Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2021 / Module NE4 / 2 / d / Solution

For the ideal guaranteed-delay model, the total untreated yearly input at $250$ operating days isSince the discharge is a decreasing function of the residence time, the minimum is obtained at equality:This is $7.642$ half-lives and corresponds to $3.444$ L of carbon-dioxide-equivalent delay volume. The three-hour choice provides margin; operating every day instead would require at least $166.22$ minutes under the same assumptions.

This minimum is conditional on a guaranteed residence time. More generally the surviving fraction iswhere $E(t)$ is the residence time distribution. Because the exponential function is convex function, a spread of transit times gives more surviving radioactive activity than plug flow at the same mean time. A perfectly mixed continuously flowing vessel has a steady-state survival fraction $1/(1+\lambda V/Q)$, not $e^{-\lambda V/Q}$. Thus no unique minimum tank filling time follows without specifying the filling, retention and emptying schedule. The boxed minimum applies to the explicitly specified delay system, and must not be represented as approval of an unspecified tank.