,. .-, .

i. .— -

\

~lc-14 REPORT COUECmW

REPRODUCTION

LAMS-2482

or

COPY c%●

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITYOF CALIFORNIAo LOSALAMOS NEW MEXICO

TURRET PRELIMINARY NUC LEAR CALCULATIONS
.

.,, .

LEGAL NOTICE

This report was prepared as an account of Government sponsored work. Neither the United States, nor the

Commission, nor any person acting on behalf of the Commission:

A. Makes any warranty or representation, expressed

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Printed in USA. Price $1.00. Available from the

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Washington 25, D. C.
.,. >

LAMs-2482

REACTORS - GENERAL

(TID-4500, 15th Ed.)

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITY OF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN: December 1, 1960

REPORT DISTRIBUTED: March 2, 1961

TURRET PRELIMINARY NUCLEAR CALCULATIONS

Work done by:

M. E. Battat

B. M. Carmichael

R. J. La Bauve ,

M. Wilson

W. Schmedt, Sandia Corporation

Written by:

B. M. Carmichael

R. E. Peterson

2 <

_

~

>~*

~e~ Contract W-7405 -ENG. 36 with the U. S. Atomic Energy Commission

$~~

s= All LAMS reports are informsl doeumente, usually prepared for a special purf~~

pose and primarily prepared for uae wftbfn the Laboratory rather than for

general distribution. This report has not been edfted, reviewed, or verffied

<—$= ,

for accuracy. All LAMS reports express the viewe of tbe authors as of the

time they were written and do not neceswm’tly reflect the opinfona of the Los

@J Alamos Scientific Laboratory or the finsl opinion of the authors on the subject.

y= (v)

~,

—

3___ 1
*.4*
ABSTRACT

A preliminary study of the nuclear parameters of

the Turret reactor is described. It is estimated that

at the design operating temperature of 1350”C and power

of 3 MW (Thermal) the critical mass will be 7.1 kg of

highly enriched uranium. An estimate of 21% Ak is

obtained for the amount of shutdown control needed,

and it is calculated that the proposed three-ring control

rod system will provide 2870 Ak shutdown control. A

slow negative temperature reactivity coefficient of

1.4 x 10-4 (Ak/k)PC is found, and a negative prompt

reactivity coefficient of the order of 10-5 (Ak/k)PC

is envisioned.

ACKNOWLEDGEMENTS

In performing Turret calculations maximum use

has been made of the generalized reactor codes, DSN1

and FIRE, 2 which have been developed by staff members of T-1, LASL. The authors wish to acknowledge

in particular the considerable assistance rendered by

B. Carlson, C. B. Mills, F. Brinkley, L. Stone, and

M. Riedel in the application of these codes to the

Turret problem.

-3-

CONTENTS

Page

ABSTRACT

ACKNOWLEDGEMENTS

I.

IL

IIL

W.

v.

Introduction

Criticality Calculations

10 Normalization

2. Neutron Group Reductions

3. First Turret Geometry

4. Procedure for First Criticality Calculations

5. Results of First Criticality Calculations

Isothermal Temperature Coefficients

Control Rod Calculations

1. Survey Calculations

2. Worths of Control Rod Rings

Miscellaneous Reactor Kinetics Considerations

APPENDICES

I. The Critical Determinant for a Ring of

Control Rods in a Bare Core

II. Excerpts From LASL Internal Memoranda

REFERENCES

-5-

3

3

7

9

9

10

12

12

15

24

27

27

28

40

43

43

45

47

—

I. Introduction

Calculations of the nuclear parameters of importance in the design of

Turret have to date been mostly of a preliminary nature, with the aim of

uncovering any gross incompatibilities between the nuclear and mechanical

requirements of the proposed systems. A program of refining the critical

calculations is now developing as material specifications, mechanical details,

and heat transfer data become available. The scope of the first calculations

will be outlined in this introductory section, and the planned refinements

noted; a more detailed presentation of the results of the calculations is given

in the following sections.

When the Turret project started, the need for critical experiments was

considered. The final decision not to do any critical mockups was based on

two factors. First, there was available data from previous LASL critical

measurements on a graphite moderated, highly enriched U-fueled system

(Honeycomb) of a size and composition comparable to the Turret conceptual

design. Second, the very earliest estimates indicated that the fuel loading

required for Turret criticality would be roughly one-half the upper limit attainable by solution impregnation of the graphite fuel-bearing elements which

were the basis of the Turret concept. Thus it appeared that if the available

experimental data could be reproduced by calculation, extrapolation of these

computational methods to Turret design should be satisfactory in view of the

safety factor available by a simple change in Turret fuel concentration during

startup tests. Experimentally verified information on control rod worths

would have been useful during Turret design, but the decision has been made

to base control element specification on a conservative evaluation of calculations.

The first step in the calculational program was to calculate keff for

the available critical assembly data. ~ff for the critical system was computed to be about 1.05, and this value was taken as the “normalized” ~ff

which would correspond to the critical configuration in Turret calculations.

A first model for the actual Turret configuration was then selected on

the basis of heat removal and mechanical criteria. Using preliminary

graphite specification data, densities and parasitic absorption cross sections

-7-
were chosen. Fuel concentration in the reactor was assigned a radial variation corresponding to about 50% burnup as fuel passed through the system.

To simplify computation, the 18 group cross sections used in the normalization of keff were collapsed to 5 groups. keff on this Turret-like system at

1350”C was then computed as a function of fuel concentration (0,73 <keff <1.08).

The critical loading &ff ~ 1.05) was found to be -0.08 g/cc, or 4.6 gin/fuel

element (7.1 kg total). Flux plots from this first calculation were used to

estimate power distribution for the isothermal core. A preliminary value

for the reactivity worth of a fuel element was also obtained; worth per element was so small that no problems were anticipated in reloading the reactor

at power.

Estimates were next made of the system!s isothermal temperature coefficients of reactivity by comparing keff at 1350”c and 20”C (isothermal temperature distributions). Variation of cross sections with temperature, as well

as the temperature-induced changes of material density, were considered.

It was found that a Ak of 0.21 must be provided to shut the reactor down to

room temperature when it was loaded for criticality at 1350”C. At this stage

of the calculations only the average isothermal temperature coefficient over

the entire range of temperatures (startup to full power) was investigated,

and no detailed prediction of the reactivity coefficient in the vicinity of operating temperature was made. It was also found, as expected, that prompt

reactivity effects associated with fuel temperature are small, the main reactivity effect being associated with changes in moderator temperature. The

transient behavior of the system will be, therefore, determined by the heatflow coupling between the fuel elements and the moderator. No temperaturedependent fission product poisoning was considered in these first estimates

of temperature coefficients.

