LA-10860-MS

UC-46Issued: July 198?

LA—10860-MS

DE87 012702

Critical Dimensions of SystemsContaining 235U,239Pu, and 233U

1986 Revision

H. C. PaxtonN. L. Pruvost

DISCLAIMER

This report *us prepared as an account ol v»oik sponsored b) an agency of the United States(iiivernment. Neither the United States (iovernment nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or rcsponsi-bihiv fur (he accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service hy trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily stale or reflect those of theUnited States Government or any agency thereof.

MASTERLos Alamos National LaboratoryLos Alamos.New Mexico 87545

DISTRIBUTION CF THIS Dt/CLifYILfiT '.S UteUW

TABLE OF CONTENTS

PREFACE viABSTRACT 1lNTHODl'CTlON 1

PART I - INDIVIDUAL UNITS 3

HISTORY OF EARLY CRITICAL EXPERIMENTS 4EXPERIMENTS RELATED TO THE WEAPONS PROGRAM 4EXPERIMENTS RELATED TO THE CASEOUS DIFFUSION PROCESS 4TRANSITION TO FACILITIES FOR CRITICAL EXPERIMENTS 12

RELATIONS FOR CONVERSION TO STANDARD CONDITIONS 13CYLINDER-SPHERE CONVERSIONS 13CORE-DENSITY CONVERSIONS 19

HOMOGENEOUS HYDROGEN-MODERATED URANIUM 23HIGHLY ENRICHED URANIUM 23URANIUM OF VARIOUS ENRICHMENTS 31

HETEROGENEOUS WATER-MODERATED URANIUM AT 44VARIOUS ENRICHMENTS

NEAR-HOMOGENEOUS HYDROGEN-MODERATED URANIUM 44ENRICHED IN 2 3 S l 'WATER-MODERATED LATTICES OF SLIGHTLY ENRICHED 44URANIUMWATER-SPACED ,NITS OF ENRICHED URANIUM 4C

Highly Enriched Uranium 49Uranium of Various Enrichments 51

METAL-SOLUTION MIXTURES 56

HOMOGENEOUS HYDROGEN-MODERATED PLUTONIUM 59PLUTONIUM SOLUTIONS, LOW 240Pu 59PLUTONIUM SOLUTIONS, UNLIMITED 240Pu 71PLUTONIUM - POLYSTYRENE COMPACTS 74HOMOGENEOUS PLUTONIUM-URANIUM SYSTEMS 74

WATER-MODERATED PLUTONIUM AND PLUTONIUM- 76URANIUM LATTICES

HOMOGENEOUS HYDROGEN-MODERATED 233U 78

IV

POISONED SYSTEMS S3Ho\Kx;t:.\Kors ABSORBKKS K:I

U r a n i u m 235 S3P l u t o n i u m S o l u t i o n s N4U r a n i u m 233 Solu t ions S7

UK lT.ROCKNKOlS AHSOklJr'.RS SS

F I S S I L E U N I T S W I T H V A R I O U S R E F L E C T O R S 91RKKl.F.rToKS AMol 'T l"UAMI \1 AND IM.I' I O N H M MI 1 A L 9-1RKr'l.KA'TOKS AHDl 'T IIYDRUCK.N MODF.UATF.D I 'NITS ][)',

S Y S T E M S W I T H N O N H Y D R O G E N E O U S DILUENTS 109

C O M P L E X SHAPES 112AXNCI.I 112

INTF.RSFiTloNS US

PART II - I N T E R A C T I N G UNITS 121

H O M O G E N E O U S L Y M O D E R A T E D UNITS 122

PLANAR AND LINEAR ARRAYS 122Cylindrical Uni t s of U( -90) 122Slab Units of U ( - 9 0 ) 142Combinat ions of Slab and Cylindrical Units of U(--90) 149Units of LI( 90) 153Plutonium Solutions 161Solutions of -'"U 169

SPATIAL ARRAYS OF J j r T SOU'TION 172H E T E R O G E N E O U S L Y M O D E R A T E D U N I T S 173

PLANAR AND SPATIAL ARRAYS 173Interacting Fuel-Rod Lattices 173Units of U(93)O 2 - Alcohol 173

METAL UNITS A N D C O M P O U N D S 176IRANITM J7(i

Planar Arrays 17(iSpatial Arrays i 7S

PI. ITONUM ' 185

RKFKRKNC'KS 187

APPENDIX 200

PRKKACK

This report i> uunlf po» ih l e by t i iuuuia l support of t h e UHice of Nuclear Safety of

the Depar tment ol Energy as tlit' result ol a proposal by D. Yi. Smi th nl thf Los Alamos

Nat ional Labora tory . 1*1 n• iiunpilor> o| th is version were ass is ted ahlv by an edi tor ia l

advi.soi'N >;n>up I'onsisl inu, ol K. \\. . lohiison, .1. i . r i io inas , and l)ixt)n Cail ihan (all ol

whoiu, wit 11 H. C. P a x t o n , ;-oiupilcd the original r epor l ) . 1). R. Smi th , K. 1). C lay ton of

Paciiic Northwest Labora to ry and ('. M. lh>pper of Oak Hid^e Natiomil Labora tory . T h e

in-dt |>th ell'tirt of [•'.. H. lolmsoii was es|)eciall\ valuable, We also wish to acknowledge the

o u t s t a n d i n g skill ol Ann Hopkins in p r epa ra t i on of this d o c u m e n t .

11. C. Paxton

N. L. I 'mvos t

VI

CRITICAL DIMENSIONS OF SYSTEMSCONTAINING - : t r r . -'-'I'll. AND -XiV

REVISION

II. ('. I'axti.n and N. L. l'ruvost

ABSTRACTThis document is a revision of T1D-702K, CRITICALDIMENSIONS OF SYSTEMS CONTAINING l ' - : t \Pii-'-1". AND 1 J:M (Kef. 1).

I.NTRQDI'CTION

This report is primarily a compilation o! critical data obtained from experiments performedin a number of laboratories during the period of 19-45 through 1985. It supplements theNuclear Safety Guide Report TID-701H (Rev. 2)." in presenting critical data on whichrecommendations of the Guide are based.

It must be emphasized that this report gives critical data without safety factors, so it isno substitute for the Guide or for the related document. The American National Standardfor Nuclear Critically Safety in Operations with Fissionable Materials Outside Reactors3.This standard is supported by publications of H. K. Clark that interpret criticality infor-mation for Pu, "33lT and 2 3 rT systems.4"6

Information for guiding the safe desigii of equipment for handling the three common fissilematerials appears to a certain exient in the Nuclear Safety Guide, and with greater detailin several handbooks.' Other publications specify conditions for unique processes with aparticular fissionable material.8

Experiments ol several types winch contribute insults applicable to nuclear design and tosalety problems have been described by Callihan.9 Critical measurements with materialsof interest in desired configurations yield information of greatest usefulness and accuracy.V\ here it is not Icasihle to obtain the desired critical data, for example, as a result of safelyrestrictions, subcritiral data may be directly applicable, and in some cases may be extrap-olated to approximate critical conditions. Critical conditions also may be approximatedfrom tlie distribution ol neutrons introduced into a subcritical assembly. These "exponen-tial experiments" may be the only alternative where the quantity of material required istoo great for a critical experiment.

In some cases, calculated results are desirable to Hll in experimental data or to extend them.Where they appear in this report, they are identified as calculations because of the uncer-tainty associated with them. In general, the IB-group Hansen-Roach cross-section set10 isused with the one-diinensional transport code, DSN," or its modern version, ONEDANT12

in one dimension and TWODANT1"1 in two dimensions. The reason for this choice isSt rat ton's extensive comparison of results oi such calculations with experimental data.14

Where this comparison is unfavorable, particularly for solution or hydrogeneous mixturesof few-percent "15l'-enriehed uranium, the Los Alamos MCNP Monte Carlo code15 andcross-sections based on ENDF-B/Y* are used instead. This combination is also applied, inthe few cases where finite-cylinder calculations are desired.

Calculated extensions of experimental data are included to show the nature of trends, notto substitute for results of experiments. They should be used with caution.

A fundamental aim of this document is 10 illustrate relationships among critical data. Thecompilation and correlation of data for this purpose, from many measurements in a numberof laboratories, require a certain amount of normalization or reduction to common terms.Frequently, for example, the effects of variations in geometry or density must be removedto show trends in data. The manner in which these alterations may be made is discussedin the early section RELATIONS FOR CONVERSION TO STANDARD CONDITIONS.

Reactor inockups and related critical assemblies are generally outside the scope of thisdocument. Many of those that are appropriate to serve as computational models are fast-ueutron s\stems that are of secondary interest in criticality safety matters.16 Nevertheless,fast-react or mockups which are used as benchmark assemblies (for comparison with calcu-lations) are instructive, for they illustrate the value of data other than criticality specifi-cations, namely, spectral information, neutron flux distributions, prompt-neutron lifetime,and reactivity coefficients. Unfortunately, data for critical assemblies suitable as safetybenchmarks are usually limited to critical compositions, dimensions and masses withoutuseful supplementary information.

Robert ('. Little. Los Alamo? National Laboratory. Los Alamos. NM 87545. 1986

PART I — INDIVIDUAL UNITS.

HISTORY OF EARLY CRITICAL EXPERIMENTS

EXPERIMENTS RELATED TO THE WEAPONS PROGRAM

The Hrst critical assembly with enriched uranium* was the Los Alamos water boiler LOPO,operated initially in 1944.17'18 This was a predecessor of HYPO that operated as a neutronsource for many years. 01 course, they were both preceded by reactor-oriented assembliesof natural uranium, namely the Chicago and Oak Ridge piles.19 LOPO was a 14.95-litersphere of 11 (14.67)O2SO.j-H2O** solution at H/

2;15U=647 in a 0.08-cm-thick containerand reflected by a roughly 90-cm-diam stack of beryllium oxide (density 2.7 g/cm3) on agraphite base.

About the same time, before U( -90) became available, Los Alamos critical experimentswere directed toward weapon design. The critical assemblies of which descriptions havebeen declassified consisted of U(~80) metal cores reflected by natural uranium, tungstencarbide, or beryllium oxide; U(-~74)HioCj cores reflected by natural uranium, tungstencarbide, beryllium oxide, or iron; and plutoniuni metal reflected by thin natural ura-nium within tungsten carbide.20 The UHJOC^J was an intimate mixture of UH3 and Styrex(CH1.75). Table 1 gives critical specifications of those assemblies with nearly regular ge-ometries. Because uncertain geometric details cannot be clarified at this late date, thesespecifications are given lor historical interest, rather than scientific value.

EXPERIMENTS RELATED TO THE GASEOUS DIFFUSION PROCESS

In 1945, Oak Ridge undertook the first of a series of critical experiments directed to-ward crit icality control of enriched uranium as UFg, the working material of the gaseousdiffusion plant.21 Critical assemblies at that time came as close as practicable to sim-ulating condensed U(24)F6 with hydrogen moderation. One-inch cubes of an intimatemixture of U(24)3O8 and as much viscous fluorocarbon, reported to be CFo.68> as wouldretain its shape were prepared in 0.013-cm-thick aluminum. The effective composition,U(24)02.83Cs.3oF3.f.oAlo.'i, did not approach the actual F/U atomic ratio 6. Because theavailable inventory. 9.5 kg 235U, was subcritical without moderation, one-inch cubes ofpolyethylene (('Hi.*-) were latticed among the uranium units to make some assembliescritical. Table 2 gives the reported description of these assemblies, with roughly cubicalcores reflected by paraffin. Because of a quest inn concerning the reported fluorocarbonlonmiia,*** these assemblies, also, arc described primarily for historical interest.

* I'ranium enriched in tin- isotope "'l3U.Numbers in parentheses indicate the 2 3 S1" content of the- uranium in weight percent.

