g/L. A t ot a l ca pa cit y of

1890 lit er s (500 g a l) is d es ir ed , a n d , beca u se of t h e pos sibilit y of lon g-t er m st or a ge a n d t h e

d ifficu lt y of in ter na l in spect ion , a sin gle vessel pa cked w it h R a sch ig r in gs is n ot select ed .

Th e pr efer r ed a r r a n g em en t is a pla n a r ba n k of cy lin d er s n ea r a 12-m -lon g, 5-m -h ig h con cr et e

w a ll, w it h a n a rrow w a lkw a y bet w een t he cylin der s a nd w a ll.

Accor din g t o Ta ble 8, t he subcr it ica l lim it on cy lin der d ia m et er for U (4) solu tion is 30 cm ;

t he n ext sm a ller com mer cia l pipe size is 10-in ch S ch ed ule-5S (26.6-cm -i.d .). At a usa ble

h eigh t of 4.6 m, t he ca pa cit y per cylin der is 250 lit ers a nd 8 cylin ders w ould be req uired.

C on st ruct ion a n d oper a t ion a l con ven ien ce w ould be m et by a on e-m et er cen t er spa cin g of

cy lin ders a nd w ould result in a ddit ion a l spa ce a t t he en ds of t he ba nk of cy lin der s.

A w a lkw a y of 0.7 m sepa r a t es t he cy lin der s fr om t he con cr et e w a ll a n d r ed uces t he effect of

t he w a ll t o t ha t of in ciden ta l r eflect ion on ea ch vessel. B eca use t he 30-cm d ia m et er limit

is ba sed on full w a t er r eflect ion , w hich is much mor e effect ive t ha n in ciden ta l r eflect ion ,

it is n ecessa ry t o sh ow t ha t t he effect of in t era ct ion a mon g t h e cylin ders is a ccept able.

Accor din g t o v a lid a t ed K E N O ca l cu la t i on s,147

k.ff = 0.725 for a sin gle cy lin der h a vin g on ly

2.5-cm -t h ick w a t e r r eflect ion , a n d k,ff = 0.785 for t h e lin ea r a r r a y spa ced fr om t h e con cr et e

w a ll, sh ow in g t h a t in ter a ct ion is a d eq ua t ely sm a ll. Th us, it is a ppr opr ia t e t o pr oceed w it h

t h e d esig n of t h is a r r a ng em en t a n d w it h d et a iled explor a t ion of con t in gen cies:

The low va lues of k,ff suggest t he rea son a blen ess of fur t her in vest iga t ion of a st ora ge

ba n k w it h sig nifica n t ly in cr ea s ed ca pa cit y . F or exa m ple, a on e-d im en sion a l ca lcu la t ion of

a 12-in ch S ch edule-5S pipe (31.5-cm-i. d .) in st ea d of t he 26.6-cm pipe r esult ed in a keff

of 0.9. Th e ca pa cit y of 8 cylin ders a t t he 4.6 m h eigh t w ould be in crea sed t o 750 ga llon s.

Of cour se, a ca r efu l com put a t ion a l st ud y a n d a n a ly sis of con tin gen cies w ould be r eq uir ed

b efor e a d opt in g t h is a p pr oa ch .

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3. Solution in Tanks Packed with Boron-Containing Raschig

Rings

In cer t a in ca ses, a s not ed before, a n a lt erna t ive t o geomet rica lly subcrit ica l t anks for

solu tion st or a ge is t he use of la r ge ca pa cit y t a nks pa cked w it h bor osilica t e-g la ss R a sch ig

r in gs. Ty pica lly , a lt hou gh on e-q ua r ter t o on e-t hir d of t he t a nk volum e is sa cr ificed t o t he

g la s s a b sor ber , t he t a n k m a y s till a ccom mod a t e la r ge volu mes of s olu tion m or e efficien t ly

t h a n lon g, lim it ed -d ia m et er cy lin der s or t hin s la b-like con t a in er s. Ot h er t ha n for pr im a ry

cr it ica l it y con t r ol, R a sch ig r in gs in a u xilia r y t a n k s m a y pr ot ect a g a in st a ccid en t a l cr it ica l it y

r es ult in g f rom in a d ver t en t d iv er sion of f is sile s olu t ion t o t h os e t a n k s.