On the basis of these preliminary estimates of control requirements,

the reactivity worth of some suggested control rod configurations was computed. The first scheme studied involved rods in the central plug of the

core, and a !?curtain~~ of neutron absorber between the rotatable core and

the stationary reflector. It appeared that this layout provided a satisfactory

amount of control, but upon re-examination, the mechanical features and

material restrictions appeared unacceptable. A method for introducing rods

into the moveable core, as well as into the stationary central plug and outer

reflector, was devised; and further calcukrtions indicated that the selected

control element material, Z rB2, would give a reasonable amount of cent rol.

Better information on materials density, mechanical design of the core

and reflector region, etc., became available as these calculations continued.

The basic critical mass estimates were repeated with new parameters. Such

factors as increased moderator and reflector densities, extension of the calculational model to include the porous carbon thermal insulation surrounding

-8-
the reflector, etc., indicated that the critical mass might be decreased by

*25’% from that first estimated; similarly, the substitution of a radially uniform fuel concentration for the “simulated-burnup” loading would produce a

flatter power distribution and a further decrease in critical mass. It is expected, however, that the effects of fission product poisons, gaps in the reflector, gas plenums, etc., will about offset these increases in keff, so that

the final value of the critical mass will not be greatly different from the

7.1 kg first obtained.

Refinements to the idealized, isothermal problems are now underway.

The temperature distribution of the reactor at rated power has been estimated, and the control requirements for this case will be computed, taking

into account fission product poisoning effects which are temperature dependent

because of cross section dependence on temperature. The power coefficient

of the system will also be estimated, and better estimates of the isothermal

temperature coefficient will be made considering in greater detail the structure of the core, reflector, and insulating carbon.

Transient behavior of the system has been given only casual attention

so far. One of the basic problems is the obtaining of data on fission product

retention as a function of fue 1 element temperature. The interrelation of

negative temperature coefficients of nuclear origin and the reactivity effects

associated with the 10Ss or migration of fission product poison during a temperature excursion can not be evaluated until reasonable assumptions as to

fission product diffusion rates can be made.

IL Criticality Calculations

1. Normalization. The Honeycomb assembly used as the reference

critical experiment consisted of a core in the form of a four foot cube; it

was reflected on all sides by one foot of graphite of density 1.55 g/cc. 3 The

critical core consisted of 7.98 kg of Oy (93.5% U235, 6.5% U238) in the form

of foils (0.001 in. by 2.8 in. by 7.5 in.) which were distributed uniformly

through graphite of density 1.50 g/cc. The structural Al in the Honeycomb

assembly had an effective density of 0.165 g/cc in the core and reflector.

The fuei region C/Oy atomic ratio of 6650 corresponded roughly to the average C/Oy ratio in Turret.

In the calculation of keff for the above experiment, C. B. Mlls ? 18

group cross-sections were used. 4 A technique for applying results from the

one-dimensional FIRE code to the experimental cubic geometry was similar

to that later used in Turret calculations. To start the calculations, artificial

g-th group absorption cross sections

-9-
Za = 2 Dg B2

g

(1)

were used in a plane calculation to mock up side leakage. Dg in Eq. 1 is

the group diffusion coefficient for the g-th group and B2 is an approximate

buckling obtained from

B2 =

()

2

m2/H +H

c r

(2)

where Hc and Hr are the core thickness and total reflector thicknesses, respectively.

From the flux distributions, P g, obtained from the first plane calculation, new leakage cross-sections were computed using

()dq

g 2D—

dz He/2

J

H /2

c

dz

o

‘g

(3)

in which the slope of the flux ~ g is evaluated at the core boundary, He/2,

for the symmetric plane case. Equation 3 represents the core leakage per

(cm3) of core transverse to the one dimension of calculation, and it may be

shown that, for solutions F (Bgz) to the diffusion equation, Eq. 3, reduces to

Za = 2DgB&

5

The Za from Eq. 3 were then inserted into a second plane

ca culation, replacing those obtained from Eq. 1. After a third plane calculation, using the fluxes obtained from the second calculation to recompute

the Eq. 3 cross-sections, keff converged to a value of 1.111.* After making

the experimentally determined correction for the Oy foil self-shielding

(~eff = - 0.067), a net l~ff of 1.044 was obtained for the Honeycomb critical

experiment. Although the disagreement may be due to other inhomogeneities

in the experiment which were neglected in the calculation, critical calculations for Turret have been normalized to a keff - 1.05.

2. Neutron Group Reductions. In order to reduce the labor involved

in the Turret calculations, the 18 group cross-sections used in the normalization were colIapsed to five groups usirqg a procedure described in A.N.L.

I

*This method of computing the transverse group leakages has been used by

other authors. See, for example, reference 5.

-1o-
5800, p. 411.6 From a preliminary 18-group Turret calculation, group

weighting factors

dg=Uq/~Ugqg (4)

g ‘gin G

were formulated. In Eq. 4, g and G refer to the 18 and 5 group sets, respectivel~ and Ug and q are lethargy widths and fluxes per unit lethargy,

respectively. The weight%g factors were used in the following formulae:

Fission and absorption cross-sections:

‘G = x da

gin Ggg

Transport cross -sections:

‘G = () 1/ ~ dg/ug

gin G

Transfer cross-sections:

oG+G? = x d

I

o

gin G ‘g’in G’ ‘-g’

Fission Spectrum, lethargy widths and fluxes:

‘G ‘Lxg

gin G

(5)

(6)

(7)

(8)

Maxwellian-averaged values for the thermal. fission and absorption

cross-sections at 1350”C were calculated using

(J(T) = (m To/4T) 1’2 g(T) a. (9)

In Eq. 9 C’. is the “cross-section at 2200 meters per see, To and T represent

-11-
I

20”C and 1350”C expressed in “K, respectively, and g(T) is the non-1/v correction factor given in ANL-5800, p. 79.6 The five-group parameters as -

computed using Eqs. 4 through 9 are listed in Table I.

