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:'3 ‘?{'}JI THE MODERATOR COOLIN s ' 8N
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ke EL‘)P‘%‘ THE REFLECTOR-MODERATED REACTOR

 

6') R. W. Bussard
' W. S. Farmer o
H.
P

A, Fox
A, P. Fraas

 

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CLASSIRICA TN cn“zi To: te o

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OAK RIDGE NATIONAL LABORATORY

OPERATED BY

CARBIDE AND CARBON CHEMICALS COMPANY

A DIVISION OF UNION CARBIDE AND CARBON CORPORATION

(4 oy >
POST OFFICE BOX P

OAK RIDGE. TENNESSEE »
¥

ORNL 1517

 

%

£ ’ »
This documeht consists of 45 pages.
Copy4 of 165 copies. Series A.

Contract No. W-T405-eng-26

THE MODERATOR COOLING SYSTEM FOR THE

REFLECTOR-MODERATED REACTOR

R. W. Bussard A. H. Fox
W. S. Farmer A, P. Fraas

September 1953

DATE ISSUED

JAN 22 1954

OAK RIDGE NATIONAL LABORATORY
Operated by
CARBTIDE AND CARBON CHEMICALS COMPANY
A Division of Union Carbide and Carbon Corporation

Post Office Box P
‘r .
- LCRTRIGY AR

Osk Ridge, Tennessee
o ' 3 4455 D349555 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 
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ORNL 1517
Reactors-Research and Power

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THE MODERATOR COOLING SYSTEM FOR THE REFLECTOR-MODERATED REACTOR

R. W. Bussard A. H. Fox
W. S. Farmer A. P. Fraas

INTRODUCTION

Cooling the reflector region of the reflector-moderated circulating
fuel reactor presents an important set of problems. While reflector mate-
rials and coolants can be chosen independently of shielding considerations
for most types of reactor, this 1s not the case for an aircraft reactor
because the shield and i1ts weight are of such great importance. Not only
mist the reactor core be as small as possible but the reflector material
mist be chosen to give both a minimum fast neutron leakage from the reflec-
tor and a minimum production of hard secondary gammas in the reflector.

A number of reflector materials were considered on the basis of fast
neutron leakage per unit of thickness to get some notion of their influence
on shield design. (See Fig. 4.11 in ORNL-1515)1. This study showed that
beryllium was far superior to any of the other common reflector materials
such as beryllium oxide, carbon, or sodium deuteroxide, and 1t is somewhat
superior to Dp0. Although a fluid reflector might simplify the heat remov-
al problem, none of the fluid reflector materials that can be used at
temperatures of the order of 1000°F were comparable with beryllium for
neutron moderation and reflection. The reflector cooling problem was there-
fore studied using beryllium in order to evaluate the sources of heat
generation and the magnitude of their effects.

The design chosen for this study employed a fuel region in the form
of a thick-walled spherical shell of fuel with a beryllium reflector
surrounding 1t and a beryllium "island” filling the interior. The power
was taken as 200 megawatts and the core diameter as 18 in. giving a power
density 1in the fuel region of 5 kw/cm3 (see ORNL-15151, page 65). This
gave a source of radiation next to the reflector greater than that in any
existing reactor. (The MTR has a power density of 0.291 kw/cm3 in the
fuel region.)

SOURCES OF HEAT GENERATION

 

Heat wi1ll be generated in the reflector by the absorption of gamma
rays coming from the fuel and heat exchanger regions, and by the slowing

 

1. "ANP Quarterly Progress Report for Period Ending March 10, 1953,"
ORNL-1515.

 
down of fast fission neutrons. Approximately 12 Mev per fission was taken
as the total energy of the gamma rays generated in the fuel region. Of
this T.35 Mev was taken as representing fission fragment decay gammas with
an average photon energy of 1.5 Mev while 4.60 Mev was taken as coming
from prompt fission gammas with an average energy of(é°5 Mey.2 This choice
of gamma ray energy per fission has the effect of lumping all the fission
product decay gammas in the core. The distribution of decay gamma energy
between the core and heat exchanger regions depends upon the relative fuel
volumes of the regions, which varies with the detail design. As a first
approximetion all of the decay gamma energy was "lumped" in the core. For
such an assumption the estimate of the power density distribution in the
reflector will be somewhat higher and require more cooling close to the
fuel region than would be the case had the fission product decay gamma
energy been balanced between the core and heat exchanger regions. At the
same time "lumping" the decay gammas in the fuel region will yield an
underestimate of power density in the outer region of the reflector. It
is easy to correct for these effects later as will be shown (Appendix F).
Neutron capture results in the emission of a photon of approximately 9
Mev energy in the case of nickel and 6.8 Mev energy in the case of beryl-
lium.3 The kinetic energy of the neutrons amounts to approximately 5 Mev
per fission.

 

CALCULATION OF HEAT GENERATION

 

The ratio of peak to average power density appears to be fairly close
to unity i1n the three-region beryllium-reflected sodium-cooled reflector-
moderated reactor design. (See Table 4.1 - ORNL-1515)1. Therefore a
uniform power density and hence a uniform gamma source was assumed for
the fuel region in the first calculations. A later check using a non-
uniform power distribution from a multigroup calculation gave essentially
the same results.

In order to evaluate the self-absorption of gammas i1n the fuel region,
a typical fuel of sodium fluoride, potassium fluoride and UF) was
chosen for evaluating the absorption coefficient. (This does not mean
that the above fuel would necessarily be that specified finally for thas
reactor.) Gemma rays emtted in the fuel region cover a fairly wide
spectrum of energies with the mean value somewhere between 1 and 2.5 Mev.
Since heat generation was of principle concern, the mean energy was taken

 

2. "Estimested Heat Production in the NRU Reflector," CRT-50L.

3. E. P. Blizard, "Introduction to Shield Design," CF 51-10-70,
Part 1.

 

rI -‘ -6
‘
as 1 Mev i1n determining the absorption coefficient in the Inconel and
beryllium. This served to maximize the heat generation rate and gives a

limiting value. The absorption coefficient in the fuel region was also
evaluated at 1 Mev.

Since the elements in the reflector and fuel are principally of low
atomic number, Compton scattering 1s the principle mechanism for degrada-
tion of the gammas in the above energy range. The gamma rays were assumed
to be attenuated exponentially in calculating the heat generation. A
build-up factor was not employed since core diameter and reflector thick-
ness were small enough (of the order of one mean free path for 1 Mev
gammas) to make the need for a scattering correction questionable. It was
assumed that scattering would be straight ahead in direction and that
Compton collisions merely degrade the photon in energy. This method over-
estimates the gamma ray intensity for large distances.

An 18 in. diameter spherical fluoride fuel region surrounding a 9 in.
diameter central beryllium island and enclosed by a 12 in. thick beryllium
reflector were chosen as a typical geometry for calculation. The fuel
region was separated from the island and outer reflector by a shell of
3/16 in. thick Inconel. The reactor power output was taken as 200 mega-
watts 1n determining the total energy release. The neutron flux for
computing neutron capture gammas was taken from the spatial flux plot for
reactor calculation number 129 (see Table 4.1 ORNL-1515)1. The specific
heat generation rate was computed at points spaced about one inch apart
along the radius from the center of the 1sland to the outside of the
reflector.

