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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS-1963
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ACTID
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CONSEQUENCES OF INTERNAL HEAT FLOW IN RADIOISOTOPE HEAT SOURCES*
LEGAL NOTICE
AUG 22 1967
J. C. Posey
Isotopes Development Center
Oak Ridge National Laboratory
Post Office Box X
Oak Ridge, Tennessee 37830
This raport me 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 or implied, with respect to the accu-
racy, completeness, or usefulness of the information coatined in this roport, or that the use
of any information, apparatus, method, or process disclosed in this repor may not Infringe
privately owned righto; or
B. Assumro any liabilities with respect to the use of, or for damages rcouting from the
use of any information, apparatus, method, or process disclosed in this report.
As used in the above, "por son acting on behalf of the Commission" includes any em-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the Commission, or employee of such contractor prepares,
illo seminates, or provides access to, any information purevant to his employment or contract
with the Commission, or his employment with such contractor.
ABSTRACT
from the heat source and the size of some sources
are limited by the maximum a lowable internal
temperature.
Equations relating the maximum internal tempera-
tures of various-shaped fuel forms and other
source parameters are given. By using calculated
results, the use of cerniets is compared with the
use of oxide-metal laminar structures as methods
of decreasing the resistance to heat flow in the
interior of heat sources and achieving higher sur-
face heat fluxes without exceeding maximum allow-
ab.le internal temperatures. The laminar composite
fuel form is moderately superior.
This report gives basic equations for the predic-
tion of these limitations for simple cases. It is
shown that the maximum available surface heat flux
can be increased by the use of cermets or laminar
metal structures when radioisotope compounds of
poor thermal conductivity are used.
The expected vapor pbase migration and equilibrium
distrivution of a volatile radioactive fuel inside
a heat source capsule is described. Redistribu-
tion does not always result in serious problems
and can reduce the cost of source fabrication.
The existence of temperature differences within the
capsule also leads to volatile transfer of radio-
active material. The factors which determine the
equilibrium distribution of the material ere dis-
cussed, and a mathematical analysis is given for
a simple case.
EQUATIONS FOR FUEL FORMS OF SIMPLE GEOMETRIC SHAPE
INTRODUCTION
A radioisotope heat source is generally considered
to consist of a quantity of radioactive fuel and
its container. The radioactive fuel can be in the
form of a single compound of the radioisotope, or
it can be associated with a matrix material of
better thermal conductivity.
Equations relating the difference between the sur-
face temperature and the maximum internal tempera-
ture to various parameters for fuel forms of simple
geometric shapes are given in Table 1. The symbols
used in the equations are defined as follows:
AT = difference between the maximum allowable
internal temperature and the surface
temperature,
P = power density (rate of heat generation
per unit volume) of the fuel,
Large temperature differences can exist between
the suriace and the interior of the radioactive
fuel (defined as the radioactive material with or
without a diluent or matrix material) in heat
sources. These gradients are created by the flow
of heat, which is generated throughout the radio-
active material, to the surface. Very high inter-
nal temperatures can result.
h = distance from surface to plane of maximum
temperature in a planar fuel form,
k = thermal conductivity of the fuel,
q = total rate of heat production by the
source,
There are no specific rules for setting the maxi-
mum allowable internal temperature for a heat,
source. It must be set on the basis of the prop-
erties and use of the particular source. The
melting point of the fuel has often been set as
the maximum allowable temperature. Certainly,
melting can in some cases result in a volume
change which can Qamage the source. In other
cases, it can ai.low an undesirable movement of
A = surface area through which heat leaves
the fuel form,
D = diameter of a cylindrical or spherical
fuel form,
..
..
..
-
TI
J. C. Posey
Isotopes Development Center
Oak Ridge National Laboratory
Post Office Box X
Oak Ridge, Tennessee 37830
Huwa ROT London
States, nor the Commission, nor any person acting on betalt of the Commission:
A. Makes my warranty or ropresentation, expressed or implied, with respect to the accu-
racy, complotoners, or wrotulnost of the laformadon contained in this roport, or was the wo
of any information, apparatus, morbod, or process disclosed lo this romi may not infrico
privately owned righta; or
B. Assumos any liabilities with respect to the use of, or for damages resulting from the
use of any information, apparatus, wethod, or 0. OCOss disclosed in this roport.
As usod in the above, "por son acting on behalf of the Commission" includes any 60-
ployoo or contractor of the Commission, or omployee of such contractor, to the extent that
such omployee or contractor of the Commission, or ployee of such contractor prepares,
disseminates, or provides access to, any information pursuant to bilo employment or contract
with the Commission, or his employment with such contractor.