3. First Turret Geometry. The calculational model first used for

Turret is illustrated in Fig. 1. The core was a graphite cylinder of height

37.5 in. and outer radius 36.32 in. containing 312 radial fuel channels. The

region from radius zero to 6.94 in. contained 79% graphite by volume and

21% void. The fuel channels were located in 13 equally spaced horizontal

planes with each plane containing 24 radial channels. Each fuel channel contained five fuel elements which were graphite cylinders of length 5-7/8 in.,

1 in. o.d., and 1/2 in. id. The channel i.d.ls were 1.1 in.

For calculational purposes the core was divided into five- radial regions

of equal width corresponding to the five fuel elements per channel. These

are designated as Regions 2 through 6, Region 2 being the innermost core

region. The inner graphite zone (r = O to r = 6.94 in.) described above was

designated as Region 1. The concentration of Oy in the fuel elements of

each radial region were selected according to the following prescription in

order to approximate 50% burnup in the fuel elements as the fuel moves

through the reactor:

Region 2 3 4 5 6

Concentration 4/6 C 5/6 C C 7/6 C 8/6 c

The concentration c (in grams of Oy per cc of graphite), and hence the

critical mass, was the parameter which was varied in the calculations for

criticality.

In the calculations all the radial regions were homogenized. This neglected the self-shielding of the fuel which was estimated to affect ]~ffl <0.02.

A perturbation that was taken into account was the effect of contaminants in

the graphite. It was assumed that type H4LM graphite would be used in fabricating Turret components. From the contaminant specifications for this

graphite7 it was estimated that the thermal absorption contributed by the contaminants was about equal to the thermal absorption due to the carbon. The

contaminants were mocked-up in the calculations by inserting an amount of

boron giving the same thermal absorption.

The seventh radial region in the problem, of radial thickness 15.7 in.,

was the radial graphite reflector. Top and bottom reflectors were 16.3 in,

thick graphite. The graphite density was assumed to be 1.6 g/cc throughout.

4. Procedure for First Criticality Calculations. The procedure used

in calculating Turret problems was similar to the method used in the normalization calculation. An infinite cylinder problem, containing the seven

-12-
I

Group

1

2

3

4

5

Element

Boron

Carbon

u

235

u

238

&***

FIVE

Energy Range

10 Mev - 900 kev

900 kev - 17 kev

1’7 kev - 61.44 ev

61.44 ev - 0.14 ev

Thermal

Group

1

2

3,

4

5

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

TABLE I

GROUP PARAMETERS

“* u **

—g —i%

2.408 2.408

6.377 3.969

12 5.623

18.09 6.09

--- 1

FIVE GROUP CROSS-SECTIONS

u

—a

5.170 - 02

3.947 - 01

5.475 + 00

4.581 + 01

2.779 + 02

0

0

0

2.083 - 04

1.178 - 03

1.303 + 00

2.393 + 00

1.433 + 01

7.372 + 01

2.311 + 02

4.295 - 01

2.839 - 01

4.711 + 00

5.715 + 01

1.013 + 00

2.278 - 01

1.225 - 01

8.081 - 02

8.132 - 02

7.310 - 02

‘TR

1.830 + 00

2.501 + 00

7.244 + 00

3.888 + 01

2.814 + 02

1.549 + 00

2.880 + 00

4.387 + 00

4.340 + 00

4.828 + 00

4.505 + 00

7.327 + 00

2.148 + 01

7.063 + 01

2.411 + 02

4.367 + 00

7.389 + 00

1.643 + 01

6.301 - 01

1.001 + 01

0

0

0

0

0

—‘f

o

0

0

0

0

0

0

0

0

0

1.312 + 00

1.988 + 00

1.008 + 01

4.004 + 01

1.908 + 02

3.968 - 01

0

0

0

0

0

0

0

0

0

Fission Spectrum

inel

u

g, I?+l

2.469 - 01

1.612 - 01

1.034 - 01

1.555 - 01

0

2.402 - 01

1.747 - 01

1.275 - 01

1.892 - 01

0

1.137 + 00

2.178 - 02

1.724 - 02

1.296 - 02

0

1.582 + 00

4.356 - 02

1.379 - 02

1.815 - 02

0

0

0

0

0

0

0.716

0.284

0

0

0

inel

u

g+I(I>l)

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*U = lethargy of lower energy limit of group

g

**U c grouplethargy width

g

***Used in conjunction with Eq. 1

-13-
w

a

a

a

E

tw

—— —

L NO193&i

I

I

I

I

,,SZ”EII —-----l

t

‘l+

In

+ :

r-p

U-J

In

.- _-l1

9 NO1’33ti

I

I

b NO193Ll

G NO193kl

I

1

Z NO193&i

I

I

2a/u16 9Z’I ,d

311 HdVM9

I NO193M

——

3LiO~ H9f10kl Hl 1 lVIXV

-14-

I

1

1-

r

0

E

N

0

z

s

-1

-4
homogenized radial regions previously described, was first computed using

the Eq. 1 approximation for the vertical leakage. From the radial fluxes,

P rw gene~ted bY the first cYlinder problem> radial lea~ge cross-sections

we~e computed for each of the five core regions

(

d(p d(p

Dr~ rg

gldr ‘1 )

- ‘2 dr r2

Za =

g ~ rO

from

(lo)

In Eq. 10, rl and r2 are the inner and outer region radii, respectively.

An infinite plane problem was next computed, using the Turret vertical

dimensions, for each of the five core regions. The radial leakage crosssections (previously generated in the cylinder problem) for each given region

were added as absorption cross-sections to the core material cross-sections

for the region. Vertical leakage cross-sections were then obtained from the

plane problems using Eq. 3. Finally, the infinite cylinder problem was recomputed using the new vertical leakage cross-sections in place of those

computed by Eq. 1.

5. Results of First Criticality Calculations. Four calculations were

performed in order to obtain a curve of keff versus total Oy mass. The

results are given in Table II.

TABLE II

CALCULATED keff VERSUS TOTAL OY MASS

Problem Number Oy Mass (kg) k

eff Temperature ~C)

42F 3.06 0.732 1350

41F 5.11 0.967 1350

43F 7.10 1.055 1350

40F 7.65 1.072 1350

The above data are plotted in Fig. 2. JII the vicinity of case 43F the relation

-15-
1.1

.0

% 0.9

x

0.8

0.7 I 1 I I

3 4 5 6 7 8

OY MASS (KGS)

Fig. 2 keff vs Oy mass at 1350”C.

-16-
‘eff’keff = 0.21 AM/M (11)

is obtained. Case 43F has been adopted as the standard critical-configuration

for flux and power plots, and for future computations based on the same

geometry. The fuel concentration, c, for this case is 0.08 g/cc. The maximum concentration which may be conveniently attained during fuel element

impregnation is about 0.15 g/cc.

