The Nuclear Reactor — Question 7

Question 7

Residual heat requires cooling even after fission power is removed

After shutdown a supplied residual-heat model is $P(t)=P_0e^{-t/\tau}$ with $P_0=2.00$ MW and $\tau=600$ s. If cooling is absent, all heat enters a body with constant heat capacity $C=20.0$ MJ/K. Start with zero temperature rise.

1. Derive the temperature rise as a function of time.

2. Calculate the rise after one time constant and its limiting value.

Question 7 - Solution

Residual heat requires cooling even after fission power is removed

(a) Method and calculation

$$C\frac{d\Delta T}{dt}=P_0e^{-t/\tau}\Rightarrow C\Delta T=P_0\tau(1-e^{-t/\tau}).$$$$\boxed{\Delta T(t)=60.0(1-e^{-t/600})\,\mathrm K}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta T(600)=60.0(1-e^{-1})=37.927\,\mathrm K}.$$$$\boxed{\Delta T(\infty)=P_0\tau/C=60.0\,\mathrm K}.$$

Physical check

Zero ongoing fission does not mean zero heat production. This single-exponential model is supplied for illustration, not as a real shutdown heat prediction.