caixuated utron multiplication factors of u ...e. the container was 0.8775 in. thick methacrylate...

ORNL-CBC-2 UC-46 - Criticadity Studi

CAIXUATED UTRON MULTIPLICATION FACTORS OF U

AQUEOUS SOLUTIONS 01: 233U AND 235 U

, J. Wallace Webster

Oak Ridge National Laboratory

LEGAL NOTICE

CALCULATED NEUTRON MULTIPLICATION FACTORS OF UNIFORX AQUEOUS SOIiU!FIONS OF '==U ANDa3'U

J. Wallace Webster Neutron Physics Ditision

Oak Ridge National Laboratory

OCTOBER 1967

OAKRIDGENATIONALL4BORATORY Oak Ridge, Tennessee

operated by UNION CARBIDE CORPORATIOiV

for the U. S. ATOMIC EWRGY COMMISSION Contract NO, W-7405-eng-26

__ . “r, *&- . - ~ . - . . . . . ^. . . .

T A R U O F C O lV 2 T N T S

A B S T R A C T ...................................................... 1

I. P m m T IO N .................................................. 3

II. C m A m O N S A N D R E S U L T S ...................................... 4

III. A C C U R A C Y O F T 'H E ~ U IIP S ....................................... 6

IV . C O N C L U S IO I'E ................................................... 0

V . A .............................................. 0

1

CALCULATEDNEUTRGNMULT~LICATIONFACTORS OFUXFORM AQUEIOUs SOLUTIONS OF ===U AND sasU

J. Wallace Webster

ABSTRACT

Ccmiputations of the effective neutron multiplication factor of single units of aqueous solutions of sss~F, and s3'IKIaFa are reported for guid- ance in the specification of limits applicable to processes, such as storage and transport, for these fissile isotopes. Graphs are presented

Of keff as a function of such parameters as the mass of fissile material, the chemical concentration, the dimensions of spheres and infinitely long

/ cylinders, and the thickness and area1 density of infini%e slabs. Trans- I i port theory (LYE) codes in the Sn approximation with Ransen-Roach cross i i

sections were utilized and the results agree with relevant experiments to' i I within 0.01 in keff.

3

I. fXI3tOIJUCTION In 1958 a Standard, entitled "Safety Standard for Operations with

Nssionable &terials Outside Reactors," was drafted by a subcamnittee of the-Standards Ccmmittce of the American Nuclear Society (ANS) and Sectional Committee N6 of the American Standards Association. In 1964 this standard vas designated an American Standard, ASA N6.1-136La.

Increased basic knowledge and operating experience have made desir- able a revision of ASA 36.1-1964 which would extend it in various ways and which would include, in particular, the specification of limits on single process variables for guidance of nucl.ear criticality safety. Such a revision is now being prepared by Subcommittee 8 of the Standards Committee of the ANS.

The computations reported here were made to provide background in- formation as an aid to the specification of the limits for uniform aqueous solutions of a33U and a3s U. Limits are established for the spherical mass and volume, the concentration of the solution, the diameter of an infinitely long cylinder, and the thickness as well as area1 density of a slab infinite in two dimensions.

In each case the value of keff was calculated for various asswned values of one parameter with all others optimized. In all cases the uranium is isotopically pure a3=U or a3= U in an aqueous solution of UC&F, ' surrounded by an infinitely thick water reflector. The shape is taken as spherical in the calculations for masses and volumes. In all cases the chemical concentration is taken as that giving greatest reactivity except of course where keff vs concentration for an infinite volume of solution is being investigated.

a. This standard is now designated as USA Standard N&l-l@.

4

II. COMPUIATIONSAEDRESUL!PS

All computations were made by the Sn method, as applied by the IYIT cOde,l using the Hansen-Roach 160group cross sections.a

For spherical and slab shapes the angular segmentation (n) was taken as six; for cylindrical shapes, n was taken as eight. A few test cases indicated that a finer quadrature (greater value of n) would not change the results significantly.

A spatial mesh of 0.2 cm was used for the thin slabs and 0.5 to 1.0 cm for the regions snd geometries of larger dimension. A smaller mesh was shown to have no significant effect on the results.