Amer ican Na t iona l S tandard Use of Borosi l i ca ie-G lass Rasch ig R ings as a Neu t ron Absorber

in Solu t ions of F issiZeM ater ia l , ANSI \ ANS-8 .5 , defines appropria t e condi t ions for cr i t ica l i ty

cont rol. Rest rict ions exclude t he use of a lka line solut ions, H F, a nd hot , concent ra ted

H 3P O q . Tem per a t ur e a n d r a dia t ion field s a ls o a r e lim it ed . Th e S t a n da r d d efin es ch em ica l

a nd physica l pr oper ties t ha t a re t ypified by P y rex t ype 7740 a nd K im ba l t ype K G -33 a nd

lim it s t he r in g s ize t o 3.81-cm -o.d . I t s pecifies pa ckin g con dit ion s a n d g ives r eq uir em en t s

f or in spect ion a n d m a in t en a n ce. F in a lly , m a xim um d en sit ies of f is sile m a t er ia l in v es sels of

u nlim it ed s iz e a r e s pecified f or t h ree d if fer en t volu me per cen t a g es of g la s s. Ty pica l ly , a s t h e

g la s s volu me fr a ct ion r a ng es fr om 0.24 t o 0.32~ d en sit y lim it s r a ng e fr om 150 t o 200 g /L for

233U ,fr om 270 t o 4’00 g /L for 235U -en rich ed ur an iu m, fr om 115 t o 180 g P u /L for 239P u ,

a nd from 140 t o 220 g P u /L for plut on ium con ta ining m ore t ha n 5 w t ~ o 240P u.

Alt hough it is unlikely t ha t t hese rea son ably gen erous limit s w ould rest rict a pra ct ica l

pr oces s, t h er e cou ld be u nu su a l cir cu ms ta n ces t h a t w ou ld r eq uir e g rea t er g la s s fr a ct ion s.

B eca use com put a tiona l models ca nn ot closely a pproxima t e ra ndomly pa cked Ra sch ig

rings,

148 t he preferred guida nce for increa sed limit s w ould be exper iment a l da ta nea r

t he d es ir ed con dit ion s or com pu ta t ion a l r es ult s ver ified by

i n si tu

neutron mult ipl ica t ion

measuremerits.29

An exa m ple of a n exper im en ta l sy st em t ha t is su bcr it ica l a t a plut on ium

densit y grea ter t ha n t ha t permit ted by t he S ta nda rd is repor t ed by Lloyd, B ierma n, a nd

Clayton.137

Th e su bcr it ica l d en sit y of plu ton iu m (8.3 w t % 240P u ) in n it r a te s olu tion w a s

391 g/L w hen a 61-cm-dia met er t a nk w a s filled t o a dept h of 99.1 cm. Ra schig r ings

cont a ining 4.0 w t~ o boron occupied 18.870 of t he volume, a nd t here w a s a n effect ively

in fin it e w a t er r eflect or on t he t a nk w a lls a n d ba se.

Nurrni149

r epor t s t he u se of bor os ilica t e-g la s s r in gs w it h en rich ed u ra n iu m s olu tion s t h a t

h a v e f ree f lu or id e-ion con t en t s g r ea t l y e xceed in g t h e lim it s pecif ied in t h e S t a n d a r d . B e ca u s e

of t his devia t ion , t here is da ily visua l inspect ion a nd semia nnua l empt ying of t anks for

d et a iled exa m in a t ion . Th is is a m or e s tr in gen t m a in t en a nce sch ed ule t ha n t h a t r eq uir ed by

t he S t a n da r d.

Anot her a pproa ch t o environm en ts t ha t a re host ile t o borosilica t e gla ss is suggest ed by

exper im en ts a t B a t t elle P a cific Nor th w est L a bor a tor ies137 w i th plut on ium solut ion s in

a t a nk pa cked w it h st a in less st eel Ra schig r ings cont a in ing 1.0 w tYo boron.