Five-group fluxes from case 43F are plotted in Figs. 3 through 8.

Radial fluxes are given in Fig. 3 and the vertical fluxes for Regions 2

through 6 are plotted in Figs. 4 through 8, respectively. The vertical fluxes

were normalized to the radial fluxes by setting the average vertical flux for

a given group and region equal to the average radial flux for the group and

region. All fluxes correspond to a power of 3 MW.

It is seen that the thermal flux strongly predominates in the Turret

system. The central peak in the radial thermal flux indicates that the central region will be the most valuable position for inserting control rods.

Table III contains a listing of the power produced per fuel element. In

the table, channel O designates a channel in the horizontal plane passing

through the core

through 6 in the

center, and the off-center channels are designated by 1

order of their proximity to the center. The radial variation

TABLE III

POWER PRODUCED PER FUEL ELEMENT (l@

Reactor Power - 3 MW

Reactor Region

Channel

o

1

2

3

4

5

6

—2

1.867

1.852

1.808

1.739

1.660

1.557

1.471

—3

2.166

2.147

2.092

2.006

1.908

1.778

1.668

—4

2.334

2.312

2.251

2.154

2.042

1.893

1.762

2.287

2.266

2.205

2.107

1.994

1.840

1.699

—6

2.077

2.058

2.001

1.909

1.802

1.651

1.507

Channel Total

10.731

10.635

10.357

9.915

9.406

8.719

8.107

-17-
m“ o

I

CL?I . . — uu

4

I

I

I

J

I II

o

0

0

0

*“ m“

d

-“

I

Io

(~~s-zW3) /u El(ll 40 SllNfl NI Xfl14

-18-
4.0

3.0

2.0

I.C

I I I I I I I I I

CORE

GP3

~

o 10 40 50 60 70 80 90

HEIGHT ABOVE CENTER (CM)

Fig. 4 Vertical fluxes, Region 2.

-19-
4.0

3 .C

I I I I I I

I I I I I I I I I 1 1

cORE

*

GP3

I .0 GP 2

GP4

Gp I

I

1 I I I I I 1 I I I I

I

40 50 60 70 80

0 10 20 30

2.

HEIGHT ABOVE CENTER (CM)

90

Fig. 5 Vertical fluxes, Region 3.

-20-
-

v

u

(n

I

N

z

v

\

c

~

o—

-

x

3

-1

IL

4.0

3.0

2.0

I .0

0

I I I I I I I I I [ I I [ I I I I

- GD3

GP2

GP4 REFLECTOR

GP I

I I 1 I I I I I I I I

10 20 30 40 50 60 70 80 90

HEIGHT ABOVE CENTER(CM)

Fig. 6 Vertical’ fluxes, Region 4.

-21-
I

I

3.0

cORE

I I I I I I I I

2.C

1.C

REFLECTOR

b

w

Gp 3

GP 2

Go 4

GP I

I I

, 1 I 1 I I I I I I 1

20 30 40 50 60 70 80 90

0 10

I

1

HEIGHT ABOVE CENTER (CM)

Fig. 7 Vertical fluxes, Region 5.

-22-
.1

3.0

2.0

1.0

0

I I I I I I I I I I [ I I I 1 I I

CORE

I REFLECTOR \

GP 3

GP 2

GP4/

GPI>

I I I I I I I I

10 20 30 40 50 60 70 80 90

HEIGHT ABOVE CENTER(CM)

Fig. 8 Vertical fluxes, Region 6.

-23-
in the fuel element power illustrates the departure from linear burnup that

occurs in the fuel channels. Since linear hurnup was assumed in order to

make the calculations, it would be expected that further calculations including adjusted fuel element burnups would lead to some corrections to the results. The variation in the total power per channel indicates that the fuel

element flow rates would have to be varied from channel to channel in order

to achieve the same total burnup of all the fuel elements.

III. Isothermal Temperature Coefficients

Isothermal temperature coefficients of reactivity were calculated for

the 7.1 kg Turret loading previously selected as the standard 1350°C critical

configuration. Contributions to the reactivity coefficients included the effects

of cross section, dimensional, and material density changes, over a temperature range of 20°C to 1350”C. In addition the prompt reactivity effects associated with a change in ~ element temperature was investigated. Calculational procedures were the same as those used in making the critical mass

estimates.

In the computations, it was assumed that the thermal absorption and

fission cross-sections vary with temperature according to Eq. 9. Other effects of temperature upon cross-sections were neglected except for those

changes brought about by the increase in the epithermal lower energy limit

with temperature increase. This energy limit was taken to be the energy

corresponding to the most probable velocity of the thermal neutrons at a

given temperature. Room temperature (20°C) cross-sections are listed in

Table IV for the epithermal and thermal groups (groups 4 and 5). These

may be compared to the cross-sections at 1350”C listed in Table L The

effect of thermal expansion upon Turret dimensions and ,material densities

was estimated using a linear expansion coefficient of 4.2 x 10-6 which is

the specification for H4LM graphite over the range 20”C to 1350”c.8

Results from these first calculations are summarized in Table V.

Problem 43F is the reference criticality calculation at 1350”C. In

problem 44 an intermediate cylinder calculation was performed in which

20”C cross-sections were used, but the vertical leakage cross-sections were

the same as those used in 43F. In 44F the vertical leakage cross-sections

were corrected to obtain room temperature values. The comparison of 44

and 44F illustrates the fact that the main contribution to the Turret temperature coefficient arises from the variation in thermal leakage with temperature change. The effect of thermal expansion is obtained from a comparison of problem 45 with problem 43F.

The temperature coefficients are listed in Table VI.

-24-
TABLE IV

FIVE GROUP CROSS-SECTIONS AT ROOM TEMPERATURE

Element Group

Boron 4

5

Carbon 4

5

u

235 4

5

u

238 4

5

0-

a —

1.878 + 02

6.538 + 02

8.380 - 04

2.771 - 03

1.774 + 02

5.809 + 02

2.907 + 01

2.832 + 00

‘TR

6.075 + 01

6.573 + 02

4.340 + 00

4.830 + 00

9.766 + 01

5.909 + 02

1.518 + 01

1.138 + 01

o

0

0

0

1.409 + 02

4.890 + 02

0

0

Cross -sections for groups 1, 2, and 3 are given in Table I.

