The Residence-time Distribution Function

The time it takes a molecule to pass through a reactor is called its residence time 6. Two properties of 6 are important the time elapsed since the molecule entered the reactor (its age) and the remaining time it will spend in the reactor (its residual lifetime). We are concerned mainly with the sum of these times, which is 6, but it is important to note that micromixing can occur only between molecules that have the same residual lifetime molecules cannot mix at some point in the reactor and then unmix at a later point in order to have different residual lifetimes. A convenient definition of residence-time distribution function is the fraction J ) of the effluent stream that has a residence time less than 0. None of the fluid can have passed through the reactor in zero time, so / = 0 at 0 = 0. Similarly, none of the fluid can remain in the reactor indefinitely, so that Japproaches 1 as 0 approaches infinity. A plot of J 6) vs 0 has the characteristics shown in Fig. 6-2a. 

Variations in density, such as those due to temperature and pressure gradients, can effect the residence time and are superimposed on effects due to velocity variations and micromixing. We are concerned with micromixing in this chapter, and we shall therefore suppose that the density of each element of fluid remains constant as it passes through the reactor. Under these conditions the mean residence time, averaged for all the elements of fluid, is given by Eq. (3-21) or by 

The shaded area in Fig. 6-2a represents B. The RTD can also be described in terms of the slope of the curve in Fig. 6-2a. This function, J B) = dJ 6)ld9, will have the shape usually associated with distribution curves, as noted in Fig. 6-2b. The quantity J 9) d9 represents the fraction of the effluent stream with a residence time between 9 and 9 -I- d9. Substituting J 9) d9 for dJ 9) in Eq. (6-1) gives an expression for B in terms of J 9), 

CHAPTER 6 deviations FROM IDEAL REACTOR PERFORMANCE 

6-3 Residence-time Distributions from Response Measurements 

---

THE RESIDENCE TIME DISTRIBUTION FUNCTIONS AND THEIR RELATIONSHIPS ... 

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. 

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... 

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. 

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... 

Note that in this case the right side of equation 11.1.68 is zero for t = 0 and unity for t = 00. Figure 11.9 contains several F(t) curves for various values of n. As n increases, the spread in residence time decreases. In the limit, as n approaches infinity the F(t) curve approaches that for an ideal plug flow reactor. If the residence time distribution function given by 11.1.69 is differentiated, one obtains an... 

The model suffers from the fact that it allows only integer values of n and that it may not be possible to obtain a match of the residence time distribution function at both high and low values of F(t) with the same value of n. Buffham and Gibilaro (16) have generalized the model to include noninteger values of n. The technique outlined by these individuals is particularly useful in obtaining better fits of the data for cases where n is less than 5. 

The variance approach may also be used to determine n. From Illustration 11.2 the variance of the response data based on dimensionless time is 30609/(374.4)2, or 0.218. From equation 11.1.76 it is evident that n is 4.59. Thus the results of the two approaches are consistent. However, a comparison of the F(t) curves for n = 4 and n = 5 with the experimental data indicates that these approaches do not provide very good representations of the data. For the reactor network in question it is difficult to model the residence time distribution function in terms of a single parameter. This is one of the potential difficulties inherent in using such simple models of reactor behavior. For more advanced methods of modeling residence time effects, consult the review article by Levenspiel and Bischoff (3) and textbooks written by these authors (2, 4). 

In Section 11.1.3.2 we considered a model of reactor performance in which the actual reactor is simulated by a cascade of equal-sized continuous stirred tank reactors operating in series. We indicated how the residence time distribution function can be used to determine the number of tanks that best model the tracer measurement data. Once this parameter has been determined, the techniques discussed in Section 8.3.2 can be used to determine the effluent conversion level. 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

In a laminar flow reactor (LFR), we assume that one-dimensional laminar flow (LF) prevails there is no mixing in the (axial) direction of flow (a characteristic of tubular flow) and also no mixing in the radial direction in a cylindrical vessel. We assume LF exists between the inlet and outlet of such a vessel, which is otherwise a closed vessel (Section 13.2.4). These and other features of LF are described in Section 2.5, and illustrated in Figure 2.5. The residence-time distribution functions E(B) and F(B) for LF are derived in Section 13.4.3, and the results are summarized in Table 13.2. 

A system of N continuous stirred-tank reactors is used to carry out a first-order isothermal reaction. A simulated pulse tracer experiment can be made on the reactor system, and the results can be used to evaluate the steady state conversion from the residence time distribution function (E-curve). A comparison can be made between reactor performance and that calculated from the simulated tracer data. 

The degree of conversion inside this volume is constant, but the MWD function qw(n, r), where n is the degree of polymerization, depends on r. This is a reflection of different reaction time in the various layers of the polymer. The residence time distribution function f(r) for the reactive mass in a reactor is determined from rheokinetic considerations, while the MWD for each microvolume qw(n,t) is found for various times t from purely kinetic arguments. The values t and r in the expressions for qw are related to each other via the radial distribution of axial velocity. 

At the instant r, the residence times of all the particles of the tracer A must be < f while the particles of B consist of two groups in the first group all the particles are fed at instant f0 = 0 or later and so also have residence times < t while the particles in the second group were fed into the device before tQ = 0 and so have residence times > t. If the particles in the first group are denoted by the superscript then, from the definition of the residence time distribution function, F, we have... 

On the other hand, one of the essential conditions for correct measurement of RTD is that the tracer particles have the properties, including RTD characteristics, very close to those of the process particles. This implies that, in the time domain of t > 0, the residence time distribution functions of particles A and B in the same device should be approximately equal to each other, i.e. 

According to the definition of the F-function, the residence time distribution functions of A and B for the case under consideration can be expressed by the corresponding ratios of the amounts of particles coming out from the device to those inputted in the time interval from 0 to t, i.e. 

This set of equations can be used to determine the residence time distribution function of the particles in the equipment, F(r), according to the data measured for... 

In all the just-mentioned examples, quantitative prediction and design require the detailed knowledge of the residence time distribution functions. Moreover, in normal operation, the time needed to purge a system, or to switch materials, is also determined by the nature of this function. Therefore the calculation and measurement of RTD functions in processing equipment have an important role in design and operation. 

If we accept the premise that the total strain is a key variable in the quality of laminar mixing, we are immediately faced with the problem that in most industrial mixers, and in processing equipment in general, different fluid particles experience different strains. This is true for both batch and continuous mixers. In the former, the different strain histories are due to the different paths the fluid particles follow in the mixer, whereas in a continuous mixer, superimposed on the different paths there is also a different residence time for every fluid particle in the mixer. To quantitatively describe the various strain histories, strain distribution functions (SDF) were defined (56), which are similar in concept to the residence time distribution functions discussed earlier. 

The residence time distribution function must satisfy the following conditions ... 

It is also possible to determine t from the residence time distribution function E(t) by examining the contents of a vessel at time t = 0. This is expressed by... 

Determine the mean residence time t and the residence time distribution function E(t). 

In the channel model, the residence-time distribution function is bimodal if the streams are not sufficiently mixed, a situation which occurs if a2/y2 > about 1/20. For larger values, local hot-spots are likely to develop, and the correlation to be described does not apply. This puts a lower limit on the parameter k of that model. ... 

The residence time distribution function is found as a result of the addition of the probabilities showing the possibility for a liquid element to leave the MWPB (see also Section 4.3.1) ... 

---