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ORNL-2353.txt
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M [EMS LIBRA
[TTATNT
3 445k 03kL181 1
CENTRAL RESEARCH LIBRARY
DOCUMENT COLLECTION
LIBRARY LOAN COPY
DO NOT TRANSFER TO ANOTHER PERSON
If you wish somecne else to see this
document, send in name with document
and the librory will arrange a loan.
ORNL-2353
C-84 — Reactors-Special Features of Aircraft Reactors
This document consists of 22 pages.
Ct:bpy/a?}Z of 273 copies. Series A.
Contract No, W-7405-eng-26
SHIELD PLUG ASSEMBLY FOR THE ART FUEL PUMPS
J. P. Page
J. H. Coobs
DATE ISSUED
APR 2 - 1958
OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee
operated by
UNION CARBIDE CORPORATION
for the
U.S. ATOMIC ENERGY COMMISSION
Wi TED
3 4456 D3L11LABL L
CONTENTS
ADSIEACT «nteeeetieeet et e st evee et eeneeteent sesesbebesbasaessarsabeatessans e st s aan e R e R e R e R Re et e b s S as b e R e e s R e e e easeaesr e a s R e R aasrasaeses 1
[EPOAUCEION 11veeveeeeeeeeeeeeee ettt etesesbbese s b assasseases saeesane s s easeseee s eabe et e st e e mtenbee Shbe e b eabbssbea bt bbb a bbb o bt s s b b e besab e st ansnis 1
Shield Plug Design and Materials Specifications ..........cooeviimeimiiiinii 1
Development and Evaluation of Materials and Fabrication of Parts......coccoiiiinn, 4
SUMMGIY <.ooovetieirieeeere e et es e b b e s e e s s b e b e b b e b ee s e e E e 04RS00 8 S 4 e bR e s e e e e s b b e bbb s b e b e bbb ke ke b et s e n b 9
ACKNOWIEAGMENTS ...ueiieiiccee e e bbb e e b bbbt e 10
Appendix A. Density and Conductivity Caleulations ........cccoevriinniniiciiireinicce e 11
Appendix B. Thermal Conductivity Determination Apparatus.........ccccovvviiicnnnninccninncennnneeesersenen. 14
thugr
1Ry
SHIELD PLUG ASSEMBLY FOR THE ART FUEL PUMPS
J. P. Page'!
J. H. Coobs
ABSTRACT
The shield plug assembly must protect the Aircraft Reactor Test fuel pump motors from gamma
and neutron radiation. It must also exhibit limited heat transfer properties. After an evaluation
was made of their metallurgical and physical properties, the materials tungsten carbide ~Hastel-
loy C (cermet), low-density stabilized zirconia, and stainless-steel-clad copper—boron carbide
were selected for use as gamma, thermal, and neutron shielding components, respectively. A full-
size assembly was fabricoted.
INTRODUCTION
One of several unique components of the Aircraft
Reactor Test (ART) is the shield plug assembly,
Two assemblies will be used in the reactor, one
for each fuel pump. The location of one of the
assemblies is shown in Fig. 1.
The shield plug has three functions:
the fuel pump motors from gamma radiation,
attenuates stray neutrons, and limits the heat
transfer. At full reactor power the heat developed
by gamma-ray and neutron absorption must be
dissipated, but at zero-power operation the lower
face of the shield plug must not cool below the
freezing point of the liquid fuel. Such freezing
would cause fouling of the impellers.
Only the materials-development aspects of the
shield plug assembly are discussed. Heat transfer
calculations and design detailing were performed
by members of the Experimental Engineering Design
Group. The gamma- and neutron-shielding specifi-
cations were established by the Power Plant
Engineering Section.
protects
SHIELD PLUG DESIGN AND MATERIALS
SPECIFICATIONS
An exploded view of the shield plug assembly is
presented in Fig. 2. This design was arrived at
after close and continued cooperation between
representatives of the Metallurgy Division and
members of the Experimental Engineering Design
Group.
'Present address: Wright Air Development Center,
Dayton, Ohie.
The fabricated assembly consists of a gamma
shield, a thermal shield, a neutron shield, and
four Inconel-clad Chromel-Alumei thermocouples,
all canned in an Inconel container, The upper
surface of the container is a heat exchanger through
which a noncracking oil is circulated. The gamma
shield is brazed directly to the face of the heat
exchanger,
The specifications for the gamma-shield material
are as follows:
1. density of at least 12.0 g/cm?,
2, thermal conductivity not greater than 0.10
cal/em+sec-°C,
3. structural integrity to at least 1500°F,
4. reasonable thermal-shock resistance,
5. brazeability to Inconel.
The thermal and neutron shields are below the
gamma shield and rest on the lower face of the
Inconel can. The thermal-shield specifications
called for a refractory material having a thermal
conductivity as low as possible and having ade-
quate strength for machining and handling. The
neutron shield had to have a minimum B'® content
of 0.01 g per square centimeter of the exposed
face.
Thermocouples, located in grooves cut in the
upper and lower faces of the thermal shield,
measure the temperature drop through this piece.
In order for there to be no gamma leakage, the
thermocouples run from the thermal shield through
helical grooves cut in the gamma shield to the
exit hole, which is near the top of the can. The
lower face of the shield plug assembly will be
exposed to temperatures of 1400 to 1500°F, while
IS,
PHOTO 27375
SHIELD PLUG
Fig. 1. Cross Section of the Aircraft Reactor Test, Showing Shield Plug Assembly.
COOLANT OUT
GAMMA SHIELD
p=12.09/cm3
k = 0.10 cal/cm-sec-°C
THERMAL SHIELD —
k, AS LOW AS POSSIBLE
(.
NEUTRON SHIELD | ”."HHY
I
B!'C CONTENT
> 0,01 g/cm?
LOWER CAN
UNCLASSIFIED
ORNL-LR-DWG 15073
T COOLANT IN
PPER CAN
BRAZED TO UPPER CAN
] THERMOCOUPLES
PR
%f.“-”’-fir.rfifiiiflnflu’
)fifiw S { 0 1 2 3 4
SCALE IN INCHES
Fig. 2. Exploded View of the Shield Plug Assembly. (Confidential with caption)
the temperature of the heat exchanger will never
exceed 250°F,
The design is especially attractive in at least
three respects: as much as possible, the functions
of the shield plug have been separated, permitting
the imposition of a minimum number of require-
ments per component; the weight of the dense
gamma shield is entirely supported at the coolest
part of the assembly; and the use of a canned
assembly eliminates the consideration of oxidation
and corrosion resistance, although high-temperature
compatibilities of the various materials of interest
were kept in mind throughout the development
program,
DEVELOPMENT AND EVALUATION OF
MATERIALS AND FABRICATION OF PARTS
Development of materials for the three com-
ponents, which was performed semiconcurrently,
is described below,
Gamma Shield
A review of the literature and correspondence
with several industrial organizations indicated
that no existing material had the unusual properties
required of the gamma-shield material. Since an
optimization of two nonrelated physical properties
{density and thermal conductivity) was desired,
rather than conventional
alloy-development techniques were employed.
In order to limit the scope of the investigation,
only two-component systems were considered; one
of these components was to be a high-density
material of fairly low conductivity which would be
dispersed in the second component, a very low-
conductivity matrix. The gamma shield, being a
hollow, right circular cylinder approximately 4 in.
powder-metallurgical
high by 6 in. in outside diameter by 2‘/2 in, in
inside diameter, could probably be fabricated in
one piece by conventional hot-pressing techniques,
From density, availability, cost, and thermal
conductivity considerations the materials tungsten,
tungsten carbide, and tantalum were tentatively
chosen as possible high-density components;
tungsten, because of its relatively high thermal
conductivity, was considered to be the least
desirable of the three materials. Selected properties
of these materials are presented in Table 1.
In the search for a low-conductivity matrix
material, it became apparent immediately that some
member of the nickel-base family of alloys might
be suitable. Nickel-base alioys, in general, have
solidus temperatures in a range favorable for hot-
pressing, yet retain their strengths to high tem-
peratures, are fairly dense (approximately 9 g/cm3),
have the lowest thermal conductivities of all the
commercial alloys, and are very likely to be
amenable to brazing. These alloys, however,
react extensively with both tungsten and tantalum
at elevated temperatures. Also, nickel reacts
with graphite (which, as will be described, is the
material used for hot-pressing dies) at 1322°C to
form a eutectic composition,
Hastelloy B, with a density of 9.24 g/cm® and
a thermal conductivity of 0.027 cal/cmesec:“C,
appeared to be the most promising matrix material.
