Question 8.14: In the prismatic block form of the graphite moderated gas-co......
In the prismatic block form of the graphite moderated gas-cooled reactor the heat passes through the moderator before reaching the coolant. Figures 4.1d and 4.2c show such a configuration. Develop a set of three coupled differential equations in forms similar to Eqs. (8.50) and (8.58) that describe the transient heat transfer in such a reactor. Assume that the heat transfer between fuel and moderator and between moderator and P(t)=[\overline{{{T}}}_{f}(t)-\overline{{{T}}}_{m}(t)]/R_{1}~\mathrm{and}~P(t)=[\overline{{{T}}}_{m}(t)-\overline{{{T}}}_{c}(t)]/R_{2} respectively. Assume that W is the mass flow rate, and that the masses, specific heats and densities of fuel, moderator and coolant are given by M_{i},\;c_{i}\;\mathrm{and}\;\rho_{i}\;\mathrm{with}\;i=f,\,m,\,c.
For the fuel region the equation is similar to Eq. (8.50)
M_{f}c_{f}\,\frac{d}{d t}\,\overline{{T}}_{f}(t)=P(t)-\frac{1}{R_{1}}\bigg[\overline{{T}}_{f}(t)-\overline{{T}}_{m}(t)\Big]
For the moderator region heat transfer both into the moderator and out of it into the coolant must be represented.
{ M}_{m}c_{m}\,\frac{d}{d t}\,\overline{{{T}}}_{m}(t)\!=\!\frac{1}{R_{1}}\bigg[\overline{{{T}}}_{f}(t)\!-\!\overline{{{T}}}_{m}(t)\bigg] -\frac{1}{R_2}\bigg[\!\overline{{{T}}}_{m}(t)\!-\!\overline{{{T}}}_{c}(t)\!\bigg]
The coolant equation looks similar to Eq. (8.58):
M_{c}c_{c}\,\frac{d}{d t}\overline{{{T}}}_{c}(t)=\frac{1}{R_{2}}\Bigl[\overline{{{T}}}_{m}(t)-\overline{{{T}}}_{c}(t)\Bigr]-2{ W}c_{c}\Bigl[\overline{{{T}}}_{c}(t)-T_{i}\Bigr]
In prism block reactors, the mass of the gas coolant is so small that it often can be ignored be setting M_c c_c to zero in the last equation. Solving for the coolant temperature, we then have
\overline{{{T}}}_{c}(t)=\frac{1}{1+2R_{2}W{c_{c}}}\Bigl[2R_{2}W{c_{c}}\overline{{{T}}}_{m}(t)+T_{i}\Bigr]
Finally we note that the moderator mass in a graphite block reactor is very large. As a result the core heat capacity is large providing a slow response of moderator and coolant transients to changes in power level. The slow moderator adiabatic heat up rate, which we define as follows, indicates this:
{\frac{d}{d t}}{\overline{{T}}}_{m}(t)={\frac{P}{M_{m}c_{m}}}