*** An expert in Hunrocarbon chemistry believer that the formula CF0.68 should represent asolid instead of a viscous fluid. The formula was provided by the laboratory of the GaseousDiffusion Plant which produced 'he fhiorocarbon.

TABLE 1. EARLY LOS ALAMOS METAL AND HYDRIDE CRITICAL ASSEMBLIES.

Core Reflect or

Composition,Shape"

I'(78.7)pseudosphere

T or FuDensity Material,( t i / c in 3 ) Form

17.8

Density Critical Mass(g/rin3") (kg2 3 5UorPu)

U(iiiitnral)48-cm-od sphere 19.11 21.9

pseudosphere

U(82.7)pseudosphere

pseudosphere

l'(75.2)H l0C4near rube-

r(7.-.2)H10C4near cube

Pu(1.35 \s-\% 240Pii)b

sphere

17.S

17.8

2.1:

2. IS

2.18

15.6

we35.6-cin cube

BeO61-cm cube

we41-cin cube

BeO48-cm cube

BeO79-cm cube

U(natural) insidesphere, 1.14-cm-thickWC outsideparallelepiped32.4x32.4x27.0 cm

14.7

2.69

14.7

2.69

2.69

19.0

20.8

10.3

7.00

3.52

2.80

6.13

"All uranium cores built of 1/2-in. cubic units.''Contained 1.0 \vt'7c C,n. coated with -0.013-cm-thick Ni.

H/-3RrAtomic Ratio

46«3129

AverageDensity (g/ciu3

2.14I .951.67

Critical Mass(kg

8.63

235 uj

.3

.1

.9

TABLE 2. CRITICAL CUBIC LATTICES OE 1-IN. CUBES OFUi24)0.,..saC5.3o K3.f.oAlo.4 (DENSITY 3.50 g/cm

3) AND CH 1 8 7(DENSITY 0.915 g,cin3) REFLECTED BY -fi-IN.-THlCK PARAFFIN.

H Cubes T Cubes

3/41/12/1

In 1946 there was suiHcient highly enriched uranium to extend the above experiments withone-inch cubes of intimately mixed U l\t and polytetrafluoroethylene (CF 2 ) n , effectivelyUF6C. These were latticed with cither one-inch-cubes or 1 x 1 x 0.5 in. half-cubes ofpolyethylene. A series of experiments witn U(95.3) was conducted by Oak Ridge person-nel at Los Alamos.22 Roughly cubic cores with 18-cm-thick paraffin reflectors consistedof hydrogen-to-uranium cubes in the ratios \:1, 1:1. 2:1, 4:1, and 7:1. For the 1:1 ratio,heterogeneity was varied such that there was reasonable extrapolation to a homogeneouscore, with the result included later in Table 10. Critical masses for all hydrogen-to-uraniumcombinations, nncorrected for heterogeneity, appear in Fig. 1 as the left-most curve. Ap-proximate densities of 2 3H in g/cmJ may be obtained by dividing 14.8 by H/235U.

Later that year, a similar series of experiments was performed at Oak Ridge with U(29.8)instead of highly enriched uranium.2'5 Again, one composition could be corrected reason-ably tor heterogeneity, with the result included in Table 10. Uncorrected critical massesalso appear in Fig. 1. The curves rise at large H/235U as a result of extreme heterogeneity.The expression relating 23SU density and H/"35U given above for U(95.3) also applies forU(29.8).

Although not belonging to this very early period, the only known experiments with ho-mogeneous hydrogen-moderated UF6 are described here to complete the account of UF6criticality.2'1 In this series of experiments, performed at Valduc, France* the primary ma-terial was liquid U(93)F(, to which various proportions of liquid HF were added whileretaining homogeneity. These mixtures were contained in 0.4-cin-thick Monel spheres sur-rounded by an effectively infinite thickness of water.

Experimental critical data appear in Table 3 and in Table 4 after adjustment to sphericalvolumes and normalization to F5"C. Figures 2 and 3 show critical volumes, diameters, andmasses as functions of H/235l" atomic ratio and 23r'U density. The available inventory ofI'Fe. 93.9 kg 2' I . could not be made critical without internal hydrogen. Consequently,the entry under H /215U -.-. 0 in Table 4 is the result of extrapolations shown in Figs. 2 and3, supported by Monte Carlo and transport calculations.

La Station ,le Criticalite de Yalduc may be called simply Valduc. or Dijon, which is nearby.

M

o

50

40

30

20

10-

5 -

Q

O

I

,/A\

11

\

i1 \

E

\

\

3

\

yri 1

\ \

\

s

\

1

•

4

\

y

V

3

© U (8

ffl U (2

c'7

|

BEL

5.3)

4

•y

rn

1

10

H/ 2 3 5 U ATOMIC RATIO

1G0 300

/. Critical masses of paraffin-reflected cubes consisting of intermixed one-inch cubes of VFQC andpolyethylene, for U(95.3) and U(29.8). The ratios of polyethylene to UF^C cubes are indicated.

TABLE 3. CRITICAL U(93)F6-HF MIXTURES IN WATER-REFLECTED SPHERICALCONTAINERS.

Inside Diameter of Container (cm)

53,9 53,9 5CK9 50J) 5JX9 53J) 5JL9 50.9 5(19 50,9 53J)H/U 5.7 9.9 10.8 11.2 35.2 16.9 21.0 26.0 38 74 82

0.1 0.2 0.2 0.2 0.3 0.3 0.4 0.7 1 2 3

T(°C) 89.0 75.0 85.8 75.1 74.8 94.3 74.3 75.0 75.6 75.1 75.3AT(°C)a 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

Densityp(g/cm3) 1.92 1.71 1.61 1.63 1.467 1.336 1.321 1.245 1.149 0.990 0.973Ap(g/cm3)a 0.02 0.01 0.01 0.01 0.009 0.008 0.008 0.007 0.007 0.006 0.006

LiquidHeight (cm) full 47.3 full full frill 51.0 46.6 46.2 46.1 full 46.1

CriticalVolumeV c (L) 85.0 78.5 76.0

i) 71.36 69.4b 81.3 67.3 67.1 67.0 71.36 77.2

"A value designated by A represents the uncertainty in the quantity.Value of critical volume from extrapolated reciprocal multiplication curves.

TABLE 4. CRITICAL DIMENSIONS OF WATER-REFLECTED U(93)F6-HFMIXTURES ADJUSTED TO SPHERICAL VOLUMES AT T5°C.n

H/ 2 3 5 U 0.0 6.1 10.6 11.6 12.0 16.3 18.2 22.6 28.0 41 88

p(g/cm 3 ) 3.B3 1.99 1.71 1.H5 1.63 1.465 1.413 1.317 1.245 1.140 0,973

235U Concentration

C(g/cm 3 ) 2.27 0.939 0.69 0.639 0.622 0.490 0.473 0.373 0.313 0.225 0.107AC(g/c in 3 ) 6 0.014 0.01 0.008 0.008 0.007 0.007 0.006 0.007 0.009 0.004

Critical Diameter

dc(crn)Adc(cm)

6

VC(L)AVc(L)

b

55C

88C

53.740.09

81.30.4

52.80.2

771

52.80.2

75.10.7

52.10.2

74.10.5

51.040.05

69.60.3

511

68.60.4

502

67.00.7

502

66.20.7

502

66.20.7

532

76.50.7

Critical MassMc(kg

235U) 200c 76 53.0 48.4 46.5 34.1 32.5 25.0 20.7 14.90 8.2AMc(kg

235U)i> 1.8 1.5 2.0 1.9 1.7 2.0 2.4 2.8 5.0 4.2

"Uncorrected for 0.4-cm-thick Monel container.''Uncertainties represented by A(quantity);A(H/U) and A(p) appear in Table 3.cExtrapolated value supported by calculations.

H/236U ATOMIC RATIO

90

OO

|

> on

•iE

RIC

/

U 7 0 -

-

6 5 -

821

o•1

\v

3B 21 11.21

0

o

3

/

5.7

/

/

/

/

01

/ _

-

-

-

--

-

H/235U ATOMIC RATIO

38 21 11.2 5.7

0.1 1

DENSITY OF a35U (kg/L) DENSITY OF 235U (kg/L)

(̂ . 2. Critical volume and diameter of U(93) as liquid UFe -HF at 75° C as a function of23S U density. Thesystem was contained in a water-reflected O.J-cm-thick Monel sphere.

82

rfA

SSTI

CAL

I

U

100

50

Ed

a,CO

0.1

H/235U ATOMIC RATIO

38 26 21 15.2 9.9 5.7I I I J _ I I

xT

0.2 0.5

DENSITY OF 235U

0I

Fig. 3. Critical mass of U(93) as liquid UFQ-HF at 75° C as a function of 235 U density. The system wascontained in a water-reflected 0.^-cm-thick Monel sphere.

As Fig. 2 indicates, the minimum critical volume occurs at H/U about 30. Further, UF6-HF mixtures at any concentration cannot be made critical in water-reflected spheres 45-cmor less in diameter provided the temperature is at least 75°C.

TRANSITION TO FACILITIES FOR CRITICAL EXPERIMENTS

Returning to the very early experiments, it may be noted that they were conducted inimprovised facilities. Two Los Alamos fatalities25 underlined the need for improved pro-tection of personnel, and permanent facilities designed specifically for the safe performanceof critical experiments were in operation at Los Alamos in 1947 and at Oak Rici^e in 1950after experiments starting in 1946 at an interim facility. There followed at Oak Ridgeextensive programs with uranium enriched in 335U and 233U, which contribute to the re-mainder of this report.

In 1951, critical experiments with plutoniuni solutions began in a Hanford facility that wastemporary but had adequate personnel protection.26 The permanent plutonium critical-mass facility at Hanford became operable in 1961.27 These facilities provided much of thedata for moderated plutonium systems that are included in this report.

Other laboratories that have provided results of critical experiments are at Aldermaston,England, operation beginning in 1952;28 Livermore (from 1955 in its appropriate facility )-,29

Dounreay, Scotland (from 1957);29 Saclay, France (from I960);30 Valduc, France (from1963);30 and Rocky Flats (from 19fi5).31 We have no information on USSR facilities atwhich criticality data have been generated.

12

RELATIONS FOR CONVERSION TO STANDARD CONDITIONS

Many of the data correlations that appear in this report required the conversion of ex-perimental information to certain "standard" conditions. Two of the more significanttypes, shape and density conversions, are considered immediately. Other types, such asthe correction for variations in 235(J enrichment, fit more naturally into later sections.

CYLINDER-SPHERE CONVERSIONS

Ratios of critical masses of cylinders (height h and diameter d) to those of spheres appearvs h/d in Fig. 4 for enriched uranium solutions* and in Fig. 5 for enriched uranium metal.The values for solutions and I!(93) metal reflected by polyethylene and Plcxiglas are derivedfrom measurements at Oak Ridge.32 "3(l Those for U(94) metal, unreHected and reflectedby paraffin or water, are from Los Alamos 3 |" •liv Early critical data for plutonium solutionsoriginated at Hanford"6 and for 2iiV solutions at Oak Ridge.40

For extrapolation of experimental critical dimensions to those of broad slabs and longcylinders, the following method is useful. The dimensions of critical cylinders of differentsizes and of a critical sphere, all of the same composition, are related to each other throughthe expression for the geometric buckling, B2, provided appropriate values of the cylinderextrapolation distances are used. Effective values of cylinder extrapolation distances wereobtained from the following relation using cylinder and sphere dimensions and sphereextrapolation distances of Table 5.

+ 26C )

where rc = tlr* radius of the cylinderh = the height of the cylinderr? — the radius of the sphereb = the effective extrapolation distance appropriate to these dimensions.

TABLE o. SPHERE EXTRAPOLATION DISTANCES FOR GENERAL APPLICATIONOF FIG. 6 WITH WATER REFLECTION.