The heat generation rate in the reflector arising from atfenuation
of the gamma rays emitted from the fuel region was computed by several
methods. In the first method (Appendix A) the attenuation was computed
taking into account numerical differences in the value of the absorption
coefficient of both fuel coolant, Inconel and beryllium. In Case A using
this method, the absorption coefficients employed were 0.09, 0.16, and
0.30 em-1, respectively, for the fuel, beryllium and Inconel regions. In
Case B values of 0.06 and 0.13 em-1 were used for the fuel and beryllium
regions, respectively, in order to evaluate the effect of the absorption
coefficient on the heat generation rate. The equation for exponential
attenuation using the above absorption coefficients was numerically
integrated to arrive at an answer for the heat generation rate at the
various space points. (Appendix A.) The results of this calculation
were checked by a method of graphical integration using Pappus' theorem.
(Appendix B.)

Another approach to the evaluation of the heat generation rate was
made by means of analytical solutions for exponential attenuation in terms

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of exponential integrals. A solution which would not involve graphical
or numerical integration could be obtained for two particular cases. By
assuming a uniform absorption coefficient throughout both the fuel and
the reflector regions, 1t is possible to solve the exponential integrals
directly. (Appendix C.) In the second case, the absorption coefficients
in the fuel region and reflector region were assumed to be different. An
approximaete solution can be obtained by solving the the problem in two
steps. The flux of gamma rays from the surface of the fuel region was
obtained using the uniform absorption coefficient solution method. The
attenuation through the Inconel was obtained by a slab source approxima-
tion. The resulting flux was then assumed to be spread over the surface
of the Inconel and the absorption in the reflector was determined for a

surface source. The results of this calculation are tabulated as Case C,
using the same absorption coefficients as in Case A.

The power density resulting from neutron moderation within the beryl-
lium of the island and reflector was obtained dire¢tly from multigroup
results by using the flux distribution §(r, M) or (¥nl)x C.F. (Correction
Factor)*. The energy loss for each lethargy group Nn/is the average energy
loss per collision times the number of collisions in that group at a given
radius or space point. The spatial distribution

_uli -u21

tn
7 W t —'}_ e -
10 1Z=1 _%_xC,F,, x f% ( e ) (1)

1s normalized to the total power lost by moderation (2 1/2% of reactor
power) by the use of the integration operator (1 (see ANP-58).

Gamma rays also result from parasitic capture of neutrons in struc-
tural materials and coolant. One particularly strong source of hard
gamma, rays is the Inconel shell separating the fuel annulus from the
outer reflector. These gammas are captured over an appreciable volume
rather than locally since the photon energy 1s high and the attenuation
length large. A minor amount of heating also results from the genera-
tion of gamma rays by neutron capture in the beryllium. The extent of
the captures in both beryllium and Inconel can be obtained directly from
the multigroup calculations or by using the integrated spatial neutron
flux distrabution weighted by the absorption probability. The latter
method was employed here. (Appendix D.)

REFLECTOR HEATING

 

The power density in the various regions of the reactor owing to
absorption of gamma rays from the core and reflector regions i1s tabulated

 

4. D. K. Holmes, "The Multigroup Method as Used by the ANP Physics

Group," ANP-58.

 
in Table I. The rate of heat generation is shown for Case A, Case B, and
Case C for the capture of gamms rays emitted in the fuel region. The last
column includes the heat generation rate caused by the capture of gamma
rays generated by parasitic neutron capture in beryllium plus those from
parasitic capture of neutrons in Inconel. Gamma rays will also result
from neutron captures in the coolant. These were not computed owing to
the uncertainty regarding the final coolant to be employed and the volume
fraction of the reflector that might be occupied by this coolant. Their
effect should be small, however. The power density resulting from the
slowing down of neutrons and gamma heating for Case A are plotted in Fig. 1.
The total integrated power in various regions from heating by gamma ray
absorption for Case A and Case B 1s shown in Table II.

 
Region

6

 

el

 

TABLE I

POWER DENSITY IN VARIOUS REGIONS

 

 

Island Beryllium

Island Inconel

Fuel

Reflector Inconel

Reflector Beryllium

Neutron
Radial Position Gamma. Heat Generation Capture
T Case A Case B Case C (n,d)
Cm. Watts/cm Watts/cm3
0 47.5 61.9 30 2.3
5.08 60 65 36 2.5
7.62 80 Th 6k 3.4
10.16 101 93 152 3.4
10.95 120 142 176 3.3
10.95 223 330 330 14,4
11.4 354 Lo7 koo 144
11.4 106 99 146
12.7 167 137
14 186 148
15.2 182 152
17.1 184 151
20.3 157 132
21.6 141 116
22.9 73 65 100
22.9 215 325 333 69.2
23.3 150 227 199 69.2
23.3 80 98 106 13.5
2k.1 23.7 27
25.4 37.7 4h .6 414.8 7.4
26.7 25.4 31.9
27.9 19.0 o4 L 17.4 5.0
30.5 10.3 1%.0
33.0 5.4 8.3 3.9 3.3
35.6 2.9 4.9
38.1 1.67 3.0 1.1 2.6
53.3 0.066 0.19

 
POWER DENSITY (watts /em®)

450

400

350

300

250

200

150

100

50

ISLAND

INCONEL SHELL
Na ANNULUS
INCONEL CAN

T ATING
TOTAL HE TOTAL HEATING

Y HEATING

NEUTRON
HEATING

Y HEATING

NEUTRON
HEATING

2 4 6 8

RADIAL DISTANCE FROM REACTOR CENTER LINE, » (1n)

Fig. 1. Radial Power Density from Neutron and Gamma Heating.

REFLECTOR

TOTAL HEATING

NEUTRON HEATING

y HEATING

10 12 14

16

18

DWG 187!!

 

20

i

«
AR
* -

£

kY
TOTAL INTEGRATED POWER IN VARIOUS REGIONS

Region

Island Beryllium
Island Inconel
Fuel

Reflector Inconel

Reflector Beryllium

— =

TABLE TI

Total Power Megawatts

 

Case A
0.46
0.24
6.71
0.57
3.14

Case B

0.51
0.29
5.58
0.85
3.44
 

The peak heat generation rate occurs in the Inconel shell separating
the 1sland beryllium from the fuel annulus. High heat generation rates
also occur in both the reflector and the i1sland immediately adjacent to
the fuel annulus. In cooling the Inconel core shells 1t 1s necessary to
take 1nto account not only the heat generation rate given in Table I, but
also the heat flowing through the Inconel from the fuel annulus when the
moderator region is designed to be operated at a lower temperature than
the fuel region. This can be computed readily by conventional methods.