Unfortunately, this alloy is not available in powder
form, Hastelloy C has the slightly lower density
of 8.94 g/cm? and the slightly higher thermai con-
ductivity of 0.030 cal/cm-sec:°C but is available
in powder form at the reasonable cost of approxi-
mately $5 per pound.
The alloy constantan (45% Ni-55% Cu), with a re-
ported thermal conductivity of 0.055 cal/cm+sec:°C,
Table 1. Selected Properties of Tungsten, Tungsten Carbide, and Tantalum
Material Densi;y Thermal Conductivity Approximate Cost
{g/em”) (cal/cmesecs°C) ($/1b)
Tungsten 19.2 0.394 12
Tungsten carbide 15.6 0.17* 5
Tantalum 16.6 0.13 56
*Estimated by the Wiedemann-Franz relationship.
The electric conductivity of tungsten carbide is reported
(P.Schwarzkopf and R, Kieffer, Refractory Hard Metals, p 161, Macmillan, New York, 1953) to be 40% of that of pure
tungsten. For lack of better data, the thermal conductivity was assumed ta be 40% of that of pure tungsten.
also appeared to be attractive. It has been shown?
that copper and nickel powders alloy readily at
elevated temperature by solid-state diffusion,
Copper and nickel powders were on hand for test-
ing, and it was felt that a relatively homogeneous
alloy could be produced during the hot-pressing
operation,
The choice of matrix, then, was between Hastel-
loy C and constantan. Selected properties of
these metals are presented in Table 2,
Table 2, Selected Properties of Constantan and
Hastelloy C
Density Thermal Approximate
Material 3 Conductivity Cost
(g/cm) o
(cal/cmesecs®C) ($/1b)
Hastelloy C 8.94 0.03 5
Constantan 8.9 0.055 1
When the number of materials to be considered
was reduced to a logical minimum, calculations
were performed to determine, semiquantitatively,
the physical properties that might be expected of
various combinations of these materials. These
calculations are described in Appendix A, and
although they suggested the limiting compositions,
the introduction of another variable, porosity,
prohibited a direct calculation from being made of
the optimum composition., This variable was
considered significant for two reasons: (1) con-
sistent hot-pressing to full theoretical density,
especially in a piece as large as the gamma shield,
is extremely difficult; (2) while voids would lower
the bulk density of a hot-pressed piece, they
would also lower the thermal conductivity; the
relative effects were unknown and could not be
predicted.
Since the thermal conductivities of the materials
could not be rigorously predicted, a fairly simple
apparatus for the determination of thermal con-
ductivity was designed and built. This apparatus
is described in Appendix B. During the fabrica-
tion and calibration of the apparatus several smalt
(]/2-in.-dic:) composites were hot-pressed. The
hot-pressing operation consisted in heat and
pressure being applied simultaneously to a weighed
and blended mixture of ceramic and/or metallic
2F’. Duwez and C. B. Jordan, Trans. Am. Soc. Metals
41, 194 (1949).
powders, Graphite dies were used because tem-
peratures greater than 800°C would be encountered.
The use of an inert layer between the die wall
and the material being compacted may be necessi-
tated by a reaction between the charge and graphite;
where there is a significant reaction, the use of
a molybdenum-foil liner or coating of aluminum
oxide powder will usually prove remedial. Typical
hot-pressing setups and techniques employed
have been described by Coobs and Bomar, 3
The results of the preliminary experiments are
presented in Table 3. The data showed that the
tantalum—Hastelloy C combination could not be
hot-pressed to a high density; compaction was
prohibited by the formation of a brittle inter-
metallic compound, A similar reaction product
was noted in the microstructure of the tantalum-
constantan specimen, but the copper evidently
retarded the reaction rate to the point that it did
not interfere with compaction,
Economic factors and the favorable results ob-
tained with tungsten carbide—constantan and
tungsten carbide—Hastelloy C indicated that these
combinations warranted further consideration. The
tungsten carbide—constantan composition was
most easily fabricated and was therefore investi-
gated first. In this system the required density
of 12.0 g/ecm3 could be obtained over a composi-
tion range extending from 60 wt % tungsten carbide
to 100 wt % tungsten carbide simply by varying
the porosity of the hot-pressed compact. The sig-
nificance of porosity has been mentioned.
