Washout Residence-Time Distribution Function

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

FIGURE 15.9 Residence time distribution for laminar flow in a circular tube (a) physical representation b) washout function. 

The completely segregated stirred tank can be modeled as a set of piston flow reactors in parallel, with the lengths of the individual piston flow elements being distributed exponentially. Any residence time distribution can be modeled as piston flow elements in parallel. Simply divide the flow evenly between the elements and then cut the tubes so that they match the shape of the washout function. See Figure 15.12. A reactor modeled in this way is said to be completely segregated. Its outlet concentration is found by averaging the concentrations of the individual PFRs ... 

The washout residence-time distribution function W(t) is defined as the fraction of the exit stream of age s t (and similarly for W(0)). It is also the probability that an element of fluid that entered a vessel at t = 0 has not left at time t. By comparison, F(t) (or F(6)) is the probability that a fluid element has left by time t (or 13) (Section 13.3.2.)... 

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. 

The RTD is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 0 will have a distribution of residence times regardless of whether the reactor is at steady state. The washout function for an unsteady reactor is defined as... 

The ideal cases are the piston flow reactor (PFR), also known as a plug flow reactor, and the continuous flow stirred tank reactor (CSTR). A third kind of ideal reactor, the completely segregated CSTR, has the same distribution of residence times as a normal, perfectly mixed CSTR. The washout function for a CSTR has the simple exponential form... 

A CSTR is said to have an exponential distribution of residence times. The washout function for a PFR is a negative step change occurring at time t ... 

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