Residence time

Term in fluid dynamics

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.^[1]

Distributions

Control volume with incoming flow rate f_in, outgoing flow rate f_out and amount stored m

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 ${\displaystyle \tau }$ in the set of all the particles that are located inside the control volume at time ${\displaystyle t}$ is quantified by means of the (internal) age distribution ${\displaystyle I}$.^[2]

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 ${\displaystyle \tau }$ in the set of all the particles that are leaving the control volume at time ${\displaystyle t}$ is quantified by means of the residence time distribution, also known as exit age distribution ${\displaystyle E}$.^[2]

Both distributions are assumed to be positive and to have unitary integral along the age:^[2]

${\displaystyle \int _{0}^{\infty }E(\tau ,t)\,d\tau =\int _{0}^{\infty }I(\tau ,t)\,d\tau =1}$

In the case of steady flow, the distributions are assumed to be independent of time, that is ${\displaystyle \partial _{t}E=\partial _{t}I=0\;\forall t}$, 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^[3]) and is conservative, then the exit age distribution and the internal age distribution can be related one to the other:^[2]

${\displaystyle \left.{\begin{aligned}{\frac {\partial I}{\partial t}}={\frac {dm}{dt}}=0&\\[4pt]f_{\text{in}}=f_{\text{out}}=f&\end{aligned}}\ \right\}\implies fE=-m{\frac {\partial I}{\partial \tau }}}$

Distributions other than ${\displaystyle E}$ and ${\displaystyle I}$ can be usually traced back to them. For example, the fraction of particles leaving the control volume at time ${\displaystyle t}$ with an age greater or equal than ${\displaystyle \tau }$ is quantified by means of the washout function ${\displaystyle W}$, that is the complementary to one of the cumulative exit age distribution:

${\displaystyle W(\tau ,t)=1-\int _{0}^{\tau }E(s,t)\,ds}$

Averages

Mean age and mean residence time

The mean age of all the particles inside the control volume at time t is the first moment of the age distribution:^[2][3]

${\displaystyle \tau _{a}(t)=\int _{0}^{\infty }\tau I(\tau ,t)\,d\tau }$

The mean residence time or mean transit time, that is the mean age of all the particles leaving the control volume at time t, is the first moment of the residence time distribution:^[2][3]

${\displaystyle \tau _{t}(t)=\int _{0}^{\infty }\tau E(\tau ,t)\,d\tau .}$

This drinking trough has ${\displaystyle \tau _{a}>\tau _{t}}$

The mean age and the mean transit time generally have different values, even in stationary conditions:^[2]

• ${\displaystyle \tau _{a}<\tau _{t}}$: examples include water in a lake with the inlet and outlet on opposite sides and radioactive material introduced high in the stratosphere by a nuclear bomb test and filtering down to the troposphere.
• ${\displaystyle \tau _{a}=\tau _{t}}$: E and I are exponential distributions. Examples include radioactive decay and first order chemical reactions (where the reaction rate is proportional to the amount of reactant).
• ${\displaystyle \tau _{a}>\tau _{t}}$: most of the particles entering the control volume pass through quickly, but most of the particles contained in the control volume pass through slowly. Examples include water in a lake with the inlet and outlet that are close together and water vapor rising from the ocean surface, which for the most part returns quickly to the ocean, while for the rest is retained in the atmosphere and returns much later in the form of rain.^[2]

Turnover time

If the flow is steady and conservative, the mean residence time equals the ratio between the amount of fluid contained in the control volume and the flow rate through it:^[2]

${\displaystyle \left.{\begin{aligned}{\frac {\partial I}{\partial t}}={\frac {dm}{dt}}=0&\\f_{\text{in}}=f_{\text{out}}=f&\end{aligned}}\ \right\}\implies \tau _{t}={\frac {m}{f}}}$

This ratio is commonly known as the turnover time or flushing time.^[4] When applied to liquids, it is also known as the hydraulic retention time (HRT), hydraulic residence time or hydraulic detention time.^[5] In the field of chemical engineering this is also known as space time.^[6]

The residence time of a specific compound in a mixture equals the turnover time (that of the compound, as well as that of the mixture) only if the compound does not take part in any chemical reaction (otherwise its flow is not conservative) and its concentration is uniform.^[3]

Although the equivalence between the residence time and the ratio ${\displaystyle m/f}$ does not hold if the flow is not stationary or it is not conservative, it does hold on average if the flow is steady and conservative on average, and not necessarily at any instant. Under such conditions, which are common in queueing theory and supply chain management, the relation is known as Little's Law.

Simple flow models

Design equations are equations relating the space time to the fractional conversion and other properties of the reactor. Different design equations have been derived for different types of the reactor and depending on the reactor the equation more or less resemble that describing the average residence time. Often design equations are used to minimize the reactor volume or volumetric flow rate required to operate a reactor.^[7]

Plug flow reactor

In an ideal plug flow reactor (PFR) the fluid particles leave in the same order they arrived, not mixing with those in front and behind. Therefore, the particles entering at time t will exit at time t + T, all spending a time T inside the reactor. The residence time distribution will be then a Dirac delta function delayed by T: