Question 9.8: 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. Figure 4.1d shows such a configuration. Assume that R_1 and R_2 are the thermal resistances between fuel and moderator, and between moderator and coolant respectively, and that W and c_p are the coolant mass flow rate and specific heat.
a. Develop a set of thee coupled equations similar to to those in Eqs. (8.40) and (8.41) to model the steady state heat transfer.
b. Determine the isothermal temperature coefficient in terms of the fuel, moderator and coolant temperature coefficients.
c. Determine the power coefficient in terms of the same temperature coefficients.
Part a. See solution to problem [8.14] for transient analysis. By setting the time derivatives to zero, we obtain the steady state heat transfer:
P={\frac{1}{R_{1}}}\left[{\overline{{T}}}_{f}-{\overline{{T}}}_{m}\right]
\frac{1}{R_{1}}[\overline{{{T}}}_{f}-\overline{{{T}}}_{m}]=\frac{1}{R_{2}}[\overline{{{T}}}_{m}-\overline{{{T}}}_{c}]
{\frac{1}{R_2}}[{\overline{{T}}}_{m}-{\overline{{T}}_{c}}]=2W c_{c}\ [{\overline{{T}}}_{c}-T_{i}]
from which we may write the steady state model as:
\overline{{{T}}}_{c}=\frac{1}{2W{\it c}_{c}}P+T_{i}
\overline{{{T}}}_{m}=\left(\,R_{2}+\frac{1}{2W c_{p}}\,\right)P+T_{i}
\overline{{{T}}}_{f}=\left(R_{1}+R_{2}+\frac{1}{2W c_{p}}\right)P+T_{i}
Part b: Let α_f , α_m and α_c be the fuel, moderator and coolant temperature coefficients. The isothermal temperature coefficient is them simply:
\alpha_{T}=\alpha_{_{f}}+{{{\alpha}}}_{m}+\alpha_{c}
Part c: The power coefficient is just a generalization of Eqs. (9.30) through (9.33) to include the third region:
\alpha_{P}=\alpha_{f}\,\frac{d}{dP}\,\overline{{{T}}}_{f}+\alpha_{m}\,\frac{d}{d P}\,\overline{{{T}}}_{m}+\alpha_{c}\,\frac{d}{d P}\,\overline{{{T}}}_{c}
assuming a constant inlet temperature. Find the power derivatives from the three final equations in part a. We then have
\alpha_{P}=\left(R_{1}+R_{2}+{\frac{1}{2W c_{p}}}\right)\alpha_{f}+\left(R_{2}+{\frac{1}{2W c_{p}}}\right)\alpha_{m}+{\frac{1}{2W c_{p}}}\alpha_{c}