Distribution of residence times

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. 

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. 

Danckwerts, P. V., Continuous flow systems distribution of residence times, Chem. Eng. ScL, 2, 1-18 (1953). 

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. 

The other case assumes that the fluid particles are well mixed. Specifically, assume that they have an exponential distribution of residence times... 

The ideal flow reactors are the CSTR and the PFR. (This chapter later introduces a third kind of ideal reactor, the segregated CSTR, but it has the same distribution of residence times as the regular, perfectly mixed CSTR.) Real reactors sometimes resemble these ideal types or they can be assembled from combinations of the ideal types. 

A CSTR has an exponential distribution of residence times. The corresponding differential distribution can be found from Equation (15.7) ... 

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. 

The residence time distribution is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 8 wiU have a distribution of residence times regardless of whether the reactor is at steady... 

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... 

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. 

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. 

The following data have been reported as a result of an effort to determine the distribution of residence times in a packed bed reactor. Use these data to generate an F(t) curve and to determine the average residence time in the reactor. 

The maintenance of uniform flow distribution in fixed bed reactors is frequently a problem. Maldistribution leads to an excessive spread in the distribution of residence times with adverse effects on the reactor performance, particularly when consecutive reactions are involved. It may aggravate problems of hot-spot formation and lead to regions of the reactor where undesired reactions predominate. Disintegration or attrition of the catalyst may lead to or may aggravate flow distribution problems. 

Backmix flow (BMF) is the flow model for a CSTR, and is described in Section 2.3.1. BMF implies perfect mixing and, hence, uniform fluid properties throughout the vessel. It also implies a continuous distribution of residence times. The stepwise or discontinuous change in properties across the point of entry, and the continuity of property behavior across the exit are illustrated in Figure 2.3. 

Laminar flow (LF) is also a form of tubular flow, and is the flow model for an LFR. It is described in Section 2.5. LF occurs at low Reynolds numbers, and is characterized by a lack of mixing in both axial and radial directions. As a consequence, fluid properties vary in both directions. There is a distribution of residence times, since the fluid velocity varies as a parabolic function of radial position. 

Although there is a distribution of residence times, the complete mixing of fluid at the microscopic and macroscopic levels leads to an averaging of properties across all fluid elements. Thus, the exit stream has a concentration (average) equivalent to that obtained as if the fluid existed as a single, large fluid element with a residence time of t = V/q (equation 2.3-1). 

Setting aside this goal of complete knowledge about the flow, let us be less ambitious and see what it is that we actually need to know. In many cases we really do not need to know very much, simply how long the individual molecules stay in the vessel, or more precisely, the distribution of residence times of the flowing fluid. This information can be determined easily and directly by a widely used method of inquiry, the stimulus-response experiment. 

The experimental technique used for finding this desired distribution of residence times of fluid in the vessel is a stimulus-response technique using tracer material in the flowing fluid. The stimulus or input signal is simply tracer introduced in a known manner into the fluid stream enter-... 

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