TABLE V

TEMPERATURE EFFECT CALCULATIONS

Problem Temperature of

identification cross-sections

numbs r (“c)

43F 1350

43G 1350

44 20

44F 20

45 1350

*Fuel elements expanded in length about 1.3%.

Temperature

of expansion

~c)

20

20*

20

20

1350

inel

o

!3,g+l

6.498 - 02

0

7.906 - 02

0

5.415 - 03

0

7.581 - 03

0

k

eff

1.0545

1.0540

1.1930

1.2542

1.0434

-25-
TABLE VI

TEMPERATURE COEFFICIENTS

Contribution Temperature Range % Ak Reactivity Coefficient*

Cross-sections 20”C - 1350”C 20.0 1.30 x 10-4

Expansion 20°c - 1350”C 1.1 0.08 X 10-4

Total slow 20”C - 1350”C 21.1 1.38 x 10

-4

Prompt 20”C - -3000”C -0 -JO

*Ak/k per “C

A comparison of 43G and 43F indicates that the thermal expansion

contribution to the prompt temperature coefficient is negligible. h making

the calculation, the effect of prompt expansion effects in the fuel was emphasized by allowing the fuel element rows to expand inward a total of 1 cm,

which is the extension corresponding to about a 300O“C temperature rise in

the fuel. (In the current Turret design a latch is placed at the outer end of

each fuel channel; fuel elements abutt against this latch. Since the latch is

attached to the bulk mwierator, the fuel elements must expand inwards in the

event of a prompt fuel temperature rise.)

A more exact estimate of the prompt fuel-temperature reactivity coefficient was subsequently carried out. A coefficient of -3 x 10-6 (Ak/k)~C

of fuel temperature rise was obtained. The moderator temperature coefficient, [-1.4 x 10-4 (Ak/k)flC] is thus about 50

prompt reactivity effects produced by variation

factor of 50 is roughly equal to the ratio

volume of graphite in mcxierator

volume of graphite in fuel

The bulk moderator in Turret does make some

times as effective as the

in fuel temperature. This

contribution to the prompt

reactivity coefficient, however, in that about 5% of the fission energy is deposited in the moderator by gamma rays and neutrons. One might estimate

therefore that a total prompt coefficient might be as large as

3 x 10-6 (Ak/k)~C + .05[1.4 X 10-4 (Ak/k)~C] ~ 10-5 (Ak/k)/”C

fuel moderator

-26-
The overall Ak = 21.1% represents the amount of shutdown control that

must be provided to overcome the temperature coefficient. It would be expected that an additional few percent in k of control would be required to

overcome the effects of fission product poisoning and the power coefficient.

Other effects besides expansion may contribute to the prompt temperature coefficient. In particular, fission products, including xenon, samarium,

and the delayed neutron precursors, may diffuse out of the fuel elements

during a large temperature surge. No estimate has yet been made of the

magnitude of reactivity changes which could be induced in this fashion.

rv. Control Rod Calculations

As previously noted, it is estimated that keff increases by 21.1% as the

Turret reactor cools from its operating te~perature of 1350”C to 20°C. Although the above estimate includes only effects due to isothermal temperature changes, it should represent the major contribution to the room temperature excess k which must be overcome by shutdown rods anwor other

means. Because of the large amount shutdown Ak that must be provided,

and because of the mechanical and materials problems associated with the

int reduction of rods into the high temperature rotating core of Turret, particular emphasis has been placed upon the control rod calculations for Turret.

In addition, conservative allowances have been made for calculational uncertainties since the re are no experimental data currently available on control

rod measurements in reactors similar to Turret in general nuclear characteristics.

The rod calculations performed to date are described in two parts below. In Part 1, some initial calculations are described which were designed

to investigate whether sufficient control could be achieved by using absorber

rods in regions outside the core, i.e., in the central graphite plug and radial

reflector. In Part 2, calculations which were performed on a system of

three rings of rods are discussed. This configuration, which corresponds

approximately to the control rod system that has been adopted for Turret,

contains a ring of rods in each of the three principal radial regions: the

central graphite plug, the core, and the radial reflector.

1. Survey Calculations. Four calculations were performed utilizing

B. Carlsonfs new Sn code (DSN)l. The Sn option used was S4. The results

are shown in Table VII. Problem 50 was used as a reference problem for

computing rod worths; thus, it contained no absorber rod materials. The

specifications for this problem were the same as those for problem 44F

(see Table V, Section III), except that a 4 in.

used and a 1 in. annular void was interposed

o.d. central void region was

between the core and radial

-27-
TABLE VII

k~ff FOR VARIOUS CONTROL ELEMENT CONFIGURATIONS

Problem No. Case Rod Material

50 No rods ---

51 4 in. o.d. central rod B4C

52 4 in. o.d. central rod plus 1 in. B4C

curtain between core and reflector

53 4 in. o.d. central rod plus 1 in. Berated

curtain between core and reflector graphite

k

eff

1.22

1.13

0.85

0.88

reflector. In problems 51 and 52, the rod material assumed was B4C of

density 2.5 g/cc; and, in problem 53, the material assumed was berated

graphite containing 10 a/o boron enriched to 95% in BIO. The BIO atom

densities for the two materials are 2.06 X 1022 per cc for B4C, and

7.62 X 1021 per cc for the berated graphite. The rod materials considered

for Turret use contain BIO densities comparable to the above densities.

Five-group radial flux plots obtained from calculations 50, 51, and 52

are illustrated in Figs. 9 and 10. Fluxes obtained from problem 53 were

very similar to those shown in Fig. 10 from problem 52. It is seen in the

figures that the absorption in the rods is strong enough to give effectively a

complete suppression of the thermal and epithermal fluxes at the rod surfaces. The results indicate that sufficient control can be obtained using rods

situated outside the core. It is expected, however, that some loss in effectiveness will be encountered in a more practical enginee-ring design which

must include clearances and some segregation of the material in the outer

control curtain. As a hedge against these losses, the use of four or five

smaller rods in the central plug rather than a large single rod would be

recommended. More effectiveness should be achievable by placing the

smaller rods as near to the inner surface of the core as is feasible. As is

seen in the figures the thermal flux rises appreciably between the central

rod and the core inner surface.