A 45.7-

cm -dia met er t a nk, w a t er reflect ed on sides a nd bot tom , w a s pa cked w it h 1.27-cm-o. d.,

103



1.Z7-cm -lon g s t eel r in gs occu py in g 27.070 of t h e volu me. At a d ept h of 99.1 cm , plu ton iu m

(8.3 w t % XOP U )solut ion s a t den sit ies of 275 g P u /L w it h 480 g No3/L a nd of 412 g P u/L

w it h 602 g N 03/L w er e s ubcr it ica l.

A fur t her exa mple in cludes da ta on plut on ium-ura nium n it ra te mixt ures in a 61-cm-

d ia m et er t a n k, w a t er r eflect ed on t he sid es a n d bot t om a n d pa cked w it h gla ss R a sch ig r in gs

con t a in in g 4 w t ~ o bor on .137–138 Th e r in gs , w h ich w e re 3. 81-cm -o.d . a n d 4.32 cm in len gt h ,

d is pl a ced 18. 8?l oof t h e s olu t ion v olu m e.

At a dept h of 90.4 cm, solut ion a t a den sit y of

180 g U /L (0.66 w t%

ZS SUin u) a nd 78.4 g P u/L (5.7 w t% 240P u in P u) con t a in in g 377 g

N 03/L w a s s ub cr it i ca l .

4. Solution Holdup Design

A cell in a U (93.2) r epr ocessin g fa cilit y h as a con cr et e floor a rea of 9 m2 a nd a n aly ses ha ve

s how n t h a t t h e n eu tr on in t er a ct ion bet w een t h e pr oces s ves sels a n d bet w een t h e ves sels a n d

t he floor is n egligible. Th e floor w it h sid ew a lls w ill ser ve a s a ca t ch ba sin for solu tion s t h a t

ma y lea k from t he vessels. An over flow lin e is t o be in st alled in t he floor , dra in in g t o a

poison ed ca t ch t a n k, t her eby lim it in g t he t h ickn ess of solut ion . Th e m a xim um expect ed

235u d en sit y in U OZ(N 03)2 is 25o

g/L A permitted

solut ion h eigh t over t he floor is t o

be det er min ed . The con figur a tion of t he solut ion is con ser va t ively a ppr oxima t ed by a n

efl?ect ivelyin fin it e un ifor m sla b w it h a t hick con cr et e r eflect or on on e sid e a n d in cid en ta l

r eflect ion on t h e ot h er s id e.

F r om Ta b le 1, t h e s pecified su bcr it ica l t h ickn es s of a n in fin it e sla b of U 02(N 03)2 r eflect ed

by 30-cm-t hick w a ter is 4.9 cm. A t hick w a ter reflect or on bot h surfa ces is expect ed t o

be mor e effect ive t ha n con cr et e r eflect ion on on e a nd in ciden ta l r eflect ion on t he ot her .

I t follow s t ha t t he specified h eigh t of t he over flow pipe sh ould n ot exceed 4.9 cm. Th e

ch osen h eigh t sh ould be m ea s ur ed fr om t he low est por tion of t he floor a s est a blish ed by a n

elevat ion survey.

104



APPENDIX

Th is Appen dix pr ovides a descript ion of t he ca lcula t iona l

st udy lea ding t o t he curves

pr esen ted in F igur es 2 t hrough 13 of t his docum ent . The m ot iva t ion for t his st udy w a s t o

pr ov id e q u a n tit a t iv e ex a m ples illu st r a t in g t h e r ela t i on sh ip b et w e en s ys t em r ea ct iv it y (k ,f i)

a n d s ys t em g eom et r y. I nfer en t ia l in t h es e cu rv es a r e t h e pa r t ia l d er iv a t iv es of g eom et r ica l

size ( ) versus k.ff. F igures from repor t LA-108t50 -M 5 ’, C r i t i ca l D im ensions of Systems

contai ni ng ’235u, 23~Pu,U -r i d233””u,g86 Revision;”w ere a da pt ed t o provide a ba sis for t he

illust ra t ion . Th is a da pt a tion a ppea rs direct ly in F igures 2 t hrough 13. The a da pt a tion

b rin gs f or w a r d r es ult s f rom L A-1086o-M S f or ex per im en t a lly d et er m in ed cr it ica l s y st em s.