COOLING SYSTEM

The heat generated in the Inconel and beryllium can be removed by
any one of several coolant passage arrangements. A liquid metal 1s the
most desirable coolant 1f the resulting heat is to be employed usefully
1n the engine air radiator circuit, since this gives the least sensitive
and highest heat transfer coefficient. Lead, bismuth or L1l might be
used 1n place of sodium because their effect on neutron moderation might
offer a certain nuclear advantage. The coolant chosen should not have
too high a neutron absorption cross-section (05.41_0.5b) and, what 1s
just as important, must be compatible with the materials of construction.
Lead, bismuth, non-uranium bearing fluorides, NaOH, sodium, and NaK were
all given serious consideration as coolants for the beryllium moderator.
Metallurgists consulted on the problem felt that lead or bismuth would be
likely to pose serious mass transfer difficulties. The relatively high
neutron absorption cross section of the potassium in the NaK made 1t quite
undesirable from the critical mass standpoint. Rubidium might be used 1in
place of potassium but because of little demand 1t 1is currently very
expensive. Thus sodium seemed to be the best choice for the moderator
coolant. Since corrosion and mass transfer might occur in a beryllium
and sodium system, 1t seemed desirable that the beryllium be clad in some
fashion. Work at Battelle5 indicates that beryllium can be chrome plated
electrolytically to give satisfactory resistance to sodium attack at
932°F. Chemical plating 1s also possible with beryllium. However, the
formation of brittle intermetallic compounds and the difficulty of elim-
inating pinholes with either chemical or electrical plating methods makes
the stability of any plating rather questionable under thermal cycling
and high temperature conditions. An alternate possibility 1s to can the
beryllium in thin-walled Inconel cans and to fill the small interstices
between the beryllium and the can with stagnant sodium to provide a
thermal bond. This arrangement appears to be the more promising of the
two, but both possibilities are being investigated. The reflector could
be constructed of two large hemispheres of beryllium if the canning
technique were used. Cooling passages could be rifle-drilled through
the beryllium and lined with thin-walled tubes, which could be welded
1nto headers at the ends. The Brush Beryllium Company has indicated that

 

5. J. G. Beach and C. L. Faust, "Electroplating on Beryllium," BMI-T732.

 
-]
el

the fabrication of these large hemispherical shells would probably be no
more difficult than the fabrication of large flat slabs. Personnel of
the Y-12 beryllium shop state that 1t would not be difficult to rifle
drill holes 3/16 to 1/4 in. in diemeter as much as 40 1n. deep, with the
hole diameter held to within 0.001 in. and the hole center location held
to within 0.010 in. These holes could be drilled at a rate of 2 1n./m1n.
This estimate was based on the experience gained in machining the beryl-
lium of the MIR reactor and in drilling small diameter holes in the
beryllium reflector of the SIR. In addation to the holes in the beryl-
lium reflector, channels could be provided between the Inconel core shells
and the canned beryllium reflector in order to remove the heat generated
in this region.

An slternate construction for the reflector region involves the use
of a large number of wedge-shaped segments shaped much like the sections
of an orange. These sections could be made with shallow grooves in their
surfaces to form passages for cooling streams of sodium. This would be
a relatively expensive arrangement since a great deal of machine work
would be required because the beryllium can be hot-pressed to uniformly
high densities only in flat slabs or spherical shells.

A design study was made using the rifle-drilled hole arrangement to
investigate the detail problems of cooling the beryllium regions. The
beryllium was assumed to be canned in Inconel and cooled by sodium flow-
1ing through Inconel tubes in rifle-drilled holes. Stagnant sodium would
be allowed to fill the interstices between the beryllium and the can to
facilitate heat flow across that boundary. In order to achieve a well-
balanced design, & number of factors must be considered. The volume of
both the sodium and, especially, the Inconel must be minimized to keep
parasitic neutron absorptions within reasonable limits. It 1s also nec-
essary to operate the reflector regions at a relatively high temperature
to keep from penalizing the engine-radiator system. Large thermal stresses
are likely to result from the high rates of heat generation to be found in
these regions. Because beryllium becomes quite ductile at temperatures
sbove LOOPF cracking ought not be a problem. It was felt that thermal
stresses should be kept within reasonable limits to reduce distortion,
however, as this might become a problem after a number of thermal and/or
povwer cycles of the system. For this reason, the temperature variation
1n the beryllium between adjacent coolant passages was held to 50°F in
this first design. The pressure drop through the various coolant passages
was limited to 4O psi to keep pressure-induced stresses low. The maximum
beryllium-sodium interface temperature was held below 1200°F to minimize
the possibility of mass transfer in the beryllium-stagnant sodium-Inconel
systen.

 

 
smeliiiny

Several detail designs were investigated that favored first one and
then another of the various requirements, that 1s, minimum poison, minimum
variation in beryllium temperature, minimum beryllium-sodium interface
temperature variation, minimum sodium system pressure drop, etc. Fig. 2
shows the hole pattern in the beryllium for a promising arrangement. The
temperature distribution for this hole pattern i1s shown in Fig. 3.

ALTERNATE COOLING SYSTEMS

 

A careful examination of alternate cooling systems was made 1n an
effort to avoid the problems involved in cooling a solid beryllium island.
The sodium-cooled, solid beryllium outer reflector was assumed 1n every
casge.

The use of a semi-fluid beryllium powder mixture with a liquid metal
in the interstices was considered. A mln%mum porosity or liquid metal
volume fraction of 12% may be attainable. This would necessitate the use
of a liquid metal with a low neutron capture cross-section and good
"moderating" properties such as lead or bismuth. These are difficult to
contain, however, owing to corrosion and mass transfer. If a satisfactory
container material could be found, a semi-fluid moderator with a reasonably
small neutron age, [, would be attractive on the basis of ease of removal
for beryllium recovery and also as a safety measure.

Replacing the fuel annulus and i1sland by a graphite block containing
perhaps 40 unlined fuel passages about 1.5 in. in diameter has been proposed
as another possibility. The success of this system would depend largely on
whether the fuel could be kept from diffusing or penetrating into the
graphite as a result of permeation and/or cracking. If this were to occur,
severe overheating would result and self-destruction of the graphite would
take place. The destruction would be abetted by the large decrease in
thermal conductivity that accofipanles a temperature increase in graphite.
Actual testing of graphite in fused-fluoride, uranium-bearing salts under
conditions of thermal and mechanical shock will have to be made to evaluate
this problem. In addition, multigroup calculations will be necessary to
determine the critical mass and power dastribution. This latter item 1s
expected to be poor. Should Inconel tubes be required to protect the
graphite, a secondary cooling system would be required for cooling the
Inconel tube wall to 1500°F owing to the volumetric heat source effect.

If this were required the major advantage expected of graphite would be
lost.

A non-viscous fluid moderator for the i1sland with desirable heat trans-
fer properties would simplify the heat removal and the fabrication problems

 

6. M. Muskat, The Flow of Homogeneous Fluids Through Porous Media,
J. W. Edwards Company.

el

 
 

3

     

N

Nl

ISLAND

Y

 

.
O

)
NS

DWG 18744
PASSAGE FOR INLET Na
TO REFLECTOR (2 REQ'D)

026-in -DIA HOLES IN Be, CONTAINING
025-in ~0OD TUBES FOR Na COOLANT FLOW

 

INCONEL SHELLS
AND Na ANNULUS

Fig. 2. Cooling Hole Distribution 1n Reflector-Moderator.

