UNCLASSIFIED

EMERCY e

* *

UNCLASSIFIED

AECD-3740

PHYSICS

UNITED STATES ATOMIC ENERGY COMMISSION

INTERACTION OF ENRICHED URANIUMASSEMBLIES

ByH. F. Henry

November 23, 1949

K-25 PlantCarbide and Carbon Chemicals CorporationOak Ridge, Tennessee

Technical Information Service Extension, Oak Ridge, Tenn.

metadc1OO 807

AEC COUll

Date Declassified: June 25, 1956.

Other issues of this report may bear the number KS-89.

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AEC, Oak Ridge, Tenn.

AECD-37k0

INTERACTION OF ENRICHED URANIUM ASSEMBIJES

By

H. F. Henry

November 23, 1949

Work performed under Contract No. W-7405-Eng-26

K-25 Plant

Carbide and Carbon Chemicals Corporation

Oak Ridge, Tennessee

1

INTERACTION OF ENRICHED URANIUM ASSEMBLIES

I. INTRODUCTION

Consideration has been given to the interaction effects of two or moresub-critical quantities of enriched uranium for the purpose of evaluatingthe conditions under which they may be safely arranged. A general inter-action relation has been successfully applied to the available data obtainedfrom critical mass experiments, but it should be noted that these data arethe results of only two sets of experiments each of which was performedwith reactors of a single size. This general interaction relation is thenused to indicate the actual degree of reactivity in arrays of other uraniumcontainers.

Essentially, the assumption is made that all neutrons leaking from onereactor and entering a second will be absorbed in the second reactor. Theinteraction probability, 14, may be defined as the fraction of neutronsleaking from one reactor which enter the other reactor. The problem issimplified if it is further assumed that the reactors are of identical geom-etry and that thus the utilization in each will be the same. In this geom-etry, the critical condition for a system of two identical uranium assembliesis described by the statement:

Of [U + V (1 - U) = 1 (1)

where { the interaction probability for a just-critical assembly. (Itis equal to the fraction of neutrons leaking from one reactorwhich would have to be retained or supplied from another reactorin order for the system to reach criticality).

U = the total non-leakage probability of neutrons from a singlecomponent of the system.

=:number of fission neutrons per thermal neutron captured by U-235.

f m fraction of all the thermal neutrons absorbed that are captured

by U-235.

Obviously, if U is the number of fission neutrons which are utilizedin the component in which they originate, (1 - U) escape. Of these, 4(1 - U)enter other components and are absorbed.

3

4

Letting = _I_, the critical average interaction probability is;Of

= -U (2)V 1 -U

II. DETERMINATION OF THE AVERAGE INTERACTION PROBABILITY

1. Assemblies in Air

For the case of uranium containers in air, the determination of theinteraction is essentially a geometrical problem, and should be re-lated to the equivalent solid angle subtended at one reactor by asecond. For evaluation of this equivalent solid angle, a method ofnumerical integration was performed and resulted in the graphs ofFigure 1 for two identical cylindrical assemblies. The theoreticalinteraction formula has been compared with experimental data on thecritical heights of two untamped 10" diameter cylinders for I/X ratiosof 169.3 and 328.7. The assumption was made that the utilization as

given by thermal diffusion theory and an empirical interpretation ofthe slowing down of neutrons in water are correct.2

a. Theory

The value of the non-leakage probability, U, is given by:

U: 1P(k2 ) (1 t k2L2) (3)

where P(k2) = 89.67 k6 ' 109.03 k 4- 25.36 k2 +j1, L is the thermal

diffusion length, and k is the buckling of the reactor. The buck-

ling is determined by:22

k2 = k2 f k2 _ 2.405 2 /m\ (4)1 2 r+ d thMd

where r is the radius of the cylinder and h is the height. Theextrapolation length, d, equals 0.71AT. Axis the transport meanfree path.

The condition for criticality of an isolated reactor, OfU = 1, hasbeen applied to experimental results from a cylinder 10" in diameterin order to obtain an effective value of Q assuming that f and U areknown. Values of 7, have been calculated by equation (1) for variouscombinations of reactor separation and critical heights using thiseffective 0 .