2. Worths of Control Rod Rings. Although the estimates of the worths

in Turret of a 4 in. o.d. central rod, and a 1 in. thick control curtain, inserted between the core and radial reflector indicated sufficient control could

be achieved by this arrangement, subsequent materials and design investigations led to the choice of a three-ring control-rod system composed of

-28-
In

z

I

v

0

a

w

I

i

/

I I 1 I I I 7

w 10 N

a

CJ

a

(93 S-ZWWLJ901 40 sll Nn NI xnlj

-29-
w In e m N c1

(03 S-ZWW901 40 sllNn NI xn74

-30-

!-4

0
1-1/4 in. o.d. ZrB2 cylindrical rods. In this system, a ring of five rods is

situated on a 3.5 in. radius in the stationary central plug, a second ring of

six rods is inserted on a 20.5 in. radius in the rotating core, and a third

ring of five rods is located in the stationary radial reflector on a radius

3-in. outside the core periphery. The calculations reported herein indicate

that the sixteen-rod, three -ring system should provide sufficient shutdown

control for Turret with reasonable allowances made for calculational error

and for the malfunction of one or two rods in the system.

Reduction to Two-group, One-region Model. Two-group diffusion theory

calculations have been used extensively for estimating the worths of rings of

cylindrical rods in thermal reactors. 6‘g Usually the two-group theory is applied to a one-region, or at most, two-region system since the analytic solutions become very unwieldly if more groups or regions are considered. For

these reasons, and since the geometry of a ring of control rods is not ,easily

adaptable to the numerical type of machine calculation, a two-group, oneregion model has been used for estimating rod worths in Turret.

To homogenize the Turret core and central plug regions, five-group

FIRE code calculations were performed on “a homogeneous core and reflector

model in which the fuel concentration in the core was determined by requiring that the simplified two-region model give the same keff as the more detailed calculational model described in Section IL3. Specifically, the room

temperature case, problem 44F in Section III, was taken as the reference

problem. The original average fuel concentration, 0.00279 g/cc Oy, turned

out to be the correct one to use in the simplified model.

Secondly, the five-group constants were collapsed to

for the two-group equations

using the

Xai, Zfi,

obtain constants

= o

(12)

collapsing procedure described in Section 11.2. In Eq. 12 the Di$

and ~i, i = 1, 2, are diffusion coefficients, absorption and fission

cross-sections, and

and i = 2 referring

21, z, was adjusted

solutions p2 and q2

fluxes, respectively, with i = 1 referring to the fast group

to the thermal group. The group transfer cross-section,

in value so that the buckling p2 of the two-group buckling

to Eq. 12

-31-
P2N?2=+ [Iia- P)2+4s]

1/2 —+(a+p) 1

o!=Za2/D2,13 = Zal + Z12 - VZ

( )

~1 /D1, S = v~f2Z12/D1D2

(13)

gave a correct JO@r) fit to the five-group thermal flux in the core. In

Eq. 13 v is treated as an eigenvalue where

v = v?/k

eff

in which v f is the number of neutrons released

The two-group constants generated by the

below:

TABLE VIII

(14)

per fission.

above procedure are tabulated

D

TURRET TWO-GROUP CONSTANTS

Core Reflector

Fast Thermal Fast Thermal

1.324 0.8730 1.179 0.8599

1.320 X 10-3

4.204 X 10

-3 4.154 x 10-4 5.844 X 10-4

3.667 X 10

-4

3.389 x 10

-3 0 0

2.155 X 10-3 2.155 X 10

-3

V1 = 2.46 k = 1.2342 eff

These constants were used to calculate the radial flux plots shown in Fig. 11.

The calculational procedure used is that described by Murray .10 Vertical

leakage allowances are included in the core Za ~s of Table VIII which include

group leakage cross-sections collapsed from the five= group case. It is seen

in Fig. 11 that adequate agreement is obtained between the five-group and

two-group fluxes.

Two-group Control-ring Code. A FORTRAN code, ROD1, was set-up

for obtaining solutions to the control ring problem on the 704 computer.

critical determinant for the case of a ring

produced in Appendix I from reference 11.

of M rwis in a bare core is

The ROD1 code solves the

The

re -

-32-
—l-

.-

lnH*,.,+-

A 11 II

— 2 s= -&J

: \

—

n

z \

>

o

co

In

—

●

—K

II

o

SN /

-

a

ii

-

-

o

N

o

*

o.

D

3

D

3

r%

b.j

l-l

!-l

al

UJ o In

N

~ u) — o

( Sll Nfl ANVUL18NV) Xflld

-33-
critical determinant from a given set of input data including:

(a) the core constants listed in Table VIII,

(b) the core extrapolated radius, R,

(c) the rod radius, a,

(d) the number of rcxls in the ring, M,

(e) the ring radius, d,

(f) two initial guesses for &f,

(g) convergence criteria,

(h) the order of approximation, L, and

(i) the fast and thermal rod extrapolation distances dl

spectively.

For the case of no control rods in the bare equivalent core,

ling is determined by

P2 = (2.405 )2/R2,

and d2, rethe radial buck-

(15)

where R is the extrapolated core radius, and kefi is then fixed by Eq. 13

snd 14. With rods inserted in the core, ROD1 iterates on ~. by linear extrapolation until the critical determinant vanishes within the desired convergence criteria. Comparison of the cases with and without rods establishes

the rod worths.

The ROD1 code is generally applicable to the ring problem except that

for rings containing 16 or more rods the high order Bessel functions obtained have values exceeding the capacity of the 704 computer. The case of

a single central rod is solved by setting L = O, M = 1, and the ring radius

equal to zero. Experience has shown that convergence will not be achieved

if the initial guesses for kef f are too low compared to the correct value.

Generally, for a noncentral rod or ring of rods, the second order approximation, L = 1, produces sufficient accuracy, but the code will compute orders

up to L = 9 if the Bessel function limitation cited above is not exceeded.

Rod Extrapolation Distances. Five-group DSN (S4) calculations were

used for estimating the rod extrapolation distances, dl and d2. The geometry

in the computations was that of a central rod in the bare equivalent core.

In the eight cases considered, the rod parameters were varied as described

in the following table:

-34-
TABLE IX

CENTRAL ROD WORTHS

Problem

Number

Rod o.d.

(in.)