®

i

Lo
oyt
-SI-

TEMPERATURE (°F)

1600

1400

1200

1000

800

600

400

200

DWG 18742

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ISLAND FUEL REFLECTOR
r ZAPPROXIMATE
AW NN/ TEMPERATURE
U U DISTRIBUTION ,
U N N |
7 X r
Na COOLANT vp
HOLES Na COOLANT HOLES e
by
= INCONEL SHELL—=— L |
Na ANNULUS %:?;,
v
e — INCONEL CAN o b
5
flf:w?
o
2 6 8 10 12 14 16 18 20

RADIAL DISTANCE FROM REACTOR CENTER LINE (in)

Fig 3 Temperature Distribution Across Midplane of 200-Megawatt Reflector-
Moderated Reactor Sodium inlet, 1000°F, sodium outlet, 1150°F.
in this region. A non-fuel bearing fluoride or sodium deuteroxide has
been suggested for this purpose. However, the heat transfer properties
of these fluids are generally poor so that cooling the Inconel shell
sufficiently that there would be no serious hot spots becomes a severe
problem. For example, 1f in the case 1n the preceding section sodium
deuteroxide were to flow through the 9 in. diameter” island with a mass
velocity giving a 100FF axial coolant temperature rise, a heat transfer
coefficient of only 312 Btu/hr f£2 OF would be obtained. In the design
cited having a 3/16 in. Inconel shell, there would be a heat flux of
1,000,000 Btu/hr ft2 from the shell, and thus a radial temperature drop
through the hydroxide boundary layer of 3200°F. A major improvement in
the sodium deuteroxide heat transfer coefficient could be obtained only
by i1ncreasing its velocity. To increase the velocity enough to give
satisfactory cooling would entail an excessive pressure drop, and even
then minor vagaries in velocity distribution would make occasional hot
spots rather likely. The same problem presents 1tself if a design
involving an i1sland filled with a fluoride is considered. Only the
liquid metals appear to have the necessary thermal properties for this
purpose. Lead, bismuth, or L7 appear to be the only possibilities.

In order to use these latter fluids successfully the mass transfer and
corrosion difficulties currently common to them must first be solved.
Nickel and 1ts alloys which are usually the best container materials for
fused fluoride salt fuels exhibit generally poor corrosion resistance to
these liquaids.

CONCLUSIONS

The heat generation rate or power density was computed for a reactor
using a fuel annulus with a beryllium reflector and island. The heat
generated in the reflector can be removed by sodium-cooling a canned
beryllium reflector-moderator. Further, the Inconel shell containing
the fuel annulus can be satisfactorily cooled by the same sodium system.
Differential thermal expansion does not appear to be a problem since the
volumetric coefficient of expansion of both the Inconel and the beryllium
are approximately the same -- (5.5 x 10~ 1nches/inch OC). Furthermore,
samples of extruded beryllium have shown negli%ible dimensional change
and warping after thermal cycling up to 930°F. At the time of writing
the sodium cooled, beryllium i1sland and reflector arrangement 1s felt to
be the best proposal. No new technological problems appear to be 1nvolved
1in the detailed mechanical design of this system.

A number of alternate cooling systems are available which may have
some advantages over the solid beryllium system above. However, all of
these involve problems that will have to be solved by experimental
investigation.

 

7. The Reactor Handbook, Vol. 3, Materials, USAEC.

>
4~

 

 

 
-15-
—“ .

APPENDIX A

Gamme.-Ray Absorption Inside and
Outside A Spherical Annular Source

The heat generation rate due to absorption in a region within and out-
side a spherical annular source can be obtained by means of numerical or
graphical integration of the integrals defining the dastribution of gamma
rays. Several approximations were used to set up the integrals. The first
approximation used involved the straight ahead theory of gamma absorption.
Here 1t can be assumed that Compton collisions merely degrade the gamms
energy, but do not scatter the photons. Only exponential attenuation with
a coefficient characteristic of the material through which the photons pass
need be considered. When a photon passes from one medium to another, no
refraction is considered.

A uniform fission rate in the: source region was assumed. Thus a
uniform power density for the core was determined by divading the total
reactor power output by the volume of the fuel region of the fuel annulus.

\

The attenuated intensity at @, distance r from a point source of density

Pos 18 glven by the formula

_Mr

MPo €
P = “— dv A-1
By

vhere podV represents the source intensity, e'/4r the aflsorptlon effect,
1/hsr2 the spreading effect from the point source, and A the probability
of absorption at the point Q. Then the total intensity at Q due to a
source extended over a region R 1s given by

b/(‘ 2
S
P = MPo Tz W A-2
R

For the reactor in question, the source was taken as an annulus be-
tween an inside sphere of reflector material and an outside spherical
shell of the same material. Because of the spherical symmetry, the
distribution of heating power along any diameter of the spherical system
was considered a sufficient answer to the problem.

 
Consider a point F in the fuel spece of the reactor as 1llustrated.

I
M~% e

    

If the power density at thais point 1is Py 1n an element of volume 4V, then
the power at a point Q a distance h from the center of the reactor may be
obtained by integration, using spherical coordinates centered at Q. Be-

cause of the symmetry sbout the axis along which h is measured, the element

of volume dV = 2xre sin 6d6dr includes the volume around all points F with
the given spherical coordinates r and 6. Since r may pass through both
reflector and fuel spaces, 1t 1s broken 1nto r = ry + ro, where the sub-
script 1 refers to the fuel and 2 refers to the reflector. Separate
reciprocal attenuation lengths, 47 and 4o must be determined for these
materials. For convenlence the fuel space may be divided 1nto three
regions (I, II, and III) bounded by cones tangent to the inner and outer
spheres. 1In section the figure 1s as follows

 

The total power generation P due to absorption at Q is given by
summing the contribution from the three source regions I, IT, and III.

 
M-l(No//L) '2\ 1 9—/0-1
e/ T /171 2
P= £y /. AN A o 8.4 A6

R
0 Lmfi—bi A~

 

i (r LR ) Aeri @ + b
’ t -/ a("’ A+‘2'q"i) "/""1("3_ - a ’“‘3) 4

 

+ @o/fij

X
0 Lm5+o-_1 1~

I

 

pYRIR)  fhkn 4b,
e"‘/’";f"a, - 2 A-3
+ f’a/“:-{ ifl‘n. s & 4(4. ffi(—é >

a8 i, A
T

where, for simplicaity,
g =Y ey~ A8

o '
bl = Ra "'j'. Al 9

 

 

It 15 possible to represent ro as a function of © by the equation:

ro = h cos © -YR® - h® sin® © A-b
and
1 /n® - ®° ‘
cos 6 = —-'I':é——+r2, A-5
and then
ry=Tr-Tp . A-6

!.‘ ! - .

