Age-Distribution Functions

The age of an element of fluid is ddined as the time elapsed since it entered the reactor. The concept of a fluid element or point was introduced by Danckwerts [5] to mean a volume small with respect to the reactor vessel size, but still large enough to contain sufiknent molecules so that continuous properties such as density and concentration can be defined. In liquids and gases at not too low a pressure, this is probably reasonable, but in vacuum systems the methods of kinetic theory would have been used, just as in transport phenomena. 

NONIDEAL FLOW PATTERNS AND POPULATION BALANCE MODELS 

The fraction of fluid in the exit stream with age less than 9i is 

It is often convenient to use a dimensionless time of 0 = Ojx, and a corresponding version of the RTD, E(9 ). The relation with (0) is found from the basis that both represent the same physical entity, the fraction of exit fluid with age 0  

It can be seen that this relation is also consistent with the normalization, Eq. 

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AGE DISTRIBUTION FUNCTIONS AND RESIDENCE TIME DISTRIBUTION FUNCTION E(t)... 

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... 

Table 8-1 gives tlie relationships between tlie age distribution functions and Figure 8-6 shows the age distribution functions of ideal reactors. 

Equations 10.142 and 10.143 give the point value of NA at time t. The average values Na can then be obtained by applying the age distribution functions obtained by Higbie and by Danckwerts, respectively, as discussed Section 10.5.2. 

In the Danekwens model of mass transfer it is assumed that the fractional rate of surface renewal s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is = If the fractional rate of surface renewal were proportional to surface age (say s — bt. where b is a constant), show that the surface age distribution function would then assume the form ... 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average, mass Iransfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is VlD/(7rt, where D is the diffusivity. 

If the probability of surface renewal is linearly related to age, as opposed to being constant, obtain the corresponding form of the age distribution function. 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

Danckwerts assumed a random surface renewal process in which the probability of surface renewal is independent of its age. If s is the fraction of the total surface renewed per unit time, obtain the age distribution function for the surface and show that the mean mass transfer rate Na over the whole surface is ... 

In a particular application, it is found that the older surface is renewed more rapidly than the recently formed surface, and that after a lime l/s, the surface renewal rate doubles, that is increases from s to 2s, Obtain the new age distribution function. 

For all likely operating conditions, (ie., for t < X), the appropriate values of the concentration and the polymerization rate constant are the values calculated at t = t ( 2). To prove this, the exit age distribution function for a backmix reactor was used to weight the functions for Cg and kj and the product was integrated over all exit ages (6). It is enlightening at this point to compare equation 18 with one that describes the yield attainable in a typical laboratory semibatch reactor at comparable conditions. ... 

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 chapter begins with a reiteration and extension of terms used, and the types of ideal flow considered. It continues with the characterization of flow in general by age-distribution functions, of which residence-time distributions are one type, and with derivations of these distribution functions for the three types of ideal flow introduced in Chapter 2. It concludes with the development of the segregated-flow model for use in subsequent chapters. 

The characterization of flow by statistical age-distribution functions applies whether the flow is ideal or nonideal. Thus, the discussion in this section applies both in Section 13.4 below for ideal flow, and in Chapter 19 for nonideal flow. 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

The exit-age distribution function is a measure of the distribution of the ages of fluid elements leaving a vessel, and hence is an RTD function. As a function of time, f, it is defined as ... 

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

The exit-age distribution function may also be expressed in terms of dimensionless time 8 defined by... 

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 ... 

The holdback FI is the fraction of fluid within a vessel of age greater than t, the mean residence time. As a fraction, it is dimensionless. It can be obtained from age-distribution functions (see problem 13-4). 

A summary of the relationships among the age-distribution functions is given in Table 13.1. This includes the results relating E(6) and F(6) from Section 13.3.2, together with those for W(6), 1(6), and H (these last provide answers to problems 13-3, -4 and -5(a), (b)). Each row in Table 13.1 relates the function shown in the first column to the others. The means of converting to results in terms of Eft), Fft), etc. is shown in the first footnote to the table. 

In this section, we derive expressions for the age-distribution functions E and F for three types of ideal flow BMF, PF, and LF, in that order. Expressions for the quantities W, I, and H are left to problems at the end of this chapter. The results are collected in Table 13.2 (Section 13.4.4). 

Table 13.2 Summary of age-distribution functions for ideal flow"...

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