Residence-time distribution step input

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

Reactive Tracer. If the tracer is reactive, the measured concentrations reflect both mixing characteristics and decay of the tracer. Therefore, the data must be adjusted, such that the residence time distribution within the reactor can be obtained. Example 19-3 illustrates how to adjust the data following a step input of a reactive tracer. 

Thus we see that the stimulus-response technique using a step or pulse input function provides a convenient experimental technique for finding the age distribution of the contents and the residence-time distribution of material passing through a closed vessel. 

In addition, several other two-parameter residence time distributions have been formulated [24]. With these or any other residence time distribution, the specific breakage rate, and the progeny array, the mill output can be determined ftnm the mill input thus the comminution step is mathematically described. 

F Residence time distribution function for step change input... 

Figure 4. Transient responses to slug (left) and step (right) inputs of reactant for first order reaction and selected kr values. The flow has the same residence time distribution as two perfectly mixed vessels in sequence. For slug input, c0 = m /V.

The experimental measurement and typical results for different residence time distributions in a continuous reactor are summarized in Fig. 3.4. The same arrangements used for determining the mixing time are appropriate for determining the residence time distribution in a reactor. A signal in the form of a pulse or step function or a periodic function is formed at the input, and the response is measured at the output. 

F(t) is a probability distribution which can be obtained directly from measurements of the system s response in the outflow to a step-up tracer input in the inflow. Consider that at time t = 0 we start introducing a red dye at the entrance of the vessel into a steady flow rate Q of white carrier fluid. The concentration of the red dye in the inlet flow is C. At the outlet we monitor the concentration of the red dye, C(t . If our system is closed, i.e. if every molecule of dye can have only one entry and exit from the system (which is equivalent to asserting that input and output occur by convection only), then QC(t)/QCQ is the residence time distribution of the dye. This is evident since all molecules of the dye appearing at the exit at time t must have entered into the system between time 0 and time t and hence have residence times less than t. Only if our red dye is a perfect tracer, i.e.. if it behaves identically to the white carrier fluid, then we have also obtained the residence time distribution for the carrier fluid and F(t) = C(t)/C. To prove that the tracer behaves ideally and that the F curve is obtained, the experiment should be repeated at different levels of C. The ratio C(t)/C at a given time should be invariant to C, i.e. the tracer response and tracer input must be linearly related. If this is not the case, then C(t)/CQ is only the step response for the tracer, which includes some nonlinear effects of tracer interactions in the system, and which does not represent the true residence time distribution for the system. 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a dynamic model of the system. Table 1-1 lists the types of RTDs that can be measured using tracer experiments. The simplest case is a negative step change. Suppose that an inert tracer has been fed to the system for an extended period, giving Ci = Cout = Q for t < 0. At time t = 0, the tracer supply is suddenly stopped so that Cm = 0 for t > 0. Then the tracer concentration at the reactor outlet will decrease with time, eventually approaching zero as the tracer is washed out of the system. This response to a negative step change defines the washout function, W(t). The responses to other standard inputs are shown in Table 1-1. Relationships between the various functions are shown in Table 1-2. 

This effect can be forecast on the basis of the retention time distribution function in continuous tank reactors, which represents the simplest approach to the analysis of reactor dynamics. In its cumulative form, this function represents, for any time t, the fraction of the exit volumetric flow rate characterized by a residence time smaller than t and can be measured experimentally by submitting the reactor to a step forcing input in the entering stream. Whereas for the ideal tank reactor, the following... 

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