The Exit-Age Distribution Function, E t

E t)dt = fraction of fluid leaving the vessel at time t that was in the vessel for a time between t and t- -dt 

There are several other ways of saying the same thing  

The exit-age distribution function is also known as the external-age distribution function. It is sometimes simply called the residence time distribution function. However, this can cause some confusion. As we shall see shortly, the exit-age distribution function is not the only function that is used to characterize the distribution of residence times. 

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The exit-age distribution function E(t) is approximately obtained by following the experimentally often used method of backward differencing ... 

The exit age distribution function E(t) is obtained from outside the vessel while the internal age distribution function I(t) is obtained from inside the vessel. I(t) can be represented in terms of the RTD or the F-curve as... 

Equation (10-3) permits the exit-age distribution function, E t), to be calculated from the tracer response curve that is measured after a pulse injection of tracer. 

Figure 13.2 Exit-age distribution function E(t) for arbitrary (nonideal) flow showing significance of area under the E(t) curve...

By definition the exit age distribution function E is such that the fraction of the exit stream with residence times between t and t + St is given by Ed/. 

This is defined as the fraction of material in the outlet stream that has been in the system for the period between t and t + dt, and is equal to E(t)dt, where E(t) is called the exit age distribution function of the fluid elements leaving the system. This is expressed as... 

The intemal-age distribution function Z(t) is a measure of the distribution of ages of elements of fluid within a vessel, and not in the exit stream. However, it is defined similarly to E(t) by ... 

Sometimes E t) is called the exit-age distribution function. If we regard the age of an atom as the time it has resided in the reaction environment, then E t) concerns the age distribution of the effluent stream. It is tbe most used of the distribution functions connected with reactor analysis because it characterizes ttie lengths of time various atoms spend at reaction conditions. 

I t) and E t) are the internal- and exit-age distribution functions, respectively. Now let us write a material balance on the tracer for a step change in its concentration from zero to 1 (the arbitrary concentration units will not affect the results here) at t = 0. After time t we have... 

The exit age distribution is expressed by the E fimction, defined such that it represents the age distribution of the material leaving the apparatus at a time between t and t + dt. This distribution can be experimentally determined by admitting a pulse of a tracer at the inlet at time r = 0, measuring the concentration at the exit as a function of time [c(t)] and normalizing the function to an area under the curve of unity ... 

Consider a continuous reactor at steady state. A single reaction A —products is taking place. The exit-age distribution E t) of the reactor is known, and this distribution function describes the behavior of the packets of fluid, as weU as the behavior of the fluid as a whole. 

One method of characterising the residence time distribution is by means of the E-curve or external-age distribution function. This defines the fraction of material in the reactor exit which has spent time between t and t -i- dt in the reactor. The response to a pulse input of tracer in the inlet flow to the reactor gives rise to an outlet response in the form of an E-curve. This is shown below in Fig. 3.20. 

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... 

Given the reaction stoichiometry and rate laws for an isothermal system, a simple representation for targeting of reactor networks is the segregated-flow model (see, e.g., Zwietering, 1959). A schematic of this model is shown in Fig. 2. Here, we assume that only molecules of the same age, t, are perfectly mixed and that molecules of different ages mix only at the reactor exit. The performance of such a model is completely determined by the residence time distribution function,/(f). By finding the optimal/(f) for a specified reactor network objective, one can solve the synthesis problem in the absence of mixing. 

In addition to the F, W and E curves, all of which are based on the sample space (population) of the exiting fluid, one can introduce an internal age density function based on the fluid within the system. I(t)dt is defined as the fraction of fluid elements in the system of age between t and t + dt. The relationships between I, E and other functions are readily obtained and are reported in all standard texts (2-4). I(t) can be obtained directly by injecting a radioactive tracer (of long half life) at time t = 0 and by monitoring the radioactivity of the whole system. Similarly, the internal age distribution (the integral of I(t)) can be obtained by measuring the tracer concentration throughout the system upon step-up tracer injection. 

After studying this chapter the reader will be able to describe the cumulative F(t), external age E(t), and internal age I(t) residence-time distribution functions and to recognize these functions for PFR, CSTR, and laminar flow reactions. The reader will also be able to apply these functions to calculate the conversion and concentrations exiting a reactor using the segregation model and the maximum mixedness model for both single and multiple reactions. 

The residence time distribution (RTD), also referred to as the distribution of ages, is based on the assumption that each element traveling through the column takes a different route and will therefore have a different residence time. Different methods are developed to determine the RTD in a module or in a reactor [190]. The RTD of a chromatographic column is defined by a function E (Figure 3.20), such that E dt is the fraction of material in the exit stream with an age between t and f - - dt. The -curve lies between the extremes of plug flow and continuously stirred tank reactor. The surface below the curve between f = 0 and t = oo has to be equal to unity E t) dt = 1, because all elements that enter the module must also exit the module. 

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