appendixC.tex

This appendix describes the analytical solution used in Section \ref{sec:tf-ver}.

\section{Verification of the thermal-fluids model}
\label{appendix:ver}

The analytical solution of the problem is
\begin{align}
    T_c (r, z) &= T_{in} + \frac{q_{ave} R_f^2 L}{2 \rho c_p v \pi R_c^2} \left[ 1 + cos \left( \frac{\pi}{L} z \right) \right] \\
    T_3(z) &= T_c(z) + \frac{q_{ave} \pi}{2} sin \left( \frac{\pi}{L} z \right) R_f^2 \frac{ln(R_i/R_m)}{2 k_i} \\
    T_2(z) &= T_3(z) + \frac{q_{ave} \pi}{2} sin \left( \frac{\pi}{L} z \right) R_f^2 \frac{ln(R_m/R_g)}{2 k_m} \\
    T_1(z) &= T_2(z) + \frac{q_{ave} \pi}{2} sin \left( \frac{\pi}{L} z \right) R_f^2 \frac{ln(R_g/R_f)}{2 k_g} \\
    T_f (r=0, z) &= T_1(z) + \frac{q_{ave} \pi}{2} sin \left( \frac{\pi}{L} z \right) R_f^2 \frac{1}{4 k_f} \\
    T_f (r, z=L/2) &= \frac{q_{ave}}{4 k_f} \left(R_f^2 - r^2\right) + T_1 (z=L/2) \\
    T_g (r, z=L/2) &= \frac{T_1 (z=L/2)-T_2 (z=L/2)}{ln (R_f/R_g)} ln (r/R_g) + T_1(z=L/2) \\
    T_m (r, z=L/2) &= \frac{T_2 (z=L/2)-T_3 (z=L/2)}{ln (R_g/R_m)} ln (r/R_m) + T_2(z=L/2) \\
    T_i (r, z=L/2) &= \frac{T_3 (z=L/2)-T_c (z=L/2)}{ln (R_m/R_i)} ln (r/R_i) + T_3(z=L/2) \\
    T_c (r, z=L/2) &= T_c(z=L/2)
    \intertext{where}
    T_{c} &= \mbox{bulk coolant temperature } [^{\circ}C] \notag \\
    T_{in} &= \mbox{inlet coolant temperature } [^{\circ}C] \notag \\
    q_{ave} &= \mbox{average power density } [W \cdot cm^{-3}] \notag \\
    R_f &= \mbox{fuel compact radius } [cm] \notag \\
    L &= \mbox{fuel column height } [cm] \notag \\
    \rho &= \mbox{helium density } [kg \cdot cm^{-3}] \notag \\
    c_p &= \mbox{helium heat capacity } [J \cdot kg^{-1} \cdot K^{-1}] \notag \\
    v &= \mbox{average helium velocity } [cm \cdot s^{-1}] \notag \\
    R_c &= \mbox{coolant channel radius } [cm] \notag \\
    R_g &= \mbox{gap radius } [cm] \notag \\
    R_m &= \mbox{moderator radius } [cm] \notag \\
    R_i &= \mbox{film radius } [cm] \notag \\
    k_f &= \mbox{fuel compact thermal conductivity } [W \cdot cm^{-1} \cdot K^{-1}] \notag \\
    k_g &= \mbox{gap thermal conductivity } [W \cdot cm^{-1} \cdot K^{-1}] \notag \\
    k_m &= \mbox{moderator thermal conductivity } [W \cdot cm^{-1} \cdot K^{-1}] \notag \\
    k_i &= \mbox{film thermal conductivity } [W \cdot cm^{-1} \cdot K^{-1}]. \notag
\end{align}