RTD-functions

An Excel spreadsheet (Example8-7.xls) was used to determine the various RTD functions and the computer program PROGS 1 was used to simulate the model response curve with the experimental data. The results show the equivalent number of ideally mixed stages (nCSTRs) for the RTD is 13.2. The Gamma distribution function from Equation 8-143 is ... 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

An ideal plug flow reactor, for example, has no spread in residence time because the fluid flows like a plug through the reactor (Westerterp etal., 1995). For an ideal continuously stirred reactor, however, the RTD function becomes a decaying exponential function with a wide spread of possible residence times for the fluid elements. 

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. 

The exit-age distribution function is a measure of the distribution of the ages of fluid elements leaving a vessel, and hence is an RTD function. As a function of time, f, it is defined as ... 

Both E and F are RTD functions. Although not apparent here, each corresponds to a different way of experimentally investigating RTD for arbitrary flow (Chapter 19 see also problem 13-1), and hence each is important. 

The general features of a PFR are outlined in Section 2.4.1, and are illustrated schematically in Figure 2.4. Residence-time distribution (RTD) functions are described in Section 13.4.2. The essential features are recapitulated as follows ... 

Each element of fluid has the same residence time t as any other (cf. CSTR) the RTD functions E and F are shown in Figures 13.6 and 13.7, respectively the former is represented by the vertical spike of the Dirac delta function, and the latter by the step function. 

We first describe features of nonideal flow qualitatively, and then in terms of mixing aspects. For the rest of the chapter, we concentrate on its characterization in terms of RTD. This involves (1) description of the experimental measurement of RTD functions (.E, F, IF), and development of techniques for characterizing nonideal flow and (2) introduction of two simple models for nonideal flow that can account for departures from ideal flow. 

The effects on the RTD function E of stagnation, dispersion (or partial backmixing), and channeling or excessive bypassing are shown schematically in Figures 19.2(a), (b), and (c), respectively,... 

Since tracer experiments are used to obtain RTD functions, we wish to establish that the response to a pulse-tracer input is related to (r) or E(0). For this purpose, c(t) must be normalized appropriately. We call c(t), in arbitrary units, the nonnormalized response, and define a normalized response C(t) by... 

In this section, we develop two simple models, each of which has one adjustable parameter the tanks-in-series (TIS) model and the axial-dispersion or dispersed-plug-flow (DPF) model. We focus on the description of flow in terms of RTD functions and related quantities. In principle, each of the two models is capable of representing flow in a single vessel between the two extremes of BMF and PF. 

The cumulative RTD function F(d) in the TIS model is the fraction of the outlet stream from the Nth tank that is of age 0 to 0, or the probability that a fluid element that entered the first tank at 9 = 0 has left the Nth tank by the time 0. It can be obtained from En(9) by means of the relation given in Table 13.1. 

However, any kinetics may be inserted as required. The RTD function E(t) may be known either from a flow model (ideal or nonideal) or from experimental tracer data. 

Comparison of the segregated-flow and maximum-mixedness models, with identical RTD functions, shows that the former gives better performance. This is consistent with the observations of Zwietering (1959), who showed that for power-law kinetics of order n > 1, the segregated-flow model produces the highest conversion. 

Response to impulse input of tracer is shown in the first two columns. Find the principal RTD functions and prepare plots of E(tr) and F(tr). Identify the times between which a specified fraction of the tracer has left the vessel. 

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... 

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 I(a x, t) = fr(a 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. 

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 non-interacting fluid elements, the RTD function is thus equivalent to the joint PDF of the concentrations. In composition space, the joint PDF would he on a one-dimensional sub-manifold (i.e., have a one-dimensional support) parameterized by the age a. The addition of micromixing (i.e., interactions between fluid elements) will cause the joint PDF to spread in composition space, thereby losing its one-dimensional support. 

CSD Functions. From the RTD function based on the tank-in-series model, both the number basis and weight basis probability density functions of final product... 

The residence time distribution (RTD) function, E, of the tracer, which is injected into stream entering into a vessel as an instantaneous pulse, based on the tank-in-series (j-equal size) model is given by (2),... 

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