With respect to the energy mesh of the Hansen-Roach 16-group scheme,

a finer mesh was investigated by replacing the first five groups (energy > lob eV) with l? groups and examining the fast leakage from the core. As with the angle and space variables, the discrete treatment of energy corresponding to 16 groups was shown to cause no significant error.

It was concluded that the method, the angular quadrature, and the spatial mesh as described gives essentially an 'exact" solution to the transport equation for the problems considered here. In other words, whatever errors there are in the values of k eff are due to uncertainties in the cross sections and not to the method. The magnitude of these errors is discussed in Section III.

The procedure leading to the est8blishment of minimum values of the various parameters at a selected value of keff Is illustrated in the fol- lowing discussion. In the determination of a recommended mass, keff of spherical volumes of solutions at three chemical concentrations were cal- culated as a function of sphere radius. The results for "'U are shown in Ng. la(235). The accornpolnying curves9 Ng. lb(235), show the same results plotted as keff as a function of the mass of 8"U in the various spheres. The latter results were cross plotted in Ng. lc(235)

1. B. G. Carlson et al., %!F Users Manual," UMC Phy8/Math-3321, Vol. I (r?ov.lg63), VZ.-TI (Mayly64).

2. G. E. Hansen and W. H. Ro8ch, %ix and Sixteen Group Cross Sections for Fast and Intermediate Critic81 Assemblies," LAMS-2543, Los Alamos Scientific Iaboratory (1961).

5

as keff vs a36U concentration at constant mass for several masses. The matima of these curves were plotted, in turn, in Fig. I(235) as maximum values of keff as a function of mass. The resulting relation which turns out to be roughly linear gives the largest value of keff attainable with a given mass of sssU in aqueous solution under optimum conditions, that is, in a fully water-reflected sphere at the most nuclearly reactive chemical concentration, This relation shows that about 820 g of 236U in U(lOO)O,F, solutionb will be critical and that aobut 710 g of 23sU will have a keff of 0.9'7.

In a l%ke manner the maximum value of keff attainable with various spherical volumes of sclution of U(lOO)&Fa, in a narrow range around the smallest possible critical volume, is obtained from the three curves of k eff as a function of concentration (R/d3'U ratio) in Fig. 2(235) and the plot of the maxima of these curves in Ng. II(235). These results give 6.1 liters as the minimum critical volume.

The variation of keff with the concentration of solution in volumes infinite in all dimensions, given in Fig. III(235), shows a "'U concen- tration of O.Oll8 g/cc as that of the most dilute solution which can be made critical.

Similarly the finite dimensions of cylindrical and slab-shaped geometries with infinite transverse dimensions and the area1 density

! i (mass of 23sU per unit area) of an infinite slab corresponding to various

>

;

madmum values of keff were obtained, respectively, from Ngs. 4(235), i and IV(235), from Figs. 5(235) and V(235), and from Figs. 6a, b, and ~(235) i and VI(235).

An identical series of operations , reportedin a similarly numbered series of curves~has established corresponding limiting dimensions for units of 233U solution.

The results are summarized in Table 1 where the critical dimensions and the dimensions corresponding to an arbitrarily selected subcritical condition (keff - 0.9) are recorded.

I I

b. U(100) signifies uranium enriched to lOO$ in the 23sU isotope. I

6

Table 1. Dimensions of Single Units of Aqueous Solutions of 23sU and 233U

All parameters optimized.

1.00 233u 0.9 Parame\ keff 1.i3sU 0.5

1hss (g) 820 710 570 510 Volume (liters) 6.1 5.4 3-7 3.3 Isotopic Solution Concentration (g/l) 11.8 11.1 11.2 10.6 Diameter of Infinite Cylinder (cm) 14.3 13.6 11.9 11.3 Thickness of Infinite Slab (cm) 4.9 4.4 3*2 2.8 Area1 Isotopic Concentration (g/c$) 0.42 0.39 0.37 0.34

III. ACCURMY OF THF. FEStTLTS

To evaluate the accuracy of the computational procedure, a number of critical experiments was analyzed in order to compare the computed values

Of keff with the experimental value (unity). Since the configurations of interest and the version of the Sn code used were both we.dimensional, comparison was made with critical experiments of a one-dimensional or close to one-dimensional shape. It was possible to select among the many ellperiments that have been done over the years at the Oak Ridge Critical Experiments Facility a group for both a33U and 23sU solutions which cover a broad range of concentrations.