Th ese d a ta pr ovid e a r efer en ce t o in ter pr et t he cu rves. Th e t hr ee w ell-est a blish ed fissile

Duclides ZSSU ,~ s s~ , a n d Zs gpu w e re s elect ed f or t h e co-n st r uct ion of t h e ex a m -pies . Th e 235U

w a s t a ken t o be present a s U (93.2). S yst em composit ions w ere t a ken t o be met a l-w a ter

m ixt ur es a n d w er e select ed t o sy st em a tica lly spa n t he en tir e r an ge fr om lim it in g cr it ica l

fissile densit y (7 t o ’13 gra ms per lit er in w a ter) t o pure met a l densit y (a pproxima tely

20 ~ilograms per lit er ). F or t h es e s ys t em s , t h e n eu t ron s pect r um v a r ies s ys t em a t ica l ly f rom

a t h er m a liz ed d is t rib ut ion f or d ilu t e fis sile d en sit ies t o a s lig ht ly s of t en ed f is sion s pect r um

for t he pure met a l syst ems. Three syst em geomet ries w ere select ed t o complet e t he set

of exa m ples: sph er ica l, in finit e circula r cylin der , a nd infin it e pla na r sla b. In ea ch ca se,

t h e f is sile-b ea r in g r eg ion is s ur rou nd ed b y a t ig ht -f it t in g pu r e. w a t e r r ef lect or of ef fect iv ely

in fin it e t h ickn es s. Th ese a r e cla s sic g eom et r ies w h ich occu r r epea t ed ly in t h e lit er a t ur e of

cr it ica lit y s a fet y . Th e fir st d ocu men t ed occu rr en ce of t h es e g eom et r ies a n d t h e a s socia t ed

ch a ra ct er ist ic cu rves, kn ow n t o t he ed it or s, is foun d in t he r epor t 2~-400,Chain React ion

of Pure F issionable”M at ;r i a l s in Solu t ion .150

Th e m et a l -w a t e r s ys t em s u sed in t h e ex a m ples h a ve n o d ir ect exper im en t a l a n a l og . U r a n iu m

met a l a nd plut onium met a l a re not , in a chemica l sen se, soluble in w a ter . H ow ever , t he

m et a l -w a t e r m ix tu r es a r e n eu t ron ica l ly a p pr oa ch ed in a n a s y mpt ot ic s en se f or d ilu t e f is sile

sy st em s. 111su ch sy st em s t he a t om ic r a tio of t he h yd rogen t o t he fissile a t om ic species is

ver y h igh (a bove 1000). I n t hese s ys tem s, t he ot her n uclea r species n eed ed for a ch em ica l

s olu t ion , s uch a s n it r og en a n d flu or in e, a r e a ls o ver y d ilu te a n d h a ve a m in im um per tu rbin g

effect . H ence, t hese dilut e syst ems a pproa ch t he idea lized met a l-w a ter mixt ure. Over

t he r em a in in g r a nge, h ow ever , t he ch em ica l con st it uen ts , such a s n it rog en a n d flu or in e,

r epr esen t a ser ious pert ur ba t ion fr om t he idea lized m et a l-w a t er m ixt ur e.

H ence, a ny

com pa r is on b et w e en ca l cu la t ion a l a n d ex per im en t a l r es ult s r eq u ir es a ca r ef ul a n d a ccu ra t e

d et er min a t ion of t h e im pa ct of t he pr es en ce of t h es e ot h er n uclea r s pecies .

C a ut ion should be exercised in t he a pplica tion of t he curves presen t ed in Figures 2

t hrough 13. F irst , t he rea der should recognize t ha t t he curves represen t ca lcula t iona l

results.

