Plug flow reactor residence time distributions

A reasonable assumption for the fluidized-bed reactor is that the fluid is partially mixed, whereas the solid is fully mixed. If we assume that the fluid is in plug flow, the residence time distribution of the solids will be given by exp(-t). As for the fixed-bed reactor, we shall first consider exponential decay. Therefore, the average rate constant to be used in solving Equation 12.46 is... 

In the previous section, we learned that not all fluid elements spend exactly the same time in a reactor, excqit for the special case of an ideal, plug-flow reactor. Residence tune distribution functions provide a quantitative way to describe how much time a flowing fluid spends in a reactor. Residence time distribution functions can be obtained from tracer response curves. 

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

The most commonly used model of a mixed vessel is the fractional tubularity (delay-lag) model in which some part of the reactor is taken as exhibiting plug-flow conditions and contributing a delay and the rest of the reactor is taken as perfectly mixed (uniform concentrations) contributing a first-order lag (/( ,). The delay and lag in series are taken as describing the reactor residence time distribution (RTD). The delay-lag representation was validated using both CFD analysis and experimental residence time distributions (Walsh, 1993). 

Ideal plug flow behavion The behavior of the reactor (residence time distribution) should be such that we can consider the fixed bed as an ideal plug flow reactor (PFR). If this condition is fidfiDed we can use the (relatively simple) equations valid for a PFR that correlate the conversion with the rate constant, residence time, and initial reactant concentration. For example, we can determine the rate constant for a reaction with order n by Eq. (4.10.26) if we have measured the conversion of reactant A at a given value of the residence time by ... 

This situation changes in liquid-phase reactions if the cross-sectional dimension of the flow reactor is reduced up to several millimeters and the dimensions of microstmctured reactors are reached. Due to shorter radial diffusion path from the bulk to the edge and vice versa, the mean diffusion time becomes substantially shorter than the required mean residence time. As a result, the typical laminar flow profile is blurred by the radial diffusion and the flow profile is similar to the turbulent flow. The residence time distribution at the reactor exit becomes narrower and approaches that of the plug flow. 

FIG. 7.50. Plot of extent of reaction aganst dimensionless reactor residence time for spherical particles made up of spherical grains in a batch or plug flow reactor. Particle size distribution given by the Gates-Gaudin-Schumann equation. Fg = 3. 

Topics that acquire special importance on the industrial scale are the quality of mixing in tanks and the residence time distribution in vessels where plug flow may be the goal. The information about agitation in tanks described for gas/liquid and slurry reactions is largely apphcable here. The relation between heat transfer and agitation also is discussed elsewhere in this Handbook. Residence time distribution is covered at length under Reactor Efficiency. A special case is that of laminar and related flow distributions characteristic of non-Newtonian fluids, which often occiu s in polymerization reactors. 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

Unlike the situation in a plug flow reactor, the various fluid elements mix with one another in a CSTR. In the limit of perfect mixing, a tracer molecule that enters at the reactor inlet has equal probability of being anywhere in the vessel after an infinitesimally small time increment. Thus all fluid elements in the reactor have equal probability of leaving in the next time increment. Consequently there will be a broad distribution of residence times for various tracer molecules. The character of the distribution is discussed in Section 11.1. Because some of the... 

Except for the case of an ideal plug flow reactor, different fluid elements will take different lengths of time to flow through a chemical reactor. In order to be able to predict the behavior of a given piece of equipment as a chemical reactor, one must be able to determine how long different fluid elements remain in the reactor. One does this by measuring the response of the effluent stream to changes in the concentration of inert species in the feed stream—the so-called stimulus-response technique. In this section we will discuss the analytical form in which the distribution of residence times is cast, derive relationships of this type for various reactor models, and illustrate how experimental data are treated in order to determine the distribution function. 

For a few highly idealized systems, the residence time distribution function can be determined a priori without the need for experimental work. These systems include our two idealized flow reactors—the plug flow reactor and the continuous stirred tank reactor—and the tubular laminar flow reactor. The F(t) and response curves for each of these three types of well-characterized flow patterns will be developed in turn. 

Note that in this case the right side of equation 11.1.68 is zero for t = 0 and unity for t = 00. Figure 11.9 contains several F(t) curves for various values of n. As n increases, the spread in residence time decreases. In the limit, as n approaches infinity the F(t) curve approaches that for an ideal plug flow reactor. If the residence time distribution function given by 11.1.69 is differentiated, one obtains an... 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Figure 8-5 Residence time distribution in a laminar flow tubular reactor. The dashed curve indicates the p t) curve expected in laminar flow after allowing for radial diffusion, which makes p(t) closer to the plug flow.

The next thing we note about chromatography is that it is equivalent to tracer injection into a PFTR. Whereas in Chapter 8 we used tracer injection to determine the residence time distribution in a reactor, here we have nearly plug flow (with the pulse spread somewhat by dispersion), but adsorption from the fluid phase onto the solid reduces the flow velocity and increases the residence time to be much longer than x. ... 

The apparatus consisted of a flow reactor containing the coal and an oxygen absorber. A fixed bed reactor was used for studies on large coal particles, while a spout reactor was used for studies on small coal particles. Both reactors had a volume of about 400 cc. and were designed so that both wall and end effects were eliminated. Experimental residence time distributions indicated that the fixed bed reactor approximated a plug flow reactor, while the fluidized bed spout reactor had perfect mixing. 

A graphical representation of the cumulative residence time distribution function is given in Figure 4.97 for a structured well, a laminar flow reactor and an ideal plug flow reactor assuming the same average residence time and mean velocity in each reactor. 

Obviously the characteristic distribution of the structured square, as expected, is much closer to the ideal plug flow reactor than to the laminar flow reactor. This desired behavior is a result of the channel walls, which are flow-guiding elements and pressure resistors to the flow at the same time. Two of the streamlines are projecting with a residence time of more than 0.4 s. These are the streamlines passing the area close to the wall of the distribution area, which introduces a larger resistance to these particles due to wall friction. This could, for example, be accounted for by a different channel width between the near wall channels and the central channels. 

Figure 4.97 Calculated cumulative residence time distribution function for a multi-channel well, a laminar flow reactor and a plug flow reactor [147] (by courtesy of VDI-Verlag GmbH).

Using these results Van de Velden et al. (2008) were able to recommend design rules for operation of such reactors in terms of the gas velocity/solids loading parameters. For example, to ensure a narrow residence time distribution (operating, in effect, in plug flow), the superficial gas velocity should exceed the transport velocity by approximately lm/s and the solids circulation rate should exceed 200kg/m2s. 

An important advantage of the use of EOF to pump liquids in a micro-channel network is that the velocity over the microchannel cross section is constant, in contrast to pressure-driven (Poisseuille) flow, which exhibits a parabolic velocity profile. EOF-based microreactors therefore are nearly ideal plug-flow reactors, with corresponding narrow residence time distribution, which improves reaction selectivity. 

Tubular reactor advantages include their well-defined residence time distributions, turbulent mixing reactants, ease of obtaining and applying kinetic data, efficient use of reactor volume, and mechanical simplicity. However, great care must be taken when applying the correct flow model (e.g., plug... 

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