P
_18-

el

The total power generation P at Q, on eliminating ry 1s then given by the
expression*

 

M‘J (No/,l. ) A ara B — oty
P:/Po/"'&f J e—w&#d/‘&fiimflx’wéfi,dfi
0

A
X.Mfi“bi

 

pion 8 A A8

1
R @VKKJ ‘&A#L5i+b
L 4"0/‘-;[ ’ [ 16‘(/“'.1‘/“'13('“&*2"'1)"/"1"'
A

0

ng-hfl-g_

o4
A

 

- L = /'va__—/u._.l/‘u
+ e Q”a /1)

M-l(fi/j\.) s 6 +by
f i B A e . A-T

‘“"""-1 ("’o/'&) Aowoa 8 ~by

Integrating this equation with respect to r yields:

 

/MM-:L("’D/’L)\ ;
& g
-1
A (R/A) :
ey TV -
+{ e &11—63&’%1)%«9%9 A8
4

Arasa (ngfi&)

 
-y

 

Let s = h cos ©. Then

A
P io_fi’; e—/“;(fl-*fq) [l _ e-—/"-i("‘i “Ai)jl s
g A
YAE - A2
X 2
-n,
. e-fla@""'i) |: _ ‘4/:1‘1] / , A-9
A2 - R
where

/Ci = VR& _J\.:L'l'fl&
/FE-I :-]/Nj‘-'/e\. + ’

This 1ntegral can be evaluated by numerical or graphical means for
the heat generation at any point Q. In several special cases the above
equation can be 1ntegrated.

For h = 0, the complete spherical symmetry of the system gives the
power at the center as:

 

R
- - - 2
P = Po/ae f e Agro e /Ml(r ro) t_z'%g
s
_ PoMa S HM2To [1 - e'fll(R'rO)] . A-10
A

Another point of special interest 1s the point at the interface of
fuel and island reflector. Here h = r, and the fuel region can be broken

 
-20-

&

-

 

 

up into two regions as follows: ™~ .
}PL_ (.]/" 2
\ }pL Qo »‘
/2 v AL +b
Fo s g et = o = Ay )
P = 2 c R 8 A E
0 alvomé

 

 

 

A
1/2. ~ o
IT
T/
_ ’f’o/"';_ f -ol/a-&/vomél: _ _/,.,1(6 —namfi):, ,04»9,4(.9
- 1/a3 :
H
4‘ - (b + A mfi) _
+ao/"‘.1j [1-—6/_13‘ 2 ]Mé’,fiéfi A-12
/1 /2

 
 

Let s = ro cos®. Then ds = ry sin 646

 

and
"
F‘ o9 e"%/"‘a’a‘ [1 _ e-/“i("'a'/")] Ao
2y 1
0
I'\./o )
- (tg—a
+ [l - e.’Ahi : ] Ao
0
—S g Yy
400/«72 _1 - e f T ———
Ze— o —— + A, ———
1/'-_1“"0 Q/";\ e
T leama)
- Ca= - s
_ e/“‘i:."(eé,/“';. +1)/& , A-13
0
where
/C»&: R&_%a "l'/cla=L .

The last integral can easily be determined by graphical or numerical
integration.

For a point at the surface of the outer reflector adjacent to the
fuel space, the fuel space can be broken into three regions as before.

 
 

The heat generation or power density P is given by the sum of the
contributions from each of the three regions.

o

b porn (00 /R) f R et 6=,

0 - M

P= A - f [ T i b de
0

- I
i o Yoy IR) 2R 22 6

* /5 f
+ X :

Rm@*‘AL

- IT

/ T/ AR o b
""”’*f f " pin b ds| , A
A e na (R) Jy

 

0

—pg (R ) ety
e )7 ® i 6.4 s

 

 

 

IT

where

R )
e R

 

Or on integrating with respect tor ,

Po
- 22 f

- - e B
+ f 0 Sy O [e 4 (K era 2
¢

 

I:l _ e-/"-.L(Rma"""&)] Ry

a8 ol6

e—/fiiaflmé -;lza-;\i‘l

A-15

 
 

Let s = Rcos®. The above integrals then become:

 

JRE R
_ a e"a/kl B

P _ 1
N A [ gy gl s
+ e e -1l-e ]Ao A-/6
R —n

Where

/v‘e-a': f"oa.""‘ fia‘ +A‘=L

The last term above can be integrated numerically or graphically.

 
-2~

APPENDIX B

Mechanical-Graphical
Gamma-Ray Heating Computation

A method of graphical integration was used to check the- approximate
values of power density obtained by numerical integration using the method
of Appendix A. In this method lines were scribed on an aluminum sheet to
simulate the fuel space boundaries. From a point P i1n the diagram below
curves were drawn showing lines of constant values of e-MT, fThese curves,
when rotated about the X-axis, form surfaces of revolution enclosing regions
of a given total power. |

 
    

/
Ao}

- . —te———

\ ) |
By choosing radii so that each of these shells encloses the same fraction,
say 10%, of the total power, 1t is possible to keep errors uniform through-
out the computations. When more than one material is involved, the radius

has to be divided into r) and rp, so that e-(H1r1 + M2r2) has the required
value.

After the curves are drawn, the areas of the irregularly shaped
regions bounded by these curves may be obtained with a planimeter. Then
the volumes of the corresponding solids of revolution may be computed
from Pappus' theorem in the form

V = A(2ny) B-1

where ¥ is the distance from the axis of rotation to the centroid of the
area. This centroid may be located by cutting the aluminum sheet along
the bounding curves, and balancing the irregular pieces on a knife edge.

A point Q 1s then chosen 1n the volume and the average power 1is
considered as generated at this point. Then the value of the integrand
(e~MT/hxr2) 1s evaluated at this point. The product of this value and
the volume of the region gives the power density contributed by this

 

.
. >
- ’ 1
.‘\
 

region to the point P. Repeated computations for all the areas in the
fuel region give the total power density as observed at P.

The arbitrary choice of a point for evaluating the integrand does
not seriously affect the result as long as points are chosen near the
geometric center of the area. Accuracy 1s limited to two significant
figures by the precision of cutting the sheet, and of locating the cen-
troids. This method, however, can be applied to very irregular geometries
in which axial symmetry allows the use of Pappus' theorem.

 
«26-

mllleny

T
APPENDIX C

Solutions 1in Terms of Exponential Integral
Functions for Radiation Heating due to a
Spherical Annular Source

The power density at any point within or outside of a fuel bearing
source region can be evaluated by approximation without recourse to numer-
ical or graphical integration. As in Appendix A the straight ahead
collision theory of Compton Scattering was assumed. The gamma rays may
be attenuated exponentially without considering a "buildup factor.” The
source was assumed to be an isotropic emitter.

The power density at a point within a spherical annular source 1is
given by the following expression for the case of the same asbsorption
coefficient 1n both source and absorber region:

R=Ro e=fl
AS
Pg = Po Mg 2xR° s1n6 WE— dodr C-1
R=Ri 8=0
where Ry - 1nner radius of the source annulus
Ro - outer radius of the source annulus
r - the distance from the center of the spherical
coordinates to the absorber
Po - source power density
/qa - absorption coefficient of the absorber
o - distance from particular source point in any given gesma
source shell to the absorber

The resulting expression can be reduced to a double integral in
terms ofJP and R by eliminating the angle & through the cosine law.

f.2 = R2 + r2 - 2r R cos® c-2
This gives the following equation:

R=Ry f = R+tr

Af
Py = I’_g}./'& R 7?'—dfd3 c-3

R=R, f = R-r
 

 

 

 

By integration by parts and using the exponential integrals tabulsted
in the WPA Tables of Sine, Cosine and Exponential integrals it is possible
to evaluate the above integral completely.