71F

71D

71K

711

71M

71L

71E

7lJ

7/8

1

7/8

1

1-1/8

1-1/4

1

1

Rod Material

ZrB2 (0.621 g/cc)

natural boron

ZrB2 (0.621 g/cc)

natural boron

ZrB2 (4.65 g/cc)

natural boron

ZrB2 (4.65 g/cc)

natural boron

ZrB2 (4.65 g/cc)

natural boron

ZrB2 (4.65 g/cc)

natural boron

B4C (2.5 g/cc)

natural boron

ZrB2 (4.65 g/cc)

Rod Worth

(% Ak)

2.66

3.03

3.14

3.37

3.55

3.66

3.64

3.87

dl (cm)

11.04

9.959

7.085

7.233

7.376

8.138

5.979

8.138

d2 (cm)

3.175

3.293

3.225

3.432

3.760

4.108

3.403

4.108

boron enriched to

85% in BIO

A second FORTRAN code, ROD2, was devised for computing dl and d2

from the DSN results. ROD2 contains a subroutine which is a two-group -

version of the FIRE code modified to solve the central rod case. Flux calculations start at the rod boundary, r = r., from the boundary condition

where d is

FIRE code

d=‘(w%) (16)

the appropriate group rod extrapolation distance. h terms of the

flux solution

Pn = @n Pn+l + Pn (17)

-35-
for the nth space point (Eq. 18 of LA-2161, p. 18), the starting conditions

at the rod boundary are

2

a. =

[)

Ar CAr2 + aAr2 21+ T-— —

rod D

PO = b a. Ar2/2D

(18)

The definitions of the terms in the above equations and the remaking procedures for the diffusion problem, as incorporated in ROD2, are described

in IA-2161.2

Using the diffusion theory solutions, ROD2 fufiher iterates on dl ~d d2

(by linear extrapolation from initial guesses) until the relative changes in the

group neutron losses, due to inserting the rod, satisfy

L21 - L

2

L1 + L2

(19)

where L1 and L2 are the nor~lized two-group neutron losses (due to absorption and escape) and 1g,g=l, ..., 5 are the normalized five group losses

obtained in the DSN calculations. The primed losses refer to the case with

the rod inserted in the bare core, and unprimed losses are for the case of

no rods in the core. Thus the two group neutron balance conditions above

give criteria for computing dl and d2 and at the same time meet the condition that the correct Ak be obtained since

AK/K =1 - 1/(1 + AL/L) = AL/L (20)

where L now represents the total losses.

The fluxes in the control rod as obtained from the S4 calculations for

cases 71D, E, I, and J are illustrated in Fig. 12. The flux discontinuities at

the rod boundary, particularly in the case of the thermal flux, illustrate the

desirability of using a transport theory approach to obtain rod extrapolation.

-36-
ii! 3x

0

IL

In

\

f=

\

ca

k

.+G

!-l

Oa
distances. Figures 13a and 13b show the effect of varying the BIO concentration and the rod radius on the extrapolation distances. It is seen that

practically all of the change in rod worth for the given range of absorber

density is due to the change in the epithermal absorption in the rod (represented by change in all). Consequently, the massive boron densities represent an uneconomic use of poison, but it is highly desirable from the mechanical standpoint to minimize the number of rods in Turret.

Ring Worths . Worths of the three rings described in Table X below

are tabulated in Table XI as computed by ROD1 using the extrapolation dis -

tances for the eight rod cases described in Table IX.

TABLE X

lUNG PARAMETERS

Description Number of Rods, M

Plug ring 5

Core ring 6

Reflector ring 5

Rod Case

TABLE XI

RING WORTHS (% ~)

Ring Worths

71F

71D

7lK

711

71M

71L

71E

7lJ

Ring Radius, d

8.89 cm

52.07 cm

98.11 cm

Plug Ring

8.38

9.13

9.31

9.74

10.00

10.17

10.19

10.56

Core Ring

9.76

11.18

11.56

12.47

13.12

13.47

13.47

14.35

-38-

Reflector Ring

3.64

4.14

4.30

4.63

4.86

4.98

4.99

5.32

Total Worth

21.78

24.45

25.17

26.84

27.98

28.62

28.65

30.23
!

u1 IL w u,

I18L

YI.L

(SUN9) s39Nvlsla

I

3Qq

bil k

bjl

E-4

.!+

-39-
The reflector rod worths as computed by ROD1 were corrected by the factor

f = @:(d) /q:(d) = 2.97

where q R(d) is the unperturbed thermal flux at the ring radius in the re -

fleeted case, and qB (d) is the same flux in the bare equivalent core.

The 1-1/4 in. rod, case 71L, has been tentatively adopted for use in

Turret since it does not require the use of enriched boron to give sufficient

shutdown control. Since it is estimated that approximately 21% N of control

will be needed to shutdown Turret from full operation at 3 MW and 1350”C to

room temperature, a 7~ & safety factor will be available as an allowance

for calculational error, or for the malfunction of some rods. Additional control may be readily achieved by increasing the number of rods in the reflector ring if future investigations indicate more control is needed.

The worths of the rings are plotted in Fig. 14 in terms of the number

of rods in the ring for case 71L. It is seen that in the central plug, negative rod shadowing (ring worth less than M times values of one rod) occurs

due to the close spacing of the rods. In the core and reflector rings, on the

other hand, positive shadowing occurs.

v. Miscellaneous Reactor Kinetics Considerations

Some rough estimates have been made which

behavior of Turret following a loss of coolant flow

indicate qualitatively

during operations at

the

3 MW.

It was assumed that the prompt temperature coefficient of reactivity associated with the fuel is N zero, so that power generation continues, without any

sudden decrease, at the 3 MW level. The heat capacity of the fuel elements

should limit the initial fuel element temperature rate of rise to 10°C/sec.

Since the neutron temperature is determined primarily by the bulk moderator

temperature and since the mode rator heat capacity is 40-50 times greater

than the capacity of the fuel elements, it is expected that the fuel element

temperatures would rise to a high level before the slowly-acting reactivity

coefficient associated with mcderator temperature would shut off the power.

However, it appears at least 10-20 seconds of time is available for inserting

control rods before the fuel element temperatures become uncomfortably high.

In connection with the above, the possibility of diffusion of fission product poisons out of the fuel elements during a fuel element temperature rise

should be considered. Jn this event a power surge would be initiated if the

poisons escaped the core. It is expected that the power rise would depend

upon the amount of reactivity insertion and upon the shut off mechanism

-40-
14

12

10

8

—

—

—

— RING WORTH

-----(M)x (WORTH OF ONE ROD)

/

G

+$/

Jo,/4-G‘

/’PLUG

4

/

/

/ RING

/-

/

/1/

/’ /

I

v I I I I I I

I 2 3 4 5 6

M(NUMBER OF RODS)

Fig. 14 Ring worths vs number

ZrB2 rods (4.65 g/cc -

of rods, M, in rings [Case 71L: 1

natural boron)].