Since the average fractional solid angles given in Figure 1 are

integrated over the lateral surface only, the interaction formula must

1 K-406 to be published

2 E. Greuling, "Theory of Water-Tamped Water Boiler", LA.-399, 9-27-45

5

be corrected for end leakage. As the cylinders are parallel,neutrons leaking through the ends of one component do not enterthe other component. Thus,- the interaction formula takes theform:

( - U)

where is the ratio of lateral leakage to total leakage and

(f=:1Vt . This ratio is obtained by integrating the neutroncurrent over both the lateral surface and the total surface.Thus,

2

1 where k2 is the lateralk2 k 1

buckling and k is the longitudinal buckling of the cylinder.

If it is assumed that the average interaction probability, 4, isproportional to the average solid angle,X1L, a constant of pro-portionality, a, may be found by dividing the experimental value of.n. by (.

b. Experimental Data for Two Untamped 10" Aluminum Cylinders

TABLE I

Separation Critical Height .H/X cm cm (steradians)

169.3 0.635 28.7 0.14642.63 30.7 0.119810.18 34.3 0.070025.28 37.2 0.0382

00 41.2 0

328.7 0.635 40.6 0.17182.63 45.0 0.1368

10.18 57.0 0.092325.28 70.6 0.0557ov >80

Note: The separation includes the thickness of the cylinder walls

(0.635 cm).

The data of Table I, for the two solution concentrations used,have been plotted in Figures II and III where the critical heightis shown as a function of the solid angle subtended at one cylinder

by the second. These solid angles were determined from Figure I.

6

c. Constants

2.19 (adjusted value of i) ) as determined from experimentswith a single 10" cylinder.

Tt=640 barns (fission cross section).

0.31 barns (hydrogen absorption cross section).

s=21 barns (hydrogen scattering cross section).

A T = 2.20 cm (transport mean free path).

d = 1.53 (extrapolation length).

e = 0.4953

d. Comparison of Theoretical Calculation With Experiments

Table II is a tabulation of critical height, buckling, interactionprobability, solid angle, etc. for representative points on Figures IIand IIL Column 9 of Table II shows the value of a which is theratio of the solid angle to the interaction probability. Column10 gives calculated values ofL.obtained from the average a andappropriate values of c . These values of1 are also plotted inFigures II and III.

TABLE II

H/X :=169.3; L2 = 0.6475

.(Le -1 a

Critical (exp.) (= aW)Height 1 2 (stera- a WTC

cm k P(k ) 1 # k2L U ,Q cc dians) :.l( /g (steradians)

29.0 0.03813 0.4694 0.9760 0.4581 0.748 0.0918 0.1420 1.55 0.153334.0 0.03570 0.4882 0.9774 0.4772 0.799 0.0433 0.0741 1.71 0.072338.0 0.03440 0.4988 0.9784 0.4879 0.830 0.0174 0.0301 1.73 0.029140.0 0.03387 0.5032 0.9785 0.4924 0.843 0.0068 0.0115 1.69 0.011441.2 0.03358 0.5056 0.9785 0.4947 - 0 - -

Ave a,: 1.67

X . 328.7; L2 = 0.406

40 0.03387 0.5032 0.986 0.4961 0.843 0.1069 0.1718 1.61 0.18150 0.03206 0.5186 0.987 0.5118 0.890 0.0683 0.1155 1.69 0.11560 0.03104 0.5278 0.987 0.5209 0.920 0.0467 0.0831 1.78 0.078980 0.02999 0.5374 0.988 0.5309 0.952 0.0237 - - 0.0400

Ave _ a,.: 1.69

7

2. Water Tamped Assemblies

a. Theory

In the case of water tamped assemblies, the average interaction prob-ability falls off rapidly with increased separation due to the ab-sorption of neutrons by hydrogen. Thus, it is possible to determinea maximum interaction probability, aside from any solid angle con-siderations, by calculating the attenuation of a beam of fissionneutrons in water.

According to the age theory of the slowing down of neutrons, theprobability that a fast neutron will be thermalized in a distanceX from a plane source in an infinite medium is:

X

22P = .... e- dy where r -:33 cm in1H200

Since the effect of the diffusion of neutrons after thermalizationon the thermal neutron density is quite small at large distancesfrom the source 3 , the above integral is approximately the same asthe probability that a neutron will be thermalized and absorbedwithin a distance X from a plane source.

