RTD theory

In the CRE literature, the residence time distribution (RTD) has been shown to be a powerful tool for handling isothermal first-order reactions in arbitrary reactor geometries. (See Nauman and Buffham (1983) for a detailed introduction to RTD theory.) The basic ideas behind RTD theory can be most easily understood in a Lagrangian framework. The residence time of a fluid element is defined to be its age a as it leaves the reactor. Thus, in a PFR, the RTD function E (a) has the simple form of a delta function  

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. 

In this book, an alternative description based on the joint probability density function (PDF) of the species concentrations will be developed. (Exact definitions of the joint PDF and related quantities are given in Chapter 3.) The RTD function is in fact the PDF of the fluid-element ages as they leave the reactor. The relationship between the PDF description and the RTD function can be made transparent by defining a fictitious chemical species 

13 The outflow of a CSTR is a Poisson process, i.e., fluid elements are randomly selected regardless of their position 

At steady state, the PDF (and thus the RTD function) will be independent of time. Moreover, the internal-age distribution at a point x inside the reactor is just /(a x, t) = fr (or, x, t). For a statistically homogeneous reactor (i.e., a CSTR), the PDF is independent of position, and hence the steady-state internal-age distribution 1(a) will be independent of time and position. 

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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. 

One of the early successes of the CRE approach was to show that RTD theory suffices to treat the special case of non-interacting fluid elements (Danckwerts 1958). For this case, each fluid element behaves as a batch reactor. 

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. 

The utility of RTD theory is best illustrated by its treatment of first-order chemical reactions. For this case, each fluid element can be treated as a batch reactor.15 The concentration... 

In RTD theory, the concentrations at the reactor outlet are found by averaging over the ages of all fluid elements leaving the reactor 16... 

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. 

In some areas, e.g. aerosol physics and crystallisation, population balance models are used in situations when a number balance equation is required as well as conventional mass and energy balances. Randolph and Larson review this theory as it applies specifically to particulate systems [15], whilst Froment and Bischoff [16] present population balance equations in the context of an extension of classical RTD theory. 

Most chemical reaction engineering textbooks contain material on residence time distribution theory. Levenspiel [17] and Hill [18] present particularly useful introductions as do refs. 9 and 16. The proceedings of a recent summer school [19] contains a brief overview of the field [8] as well as papers describing many specific applications of RTD theory in chemical engineering contexts. Nauman s comprehensive invited review cited earlier [4] is an extremely thorough and yet highly readable contribution to the literature. The book by Nauman and Buffham [20], will no doubt fill a most important gap in the literature on mixing in continuous flow systems. 

Recent work has attempted to generalize RTD theory to less restrictive assumptions, so that it might apply to more realistic situations encountered in practice. The type of flow and the operating conditions of the reactor are given in Table II below, together with the corresponding references ... 

The theory of RTD was first enunciated by Danckwerts (1953). It is a useful technique for identifying the nonidealities in chemical reactors, but as with any theory, it comes with its own limitations. In this section, we describe the RTD theory with its merits and demerits. In the next section, we give examples of more recent theories of mixing and nonideal reactor modeling. 

As shown in Figure 3.8, the RTDs of the reactors are identical. On the one hand, the conversions at the exit of these reactors will be substantially different for reaction orders different from 1 especially when a complex reaction scheme is involved. On the other hand, RTD theory predicts that the reactor behavior should be identical whether the PFR or the CSTR is the first reactor in the combination. Therefore, with this example, the need for a more sophisticated theory than RTD is clearly demonstrated. In the next section, we will review some of the existing theories of mixing. 

RTD theory fails to describe reactor behavior for interacting fluid elements ... 

Briefly we treated the perfectly mixed reactor RTD in the mathematical analysis provided above. It is important to note from Example 3.1 that the RTD theory is not fully capable of explaining the behavior of the reactors, especially when the fluid elements are interacting. Thus, we give examples for a few other models here and refer the reader to an excellent text by Fox (2003) for a more in-depth analysis of these models in the turbulent flow regime. Four broad classes of micromixing models are sketched in Figure 110. 

The first of these is the residence time distribntion (RTD) method. This technique allows us to classify the dispersion properties of a constituent in a chemical reactor, with reference to ideal behaviors of simple reactors. The RTD theory does not explicitly associate a RTD with a flow configuration inside the reactor. We examine this particular issue when the flow is turbulent, by considering successively the cases of a tubular reactor with axial dispersion and of a continuous stirred tank reactor (CSTR). Matching up the dimertsiorts of the reactor with the mean residence time and the velocity and length scales of turbulence allows the determirration of the hydrodynamic conditions associated with either type of reactor, for which the RTD laws are recovered, using the trrrbulent dispersion concepts introduced in Chapter 8. 

Macromixing is the process whereby parts of the fluid having different histories come into contact on the macroscopic scale. Danckwerts [l] proposed to characterize macromixing in continuous flow systems by the Residence Time Distribution (RTD) of the fluid, a concept which became very famous and is now the basis of most reactor models. RTD theory can be found in many classical textbooks [2] [ 3 [43. Usually, simplifying assumptions are made such as... 

With these assumptions, which may appear somewhat restrictive, RTD theory makes it possible to solve many industrial problems. 

As pointed out in the introduction to this section, the assumptions on which the classical RTD theory are based, are restrictive. However, it is still possible to use concepts like RTD or lAD when these assumptions are no longer valid - i.e. at unsteady state, in a compressible fluid, or in a variable volume reactor having large inlet and outlet ports - by using the population balance method. 

The concept of residence time distribution (RTD) and its importance in flow processes first developed by Danckwerts (1953) was a seminal contribution to the emergence of chemical engineering science. An introduction to RTD theory is now included in standard texts on chemical reaction engineering. There is also an extensive literature on the measmement, theory, and application of residence time distributions. A literature search returns nearly 5000 references containing the concept of residence time distribution and some 30 000 references dealing with residence time in general. This chapter necessarily provides only a brief introduction the references provide more comprehensive treatments. 

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