.. _section-A10:
Appendices
----------
.. _section-A10.1:
APPENDIX 10.1: Equivalency of Interpolation in Terms of Weight Fractions and Atom Fractions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As described in :numref:`section-10.3.2.1`, the enthalpy of a desired U-Pu-Zr fuel
composition is evaluated by mixing 3 basic alloys with enthalpy data
known and reported in the Metallic Fuels Handbook [10-8]. The
proportions or weight fractions of the 3 database alloys in mixing are
so calculated that the resulting mixture has the composition of the
desired fuel. The purpose of this appendix is to show that it makes no
difference whether this mixing or linear interpolation is carried out by
expressing composition in terms of weight fractions or atom fractions,
and correspondingly enthalpy in terms of J/unit mass or J/mol.
Let
:math:`a_{1}, a_{2}, a_{3}` = average atomic weights of
the 3 database alloys,
:math:`h_{1}, h_{2}, h_{3}` = enthalpy (J/gm) of the 3
database alloys at a desired temperature.
:math:`H_{1}, H_{2}, H_{3}` = enthalpy (J/mol) of the 3
database alloys at the desired temperature,
:math:`X_{1}, X_{2}, X_{3}` = weight fractions of the 3
database alloys to be mixed to produce the desired fuel composition.
:math:`Y_{1}, Y_{2}, Y_{3}` = atom fractions of the
database alloys to be mixed to produce the desired fuel composition.
The desired fuel enthalpy interpolated in terms of weight fractions is
given by
.. math::
:label: A10.1-1
h = X_{1} h_{1} + X_{2} h_{2} + X_{3} h_{3}
where :math:`h` is in units of J/gm. The desired fuel enthalpy interpolated in
terms of atom fractions is given by
.. math::
:label: A10.1-2
H = Y_{1} H_{1} + Y_{2} H_{2} + Y_{3} H_{3}
where :math:`H` is in units of J/mol. The enthalpies given by :eq:`A10.1-1`
and :eq:`A10.1-2` will be the same if it is shown that
.. math::
:label: A10.1-3
h = \frac{H}{a}
where
:math:`a` = average atomic weight of the desired fuel composition.
The average atomic weight of the desired fuel composition is related to
the atomic weight of the 3 database alloys and their weight fractions in
the desired fuel as follows:
.. math::
:label: A10.1-4
a = \frac{1}{\left( \frac{X_{1}}{a_{1}} + \frac{X_{2}}{a_{2}} + \frac{X_{3}}{a_{3}} \right)}
The atom fraction :math:`Y_{1}` of database alloy 1 in the desired fuel
is related to the weight fractions :math:`X_{1}, X_{2}`,
:math:`X_{3}` of the 3 database alloys in the desired fuel as follows:
.. math::
:label: A10.1-5
Y_{1} = \frac{\left( \frac{X_{1}}{a_{1}} \right)}{\left( \frac{X_{1}}{a_{1}} + \frac{X_{2}}{a_{2}} + \frac{X_{3}}{a_{3}} \right)}
With the help of :eq:`A10.1-4`, the atom fraction :math:`Y_{1}` given by
:eq:`A10.1-5` simplifies to
.. math::
:label: A10.1-6
Y_{1} = \frac{a X_{1}}{a_{1}}
Similarly, the atom fractions :math:`Y_{2}` and :math:`Y_{3}` can be
written as
.. math::
:label: A10.1-7
Y_{2} = \frac{a X_{2}}{a_{2}}
.. math::
:label: A10.1-8
Y_{3} = \frac{a X_{3}}{a_{3}}
The enthalpies of the 3 database alloys in units of J/mol are related to
their enthalpies in units of J/gm and their atomic weights as follows:
.. math::
:label: A10.1-9
\begin{aligned}
H_{1} = a_{1} h_{1}, && H_{2} = a_{2} h_{2}, && H_{3} = a_{3} h_{3}
\end{aligned}
Substituting the values of :math:`Y_{1}, Y_{2}, Y_{3}`
and :math:`H_{1}, H_{2}, H_{3}` from :eq:`A10.1-6` to
:eq:`A10.1-9` into :eq:`A10.1-2`, one obtains
.. math::
:label: A10.1-10
H = a X_{1} h_{1} + a X_{2} h_{2} + a X_{3} h_{3}
With the help of :eq:`A10.1-1`, the desired fuel enthalpy interpolated
in terms of atom fractions given by :eq:`A10.1-10` becomes
.. math::
:label: A10.1-11
H = ah
:eq:`A10.1-11` proves what :eq:`A10.1-3` required.
.. _section-A10.2:
APPENDIX 10.2: Correlations in the IFR Handbook and Some Other Sources
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The following 4 correlations are available to estimate 100% dense solid
U-Pu-Zr fuel thermal conductivity, :math:`K_{\text{o}}` (W/m-K):
1. The correlation in the IFR Metallic Fuels Handbook [10-8, 10-9] is
written as
.. math::
:label: A10.2-1
K_{\text{o}} = B_{1} + B_{2} T + B_{3} T^{2}
where
:math:`B_{1} = 17.5 \left\lbrack \frac{\left( 1 - 2.23 W_{\text{z}} \right)}{\left( 1 + 1.61 W_{\text{z}} \right)} - 2.62 W_{\text{p}} \right\rbrack`
:math:`B_{2} = 0.0154 \left\lbrack \frac{\left( 1 + 0.061 W_{\text{z}} \right)}{ \left( 1 + 1.61 W_{\text{z}} \right)} + 0.90 W_{\text{p}} \right\rbrack`
:math:`B_{3} = 9.38 \times 10^{-6} \left( 1 - 2.70 W_{\text{p}} \right)`
:math:`W_{\text{p}}, W_{\text{z}}` = weight fractions of Pu and Zr
:math:`T` = fuel temperature, K.
