User:RockMagnetist (DCO visiting scholar)/Drafts/Residence time

For material flowing through a volume, the residence time is a measure of how much time the matter spends in it. Examples include fluids in a chemical reactor, specific elements in a geochemical reservoir, water in a catchment, bacteria in a culture vessel and drugs in human body. A molecule or small parcel of fluid has a single residence time, but more complex systems have a residence time distribution.

There are at least three time constants that are used to represent a residence time distribution. The turn-over time or flushing time is the ratio of the material in the volume to the rate at which it passes through; the mean age is the mean length of time the material in the reservoir has spent there; and the mean transit time is the mean length of time the material spends in the reservoir.

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
Basic residence time theory treats a system with an input and an output, both of which have flow only in one direction. The system is homogeneous and the substance that is flowing through is conserved (neither created nor destroyed). A small particle entering the system will eventually leave, and the time spent there is its residence time. In a particularly simple model of flow, plug flow, particles that enter at the same time continue to move at the same rate and leave together. In this case, there is only one residence time. Generally, though, their rates vary and there is a distribution of exit times. One measure of this is the washout function $$W(t)$$, the fraction of particles leaving the system after having been there for a time $$t$$ or greater. Its complement, $$F(t) = 1-W(t)$$, is the cumulative distribution function. The differential distribution, also known as the residence time distribution or exit age distribution, is given by
 * $$E(t) = dF(t)/dt.$$

This has the properties of a probability distribution: it is always nonnegative and
 * $$\int_0^\infty E(t)dt = 1.$$

One can also define a density function based on the flux (mass per unit time) out of the system. The transit time function is the the fraction of particles leaving the system that have been in it for up to a given time. It is the integral of a distribution $$I(t)$$. If, in a steady state, the mass in the system is $$M_0$$ and the outgoing flux is $$F_0$$, the distributions are related by
 * $$F_0 I(t) = -M_0 \frac{d E(t)}{dt}.$$

As an illustration, for a human population to be in a steady state, the deaths per year of people older than $$t$$ years (the left hand side of the equation) must be balanced by the number of people per year reaching age $$t$$ (the right hand side).

Some statistical properties of the residence time distribution are frequently used. The mean residence time, or mean age, is given by the first moment of the residence time distribution:
 * $$ \overline{t} = \int_0^\infty t E(t) dt$$,

and the variance is given by
 * $$ \sigma_t^2 = \int_0^\infty \left(t-\overline{t}\right)^2 E(t) dt$$

or by the dimensionless form $$\sigma^2 = \sigma_t^2/\overline{t}^2$$.

Simple models
In an ideal plug flow reactor there is no axial mixing and the fluid elements leave in the same order they arrived. Therefore, fluid entering the reactor at time $$t$$ will exit the reactor at time $$t+\overline{t}$$, where $$\overline{t}$$ is the residence time of the reactor. The fraction $$F(t)$$ leaving is a step function, going from 0 to 1 at time $$\overline{t}$$ The distribution function is therefore a Dirac delta function at $$\overline{t}$$.
 * $$E(t) = \delta(t-\overline{t})\,$$

The mean is $$\overline{t}$$ and the variance is zero.

In an ideal continuous stirred-tank reactor (CSTR), the flow at the inlet is completely and instantly mixed into the bulk of the reactor. The reactor and the outlet fluid have identical, homogeneous compositions at all times. The residence time distribution is exponential:
 * $$E(t) = \frac{1}{\overline{t}} \exp\left(\frac{-t}{\overline{t}}\right).$$

The mean is $$\overline{t}$$ and the variance is 1.

Time constants
The mean residence time is just one of the time constants used to represent the distribution. The mean transit time is the first moment of the transit time distribution:
 * $$ t_t = \int_0^\infty t I(t) dt,$$

and the turnover time is simply the ratio of mass to flux:
 * $$ t_0 = M_0 / F_0.$$

It can be shown that, in a steady state, $$t_t = t_0.$$