Reactor laminar flow

Kirelic/lranspon for Isothermal Laminar-Flow Reactor with no Axial Dispersion [See Shinohara and Christiansen (I974J for ilie non-isoihermul... 

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

Solution This is the simplest, nontrivial example of a laminar flow reactor. The solution begins by integrating Equation (8.2) for a specific streamline that corresponds to radial position r. The result is... 

This integral is a special function related to the incomplete gamma function. The solution can be considered to be analytical even though the function may be unfamiliar. Figure 8.1 illustrates the behavior of Equation (8.8) as compared with CSTRs, PFRs, and laminar flow reactors with diffusion. 

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is... 

The hnal step in the design calculations for a laminar flow reactor is determination of mixing-cup averages based on Equation (8.4). The trapezoidal rule is recommended for this numerical integration because it is easy to implement and because it converges O(Ar ) in keeping with the rest of the calculations. 

The next few examples show the various numerical methods for a simple laminar flow reactor, gradually adding complications. 

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. 

Example 8.6 Generalize Example 8.5 to determine the fraction unreacted for a first-order reaction in a laminar flow reactor as a function of the dimensionless groups and kt. Treat the case of a parabolic velocity profile. 

Figure 8.1 includes a curve for laminar flow with 3>AtlR = 0.1. The performance of a laminar flow reactor with diffusion is intermediate between piston flow and laminar flow without diffusion, aVI = 0. Laminar flow reactors give better conversion than CSTRs, but do not generalize this result too far It is restricted to a parabolic velocity profile. Laminar velocity profiles exist that, in the absence of diffusion, give reactor performance far worse than a CSTR. 

Chapter 3 introduced the basic concepts of scaleup for tubular reactors. The theory developed in this chapter allows scaleup of laminar flow reactors on a more substantive basis. Model-based scaleup supposes that the reactor is reasonably well understood at the pilot scale and that a model of the proposed plant-scale reactor predicts performance that is acceptable, although possibly worse than that achieved in the pilot reactor. So be it. If you trust the model, go for it. The alternative is blind scaleup, where the pilot reactor produces good product and where the scaleup is based on general principles and high hopes. There are situations where blind scaleup is the best choice based on business considerations but given your druthers, go for model-based scaleup. 

The temperature counterpart of Q>aVR ccj-F/R and if ccj-F/R is low enough, then the reactor will be adiabatic. Since aj 3>a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T i, will be the same in the small and large reactors, and blind scaleup is possible. More commonly, ari/R wiU be so large that radial diffusion of heat will be significant in the small reactor. The extent of radial diffusion will lessen upon scaleup, leading to the possibility of thermal runaway. If model-based scaleup predicts a reasonable outcome, go for it. Otherwise, consider scaling in series or parallel. 

Consider an isothermal, laminar flow reactor with a parabolic velocity profile. Suppose an elementary, second-order reaction of the form A -h B P with rate SR- = kab is occurring with kui 1=2. Assume aj = bi . Find Uoutlam for the following cases ... 

Determine the opposite of the Merrill and Hamlin criterion. That is, find the value of QIaVR above which a laminar flow reactor closely... 

Practical applications to laminar flow reactors are still mainly in the research literature. The first good treatment of a variable-viscosity reactor is... 

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. 

Turbulent flow reactors are modeled quite differently from laminar flow reactors. In a turbulent flow field, nonzero velocity components exist in all three coordinate directions, and they fluctuate with time. Statistical methods must be used to obtain time average values for the various components and to characterize the instantaneous fluctuations about these averages. We divide the velocity into time average and fluctuating parts ... 

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R > 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

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

Example 15.6 Determine the washout function if a diffusion-free, laminar flow reactor is put in a recycle loop. Assume that 75% of the reactor effluent is recycled per pass. 

FIGURE 15.3 Effect of recycle on a laminar flow reactor. 

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. 

Calculate the volumes of a plug flow reactor and a laminar flow reactor required to process 0.5 m3/ksec of feed containing 1.0 kmole/m3 of species A to 95% conversion. The liquid phase... 

Laminar-flow reactor (LFR), based on laminar flow. 

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