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The useful mechanism is radioactive decay during retention, not chemical destruction of carbon dioxide. With the time in seconds,Here $a$ is a radioactive activity concentration; the supplied units are per volume rather than per mass. The original plot follows this concentration over four hours.

A practical illustrative choice is a three-hour guaranteed residence time, at which $a/a_0=0.00214175$, giving about a factor $467$ reduction. On a plug-flow basis, the carbon-dioxide-equivalent hold-up volume isUse a contained delay line or a validated retention arrangement that prevents early breakthrough, followed by monitored discharge. A retention bed could instead hold the radioactive carbon dioxide, but its capacity and breakthrough performance would have to be established. Gas retention and monitored discharge are waste-treatment functions, independent of the eventual use of the isotope.^[ref]

The $4.00$ L result is a flow-delay volume, not a fill-and-immediately-dump tank specification. It credits radioactive decay during the guaranteed transit time and no subsequent storage after discharge from the delay system. Gas entering a filling tank near the end of filling has not aged for the entire filling time. For a literal tank filled for $T$ and immediately emptied, the radioactive activity at emptying is insteadThat alternative must not be analysed by multiplying the whole collected inventory by $e^{-\lambda T}$. If “no post filling radioactive decay” is intended to prescribe this literal fill-and-dump arrangement, the three-hour calculation does not establish compliance for it.

The nitrogen fraction is also unspecified. If $y$ is the carbon dioxide volume fraction at the stated volume reference conditions, a gas delay line requires $V_{\mathrm{mixture}}=V_{\mathrm{C}{\mathrm{O}}_2}/y$, not necessarily $4.00$ L of physical vessel volume. The graph alone does not define a unique optimum: a cost objective, operating calendar and residence-time model are needed. Parts (c) and (d) quantify the discharge constraint for the stated guaranteed-delay design.

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.

Temporary retention of radioactive gas so that nuclear radioactive decay lowers the radioactive activity released. A guaranteed minimum residence time, an average residence time and a tank filling time are different design quantities.

The probability density $E(t)$ of times spent in a flowing system. The surviving fraction of a first-order decaying solute is ${\int }_0^{\infty }E(t)e^{-\lambda t}dt$; it is generally not the exponential function evaluated at the mean residence time.

A continuous reactor with uniform contents. At steady state for first-order radioactive decay, $a_{\mathrm{out}}/a_{\mathrm{in}}=1/(1+\lambda V/Q)$; young material can leave before the mean residence time.