RTD in Ideal Reactors

This section introduces how tracers are used to establish the RTD in a reactor and to contrast against RTDs of ideal reactors. The section... 

In practice, especially in large-scale reactors, plug-flow or complete mixing are rarely achieved, and it is desirable to quantify the deviation from those idealized flow conditions. Also, when a chemical reactor does not perform at the expected level, it is necessary to identify the reason. A diagnostic method that is applied in such situations is based on measuring the residence time distribution (RTD) in the reactor. An inert tracer is injected at the reactor inlet, and its concentration at the reactor outlet is measured with time. By comparing the outlet concentration curve to the inlet concentration curve, the RTD curve of the reacting fluid in the reactor can be constructed [1,7,10,43]. 

Plug flow reactors are characterized by a unique residence time for all molecules and can be operated at t = It is evident that any residence time distribution (RTD) in the reactor will diminish the yield of the intermediate. Therefore, the highest yield is always obtained for a uniform residence time corresponding to an ideal plug flow reactor [7]. Evidently, the yield increases with decreasing ratios of the two rate constants /c. 

In Figures 3.4 and 3.5, the RTDs of ideal reactors are presented together with the RTD of a real reactor. The ideal, continuously operated stirred tank reactor (CSTR) has the broadest RTD between all reactor types. The most probable residence time for an entering volume element is t = 0. After a mean residence time t = t), 37% of the tracer injected at time t = 0 is still present in the reactor. After five mean residence times, a residue of about 1% still remains in the reactor. This means that at least five mean residence times must pass after a change in the inlet conditions before the CSTR effectively reaches its new stationary state. 

In the case of known formal kinetics, the reactor performance can be determined directly from the RTD. We can imagine, for example, that the RTD in the reactor under consideration can be represented by a series of ideal plug flow reactors of different lengths arranged in parallel through which the reaction mass flows at equal rates (see Figure 3.17). 

In the case of identical mean residence times for different tubular reactors, the conversion and selectivity of a complex reaction will depend on the RTD in the reactor. With increasing backmixing, the reactor approaches the behavior of an ideal CSTR. Accordingly, the performance of any tubular reactor will decrease with increasing RTD at a constant mean residence time for reactions with formally positive reaction orders. 

In Section 3.4.2, it was shown that the RTD in real reactors can be described with a series of ideally continuous stirred tank reactors. The scheme of such a cascade of continuous stirred tanks is shown in Figure 3.21. The total volume is divided in N equal sized stirred vessels. 

Diagnosing the non-ideality does not stop with finding out if the reaction vessel is ideal or non-ideal. On knowing that the reactor is non-ideal, it is necessary to predict the impact of non-ideality on the reactor performance, which is the conversion achievable in the reactor. For this, the non-ideality has to be quantified first. Quantification of non-ideality involves assigning some kind of metric or measure for the extent of deviation from ideality. By comparing the RTD of the reaction vessel with the RTD of ideal reactors, one can get a qualitative idea about the gap or deviation between the real and the ideal reactors. Fiowever, one has to come up with an appropriate quantification of this gap in such a manner that this quantification will be useful for predicting the conversion achievable in the reactor. 

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

Equation 13.5-2 is the segregated-flow model (SFM) with a continuous RTD, E(t). To what extent does it give valid results for the performance of a reactor To answer this question, we apply it first to ideal-reactor models (Chapters 14 to 16), for which we have derived the exact form of E(t), and for which exact performance results can be compared with those obtained independently by material balances. The utility of the SFM lies eventually in its potential use in situations involving nonideal flow, wheic results cannot be predicted a priori, in conjunction with an experimentally measured RTD (Chapters 19 and 20) in this case, confirmation must be done by comparison with experimental results. 

As discussed in Section 17.2.3.1, reactor performance in general depends on (1) the kinetics of reaction, (2) the flow pattern as represented by the RTD, and (3) mixing characteristics within the vessel. The performance predicted by ideal reactor models (CSTR, PFR, and LFR) is determined entirely by (1) and (2), and they do not take (3)... 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

Figure 1.4. Sketch of the residence time distribution (RTD) in a non-ideal reactor. 