At omic Rat io H / X": .50 2002 JHJO 2F- 2 5.8 5.4•^UO.F.. 5.6 5.2Pu(NO.s)., 6.3 5.8

•X-23SU. 2 3 3Uor Pu.

ice b, (cm)

5005.25.05.4

10005.25.05.2

Aqueous solutions are implied throughout this document .

13

en<

Cd

W

Q-,CO

<

a2

CO

<

OSEdQ

<

2

6-

o

4 .

Q _iO 1

pc

1

1

o-

-

-

\

A

- •

I

\

wRE

T

V

\

\

ATER ^ \FLECTOR

UNRE\

\

\

\

IV23

44.3 O 389

FLECT

°\

ED

n-—

ATOliAlC RATIO

O 52.9 A 43.9 D32C

1

\

i

! /

111

Ii

/]/-

/-/

-

-

_

-

0.05 0.1 1

CYLINDER HEIGHT / CYLINDER DIAMETER

10

Fig. 4- The ratio of cylindrical to spherical critical masses of U(93)O2p2 solutions, unreflected and withwater reflector, as a function of cylinder height to cylinder diameter ratio.

vfDE

R M

ASS

/ CR

ITIC

AL

SPH

ERE

MA

SS

£3 ^>-U

<

in 1

s

o-

\

\

\

UNRE

\

t

?EFLEC

_ \J'LLLILU V11

\

\

\

L l

TED L l

C

\

\

—

\

A

f.~VA\

V PLEXIGLAS-REFLECTED U (93)

• POLYETHYLENE-REFLECTED U (93)

A PARAFFIN-REFLECTED U (94)

O WAr

—A

rER-REFLECTE D U(94)

/

h

1/ ;A

-

-

0.02 0.1 1 10 20

CYLINDER HEIGHT / CYLINDER DIAMETER

Fig. 5. The ratio of cylindrical to spherical critical masses, for U(>90) metal, unrefleeted and with hydroge-nous reflector, as a function of cylinder height to cylinder diameter ratio.

The resulting ratios of cylinder to sphere extrapolation distances appear in Fig. 6, inwhich some dimensions were obtained experimentally and some were computed by theTWODANT13 code with Hansen-Roach cross sections.30 Figure 7 gives similar extrapo-lation distances for reflected and unreflected U(93.5) metal disks, where 6a = 2.0 cm isassumed for an unreflected sphere and reflector savings are consistent with Fig. 5.

In Figs. 6 and 7 the abscissa was chosen such that at zero the value of bc determinesthe thickness of an infinite slab j= (n/B) - 2bc] and at unity the value determines theradius of an infinite cylinder 1= (2.405/6) - 6C]. The calculated end points of Fig. 6 wereobtained by means of the ONEDANT code12 and Hansen-Roach cross sections.10 In Table.">. the value of the sphere extrapolation distance for 235UO2F2 solution at H/'

!35U = 50was obtained from St rat ton's report LA-361214 by combining the extrapolation distancewithout reflector from his Eq. 2 and the reflector savings from his Table V. Results are ba =5.8 cm with a water reflector and t\, •= 2.2 cm with a 0.13-cm-thick stainless steel reflector.Other extrapolation distances in Table 5 (for more dilute 23!lU solutions, 233U solutions,and Pu solutions) were obtained from sphere, infinite cylinder and infinite slab riun"jisionscalculated by ONEDANT.10'12 The listed sphere extrapolation distances, then, are thoserequired to bring both calculated end point extrapolation distances into coincidence withthe end points of Fig. 6.

With 0.13 cm stainless steel reflection, 6 — 2.2 cm from Stratton's spheres applies uni-versally to the transformation of solution cylinders. It has been confirmed empiricallywithin ± 2% for U(93.2)O2F2,

233UO2F2, and Pu(NO3)4 + 1 N HNO3 solutions over theexperimentally available ranges of height-to-diameter ratios.

to

I2

2<OSE->X

<

XCO

<y5

o

0.9

0-S J

0.7

0.6

INFINITE SLAB

O EXPERIMENTAL CYLINDERS, H/ 2 3 S U = 44 TO 57

• EXPERIMENTAL SLABS. H / " s U = 45

A EXPERIMENTAL SLABS. H / " ° U = 4 3 0

X CALCULATED

I

INFINITE CYLINDER

0 0.2 0.4 0.6

h/d / (1 + h/d)

0.8

Fig. 6. Ratio nf cylindrical extrapolation distance to that of sphere for water reflected U(93)O2p2 solutions.Cylinder height and diameter are h and d respectively.

EdU2

£9

o

Cd

W

ECdd-

Cd

UNHEFLECTED

o- Oo

O UNHEFLECTED U (94)

V PLEXIGLAS-REFLECTED U (93)

D POLYETHYLENE -REFLECTED U (93)

A PARAFFIN-REFLECTED U (94)

O WATER-REFLECTED U (94)

£L

0

INFINITE SLAB INFINITE CYLINDER

0 0.1 0.2 0.3 0.4 0.5 0.6

h/d / (1 + h/d)

0.7 0.8 0.9

Fig. 7. Effective extrapolation distances of U(>90) metal cylinders, unreflected and with hydrogenous reflec-tors. Cylinder height and diameter are h and d respectively.

CORE-DENSITY CONVERSIONS

A change in the density of a fissile sphere by the ratio p/p,, leads to a changed criticalmass, rnc, that may be expressed as

r n c j m r l , - ( f> / f>o )

where n is constant over a considerable range of density ratios. In fact, where density ofboth spherical core and reflector is changed by the same ratio and the ratio of reflectorthickness to core radius is maintained, then n — 2 (the value for an unreflected sphere).Similarly, in the case of an infinite slab, the critical mass per unit area is necessarilyindependent of p (i.t., n -: 0).

Where- reflector characteristics remain constant, however, the value of n associated withthe density change of a spherical core depends considerably upon the system. CombinedLos Alamos, Livermore, and Rocky Flats data for U(93.5) metal and 6-phase plutoniumcores20'41'42 seem to follow a unique relation between the density exponent and the degreeof reflection (see Fig. 8). The scatter associated with subcritical plutonium measurementswould mask any small differences between the two fissile materials.

The experimental values of n (as determined by the UKAEA Atomic Weapons ResearchEstablishment at Aldermaston, ORNL, and Los Alamos) for near-equilateral nonmetalcores are given in Table 6.'>3~45

19

c&-•

Ed

OCL.XCd

CdQCdOS

8

1.8

i.e

D U(93.5) COPFS (FROM CRITICAL DATA)

O d - PHASE "*PU CORES (FROM SUBCRITICAL DATA)

I

-a

o

-o-

0.6 0.7 0.8 0.9

REFLECTED CRITICAL RADIUS / UNREFLECTED CRITICAL RADIUS

Fig. 8. Density exponents of unmoderated spherical cores in constant-density reflectors. Critical mass =constant (core density)~n.

TABLE 6. EXPERIMENTAL VALUES OF THE NEGATIVE DENSITY EXPONENT,n, FOR NONMETAL CORES.

CoreComposition H/2 3 5U Reflector Reference

U(30)O2-paraffinU(30)O2-paraffinU(30)O2-parafrmU(30)O2-paraffinU( 30 )O2-paraffinU(30)O2-paraffinU(93)O2(NO3)2U(93)H3C

8.2616.516.58282822303.2

"Methacrylate plastic, called

20.3-cin-thick Perspex0

20.3-cm-thick Perspex20.3-cm-thick polyethylene20.3-cm-thick Perspex20.3-cm-thick polyethylenethick waterthick water22.2-cm-thick U(0.7)

Plexiglas in the U.S.

1.461.501.691.561.671.65l,88b

1.57

'Possibly influenced by the manner in which voids were introduced.

4343434343434445

The lack of experimental core-density exponents for solutions forces the use of computedvalues. Figure 9 shows such exponents for 235U calculated by the DSN code11 usingHansen-Roach cross sections.10 Hanford calculations for 239Pu used a similar code (DTK)but different cross sections (from GAMTEC-II).46

21

1 0 2

1.9

rj 1 a

t~; l .owoCLX

>« l . r

CO

DEI

UJK 1 RO l b

u

1 ^l .O

o•

1.4-

^—

^ -

O 235y

O 239Pu

x Vi

y

/

//

A/s

1

/

/

/

/

i

/-^ -

-

-

-

-

-

10 100

H/235U OR H/239Pu ATOMIC RATIO

1000 4000

jg. 9. Calculated core-density exponents for water-reflected spheres of homogeneous metal-water mixtures.The solid symbols represent limiting conditions.

H O M O G E N E O U S H Y D R O G E N - M O D E R A T E D U R A N I U M

HIGHLY ENRICHED URANIUM

Figures 10 and 11 represent critical masses and critical volumes of homogeneous water-moderated spheres of U(93.2), both bare (except for the thin-wall container) and waterreflected. For a water-reflected sphere of U(93.2) metal at a density of 18.75 g/cm3, thecritical mass is 22.8 kg 235U and the critical volume is 1.30 L. Estimates of diametersof infinite critical cylinders of U(93.2)O2F2 solution appear ir. Fig. 12, and correspondingestimates of thicknesses of infinite critical slabs appear in Fig. 13. Values for water-reflectedU(93.2)metal are 7.5 cm infinite cylinder diameter and 1.4 cm infinite slab thickness. Itshould be noted that the curve for bare infinite slabs is fictitious because a slab of infiniteextent would intercept neutrons returned from material at any distance. Nevertheless, it-may be useful for comparison with similarly fictitious calculations.

The branched curves of the figures show how concentrated UO2F2 solutions depart fromideal metal-water mixtures. As indicated by the computed curve for UO2-water in Fig. 10,densities of the fissile isotope in metal-water mixtures are greater th^i actually foundin practice; hence, critical values are lower limits that may be quite o-noervative. Forconvenience the assumed relations between the density of 235U and the atomic ratio H/235Ufor solutions and metal-water mixtures are given in Table 7.47 Similar relations for 233Uand Pu (Ref. 26) are also included.

The appendix gives theoretical densities of unmoderated common uranium compounds.

23

to

<

o<o2Ed

a,CO

100

m -1U

ii

n K\J.D

D

\\\\\

\

\

Zi

O D

a D

x c

*• c

ATA DERIVED FROM CYLIN1

ALCULATED METAL-WATER

MCULATED UOZ-WATER MI

i

- - •

i LIMITING CRITICAL DENJ

3ERS

MIXTURES

XTURES

-cm STAINLESS STEELREFLECTOR

D""P~"|~ "

1 1

r

/

/

F/

RE

1TATIFLE

LJ

• 0

/ • '

'CTOR

» X

+ • '

< • '

. + •

1

1

•i- ;

. . . jx : 6

HETAL — • ; -

0.01 0.1 1

DENSITY OF 235U (kg/L)

10 20

/g. 10. Critical masses of homogeneous water-moderated U(93.2) spheres. Solution data appear unlessindicated otherwise.

DENSITY OF 235U (kg/L)

Fiy. 11. Critical volumes of homogeneous water-moderated V(93.2) spheres. Solution data appear urilessindicated otherwise.

toO2

OS

aas

Cd

J U U

i n n

1 A1U

o

•

i \• \

V\

«— u>

\

I

SITING

\

WATIEFI

CHIT

ER ""ECTOR

1I

0.16-cm STREFLE

AINLEECTOR

3—C

-\-

SST

—C

EEL

] — D

to- I

100

U1Ed

10

a-

H

\

\

_i

0 . 1 6 - c m STAINLESS STEELREFLECTOR

I!

WATERREFLECTOR

P

LIMITING CRITICAL DENSITY

- ) 1 1 1 1 1 I 1 I

T~!

O DAW DERIVED FROM :

n DATA DERIVED FROM CYLINDERS

A DATA FROM SLABS

X CALCULATED METAL-WATER MIXTURES

%:^3 ^x ---X^T7X.

r-

METAL

0.01 0.1 1

DENSITY OF 235U (kg/L)

10 20

Fig. 1:1. Estimated critical thicknesses of slabs, infinite in other dimensions, of homogeneous water-moderatedU(9J.2). Solution data appear unless indicated otherwise. Unreflected infinite slabs are fictitious.