ABSTRACT
from the heat source and the size of some sources
are limited by the maximum allowable internal
temperature.
Equations relating the maximum internal tempera-
tures of various-shaped fuel forms and other.
source parameters are given. By using calculated
results, the use of cermets is compared with the
use of oxide-metal laminar structures as methods
of decreasing the resistance to heat flow in the
interior of heat sources and achieving higher sur-
face heat fluxes without exceeding maximum allow-
ab.le internal temperatures. The laminar composite
fuel form is moderately superior.
This report gives basic equations for the predic-
tion of these limitations for simple cases. It is
shown that the maximum available surface heat flux
can be increased by the use of cermets or laminar
metal structures when radioisotope compounds of
poor thermal conductivity are used.
The expected vapor phase migration and equilibrium
distribution of a volatile radioactive fuel inside
a heat source capsule is describea. Redistribu-
tion does not always result in serious problems
and can reduce the cost of source abrication.
The existence of temperature dil'ferences within the
capsule also leads to volatile transfer of radio-
active material. The factors which determine the
equilibriun distribution of the material are dis-
cussed, and a mathematical analysis 18 given for
a simple case.
EQUATIONS FOR FUEL FORMS OF SIMPLE GEOMETRIC SHAPE
INTRODUCTION
A radioisotope heat source is generally considered
to consist of a quantity of radioactive fuel and
its container. The radioautive fuel can be in the
form of a single compound of the radioisotope, or
it can be associated wita a matrix material of
better thermal conductivity.
Equations relating the difference between the sur-
face temperature and the maximum internal tempera-
ture to various parameters for fuel forms of simple
geometric shapes are given in Table 1. The symbols
used in the equations are defined as follows:
AT = difference between the maximum allcwable
internal temperature and the surface
temperature
-
P = power density (rate of neat generation
per unit volume) of the fuel,
Large temperature differences can exist between
the surisce and the interior of the radioactive
friel (defined as the radioactive material with or
without a diluent or matrix material) in heat
sources. These gradients are created by the flow
of heat, which is generated throughout the radio-
active material, to the surface. Very high inter-
nal temperatures can result.
h = distance from surface to plane of maximum
temperature in a planar fuel form,
k = thermal conductivity of the fuel,
q = total rate of heat production by the
source,
A = surface area through which heat leaves
the fuel form,
There are no specific rules for setting the maxi-
mum ai.lowable internal temperature for a heat
source. It must be set on the basis of the prop-
erties and use of the particular source. The
melting point of the fuel has often been set as
the maximum allowable temperature. Certainly,
melting can in some cases result in a volume
change which can damage the source. In other
cases, it can allow an undesirable movement of
material within the capsule. However, in other
applications, melting in the interior of the
radioactive mass of material does no barm. The
maximum surface heat flux which can be obtained
D = diameter of a cylindrical or spherical
fuel form,
1 = length of a cylindrical fuel form,
q/A = surface heat flux.
*Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation.
DISTRIBUTION OF THIS DOCUMENI S UNDIALULUI
per
These equations were derived under the assumption
that P and k are constant throughout the fuel and
that the surface temperature is constant. Heat
losses from the edges of planar fuel fcrus and from
the ends of cylindrical fuel forms are neglected.
It can be seen that the maximum heat output of a
spherical fuel form is limited by ATmax, The total
power is not limited for flat or cylindrical fuel
forms since the area or length respectively can be
increased. The surface heat flux, q/A, is limited
for all fuel forms.
Table 2 gives the maximum surface heat flux for
fuel forms composed of an optimum laminar combina-
tion of tungsten metal and various radioactive
materials. These data were calculated using the
thermal conductivity of Jun and Hoch(4) for tung-
sten. Power densities were taken from references
5 and 6. · The thermal conductivities of the radio-
active materials were all assuned to be 0.03
watt/cm°c. This value has little influence on
the thermal conductivity of the composite fuel and
exact values for many radioactive materials are
not obtainable at the temperatures under considera-
tion.
The value of ATmax is determined by the application
in which the source is useä as well as ty the na-
ture of the fuel materials. The surface tempera-
ture of the source is an important design parameter
in such applications as thermionic and thermo-
electric generators and is set in the design of
the equipment. Since the required surface tempera-
ture may be quite high, ATmax will be seriously
restricted unless the melting point of the fuel
material is very high or interior melting is
allowed.
