Age distribution internal

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

For the perfeetly mixed CSTR, the internal age distribution funetion is... 

Equation 8-13 shows that in a perfeetly mixed vessel, the internal and exit eonditions are identieal. Figure 8-4 illustrates the internal age distribution eharaeteristies. 

Table 8-4 shows tlie details of tlie suimuation, whieh is perfonued using the trapezoidal rule. The internal age distribution 1(6) is obtained from... 

Table 8-5 shows the details of the summation, whieh is performed using tlie trapezoidal rule. The internal age distribution 1(6) (Table 8-6) is obtained from 1(6) = 1 - E(6) = 1 - LE56. An Exeel spreadsheet (Example8-2.xls) was developed and Eigure 8-18 shows plots of E(6), E(6), and 1(6) versus 6. 

At steady state, the PDF (and thus the RTD function) will be independent of time. Moreover, the internal-age distribution at a point x inside the reactor is just I(a x, t) = fr(a x, t). For a statistically homogeneous reactor (i.e., a CSTR), the PDF is independent of position, and hence the steady-state internal-age distribution 1(a) will be independent of time and position. 

Where time rather than reduced time is used let the internal age distribution function be 1(0- Then... 

In a manner similar to the internal age distribution function, let E be the measure of the distribution of ages of all elements of the fluid stream leaving a vessel. Thus E is a measure of the distribution of residence times of the fluid within the vessel. Again the age is measured from the time that the fluid elements enter the vessel. Let E be deflned in such a way that E dd is the fraction of material in the exit stream which has an age between 6 and 6 -I- dO. Referring to Fig. 4, the area under the E vs. 6 curve is... 

Internal circulation patterns and turbulence. Sasakura et al. (138) and Rachez et al. (139) investigated internal age distributions by means of tracers and proposed representative models of the circulation pattern. Bryant and coworkers (135, 140, 141) and Reuss studied more specifically circulation times in large stirred tanks by using radio flow-followers (aiming at applications for fermenters > 0.5 m ). They showed that circulation times tQ were log-normally distributed ... 

Although this section is not a prerequisite to the remaiiting sections, the internal-age distribution is introduced here because of its close analogy to the external-age distribution. We shall let a represent the-age of a molecule inside the reactor. The internal-age distribution function I (a) is a function such that /(a) Aa is the fi action of material inside the reactor that has been inside the reactor for a period of time between a and a + Aa. It may be contrasted with E(oi) Aa, which is used to represent the material Eeaving the reactor that has spent a time between a and a Aa m the reaction zone /(a) characterizes the time the material has been (and still is) in the reactor at a particular time. The function (a) is viewed outside the reactor and /(a) is viewed inside the reactor. In unsteady-state problems it can be important to know what the particular state of a reaction mixture is, and /(a) supplies this information. For example, in a catalytic reaction using a catalyst whose activity decays with time, the age distribution of the catalyst in the reactor is of importance and the intemal-age distribution can be of use in modeling the reactor. 

As a brief exercise, the internal-age distribution of a perfectly mixed CSTR will be calculated. Equation ( 13-27) gives the RTD of the reactor, which upon substitution into Equation (13-32) gives... 

Thus the internal-age distribution of a perfectly mixed CSTR is identical to the exit-age distribution, or RTD, because the composition of the effluent is identical to the composition of the material anywhere within the CSTR when it is perfectly mixed. 

The internal-age distribution, [/(a) da], gives the fraction of material inside the reactor that has been inside between a time a and a time (a "I" da). 

The age of the fluid particle located at a specific point in the reactor is now treated as a random variable. In the CRE literature, [a ( ) is known as the internal-age distribution function. 

The intemai-age distribution is discus.sed further on the Professional Reference Shelf where the following relationships between the cumulative internal age distribution /(a) and the cumulative external age distribution F(a)... 

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