 

 

 

 

 

 

 

-
Pa, = %{‘E Roe-r [El| /\' (Ro—r) - El' I‘“(Ro"’r) J
RiE_rE = ) (® )
iy l‘ M (Ry-r - E3] M(Ri+r
- M(Ro-r) =2
-(MRo+1+Mr) E%IE 0 (MRi+1+pM 1) #M(Ri ¥}
+(MRBo+l-Mr) -2‘“"—';' ) (MRi+1l-Mr) ;H: KEL v C-4

o ber is located a.t a radi&l m point r (r ZR,) out-
side the source annulus , the absorption or power density for the same
absorption coefficient in both source and sbsorber region is given by the
following double integral.

 

-r+R

R=R,
PoM
P, = 3528 f f R—-—dfrm c-5

R=Rj -I‘-R

 
 

 

 

 

-28-

 

—

Integrating by parts and using exponential integrals again this becomes

 

2 2
P, = Po Aa Ro-r [ Ell/u(r-Ro)! - E]_’ (I‘+Ro)l]
2r e
|
312~r2

- AT [ ElI/L{(r—Ri)I - Ell ,U(r+R1)U |

 

~-M(r-Ro) ~-M(r-Ri)
+ (MRg-l+Mur) 2—%?- - (MRy-1+pmr) _2___}(9_2 i
- M(Ro+T) - M (Ry+r)
+ (MBoHl-Mr) 5z - URL-MT) o c-6

When the source region has an absorption coefficient that i1s dif-
ferent from that of the absorber (M #A445), or in the present case the
reflector region, the previous solutions are not exact outside the
boundaries of the source region. It 1s necessary then to use the above
solutions 1n the case of a thick annulus to compute the flux Pa//t( g at
the boundaries of the source region, r= R; and r= Ro. For velues of r
much greater than Ry or much less than R, the resulting power density at
r can be obtained by treating the flux (Fg/Mg) computed from our previous
expression as the source term of an isotropic spherical surface emitter.

The power density for values of réRi due to an isotropic spherical
surface emitter would then be given by:

Py = ‘;%)Rifla g.": Ell/,{(Rl—r)l - El} M(Rl+r)I} C-T

2

At the center of the sphere where r=o0 this becomes:

P
- M. R:
P, = [2 Ma e Ta fi c-8

s

andk

 
 

A‘%

The power density for values of r > R, due to an i1sotropic spherical
surface emitter would then be given by:

PB.
Py = ‘/‘7521230#& fi_% { Ell,u(r-Ro)l - El’ M(r+Ro)’} C-9

For values of r just outside the boundaries of the source region but
close to R; or Ry, the power density in the case of U ,#Ag cen be obtained
by treating the source as a slab and introducing a radius ratio correction.
Thus for r = Ry but near Rg,:

 

2
Py = (13%) PZ:Z: {C/Q(RO—R:L) + Mg (r - Rofl EljflS(Ro-Rl) + Hg(r-Ro)|

+ e“/aa(r-Ro) [1 _ e'/'/S(Ro'R:L)J

- U, (r-Ro) Ell /Ia(r-Ro)l} c-10

A similar result can be obtained for r < R; but near R;.

By Judiciously solving the spherical annulus, infinite slab, and
spherical shell absorption equations in series, it 1s possible to obtain
a good approximation of the true” absorption in a compdsite solid system
such as the reflector-moderated reactor. The flux from the surface of the
spherical annulus source 1s used as the boundary condition in solving the
slab source case for the attenuation through the thin Inconel shell on
either side of the source annulus. The flux leaving the face of the
Inconel 1s corrected for angular distraibution and i1s used as the boundary
condition in the spherical surface source solution.

 
 

APPENDIX D

Gamma Heating from Parasitic Capture of Neutrons

Gamms rays would be emitted as a result of neutron capture in the
beryllium reflector, beryllium island, Inconel shells and sodium coolant.
The source term can be evaluated approximately by using the spatial flux
plot (1n ORNL-1515,l page 54) for Reactor Calculation Number 129,

The reactor power was taken as 200 megawatts and a fuel space of
43,700 cm3 was computed for an annular fuel space bounded by spheres of
22.9 and 11.4 cm radius. Thus the thermal neutron flux at the outer
Inconel shell is approximately 5 neutrons/cm2 sec per f1351on/sec em3
of core volume. Since there are 6.2 x 1018 fissions per second for opera-
tion at 200 megawatts, the thermal flux is given as 7.1l x 101k neutrons/cm?
per second in this region. The resulting source of gamma rays 1s treated
as a slab source in computing the heat generation in the external reflector.
Thus

P, = P08 ) (MgtgrMgty) Ep(MgtgiMgts)
2 Mg
+ o Meata [1 - e stsj -MgtgE] (M ata)} D-1

The absorption was corrected in turn by the radius ratio squared.

The heat generation in the beryllium reflector and island due to
gamma absorption from neutron capture by beryllium was assumed to be
uniform. The magnitude of the heating was assumed to be equal therefore
to the source term.

 

1 "ANP Quarterly Progress Report for Period Ending March 10, 1953."
ORNL-1515

 
 

APPENDIX E

A Possible Cooling System Based on Sodium
Cooling of a Beryllium Reflector-Moderator

In order to demonstrate the feasibility of cooling a reflector-
moderator of solid beryllium, a study based on cooling with a liquid
sodium coolant flowing through a number of holes was made. The heat
removed from the moderator was to be transferred to liquid NaK in the
reactor intermediate heat exchanger-air radiator circuit. In order to
avoid compromising the air radiator operation it is desirable to operate
the moderator coolant system at temperatures comparsble to those in the
external NaK systems (~~ 1400°F). Consideration must also be given in
setting a temperature to corrosion and mass transfer in the coolant system
and to the strength of the structural materials. The chosen maximum allow-
able Be-Na interface temperature was approximstely 1200CF. At this
temperature thermal stresses are no longer controlling since the beryllium
w1lll be stress relieved by plastic flow,

The number of cooling holes of & given size required at any section
of the moderator will depend on the power density and allowasble temperature
difference within the beryllium. The hole size 1s dependent upon the allow-
able maximum pressure drop in the coolant system, the alloweble neutron
poison fraction within the moderator, and the heat transfer characterisites
of the system. Coolant flow rates are determined by the allowable pressure
drops, coolant temperature rise, and system heat transfer characteristics.
Sodium was chosen as a coolant because 1t has excellent heat transfer
properties combined with reasonably low neutron capture cross-section and
can be contained without excessive corrosion in acceptable materials of
construction.