Some calculated values of P are:

X cm p

17.07 0.965920.92 0.9900

26.70 0.999031.57 0.99990

It is evident from the interaction equation that if the utilizationof a single component is known, a safe spacing distance for the watertamped case may be obtained by setting (1 - P) equal to (/{.

b. Theoretical Application to Actual Assemblies

As will be shown in the section on "always-safe" components, theutilization of an infinitely long 5 inch pipe is 0.472 based upon avalue of 2.09 for i) . From this, the critical average interactionprobability is found to be 0.0341 at H/X = 50. This value of H/Xwas chosen to minimize the value of (( - U) since experimental dataindicate that U decreases rapidly with decreasing H/X below thatvalue. Thus, a separation much less than 17.07 cm. will be requiredfor two infinitely long 5 inch cylinders to be critical. The distanceof 17.07 cm. is determined from ,4 (1 - P) - 0.0341.

3 Wallace, P. R. and LeCaine, I., "Diffusion of Thermal Neutrons withSources Determined by the Slowing Down Theory" - 1-MT-12, Figure 5.12.b.

8

Since two infinite 5" cylinders, in contact, mutually subtend anaverage fractional solid angle of 0.1925 steradians, P may be moreaccurately determined by the equation 0.1925 x (1 - P) _c= 0.0341.From this statement, P = 0.8229 and the corresponding average sep-aration is approximately 11 cm. This distance may be interpretedto be the minimum safe edge to edge distance between two cylinders.

c. Experimental Data

Experimental data indicate that two infinitely long 5" cylinders arecritical at a separation of 6 cm. or less. Furthermore, experimentshave shown that if two cylinders which individually may be criticalare separated by a distance of 30 cm., edge to edge, or greater,they will become critical only when each is individually critical.

III. ASSEMBLIES OF "ALWAYS-S FE" COMPONENTS

If the relations given in Part II above are applied to a geometrical con-figuration each component of which cannot be made singly critical, thevalues of We given below may be determined. In this case, (R representsthe fraction of the escaped neutrons which would have to be suppliedfrom another reactor for criticality to be attained and may thus be re-lated to the solid angle of interaction of the component in question.

1. Untamped:

Infinite Cylinders

Assume f 1.'(= -L * 0.465 () = 2.15)

Diameter

(inches) k2 U 1 l/P(h2 )

5 0.0924 0.230 0.3054 0.1312 0.156 0.3663 0.2005 0.0896 0.412

Infinite Slab

110.2055 0.086 0.415

In the case of the 5" cylinder it will be noticed that the value ofis 0.305. Thus, for a 5" cylinder to become critical, it is necess-ary that it receive from other reactors 30.5% of its total neutronleakage. If it is assumed that this 4' is identical with the solidangle instead of being related to it by a factor of 1.67 as foundexperimentally, a solid angle of 30% of 4 1r will be the criticalinteraction angle for a 5" cylinder. The same consideration appliesto the other components given above.

9

2. Tamped:

The utilization of tamped cylinders and slabs has been calculated bymeans of the water boiler theory, where the utilization is given bys

U= OH -G -1 2

Infinite Cylinders

Diameter(inches) H G U Vc

5 0.275 0.589 0.472 0.01234 0.207 0.510 0.442 0.0653 0.133 0.410 0.415 0.108

Infinite Slab

120.207 0.447 0.472 0.0123

1 0.141 0.339 0.448 0.0517

Although the interaction probability is very small in the tamped case,it should be noted that the effect of absorption by the water wasnot considered in these calculations and that thus the spacing forthe tamped case is not entirely dependent upon the geometrical con-figuration. In fact, as stated previously, experiments have shownthat containers of enriched uranium may be considered to be isolatedif they are separated by 30 cm. of water.

3. Interaction of Unlimited Geometries with "Always-Safe" Amounts

An endeavor has been made to evaluate conservatively the criticalinteraction probability of two vessels of unlimited dimensions buteach supposedly containing 350 grams of U-235. Two cylinders 10"in diameter and assumed to contain 700 grams of U-235 were considered, andthe critical interaction probability calculated and compared withexperiments at several values of H/X. The non-leakage probability,U, was evaluated from data obtained from critical experiments withuntamped single 10" reactors of various moderations and heights.

The results, plotted in Figure 4, indicate a minimum critical inter-action probability of 0.203 at an H/X of about 500. This value occurswhen the height is approximately 10", which forms an equilateralcylinder. This indicates that two optimum cylinders, each with doublethe "always-safe" amount of U-235 will have a critical interactionprobability at an interaction solid angle no less than approximately20% of 4rT.

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