TABLE 7. DENSITY OF X VS. H/X ATOMIC RATIO[X ->35l' as V(93.2), 23:iV as 233U(!>8.7 \vt%' or I'n,

"JS Density (g/cm3) Pu Density(g/cm3

H/X Metal-HjO Solution" Metal-IijO Solution" nPu-H2O 6Pu-H2O Solution6

[.)1•)

35

1020

30

50100200300

5001000

1500

2000

3000

17.53

10.48

7.4S

5.S1

4.022.271.210.83

0.51

0.257

0.129

0.086

1.06

0.76

0.48

0.252

0.128

0.086

0.052

0.0260

0.0175

0.0132

18.28

10.717.575.86

4.03

2.27

1.21

0.82

0.50

0.255

0.128

0.086

1.070.760.480.2500.1270.0850.0510.02580.01790.0134

19.611.277.916.094.182.341.240.850.520.2630.1320.088

0.857.185.663.962.271.220.840.510.261

(0.429)c

0.2340.1220.0830.0500.02540.01700.0280.0085

u l 'O 2 F 2 solution.f>Pu(NO3)4 solution with 1 N HNO3, Pu contains 3%

240Pu. Water densities from therelations on page 69, Ref. 4.

rSolution probably unstable.

28

Sources of experimental data and the nature of conversions to the conditions of Figs. 10 to13 are as follows. The portions of those figures for the H/"3oU range greater than 20 arebased on many critical-solution measurements of a variety of cylinders,32'33 some spheresor cubes,"'48 51 and a slab.3"1

The value of thickness of a critical infinite solution slab reported in Ref. 34 was obtainedby extrapolating the reciprocal critical height of vertical finite slabs to zero. Alternatively,the infinite slab thickness may be obtained from the least subcritical finite slab by meansof Table 5 and Fig. 6. The result at H/"35U - 44.7, 5.0 cm (including Plexiglas correctionof 0.2.r> cm), compares with 4.5 cm reported in Ref. 34. There was a similar reexarainationof the critical infinite slab thickness reported for Plexiglas-rellected U(93) metal.35 Theresult, 1.38 cm, compares with the reported value of 1.52 cm (0.6 in.).

The most nearly equilateral critical cylinders are generally selected for conversion tospheres, elongated for infinite cylinders, and squat for infinite slabs. Conversions to therequired shape make use of Fig. 6 and Table 5. Extrapolation of critical solutions con-centration data to zero buckling gave 12.30 ± O.lOg of 235U/L as the limiting criticalconcentration,51 and 12.05 ± 0.03 resulted from measurements at the Hanford PhysicalConstants Testing Reactor (PCTR).52

Although they do not apply directly to the curves, critical data for slightly moderatedsolids are available for checking calculated points, for example, cores of effective compo-sition U(93.15)H7.97 Cj.iiO.25 were reflected by natural uranium or iron.45 More nearlyappropriate is the critical mass of paraffin-reflected U(95.3)F6C mixed with polyethy-lene, of H/2'l5V — 10 (Ref. 22). The enriched-uranium-metal points were based on LosAlamos values"0'53-M and ORNL slab data" (with DSN correction from Plexiglas to wa-ter reflector), supported by measurements at Lawrence Livermore National Laboratory(LLNL).56'5" Shape conversions for the metal made use of extrapolation distances fromFig. 7. Core-density corrections for water-reflected spheres were made by using the com-puted relations of Fig. 9 and. for metal, the experimental points of Fig. 8. In regions ofscanty or uncertain data, the curves of Figs, 10 to 13 are guided in form by results of DSNcalculations.

Points of Fig. 10 for water-reflected U(93)O2-H2O (Table 8) mixtures are those reportedby Magnuson of Oak Ridge.58 They result from calculations validated by comparison withdense crit ical arrays of U O J - C S O 2 H H units reflected by polyethylene and containing internalniethvl methacrvlatr-.

29

TABLE 8.

UraniumOxideFraction

1.0.90.80.7U.ti0.50.10.30.20.1

CALCULATED CRITICAL PARAMETERS FOR17(93.3)0;.-]

H-.0Fract ion

0.00.10.20.30.10.50.60.70.80.9

:I_>G MIXTURES

H/2 3 5UAtomRat io

0.00.32*0.738L2661.9692.9544.4316.89211.8226.58

Density

(g/cm3)

8.8177.9367.0546.1725.29!)4.4093.5272.6451.7630.882

Sphere

Radius(cm)

10.1910.3310.4710.6410. S311.0411.2311.3611.4411.34

235 , ,

Mass

(kg)

39.0836.6433.9131.1428.1524.8520.9216.2411.065.39

WATER REFLECTED

SlabThickness

(cm)

3.3763.5023.6383.7803.942•1.1144.2924.4864.6404.842

CylinderRadius

(cm)

6.226.326.436.556.696.846.977.087.157.18

Outside ORNL, there have been limited critical experiments with highly enriched uraniumsolutions. Results do not appear in Figs. 10-13 because they are less easily interpretedthan ihe ORNL data. The ALECTO series in France was primarily a comparision of-3SU, 2l5F'Pu. and 233U critical solutions in similar geometries not necessarily as clean aspossible.S9 U(9r!!O-J(NO3)2 solutions ranging from about 30-300 g

2 3 5 U/L were critical in25- f r 30-cm-diani cylinders with partial water or paraffin reflector and 30- or 42-cm-diamcylinders without reflector. Further French experiments directed toward excursions studiesyielded critical masses of U(93)Oj(NO3)2 solution ranging from 22 to 380 g U/L, in 30-and 80-cm-diam cylinders.60

Critical experiments in the USSR used UO 2 (NO 3 ) 2 solutions with 2 3 5U enrichments of

90%. 10%, and 5C/J. The U(90) solutions were parallelepipeds, without reflector, and withpartial water and water-steel reflectors.61 The range of concentrations was similar to thatof ALECTO.

Two critical unr^flect ed spheres of U(92.14)O2F,, solution have been reported informallyIrotii the I K.* Results, for which there is no elaboration, appear in Table 9.

.John G. 'Watford and .1. C. Smith. Dounreay rLxperiinental Reactor Establishment, Doun-reay, ( nited Kingdom Atomic Energy Authority. 1963.

URANIUM OF VARIOUS ENRICHMENTS

Tables 9 and 10 give critical data lor homogeneous hydrogen-moderated units of ura-nium enriched in 23513 to various degrees. In addition to UO-JF-J solutions, there are soliduranium-bearing mixtures in which the hydrogenous material is polyethylene, paraffin, orSterotex (glycerol tristearate, [(CnHasCOOtaCaHs]. In the tables there has been no at-tempt to convert to a common composition. In Table 9. critical values of one-dimensionalforms (spheres, infinite cylinders and infinite slabs) are obtained from Oak Ridge reportsof critical experiments. They are derived from quoted values of buckling, extrapolationdistance and reflector savings. Results of other experiments at various enrichments appearin Table 10.

Reports of U(4.89), U(3.00) and U(2.00) systems give values of buckling, extrapolationdistance, and reflector saving, from which spherical equivalents are derived (as well a,sequivalent infinite cylinders and slabs). Otherwise, listed spherical equivalents are as re-ported with the original data or as they appear in Ref. 14. The entries for U(95.3) andU(29.8) are from the only heterogeneous systems for which there is sufficient experimentalinformation to permit correction to homogeneous compositions.

Critical masses and critical volumes of hydrogen-moderated spheres of U(93), U(30.3),U(4.89), U(3.00). and U(2.00) are displayed in Figs. 14 and 15. Values for U(3.0U) andU(2.00) include those listed in Table 9 for U(3.00)F4-paraffin and U(2.00)F4-paraffin com-pacts. Dashed curves that extend beyond experimental ranges follow points computedby the MCNP Monte Carlo code15 and the associated cross section set.* Critical datafor uranyl fluoride-water mixtures at the four smaller enrichments are reproduced by thismeans within 0.01 kc// and the method is used for conversion to U(2.00) and U(3.00)solutions.** It may be noted that the use of Hansen-Roach cross sections leads to 3% to4% overestimates of U(2.00)F4-paraffin critical sphere radii.

14

Figures 16 and 17 give estimated ^finite cylinder diameters and infinite slab thicknesses forU(93), U(30.3), U(4.89). U(3.00), and U(2.00). Values for the latter four 235U enrichmentswere obtained from sphere radii by means of extrapolation distances consistent with thesphere radii, infinite cylinder diameters, and infinite slab thicknesses tabulated in Ref. 14.

* Robert C. Little. Los Alamos National Laboratory. Los Alamos. NM 87545, 198P.** N. L. Prnvost, Los Alann-.s National Laboratory. Los Alamos. NM 87545 (1986).

TABLE 9.

H,/335UAtomicRatio

102124147172199245320396449503757

102124147172199245320396449503757

524643735

CRITIAL SPECIFICATIONS OF ONE-DIMENSIONAL FORMS OFURANIUM AT

235 r

Density(g/cm3)

U

0.0940.0890.0830.0700.0650.0560.0480.0400.0370.0340.022

U(4

0.0940.0890.0830.0700.0650.0560.0480.0400.0370.0340.022

U(4.89)O2F

0.04250.03560.0318

SEVERAL ENRICHMENTS

SphereVolume

(L)

(4.89)3O8 - Sterotex.

271208194176164152136135140152273

InfiniteCylinder

Diameter (cm)

. Uureflected62

60.255.253.952,251.049.748.047.948.749.860.8

.89)3O8 - Sterotex, Water Reflected62

15211210595918377808595195

46.341.540.539.238.337.736.937.538.740.352.2

InfiniteSlab

Thickness (cm)

38.935.634.733.732.932.030.930.931.332.239.3

23.420.620.319.619.119.018.919.720.622.030.0

2 Solution, Aluminum Container, Unreflected62

698094

38.640.642.8

24.726.027.5

32

TABLE 9. (cent.)

H -'••'••'IT - • J . - ' C Sphere Infinite InfiniteAtomic Density Volume Cylinder SlabRatio (u, cm'1) (L) Diameter (cm) Thickness (cm)

U(4.S9)O-JFJ Solution. Aluminum Container, Water Reflected62

5246437351000

0.04250.03560.03 IS

0.0240

4553

65

132

31.833.8

36.316.7

l/(3.nO)F4 - Paraffin, rnreflected03

l'(2.U0)F4 - Paraffin, I nreflected63

195291406496

614972

0.06270.0528

0.04440.03940.03450.0245

379239202201224513

67.657.854.654.7

56.6

74.8

V(2.00)F4 - Paraffin. Plexigles-Paraffinor Plexinlas-Polvetlivleiir Reflect edfi!

195294406496

614972

0.06270.052*0.04440.03940.0345

0.0245

17.919.320.9280

133 0.093 208 55.3 34.6277 0.066 100 43.0 26.5

l'(3.00)F4 - Paraffin. Plexiglas-Paraffinor Plexiglas-Polyethylene Reflected63

133 0.093 124 43.6 22.9277 0.066 60 34.1 17.6

42.436.134.034.235.547.1

257161139

142163113

56.648.346.046.6

49.067.9

31.526.5

25.426.2

27. x40.2

33

TABLE 10. CRITICAL MASSES AND VOLUMES OF HOMOGENEOUSHYDROGEN-MODERATED SPHERES, HEMISPHERES, AND CUBESOF URANIUM AT SEVERAL ENRICHMENTS

Reflected" UnreflectedH/335UAtomicRatio

235^'

Density(g/crr.3)

Volume(L)

Mass(kg235U)

Volume(L)

Mass(kg 235U)

10 1.48

U(95.3)F4CF2 - Polyethylene Cube22

Paraffin Reflected

10.2 15.1 —

U(44.6)O2F2, Water Solution, Spheres'1

365635

258493678

0.07050.0407

U(44.6)O2F2, Water Solution,

0.09740.05180.0379

——

Sphere

11.519.529.5

——

Dimensions

1.121.011.12

22.2734.91

Transformed from

17.6

1.5701.421

Cylin

1.71

0.1

7621835153457378310371193277488

1.18

0.2880.11300.07160.04780.04450.03280.02480.02160.09000.0520

U(37.5)F4-CF2 Cube62

156 184 -408

U(30.45)O2F2, Water Solution. Spheres'1

14.85

22.2734.91

91.3417.44

1.063

0.9911.143

1.9731.571

22.2722.27

34.91

91.34

45.61

-482

6.402.517

1.667

2.265

2.375

34

TABLE 10.

