Mean residence times in steady-flow and some non-flow systems

Abstract

WHEN an incompressible fluid flows at a constant rate through a constant-volume system, the mean residence time of the fluid elements is equal to the ratio of the volume to the volumetric flow-rate and does not depend on the nature of the flow pattern. This possibly surprising result is of value in several branches of science and technology: it was brought to the attention of chemical engineers by Danckwerts, and is associated in the biological literature with Stewart and Hamilton^2,3. Early proofs assume that the fluid, having entered the system, may leave only through the exit and, having left, cannot return. Such a system is said by chemical engineers to be closed. Gibilaro has recently found^4,5 that the mean residence time in a region of a flow system is the ratio of the volume of that region to the volumetric flow-rate through the entire system. The new result differs from the previous one in that fluid may pass through the region several times, or not at all, before leaving the system, and in that the region may be connected to the system in any way. Residence time is counted as the total time a molecule spends in the region before ultimately leaving the system, and the average is taken over all molecules entering the system, including those which never enter the region. Gibilaro also emphasised the diversity of possible applications: the mean residence time is of interest in the study of continuous chemical reactors, respiratory systems, rivers and so on. Washout experiments are used in biological studies⁶ but rarely in engineering. In a washout test, tracer is added continuously to the feed stream until steady conditions prevail. Tracer flow is then stopped; the inventory of tracer in the system recorded as a function of time is the washout function. This is related to the residence-time density and distribution functions of closed systems⁷ and may also be used in studies of non-closed (open) systems⁸. Here we show how the washout-function idea may be used as the basis of a mean-residence-time theorem of the same type as Gibilaro's but of wider compass: it can apply to non-flow, as well as continuous-flow systems.