Table 2 presents the results of this comparison of theory and ex- periment. It is concluded that, overall, the ccmputed values of keff are accurate to about 2 1s; there is a bias of about + 0.5s in those for the 233U solutions.

7

Table 2. Comparison of Computed Values of the Effective Neutron Multi- plication Factor for Aqueous Solution of U02F2 with Experiment.

All units were surrounded by an effectively infinite water reflector except as noted.

Dimensionsa I@drop;en Atoms Computed Bcperiment Shape (cd Nssile Atoms k Reference eff

233 u

Cylinder Diam = 12.7 .74.1 1.005 3 (PO J-s> Height = 56.5

Sphere 26.4 378.1 1.007 4 Sphere 32.0 663.1 1.009 4 SphereC 69.2 1533 -0 1.000 5

23Su

Thin slabe 1.995 in.. x 44.7 1.005 5 47.20 in. x 58 in.

Sphere 23.0 76.1 0.995 7 (P* 42) Sphere gg 126.5 0.9 ; (P. 42) c Sphere 239.3 y-2 Sphere ;;:; 515.1 4 Sphere . 1270.0 1:008 7 (p. 42)

a. Diameter unless otherwise specified. b. The uranium contained about 96% 233U. c. This was an unreflected solution of UOe(N03 &. d. The uranium contained about 93.2$ 23sU. e. The container was 0.8775 in. thick methacrylate plastic surrounded

by effectively infinitely thick water.

3. J. K. Fox et al., "Critical Mass Studies, part VIII, Aqueous Solutions of 233U,W 062143, Oak Ridge National Laboratory (Sept..1959).

4. J. T. Thomas et al., "A Mrect Comparison of Some Nuclear properties of U-233 and r23," ORNL-1992, Oak Ridge National laboratory (Nov. 1955); Nucl. Sci. EIIJ. 1, 20 (1956).

5. R. Wins Dx. MagnuFon, Nucl. Sci. 3

12, 364 (1962). 6. J. K. Fox et al., NUCL. s&.=. 3,6g i9m. 7. J. K. Fox zz., stron Physics%Ivision Annual progress Report for

the Period%xw Sept. 1, 1958,” 0~~~2609, pa 42, Oak Ridge 8ational Laboratory (1958).

8

',"f?e calculated values of the dimensions of single units of aqueous soliltions of 235U and 233 U *under optim*um reactivity conditions afford establishment of specifications for chemical and other processes whereby nuclear criticality may be avoided. The ,xxertainty in the calculation of k e9f, shoh?l by comparison btith experiment to be no more than 0.01, a s%o.~ld be considered w'nen specifying the dimension. In the example shaJn in Table 1, a safety factor of 0.02 was first stipulated as an arbitrary condition; to it was added 0.01 to cover the uncertainty in the calculation, hence the selection of the dimensions corresponding to keff = 0.97. The span of values of kecf given in the graphs, however, permit selection of dimensions corresponding to other degrees of subcriticality to conforu to particular operating conditions. They may be easily obtained from the graphs designated by Roman numerals. The bias apparent in the "'U calculations adds a bit of additional contiervatism.

The autiior gratefully acknowledges the consulting help of John Knight and Elliot Whitesides of the Computer Sciences Center with respect to the computations; and of J. T. Thomas , Dale Magnuson, and E. B. Johnson at the Oak Ridge Critical Experiments Facility with respect to inditidual experiments.

9

23sU Results

.

.

10

m5

wo k ‘” 095

0.90

un

090

(d (6 I7 48. 49 20600 700 800 wo 4ooo Htxl

SPHERE RADIUS hi) (b) MASS (g)

3%. h(235)a k vs Radius of Sphere for Solutions of Thre:" assU Concentlrtiorm.

