Danckwerts age distribution function

S stoichiometric coefficient also parameter in Danckwerts age distribution function hr" s ... 

Danckwerts age distribution function also Laplace transform variable pure error variance... 

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. 

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. 

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 ... 

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. 

Surface renewal theory (Danckwerts, 1951) proposes that there is an infinite range of ages for elements of the surface and the surface age distribution function (t) can be expressed as... 

In the Danckwerts 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 s st. 

The original Danckwerts model has later been extended to relate the renewal rate s to many flow parameters, and to account for the existence of the micro scale flow of the fluid within the individual eddies. Further modifications relate to the fact that not all the penetrating eddies reach the whole way to the interface. Many model extensions have thus been developed based on the basic surface-renewal concept. A review of these models is given by Sideman and Pinczewski [135]. The various extensions are motivated by the inherent assumptions regarding, the governing equations, the boundary conditions, the age distribution function and/or the mean contact time. 

With Higbie s distribution function all elements at the surface have the same age. Such a situation could be encountered with a quiescent liquid or with completely laminar flow. In that case is simply given by Eq. 6.4.b-4 in which t takes a definite value I, the uniform time of exposure. With Danckwert s age distribution function Eq. 5.4-1 the average rate of absorption per unit surface, is given by ... 

DatKkwerts theory cf penetration with random surface renewal modifies diis picture by proposing an infinite range of ages for elements of the surface. The probability of an element of surface being rqrlaced by a fresh eddy is considered to be independem of the age of that element. Danckwerts introduced this modification by defining a surfiKe age distribution function, (r), such that the fraction of surface with ages between r and r -l- is 6(r)dir. If die probability of replacement of a surface element is independem of its age, Danckwetts showed that k = (sD), where s is the fractional rate of surface renewal. 

The classical Danckwerts surface-renewal model is analogous to the penetration theory. The improvement is in the view of the eddy replacement process. Instead of Higbies assumption that all elements have the same recidence time at the interface, Danckwerts [29] proposed to use an averaged exposure time determined from a postulated time distribution. The recidence time distribution of the surface elements is described by a statistical distribution function E(t), defined so that E(t)d,t is the fraction of the interface elements with age between t and t + dt. The rest of the formulation procedure is similar to that of the penetration model. 

In surface renewal models the liquid surface is assumed to consist of a mosaic of elements with different age at the surface. The rate of absorption at the surface is then an average of the rates of absorption in each element, weighted with respect to a distribution function (t)—see Eq. 6.2-5. Under this heading of surface renewal theory we will also occasionally mention results of Higbie s [23] so-called penetration-theory, which can be considered as a special case in which every element is exposed to the gas for the same length of time before being replaced. The main emphasis of this section is on the Danckwerts [24] approach using the distribution function for completely random replacement of surface elements ... 

This type of curve, then, has an ordinate that gives the fraction of fluid that has a certain residence time, which is plotted on the abscissa. In more formal terms, the curve defines the residence time distribution or exit age distribution. The exact definition uses the common symbol (0) for the exit age-distribution frequency function as defined by Danckwerts [6] (see Himmelblau and BischofT [4] for more details) ... 

The age of a fluid element is defined as the time it has resided within the reactor. The concept of a fluid element being a small volume relative to the size of the reactor yet sufficiently large to exhibit continuous properties such as density and concentration was first put forth by Danckwerts in 1953. Consider the following experiment a tracer (could be a particular chemical or radioactive species) is injected into a reactor, and the outlet stream is monitored as a function of time. The results of these experiments for an ideal PFR and CSTR are illustrated in Figure 8.2.1. If an impulse is injected into a PFR, an impulse will appear in the outlet because there is no fluid mixing. The pulse will appear at a time ti = to + t, where t is the space time (r = V/v). However, with the CSTR, the pulse emerges as an exponential decay in tracer concentration, since there is an exponential distribution in residence times [see Equation (3.3.11)]. For all nonideal reactors, the results must lie between these two limiting cases. 

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