The Nuclear Reactor — Question 9

Question 9

A neutron-absorbing product approaches a source-removal equilibrium

A generic absorbing fission product has inventory X obeying $dX/dt=S-(\lambda+a)X$, with constant production S=1200/s, radioactive removal $\lambda=.010\,\mathrm{s^{-1}}$ and neutron-absorption removal $a=.020\,\mathrm{s^{-1}}$. Initially X=0.

1. Derive the equilibrium inventory and the time-dependent approach to it.

2. Find X after one relaxation time and the fractions of equilibrium removals due to decay and absorption.

Question 9 - Solution

A neutron-absorbing product approaches a source-removal equilibrium

(a) Method and calculation

$$\boxed{X_{\mathrm{eq}}=S/(\lambda+a)=1200/.030=40000}.$$$$\boxed{X(t)=X_{\mathrm{eq}}(1-e^{-(\lambda+a)t}),\quad\tau=33.333\,\mathrm s}.$$

(b) Interpretation and further calculation

$$\boxed{X(\tau)=40000(1-e^{-1})=25284.8}.$$$$\boxed{f_{\mathrm{decay}}=\lambda/(\lambda+a)=1/3,\quad f_{\mathrm{absorption}}=2/3}.$$

Physical check

The absorption rate coefficient is held fixed here. In a coupled reactor model it depends on the neutron field, so this solution is conditional.