H/235U

AtomicRatio

(cont.)

235 u

Density(g/cm*)

Reflected"

Volume Mass(L) (kg23sU)

Unreflectcd

Volume Mass(L) (kg23sU)

U(30.45)O2F2, Water Solution, Hemispheres,0

728 0.0352 45.67 1.608 — —

U(30.3)C>2F2, Water Solution, Sphere Dimensions Transformed from Cylinders6

76.710"167257378439

0.2880.2200.1460.0978

• 0.06750.0584

11.311.611.613.016.117.1

3.262.541.701.281.081.00

19.520.020.022.126.327.7

5.624.382.932.161.771.62

657815

0.03940.0317

27.838.1

1.101.24

42.155.5

1.661.76

U(30.14)O2-CH2 Compacts, Cube Dimensions Transformed from Parallelepipeds43

Perspex Reflected

8.148.1416.316.339.281.381.381.4

8.1416.316.339.281.381.381.4

1.5701.190c

1.1300.845c

0.6680.3320.2480.244'

1.1901.1300.8450.6680.3320.2486

0.244d

13.927.69.3719.376.665.8312.1y.58

21.832.910.5916.404.451.943.00d

2.38d

31 ±2—26.5—

18.4015.3438.3rf

25.6rf

Polyethylene Reflected

32.410.2122.37.216.3213.710.58

38.611.5418.94.822.103.392.58

26.5—

18.4015.3438.3C

25.6C

49±3—30.0

12.295.099.51d

6.24rf

__

30.0

12.295.099.51C

6.24C

35

TABLE 10. (cont.)

Reflected" Unreflected

H / 2 3 5 I ; -3 5uAtomic Density Volume Mass Volume MassRatio (g/cm*) (L) (kg 235U) (L) (kg235U)

U(29.S3)F4CF2-Polyethyleue Cube,23 Paraffin Reflected

32 0.542 2.66 7.45

U(1S.8)F4-CF2 Cube, heterogeneous02

0.14 0.591 733 433

U(14.7)O2SO.,, Watei Solution, Sphere65

-0.081 14.8 -1.2

U(4.9S)O2F2, Water Solution, Sphere66'67''

490 45.3 44.4 2.01 68.5 3.11

l'(l-42)F4-Par8.ffin Compacts, Transformed to Sphere14'6

Water Reflected

418 0.0353 697 24.6 847 29.9

562 0.0305 728 22.2 873 26.6

Polyethylene Reflected

418 0.0353 665 23.5 — —662 0.0305 717 21.9 — —"Water reflected unless indicated otherwise.6John G. Walford and J.C. Smith. Dounreay Experimental Reactor Establishment,

Dounreay, United Kingdom Atomic Energy Authority, 1963.CJ. R. Dominey and A. F. Thomas, Atomic Weapons Research Establishment,

Aldermaston, United Kingdom Atomic Energy Authority, 1962.^Estimated from replacement measurements on one face.rExtra graphite wa:. tided to change the effective composition of the wax to CH.'See Fig. 18.

36

100-

A U(2)F4-PARAFFIN

0 U(3)F4-PARAFFIN

U(5)0aFs SOLUTION

O U(30.3)02F2 SOLUTION

V U(93)0EFE SOLUTION

X CALCULATED U O J F J - WATER

4- CALCULATED UF4-PARAFFIN

100 500

H/235U ATOMIC RATIO

1000 2000

/•'/(/• H- Critical masses uf water-reflected spheres of hydmgen-moderated U(93j, U(J0.3). U(5.00), U(J.QO)and U(2.00).

so

u

OS

ca

en

ouuu

1000-

100-

i nIV

5-

V

X

• •

V

X—

-. .

X

-- x

X

^

V

X

--.

A u(a)F«-

0 U(3)F4-

r • ~

••IK1

-PARAFFIN

D U(5)08Fj SOLUTION

O U(30.3)OjFs SOLUTION

V U(93)0zFz SOLUTION

X CALCULATED UO2F2-WATE

<

>

->

< x

/ ' XJ /

—?-y/ /

'' //

20 100 1000 2000

H / 2 3 5 U ATOMIC RATIO

Fig. 15. Critical volumes of water-reflected spheres of hydrogen-moderated U(93), U(30.3) U(5.00), U(3.00)and U(2.00).

200

BascaCd

OS

Q

oEdE-

<

U(2)F« -PARAFFIN

U(3)F4-PARAFFIN

U(5)0jF2 SOLUTION

U(30.3)OzFj SOLUTION

U(93)0tFi SOLUTION

CALCULATED l3OsFs.-WATER

ioooH/235U ATOMIC RATIO

2000

COCO

Fig. 16. Estimated critical diameters of infinitely long water-reflected cylinders of hydrogen-moderated U(93),U(30.3), U(5.00). U(3.00) and U(2.00).

100

100 1000 2000

H/335U ATOMIC RATIO

Fig. 17. Estimated critical thicknesses of water-reflected slabs, infinite in other dimensions, of hydrogen-moderated U(93), U(30.S} U(5.00), U(3.00) and U(2.00).

Fig. IS. Spherical container, suspended by stainless-steel cables, for establishing unreflecfed critical specifica-tions of 1'(.J.00)O2F2 solution.

66 The stainless-.tterl-wall thickness .sas 0.05 cm. and the inside diameter was.',0.77 ± 0.03 cm.

Instead of being listed in Table 10, data from Los Alamos for unmoderated uranium metalat various enrichments appear in Fig. 19.68""'1 Because the curves for unreflected uraniumand uranium with a natural uranium rellector appear to be parallel, it is assumed thatthe curve for water reflection would also be parallel as indicated in Fig. 19. Exponentialexperiments, which supplement critical experiments, indicate that unmoderated uraniumcannot become, critical if the J35U content is below 5 or 6 wt%. A cooperative Europeanreactor physics program has narrowed this limiting critical enrichment to 5.56 ± 0.02 at%235U, i.e. U(5.49)7-

This 235U enrichment limit applies to single undiluted uranium units. As shown by thefollowing Oak Ridge critical data, it does not apply to clusters of metal units in water. Forexample, ten 442-kg U( 1.95) metal annuli were critical as a triangular lattice with 2.54-cmoptimum separation of aunuli.73"1 The dimensions of these units were 18.3-cm-o.d., 6.60-cm-i.d. and 101.6-cm-long. The critical mass. Sfi kg 2i*V compares with 54 kg 235TJ in asimilar 9-unit critical array of U(1.95) atmuli with the outside diameter reduced to 15.7cm and other dimensions the same.

Optimum critical lattices of massive U(3.85) annuli, and rods consisting of annuli withwater-filled interiors, again were triangular with 2.5-cm separation of units. As examples,the interpolated critical mass of 332-kg aimuli, 18.3-cm-o.d., 6.6-cm-i.d. and 76-cm-long,was 57-kg 235U. and that of 380-kg rods of the same outside diameter and length was 90kg 235 lT ,73.74

42

enCOC

2

<

5

cuuu

1000

1 r\r\LUU

i nIV

•

A

5.56

w

g

in

E-Zbl

CR

ITIC

AL

EN

RIC

HM

MIT

INC

\\

\A

XQJ UNREFLECT

O FROM CRI'riCAL SPl• I L I R E S

- - D FROM CRITICAL CYLINDERS

A

ED

FROM EXPONENTIAL EXPERIMENTS

^—o

1

I

NATURALJRANIUMiEFLECTO

-n

R

WATERREFLECTOR

O

•o

10 100235U ENRICHMENT OF URANIUM METAL (wt%)

Fii/. 19. Critical mass vs 23r>(r enrichment of iirxmiunt metal. The dashed line represents the enrichmenthi low which a puce of uranium metal cannot become critical, as determined by an international react or physicsprngmm.

HETEROGENEOUS WATER-MODERATED URANIUM AT VARIOUSENRICHMENTS

NEAR-HOMOCiENEOlS HYDROCF.X MODERATED IRANIVM ENRICHED IN 235U

Parallelepipeds and a pseudocylinder of U(lJ3.16) metal foil (0.005- to 0.030-ciu-thick)interleaved with various combinations of 0.16-cni-thick Plexiglas plates and 0.46-cnt- or0.71-cvn tliirk graphite plates were effectively homogeneous.'5 An aluminum matrix ata mean density of O.lf.io g,'cm3 was distributed throughout the core and comprised thereflector. Critical conditions appear in Table 11. Volumes of equivalent spheres are quotedin the reference.

WATER-MODERATED LATTICES OF SLIGHTLY ENRICHED URANIUM

Early lueiisurements on lattices of slightly enriched uranium metal or oxide rods in waterwere summarized by Kouts et al. at the (ieneva Conferences of 19?»!J and 1957.'°''' Exper-iments with exponential columns established values of buckling and extrapolation distancefor a number of lattice spacings at each rod diameter. For each diameter of U(1.027),U( 1.143), and U(1.299) metal rods, a spacing leading to the largest value of buckling, thussmallest value of critical volume, was spanned. Such volumes for equilateral cylindrical lat-tices are listed in Table 12. (Equilateral cylindrical geometry is chosen as more appropriatefor arrays of rods than spherical geometry.) There are less extensive data for lattices ofaluminum-clad U( 1.3)0.. rods.77 r(3.f)'.)O;2 rods clad with Inconel78 and U(4.02)O2 rodschid with stainless steel.''

T A B L E 11. A S S E M B L I E S t )E l'|!>:{) WITH I I Y D R O C E N AND C A R B O NM O D E R A T O R S "

- TDensity

(B.-"..:I)

2.3032.0900.9170.52!1.3170,2580.2580.2580.4800.3360.223

Atomic

fi.O

fi.O

(i.O

(i.O

12.112.312,412.435.135.235.5

Ratio

c -'••'•r'r

3.7(i

3.712-1.3•18.57.(i9S.798.298.221.948.299.4

Dimensions(cm)

59.7 x 30.5 x 23.559.7 x 30,5 x 25.79.7 x 15.7 x 15.759 .7 x ( i l . l t x (i9.!i

38,! x 30.5 x 2*.959.7 x 72.4 X 72.9(il.3 x 62.9 x f>2.601.3 long x HV1.1 diani38.1 x 340.5 x 30.538.1 x 38.1 x 42.438.1 x 53.3 x 53.3

Critical

Observed

-12.N50. S12525133.(i31532030535.461.5108

VolumeL)

SplilTt

25. cS

33.3

10020.X

27.(3

25.5

25326329.3

51.585

" A c o m p l r t c c o m p n l a t iniinl s u r v e y of cr i t ic i i ! m a s s e s of V ( 9 3 . . r ) ) - \ v a t e r - g r a p h i U ' sp l i c r c s

r t - su l t fd in K i ^ . 7 of Rcf. I I .

45

TABLE 12.