S econd, t he rea der should not e t ha t t hese ca lcula tions do not conform t o

current va lida tion a nd verifica t ion crit er ia . No a t t empt ha s been ma de t o document a

r igor ous com plia nce w it h such crit er ia . Tha t is , soft w a re a nd pla t for m ver ifica t ion a nd

t he com pa rison of ca lcula t ion al r esult s w it h exper im en ta l r esult s ha ve n ot been ca rr ied

out a s descr ibed in C ha pt er I , S ect ion B -4 of t his docum en t, The Role of Calculational -

105



Validation. Instead,wecomplywiththetraditionalcriteriafor reportingscientificresults

by providingsufficientdetail to allowfor independentreproducibilityandconfirmationof

results.

Thevalueofk,fiwascalculatedoraspecificnuclidetype,density,andsystemdimensionr .

Th e d im en sion x cor respon ds t o a sph er ica l d ia m et er , a n in fin it e cy lin der d ia m et er , or a n

in fin it e sla b t hickn ess. F or ea ch n uclide t ype, d en sit y, a n d sy st em geom et ry , four t o five

va lues of r w ere select ed w hich result ed in ca lcula ted k.ff’s in t he ra nge 0.5 t o -1.2. In

a d dit ion t he va lue of km for a n in fin it e m et a l-w a t er m ixt ur e w a s ca lcu la t ed . To d et er min e

t he va lue of x for a pa r ticula r va lue of k,ff, t he a ppr opr ia t e set of ca lcula t ion a l r esu lt s w er e

fit t ed t o a con t in uou s cu rve h a vin g t h e follow in g a lg ebr a ic for m.

In t h e a bove expression a , ~ , a n d y a re fit t in g pa ra met ers. Th is form provides a

m on ot on ica lly in crea sin g k,ff ver sus r w hich a sy mpt ot ica lly a ppr oa ch es km for la rge x

Th e curves sh ow n in F igures 2 t h rough 13 w ere gen era ted by fit t in g a splin e t h rough

t he ca lcu la t ed va lu es of x for ea ch select ed fissile d en sit y. Th e ca lcula t ion a l r esu lt s w er e

produced using t h e MC NP Mon t e C a rlo code (see Ref. 13). The cross-sect ion s w ere

ba s ed on E N DF /B -V cr os s-sect ion eva lu a t ion s pr ovid ed by t h e XTM g rou p a t L os Ala m os.

S pecifica l ly , t h e M C NP n ucli.d e id en t ifier s (ZAI D S ) s how n in Ta b le 16 w er e u sed .

Ta ble 16

Nuclides, C ross-S ect ion E va lua t ion s, a nd At omic

Weigh t s U sed for C a lcula tion a l Result s

106

Nuclide

IH

160

233

u

235u

238u

239pu

Avogac

ZAID

1001.5OC

8016.50c

92233.50c

92235.50c

92238.50c

9~ 239.55c

)’s n umber

(a t oms /b-cm)

Atomic Weight

1.00782475

15.994914,80

233.03962900

235.04392497

238.05078549

23 3.05215781

0.602204345



The lw t r .Olt version of t he S (a ,fl) sca t t er ing model w a s used for t he w a t er in t he

m et a l-w a t er m ixt ur e a n d for t he w a t er in t he r eflect or .

Ta ble 17 g ives va lu es of t he m a ss d en sit ies a s su med for w a t er a n d for t he m et a l s ta t e of ea ch

n uclid e. Ta ble 18 g ives t h e n um ber d en sit ies of h yd rog en a n d oxy gen u sed for t he 15.2-cm

w a t er r eflect or . Ta b les 19 t hr ou gh 21 g ive t h e n um ber d en sit ies ca lcu la t ed for 22 s elect ed

f is sile m a s s d en s it i es f or t h e t h r ee f is sile n u clid es ZSSU ,ZSSU ,a nd 239P U .Fi na ll y, Ta bl es 22

t h rou gh 24 lis t t h e fin a l ca lcu la t ed g eom et r ica l d im en sion s (x va lu es) u sed t o pr od uce t h e

cu rves s how n in F ig ur es 2 t h rou gh 13.