In order for variations in porosity to be taken
into account, and even for the porosity to be used
toe advantage if possible, six tungsten carbide-
constantan charges were pressed into I-in,-dia by
2]/2-in.-|ong slugs for thermal conductivity de-
termination. Figure 3 presents the conductivities
of the specimens as a function of the volume per-
centages of the three phases, tungsten carbide,
constantan, and porosity, present in each compact;
the closed circles indicate the compositions of
the compacts after they were hot-pressed. The
conductivities were found to be essentially linear
with temperature to 500°C, the upper limit of the
apparatus. These data proved conclusively that
no combination of tungsten carbide, constantan,
and porosity would simultaneously fulfill the
3J. H. Coobs and E. 5. Bomar, Methods of Fabrication
of Control and Safety Element Components for the
Aircraft and Homogeneous Reactor Experiments, ORNL-
1463 (March 15, 1953),
Table 3. Results of Preliminary Hot«Pressing Experiments
. 3 Maxi
COmPOSifiOI’I Dens”.y (g/cm ) axtmum R l'(
- - Temperature emarks
(wt %) Theoretical Attained o
(- C)
60 tantalum— 12.3 12.1 1275 Compacted readily; definite Ta-Ni
40 constantan reaction; no Ni-graphite reaction
60 tantglum— 12.4 8.2 1300 Would not compact; definite Ta-Ni
40 Hastelloy C reaction; slight Ni-graphite re-
action
60 tungsten carbide — 12.0 11.9 1275 Compacted readily; no WC-Ni
40 constantan reaction; no Ni-graphite reaction
60 tungsten carbide — 12,0 11.8 1300 Compacted with some difficulty; no
40 Hastelloy C
WC-Ni reaction; slight Ni-graphite
reaction
UNCLASSIFIED
ORNL-LR-DWG 15034
VALUES SHOWN ARE
THERMAL CONDUC;TIVITY
{cal/em-sec- C)
o,
~Itns .
TR
IR
TUNGSTEN 10 200 30 40 50 60 70
RBI
CARBIDE CONSTANTAN (vol %)
Fig. 3. Thermal Conductivity of Tungsten Carbide-~
Constantan,
thermal conductivity and density specifications
(£ < 0.10 cal/cmesec:°C, p > 12.0 g/cm?®) set by
the Experimental Engineering Design Group.
Investigation of the tungsten carbide—constantan
system was therefore discontinued.
The tungsten carbide—Hastelloy C system was
then investigated in a similar manner. Four
specimens were hot-pressed and their thermal con-
ductivities determined. The results are presented
in Fig. 4. It is evident that the conductivity is
almost constant for compositions with a density
of 12.0 g/cm3, probably because of the opposing
effects of tungsten carbide and porosity. With
the density held constant, an increase in the
volume percentage of tungsten carbide produces
an increase in the volume percentage of pore
space. The tendency of tungsten carbide to raise
the conductivity of the compact is therefore
opposed by the insulating effect of pore space.
UNCLASSIFIED
ORNL-LR-DWG 15030
VALUES SHOWN ARE
THERMAL CONDUCTIVITY
¥ }*fi, . {cal/em-sec- C)
= -1y
. Tt
LN f’ h ;
; o S
IS0 N SN .
> {%fi/ ??‘\'0.0T, S <
TUNGSTEN 10 20 30 40 50 60 70
CARBIDE
HASTELLOY C (vol %)
Fig. 4. Thermal Conductivity of Tungsten Carbide~
Hastelloy C.
The composition that was tentatively chosen for
use in the gamma shield was 75 wt % tungsten
carbide—25 wt % Hastelloy C. The microstructure
of this composition, at the specified density of
12.0 g/cm?, is shown in Fig. 5. The feasibility of
hot-pressing a full-size gamma shield of this
composition and density was confirmed by the
successful pressing of two models of the cylinders,
Because existing dies were utilized, the models
were not true miniatures of the gamma shield but
had charges calculated to give the same ratio of
cross-sectional area to die wall area as would
exist in the gamma shield. This ratio is a domi-
nant consideration in hot-pressing practice. The
two models were 1.20 and 2.36 in. in inside
diameter, 2.23 and 3.86 in. in outside diameter,
1.25 and 174 in. high, and were pressed to
densities of 11.9 and 12.4 g/cm?, respectively.