The proportional increase in the maximum surface
heat flux of a laninar composite fuel form as com-
pared to a pure radioactive material fuel form
depends on the relative thermal conductivities of
the metal and radioactive material. As an example,
an increase of 3.5 fold could be achieved with a
fuel of strontium oxide in combination with tung-
sten while an increase of only 2.9 fold could be
achieved with one of curium sesquioxide and tung-
sten. This calculation assuues thernal conduc..
tivities of 0.02 and 0.03 watt/cm °C, respectively,
for the oxides. The value for strontium oxide was
obtained from E. E. Ketchen and that for curium
sesquioxide was estimated.
The maximum curface beat flux is an important
parameter in certain beat source applications. It
has been a limiting factor in the design of thermi-
onic generating units. These units require a high
heat flux delivered at a high temperature. The“
surface temperature of the fuel must be at least
a little greater than the temperature at which the
heat is delivered.
THE EFFECTS OF VOLATILITY
IN THERMIONIC HEAT SOURCES
It can be seen that the maximum surface heat flux,
9/A, is proportional to (KP)1/2. Consequently,
high thermal conductivity and high power density
are desirable,
There are limited data which indicate that certain
radioactive fuels are moderately volatile under
the proposed conditions of use. Consequently,
there has been concern that these materials may
leave the desired location by volatile transfer
to other parts of the capsule.
IMPROVEMENT OF THE THERMAL PROPERTIES
OF HEAT SOURCES
TE
.
.
.
.
It will be shown that this volatility will cause
no severe probleins with regard to the heat flow
characteristics of a well designed source form
and that volatile transfer will, in fact, tend to
create a distribution of radioactive material with-
in the capsule similar to the ideal distribution
regardless of the initial physical form of the
radioactive material. However, this does not pre-
clude other problems caused by volatility such as
the plugging of helium vents.
The product KP can be improved for fuel of poor
thermal condutivit bestibo addition of a metal of
high conductivity. This can be cone in two ways: .
(1) a cermet can be produced, drià (2) oriented
conductors of metal can be introduced into the
body of the radioactive material.
.! Ei
The effectiveness of these methods in increasing
the thermal conductivity is shown by the calculated
curves in Figure 1. These curves assume that the
thermal conductivity of the radioactive material is
very small compared to that of the metal. Both
phases are assumed to be 100% dense. The thermai
conductivity of the ceimets was calculated using
three different equations to chosen from the
many that various people have derived for the con-
ductivity of aggregates. The oriented conductors
give moderately better results.
Recognition of this phenomenon can lead to a worth-
while reduction in the cost of fabrication of many
sources. The material can be encapsulated as a
loose powder instead of as a fabricated form. It
will distribute itself in approximately the desired
manner.
Figure 2 is a chart which aids in the rapid calcu-
lation of KP for any coübination of a highly con-
ducting metal and a poorly conducting radioactive
material providing the values of k and P for the
Figure 3-B shows a thermionic heat source which is
typical of many proposed designs. The useful heat
flow leaves through one end of the cylindrical
container, The rest of the container is assumed
to be insulated so as to reduce extraneous heat
losses to a low level. The radioactive material
is attached to the inner surface of the cylinder
at the end from which the useful heat flow passes.
a
.
::
.
::*
*
:
----
It can be seen that the maximum heat output of a
spherical fuel form is limited by ATmax. The total
power is not limited for flat or cylindrical fuel
forms since the area or length respective.ly can be
increased. The surface heat flux, q/A, is limited
5 and ú. The thermal conductivities of the radio-
active materials were all assumed to be 0.03
watt/cm °C. This value has little influence on
the thermal conductivity of the composite fuel and
exact values for many radioactive materials are
not obtainable at the temperatures under considera-
tion.
The proportional increase in the maximum surface
heat flux of a laminar composite fuel form as com-
pared to a pure radioactive material fuel form
depends on the relative thermal conductivities of
The value of ATmay is determined by the application
in which the source is used as well as by the na-
ture of the fuel materials. The surface tempera-
ture of the source is an important design parameter
in such applications as thermionic and thermo-
electric generators and is set in the design of
the equipment. Since the required surface tempera-
ture may be quite high, ATmax will be seriously
restricted unless the melting point of the fiel
material is very high or interior melting is
allowed.
an increase of 3.5 fold could be achieved with a
fuel of strontium oxide in combination with tung-
sten while an increase of only 2.9 fold could be
achieved with one of curium sesquioxide and tung-
sten. This calculation assumes thermal conduc-
tivities of 0.02 and 0.03 watt/cm °C, respectively,
for the oxides. The value for strontium oxide was
obtained from E. E. Ketchen (T) and that for curiwn
sesquioxide was estimated.