Table

17

Mass Densities Assumed for

Water and Fissile Metal

M a ss D en sit y

Materia l

(g/cm 3)

Water

0.997801

233UMetal

18.05

235UMetal

(93.2 Wt%23’U )

18.76

II

239PuMetal

I

19.74

Table

18

Calculated Number Densities for

the 15.2 cm Water Reflector

NumberDensity

Nuclide

atoms/b-cm

IH

0.066725294

160

0.033362647

.

1 7



108

Table

19

Fissile Mass Densities and Calculated Number

Densities for 233UMetal-Water Mixtures

233u

Mas s

Densi ty

(kg/L)

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.020

0.030

0.050

0.100

0.200

0.500

1.000

2.000

5.000

10.000

14.000

18.050

N um ber D en sit v (a t om s

233u

.000012921

.000015505

.000018089

.000020673

.000023257

.000025841

.000028425

.000031010

.000033594

.000036178

.000051683

.000077524

.000129206

.000258413

.000516826

.001292064

.002584128

.005168257

.012920642

.025841285

.036177799

.046643517

“\

1~

.066706810

.066703114

.066699417

.066695720

.066692024

.066688327

.066684630

.066680934

.066677237

.066673540

.066651360

.066614393

.066540459

.066355625

.065985955

.064876948

.063028602

.059331909

.04.8241833

.029758372

.014971603

.000000004

160

.033353405

.033351557

.033349709

.033347860

.033346012

.033344163

.033342315

.033340467

.033338618

.033336770

.033325680

.033307197

.033270230

.033177812

.032992978

.032438474

.031514301

.029665955

.024120916

.014879186

.007485802

.000000002



Table 20

I?issileMass Densities and Calculated Number

Densities for

U(93.2)

Metal-Water Mixtures

235u

Mas s

Number Densi ty (at oms/barn -cm)

Densi ty

(l{g /L)

235

u

238

u

IH

160

0.005

.000012810

.000000923

.066706212

.033353106

0.006

.000015373

.000001107

.066702396

.033351198

0.007

.000017935

.000001292

.066698580

.033349290

0.008

.000020497

.000001477

.066694764

.033347382

0.009

.000023059

.000001661

.066690947

.033345474

0.010

.000025621

.000001846

.066687131

.033343565

0.011

.000028183

.000002030

.066683315

.033341657

0.012

.000030745

.000002215

.066679498. .03.3339749

0.013

.000033307

.000002399

.066675682

.033337841

0.014 .000035869 .000002584 .066671866 .033335933

0.020

.000051242’

.000003691

.066648968

.033324484

0.030

.000076863

.000005537 .066610805

.033305403

0.050

.000128105

.000009229 .066534479

.033267240

0.100

.000256209

.000018457 .066343665

.033171832

0.200

.000512419

.000036915 .065962035

.032981018

0.500

.001281046

.000092286

.064817147

.032408574

1.000

.002562093

.000184573

.062909001

.031454500

2.000

.005124186

.000369145 .059092707

.029546354

5.000

.012810464

.000922863 .047643827

.023821914

10.000. .025620928 .001845726 .028562361 .014281180

14.000

.035869299’

.002584017

.013297187

.006648594

17.484

.044796448 .003227126

.000000004

.000000002

109



110

Table 21

Fissile Mass Densities and Calculated Number

Densities for 239PUMetal-Water Mixtures

239pu

h fas s

Densi ty

(kg/L)

0.00.5

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.020

0.030

0.050

0.100

().2()()

0.500

1.000

2.000

5.000

10.000

14.000

19.740

N um ber D en sit v (a t o ms

239pu

.000012596

.000015115

.000017634

.000020153

.000022672

.0000251S1

.000027710

.000030230

.000032749

.000035268

.000050383

.000075574

.000125957

.000251913

.000503827

.001259567

.002519134

.005038267

.012595668

.025191337

.035267872

.049727697

U

IH

.066708393

.066705013

.066701632

.066698252

.066694872

.066691492

.066688112

.066684731

.066681351

.066677971

.066657690

.066623888

.066556283

.066387273

.066049252

.065035190

.063345086

.059964879

.049824257

.032923220

.019402390

.000000003

160

.033354196

.033352506

.033350816

.033349126

.033347436

.033345746

.033344056

.033342366

.033340676

.033338985

.033328845

.033311944

.033278142

.033193637

.033024626

.032517.595

.031672543

.029982440

.024912128

.016461610

.009701195

.000000002



233

u

Mas s

Densi ty

(kg/L)