Enrichmentin -3H'

U| 1.027)

V( 1.143)

U( 1.299)

U(2.0)

I' (3.063)

LATTICES OF SLIGHTLYWATER

RodDiameter

(cm)

0.9S1.521.90

0.9S1.52

0.981.52

1.522.35

0.4450.7621.522.35

ENRICHED I

A verage2tflU Density

(S/cin;t)

0.0550.060.065

0.0550.065

0.060.075

0.0950.12

0.090.1050.150.175

TRANIUM METAL RODS IN

Critical Volume of EquilateralCylinder at Optimum Lattice

Spacing (L)

524430393

274238

175155

58.256.6

32.029.830.135.1

Subsequent exponential experiments at Hanford provided the further optimum criticalvolumes for U(2.0) and U(3.063) that appear in Table 12.'9 Savannah River personnelextended the U(3) data to 5.1-cm- and T.H-cni-diani rods.80 It is illuminating to plotthe U(3.0t>3) critical volumes of Table 12 and corresponding critical masses against roddiameter, as in Fig. 20. The minimum critical volume of 29.5 L occurs with a rod diameterof about 1.1 cm. The critical mass minimum is far below the smallest rod diameter shown;in fact, computations indicate that it should occur at a diameter of about 0.35 cm. Thecorresponding volume would be extremely large.

Oak Ridge reported critical lattices in water* of U(4.9,ri)**inetal rods at a range ofdiameters f" ' '8 Spherical critical masses were calculated from experimental values ol buck-ling and corresponding extrapolation distance. A minimum spherical critical mass of 1.6kg J I 5 1 ! at a rod diameter of about 0.17-cm compares with a minimum spherical criticalmass of 2.0 kg 2 J 5 lT for l;(5.00) solution with water reflector.

Experiments at Valduc established critical heights of water in lattices of U(4.75)O2 rodsat various pitches.81 Each rod contained 0.79-cm-diam by 90-cm-long UO2 at a density of10.4 g/cm3 and was clad with 0.06-cm thick aluminum. Extrapolation to fully immersedand reflected conditions led to a critical square lattice of 217 rods at an optimum pitch of2.1 cm.

Boron concentration of 0.1-ID R,'L increased critical masses about 2"//f.Corrected by the author from r(-l.S9) reported in Ref. 7

7

6

00 c

eg 5

o

I

o'

1 I 1 . 1 1 1 1 1 1 1 1

36

0 1 2

ROD DIAMETER (cm)

34

32

30

28—

I—I—

r—i—

i—i—

i—i—

i—

-

\

1i

!

]

J0

/ i/;--

0 1 2

ROD DIAMETER (cm)

Fig. 20. Critical mass and volume for lattices of U(3.06J) metal rods in water. At each rod diameter, thecritical volume is that of an equilatcml cylinder nt optimum lattice spacing, and the critical mass corres]x>nds.

WATER-SPACED UNITS OF ENKICHED URANIUM

Highly Enriched Uranium

Unlike slightly enriched uranium lattices, water lattices of uranium enriched to more than10% 23-">U have critical masses that are greater than the corresponding homogeneous sys-tems. This is illustrated by results in Fig. 21 of Los Alamos measurements on lattices ofU(93.5) metal as 2.54-cm cubes, 1.27-cm cubes and 0.318-cin-diam rods."" Surface spacingfor minima in critical mass varies from 1.8 cm lor the 2.54-cni cubes to 1.5 cm for the0.318-cm rods.

A limited number of Oak Ridge experiments with MTR-type fuel elements provide guidancefor pool storage and recovery operations.8184 These elements were roughiy 7.6 cm-squarein cross section and the fissile material. U(---93), was contained in a number of closelyspaced clad plates with the active fuel length about (SO cm. The ~3SV content was 140.168 or 306 g per element. Measurements established, for each -•'H> loading, the elementseparation in water required for optimum moderation and effects of greater separation onthe critical number of elements.

As an example. 47 of the 306 g elements spaced about 4 cm apart were critical latticed inwater. Further, 4 g 235U/L as U(93)02|NO)o dissolved in the reflector-moderator reducedthe critical mass of a compact lattice by about one-third, and 1.12 g B/L dissolved in thatsolution increased the critical mass by a factor of five. Influences of cadmium interspersedin various flooded storage patterns also were established.

60

50

in

S8

5

40

30

20

10

-i 1 1 r-

O 2.54- cm CUBES

n 1-27- cm CUBES

A 0.318- cm DIAM BY 30.5- cm LONG RODS

(3.5)

•—D •( 10.3 ) ( 17.5 )

( 38.4 )

- i 1 r~

•_ ( 1 3 3 ) _(SUBCRITICAL)

0

£ (SUBCRITICAL)

: £ \ (132)

-1 1 1 r- 1 1 r- 1 1 r

0 1 2 3 4

SURFACE SEPARATION OF URANIUM PIECES (cm)

F'g. 21. Critical masses of water-immersed U(94) metal lattices as functions of spacing between units. MeanA/235 U ratios are in parentheses.

Uranium of Various Enrichments

Over the range of 235U enrichments, minimum critical masses of heterogeneous uraniumin water compare with homogeneous values as shown in Fig. 22. Below about 10% J35Uenrichment, heterogeneous critical masses are smaller than corresponding homogeneousvalues. Heterogeneous minimum critical volumes (Fig. 23), diameters of infinite cylinders(Fig. 24), and thicknesses of finite slabs (Fig. 25), however, are less than correspondinghomogeneous values throughout the entire enrichment range.

These figures, originally from Callihan, Ret". 9, have been modified to include the subse-quent Oak Ridge experiments, in particular, the comparison at U(~ 5) noted earlier. Asa reminder, the minimum critical nuiss of U(4.95) metal rods latticed in water was 1.6 kg235^67,78 . i n d t h a t of wat,er-reflected U(5.00) solution was 2.0 kg

235U. Further, homoge-neous values at 2.00 and 3.00 wt% enrichment are derived from ORNL experiments withU(3.00)F4-paraffin and U(2.00)F4-pararh'n compacts.

6:t

51

3O

235U MINIMUM SPHERICAL CRITICAL MASS (kg)

Ifnj33

II'r,•j£*to< • • * .

n

3

3e3

3gnt

3 ?*

toCo

mCo

OCo

o'

tow01

ft

1

s

S3

g

enc

t - c

i—»

o ~

Oi .o

1—' CJl

?ITICAL HETEROGENEOUS LIMIT

111 1RITICAL HOMOGENEOUS LIMIT

/

/ /

/ // /

: /i I11:

i]—i

/ /

/ /

1 / /1/ /

/j /r

I T_T !

i

—/u-

o

I2ouwoG

5̂SJ

awPI

o

•3onn

md

- 1

ra

mGH

---

---

0

UO

- - -

- - -

a§c»i

wo

MINIMUM SPHERICAL CRITICAL VOLUME (L)

S 2

S 3

1

s0_

i

3

a-3

3-O3

;S3Oa:

3~0)

3

—̂'Ul

H-fc —

CJI -

o

mo

i—••o

o1 • |1 1CRITICAL HETEROGENEOUS LIMIT

11 M lCRITICAL HOMOGENEOUS LIMIT

/

//

/

/

/

/

7////

I1////

/1

\

//

1o

1znf

ii/

/

/

/

r/ XLJ U/ /

1

I1

oo

Pi r̂/ " /I

/

OO

. - • - _ : -

D> D O

- X X n:w o os § §5 o oo w m2 2 25 m mg o oM C CO 03 tfl

I i 1w » 5

r- oP3 -4O

300

5 10

235U ENRICHMENT (wt%)

50 100

Fig. 24- Minimum critical infinite cylinder diameters as functions of23sU enrichment in homogeneous andheterogeneous hydrogen-moderated systems.

MINIMUM CRITICAL INFINITE SLAB THICKNESS (cm)

P

3 J"a. 5.

n'

"> oto 3

a* *

a to3 u

33.ft3-3"I

3-Q

3

ssnPIo

osH

Ui

t—k —

en -

o

g

o -

cn

CRITICAL HETEROGENEOUS LIMIT

CRITICAL 1HOMOGENEOUS LIMIT

/

/

/

r

g3wo N

CTED

/

/

3ii

////

o

/

/

/ /

/ / ^/ /

/ // /

_j J.

' / /

r' TZ_ ^

/

/

/ /

//1

QgHeDM

ooo

*

1

5l

M r

"8

METAL-SOLUTION MIXTURES

All experimental information about another practical type of inhoinogeneons system, com-binations of massive fissile met til an:! iissile solution, is derived from subcritical measure-ments at Rocky Flats. One set of measurements applies to spaced 0.15-cm-thick disks ofU(93) metal in solutions containing 102 and 308 g U(93)/L.85 Other measurements areof more general interest because they may apply to U(93) billets in cleaning solution.86'87

They relate concentrations of j a5U in solution to the critical thickness of a U(93) metalslab within the solution. Figure 26 gives results for 12.7- by 20.3-cm slabs in a 24-cm-diam,41-cm-high solution, and Fig. 27 does the same for 25.4- by 41-cin slabs in a 76-cm-diam,71-cni-high solution.

56

COco~co

o

E-

<

CO

62o

STA1NLESS STEEL CYLINDER

0 15-cm-THICK WALLS

24.0-cm INSIDE DIAMETER

A = 12.7 cm; 8 = 20.3 cm; C = 40.6 cm

0 20 40 60 80 100

DENSITY OF 235U (g/L)

120 140

F»

STAINLESS STEEL CYLINDER

27-cm-THICK WALLS

"6 2-cm-INSIDE DIAMETER

A = 25 4 cm; B = 40 6 cm.

C = 71 l cm: D = 15 2 cm

0 2 6 8 10 12DENSITY OF 235U ( g/L)

Fig.27. Critical thicknesses of 25.4- by ^0.6-cm l'(93)mctal slabs immersed m 16-cm-diam U(93)O-2(NO-s)2solutions at various 235 U densities.

HOMOGENEOUS HYDROGEN-MODERATED PLUTONIUM

PLUTONIUM SOLUTIONS. LOW -MUPu

Plutonium solutions, nominally Pu^NO^).,. are not as stable as uranyl fluoride and uranylnitrate solutions.* The addition of nitric acid, one-hall normal or greater, is necessaryto prevent the formation of polymer over most of the plutoniuin concentration iange. At300 g Pu/L ami greater, polymer can appear even in the presence of nitric acid. Forcomparison, uranyl fluoride solutions without excess acid may attain almost 1 kg U/Lwithout precipitation.

Karly Hanford critical experiments with plutoniuni solutions, both spherical and cylin-drical, are reported in Ref. 26. Results for waier-rellected solutions are summarized inFii;. 28 where there have been empirical adjustments for J"10Pu and nitrate contents. Therelations between H/Pu and Pu density of Table 7 are supplemented by Fig. 29, whichshows how that relation changes with HNO3 molarity.

NlS Results axe supported by Harwellmeasurements on partially reflected solution.H''

An extensive program at Sac lay led, among other results, to the data summarized in Tablej3_59,so-92 Further, supplementary Hanford critical data for solution bpheres with variousreflectors are reported in Ref. 93 (see Fig. 30). Table 14 contains the items of Table 1 ofthat report for which corrections for neck and support of spheres are available. Data forreflectors of 1.27-cm-thick paraffin. 10.2-cin-thick concrete, and concrete with gaps are notincluded because such corrections are lacking. Reference 93 also includes critical massesoi hypothetical x-php-se'^'Pu-water mixtures over a range of phitonium densities, fromcalculations that give near agreement with experimental data.

Critical masses and volumes of water-moderated plutonium appear in Figs. 31 and 32,diameters of infinite cylinders in Fig. 33, and thicknesses of infinite slabs in Fig. 34.26>M

1050

1000-

Ofl

ya:

950-

900-

850-

3C

OT 800

g ^50 HJ 1

700-

65014

TOTAL

16 18— | r- -] 1 [—

20 22 24

SPHERICAL VOLUME (L)

26 28 30

Fig. 2$. Spherical critical masses and rnhinifs of water-reflected Pu(NOz)4 solutions with various nitratedensities.

