Question 9.7: For the reactor specified in problem [9.4] the power is main......

For the simple model of problem [9.4] the reactivity is found in Eq. (9.36) , with the absolute value signs removed to allow positive as well as negative temperature coefficients . Since the reactor must be maintained in a critical state for the power to be constant. We have ρ(t) = 0, and hence

\rho_{i}(t)+\alpha_{f}\biggl[\overline{{{T}}}_{f}(t)-\overline{{{T}}}_{f}(0)\biggr]+\alpha_{c}\biggl[\overline{{{T}}}_{c}(t)-\overline{{{T}}}_{c}(0)\biggr]=0

where ρ_i(t) is the control activity, which must be added or subtracted to keep the reactor at constant power Under quasi static conditions the fuel and moderator temperatures are determined by Eqs. (9.45) and (9.46), which we modify to allow slow changes in the flow rate and inlet temperature :

\overline{{{T}}}_{c}(t)\!=\!\frac{1}{2W(t)c_{p}}P\!+\!T_{i}(t)\,.

\overline{{{T}}}_{f}(t)=\left(R_{f}+\frac{1}{2W(t)c_{p}}\right)P+T_{i}(t)\,.

Hence

\left[\overline{{{T}}}_{c}(t)-\overline{{{T}}}_{c}(0)\right]\!=\!\left[\frac{1}{2W(t)c_{p}}-\frac{1}{2W(0)c_{p}}\right]\!P\!+\!\left[\overline{{{T}}}_{i}(t)-\overline{{{T}}}_{c}(0)_{i}\right]

and likewise

\left[\overline{{{T}}}_{f}(t)-\overline{{{T}}}_{f}(0)\right]=\left[\frac{1}{2W(t)c_{p}}-\frac{1}{2W(0)c_{p}}\right]P+\left[\overline{{{T}}}_{i}(t)-\overline{{{T}}}_{c}(0)_{i}\right]

The reactivity equation then becomes

\rho_{i}(t)=-\left(\alpha_{f}+\alpha_{c}\right)\left \{\left[{\frac{W(0)}{ W(t)}}-1\right]{\frac{P}{2 W(0)c_{p}}}+\overline{{{T}}}_{i}(t)-\overline{{{T}}}_{c}(0)\right\}

Before proceeding we must calculate the flow rate: Form Eq. (8.37) the mass flow rate is

{W}(0)=\frac{P}{c_{p}\left[\overline{{{T}}}_{o}(0)-T(0_{i})\right]}=\frac{1000\cdot10^{6}}{1250(500-350)}=5,333\,\mathrm{kg/s}

Plugging in the parameters from problem [9.4] into the reactivity equation:

\rho_{i}(t)=1.35\cdot10^{-5}\left\{\left[{\frac{W(0)}{W(t)}}-1\right]\!225+\overline{{{T}}}_{i}(t)-\overline{{{T}}}_{i}(0)\right\}

Part a. \overline{{{T}}}_{i}(t)-\overline{{{T}}}_{i}(0)=-\ 5.0

\rho_{i}(t)=1.35\cdot10^{-5}\left\{\overline{{{T_{i}}}}(t)-\overline{{{T_{i}}}}(0)\right\}=-6.75\cdot10^{-5}

If the inlet temperature decreases, the feedback reactivity increases, thus the control system must subtract reactivity.

Part b: W(0) /W(t) = 1/1.1 = 0.909

\rho_{i}(t)=1.35\cdot10^{-5}\left[{\frac{W(0)}{W(t)}}-1\right]225=-27.6\cdot10^{-5}

Increasing the flow rate, decreases fuel and coolant temperature. Thus the feedback reactivity positive, and the control system must subtract reactivity to maintain criticality.