Ng. lb(235). keii vs Mass of a3sU in Spheres for Solutions of Three Concentrations.

..-. _... _.___._.___ ._. I

k eff

ORNL-DW6 67-546R 1.00 ’

Ux.

0.90 0 0.02 0.04 0.06 0.08 0.40

235U CONCENTRATION (g/cc)

lc(235). k,,, vs Concentration for Various Messes of a3eU.

oRNl.-WC? 67-SO!bR

235U MASS (01

Fig. I(235 )* k l it v8 Mm8 of 0=6u.

ORNL DWG 67-526f? 1.02

& off 0.98

a94 0.99

bff 0.95

0.94

0.93 trt

0.89 20 40 60 80 mo

H/=%J

Ha. 2(235). kerr va @"tJ for Three Spherical Volumes.

14

ORNL-DWG 67-515R 1.01 1.01

1.00 1.00

0.99 0.99

0.98 0.98 r: r:

z-0.97 z-0.97

1 1 g E 0.96 0.96 $ 0.95 $ 0.95

0.94 0.94

0.93 0.93

0.9212 09212 . 4.0 4.5 5.0 5.5 6.0 6.5 4.0 4.5 5.0 5.5 6.0 6.5

VOLUME OF 235 VOLUME OF 235 U02F2 U02F2 SOLUTION (liters) SOLUTION (liters)

Ng. 1X(235). Maximum kett vs Spherical Volume of a3sU.

0.98

%rr 0.97

0.96

0.95 .

0.94

493

092 .& ..!

or0 O9 oqe ,F1””

235U CONCENTRATION (g/cc,

Fig. III(235). kai, vs Concentration of 336U.

16

ORNL DWG 67-525R

0.96

bf f 0.92

0.88

hlff 0.84

20 40 60 60 100

Ng. 4(235b karr vs Il/da6U for Cy3.lnders of Three Diematers.

17

ORNL-DWG 67-so0J3

I I I

'c c.00 2 I 2 0.98

a = 0.96

0.94

0.92

0.90

0.88 - 42 43 44 45 16

DIAMETER OF 23sU02F2 SOLUTION (cm)

Fig. IV(235). Matimum k,rr w3 “‘U Cylinder Y&meter.

1.01

k off 0.97

ORNL DWG 67-524R

0.96

0.92

bff 0.88

20 40 60 80 iO0 H/235 U

Ng. 5(235). kefi vs H/as% for Infinite Slabs of Three Thicknesses.

19

--.. - -- IR

0.99 I/

4.0 4.2 4.4 4.6 4.8 THtCKNESS OF 235 U02F2 SOLUTtON (cm)

Ng. V(235). Maximum kerf vs Infinite a3sU Slab Thickness.

20

6 7 8 a33 a35 a37 a3!3 OAl a43 0.45 1.05 coo

kdzE 0.02m

I3 15 17 0.36 0.38 0.40 Q42 Q44 a46 a48

20 24 28 a35 a37 a39 0.44 Q43 0.45 a47 (4

HEKMT OF SOWTlON (cm) (b) =?J AREAL DENSITY (g/cm2) IN AN INFINITE SLAE

Ng. 6a(235). k,,, vs Height of "'U Solution in Inrinite Slabs for Solutiom of Three Concentrations.

Ng. 6b(235). k vs Density of a3sU per Unit Area of Infiat: Slabs for Three Concentrations.

21

ORNL-DWG 67-548R

a95

0.93 6 eff

Q94

Q89

Q87

a85

I_\ 0.39 I ‘III I I(c) I I I I \ 0.37 I

QO4 a02 QO3 0.04 a05 Cl06 0.08 235U SOLUTION CONCENTRATION (g/cc)

U. 6d235). kerr vs Concentration of a3sU for Various Densities per Unit Area.

22

1.00 6’ 7-507R /

I I I I I I

0.94 ’ I I I I I I 0.37 0.38 0.39 0.40 0.44 0.42 0.43

AREAL DENSITY (g/cm2)

Fig. VI(235). bximum k*,, vs "'U Mess per Unit Area.