1000 -r

P u METAL-WATER(No HNO

80 100

PLUTONIUM DENSITY (g/L)

200 400

C!/•'if/. 29. Ator)iic ratios ll/Pu as functions of plutoniutn dtnsity forM. 7.5 i\[ and 10 M HXO3, and for idealized plutonium-water mixtures

.solutions containing 4 M, 5.6

TABLE

H Pu

560

h 10

660

710

780

820

880

13.

s 1-

44,

41

38

35

32

30

28

Sc

, 1

.0

.0

.6

. i

.9

CRITICAL DIMENSIONS OFPu(NO3)4, FROM

ilution ConipositioiDensity

I- Is, cinJ)

FRANCE

i

Total XO,

m D

Cylinder o.d. 30.0

1.131.131.121.121.111.111.11

174

170

162

Hi I

157

157

153

' CYLINDERS

Height ( : 0.05

; OF

Critcm)

cm. Wa te r Retlectc

31.2

33.135.4

37. !t

41.1

14.N

49.6

SOLUTIONS OF

ual Du.ie.isionsVolume (L)

>r

21.3

22.6

24.1

25.928.130.633.9

Mass

0.950.930.920.910.920.940.98

(kg

- 0.± o.x 0.x 0,± o± 0± o

Pu)

012

012

012.012

.012

.012

.012

Cylinder o.d. 30.0 cm. Conc re t e Reflector

540H70

770

900

940

580

640

710

740

8 1 0

880

47.037.5

32.8

28.2

27.0

43.6

39.4

35.4

34.0

31.228.7

1.131.12

1.11

1.10

1.11

Cylinder

1.13

1.12

1.11

1.111.10

1.10

170166

155

142155

o.d. 30.0 cm.

16215S

153152145

151

30.735.039.045.449.0

Beech Wood

33.736.140.141.146.651.9

20.

23 .

26.

31 .

33.

Reflector

23

24

27

28

3135

99

i

1

5

.0

.9

.3

.1

.9

.6

0.98 ±0.90 ±0.88 x0.88 x0.90 x

1.00 x0.98 x0.97 x0.96 ±0.99 x1.02 -

0.0150.0130.0120.0120.012

0.014

0.014

0.0130.0130.013

0.013

Cyl inder o.d. 33.0 cm. Wa te r Reflector

510650

751)

X30

X!M)

9*0

1071)

Nut..

wall

45.5 13.x. 233.3

31). 1

2.x.0

25.7

23.4

•: J ' " ' Pu cunt cut --

thickness 0.3 cm.

1.131.11

1.12

1.11

1.10

1.111.09

. i . : . ' • ; :

177176

;«s161

152146

145

reflector tliickm

26.329.032.735.638.543.751.6

•ss 40 cm mi lateral

20.5

25.728.130.434.741.7

1 surface only.

0.93 -_0.87 x0.86 x0.85 ~0.K5 -0.89 :0.96 :

stainless

0.0090.0080.0080.0070.0070.0070.007

steel

62

Fig. 30. Aluminum spherical container, of 9^9 L capacity, used at Han ford for measurements establishingtin minimum critical concentration of plufonium in solution.96 The same sphere had been used at Oak Ridgefor similar measurements with 233 U and 23ri U solutions.51

OS

TABLE 14. CRIT1CALITY OF Pu(NO3)4 SOLUTIONS (4.6% 240Pu)

Density HNO3 Total NO.-, Critical Critical MassReflector H/Pu (gPu/L) Normality (g/L) Volume (L)a (kg Pu)n

29

WaterWaterWat erWaterWaterWaterWater28.6-cm Concrete2S.6-cni Concrete

.3-em-diam

354344243192878154341100

Stainless

73741001262692956

435b

75236

Steel Sphere,

0.20.41.92.21.10.80.80.441.16

0.124-cm-thick Wall

86105230262346303372109318

12.912.912.912.912.912.912.912.912.9

0.940.961.291.633.473.815.610.973.04

35.6-cm-dia«i Stainless Steel Sphere, 0.112-cm-thick Wall

2.54-cm Paraffin2.54-cm Paraffin2.54-cm Paraffin25.4-cm Concrete25.4-cm Concrete25.4-cm ConcreteWaterWaterWaterWater 4 0.183cm

Stainless Steel

44325622384865151S754618466452

54859829.636.643.4;;3.238.647.549.5

4.36.46.11.074.396.372.084.076.666.01

323482473118310445164292462424

23.423.423.423.123.123.123.123.123.123.1

1.251.992.280.680.851.000.770.891.101.14

38.6-cm-diam Stainless Steel Sphere, 0.122-cm-thick Wall

NoneNone0.66-cm

Stainless SteelWaterWaterWater + 0.203-cm

Stainless Steel

668125758

10685531032

39.017234.3

24.438.725.2

"Corrected for neck and support.''Contains Pu(VI) ;and plutonimn poly

0.44.90.5

0.57.70.5

mer. mere

6448666

5851760

30.830.830.8

30,130.130.1

using the uncertainty to i 5%

1.5.1.

0.1.0.

203106

731676

en

59

E -

2

a:

CLCO

CL

12 |1 r\1U

1l

n AU.4

II: U

I \

: O N

C

\\

'—c

i

>

—i—r

UNI

0/

/>

REFLECTED| /

/

/ '

O/

o /

T //

- • - LIMITING CRITICAL DENSITYi i i i

I i i

.̂/

o/ c

c

c

/

] D

^ F

/

ATA

ATA

' P iROli

/

/

1

FRO1

DERI

i METAk POLY

WAT C D

/ REFLECTEE

I WATER REF

VED FROM V

L-WATER MIXSTYRENE COJ

LECTE

FATER

T U R E^PACT

1

:D s

RE

SDAT

A -

PH

FLE

A

—__

ER

CT

**>

E3

ED C

7 UNREFLECTED SPHERE DATA

> z 3 ' P u METAL-WATER MIXTURESFROM SOLUTION DATA

11

I

11

I1

!

Si

\ j\)Ag

CYLINDERS ;

3

a-PHASEPu METAL —*i1 i t i t i i i !

i i i

0.005 0.01 0.1 1DENSITY OF Pu (kg/1)

10 20

Fig. Si. Critical masses of homogeneous water-moderated plutonium spheres. The points suggesting o'

O

g f gintermediate curve apply to wafer-reflected Puthe plutonium.

p p p g g gsolution with I N HNO3 and 3.1% Pu content of

_D

O

<ya:o

>O DATA DERIVED FROM WATER REFLECTED SPHERES

D DATA DERIVED FROM WATER REFLECTED CYLINDERS

A 2 3 SPu METAL-WATER MIXTURESFROM POLYSTYRENE COMPACT DATA

O 2 3 BPu METAL-WATER MIXTURESFROM SOLUTION DATA

• LIMITING CRITICAL DENSITY a— METAL

0.005 0.01 0.1 1

DENSITY OF Pu (kg/I)

10 20

Fig. JJ. Estimated critical diameters of infinitely long cylinders of homogeneous water-moderated plutomum.The points suggesting an intermediate curve apply to water-re flee ted Pu (XO^j^ solution with 1 S HNO3 and3.1% 2A0Pu content of the plutomum.

~S

<

0.5-

O DATA DERIVED FROM WATER REFLECTED SPHERES

D DATA DERIVED FROM WATER REFLECTED CYLINDERS

V DATA FROM WATER REFLECTED SLAB

A s 3 8 Pu METAL-WATER MIXTURESFROM POLYSTYRENE COMPACT DATA

| \

2 3 9Pu METAL-WATER MIXTURESFROM SOLUTION DATA

•LIMITING CRITICAL DENSITY

t - l l - i

I Ia-PHASE Pu METAL

0.005 0.01 0.1 1

DENSITY OF Pu (kg/L)

10 20

Fig. 34- Estimated critical thicknesses of slabs, infinite in other dimensions, of homogeneous water-niodemtedphifonium. Unrefiected infinite slabs are fictitious. The points suggesting an intermediate curve apply towafer-reflected Pu (N03)4 solution with 1 N HN03 and 3.1%

240 Pu content of the plutonium.

II /PuAtomic

RatioSpline Radius

( f i n )

Infinite CvlindeiDiameter

(cm)

TABLE 15. CRITICAL DIMENSIONS OE REELECTED SPHERES, INFINITECYLINDERS AND INFINITE SLABS DEIMCED FROM EXPERIMENTSWITH PuO... POLYSTYRENE COMPACTS

Infinite SlaliThickness

(cm)

5 PuO.. - Polystyrene, 2.30 g PU/CIII3 , 1 1.46'',' -'"'Pu:"14.15 ; 0.4 17.9 -i 0.7 5.9 -t. 0.2

-:i!lPu - Water, 1.16 g Pu/rm3 , 0",' -M"Pu:''S,| : 0.1 10.2 : 0.4 2.7 i 0.1

5 -:it'Pu0-.. - Water, 3.46 g P n c m ' , OS'T -'"'Pu:'1

9.3 r 0.1 11.5 : 0.5 3.24 ± 0.1

15 PuO2 - Polystyrene, 1.12 g Pu/cm3 , 2.2% 240Pu:"

13.5 -1 0.2 17.2 ± 0.4 6.0 ± 0.1

15 PUO2 - Polystyrene, 1.05 g Pu/cm3 , 8.0% 210Pu:a

15.6 i 0.2 20.1 ± 0.4 7.4 ± 0.1

15 -M;'Pu - Water. 1.62 g Pu/cm3, 0% 240Pu:6

10.fi r 0.2 13.4 + 0.3 4.5 x 0.1

15 ->i"PuO. - Water. 1.50 g Pu/cm3, 0%, 2"°Pu:'1

11.0 .: 0.2 14.0 i 0.3 4.S - 0.1

50 P)iO2 - Polystyrene. 0.367 g Pu/cm3 , 18.35% 24oPu:a

20.0 : 1. 26.5 ± 0.2 10.9 ± 0.1

50 -1!'Pu - Water, 0.521 g Pu/cm3, 0% 24OPu:(/

11.6 r 0.1 14.S ± 0.2 5.2 - 0.1

50 2ai lPuO2 - Water, 0.50S g Pu/cm3 . 0LA '-""'Pu:6

ll.N : 0.1 15.0 - 0.2 5.3 :: .1

50 Pu - Water. 0.521 g Pu/cm3. 1^.35% ->10Pu:1'17.2 : 0.1 22.S : 0.2 9.4 - 0.1

50 PuOj - Water. 0.50S R p,, n n: i . 18.35'"; 2'lllPu:''

17.4 : 0.1 23.2 : 0.2 9.6 z 0.1

" Plexiglas reHeetor.''Water reHector.

TAB1.K It.. CKI1 1 c'A1. DIMENSIONS OK REFLECTED SPHERES. INFINITE

CYLINDERS. AND INF1NLI M SLABS DEIM'CED FROM EXPERIMENTS

WITH PuO, COMPAC IS AT 11 1'u 0.01

Composit ion (g cm

PuO-j-water" .r..Sj : t ( 'Pu-water ' ' lit. Hij : i "PuOj-water ' ' !>.!•»()

PuOj-water '1 1CU (i

P\ iO.-water K.H(i

" Ple.\ij;la> reflect i>r.

' 'Water relied«>r.

•-'•"'Pi.

( W t ' ' ( ' )

is.:r.0

ii

is.;r.1 s.;r.

