A Natural Nuclear Reactor — Question 6

Question 6

Water feedback can produce a repeating heating and cooling cycle

A schematic water-feedback model heats rock from temperature excess 10 K to 40 K at net rate 20 kW with heat capacity 1.00 MJ/K. Heating then switches off. Cooling follows $\theta=40e^{-t/(1000\,\mathrm s)}$ K until 10 K, when heating resumes. These are illustrative parameters.

1. Find heating and cooling durations.

2. Find the fraction of each cycle spent heating, and identify the negative-feedback sequence.

Question 6 - Solution

Water feedback can produce a repeating heating and cooling cycle

(a) Method and calculation

$$t_h=C(40-10)/P\Rightarrow\boxed{t_h=10^6(30)/(2.00\times10^4)=1500\,\mathrm s}.$$$$10=40e^{-t_c/1000}\Rightarrow\boxed{t_c=1000\ln4=1386.29\,\mathrm s}.$$

(b) Interpretation and further calculation

$$\boxed{d=\frac{t_h}{t_h+t_c}=.51970}.$$$$\boxed{\text{heating}\to\text{water loss}\to\text{reduced moderation}\to\text{less heating}}.$$

Physical check

The model illustrates possible water-mediated cycling, not measured Oklo timing. The feedback sign depends on the system and is supplied here.