Laminar flow in tubular reactors

Similar balance equations with purely laminar diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.353) might suffice. 

Before proceeding to show how the RTD can be used to estimate conversion i a reactor, we shall derive (/) for a laminar flow reactor. For laminar flow in tubular reactor, the velocity proflie is parabolic, with the fluid in the center c the tube spending the shortest time in the reactor. A schematic diagram of th fluid movement after a time t is shown in Figure 13-8. The figure at the lei shows how far down the reactor each concentric fluid element has traveler after a time /,... 

By changing D x, one may vary the reactor performance from plug flo v [Dax/(wl) = 0 or Bo = oo] to a CSTR [Dax/(w I) = oo or Bo = 0]. At first sight, this simple model appears to account only for axial mixing effects. However, this approach not only compensates for problems caused by axial mixing but also for those related to radial mixing and non-uniform velocity profiles (Aris, 1956), as shown in Section 4.10.6.3 for laminar flow in tubular reactors. 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

Flow in tubular reactors can be laminar, as with viscous fluids in small-diameter tubes, and greatly deviate from ideal plug-flow behavior, or turbulent, as with gases, and consequently closer to the ideal (Fig. 2). Turbulent flow generally is preferred to laminar flow, because mixing and heat transfer... 

Experimental work with styrene in tubular reactors has been reported (39) where viscosities were relatively low due to conversions below 32%. However, Lynn ( ) has concluded that a laminar flow tubular reactor for styrene polymerization is probably technically infeasible due to the distortion in velocity... 

The final idealized flow situation that we will consider is laminar flow in a tubular reactor in the absence of either radial or longitudinal diffusion. The velocity profile in such a reactor is given by... 

For laminar flow in an isothermal tubular reactor of length h and velocity v, calculate the... 

To consolidate the experimental screening data quantitatively it is desirable to obtain information on the fluid mechanics of the reactant flow in the reactor. Experimental data are difficult to evaluate if the experimental conditions and, especially, the fluid dynamic behavior of the reactants flow are not known. This is, for example, the case in a typical tubular reactor filled with a packed bed of porous beads. The porosity of the beads in combination with the unknown flow of the reactants around the beads makes it difficult to describe the flow close to the catalyst surface. A way to achieve a well-described flow in the reactor is to reduce its dimensions. This reduces the Reynolds number to a region of laminar flow conditions, which can be described analytically. 

Figure 8-22 shows the F(0) curves for laminar flow in a tubular reactor and for other idealized flow patterns. 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

The polymerization time in continuous processes depends on the time the reactants spend in the reactor. The contents of a batch reactor will all have the same residence time, since they are introduced and removed from the vessel at the same times. The continuous flow tubular reactor has the next narrowest residence time distribution, if flow in the reactor is truly plug-like (i.e., not laminar). These two reactors are best adapted for achieving high conversions, while a CSTR cannot provide high conversion, by definition of its operation. The residence time distribution of the CSTR contents is broader than those of the former types. A cascade of CSTR s will approach the behavior of a plug flow continuous reactor. 

Gas-phase reacdotis are carried out primarily in tubular reactors where the flow is generally turbulent. By assuming that there is no dispersion and ttiere are no radial gradients in either temperature, velocity, or concentration, we can model the flow in the reactor as plug-flow. Laminar reactors are discussed in Chapter 13 and dispersion effects in Chapter 14. The differential form of the design equation... 

Dispersion in a Tubular Reactor with Laminar Flow. In a laminar flow reaetor we know that the axial veloeity varies in the radial direetion according to the Hagen-Poiseuille equation ... 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

Determine the RTD for an isothermal tubular-flow reactor in which the liquid is in laminar flow in an annulus of inner radius and outer radius r2 rjr2 = a). Neglect molecular diffusion. The velocity in the axial direction at any radius r (between i and r2) is given by... 

Fig. 2.2-4 Cumulative residence-time function F t/z) for an ideal continuous stirred tank (1), an ideal tubular reactor with plug flow (2), and laminar flow in a tubular reactor (3).

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

Figure 3.15 Axial dispersion in tubular reactors (a) laminar flow and (b) turbulent flow. Gray area represents experimental results. (Adapted from [6], Figure 27.25 Copyright 2012, Wiley-VCH GmbH Co. KGaA.)...

As mentioned above, radial mixing is crucial to get narrow RTD. Therefore, the use of passive mixer helps equalizing the radial concentration in the laminar flow domain. It is known that static mixer allows to obtain narrow RTD in tubular reactors even with high viscous media [18]. The beneflcial effect of radial mixing can also be expected in microstructured mixers. Boskovic et al. [19,... 

The study of polymerization in a tubular reactor involves simultaneous consideration of the velocity profile, heat release and conduction, as well as the kinetics of the polymerization. This work was undertaken to determine the effect of this type of reactor upon the molecular weight distribution of the polymer produced. The system studied was the copolymerization of an elastomer from ethylene, propylene, and 1,4-hexadlene In solution using Ziegler catalysis. Laminar flow polymerization similar to previously described work involving polymerization of sytrene in tubular reactors (1, 2, was analyzed, but the numerical... 

Under some circumstances, the Dispersion model can be applied to laminar flow tubular reactors. If either 4Df/(Din) > O.BOorD/wDin < 1, the Dispersion number for laminar flow in long, empty tubes is given by... 

A summary of the nine batch reactor emulsion polymerizations and fifteen tubular reactor emulsion polymerizations are presented in Tables III IV. Also, many tubular reactor pressure drop measurements were performed at different Reynolds numbers using distilled water to determined the laminar-turbulent transitional flow regime. 

The styrene conversion versus reaction time results for runs in the laminar flow regime are plotted in Figure 8. Both the rate of polymerization and the styrene conversion increase with increasing flow rate as noted previously (7). The conversion profile for the batch experimental run (B-3) is presented as a dashed line for comparison. It can be seen that the polymerization rates for runs with (Nj e e 2850 are greater than the corresponding batch polymerization with a conversion plateau being reached after about thirty minutes of reaction. This behavior is similar to the results obtained in a closed loop tubular reactor (7J) and is probably due to an excessively rapid consumption of initiator in a... 

Figure 7. Monomer conversion vs, polymerization time in the helical tubular reactor laminar flow regime...

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