The laminar flow reactor

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 

Note that, in a laminar-flow tubular reactor, the material on the reactor centre line has the highest velocity, this being exactly twice the average velocity, Q/A, for the whole reactor. This means that, following any tracer test, no response will be observed until the elapsed time exceeds one half of the reactor space time or mean residence time. The following values for 0 and F(0) emphasise the form of the cumulative RTD and the fact that, even up to 10 residence times after a tracer impulse test, 0.25% of the tracer will not have been eluted from the system. 

By definition, the laminar-flow reactor is segregated. Each radial element of fluid is assumed to slide past its adjacent elements with no mixing. Thus, eqn. (34) may be used to predict reactor conversion. The case of a first-order reaction has been analysed by Cleland and Wilhelm [43]. As we have seen previously 

The exponential integral in eqn. (56) is a standard tabulated function [33], so predictions of conversion can be made and are plainly a function of Tk alone. Such calculations have been performed by various authors [43—46]. Hilder [47], when repeating these calculations, found eqn. (57) to be a simple and adequate approximation to eqn. (56). 

Data on residual reactant concentration as a function of Damkohler number for the PER and laminar flow reactors. 

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The performance of the laminar flow reactor is appreciably worse than that of a PFR, but remains better than that of a CSTR (which gives T=0.5 for kt= 1). The computed value of 0.4432 may be useful in validating more complicated codes that include diffusion. 

Example 8.4 Suppose that the reactive component in the laminar flow reactor of Example 8.2 has a diffusivity of 5x 10 m /s. Calculate the minimum number of axial steps, J, needed for discretization stability when the radial increments are sized using 7=4, 8, 16, 32, 64, and 128. Also, suggest some actual step sizes that would be reasonable to use. 

The models of Chapter 9 contain at least one empirical parameter. This parameter is used to account for complex flow fields that are not deterministic, time-invariant, and calculable. We are specifically concerned with packed-bed reactors, turbulent-flow reactors, and static mixers (also known as motionless mixers). We begin with packed-bed reactors because they are ubiquitous within the petrochemical industry and because their mathematical treatment closely parallels that of the laminar flow reactors in Chapter 8. 

FIGURE 13.9 Curved streamlines resulting from a polycondensation in the laminar flow reactor of Example 13.10. 

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. 

In addition to the CSTR and batch reactors, another type of reactor commonly used in industry is the tubular reactor. It consists of a cylindrical pipe and is normally operated at steady state, as is the CSTR. For the purposes of the material presented here, we consider systems in which the flow is highly turbulent and the flow field may be modeled by that of plug flow. That is, there is no radial variation in concentration and the reactor is referred to as a plug-flow reactor (PFR). (The laminar flow reactor is discussed in Chapter 13.)... 

In Example 6-1 DJuL is evaluated with the response curve for the laminar-flow reactor described in Sec. 6-4 (Fig. 6-7). Example 6-2 treats the same problem, but with other response data. 

The series-of-stirred-tanks model could not represent the RTD for the laminar-flow reactor shown in Fig. 6-7. However, the RTD data given in Example 6-2 can be simulated approximately. The dashed curve in Fig. 6-1 Oh is a plot of this RTD. While no integer value of n coincides with this curve for all 6/6, the curve for = 5 gives approximately the correct shape. Comparison of the fit in Figs. 6-9 and 6-lOh indicates that about the same... 

Example 6-4 Consider the laminar-flow reactor described in Sec. 6-4 and calculate the conversion for a first-order reaction for which k- = 0.1. sec and 0 = 10 sec. 

Calculate the conversion for the laminar-flow reactor of Prob. 6-7, using the dispersion model to represent the actual RTD. 

Compare this result with the exact analytical formula for the laminar flow reactor with a second-order reaction... 

Figure 8.1 gives conversion curves for an isothermal, first-order reaction in various types of reactor. The curves for a PFR and CSTR are from Equations 1.38 and 1.49. The curve for laminar flow without diffusion is obtained from Equation 8.14 and the software of Example 8.2. Without diffusion, the laminar flow reactor performs better than a CSTR but worse that a PFR. Add radial diffusion and the performance improves. This is illustrated by the curve in Figure 8.1 that is between those for laminar flow without di ffusion and piston flow. The intermediate curve is one member of a family of such curves that depends on theparameter f// . IfL // is small, <... 

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. 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

In general, analytical solutions to the laminar-flow reactor are not convenient to work with because of the awkward forms that arise from the concentration averaging of equation (4-58). For first-order irreversible kinetics, the design equation (4-57), becomes... 

Compare the experimental (Ca)/Caj, data with the predictions of equation (4-107) for the laminar-flow reactor. 

If k /k is 0.8, the laminar-flow reactor would have to be 1/0.8, or 1.25, times longer than a plug flow reactor with the same conversion. Values of k jk are given in Table 6.1. 

This represents the laminar-flow reactor which is characterized by a parabolic radial velocity profile and molecular radial diffusion. Hence the radial diffusivity is not exactly zero, and the corresponding Peclet number is finite (see Chapter 13). 

A plot of E(0) versus 0 for the laminar flow reactor is shown in Figure 3.65. 

Each one of the fluid elements, which is a completely segregated cluster of fluid molecules, can be treated as a micro-batch reactor. The residence time 0 of a fluid element is taken as the batch reaction time to determine the conversion achieved in the fluid element. Consider a first-order reaction A—carried out in the laminar flow reactor, (-ni) = kCA is the kinetic rate equation. The rate of change of reactant concentration in a single fluid element (treated as a batch reactor) is given by... 

Thus, the conversion in the laminar flow reactor is lower than the conversion (80%) in the ideal PFR of the same size. 

Fig. 7.1. Measured hysteresis in our reaction system [1]. Plot of the redox potential of Br, Fpt, as a function of the flow rate coefficient fct (in units of reciprocal residence times, the time spent by a volume element in the laminar flow reactor (LFR)). Filled dots represent one of the stable stationary states (the oxidized state) and empty dots the other stable state, the reduced state. Prom [1]...

Fig. 7.2. Schematic diagram of the apparatus. Each solution, one corresponding to one stable stationary state and the other to the other stationary state, is stored in one of two continuous-stirred tank reactors (CSTR) and pumped at a determined and variable rates through the laminar flow reactor (LFR), where they are brought in contact with each other in a sharp well-defined boundary. For the remainder of the definitions see the text. Prom [1]...

Fig. 7.3. Intensity profile aeross the interface of the two stationary states brought in contact in the laminar flow reactor. Both solutions flowed through the reactor at a rate such that there is no time for diffusion to occur. This measurement is taken prior to the start of the experiment itself. Prom [1]...

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