The residence time of a fluid parcel is the total time that the parcel has spent inside a control volume (e.g.: a chemical reactor, a lake, a human body). The residence time of a set of parcels is quantified in terms of the frequency distribution of the residence time in the set, which is known as residence time distribution (RTD), or in terms of its average, known as mean residence time. Residence time plays an important role in chemistry and especially in environmental science and pharmacology. Under the name lead time or waiting time it plays a central role respectively in supply chain management and queueing theory, where the material that flows is usually discrete instead of continuous.
History The concept of residence time originated in models of chemical reactors. The first such model was an axial dispersion model by Irving Langmuir in 1908. This received little attention for 45 years; other models were developed such as the plug flow reactor model and the continuous stirred-tank reactor, and the concept of a washout function (representing the response to a sudden change in the input) was introduced. Then, in 1953, Peter Danckwerts resurrected the axial dispersion model and formulated the modern concept of residence time.
Distributions
The time that a particle of fluid has been in a control volume (e.g. a reservoir) is known as its age. In general, each particle has a different age. The frequency of occurrence of the age τ {\displaystyle \tau } in the set of all the particles that are located inside the control volume at time t {\displaystyle t} is quantified by means of the (internal) age distribution I {\displaystyle I} . At the moment a particle leaves the control volume, its age is the total time that the particle has spent inside the control volume, which is known as its residence time. The frequency of occurrence of the age τ {\displaystyle \tau } in the set of all the particles that are leaving the control volume at time t {\displaystyle t} is quantified by means of the residence time distribution, also known as exit age distribution E {\displaystyle E} . Both distributions are positive and have by definition unitary integrals along the age:
∫ 0 ∞ E ( τ , t ) d τ = ∫ 0 ∞ I ( τ , t ) d τ = 1 {\displaystyle \int _{0}^{\infty }E(\tau ,t)\,d\tau =\int _{0}^{\infty }I(\tau ,t)\,d\tau =1}
In the case of steady flow, the distributions are assumed to be independent of time, that is ∂ t E = ∂ t I = 0 ∀ t {\displaystyle \partial _{t}E=\partial _{t}I=0\;\forall t} , which may allow to redefine the distributions as simple functions of the age only. If the flow is steady (but a generalization to non-steady flow is possible) and is conservative, then the exit age distribution and the internal age distribution can be related one to the other:
∂ I ∂ t = d m d t = 0 f in = f out = f } ⟹ f E = − m ∂ I ∂ τ {\displaystyle \left.{\begin{aligned}{\frac {\partial I}{\partial t}}={\frac {dm}{dt}}=0&\\[4pt]f_{\text{in}}=f_{\text{out}}=f&\end{aligned}}\ \right\}\implies fE=-m{\frac {\partial I}{\partial \tau }}}
Distributions other than E {\displaystyle E} and I {\displaystyle I} can be usually traced back to them. For example, the fraction of particles leaving the control volume at time t {\displaystyle t} with an age greater or equal than τ {\displaystyle \tau } is quantified by means of the washout function W {\displaystyle W} , that is the complementary to one of the cumulative exit age distribution:
W ( τ , t ) = 1 − ∫ 0 τ E ( s , t ) d s {\displaystyle W(\tau ,t)=1-\int _{0}^{\tau }E(s,t)\,ds}
Averages
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