The Nuclear Reactor — Question 8

Question 8

A thermal-feedback model distinguishes stable and unstable responses

Near equilibrium, let a temperature perturbation $\theta$ change generation by $a\theta$ and heat removal by $h\theta$. The heat capacity is C=10 MJ/K, with $a=.10$ MW/K and $h=.30$ MW/K.

1. Derive the linear perturbation equation and its relaxation time.

2. Find the fraction of an initial perturbation remaining after 100 s, and state the condition for instability.

Question 8 - Solution

A thermal-feedback model distinguishes stable and unstable responses

(a) Method and calculation

$$C\dot\theta=(a-h)\theta\Rightarrow\boxed{\theta(t)=\theta_0e^{-(h-a)t/C}}.$$$$\boxed{\tau=C/(h-a)=10/.20=50.0\,\mathrm s}.$$

(b) Interpretation and further calculation

$$\boxed{\theta(100)/\theta_0=e^{-2}=.135335}.$$$$\boxed{h>a:\text{ decay};\quad h=a:\text{ neutral linear response};\quad a>h:\text{ growth}}.$$

Physical check

This scalar thermal model isolates the sign of feedback. It omits the neutron and delayed-precursor dynamics needed for an actual stability analysis.