= Solution
\b[Each equation is stored thermal energy changing at the rate “input minus output”.] Define the three <heat capacities> $C_j=M_jc_j$, and introduce the internal heat-transfer rates
$$
Q_{FM}=\frac{\overline T_F-\overline T_M}{R_1},
\qquad
Q_{MC}=\frac{\overline T_M-\overline T_C}{R_2}.
$$
In the fuel balance, $C_F\dot{\overline T}_F$ is accumulation, $P$ is thermal <power> generation and $Q_{FM}$ is transfer to the <neutron moderator>. In the moderator balance, $Q_{FM}$ enters, $Q_{MC}$ leaves and $C_M\dot{\overline T}_M$ is accumulation. In the coolant balance, $Q_{MC}$ enters, $C_C\dot{\overline T}_C$ is accumulation and $2W_Cc_C(\overline T_C-T_i)$ is net energy carried away by flowing <coolant>.
The factor two follows the model's arithmetic-mean approximation $\overline T_C=(T_i+T_o)/2$, since the advective energy removal is $W_Cc_C(T_o-T_i)$. The parameters $R_1$ and $R_2$ are effective <thermal resistances>, in $\mathrm{K\,W^{-1}}$, for fuel-to-moderator and moderator-to-coolant transfer. They summarize the relevant geometry and heat-transfer mechanisms; they are not <heat capacities>.
\Image[/past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2019/paper-ne1-thermal-balance.png]
{title=Original three-reservoir energy-balance diagram}
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