The maximum surface beat flux is an important
parameter in certain heat source applications. It
has been a limiting factor in the design of thermi-
onic generating units. These units require a high
beat flux delivered at a high temperature. The“
surface temperature of the fuel must be at least
a little greater than the temperature at which the
heat is delivered.
THE EFFECTS OF VOLATILITY
IN THERMIONIC HEAT SOURCES
There are limited data which indicate that certain
It can be seen that the maximum surface heat flux,
q/A, is proportional to (kP)2/2. Consequently,
high thermal conductivity and high power density
are desirable.
the proposed conditions of use. Consequently,
there has been concern that these materials may
leave the desired location by volatile transfer
to other parts of the capsule.
TMPROVEMENT OF THE THERMAL PROPERTIES
OF HEAT SOURCES
p
U
:
It will be shown that this vclatility will cause
no severe problems with regard to the heat flow
characteristics of a well designed source form
and that volatile transfer will, in fact, tend to
create a distribution of radioactive material with-
in the capsule similar to the ideal distrivution
regardless of the initial physical form of the
radioactive material. However, this does not pre-
clude other problems caused by volatility such as
the plugging of helium vents.
++
+2
I
aht: V
1.. A
?",-*
..
..*.
*
*76...
.
. .
.
The product KP can be improved for fuel of poor
thermal conduntivi da tela. Addition of a metal of
high conductivity. This can be done in two ways::
(1) a cermet can be produced, and (2) oriented
conductors of metal can be introduced into the
body of the radioactive material.
.
The effectiveness of these methods in increasing
the thermal conductivity is shown by the calculated
curves in Figure 1. These curves assume that the
thermal conductivity of the radioactive material is
very small compared to that of the metal. Both
phases are assumed to be 100% dense. The thermal
conductivity of the ceimets was calculated using
three different equations (+2) chosen írom the
many that various people have derived for the con-
ductivity of aggregates. The oriented conductors
give moderately better results.
Recognition of this phenomenon can lead to a worth-
while reduction in the cost of fabrication of many
sources. The material can be encapsulated as a
loose powder instead of as a fabricated form. It;
will distribute itself in approximately the desired
manner.
Figure 2 is a chart which aids in the rapid calcu-
lation of kP for any combination of a highly con-
ducting metal and a poorly conducting radioactive
material providing tbe values of k and P for the
pure materials are known.
Figure 3-B shows a thermionic heat source which is
typical of many proposed designs. The useful heat
flow leaves through one end of the cylindrical
container. The rest of the container is assumed
to be insulated so as to reduce extraneous heat
losses to a low level. The radioactive material
is attached to the inner surface of the cylinder
at the end from which the useful heat flow passes.
The temperature of this end of the container is
"Superior numbers refer to similarly-numbered references at the end of this paper".
2 - PAh.
set in the design of the apparatus. The volume of
the container is appreciably greater than the vol-
ume of the radioactive material in order to allow
space for the helium released by the radioactive
decay of an alpha emitting radioisotope.
@ @
or
h = 9/
where q = rate of heat flor,
A = area,
It is assumed that the radioactive material vapor-
izes and, condenses without change in its proper-
ties. It is also assumed that all of the heat is
generated within the radioactive material. Thus,
the reasoning will not necessarily apply if an
appreciable fraction of the energy is emitted as
gamma rays.
P = power density,
h = thickness of radioactive layer.
The thickness is directly proportional to the rate
of heat loss per unit area.
The volatile radioactive material will distribute
itself in a continuous layer covering the inner
surface of the capsule. No part of the surface
OI' the capsule can remain bare so long as at least
some heat escapes from all parts of the outer sur-
fece. This results from the fact that heat can
flow only from a higher to a lower tempereture.
Furthermore, the heat leaving a bare surface can
cožie only from radioactive material which must,
therefore, be at a higher temperature than the
bare surface. Transport of volatile material from
the hotter to the cooler surface will occur.
Application of the equation from Table 1 for the
maximum internal. temperature of a planar source
form leads to Equation 3, which relates the outer
surface of the radioactive layer to the constant
inner surface temperature.
