The thermal conductivity of $U{O}_2$ has been of extensive research due to its importance in nuclear

fuel design. Empirical models have been suggested for the two major variables, temperature $T$

and fuel burnup B by Halden $(1997)$ or Fink $(2000).$ Answer the following questions regarding

the Halden model $($in the slides of lecture $18).$

a$)$ If the general form of thermal conductivity can be expressed as $k=\frac{1}{a+bT},$ write down the units

for a and $b.$

b$)$ Using your favorite plotting programs $($Excel$,$ Matlab, Origin, etc.$),$ plot the thermal conductivity

of $U{O}_2$ between $300$ and $700K,$ for the fuel burnup of $10,30,$ and $50MW\frac{d}{M}tU.$ Describe what

happens to the thermal conductivity as $T$ increases. Describe the trend in thermal conductivity

as $B$ increases.

c$)$ If there exists porosity inside the $U{O}_2$ pellet, the thermal conductivity decreases with the given

equation:

$k_p=k_0(1-P)$ for $P0\mathrm{.}1$

Using the Halden model, plot the thermal conductivity of UO$2$ between $300$ and $700K$ for the

constant fuel burnup of $10MW\frac{d}{M}tU,$ with varied porosity of $0\mathrm{.}01,0\mathrm{.}1$ and $0\mathrm{.}2,$ assuming the

above equation holds up to the porosity of $0\mathrm{.}2.$