Residence time distribution ($E(t)$)

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.

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.

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.

An ideal flow model in which all fluid elements have the same transit time. For first-order radioactive decay the outlet-to-inlet radioactive activity concentration ratio is $e^{-\lambda \tau }$.