To = Te - Pn2/2 k
(3)
where T - tenperature of the outer surface of
the radioactive layer,
Tg = temperature of the inner surface,
The temperature of the inner surface of this layer
wiil be uniform at equilibrium. The reason for
this uniformity is apparent when one considers
that if a difference in temperature did exist
material would vaporize from the hotter surface
and condense on the cooler surface.
k = thermal conductivity of the fuel.
The other symbols have been defined. Equations 2
and 3 can be combined to produce Equation 4.
(4)
2 Pk
For thin layers essentially all of the beat loss
from any part of the radioactive layer passes .
through the outer surface of the layer. This rate
of loss is essentially equal to the rate at which
heat is generated within that part of the layer.
It can be seen that T will vary and will be at a
maximum when (g/A) is at a minimum. Thus, the
outer surface of the radioactive material layer
will be botter for the well insulated parts of the
capsule than at the end from which the thermionic
heat flow occurs,
This equality exists because of the lack of other
significant modes of heat loss or gain. The
radioactive material layer cannot gain or lose
heat by conduction or thermal radiation across the
interior of the capsule because of the uniform
inner surface temperature. Conduction within the
layer parallel to the surface will be small as
compared to flow from the outer surface of the
layer so long as the thickness of the layer is
small compared to the dimensions of the capsule.
One consequence of the uniformity of the interior
temperature, Te, is that the lowering of T, at
any point will result in a general lowering of T,
throughout the source at equilibrium. One method
of accomplishing this is to introduce oriented
metallic conductors into the radioactive material
layer on the thermionic end of the capsule (see
Figure 4).
It is apparent that the layer will be thick at the
end from which the thermionic heat flow occurs and
thin on the surfaces corresponding to the well
insulated parts of the capsule. Thus, the radio-
active material will tend to locate itself in the
desired manner.
The change in T, produced by this method can be
of great practical importance as is shown by the
following example. If the radioactive material
layer at the thermionic end of the capeule is
assumed to consist of pure 244 Cm203 and the values
of 50 watts/cm2, 26.4 watts/cm®, and 0.03 watts/cm
°C are used for q/A, P and k, respectively, Ti
will exceed To by 1580°C according to Equation 4.
If To is fairly high, part of the Cm203 will be
The thickness of this laver can be predicted mathe-
matically by simple equations for the case in which
the thickness of the layer of radioactive material
is small as compared to the diameter and length of
the capsule. This will se tho 0250 wie??. ?oore-
-
.
.
.
4
+
1
17
A = area,
***IT 18 assumed "that the radioactive material vapor
izes and, condenses without change in its proper-
ties. It is also assuned that all of the heat is
generated within the radioactive material. Thus,
the reasoning will not necessarily apply if an
appreciable fraction of the energy is emitted as
gamma rays.
P = power density,
h = thickness of radioactive layer.
The thickness is directly proportional to the rate
of heat loss per unit area.
:
The volatile radioactive material will distribute
itself in a continuous layer covering the inner
surface of the capsule. No part of the surface
of the capsule can remain bare so long as at least
some heat escapes from all parts of the outer sur-
face. This results from the fact that heat can
flow only from a higher to a lower temperature.
Furthermore, the heat leaving a bare surface can
come only from radioactive material which must,
therefore, be at a higher temperature than the
bare surface. Transport of volatile material from
the hotter to the cooler surface will occur.
Application of the equation from Table 1 for the
maximum internal temperature of a planar source
form leads to Equation 3, which relates the outer
surface of the radioactive layer to the constant
inner surface temperature.
T. = T; - Ph2/2 k
(3)
where I = temperature of the outer surface of
the radioactive layer,
T, = temperature of the inner surface,
The temperature of the inner surface of this layer
will be uniform at equilibrium. The reason for
this uniformity is apparent when one considers
that if a difference in temperature did exist
material would vaporize from the hotter surface
and condense on the cooler surface.
k = thermal conductivity of the fuel.
The other symbols have been defined. Equations 2
and 3 can be combined to produce Equation 4.
To = Ti - 19/4)
(4)
..
For thin layers essentially all of the heat loss
from any part of the radioactive layer passes.
through the outer surface of the layer. This rate
of loss is essentially equal to the rate at which
heat is generated within that part of the layer.
2 Pk
It can be seen that I will vary and will be at a
maximum when (a/A) is at a minimum. Thus, the
outer surface of the radioactive material layer
will be hotter for the well insulated parts of the
capsule than at the end from which the thermionic
heat flow occurs.