- o
ey
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-2y
ot s
sAad e
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Ve
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Fig. 5. Microstructure of Tungsten Carbide —Hastelloy C. 500X,
These models and smaller trial pieces were
given to the Welding and Brazing Group for brozing
evaluation. The tungsten carbide—Hastelloy C
material was successfully copper-brazed to Inconel
in a dry-hydrogen atmosphere; the copper readily
wet both materials.
The models were brazed to ]fiz-in.-thick Inconel
plate and subjected to several thermal shocks.
They were heated in an argon-filled muffle furnace
at 1500°F, then air-cooled to room temperature,
No cracking or thermal-shock sensitivity was
noted. Figure 6 shows a small trial piece, a
typical thermal conductivity specimen, and the
two models after they were brazed. The only
unattractive property exhibited by this material
was its extremely poor machinability. All finishing
operations had to be performed by grinding, pref-
erably with a diamond wheel.
The fabrication of two full-size gamma shields
was accomplished ofter several problems had been
solved in upscaling from the 3.86-in.-dia model
to the 6-in,-dia gomma shield. These problems
arose from the necessity of changing die materials
and as a result of the difficulty of setting up the
large dies in a confined space. The grade C-18
graphite which had been used as die material for
the trial pieces was not available in the large
(15-in,) diameter required; therefore the substan-
tially weaker grade CS-312 was used for the large
dies. As a result, only two runs were made be-
cause the dies cracked open. A longer pressing
time at a lower pressure (1500 psi as against
2500 psi) and elimination of the stress-raising
sight hole used for optical pyrometer temperature
measurement prevented similar difficulty in sub-
sequent runs made with a new die. The second
run was terminated just before the end of the
pressing cycle, and even though the outside di-
ameter of the piece had swelled after the die
broke, the inside diameter remained true, After
this was accurately
as was the diameter of the mandrel
cooling, inside diameter
measured,
UNCLASSIFIED
¥-19835
Fig. 6. Hot-PressingDevelopment Series. (a) Trial piece, (b) typical thermal conductivity specimen, (c) 2.23-in.-0D
model, (d) 3.86-in.-0D model.
which had formed it. A relative expansion co-
efficient was then calculated which was utilized
in the design of the new die.
The new die produced two gamma shields which
required little or no grinding of the cylindrical
surfaces. The gamma shields were merely faced
by grinding and the helical grooves cut with a
Efi-in.-wide diamond wheel, The grooves were cut
by means of an ingenious grinding technique in
which a slab-mill cutter was used as a guide.
The gamma shields were then copper-brazed to
the heat exchanger faces of two Inconel cans.
Figure 7 shows a brazed assembly before the intro-
duction of the thermocouples and final sealing of
the can.
Thermal Shield
A survey of the literature showed zirconium
dioxide (zirconia) to have the lowest thermal con-
ductivity of the more common ceramic materials,
The thermal conductivity of zirconia is reported*
4E. H. Norton et al., The Measurement of Thermal
Conductivity of Refractory Materials, NYO-3643 (March
15, 1953).
UNCLASSIFIED
¥-22865
NEUTRON SHIELD
THERMAL SHIELD
GAMMA SHIELD
Fig. 7. Fabricated Shield Plug Assembly.
to be 0.004 cal/cm-sec:°C, The effect of density
on thermal conductivity was not reported, nor was
the effect of stabilization (addition of a small
amount of calcium oxide for the stabilization of
the cubic phase).
Arrangements were made with Battelle Memorial
Institute for them to determine the thermal con-
ductivities of zirconia specimens supplied by the
Oak Ridge National Laboratory. Three specimens
of stabilized zirconia, one each of density 3.08,
3,52, and 4.41 g/cm®, were cold-pressed and
sintered by the Ceramics Group of the Metallurgy
Division at ORNL. These were machined to BMI
specifications and sent there for thermal conduc-
tivity determination. The thermal conductivity of
these specimens is shown as a function of tem-
perature and density in Fig. 8,
After the thermal conductivities were determined
by BMI, the Ceramics Group fabricated the thermal
shields for the shield plug assembly to a density
of 3.25 g/cm>; one of the shields is shown in
Fig. 7.
UNCLASSIFIED
DRNL -LR-DWG 18704
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0.0038
0.0034
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20
L
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0.0030
o{} o
o
3 ol
£ &
< /
£ 0002 /
> -
=
- /
=
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Z 0.0022 ,/
o
=
<{
=
o
I
" 0.0018 / /
0.0044 /
0.0040
3.00 3.50 400 4.50 5.00
DENSITY (g/cm3)
Fig. 8. Thermal Conductivity of Zirconia as a Func-
tion of Temperature and Density.
