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Chemical reaction engineering designs and analyses reactors by combining chemical kinetics with mass balance, energy balance and transport. A desired molecular reaction must be achieved in a vessel where mixing, temperature and residence time affect conversion.

For a steady well-mixed flow reactor with constant volume $V$, volumetric flow $Q$ and a first-order consumption rate $kC$, the species balance is $0=Q{C}_{\mathrm{in}}-QC-kVC$. Hence $C/C_{\mathrm{in}}=1/(1+kV/Q)$. This derivation assumes uniform concentration, equal inlet/outlet volumetric flow and constant rate coefficient. An ideal plug flow reactor instead gives $C_{\mathrm{out}}/C_{\mathrm{in}}=e^{-kV/Q}$. Different flow histories can thus yield different conversion at the same mean residence time.

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.

The time a material element spends in a system. In ideal plug flow it equals the volume divided by volumetric flow rate; in a mixed vessel that ratio is a mean and does not guarantee the age of every discharged element.