This equality exists because of the lack of other
significant modes of heat loss or gain. The
radioactive material layer cannot gain or lose
heat by conduction or thermal radiation across the
interior of the capsule because of the uniform
inner surface temperature. Conduction within the
layer parallel to the surface will be small as
compared to flow from the outer surface of the
layer so long as the thickness of the layer is
small compared to the dimensions of the capsule.
40 that the lowering OI twe
One consequence of the uniformity of the interior
temperature, T., is that the lowering of T. at
any point will result in a general lowering of T.
throughout the source at equilibrium. One method
of accomplishing this is to introduce oriented
metallic conductors into the radioactive material
layer on the thermionic end of the capsule (see
Figure 4).
It is apparent that the layer will be thick at the
end from which the thermionic heat flow occurs and
thin on the surfaces corresponding to the well
insulated parts of the capsule. Thus, the radio-
active material will tend to locate itself in the
desired manner.
The thickness of this layer can be predicted mathe-
matically by simple equations for the case in which
the thickness of the layer of radioactive material
is small as compared to the diameter and length of
the capsule. This will be the case won an appre-
ciable void volume has been allowed to contain
helium generated by alpha decay. In this case
the layer can be treated as a flat plate and the .
Iollowing equations from Table I will apply to
any small area of the layer of radioactive mate-
rial:
The change in 1, produced by this method can be
of great practical importance as is shown by the
following example. If the radioactive material
layer at the thermionic end of the capsule is
assumed to consist of pure 244cm203 and the values
of 50 watts/cm², 26.4 watts/cm®, and 0.03 watts/cm
°C are used for q/A, P and k, respectively, Ti
will exceed T, by 1580°C according to Equation 4.
If T, is fairly high, part of the Cm203 will be
molten. If an optimum combination of tungsten
metal and 244cm203 is used, the temperature differ-
ence will be only 187°C.
A high value of Tz has three undesirable conse-
quences: (1) The stored helium gas is at the
temperature Ty and its pressure increases with
temperature. (2) The metal wall of the capsule .
.at a distance froin the thermionic surface tends to
leads to weakening and accelerated deterioration.
(3) The high temperature of the capsule walls also
will cause an increase in the heat loss through the
insulation.
untoward situation will occur during the luigra-
tion although the temperature in the center of a
charge of loose powder might become very high.
This would ordinarily do no more than cause rapid
vaporization and migration to the cooler surfaces
of the capsule. The material in contact with the
capsule cannot reach a temperature much above the
operating temperature because of rapid heat trans-
fer to the capsule surface.
So far, the temperatures and heat flow have rew
ferred only to the layer of radioactive material.
When it can be assumed that the flow of heat along
ühe capsule walls parallel to the surface is negli-
gible, (g/A) can be considered to be the heat flow
from the surface of the capsule. However, in many
cases this assumption will involve considerable
error.
In many sources, the rate of volatile migration
will not be rapid enough for an equilibrium dis-
tribution to be established during the life of
the source, although at least some movement toward
this distribution will always occur. The avail-
able vapor pressure data for most heat source
compounds are not adequate for good estimations
of migration rates.
In the common case of a thick-walled metal capsule,
the conduction of heat by the walls to the low
temperature end of the capsule will have a sub-
stantial leveling effect on To, 9/A, and h. The
heat flow, q/A, from the radioactive material
layer in any part of the hotter (insulated) part
of the system will now equal the sum of the heat
lost through the insulation and the heat conducted
along the capsule walls. This greater flow of heat
requires a thicker deposit of radioactive material
to hold T, constant. This is illustrated in Fig-
ure 4.
It should be noted, however, that the overall
effects of the thermal conductivity of thick cap-
sule walls are beneficial. The heat flowing along
the wails flows to thermionic surface and is not
lost. Heat losses are, in fact, reduced since the
temperature of the insulated parts of the capsule
are lowered towaad that of the thermionic end
which is fixed by the design of the device. This
lowering of the temperature will also result in
greater strength and less deterioration for the
capsule metal. The movement of radioactive mate-
ria away from the thermionic surface also reduces
the thickness of the layer on this surface. Since
To is fixed, this thinning of the radioactive lever
causes a lowering of T, which is the temperature
of the helium storage volume. Thus, the pressure
exerted by a given quantit
reduced.
iven quantity of heli
Although this discussion was directed toward
thermionic heat sources, the basic conclusions wil).
apply qualitatively to any capsule geometry. The
material will tend to concentrate on the surfaces
from wnici the greatest heat flow occurs and T.,
will be constant at equilibrium.