23

a33U Results

24

a90 I2 I3 Is

I@360 450 880 650 7so -0

(0) w

Fig. Ia(233). k.,, VB Radius of SplMwes forsolutlcm#3 ovI!ilrdUu concentrrtione.

Pre. lb(233). k ?J in Sphere for Sol'ui~~ z'k!k concentrations.

25

k l ff

ORNL-DWG 6?-547R

0 0.02 a04 a08 Q08 CM0 2% CONCENTRATION (g/cc)

Fig. ac(233). kar, VB Concentration for Various Messes of 9.J.

26

ORNL-DWG 67-!506R

= 0.98 ** 5 0.96 I

ij 0.94

0.92

1 .oo

0.90 400 450 500 550 600 650

2331J MASS (gl

Ng. X(233). I4xdmum k,rr va Mass of a=3~.

27

ORNL DWG 67- 522R 4 1.03 R

keff

1.01 I \

0.950

*err 0.930

0.940 0.9io

&err

0.890

0.870

Ng. 2(233). keri vs v3'U for 7!hree Spherical Volumes.

28

ORNL-DWG 67-512R

.

- -2.0 2.5 3.0 3.5 4.0 4.5

VOLUME OF 233 lJ02F2 SOLUTION (liters)

Mg. II(233). !&dmum keir vs Spherical Volume of aasU.

0.98

k 8ff

0.96

0.95

. 0.94

1.02 ORNL-DWG 67-b43R

o.oioo 0.0104 O.OtO8 0.0442 O.Oli6 233U CONCENTRATION (g/cc)

Me. ffI(233). k8,f vs Concentration of a33U.

30

m65

%ff

1.045

1.025

0.950

k8ff

0.930

0.910

0.820

‘+8ff

0.800

ORNL DWG 67-523R

20 40 60 80 100 H/233U

me. ‘G33). k8,r vs @33U for Cylinder of Three Diameters.

31

0.85

0.80

ORNL-DWG 67-509R _ ..- -..- _.

9 10 11 12 13 IXAMETER OF 233U02F2 SOLUTION (cm)

Fig. XV(233). l4aimm k#,, vs a33U Cylinder Dhneter.

ORNL DWG 67-521R

3.2 cm = 1.26 in.

20 40 60 80 100 H/233U

Ng. 5@33). k,,ir vs H;lassU for Infinite Slabs of !Three Thicknesses.

33

ORNL-DWG 67-SllR ORNL-DWG 67-SllR 1.01 J I I I

2.2 2.2 2.4 2.4 2.6 2.6 2.8 2.8 3.0 3.0 3.2 3.2 34 . 34 . THICKNESS OF 233U02 F2 SOLUTION

Fig* V(233). bb&mxu ke il vs Infinite a33U Slab Thickness.

34

090 k df

0.85

ORNL-DWO 67- 52DR

81

5 6 7 8 0.30 0.32 0.34 0.38 0.38 0.39

f.00

0.98

/'

1'

A,,, O.S6

0.94 0.0(848

' (a) 0.92

17 19 2i 23 0.30 0.32 0.34 0.36 0.38 0.39 HEIGHT OF SOLUTION (cm 1

IN AN INFINITE SLAB 233U AREAL DENSITY (q/cm21

Ng. 6a(233). k8rr vs Height of 233U Solution in Infinite Slabs for Solutions of Three Concentrations.

Fig. 6b(233). k vs Density of 233U per Unit Area of Infin!%~ Slabs for Solutions of Three Concentrations.

35

K2 , ORNL-DWG

I 1 I I

4.00

Q98

Q96

krff Q94

0.92

67-4377R

---I

(cl a86 I I 1 \ 0.33 I I

0.04 0.02 0.03 0.04 0.05 0.06 233U SOLUTION CONCENTRATION (g/cc)

008

=g. W233). kafr per Unit Area.

vs Concentrat.ion of a33 U for Various Densities

I z) 0.98

E $ 0.97

0.96

36

ORNL-DWG 67- 527R

P-i - 0.95 - 0.33 0.34 0.35 0.36 0.37 0.38

AREAL DENSITY ( g/cm2 1

-1

Fits- vI(233). Maximum keif vs a33U Mesa per Unit Area.