PFR and CSTR models

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 (Uz) and the length of the reactor Lpfr  

Defining the dimensionless axial position by z = z/Lpfr, 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 = rpfrz , 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 notation is chosen to be consistent with that used in the remainder of the book. Alternative notation is employed in most CRE textbooks. 

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The PFR and CSTR models encompass the extremes of the residence-time distributions shown in Figure 4.3 however the batch reactor and the laminar-flow reactor, both of which we have already mentioned in this chapter, are also types exhibiting a well-defined mixing behavior. The batch reactor is straightforward, since it is simply represented by the perfect mixing model with no flow into or out of the system, and has been treated extensively in Chapter 1. 

Packed beds usually deviate substantially from plug flow behavior. The dispersion model and some combinations of PFRs and CSTRs or of multiple CSTRs in series may approximate their behavior. 

One might intuitively expect that infinite recycle rates associated with a system as described by eqn. (61) would produce a completely well-mixed volume with concentration independent of location. This is indeed so and under these conditions, the performance tends to that of an equal sized CSTR. At the other extreme, when R is zero, PFR performance pertains. Fractional conversions at intermediate values of R may be determined from Fig. 14. The specific form of recycle model considered is thus seen to be continuously flexible in describing flow mixing between the PFR and CSTR extremes just as was the tanks-in-series model. The mean and variance of this model are given by eqns. (62) and (63) and these may be used for moments matching purposes of the type illustrated in Example 6. 

The PFR and CSTR are readily imagined and widely applicable descriptions of how real large scale reactors actually behave. These models of the patterns of reactant contacting form the basis for much of present day chemical reaction engineering. 

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. 

The plug flow reactor has been mainly utilized for the removal of phenol in waste streams by HRP [76, 83] and Coprinus cinereus peroxidase [2]. According to Buchanan et al. [83], who modeled the kinetics of the HRP-aromatic substrate system and applied to PFR and CSTR, plug-flow configuration is recommended when working with low HRT, since considerably less enzyme would be required for equal phenol removal. However, for long HRTs, a multiple-stage CSTR would be more efficient than a PFR, due to the lower rate of enzyme inactivation. 

In Chapter 3, it was stated that the ideal PFR and CSTR are the theoretical limits of fluid mixing in that they have no mixing and complete mixing, respectively. Although these two flow behaviors can be easily described, flow fields that deviate from these limits are extremely complex and become impractical to completely model. However, it is often not necessary to know the details of the entire flow field but rather only how long fluid elements reside in the reactor (i.e., the distribution of residence times). This information can be used as a diagnostic tool to ascertain flow characteristics of a particular reactor. 

In addition to the one-parameter models of tanks-in-series and dispersion, many other one-parameter models exist when a combination of ideal reactors is to model the real reactor. For example, if the real reactor were modeled as a PFR and CSTR in series, the parameter would be the fi action,/, of the total reactor volume that behaves as a CSTR Another one-parameter model would be the fi action of fluid that bypasses the ideal reactor. We can dream up many other situations which would alter the behavior of ideal reactors in a way that adequately describes a real reactor. However, it m be that one parameter is not sufficient to yield an adequate comparison between theoiy... 

Figure 14-14 Combinations of ideal reactors used to model real PFRs. (a) two PFRs in parallel (b) PFR and CSTR in parallel.

Comparison of conversions for a PFR and CSTR with the zero-parameter and two-parameter models. symbolizes the conversion... 

A real reactor is modeled as a combination of ideal PFRs and CSTRs. 

As expected, using the E(l) for an ideal PFR and CSTR wtih the segregation model gives a mean conversion X identical to that obtained bv using the algorithm in Ch. 4. 

The model in Fig 14-14b is PFR and CSTR in parallel. The exit age distribution for the eSTR is a negative gradient curve, intcmipted by the distinct exit age distribution pulse of the PFR. The CSTR will always provide a fraction of eflluenc which has been inside the reactor for less than time t, which incicases with time. But when (he effluent exits the PFR at a specified time after zero, this increased effluent is superimposed upon F9t) of the CSTR. giving a combined F(t). 

Consider the feed and effluent tracer concentrations from a pulse test shown in Figure 8.39. Draw the simplest reactor configuration consisting of PFRs and CSTRs similar to those shown in Figure 8.38 that you would use to model the reactor based on these test data. Determine t = Vr/Q/ for each reactor in yoiir configuration. 

A second approach is to use various combinations of the ideal reactor models in simulation of nonideal behavior. This may seem a bit contradictory at first, but hopefully our later discussion will be sufficient to illuminate the reasoning behind this method. A hint of this approach is given by the discussion in Chapter 4 on the comparisons between the conversions in PFR and CSTR sequences of the same total... 

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