Residence time distribution function

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. 

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. 

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. 

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. 

In a manner similar to the internal age distribution function, let E be the measure of the distribution of ages of all elements of the fluid stream leaving a vessel. Thus E is a measure of the distribution of residence times of the fluid within the vessel. Again the age is measured from the time that the fluid elements enter the vessel. Let E be deflned in such a way that E dd is the fraction of material in the exit stream which has an age between 6 and 6 -I- dO. Referring to Fig. 4, the area under the E vs. 6 curve is... 

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

Residence time distribution (RTD). The curve of concentration of tracer, C(t). as a function of time, recorded at the exit, in effect displays the distribution of residence times or holding times of the vessel. 

The distribution of residence times for a stream of fluid leaving a vessel is called the exit age distribution function E (synonymous with residence time distribution or... 

Similarly, in continuous operations the residence times that exiting fluid elements experienced in the system are not necessarily uniform, but there is a distribution of residence times that we must take into account, and we must define residence time distribution (RTD) functions. 

Using Eq. 7.3.16, it is possible to derive all the interrelationships of the RTD functions, which are listed in Table 7.1. The two extreme flow systems with respect to RTD are the plug flow system, which exhibits no distribution of residence times, and the continuous stirred tank (CST), which exhibits perfect back-mixing and has the following RTD function ... 

E E(t) E(tr) f. Fit) Activation energy Residence time distribution Normalized residence time distribution Fraction of A remaining unconverted, Ca /Ca0 or nja0 Age function of tracer kJ/(kgmol) Btu/(lb-mol)... 

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. 

One measure of the distribution of residence times (ages) of the fluid elements within a reactor is the -function, defined so that E d0 is the fraction of material in the exit stream with age between h and h + dO (Levenspiel, 1972). It can be shown (Levetispiel, 1972) that the C and E functions are identical, and that for an isothermal process the ratio of the final (C) to initial (Co) concentrations of either microorganisms or nutrients can be determined from the expression ... 

Here C and r refer to conditions in the product stream (or in the reactor) and B is the average residence time. It is equal to V/Q and also equal to the average of the distribution of residence times for an ideal stirred-tank reactor. This distribution function is developed in Chap. 6. 

In a continuous reactor the solid particles may not be in plug flow but may have a distribution of residence times. In general, the conversion for a given particle size will be a function of and t, so that x = x(r, )-However, the residence-time distribution is likely to be caused by the distribution of particle sizes that is, there may be a unique relation between residence time and particle size. For this situation, Eq. (14-26) is applicable, but the conversion for a given particle size will be evaluated from Eqs. (14-19) and (14-20) by using the residence time applicable for that size. Example 14-2 illustrates these concepts. 

The RTD is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 0 will have a distribution of residence times regardless of whether the reactor is at steady state. The washout function for an unsteady reactor is defined as... 

In general, the larger the breadth of the distribution of residence times, the greater the discrepancy between the conversion levels predicted on the basis of the segregated flow model and those predicted by the various mixing models. For narrow distribution functions, the conversions predicted by both models will be in good agreement with one another. 

The distribution of residence times of fluid elements in a vessel is expressed in terms of a function called RTD function denoted as F(0). f(0) is defined as the fraction of the fluid elements (leaving the vessel at any time) whose residence time is less than or equal to 0. f (0) = 0 at 0 = 0 as no fluid element can have zero residence time and f(0) = 1 as 0 as all the fluid elements reside in the vessel only for finite time duration. Further, F(0) is mono-tonically increasing function of 0 as F(0 + A0) > F(0) for all 0 -F A0 > 0. A sketch of a typical RTD function F(0) is shown in Figure 3.44. 

The RTD function E(0) or E(0), obtained from the tracer experiment conducted on the reaction vessel, can be used to characterise the non-ideality as the fluid mixing pattern in the vessel has a strong influence on the distribution of residence time. Given the RTD for a reaction vessel, we would first like to know if the mixing patterns in the reaction vessel match well with the mixing patterns assumed for ideal reactors (ideal CSTR or ideal PER). This can be done by comparing the RTD function (F-curve or E-curve) for the given reactors with the RTD functions for the ideal CSTR or ideal PER. For this, we should know the RTD functions for ideal reactors. As the ideal CSTR and ideal PER are theoretical reactors, the RTD function equations for these reactors are derived theoretically. 

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