Residence distribution functions

The residence distribution function/(t) dt, which was defined as the fraction of exiting flow rate with a residence time between t and t + dt, is exactly the fraction of flow rate between r and r + dr. Thus... 

The fraction of the fluid elements with a residence time of less than t is given by the cumulative residence distribution function F(t) given by ... 

It follows that the residence distribution function can be modified by the conversion factor [1 - exp(-fc r)] to obtain the vertical readout for any given residence time. For peak height and delta injection (5, Vr) Reijn et al. [554] found for the SBSR reactor ... 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

THE RESIDENCE TIME DISTRIBUTION FUNCTIONS AND THEIR RELATIONSHIPS ... 

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

Figure 8-3. Cumulative residence time distribution function.

Figure 8-17. Residence time distribution functions E(0), F(0), and i(0) versus 0.

Residence time distribution functions These give information about the fraction of the fluid that spends a certain time in a process vessel. 

As we have said, the key to the analysis of asystemlike this one is tohave a function that approximates to the actual residence time distribution. The tracer experiment is used to find that distribution function,butwewillworkfroman assumed function to the tracer concentration-timecurvetoseewhattheexperimentaloutcomemightlooklike. 

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

The monomer conversion in this seeded polymerization system is independent of the degree of segregation as long as an exponential residence time distribution function is maintained. 

Positive Step Changes and the Cumulative Distribution. Residence time distributions can also be measured by applying a positive step change to the inlet of the reactor Cm = Cout = 0 for r<0 and C = Co for r>0. Then the outlet response, F i) = CouMICq, gives the cumulative distribution function. ... 

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

Material flowing at a position less than r has a residence time less than t because the velocity will be higher closer to the centerline. Thus, F(r) = F t) gives the fraction of material leaving the reactor with a residence time less that t where Equation (15.31) relates to r to t. F i) satisfies the definition. Equation (15.3), of a cumulative distribution function. Integrate Equation (15.30) to get F r). Then solve Equation (15.31) for r and substitute the result to replace r with t. When the velocity profile is parabolic, the equations become... 

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. 

Fig. 5.1.7 (a) Propagators in both a clean square capillary (dotted) and for flow around a biofilm structure (dashed) for an observation time, A = 15 ms. (b) Residence time distribution functions calculated from the propagator data shown in (a). The induction of a high... 

Molecular weight distribution function for the case where the length of the growth stage is short compared to the residence time in reactor. (Reprinted with permission from Chemical Reactor Theory, by K. G. Denbigh and J. C. R. Turner. Copyright 1971 by Cambridge University Press.)... 

Except for the case of an ideal plug flow reactor, different fluid elements will take different lengths of time to flow through a chemical reactor. In order to be able to predict the behavior of a given piece of equipment as a chemical reactor, one must be able to determine how long different fluid elements remain in the reactor. One does this by measuring the response of the effluent stream to changes in the concentration of inert species in the feed stream—the so-called stimulus-response technique. In this section we will discuss the analytical form in which the distribution of residence times is cast, derive relationships of this type for various reactor models, and illustrate how experimental data are treated in order to determine the distribution function. 

Since F(t + dt) represents the volume fraction of the fluid having a residence time less than t + dt, and F(t) represents that having a residence time less than r, the differential of F(t dF(t will be the volume fraction of the effluent stream having a residence time between t and t + dt. Hence dF(t) is known as the residence time distribution function. From the principles of probability the average residence time (t) of a fluid element is given by... 

Experimental Determination of Residence Time Distribution Functions... 

For a few highly idealized systems, the residence time distribution function can be determined a priori without the need for experimental work. These systems include our two idealized flow reactors—the plug flow reactor and the continuous stirred tank reactor—and the tubular laminar flow reactor. The F(t) and response curves for each of these three types of well-characterized flow patterns will be developed in turn. 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

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