# Laboratory experiment: Residence Time Distribution (Cascade)

## Experiment Analysis

### Analysis

The measured residence time function makes it possible to draw conclusions about the technical flow ragime of the reactors that have been studied. Possible variances from the ideal behaviour (dead zones, short circuits) can be identified in this manner.

### Determination of the Mean Residence Time

In the experiment, a concentration c corresponds to a voltage U (conductivity). The mean residence time t can be determined for the continually stirred tank reactor based on the experimental data (volume flow, tank volume, conductivity-time-curve).The (theoretical) mean residence time with the following formula is arrived at:

- $$\tau =\frac{\phantom{\rule{0.7999999999999999ex}{0ex}}{V}_{R}}{\phantom{\rule{0.7999999999999999ex}{0ex}}{\dot{V}}_{A}}\text{\hspace{1em}}\mathrm{where:}\phantom{\rule{0.7999999999999999ex}{0ex}}{V}_{R}=\mathrm{Tank\; Volume,}\phantom{\rule{0.7999999999999999ex}{0ex}}{\dot{V}}_{A}=\mathrm{Volume\; Flow}$$

The actual mean residence time can be determined using different methods. In each case, please keep in mind that the blank value of the measuring cell (base conductivity) is to be subtracted from the measured value before calculation.

### Graphic Determination of the Mean Residence Time

- $$\begin{array}{l}U\left(t\right)=-\phantom{\rule{0.7999999999999999ex}{0ex}}\frac{d{c}_{A}}{{c}_{\mathrm{A,0}}}=\frac{{\dot{V}}_{A}}{{V}_{R}}dt\phantom{\rule{0.7999999999999999ex}{0ex}}\\ \mathrm{Is\; as\; follows\; after\; integration:}\frac{{c}_{A}}{\phantom{\rule{0.7999999999999999ex}{0ex}}{c}_{\mathrm{A,0}}}=\frac{U}{\phantom{\rule{0.7999999999999999ex}{0ex}}{U}_{0}}={e}^{-\phantom{\rule{0.7999999999999999ex}{0ex}}\frac{t}{\tau}}\\ \mathrm{After\; taking\; the\; logarythm\; and\; convertion\; one\; gets:}\phantom{\rule{0.7999999999999999ex}{0ex}}lnU=ln{U}_{0}-\phantom{\rule{0.7999999999999999ex}{0ex}}\frac{1}{\tau}t\\ \mathrm{When\; using\; the\; linear\; equation:}\phantom{\rule{0.7999999999999999ex}{0ex}}y=mx+b\phantom{\rule{0.7999999999999999ex}{0ex}}\mathrm{with:}\phantom{\rule{0.7999999999999999ex}{0ex}}m=-\frac{1}{\tau}\text{\hspace{1em}}\to \text{\hspace{1em}}\tau =-\frac{1}{m}\phantom{\rule{0.7999999999999999ex}{0ex}},\phantom{\rule{0.7999999999999999ex}{0ex}}\phantom{\rule{0.7999999999999999ex}{0ex}}b=ln{U}_{0}\end{array}$$

### Determination of the Mean Residence Time Using the Residence Time Function

This procedure is a graphical analysis which can also be used to check the idealness of the corresponding reactor.

Determination of the mean residence time τ:

Testing for Idealness:

### Determination of the Numbers of Equivalent Stir Levels

The experimentally determined residence time function (conductivity-time-curve) of the 3rd and 4th tank of the cascade of continuously stirred tank reactors are used to determine the equivalent stir level number (n). The values derived at are to be compared to the flow reactor. The number shows how many connected stirred tank reactors of equal size have a residence time function analogous to that of the measured ones.

The adaptation can be performed using different methods. In the training exercise the relationship is determined to be ta/tb. The abscissa value of the curve maximum (H=1) is tw.

Determination of the equivalent stir level number:

A mean value is to be determined from the three determined stir levels. This is the equivalent stir level number of the studied tank.

Values for the Auxiliary Function m_{H}=n-1=f(t_{a}/t_{b})H

### Determination of the Summation Curve

The determination curves is determined by integration of the residence time distribution curves. For the first tank and the flow reactor F(t) it can be produced according to the following pattern:

U[V] | t(min) | F(t) |
---|---|---|

1,1 | 0 | 1,1 |

1,4 | 1 | 1,1+1,4 |

1,6 | 2 | 1,1+1,4+1,6 |

The residence time distribution curves that have been determined in this manner are to be illustrated diagrammatically in a graph and the differences between flow reactor and cascade are to be discussed.

Hint: For a number of tanks > 1 a numerical integration is to be performed!