The characteristics of the ideal capsule for a
volatile radioactive material in applications where .
internal temperature is a limiting factor can be
summarized as follows: (1) It should have thick
wails of a highly conducting metal. (2) Internal
metallic heat paths should be used to conduct the
heat to the desired surface.
A desirable distribution can be obtained even if
So far, the temperatures and heat flow have re-
ferred only to the layer of radioactive material.
When it can be assumed that the flow of heat along
the capsule walls parallel to the surface is negli-
gible, (q/A) can be considered to be the beat flow
from the surface of the capsule. However, in many
cases this assumption will involve considerable
error.
In many sources, the rate of volatile migration
will not be rapid enough for an equilibrium dis-
tribution to be established during the life of
the source, although at least some movement toward
this distribution will always occur. The avail-
able vapor pressure data for most beat source
compounds are not adequate for good estimations
of migration rates.
.
I
is
In the common case of a thick-walled metai capsule,
the conduction of heat by the walls to the low
temperature end of the capsule will have a sub-
stantial leveling effect on To, q/A, and h. The
heat flow, q/A, from the radioactive material
layer in any part of the hotter (insulated) part
of the system will now equal the sum of the heat
lost through the insulation and the heat conducted
along the capsule walls. This greater flow of heat
requires a thicker deposit of radioactive material
to hold Ti constant. This is illustrated in Fig-
ure 4.
is
.
.
think
t
It should be noted, however, that the overall
effects of the thermal conductivity of thick cap-
sule walls are beneficial. The heat flowing along
the walls flows to thermionic surface and is not
lost. Heat losses are, in fact, reduced since the
temperature of the insulated parts of the capsule
are lowered toward that of the thermionic end
which is fixed by the design of the device. This
lowering of the temperature will also result in
greater strength and less deterioration for the
capsule metal. The movement of radioactive mate-
rial away from the thermionic surface also reduces
the thickness of the layer on this surface. Since
To is fixed, this thinning of the radioactive laver
causes a lowering of T; which is the temperature
of the helium storage volume. Thus, the pressure
exerted by a given quantity of helium will be.
reduced.
na
l idad
necesibi
Although this discussion was directed toward
thermionic heat sources, the basic conclusions will
apply qualitatively to any capsule geometry. The
material will tend to concentrate on the surfaces
from which the greatest heat flow occurs and Ti
will be constant at equilibrium.
highest sisteminin insatisfacer
1
The characteristics of the ideal capsule for a
volatile radioactive material in applications where ·
internal temperature is a limiting factor can be
summarized as follows: (1) It should have thick
walls of a highly conducting metal. (2) Internal
metallic heat paths should be used to conduct the
heat to the desired surface.
is there mandamientation with
now and
A desirable distribution can be obtained even if
the radioactive material is encapsulated as loose
lumps or powder when comparatively fast migration
toward the equilibrium distribution occurs. No
initiation artistice is
isitioning
m
REFERENCES
Russell, H. W., Principals of Heat Flow in
Porous Insulators, J. Am. Cer. Soc. 18: 1-5
(January 1935).
Rayleigh, Lord, On the Influence of Obstacles
Arranged in Rectangular Order Upon the Prop-
erties of a Medium, Phil. Mag. 34: 481-507
(1892).
Bruggeman, D. A. G., Dielectric Constant and
Conductivity of Mixtures of Isotropic Mate-
rials, Ann. Physik 24: 636-79 (1935).
(4)
Jun, C. K. and Hoch, M., Thermal Conductivity
of Tantalum, Tungsten, and Tantalum-Tungsten
Alloys, Technical Rpt. AFML-TR-65-191,
University of Cincinnati (May 1965).
Rohrman, C. A., Radioisotopic Heat Sources,
USAEC Rpt. HW-76323, Hanford Atomic Products
Operation (February 1963).
(6)
Rimshaw, S. J., Ketchen, E. E., and Burnett,
J. L., private communication, Oak Ridge
National Laboratory (March 1966).
(7) Ketchen, E. E., private communication, Oak
Ridge National Laboratory (June 14, 1967).
ILLUSTRATIONS
Figure 1. Comparison of the Thermal Conductivities
of Laminar Composite Fuels and Cernets.
Figure Comparison of Surface Heat Flux Factors
of Laminar Composites and Cermets Fuels.
Figure 3. Distribution of Radioactive Material
Inside Capsule.
