Model PFR

The PFR model is frequently used for a reactor in which the reacting system (gas or liquid) flows at relatively high velocity (high Re, to approach PF) through an otherwise empty vessel or one that may be packed with solid particles. There is no device, such as a stirrer, to promote backmixing. The reactor may be used in large-scale operation... 

In the case of an LFR, it is important to distinguish between its use as a model and its occurrence in any actual case. As a model, the LFR can be treated exactly as far as the consequences for performance are concerned, but there are not many cases in which the model serves as a close approximation. In contrast, the CSTR and PFR models serve as useful and close approximations in many actual situations. 

We focus attention in this chapter on simple, isothermal reacting systems, and on the four types BR, CSTR, PFR, and LFR for single-vessel comparisons, and on CSTR and PFR models for multiple-vessel configurations in flow systems. We use residence-time-distribution (RTD) analysis in some of the multiple-vessel situations, to illustrate some aspects of both performance and mixing. 

Comparison of the CSTR and PFR models shows that the latter gives better performance. 

The PFR model is based on turbulent pipe flow in the limit where axial dispersion can be assumed to be negligible (see Fig. 1.1). The mean residence time rpfr in a PFR depends only on the mean axial fluid velocity (U-) and the length of the reactor Lpfr ... 

Defining the dimensionless axial position by z = z/LVfT, the PFR model for the species concentrations 0 becomes7... 

The PFR model ignores mixing between fluid elements at different axial locations. It can thus be rewritten in a Lagrangian framework by substituting a = Tpfrz, where a denotes the elapsed time (or age) that the fluid element has spent in the reactor. At the end of the PFR, all fluid elements have the same age, i.e., a = rpfr. Moreover, at every point in the PFR, the species concentrations are uniquely determined by the age of the fluid particles at that point through the solution to (1.2). 

In addition, the PFR model assumes that mixing between fluid elements at the same axial location is infinitely fast. In CRE parlance, all fluid elements are said to be well micromixed. In a tubular reactor, this assumption implies that the inlet concentrations are uniform over the cross-section of the reactor. However, in real reactors, the inlet streams are often segregated (non-premixed) at the inlet, and a finite time is required as they move down the reactor before they become well micromixed. The PFR model can be easily... 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

Remark 1 If no approximation is introduced in the PFR model, then the mathematical model will consist of both algebraic and differential equations with their related boundary conditions (Horn and Tsai, 1967 Jackson, 1968). If in addition local mixing effects are considered, then binary variables need to be introduced (Ravimohan, 1971), and as a result the mathematical model will be a mixed-integer optimization problem with both algebraic and differential equations. Note, however, that there do not exist at present algorithmic procedures for solving this class of problems. 

The same example was solved using MINOPT (Rojnuckarin and Floudas, 1994) by treating the PFR model as a differential model. The required input files are shown in the MINOPT manual. Kokossis and Floudas (1990) applied the presented approach for large-scale systems in which the reactor network superstructure consisted of four CSTRs and four PFR units interconnected in all possible ways. Each PFR unit was approximated by a cascade of equal volume CSTRs (up to 200-300 CSTRs in testing the approximation). Complex reactions taking place in continuous and semibatch reactors were studied. It is important to emphasize that despite the complexity of the postulated superstructure, relatively simple structure solutions were obtained with the proposed algorithmic strategy. 

The main physicochemical processes in thin-film deposition are chemical reactions in the gas phase and on the film surface and heat-mass transfer processes in the reactor chamber. Laboratory deposition reactors have usually a simple geometry to reduce heat-mass transfer limitations and, hence, to simplify the study of film deposition kinetics and optimize process parameters. In this case, one can use simplified gas-dynamics reactor such as well stirred reactor (WSR), calorimetric bomb reactor (CBR, batch reactor), and plug flow reactor (PFR) models to simulate deposition kinetics and compare theoretical data with experimental results. 

A reactor for thin-film deposition can have a tubular structure with a flowing chemically active gas. In this case, the PFR model can be used. This one-dimensional model uses the assumption of a uniform component distribution in a reactor section. This means that the boundary layer formation processes are not taken into account. 

PFR models are limited, however, because of the slow velocities encountered in groundwater aquifers and the tendency for many contaminants (particularly hydrophobic organic compounds) to sorb. More appropriate but more complex models based on various forms of the advection-dispersion equation (ADE) have been used by several researchers to incorporate more processes, such as dispersion, sorption, mass transfer, sequential degradation, and coupled chemical reactions. 