RTD functions for combinations of ideal reactors can be constructed (Wen and Fan 1975) based on (1.6) and (1.7). For non-ideal reactors, the RTD function (see example in Fig. 1.4) can be measured experimentally using passive tracers (Levenspiel 1998 Fogler 1999), or extracted numerically from CFD simulations of time-dependent passive scalar mixing. 

For the general case of interacting fluid elements, (1.9) and (1.10) no longer hold. Indeed, the correspondence between the RTD function and the composition PDF breaks down because the species concentrations inside each fluid element can no longer be uniquely parameterized in terms of the fluid element s age. Thus, for the general case of complex chemistry in non-ideal reactors, a mixing theory based on the composition PDF will be more powerful than one based on RTD theory. 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

An alternative method to RTD theory for treating non-ideal reactors is the use of zone models. In this approach, the reactor volume is broken down into well mixed zones (see the example in Fig. 1.5). Unlike RTD theory, zone models employ an Eulerian framework that ignores the age distribution of fluid elements inside each zone. Thus, zone models ignore micromixing, but provide a model for macromixing or large-scale inhomogeneity inside the reactor. 

The RTD in a system is a measure of the degree to which fluid elements mix. In an ideal plug flow reactor, there is no mixing, while in a perfect mixer, the elements of different ages are uniformly mixed. A real process fluid is neither a macrofluid nor a microfluid, but tends toward one or the other of these extremes. Fluid mixing in a vessel, as reviewed in Chapter 7, is a complex process and can be analyzed on both macroscopic and microscopic scales. In a non-ideal system, there are irregularities that account for the fluid mixing of different... 

Understanding Reactor Flow Patterns As discussed above, a RTD obtained using a nonreactive tracer may not uniquely represent the flow behavior within a reactor. For diagnostic and simulation purposes, however, tracer results may be explained by combining the expected tracer responses of ideal reactors combined in series, in parallel, or both, to provide an RTD that matches the observed reactor response. The most commonly used ideal models for matching an actual RTD are PRF and CSTR models. Figure 19-9 illustrates the responses of CSTRs and PFRs to impulse or step inputs of tracers. 

Table 1 lists the characteristics of the measured RTD for five different conditions. The first one is shown in Figure 2. The evolution of this curve can be explained by equation (1), although the peaks are not ideal Dirac pulses, because the flow inside the reactor (i.e. the reactor tube (c) and the recirculation pipe (d) in Figure 1) is not of the ideal plug flow type. Therefore, the tracer pulse broadens and eventually spreads throughout the reactor. Nevertheless, the distance between two peaks is a reasonably accurate estimate of the circulation time r/(R+1) in the reactor, and from this the flow through the reactor can be calculated. The recycle ratio R is calculated from the mean residence time r and the circulation time r/(R+l). 

The state of mixing in a given reactor can be evaluated by RTD experiments by means of inert tracers, by temperature measurements, by flow visualization and, finally, by studying in the reactor under consideration the kinetics of an otherwise well-known reaction (because its mechanism has been carefully elucidated from experiments carried out in an ideal reactor, the batch reactor being generally chosen as a reference for this purpose). From these experimental results, a reactor model may be deduced. Very often, in the laboratory but also even in industrial practice, the real reactor is not far from ideal or can be modelled successfully by simple combinations of ideal reactors this last approach is of frequent use in chemical reaction engineering. But... 

Allowable Spread in Residence Time. Other ways of stating the requirement of equal residence time of all parts of the reactant is that the flow through the reactor should approach plug flow or that the residence time distribution (RTD) should be equivalent to that in a large number of mixers in series. An often used rule of thumb is that this requirement is met when the equivalent number of mixers (N ) exceeds a certain value, say 5. However, this criterion is at best a semi-quantitative one, since the minimum value of is dependent upon the accepted deviation from the ideal reactor, and on the degree of conversion and the reaction order. 

In previous chapters treating ideal reactors, a parameter frequently used was the space-time or average residence time x, which was defined as being equal to V/v. It will be shown that no matter what RTD exists for a particular reactor, ideal or nonideal, this nominal holding time, x, is equal to the mean residence time.,r . 

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