In view of the present lack of information on mass transfer and
corrosion of beryllium by sodium at high temperatures, the present system
was designed based on canning the beryllium with Inconel. The cooling
holes were to be lined with Inconel tubes which could be welded to the
moderator can. To insure good thermal contact between the beryllium and
the can, stagnant sodium would be introduced into the can along with the
beryllium.

The reflector-moderator can be constructed in two canned hemispheres,
accurately aligned by tapered positioning pins. The coolant tube holes
would be rifle-drilled through each hemisphere in a conical pattern con-
forming with the shape of the fuel passage.

i{“n
ey
hY g
.
»
 

-
. -
-
~

Equations describing the power density dastribution in the Be of
the i1sland and reflector were fitted within 10% to the curves in Fig. 2.
These were found to be (see Nomenclature, p. 38)

- -7.06
Pp (r) = 180 (2) watts/ce E-1

and

P; (r) = 100 e0'907(r/r1) watts/ce E-2

The total power generated within the reflector beryllium was determined
to be 6.46 megawatts by

Piot, = f Pp (r) av

\Y

The 1sland 1s not physically spherical, but for purposes of analysis 1t

is closely approximated by a Be sphere symmetrically located on the axis
of a Be cylinder. In order to determine the power generated within the
cooled 1sland structure 1t was necessary to determine the total power
generated within the cylindrical "end caps" as well as 1n the central
sphere. For the cylindrical end caps 1t was assumed that the average
power density was one-half of that in the central sphere. Thus the power
could be calculated from Eq. E-2 for any given value of (r/r ), where

r1 for the central sphere was taken as 4.2 inches and for the cylindrical
end caps as 2.7 1nches. The over-all cooled cylindrical length was assumed
to be 12 inches. The total i1sland power generation was then determined to
be 1.42 megawatts by:

1
Piot. =fi1(r) dVsphere E[PI(I') WVoyl.

The total power to be removed from the beryllium is thus 7.88 megawatts
or 3.9% of the total reactor power (200 megawatts).

The peak heating occurs in the Inconel shells containing the fuel
in the core region. The total power to be removed was determined from
the average power density within the Inconel and the shell thicknesses
and sizes. It was found to be 0.21 megawatts for the island outer shell
and O0.73 megawatts for the reflector inner shell. The total power to be
removed from the island and reflector, including the Inconel core shells
thus becomes 8.82 megawatts or 4.4% of the total reactor power.

a
A

 
The power generated in the reflector beryllium was broken up into
eight spherical shells of equal total power in order to facilitate the
numerical calculations for the coolant system design. Similarly, the
total power in the island exclusive of that absorbed by the annular
sodium layer was divided into three equal power spherical (i1sland sphere)
and 2 equal power cylindrical (i1sland end caps) shells. In order to
remove the heat generated in the Inconel shell and that transferred from
the fuel region, sodium mist flow in an annulus between the Inconel shell
surrounding the fuel region and the Inconel can containing the beryllium
reflector.

The number, length and diameter of the passages for cooling the
system can be defined by the following relationships and specifications.

(a) Tube surface area heat transfer requirements-

U
3 = 3500 Atot L6
where A 8y pax, & 100CF E-3

(b) TFlow heat capacity:

Wiot. Cp ATyg = 9

b

where llTNa mnex. 200°F E-4

(c) Flow pressure drop:

L fv2

= )-I- = T m—

AP f D 2g,

where AP, = 2880 psf. (20 psi) E-5

(d) Temperature difference within the berylllum:8

2 2 2
ATg, = 6.68 x 103 E:—e L—ln (]—}%—e) + 0.908 (ESBE - o.9ozj E-6

where A Tge max = 50°F

8, W. S. Farmer, "Cooling Hole Distribution for Some Reactor Reflectors,"
CF 52-9-201. e ol

X

 
(e) Auxaliary relations:

Weot, = N DPPv E-T

 

f = 0.046/Re® 2 where Re = ?::P

Atot . = NnDL

Material properties were taken at 1100°F for sodium snd at 1200°F
for beryllium and Inconel.

e (1b/1t3) pa(1b/sec £t)  Cp(Btu/1b OF) k[Btu/hr ££2(OF /1))

 

Sodium 50.4 1.4 x ]_O'lL 0.30 37
Inconel -- - -- 12
Beryllium -—- -- -- 50

In determining a tube diameter, consideration was given to the poison
volume fraction. _The poison volume fraction is defined as the ratio of the
total tube hole cross-sectional area, in any given moderator shell, to-the
total moderator cross-sectional ares of the same shell. The poison fraction
1s thus proportional to ND2. For a fixed tube length L and fixed beryllium
to sodium coolant temperature drop, A 6y, N will be inversely proportional
to D. Hence the poison fraction is directly proportional to tube diameter.

The total thermal resistance for heat transfer from the beryllium to
the sodium 1s the sum of the thermasl resistances across the stagnant sodium
layer within the Inconel tube, the tube wall, and the sodium coolant film.
Due to the low conductivity of the Inconel, the tube wall is the controlling
resistance and variations in the sodium film resistance owing to varying
velocities have only a small effect on the over-all heat transfer coefficient
U. The value of U used throughout the calculations for heat transfer to
coolant flowing in the cooling holes was 8,000 Btu/hr £t OF. This value
18 within 10% of .the values obtained by evaluation of the heat transfer
coefficients for the various different coolant velocities existing in dif-
ferent tubes. The heat transfer coefficients for heat transfer from the
Inconel shells to coolant flowing in the annulii around the fuel region
depend almost entirely on the coolant velocity, hence i1t was necessary to
evalugte U for the actual coolant velocities expected in the annulii.

The number of holes and the mass flow rate per hole was adjusted to
satisfy not only the thermal specifications but also the pressure drop

 

D
 

requirements. Repetitive trial and error calculations were performed in
order to determine a satisfactory, specific coolant system design. The
results of these calculations show that a satisfactory balance between
poison fraction, pressure drop, heat transfer requirements, and beryl-
lium temperature drop can be achieved with the use of 0.250 in. 0.D.
tubes with 0.010 in. thick walls, separated from the beryllium by a 0.005
in. annular gap filled with stagnant sodium. The Inconel modersator can
was taken to be 0.025 in. thick with a 0.025 1in. thick sodium-filled gap
between the can and the beryllium. The results of these calculations are
summarized in Table III. The resultant moderator poison fractions are
shown in Fig. k.