Reactor Selection Ideal CSTR and PFR models are extreme cases of complete axial dispersion (De = oo) and no axial dispersion (De = 0), respectively. As discussed earlier, staged ideal CSTRs may be used to represent intermediate axial dispersion. Alternatively, within the context of a PFR, the dispersion (or a PFR with recycle) model may be used to represent increased dispersion. Real reactors inevitably have a level of dispersion in between that for a PFR or an ideal CSTR. The level of dispersion may depend on fluid properties (e.g., is the fluid newtonian),... 

Examples of values of Pe are provided in Fig. 19-8. When Pe is large, n =k Pe/2 and the dispersion model reduces to the PFR model. For small values of Pe, the above equation breaks down since the lower limit on n is n = 1 for a single CSTR. To better represent dispersion behavior, a series of CSTRs with backmixing may be used e.g., see Froment and Rischoff (Chemical Reactor Analysis and Design, Wiley, 1990). A model analogous to the dispersion model may be used when there are velocity profiles across the reactor cross-section (eg., for laminar flow). In this case, the equation above will contain terms associated with the radial position in the reactor. 

This implies that the necessary recirculation ratio is not a fixed value, but depends on the reaction under consideration and the conversion level. At low conversions, the recirculation rate need not be high according to this criterion (good mixing, eq 5, is still required to avoid dead zones in the reactor), and a differential PFR model can be used. At high conversions, the recirculation rate must increase. It can easily be seen that a recirculation ratio of 20 limits the conversion for a first-order and a second-order reaction to 50% and 25%, respectively. 

Irving Langmuir (1908) first replaced the assumption of no axial mixing of the PFR model with finite axial mixing and the accompanying Dirichlet boundary condition ((Cy) = Cyyn at x = 0) by a flux-type boundary condition... 

Needless to mention that the Danckwerts model [Eqs. (146)-(147)] reduces to the ideal PFR and the ideal CSTR models in the limits of Pe —> oo and Pe —> 0, respectively. As seen in Fig. 9, the ideal PFR model and the ideal CSTR model bound the solution of the Danckwerts model [Eqs. (146)—(147)] for the case of any finite Pe number. As shown in Fig. 9 (and from Eq. (150)), for Pe 10, the two-mode model predicts very close to the Danckwerts model when the variances of the two models are matched using Eqs. (71) and (63). However, the two-mode models can predict conversions in regions in the parameter space which are inaccessible to the Danckwerts models. These are the micromixing limited conversions which are even below conversions predicted by the ideal... 

For p — /Pe — 0, this model reduces to the ideal PFR model, for p — 0, it reduces to the Danckwerts model, for l/Pe — 0, it reduces to the two-mode plug-flow model while for Pe — 0, it reduces to the two-mode CSTR model (discussed below). Thus, we have shown that the hyperbolic two-mode model given by Eqs. (130)—(134) has a much larger region of validity than the traditional homogeneous tubular reactor models. 

The transient response experiments were analyzed by a dynamic isothermal PFR model, and estimates of the relevant kinetic parameters were obtained by global nonlinear regression over all runs. It was found that a simple Langmuir approach could not represent the data accurately, and surface heterogeneity had to be invoked. The best fit was obtained using a Temkin-type adsorption isotherm with coverage-dependent desorption energy ... 

P14-1b Make up and solve an original problem. The guidelines are given in Problem P4-1. However, make up a problem in reverse by first choosing a model system such as a CSTR in parallel with a CSTR and PFR [with the PFR modeled as four small CSTRs in series Figure P14-l(a)] or a CSTR with recycle and bypass [Figure P14-l(b)]. Write tracer mass balances and use an ODE solver to predict the effluent concentrations. In fact, you could build up an arsenal cf tracer curves for different model systems to compare against real reactor RTD data. In this way you could deduce which model best describes the real reactor. 

Industrial reactors are usually more complex than the simple simulator library models. Real reactors usually involve multiple phases and have strong mass transfer, heat transfer, and mixing effects. The residence time distributions of real reactors can be determined by tracer studies and seldom exactly match the simple CSTR or PFR models. 

Sometimes a combination of library models can be used to model the reaction system. For example, a conversion reactor can be used to establish the conversion of main feeds, followed by an equilibrium reactor that establishes an equilibrium distribution among specified products. Similarly, reactors with complex mixing patterns can be modeled as networks of CSTR and PFR models, as described in Section 1.9.10 and illustrated in Figure 1.19. 

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