Ideal plug flow reactor

The ideal plug flow reactor acts solely as a delaying element without changing the form of the input signal. In the case of an impulse function at the inlet, the same pulse function is obtained at the outlet after a time delay corresponding to the mean residence time 7. 

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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. 

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. 

The responses of this system to ideal step and pulse inputs are shown in Figure 11.3. Because the flow patterns in real tubular reactors will always involve some axial mixing and boundary layer flow near the walls of the vessels, they will distort the response curves for the ideal plug flow reactor. Consequently, the responses of a real tubular reactor to these inputs may look like those shown in Figure 11.3. 

Response of ideal plug flow reactor and real tubular reactor to step and impulse inputs. 

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-... 

To illustrate the analytical procedures, we can consider the reaction scheme in (17). For a batch reactor or an ideal plug-flow reactor, substitution of eqn. (25) into eqn. (23) shows that, when Cb =0... 

Note that the difference between this material balance and that for the ideal plug flow reactors of Chapter 5 is the inclusion of the two dispersion terms, because material enters and leaves the differential section not only by bulk flow but by dispersion as well. Entering all these terms into Eq. 17 and dividing by S AZ gives... 

In the ideal plug flow reactor, the flow traverses through the reactor hke a plug, with a uniform velocity profile and no diffusion in the longitudinal direction, as illustrated in Figure 6.2. A nonreactive tracer would travel through the reactor and leave with the same concentration versus time curve, except later. The mass transport equation is... 

In the ideal plug-flow reactor (Figure 11.16) the continuous phase flows as a plug through the reactor i.e., there is no mixing or, in other words, no axial dispersion. Consequently, if a compound is consumed or produced, a concentration gradient will exist in the direction of flow. The mass balance is therefore first set up over an infinite small slice perpendicular to the direction of the flow with volume dV of the bioreactor. Assuming steady state and F =Fq=F, Equation (11.5) then is reduced to ... 

In real tubular (or column) reactors there is, usually, a back-mixing effect which influences the performance of the ideal plug-flow reactor. This axial dispersion is higher for fluidized-bed reactors than for packed-bed reactors, although comparatively lower than for continuous-feed stirred-tank reactors, where the mixing is complete. 

The modeling of real immobilized-enzyme column reactors, mainly the fluidized-bed type, has been described (Emeiy and Cardoso, 1978 Allen, Charles and Coughlin, 1979 Kobayashi and Moo-Young, 1971) by mathematical models based on the dispersion concept (Levenspiel, 1972), by incorporation of an additional term to account for back-mixing in the ideal plug-flow reactor. This term describes the non-ideal effects in terms of a dispersion coefficient. 

Plug-flow tubular reactor (PFTR) This reactor is operated under steady-state condition. The reactor is of tubular shape, the reactants enter at the inlet and the composition is a function of the distance from the inlet. However, the composition is not a function of time. The ideal plug-flow reactor is characterized by the absence of mixing in the direction of flow and complete mixing in the transverse direction. 

Finally, for the ideal plug-flow reactor, die equations reduce to... 

For a typical packed bed, take the value of the Peclet number (udp/eDl) as 2. Dl is the dispersion coefficient in a packed tube. Then show that for a fractional conversion of 0.99 and a ratio of dp L of 0.02, the length L of the reactor with axial dispersion exceeds the length of the simple ideal plug-flow reactor by only 4.6 per cent where dp is the diameter of the particles. 

Thus, since FA0 = i>0- CA0, the performance equation of the ideal plug flow reactor can be written as... 

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. 

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. 

Upon simplification the resulting ideal plug flow reactor equation is... 

The RTD in a system is a measure of the degree to which fluid elements mix. In an ideal plug flow reactor, there is no mixing, while in a perfect mixer, the elements of different ages are uniformly mixed. A real process fluid is neither a macrofluid nor a microfluid, but tends toward one or the other of these extremes. Fluid mixing in a vessel, as reviewed in Chapter 7, is a complex process and can be analyzed on both macroscopic and microscopic scales. In a non-ideal system, there are irregularities that account for the fluid mixing of different... 

The reactive transport of contaminants in FePRBs has been modeled using several approaches [179,184,186,205-208]. The simplest approach treats the FePRB as an ideal plug-flow reactor (PFR), which is a steady-state flow reactor in which mixing (i.e., dispersion) and sorption are negligible. Removal rates (and therefore required wall widths, W) can be estimated based on first-order contaminant degradation and residence times calculated from the average linear groundwater velocity [Eq. (27)]. The usefulness of... 

In the literature many studies on LDPE tubular reactors are found (2-6).All these studies present models of the tubular reactor, able to predict the influence, on monomer conversion and temperature profiles, of selected variables such as initiator concentration and jacket temperature. With the exception of the models of Mullikin, that is an analog computer model of an idealized plug-flow reactor, and of Schoenemann and Thies, for which insufficient details are given, all the other models developed so far appear to have some limitations either in the basic hypotheses or in the fields of application. 

The reactor was considered being a ideal plug flow reactor when the rate constant for the HD formation was calculated. Because of experimental restrictions it was difficult to determine the HD formation rate when the H2-D2 reaction was close to equilibrium This will... 

For a sufficiently close approach to an ideal plug flow reactor, N or Pe should exceed a certain value which depends on the degree of conversion and reaction order. Gierman [20] refined the criterion of Mears [21] and arrived at eq. 9, which can also be used for monoliths. In the latter case the axial dispersion coefficient should be replaced by the smaller molecular diffusivity, thus relaxing the criterion for monolithic elements as compared to packed beds ... 

For an ideal plug flow reactor the mass balance in the steady state gives... 

Equations (59) and (60) are in a form often used for tubular, idealized, plug-flow reactors when the reaction rate is based on the volume of the reactor. When the reaction is heterogeneous, such as one occurring on the surface of a catalyst, it is common practice to base the reaction rate on the mass of the catalyst rather than on the volume of the reactor and substitute ric for r,. The resulting design equation equivalent to Eq. (60) is... 

The fluidized-bed reactor involves a rapid movement of the solid catalytic particles throughout the bed so that the operation can come close to one of uniform temperature throughout the reactor. The actual flow pattern for the operation of a fluidized bed is very complex and is between that for the ideal back-mix reactor and the ideal plug-flow reactor so that special methods for design may be required to approximate the real situation. 

These were calculated by numerical integration of Equation (8). SD must be less than SD if an ideal plug-flow reactor is to be designed and operated. 

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