 

Aeaps
 

 

 

 

TABLE III
Reflector Moderator
q per
Region r,/rg region L N S ATge
1 9.3/9.75 765 1.4 69 0.67 50°F
2 9.75/10.3 765 1.6 69 0.76 50
3 10.3/10.85 765 1.8 65 0.81 50
L 10.85/11.5 765 2.0 65 0.90 50
5 11.5/12.4 765 2.2 76 1.01 50
6 12.4/13.9 765 2.6 76 1.35 50
7 13 9/16.2 765 3.1 69 1.93 32
8 16.2/21.3 765 3.5 65 3.27 LW1°F
Sodium Annulus Around Spherical Core
Thickness q A ey Wya,
0.070/0.100 930 280F 22.6
Totals for reflector system q = 7050 Btu/sec N
Over-all AP = 810 psf ATy,
Island Moderator
q per
Region r,/rp region L N S ATpe
1 2.8/4.2 450 (ave)0.6 88 0.58 500F
2 2.0/2.8 450 1.7 38 0.61 50
3 0.5/2.0 450 1.7 35 0.62 36CF
Sodium Annulus Around Island
Region Thickness q per region A8y Wy
End caps 0.300" 200 520F 40.
Central
Sphere 0.300/0.185 170 106°F 30.
Totals for island system q = 1720 Btu/sec N
Over-all AP = 1685 psf ATy,

Aby

56°F
51
L6
Lo
28
28
25
24 Op

295

554
150°F

A6y,

a

6
6

‘ ‘_

99°F
920F

A

161

150CF

WNa A PTube A TNa
17.0 295 150°F
17.0 322 150
17.0 423 150
17.0 463 150
20.1 532 150
19.5 570 150
17.0 650 150
17.0 810 1500F
A TNa.
137OF

Wy, = 164.2 1b/sec
WNg APpppe ATy,
10.0 30 150°F
21.2 1485 71
19.4 1485 96°F
P ATy
170 16%F

30 1.9°F

Wy, = 40.6 1b/sec
st

DWG 22520

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 24
S 020 AL
5 ZE
g o
i z
Ll
= 016 v 2\; -
L
I
> -
- - wl !
w wl a -
z 012 % 3 T
O [ 3
= o '; i
O O 1?@12
= © L3
< 008
= »
2
Q \ 1os
@
@ \
0 04 \
0
0 2 4 6 8 10 12 14 16 18 20

Fig 4. Volume Fraction of Sodium and Inconel at the Reactor Mid-plane as a Function of Radius. The Volume

RADIAL DISTANCE FROM REACTOR CENTERLI

NE (in)

of Inconel 1n any Region is 4149, of the "Sodium plus Inconel” Volume.
Symbol

Atot.

 

NOMENCLATURE FOR APPENDIX E

 

Meaning
Heat transfer surface area inside tubes
Coolant specific heat
Coolant tube 1inside diameter
Coolant hole diameter through the moderator
Flow fraiction factor
Thermal conductivity
Coolant tube length
Number of coolant holes
Power density at any point in the moderator
Power density in the reflector moderator
Power density in the island moderator
Total power
Coolant flow pressure drop
Heat transfer rate

Radial distance from reactor centerline
to any point in the moderator

Outer radius of i1sland moderator

Inner radius of reflector moderator

Inner radius of an arbitrary moderator shell
Outer radius of an arbitrary moderator shell

Reynolds number

 

mliienny

Units

Btu/1b OF
ft

ft

Btu/hr £t° OF/ft

ft

Watts/cm3
Watts/cm3
Watts/cm3
Megawatts
1b/fte

Btu/sec

in.
in.
in.
in.

in.
Symbol

ATNa
iésTBe

 

Meaning

Coolant hole center to center spacing
holes in triangular array

Temperature rise in the coolant fluid

Temperature difference within the moderator
material

Over-all heat transfer coefficient
Coolant velocity within the cooling passages
Coolant weaight flow rate

Mean temperature drop from internally
heated material to coolant stream

Coolant weight density

Coolant viscosity

 

Units

ft

OF

OF

Btu/hr f£t2 OF

ft/sec

lb/sec

OF

1b/ft3

lb/sec ft
«40-
*fi

APPENDIX F

The primary effect on the moderator coolant system of the existence
of fission products in the heat exchanger region will be heating of the
outer regions of the Be reflector by decay gammas.

For a 200 MW reactor the heat exchanger thickness will be approx-
imetely 6 inches. The volume of the heat exchanger shell region, for a
reactor core diameter of 18 in., is thus sbout 25 cubic feet. Of this,
only some T0% is occupied by the heat exchanger, the rest being void
space or pump and flow channel locations. Since only 40% of the heat
exchanger volume 1s occupied by fuel the volume of the fuel is approx-
imately (25) (0.7) (0.4) = 7 cubic feet. The fuel volume within the
18 1n. core 1s asbout 1 1/2 cubic feet, thus the amount of uranium in the
heat exchanger region will be about 0.82 of the total system investment,
while that in the core will be 0.18 of the total.

It can be shown that heating due to absorption of garmmas from a
plane source of finite thickness (bj) and bulk power demsity (Sp) will
be given by

 

T, - zzjo [ Bo(Mrx) - Bo(Meby +prx)]

s

T . = attenuation coefficient for heating in the heated
region (cm-1)

A = attenuation coefficient for total interaction in the
source region (cm-1)

by = thickness of source (cm)

x = dastance from source into heated region

S, = source power density (watts/cc)

P, = power density produced at point x in region watts/cm3

and oo

e's¢
Eo(f) = J//’ o ds

 

 

iy

 
41—

iy

By use of the foregoing equation the reflector heeting due to absorp-
tion of decay gammas from the heat exchanger region of a 200 MW reactor
was computed and 1s shown in Fig. 5. This computation was based upon an
assumed average decay gamms energy of 1.5 Mev and assumed attenuation
coefficients of 0.16 em-1 and 0.30 cm-1 for beryllium and Inconel, respec-
tively, as used previously in the body of this report. The attenuation
coefficient in the source (heat exchanger) region was teken as 0.13 cm*l,
and the coefficients for heating and total interaction were assumed to be
the same. It is readily sapparent that this heating is small compared to
that to be expected from the core gammas and from moderation of fast
neutrons. The local power density in the outer four inches of the beryl-
lium reflector 1s shown to be approximately 2.5 watts/cc, as compared with
1.0 watts/cc shown 1n Fig. I. This increase necessitates a reduction 1in
the cooling tube spacing in this region from 3.27 in. on centers as 1in
the design presented in Appendix E to 2.42 1in. on centers for the higher
power density, if the allowable AT within the beryllium 1s to be held to
50°F or less. This corresponds to an increase of the number of tubes in
region 8 of the design in Appendix E from 65 to 120 tubes. The total
power to be removed from this region i1s higher by a factor of about 2.5,
thus the temperature drop from the beryllium to the coolant sodium will
become 33° and, for a coolant temperature rise of 150°F, as previously,
the over-all pressure drop will be 1430 psf.

oF

It should be obvious from the foregoing that the effect of decay
gamma heating of the reflector on reflector-moderator cooling system
design wi1ll be minor and in any event will be well within the uncer-
tainties of the neutron and gamma heating and cooling system design
calculations.

 
  

DWG 22672

 

{00

 

—

 

 

@
o

 

-

HEATING FROM CORE ONLY

 

o

o
—
N

 

5
O
/

 

POWER DENSITY (watts/cm3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
N
20 <
N HEATING FROM CORE PLUS HEAT EXCHANGER\
~|— )
e — e
O i e Eseash e
10 12 14 16 18 20

DISTANCE FROM REACTOR CENTERLINE (in)

Fig 5 Effect of Heat Exchanger Decay Gammas on Power Generation in the Reflector

— 42—

-~
o