SphereRadius

(cm)

1 1.2 r 0.1

•1.22 : 0.1-1

fi.fi : 0 .2

•1.70 :; 0 Ifi

7.-1 : 0.3

Infinite

Cylinder

(cm)

I3.-1 i O.ti••l.Tti i 0 .22

7.7 :1 0.3

'

PLUTONIUM SOLUTIONS, UNLIMITED --"'Pu

T h e critical int ini t f slabs repor ted ln

TABLE 17. CRITICAL WATER-REFLECTED PLUTONIUM SPHERES FROMINFINITE SLAB THICKNESSES OVER A RANCE OF *>40Pu CONTENT

Infinite SlaU ._. Sphere

Thickness Calc.(Obs.) Calc.(Obs.) Critical Critical Critical Mass(cm) (

cj

u

a:wDL

30

20

10-

5-

2-

n f\u.o

20

15

5

t

i

\

\

-^

'

-^

1

+T -

/

7/

W/

/i' /

•'' / A

/ ' /

/

-1

//

/

/

//

/ //

/

/

/

/

//

/ //

—̂

L

{i i

i

L_1 —

—̂—'

j

Kk

1

i

i

3HASE

i ^ ' v ' , ' > ' • • ' , '

• v . '•-,•..;

i1

METAL - * i

0.01 0.1 1

DENSITY OF Pu (kg/1.)

10 20

FH). •>.'). Cnlculatfd effect of ~'w Pn content on the critical mass of watt r-rcflee ted homogeneous plutoniumrn

PLUTONIUM — POLYSTYRENE COMPACTS

Three reports from Hanford99" l u l describe critical experiments with homogeneous PuO2-polystyrene mixtures at H/Pu ratios of 5, 15 and 50. Each set of experiments applies to hareand Plexiglas-reflected parallelepipeds ranging from elongated to squat shapes. For eachcomposition, critical dimensions are extrapolated to the infinite cylinder and infinite slaband converted to the equivalent sphere. Calculations based on these dimensions also givecorresponding dimensions for plutonium-water and PuO2-water mixtures. Reported sphereradii, infinite cylinder diameters, find infinite slab thicknesses for the reflected systems aregiven in Table 15.

The PuO-j-polystyrene experiments were supplemented by similar measurements withslightly moderated compressed PuOj compacts.10"' The deduced critical dimensions ofreflected spheres, infinite cylinders, and infinite slabs which were reported appear in Table16.

HOMOGENEOUS PLUTONIUM-URANIUM SYSTEMS

Liquid Metal Fast Breeder Reactor (LMFBR) fuel cycle operations have led to crit-ical experiments with moderated plutonium-uranium mixtures at Hanford and Alder-maston. Hanford measurements with bare and Plexiglas-reflected Pu02-U02-polystyrenecompacts, which were readily iuterpretable,103"105 were supplemented by others with par-tially reflected plutonium-uranium nitrate solutions.106 Aldermaston reported criticality ofpolyethylene-reflected PuO2-UO2-polyethylene compacts of various shapes,

10' and water-retlected plutonium-uraiihun nitrate solutions at various concentrations of plutonium.108

Table 18 gives critical dimensions derived from the above experiments for various reflectedshapes. The Hanford reports also give critical sphere, cylinder, and slab dimensions foridealized 239PuO2-U(0.72)O2- water mixtures.

Calculations109 based on the Hanford and Aldermaston data led to subcritical limits forplutonium-uranium mixtures that appear in the American National Standard for NuclearCriticality Control and Safety of Plutonium-Uranium Fuel Mixtures Outside Reactors.110

TABLE IS. CRITICAL REFLECTED Fn-l" SYSTEMS. DIMENSIONS DEDUCEDFROM CRITICAL EXPERIMENTS

Pu Pu J 1 (Tuin P u - \ ' HNO.i Density in Pu Critical(Wt cl) H Pu (N) (R'L) IWt V;.) Sha|>e Dimension (cm)

PuO2 - r(0.1-.l)O.. Polystyrene. Plexiglas Reflect,

7.67.6

7.97.9

7.9S.IM.6

14.614.615.029.329.330.030.030.0

259

259659

K59

659

93 —210 —210 —210

20.29.7

9.715*

15*

15*

6-164

2*.42*.4

•28.4

1608585

85

510

10101010M2112112

23.023.0

s.oN.O

K.O

11.58.0

8.0

8.011.511.511.5K.O

N.O

8.0

tubeinl. slal)Spl l t ' IV

inf. cyl.inf. slabcubesphereinf. cyl.inf. slabcubecubeinf. slabsphereinf. cyl.inf. slab

47.320.0

22.fi

31.21-i.s47.s19.726.411.652.135.712.918.524.610.8

: 0.2 edge

r 0.1 r: 0.2 diam: 0.6edge"~ 0.2 r- 0.2 diani- 0.1edge"edge"

r 0.3 rr 0.2 diani-r 0.1

PuOj - 1(0.72) - Polyethylene. Polyethylene Reflector:107

25.1 ls.fi 950 5.9 cube 27.7rO.l edge

PuO2 - T(0.72) Nitrate Solution. Water Reflector:10*1

50.6'.0.6

.40.6

iO.fi

224795

13721461

1.601.100.95

11.93

10131.618.617.5

5.65.6

5.6

5.6

cyl.rvl.cvl.rvl.

,h-.h---..h-,h =

dddd

300 i338 -4 11456

0.3 diain0.3 diam

diamdiam

' [ ' lorn near-cubic parallelepiped.

WATER-MODERATED PLUTONIUM AND PLUTONIUM-URANIUMLATTICES

Consideration of the Plutonium Recycle Program led to Hautord critical experiments withwater lattices of plutouium-aluntimun rods.111 To maintain a reasonable number of ele-ments, 5 \vtl'ii 1'u in aluminum was chosen instead of the 1.8 wt.% Pu considered for recycle.Hods of 1.29-cm- diam and 61-cm length were clad with 0.078-cm-thick Zircaloy-2. Mea-sured buddings and extrapolation distances for various spacings of rods in a hexagonallattice are given in Table 19, with critical volumes and masses of corresponding water-reHected equilateral cylinders.

TABU:: 19. HKXACiONAL WATKR LATT1CKS OF 1.29-cm- BY 61-cm- RODS OF5 Wt'ri. Pu (.r»1;; -4llPu) - ALTMINT \I CLAD WITH 0.070-cm-thickZIRCALOY-2.

_ Equilateral CylinderCenterSpacing(cm)

1.902.162.292,512.793.053.303.81

H Pu

218354127580

76094011501610

Buckling(10 A cm'2)

96.05108.39112.61108.53101.0788.4076.8153.32

ExtrapolationDistance

i (cm)

8.107.927.937.737.347.036.035.80

CriticalVolume

(L)

6047.744.148.860

82120234

Critical Mass(kgPu)

3.452.141.761.581.651.832.423.35

Also at Hanford, Fast lest Reactor tuel pins became available for measurements of square-pitch water-flooded lattices.112 The fuel of each pin consisted of a PuO'.- '*(natural)Oomixture containing 19.8 1 wtf,'i plutonium and had dimensions 0.494-cm diam by 91.4-ciiilength. The plutonium content of a pin averaged 34.2 g and contained 11.5% 24nlJu. 'I hetyj>e 316 stainless-steel cladding was 0.038-cm thick. Lattice characteristics appear inTable 20.

76

TABLE 20. SQUARE WATER LATTICES OF O.-194-cm-diam BY 91.4-cm-longPuO2+UO2 AT O.S^cm' CONTAINING 19.8 wt% PLUTONIUMCLAD WITH 0.03S-cm-thick STAINLESS STEEL

CenterSpacing

0.767 ±0.952 ±0.968 ±1.259 ±1.534 zt1.905 ±

(cm)

0.0130.0130.0130.0130.0130.013

"Included 65 rodswhich contained

Water/FuelVol Ratio

1.673.333.496.S710.8817.53

containing 24.4 wt% Pu,19.8 wt% Pu.

Lattice WidthNo. of Pins

36282818IS14

which was corrected

Critical Numberof Pins

1037 ± l a

605 ± 1579 x 1279 ± 1205 ± 1162 ± 1

to 1268 rods all of

HOMOGENEOUS HYDROGEN-MODERATED 2iiV

l:nlike the survey uf U(93) solution cylinders with a large height /diameter range, nil critical->:l3U solution cylinders were nearly equilateral.'1" Tims deduced infinite cylinder and slabdimensions are more dependent upon the calculations of Table 5 and Fig. 6. With oneexception, the reflector, where present, was paraffin instead of water. A single measurementwith water reflection gave essentially the same critical dimensions as paraffin. France alsohas reported data for partially reflected J " u l ' solution in cylinders.50'9'2

The initial ~A>iV solution survey was supplemented by spherical measurements, all at OakHidge,"18"ri'" ' ' Critical parameters for •' l tU, similar to data presented earlier for U(93) andplutouiuin, appear in Figs. 3(i-39.

In addition to the J'ul'OjF''-j solutions represented in Figs. 36-39, there were data for asimilar range of critical J3"Jl;O2(NO3)j sohitions. Ratios of uranyl nitrate to uranyl fluoridecritical masses, influenced by nitrogen capture, appear in Table 21. B'."cause of significantdifferences for concentrated solutions, ratios are given its functions of both 2 'UU densityand H/J33U. Table 21 may be presumed to apply also to U(93) solutions, for which nodirect comparison exists, but see Ref. 44.

FABLE 21. SPHERICAL WATER-KEFLECmr;-

>33uAND II

233 Density(kg/L)

0.03380.04920,0640.1000.1600.2740.3X10.491

O2(NO3)2 /iiir'->?3UO3Fa

Criti'al MussRatio

1.001.001.031 .()(>1.041.14i .231.43

FED CRITICAL MASS RATIO,] SOLUTIONS, vs

II "•" U

76051039124715184",*

42

233U DENSITY

Critical McissRatio

1.021.011.031.021.001.091.1!1.26

78

c/j

S3

d5a

cc;wDC

to

40

' oOn

1~i

1

1 -

i

'• 1

:

- M -: \i \! \

}

1 ! j :\ \

i i i

D DATA DERIVED FROM SOLUTION CYLINDERS

A UNREFLECTED METAL SPHERE

X CALCULATED METAL-WATER MIXTURES

0.127-cm ALUMREFLECTOR

\\

\

-cr t i

NUM

P/

/

~| D7

•*— LIMITING CRITICAL DENSITYii

11

i i

1

• / '

VA

1REFLECTO

J T

1

—i - I

i

i

—i—,—_i—1._—

, i /

w ,fer ^1

1 , 1

! j j i

R

I

;

•

1 J

1 'i

: ;

i

METAL •-1I

0.01 0.1 1

DENSITY OF 233U (kg/1)

10 20

/• 13. I ' I . C'vihftil nttt.s.sc.c omo

S

3

g

CLCO

as

Iff

1 0 s -

Irf-

1ff-

irf1IU

:G

Jl_.T' \ ' "

• \

: \

: \

; \

i \

<

*•— LIMIT

-A-\

— '

R:

o.i;

i

44tt"ALUMINUM

\

Q

1

WATER2FLECT(

rvir* nniTM

1

:AL

\* j

3R

DENS

: = • - • -

oaA

— x—i—

i

DATA FROM SOLUTION SPHERES

DATA DERIVED FROM SOLUTION CYLINDERS :

UNREFLECTED METAL SPHERE \

CALCULATED METAL-WATER MIXTURES • ~b\

_ILJ_ _

X - - .

1

. . . .

•x..

•

•A.

M E T A L •-; •

0.01 0.1 1

DENSITY OF 233U (kg/L)

10 20

Fij 31. Critical volumes of homogeneous water-moderated 2 3 3 U spheres. Solution data, appear unless indi-

cated otherwise.

200

100c

DC

OS

a

>~u

<

5

10

D

-D-\

~M~

~ T

- L _ _

I I

Ui-

0 127-cm ALUMINUMREFLECTOR ,

X

tfcWATER

REFLECTOR~a~

=D

- D -

•"—LIMITING CRITICAL DENSITY

_L

O DATA FROM SOLUTION SPHERES

D DATA DERIVED FROM SOLUTION CYLINDERS

X CALCULATED METAL-WATER MIXTURES

1_ J_UJJJ-

^QDX j - ( . . . . .

H4----\

y

METAL

0.01 0.1 1

DENSITY