-41-
associated primarily with the 5% of fission energy which is deposited promptly

in the bulk moderator via elastic scattering of the prompt neutrons and attenuation df the prompt fission. gammas.

Some calculations have been made in an attempt to predict the fission

product inventory as a function of operating time, and hence the poison level,

in Turret, but these estimates are of limited reliability because diffusion

rates of fission products as a function of temperature are not known. The

wide range of temperatures existing in the Turret core is a further complication, since important poisons may be effectively held up in the cooler portions during normal operations, while in the hotter portions the concentration

of these same poisons may be much lower.

-42-
.,....

I APPENDIX I

THE CRITICAL DETERMINANT FOR A RING OF CONTROL RODS

IN A BARE CORE

The application of the condition that the fluxes vanish at the extrapolated

radius of the bare core, and the rod boundary conditions (Eq. 16), to the twogroup diffusion equations (Eq. 12) yields the critical determinant for a ring

of rods. The detailed method of analysis is given in reference 12, and the

critical determinant in the form described below is reproduced from reference 11. To the Lth approximation the determinant is

L

1( ‘kn%n- ‘kTkn ) = 0

k=0,1,2, . . ..L

n=o

where

‘kk=a32ka44k - a34ka42k

Bk=a 31ka44k - a34ka41k

d

kn

= 1, n+

= O, otherwise

[ 1

(-l)k (l/dl - k/a) Jk(pa) + Jk+l@)

a31k =

a32k

=ld-(/ )

k/a Yk@a) + Yk+l(pa)

1

a34k =ld(/ 1

- k/a) K#a, + Kk+l(~a)

a41k [

= (-l)k S1 (1/d2 - k/a) Jk@a) + Jk+l(w

I

-43-
s

a42k = 1 [( l/d2

s

a44k = 2 [(

l/d2

with the coupling coefficients

- k/a) Yk@a) + Yk+l @a)

1

)

- k/a K#a) + Kk+l (qa)1

defined as

D2p2

)

D2q2

)

and

al

nr

q ‘kn = sa ‘JMs+k

@d) JW_n 1

@d) Yw@R)/JMs@R) - Lh

‘kn ‘ ~=2 2 ‘k+n ()‘rim Cos ‘kn

(

m- 11

Cos p

kn

=cosr(n-k)—

M ‘F )

in which the bucklings p2 and q2 are defined by Eq. 13, rlm is the linear

distance from a reference rod to the mth rod in the ring, and the other

constants are defined in the text. In the expansion of T~, ROD1 applies a

term ratio test for selecting the number of terms required to meet a convergence criteria which is specified on input.

-44-
APPENDIX II

EXCERPTS FROM LASL INTERNAL MEMORANDA

Pertinent data from LASL internal reports are reproduced herein for

the benefit of readers who do not have access to these reports:

3. Byers, C. C., 1?Graphite - Mode rated, Graphite-Reflected Critical

Assemblies, f’ N-2-529.

. . . . . . . . The first system built up on the assembly machine utilized the maximum volume possible and consisted of a 4 ft. cubic core

surrounded by a 1 ft. thick graphite reflector. k this configuration,

the critical mass is 7.985 kg of oralloy (uranium enriched to 93.2 w/o

U235) distributed in the form of 0.001 in. by 2.8 in. by 7.5 in. foils

throughout the moderating graphite to give a C/Oy atomic ratio of 6650.

The effective density of the graphite in the reflector is 1.55 gm/cm3

while that of the core is 1.50 gm/cm3. The building up of the fuel

subassemblies from a number of graphite plates results in a lower

average density than for the single pieces used in the reflector. . . . ...”.

. . . . . . . . The effect of self shielding in the oralloy foils may be

determined by replacing a number of the 0.001 in. foils by an equivalent weight of 0.002 in. foils and measuring the effect of such an interchange on the reactivity of the system. This was done for a number

of foils and the effect from a 100% interchange was estimated to be

approximately -890 cents. . . . . . . .

7. MacMillan, D. P., ~Unterim Data on H4LM Graphite,’1 N-1-158.

. . . . . . . . No detailed analyses of molded H4LM are available, but

since extruded and molded H4LM use the same raw materials, an ash

analysis of extruded H4LM is given below:

-45-
s .05%

Fe203 .2

NiO .002

CaO .04

A1203 .01

Ti02 .01

V (Believed to be less than 100 ppm and

guessed to be 70 or 80 ppm)

sio2 .1

B Approximately 1 ppm

Total .41+

-46-
REFERENCES

10

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Carlson, B., Lee, C., and Worlton, J., ~!The DSN and TDC Neutron

Transport Codes, rr LAMS-2346, February 12, 1960.

IWlls, C. B., and Brinkley, F., !IA one .~imensio~l ~te rmediate Re _

actor Computing Program, t~ LA-2161, March 27, 1959.

Byers, C. C., ‘Graphite -Mderated, Graphite-Reflected Critical Assemblies, ‘I Internal Report, N-2-529.*

Mills, C. B., llPhysics of Intermediate Reactors, II LAMS-2288, March 31,

1959.

Fmleraro, A., !~An Iteration Method for the Specification of Multigroup

Bucklings, ” Nuclear Science and Engineering —~6 6 (1959).

?!Reactor Physics Constants, ff ANL-5800, July 1, 1958.

MacMillan, D. P., Whterim Data on H4LM Graphite, !! Internal Report,

N-1-158.**

Eatherly, W. P. et al., !Physical Properties of Graphite Materials for

Special Nuclear Applications, ” Paper l?/708, Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958.

‘Reactor Control Meeting Held in Los Angeles, March 6-8, 1957.ff

TID-7532, Part 1, October 1957.

Murray, R. L., Nuclear Reactor Physics, Prentice Hall, Inc., Englewood

Cliffs, N. J., 1957, p. 110.

Murray, R. L., ~fReactivity Values of Multiple Reactor Control Rods, ~!

Paper presented at the Nuclear Engineering and Science Congress, sponsored by Engineers Joint Council, December 12-16, 1955, at Cleveland,

Ohio. Preprint 83, published by American Institute of Chemical Engineers, New York, N. Y.

Garabedian, H. L., ~~control R&i Theory for a Cylindrical Reactor,’1

AECD-3666, Westinghouse Atomic Power Division, Pittsburgh, Pa.,

August 9, 1950.

*Pertinent information from this internal report can be found in Appendix II.

**Pertinent information from this internal report can be found in Appendix II.

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