Neutron Shield
A stainless-steel-clad copper—boron carbide
neutron shielding material has been developed at
ORNL for the ART.®> This material is directly
applicable for use as the neutron shield in the
shield plug assembly,
This neutron shield is a laminated sheet con-
sisting of a copper—boron carbide core clad with
copper foil and stainless steel, in that order.
The use of a copper matrix in the core lends a
certain amount of ductility to the sheet. This
core is enclosed in a copper diffusion barrier
which separates the boron carbide particles on the
surface of the core from the stainless steel
cladding, The stainless steel, in turn, acts as a
diffusion barrier between the copper and Inconel.
The composite sheet is fabricated by the well-
known technique; the
cold-pressed and sintered core is wrapped with a
copper foil and inserted in a stainless steel
envelope, or billet, The billet is evacuated and
sealed, then hot- and cold-rolled, The neutron
shields for the shield plug assembly were cut from
a 0.106-in.-thick sheet of this type of material, as
manufactured by the Allegheny Ludlum Steel
Corporation, The material has 6.6 wt % normal
boron carbide in a 0.080-in.-thick core., The B0
content of the sheet, at 0,011 g/cm?, adequately
fulfills the B'® specifications (B1% > 0.01 g/cm?)
of the shield plug assembly. A microstructure of
the neutron shield disk shown in Fig. 7 is given
in Fig. 9.
evacuated picture-frame
SUMMARY
The shield plug assembly protects the fuel pump
motors of the ART from gamma and neutron radia-
tion. |t also has limited heat transfer properties,
allowing heat dissipation at full reactor power
operation, yet acting as an insulator during zero-
power operation to prevent solidification of the
liquid fuel. The shield plug assembly consists
essentially of three layers stacked in an Inconel
container, Each layer, or component, has a spe-
cific function; each has specific materials require-
ments,
After evaluation of its metallurgical and physical
properties, a 75 wt % tungsten carbide—25 wt %
Hastelloy C cermet was selected for use as the
SH. Inouye, M. R. D'Amore, and J. H. Coobs, The
Neutron Shield for the ART (to be published).
Fig. 9. Microstructure of Neutron-Shield Material.
gamma-shield material. The gamma-shield com-
ponent was fabricated by hot-pressing a mixture
of tungsten carbide and Hastelloy C powders to a
density of slightly greater than 12,0 g/em>, This
material fulfills the specifications of density
greater than 12.0 g/cm®, thermal conductivity of
less than 0.10 cal/cm:sec:°C, brazeability to
Inconel, moderate thermal-shock resistance, and
structural integrity to at least 1500°F,
The thermal shield requires a refractory material
of very low thermal conductivity., Low-density
stabilized zirconia was selected for use in this
component after its thermal conductivity was de-
termined as a function of density and temperature,
The neutron-shield specifications call for a
material with o B'? content greater than 0,01 g/cm?
10
UNCLASSIFIED
Y-19509
Cols
55430 T
o
=
0.02
Q.03
>
- O
2
100 X,
of the exposed foce. Stainless-steel-clad copper—
boron carbide, with a B'? concentration of 0.011
a/em?, was utilized in this component of the
shield plug assembly.
ACKNOWLEDGMENTS
The authors are indebted to o great number of
Qak Ridge National Laboratory personnel for their
assistance throughout this investigation and
especially to the following for the work indicated:
A. G, Grindell oand W. K. Stair, design; W. R.
Johnson, powder metallurgy; D. H. Stafford, thermal
conductivity; R. L. Homner aond J. A, Griffin,
ceramics; R. L. Newbert, grinding; and G. M.
Slaughter and C. E. Shubert, brazing.
Appendix A
DENSITY AND CONDUCTIVITY CALCULATIONS
The bulk density of a heterogeneous material The densities of two-phase composites of the
may be rigorously calculated by
where
It
materials of interest are shown as a function of
composition in Fig, A. 1.
The thermal conductivity may be approximated
by the analogy of heat flow to electric current.
Limiting cases of parallel flow and series flow
(assuming a purely two-phase solid) are repre-
P = LV
i
bulk density (mass/length?), sented by Eqs. 1 and 2, respectively (see Fig. A.2