Cumulative Residence-Time Distribution Function

The cumulative residence-time distribution function F(t) is defined as the fraction of exit stream that is of age 0 to t (i.e., of age t) it is also the probability that a fluid element that entered at t = 0 has left at or by time t. Since it is defined as a fraction, it is dimensionless. Furthermore, since F(O) = 0, that is, no fluid (of age 0) leaves the vessel before time 0 and F( ) = 1, that is, all fluid leaving the vessel is of age 0 to or all fluid entering at time 0 has left by time then 

From one point of view, the definition of Fit) corresponds to the probability that a fluid element has left the vessel between time 0 and t. This is equivalent to the area under the E(t) curve given by equation 13.3-2. That is, J  

Both E and F are RTD functions. Although not apparent here, each corresponds to a different way of experimentally investigating RTD for arbitrary flow (Chapter 19 see also problem 13-1), and hence each is important. 

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In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Figure 8-3. Cumulative residence time distribution function.

The cumulative residence time distribution function for a perfectly mixed vessel is ... 

The derivation of the residence time behavior of the single streamlines now allows the formulation of the cumulative residence-time distribution function F(t) according to the following formula ... 

A graphical representation of the cumulative residence time distribution function is given in Figure 4.97 for a structured well, a laminar flow reactor and an ideal plug flow reactor assuming the same average residence time and mean velocity in each reactor. 

Figure 4.97 Calculated cumulative residence time distribution function for a multi-channel well, a laminar flow reactor and a plug flow reactor [147] (by courtesy of VDI-Verlag GmbH).

In Section 11.1.3.2 we considered a model of reactor performance in which the actual reactor is simulated by a cascade of equal-sized continuous-flow stirred-tank reactors. We indicated how the cumulative residence-time distribution function can be used to determine the number of tanks that best model the tracer measurement data. Once this parameter has been determined, the techniques discussed in Section 8.3.2 can be used to determine the effluent conversion level. 

The cumulative residence time distribution function [F(f)] for a cascade consisting of two nonidentical stirred-tank reactors can be represented by a mathematical expression of the form... 

Tracer studies indicate that at an inlet volumetric flow rate of 25 m / min, the cumulative residence time distribution function for a reactor network is... 

Therefore, the cumulative residence time distribution function is determined by measuring the concentration versus time at the reactor outlet. The function is represented graphically as follows ... 

To determine the function F(t) from experimental data, we use the property G, as shown in Chapter 1. If G is any property (conductivity, ionization, wavelength, etc.) proportional to the concentration, in which Gi is the magnitude at the inlet and Gi at the outlet, then the cumulative residence time distribution function, which remained in the reactor at an instant shorter than t, will be ... 

The reactor volume is 1 and the feed volumetric flow is 0.2 m /min. Determine the cumulative residence time distribution function (F) and the mean residence time. 

The variation of the cumulative residence time distribution function is represented by a Gauss curve, indicating the concentration variation C t) as a function of time according to Figure 14.5 ... 

FIGURE 6,15 Cumulative residence time distribution function for Poiseuille flow of a power-law fluid in a circular pipe (CPPF). 

FIGURE 8 9 Cumulative residence time distribution function versus reduced time for flow in an extruder, plug flow, flow of a Newtonian fluid in a pipe, and a continuously stirred tank (CST) vessel. 

FIGURE 8.30 Cumulative residence time distribution function versus reduced time in an extender for fluids with various values of the power-law index and compared to values for plug flow and complete back-mixing continnons stirred tank reactor (Bigg and Middlemann, 1974). 

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