Figure 4. Qualitative Representation of Fuel
Distribution in Three Models,
Table 1. Equations for Fue). Forms of Simple Geometric Shape
Parameter
Flat Plate
Solid Cylinder
Solid Sphere
Pha
PD2
PD2
16k
2k
24K
IPD3
PAN
A(zkParmax) 1/2
414IKATmax
max =
(a/A)max =
(2kPaTmax) 1/2
(KPATmax) 1/2
(akoppimax )/2
Table 2. Maximum Surface Heat Fluxes of Laminar
Composite Planar Fuel Forms
Radioactive
Compound
Surface Temp.,
Power Density,
watts/cc
Melting Point,
Maximum Surface
Heat Flux,
watts/cm2
90sro
2.08
2430
1250
1500
1800
147Pm203
2.36
2350
1250
1500
1800
244cm203
26.4
1950
1250
1500
1800
232002
2176
1250
1500
1800
2000
228Thoz
1270
3050
1250
1500
1800
2000
2050
070
860
780
ORNL-DWG 66-11423
!!
SELL'S
-LAMINAR COMPOSITE OF METAL AND OXIDE, HEAT FLOW –
PARALLEL TO STRATA I
CERMET WITH METAL PHASE CONTINUOUS, FROM RUSSEL'S
EQUATION -
CERMET WITH METAL PHASE CONTINUOUS, FROM RAYLEIGH-
| MAXWELL OR EUCKIN EQUATION+
CERMET WITH METAL PHASE CONTINUOUS FROM
BRUGGEMAN'S EQUATION,
r
-CERMET WITH OXIDE PHASE CONTINUOUS,
FROM BRUGGEMAN'S EQUATION
Et
RELATIVE THERMAL CONDUCTIVITY, k/KMETAL
0 .0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
VOLUME FRACTION OXIDE
Comparison of the Thermal Conductivities of Laminar Composite
Structu.ces and Cermets.
FUELS
.
.

.-... ". "
t
..
-
h.com
.---
.
.
w
-
.
.....
h
the
mou
i
.
..
e
thin
..
.
..
. .
.
.
-
.
.
.
ORNL-DWG 66-49422
PARALLEL
STRATA OF METAL AND OXIDE 7
MAXWELL-RAYLEIGH OR
EUCKEN EQUATION, CERMET
GWITH METAL PHASE+
CONTINUOUS
0.2
RUSSEL EQUATION,
CERMET WITH METAL
PHASE
CONTINUOUS
SURFACE HEAT FLUX FACTOR
(kP) = [KMETAL * Poxide)
BRUGGEMAN EQUATION,
CERMET WITH METAL
PHASE CONTINUOUS
BRUGGEMAN EQUATION, CERMET
WITH OXIDE PHASE CONTINUOUS
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 4.0
VOLUME FRACTION OXIDE
Comparison of Surface Heat Flux Factors of Laminar Composites
and Cermets, Fw

·
·
-
:
:
..
.....
.
.
.
..
.
..
.......
............
A. IDEAL - DISTUBUTION
..
..
VR! HEAT FLOT
..commons
Tweede
IT
..
.
........
k CAPSULE
MATERIAL
RADIOACTIVE.
:
Fig. 3.
MATERIAL INSIDE CAPSULE
DISTRIBUTION OF RADIOACTIVE
·
..
'
juccana.comcommon.
mind
..
s
·licacione
......
C
C
B. EGUILIBRUM DISTRIBUT/01/
Pallatih? HVERT
201!


-
-
.
ARRO:YS IND:CATE DIRECTION OF HEAT FLOW.
OXNL-DWG. 67-268
To:TEMPERATURE AT TSE FUEL-CAPSULATE FACE (MAXI.. AT VEDICATED 2011T)
T::TEMPERATURE AT THE INTER SURFACE OF THE FUEL AT EQUIL:3RIUM
To (4) > , < > a C ; 7; a) x 1; ) > 7 C)
- CONSTANT
*COMSTA:!T
. Toa MAXIMUM
Ti *50::START
T
AXIMUM
COSTANT
Work card
V
nennamen
IL.
PP
dawkwambambamba
- INSULATION
SILT
- CAPSULE
CONDUCTORS
A
IN
ww.rivite
conductors
FUEL

1
A. LATERAL HEAT
FLOKI
.
S. LATERAL 1.2
LOMO!TLOKAL
C.LATERAL AND
10RGITUDIA .
HEAT FLOW WITH
CONDUCTORS ::UEL
Fig. Qualitative Representation of Fuel Distribution
in Three Models.


!!
!!
!
!
N
.
.
.
H
END
2
.
2
t
.
DATE FILMED
10 / 5 / 67



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