Theory
The theory of residence time distributions generally begins with three assumptions:
- the reactor is at steady-state,
- transports at the inlet and the outlet takes place only by advection, and
- the fluid is incompressible.
The incompressibility assumption is not required, but compressible flows are more difficult to work with and less common in chemical processes. A further level of complexity is required for multi-phase reactors, where a separate RTD will describe the flow of each phase, for example bubbling air through a liquid.
The distribution of residence times is represented by an exit age distribution, . The function has the units of time-1 and is defined such that
- .
The fraction of the fluid that spends a given duration, inside the reactor is given by the value of .
The fraction of the fluid that leaves the reactor with an age less than is
- .
The fraction of the fluid that leaves the reactor with an age greater than is
- .
The average residence time is given by the first moment of the age distribution:
- .
If there are no dead, or stagnant, zones within the reactor then will be equal to, the residence time calculated from the total reactor volume and the volumetric flow rate of the fluid:
- .
The higher order central moments can provide significant information about the behavior of the function . For example, the second central moment indicates the variance, the degree of dispersion around the mean.
The third central moment indicates the skewness of the RTD and the fourth central moment indicates the kurtosis (the "peakedness").
One can also define an internal age distribution that describes the reactor contents. This function has a similar definition as : the fraction of fluid within the reactor with an age of is . As shown by Danckwerts, the relation between and can be found from the mass balance:
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