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.020

Iiefl

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0,8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

Table 22

Calculated Dimensions for

233u Metal water Mixtures

Sphere

Diameter

(cm)

—

—

—

—

—

—

—

317.12

—

—

105.78

—

—

76.12

179.13

62.29

104.73

54.16

80.25

177.95

48.55

67.26

112.46

44.71

59.44

88.23

32.93

39.88

49.50

Infinite

Cylinder

Diameter

(cm)

—

—

—

—

—

238.93

—

78.38

—

55.63

135.31—

45.22

76.90

38.85

58.24

135.68

34.78

48.71

83.69

31.68

42.72

64.94

22.72

27.93

35.19

Infinite

Slab

Thickness

(cm)

—

—

—

—

—

—

—

149.53

—

47.43

—

32.51”

84.71

—

25.58

46.12

21.67

34.28

84.06

18.92

28.02

50.54

16.96

24.04

38.59

11.20

14.47

19.10

233u

Mas s

Densi ty

(kg/L)

0.030

0.050

0.100

0.200

0.500

1.000

2.000

5.000

10.000

14.000

18.050

k,ff

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

0.8

0.9

1.0

Sphere

Diameter

(cm)

26.10

30.40

35.65

21.26

24.30

27.85

17.76

20.08

22.71

15.74

17.80

20.09

14.20

16.11

18.19

13.27

15.11

17.12

12.34

14.07

15.99

10.75

12.32

14.04

9.04

10.40

11.88

7.94

9.19

10.52

7.00

8.11

9.31

Infinite

Cylinder

Diameter

(cm)

17.50

20.72

24.74

13.82

16.09

18.72

11.07

12.79

14.75

9.50

11.00

12.69

8.25

9.63

11.18

7.53

8.87

10.35

6.85

8.11

9.50

5.77

6.89

8.12

4.71

5.66

6.71

4.08

4.91

5.83

3.58

4.31

5.11

Infinite

Slab

Thickness

(cm)

7.84

9.87

12.36

5.46

6.84

8.45

3.60

4.64

5.83

2.50

3.38

4.38

1.60

2.37

3.27

1.14

1.84

2.68

0.79

1.39

2.15

0.43

0.87

1.47

0.25

0.54

0.97

0.18

0.41

0.75

0.14

0.33

0.60

111



5u

Mas s

)ensity

‘kg /L )-

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.020

112

Table 23

Calculated Dimensions for U 93.2 Metal-Water Mixtures

kefl

8

9

1.0

8

9

1.0

8

9

1

8

9

1

8

9

1.0

8

9

1

8

9

1

8

9

1.0

8

9

1

8

9

1

8

9

1

Sphere

Diameter

(cm)

—

—

—.

——

—

—

—

130.68

—

87.11

1273.95

69.41

137.49

59.51

95.99

—

53.16

77.93

173.45

48.64

67.72

115.37

35.51

43.94

56.68

Infinite

Cylinder

Diameter

(cIn)

—

—

—

—

—

—

—

98.23

63.90

833.96

—

50.49

103.27

—

42.83

70.72

38.10

57.02

129.51

34.63

49.04

85.54

24.62

31.03

40.69

Infinite

S l a b

I’hickness

(cm)

—

—

—

—

—.

—

—

—

—

—

60.42

—

37.71

562.43

—

29.03

62.92

24.20

42.25

—

21.03

33.35

80.40

18.77

28.09

52.14

12.37

16.47

22.63

5u

Mass

Density

(kg/L )

0.030

0.050

0.100

0.200

0.500

1.000

2.000

5.000

10.000

14.000

17.484

k,ff

8

9

1.0

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1.0

0.8

9

1

8

9

1.0

8

9

1

8

9

1

8

9

1

Sphere

Diameter

(cm)

28.17

33.37

40.18

23.20

26.85

31.22

19.60

22.51

25.82

17.78

20.36

23.32

16.63

19.15

22.06

16.20

18.81

21.76

15.72

18.41

21.53

14.30

16.78

19.64

12.31

14.37

16.68

11.08

12.92

14.97

10.08

11.76

13.62

Infinite

Cylinder

Diameter

(cm)

19.01

22.94

27.98

15.20

17.96

21.28

12.44

14.56

17.02

10.96

12.88

15.06

9.99

11.83

13.98

9.55

11.47

13.72

9.17

11.14

13.41

8.15

9.98

12.10

6.90

8.39

10.09

6.12.

7.44

8.92

5.52.

6.72

8.05

Infinite

S l a b

Thickness

(cm)

8.77

11.20

14.42

6.28

7,96

9.99

4.37

5.68

7.19

3.30

4.45

5.79

2.47

3.58

4.88

2.06

3.19

4.52

1.71

2.83

4.19

1.24

2.22

3.45

0.85

1.60

2.56

0.67

1.28

2.07

0.56

1.09

1.77



9pu

Ma ss

Densi ty

(kg /L)

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013

0.014

0.020

Table 24

Calculated Dimensions for

ZagpuMetal.Water Mixtures

k,ff

8

9

1

8

9

1

8

9

1

8

9

1.0

8

9

1

8

9

1

8

9”

1

8

9

1

8

9

1

8

9

1

8

9

1

Sphere

Diameter

(cm)

235.44

84.59

344.79

61.59

100.75

50.95

72.49

135.53

44.84

59.55

89.85

40.65

52.20

72.23

37.65

47.26

62.44

35.38.

43.69

56.03

33.54

40.96

51.55

32.06

38.81

48.15

26.90

31.67

37.68

Infinite

Cylinder

Diameter

(cm)

178.13

—

—

61.77

264.35

—

44.46

74.83

36.39

52.66

101.00

31.70

42.95

66.19

28.53

37.36

52.75

26.24

33.55

45.03

24.51

30.77

40.18

23.12

28.74

36.76

21.99

27.11

34.15

18.05

21.61

26.22

Infinite

Sl a b

Thickness

(cm)

114.24

.

—

36.54

171.68

—

25.04

44.68

—

19.95

30.46

6.1.68

16.89

24.16

39.16

14.89

20.49

30.27

13.38

18.02

25.40

12.27

16.28

22.32

11.39

14.91

20.01

10.66

13.87

18.34

8.10

10.33

13.16

9pu

Ma ss

Densi ty

(kg /L)

0.030

0.050

0.100

0.200

0.500

1.000

2.000

5.000

10.000

14.000

19.740

k,fi

8

9

1

8

9

1.0

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1

8

9

1

Sphere

Diameter

(cm)

23.31

27.04

31.52

20.63

23.74

27.37

18.66

21.46

24.68

17.66

20.37

23.48

16.75

19.40

22.44

15.86

18.38

21.26

14.56

16.84

19.42

11.98

13.80

15.82

9.27

10,65

12.14

7.76

8.91

10.13

6.23

7.13

8.08

Infinite

Cylinder

Diameter

(cm)

15.29

18.04

21.42

13.16

15.49

18.19

11.58

13.68

16.09

10.73

12.73

15.07

10.00

11.93

14.18

9.33

11.20

13.35

8.45

10.12

12.03

6.74

8.05

9.53

5.07

6.04

7.09

4.19

4.98

5.84

3.34

3.95

4.60

Infinite

Sl a b

Thickness

(cm)

6.26

7.98

10.08

4.81

6.22

7.90

3.63

4.89

6.34

2.94

4.14

5.55

2.35

3.52

4.88

1.97

3.06

4.33

1.55

2.50

3.62

0.96

1.61

2.42

0.57

0.99

1.49

0.42

0.73

1.12

0.31

0.53

0.81

113



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