Distribution in residence times

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. 

Separation is normally a frinetion of time, as either the settling or the eoaleseenee proeess is limiting. It is, therefore, imperative that the flow flirough a separator is controlled in some manner. The design equations presented in Sec. IV are all based on an even distribution of flow over the cross-section of the separator. A skewed inlet distribution will lead to a distribution in residence times and impaired separation. 

The extent of axial mixing can be expressed by the distribution in residence times. Small distributions in residence times will lead to a uniform product and is generally considered to be favorable. However, the associated small axial mixing makes it nessecary to assure a very constant feed flow. An important parameter in the analysis of the residence time distribution is the mean residence time 7, which can easily be obtained from... 

The distribution in residence times can be characterized by the standard deviation at or the variance a ... 

Other models to characterize residence time distributions are based on fitting the measured distribution to models for a plug flow with axial dispersion or for series of continuously ideally stirred tank reactors in series. For the first model the Peclet number is the characteristic parameter, for the second model the number of ideally stirred tank reactors needed to fit the residence time distribution typifies the distribution. However, these models should be used with care because they assume a standard distribution in residence times. Most distributions in extruders show a distinct scewness, which could lead to erroneous results at very short and very long residence times. The only exception is the co-kneader the high amount of back mixing in this type of machine leads to a nearly perfect normal distribution. 

Equal distributions in residence times are obtained if hydrodynamic similarity also exists, indicating a constant rotation rate of the serews. When scaling up an extruder geometrically, using materials with equal viscosities, both hydrodynamic similarity and equal residence times ean be obtained if... 

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

The distribution of residence times of reactants or tracers in a flow vessel, the RTD, is a key datum for determining reactor performance, either the expected conversion or the range in which the conversion must fall. In this section it is shown how tracer tests may be used to estabhsh how nearly a particular vessel approaches some standard ideal behavior, or what its efficiency is. The most useful comparisons are with complete mixing and with plug flow. A glossary of special terms is given in Table 23-3, and major relations of tracer response functions are shown in Table 23-4. 

In the holding section of a continuous sterilizer, correct exposure time and temperature must be maintained. Because of the distribution of residence times, the actual reduction of microbial contaminants in the holding section is significantly lower than that predicted from plug flow assumption. The difference between actual and predicted reduction in viable microorganisms can be several orders of magnitude therefore, a design based on ideal flow conditions may fail. 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

The particles in the latex stream leaving a continuous stirred-tank reactor (CSTR) would have a broad distribution of residence times in the reactor. This age distribution, given by Equation 5, comes about because of the rapid mixing of the feed stream with the contents of the stirred reactor. 

Figure 3. Molecular weight distribution of polymer used in residence time distribution tests...

When solid particles are subject to noncatalytic reactions, the effects of the reaction on individual particles are derived and then the results are averaged to determine overall properties. The general techniques for this averaging are called population balance methods. They are important in mass transfer operations such as crystallization, drop coagulation, and drop breakup. Chapter 15 uses these methods to analyze the distribution of residence times in flow systems. The following example shows how the methods can be applied to a collection of solid particles undergoing a consumptive surface reaction. 

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

The molecules in the system are carried along by the balls and will also have an exponential distribution of residence time, but they are far from perfectly mixed. Molecules that entered together stay together, and the only time they mix with other molecules is at the reactor outlet. The composition within each ball evolves with time spent in the system as though the ball was a small batch reactor. The exit concentration within a ball is the same as that in a batch reactor after reaction time tf,. 

We have just described a completely segregated stirred tank reactor. It is one of the ideal flow reactors discussed in Section 1.4. It has an exponential distribution of residence times but a reaction environment that is very different from that within a perfectly mixed stirred tank. 

Unlike the situation in a plug flow reactor, the various fluid elements mix with one another in a CSTR. In the limit of perfect mixing, a tracer molecule that enters at the reactor inlet has equal probability of being anywhere in the vessel after an infinitesimally small time increment. Thus all fluid elements in the reactor have equal probability of leaving in the next time increment. Consequently there will be a broad distribution of residence times for various tracer molecules. The character of the distribution is discussed in Section 11.1. Because some of the... 

The Residence Time Distribution. All fluid elements have the same residence time in a batch reactor, but there will be a wide spread in residence times in a CSTR. 

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. 

From this equation it is evident that there is a wide distribution of residence times in a stirred tank reactor. 

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... 

Since these two types of processes have drastically different effects on the conversion levels achieved in chemical reactions, they provide the basis for the development of mathematical models that can be used to provide approximate limits within which one can expect actual isothermal reactors to perform. In the development of these models we will define a segregated system as one in which the first effect is entirely responsible for the spread in residence times. When the distribution of residence times is established by the second effect, we will refer to the system as mixed. In practice one encounters various combinations of these two limiting effects. 

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