2. The correlation in Billone's memorandum [10-21] of March 8, 1991 is
written as
.. math::
:label: A10.2-2
K_{\text{o}} = B_{4} + B_{5} T + B_{6} T^{2}
where
:math:`B_{4} = 17.5 \left\lbrack 1 - 1.471 A_{\text{z}} + 44.083 A_{\text{p}} \left( 1 - 16.198 A_{\text{p}} + 55.290 A_{\text{p}}^{2} \right) \right\rbrack`
:math:`B_{5} = 0.0154 \left\lbrack 1 - 0.5935 A_{\text{z}} - 158.88 A_{\text{p}} \left( 1 - 14.865 A_{\text{p}} + 49.478 A_{\text{p}}^{2} \right) \right\rbrack`
:math:`B_{6} = 9.38 \times 10^{-6} \left\lbrack 1 + 215.78 A_{\text{p}} \left( 1 - 14.421 A_{\text{p}} + 47.158 A_{\text{p}}^{2} \right) \right\rbrack`
:math:`A_{\text{p}}, A_{\text{z}}` = atom fractions of Pu and Zr.
3. The correlation in Billone's memorandum [10-22] of October 21, 1991
is written as
.. math::
:label: A10.2-3
K_{\text{o}} = B_{7} + B_{8} T + B_{9} T^{2}
where
:math:`B_{7} = 17.5 \left( 1 - 1.417 A_{\text{z}} - 6.8174 A_{\text{p}} \right)`
:math:`B_{8} = 0.0154 \left(1 - 0.5935 A_{\text{z}} + 15.108 A_{\text{p}} \right)`
:math:`B_{9} = 9.38 \times 10^{-6} \left( 1 - 18.778 A_{\text{p}} \right)`
4. The correlation reported in ANL/RAS 85-19 can be written as
.. math::
:label: A10.2-4
K_{\text{o}} = K_{\text{u}} + 2A_{\text{z}} \frac{\left( K_{\text{u}50} - K_{\text{u}} \right)}{\left( A_{\text{u}} + A_{\text{z}} \right)} - CA_{\text{p}}
where
:math:`K_{\text{u}} = 22.173 + 0.018562 T_{\text{c}} + 1.3278 \times 10^{-5} T_{\text{c}}^{2}`
:math:`K_{\text{u}50} = 7.4766 + 0.016738 T_{\text{c}} + 1.1599 \times 10^{-5} T_{\text{c}}^{2}`
:math:`C = \left( 89.079 - 167.57 A_{\text{z}} \right) - T_{\text{c}} \left( 0.17437 - 0.4008 A_{\text{z}} \right) + \
T_{\text{c}}^{2} \left( 1.9608 \times 10^{-4} - 3.1043 \times 10^{-4} A_{\text{z}} \right), \ \ 100 \leq T_{\text{c}} \leq 1500`,
.. math::
:label: A10.2-5
\begin{aligned}
0.0 \leq A_{\text{p}} \leq 0.2, && 0.0 \leq A_{\text{z}} \leq 0.5
\end{aligned}
:math:`T_{\text{c}}` = fuel temperature, °C
Correlations 1, 2 and 3 have been developed based on the thermal
conductivity data presented in the Metallic Fuels Handbook. Detailed
plots of these correlations are available in Ref. [10-17]. Correlation 2
should not be used because it is ill-behaved, and so recommended by
Billone [10-22]. A drawback of correlations 1 and 3 is that the thermal
conductivities become negative (unphysical) at room temperature for Pu
and Zr contents greater than 20 wt %. In addition, the thermal
conductivity obtained from correlation 3 shows a maximum when plotted as
a function of temperature (an unexpected behavior [10-18] for
temperatures above 300 K).
The comparison of these 3 correlations with the Handbook data for 3
ternary alloys (U-16.2Pu-6.2Zr, U-14.7Pu9.7Zr and U-18.4Pu-11.5Zr) shows
that there are significantly large differences among the 3 correlations
at temperatures above 1200 K where there is no thermal conductivity
data. Below 1200 K, the difference among the data and the 3 correlations
are smaller.
The comparison of these 3 correlations with the Handbook data for 3 U-Zr
binary alloys (U-1.5Zr, U-5Zr and U-20Zr) shows that the 3 correlations
are almost identical at all temperatures, and that they agree with the
data closely. The agreement is not so good with the data for U-40Zr, as
shown by a plot given in the Handbook [10-8, 10-9]. The Handbook
correlation for U-Zr binary alloys, i.e., :eq:`A10.2-1` with
:math:`W_{\text{p}}` set to zero, is found to be reliable for Zr content in the
1.5 to 20 wt % range.
The thermal conductivity obtained from correlation 4 for the some alloy
compositions becomes negative (unphysical) at room temperature, and
shows an unexpected maximum when plotted as a function of temperature.
Although the alloy compositions showing this unphysical/unexpected
behavior are out of the noted region of validity, the correlation has
another drawback that it was derived using data from sources [10-14,
10-19, 10-20] other than the IFR Metallic Fuels Handbook [10-8, 10-9].