# Laboratory experiment: Residence Time Distribution (Cascade)

## Introduction to the Residence Time Distribution Experiment

### Terms and Definitions

In the experiment, the residence time distribution is to be determined for different reactors. The terms hydrodynamic residence time or mean residence time are of central importance for the understanding of residence time function t. It is defined as the relationship of the reaction volume (of a tank reaktor, of a cascade of a tank reaktor or a flow reactor) to the volume flow of the reaction mixture at the input of the respective reactors:

$\begin{array}{l}\tau =\frac{{V}_{R}}{\stackrel{.}{\phantom{\rule{0.7999999999999999ex}{0ex}}{V}_{A}}}\text{\hspace{1em}}\stackrel{.}{\phantom{\rule{0.7999999999999999ex}{0ex}}{V}_{E}}=\stackrel{.}{{V}_{A}}\text{\hspace{1em}}\mathrm{Volume\; Flow}\phantom{\rule{0.7999999999999999ex}{0ex}}(\mathrm{Input}=\mathrm{Output})\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{V}_{R}\text{\hspace{1em}}\phantom{\rule{0.7999999999999999ex}{0ex}}Reaction\; Volume\; of\; Tank\; or\; Cascade/Reactor\; Volume\; in\; Flow\; Reactor\phantom{\rule{0.7999999999999999ex}{0ex}}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\phantom{\rule{0.7999999999999999ex}{0ex}}\tau \text{\hspace{1em}}\phantom{\rule{0.7999999999999999ex}{0ex}}\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{Hydrodynamic\; or\; Mean\; Residence\; Time}\phantom{\rule{0.7999999999999999ex}{0ex}}\end{array}$

The definition is valid requiring that a density change does not occur during transit through the reactor, meaningthat possible chemical reactions are volume constant and no back-mixing beyond the feed occurs and no product returns into the tank.

Regarding the actual residence time distribution of the molecules or dispersed particles, the mean residence time is of great importance only can only in an ideal plug flow reactor because it is only in the plug flow reactor that all particles are moving through the whole reactor at the same speed. Therefore, the actual residence time in the ideal plug flow reactor is the same for all particles and identical to the mean residence time (assuming volume constancy). In a continuously stirred tank reactor or a cascade of continuously stirred tanks, the residence times of the individual particles are not identical. This is because the residence time of the individual particles that enter the reactor are distributed across a more or less wide residence time distribution in these reactor forms.

### Different Residence Time Definitions

Several possibilities exist to define residence time. The inflow or outflow of the reactor can be chosen as a reference point. However, the outflow is the widespread reference. The individual residence times can be defined as follows:

Residence time distribution E(t) or w(t): fraction of the outflow that resided in the reactor between the times t and t + dt (exit age distribution curve). Residence time summation function F(t) or W(t): fraction of the outflow that resided in the reactor between the times (0 to t). For the case that t=infinite then F(t)=1 which means that no volume element should reside in the reactor infinitely. The residence time summation function can be obtained by integrating the residence time spectrum. Internal age distribution I(t): portion of the reactor contents (not the outflow) that resided in the reactor between the times t and t + dt.

$\begin{array}{l}\mathrm{Residence\; Time\; Function}\phantom{\rule{0.7999999999999999ex}{0ex}}E(t)\phantom{\rule{0.7999999999999999ex}{0ex}}\text{\hspace{1em}}\underset{0}{\overset{\infty}{\int}}\phantom{\rule{0.7999999999999999ex}{0ex}}E\left(t\right)\phantom{\rule{0.7999999999999999ex}{0ex}}dt=\underset{0}{\overset{\infty}{\int}}\phantom{\rule{0.7999999999999999ex}{0ex}}I(t)\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{dt}=1\\ \mathrm{Sum\; Function}\phantom{\rule{0.7999999999999999ex}{0ex}}F(t)\text{\hspace{1em}}\text{\hspace{1em}}F(t)=\underset{0}{\overset{t}{\int}}E\left({t}^{\prime}\right)\phantom{\rule{0.7999999999999999ex}{0ex}}d{t}^{\prime}\phantom{\rule{0.7999999999999999ex}{0ex}}\phantom{\rule{0.7999999999999999ex}{0ex}},\phantom{\rule{0.7999999999999999ex}{0ex}}\phantom{\rule{0.7999999999999999ex}{0ex}}E(t)=\frac{dF\left(t\right)}{dt}\\ \mathrm{Mean}\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{Residence\; Time}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\tau =\underset{0}{\overset{\infty}{\int}}t\cdot E\left(t\right)dt\end{array}$

The mean residence time t is based on the definition of the residence time distribution. In order to compare reactors of differing size and differing throughput, the residence time distribution will be referred to the standardized mean residence time. A dimensionless standardized time is usually introduced in this case.

$\Theta =\frac{t}{\tau}\text{\hspace{1em}}t=\mathrm{Measured}\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{Time}\text{\hspace{1em}}\tau =\mathrm{Mean}\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{Residence\; Time}$

### Further Readings

Denbigh, K.G.;Turner, J.C.R. ; Einführung in die Reaktionstechnik, Verlag Chemie, Weinheim 1973 Baerns, M.; Hofmann, A.; Renken, A.; Chemische Reaktionstechnik, 2. durchgesehene Auflage, G. Thieme Verlag Stuttgart, New York 1992 Patat, Kirchner; Praktikum der Technischen Chemie; 4.Auflage; W.de Gruyter Berlin, New York 1986, S.185/186 Pippel, W., Iseke, K.; Technisch-chemisches Praktikum, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig 1977