PFR

A functional is a function of a function. Electron probability density p is a function p(r) of a point in space located by radius vector r measured from an origin (possibly an atomic mi dens), and the energy E of an electron distribution is a function of its probability density. E /(p). Therefore E is a functional of r denoted E [pfr). ... 

In another land of ideal flow reactor, all portions of the feed stream have the same residence time that is, there is no mixing in the axial direction but complete mixing radially. It is called a.plugflow reactor (PFR), or a tubular flow reactor (TFR), because this flow pattern is characteristic of tubes and pipes. As the reaction proceeds, the concentration falls off with distance. 

FIG. 7-3 Concentration profiles in fiatch and continuous flow a) fiatch time profile, (h) semifiatcli time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas. 

Selectivity A significant respect in which CSTRs may differ from batch (or PFR) reaclors is in the product distribution of complex reactions. However, each particular set of reactions must be treated individually to find the superiority. For the consecutive reactions A B C, Fig. 7-5b shows that a higher peak value of B is reached in batch reactors than in CSTRs as the number of stages increases the batch performance is approached. 

TABLE 7-7 Material and Energy Balances of a Plug Flow Reactor (PFR)... 

Say the recycle flow rate in a PFR is V and the fresh feed rate is Vq, with the ratio R = V /Vq. With a fresh feed concentration of Cq and a produc t of Co the composite feed concentration is... 

TABLE 7-9 Integration of Rate Equations of a PFR at Constant Pressure... 

Reac tors that are nominally CSTRs or PFRs may in practice deviate substantially from ideal mixing or nonmixing. This topic is developed at length in Sec. 23, so only a few summary statements are made here. More information about this topic also may be found in Nauman and Buffham (Mixing in Continuous Flow Systems, Wiley, 1983). 

The RTD is a distinctive characteristic of mixing behavior. In Fig. 7-2>e, the CSTR has an RTD that varies as the negative exponential of the time and the PFR is represented by a vertical line at = 1. Multistage units and many packed beds have beU-shaped RTDs, like that of... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

In a sequence of PFR and CSTR, better performance is obtained with the PFR last. Performance of reversible reactions is improved with the CSTR at a higher temperature. 

For the consecutive reactions A B C, a higher yield of intermediate B is obtained in batch reac tors or PFRs than in CSTRs. 

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with re.sidence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

Simple combinations of reactor elements can be solved direc tly. Figure 23-8, for instance, shows two CSTRs in series and with recycle through a PFR. The material balances with an /i-order reaction / = /cC are... 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

The concentrations of all stages are found in succession when Cq is known. For a PFR,... 

PFRs, under isothermal, adiabatic, or heat transfer conditions in one or two phases. Outputs can provide profiles of composition, pressure, and temperature as well as vessel size. 

Figure 23-7 illustrates the responses of CSTRs and PFRs to impmse or step inputs of tracers. 

Figure 23-8 develops the overall transform of a process with a PFR in parallel with two CSTRs in series. C(t) is found from C(.s) by inversion of the output transform. 

FIG. 23-7 Imp ulse and step inputs and responses. Typical, PFR and CSTR. (a) Experiment with impulse input of tracer, (h) Typical behavior area between ordinates at tg and ty equals the fraction of the tracer with residence time in that range, (c) Plug flow behavior all molecules have the same residence time, (d) Completely mixed vessel residence times range between zero and infinity, e) Experiment with step input of tracer initial concentration zero. (/) Typical behavior fraction with ages between and ty equals the difference between the ordinates, h — a. (g) Plug flow behavior zero response until t =t has elapsed, then constant concentration Cy. (h) Completely mixed behavior response begins at once, and ultimately reaches feed concentration. 

Plug Flow Reactor (PFR) The material balance over a differential vohime dV) is... 

Instances of multiplicities in CSTR batteries and in PFRs also can be developed. 

Plug Flow Reactor (PFR) A plug flow reactor is a tubular reactor where the feed is continuously introduced at one end and the products continuously removed from the other end. The concentration/temperature profile in the reactor varies with position. 

Figure 14. Koch Plastic Flexiring. (Courtesy of Koch Engineering Co., Inc., Bui. PFR-1.)...

Wecanseethafwiththisparticular flowratetheconcentrationsexitingattheendofthe PFR correspond to approximately 10%conversion. Would thishavechangedif the radius were smaller,say, only l Theanswerisno. The reasonisthattheholdingtimewillbethesame becausethevolumeandflowratearethesame. 

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