Packed beds laminar flow

Ca.rma.n-KozenyEfjua.tion, Flow through packed beds under laminar conditions can be described by the Carman-Kozeny equation in the... 

Flow in empty tubes has a relatively narrow band of velocities—or Reynolds numbers from 2000 to 10000—wherein the character changes from laminar to turbulent. In packed beds, even the laminar flow does not mean that motion is linear or parallel to the surface. Due to the many turns between particles, stable eddies develop and therefore the difference between laminar and turbulent flow is not as pronounced as in empty tubes. 

However, on die basis of the relation between pressure drop and die minimum fluidisation velocity of particles, the point of transition between a packed bed and a fluidised bed has been correlated by Ergun41 using (17.7.2.3). This is obtained by summing the pressure drop terms for laminar and turbulent flow regions. 

For packed beds in either turbulent or laminar flow, the Ergun equation is often satisfactory ... 

This equation has the same functional dependence on p (namely none) and m as the Poiseuille equation that governs laminar flow in an empty tube. Thus, laminar flow packed beds scale in series exactly like laminar flow in empty tubes. See the previous sections on series scaleup of liquids and gases in laminar flow. 

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. 

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

When running a CFD simulation, a decision must be made as to whether to use a laminar-flow or a turbulent-flow model. For many flow situations, the transition from laminar to turbulent flow with increasing flow rate is quite sharp, for example, at Re — 2100 for flow in an empty tube. For flow in a fixed bed, the situation is more complicated, with the laminar to turbulent transition taking place over a range of Re, which is dependent on the type of packing and on the position within the bed. 

If a fluid is passed upwards in laminar flow through a packed bed of solid particles the superficial velocity u is related to the pressure drop AP by equation 9.33 ... 

The value of the permeability coefficient is frequently used to give an indication of the ease with which a fluid will flow through a bed of particles or a filter medium. Some values of B for various packings, taken from Eisenklam(2), are shown in Table 4.1, and it can be seen that B can vary over a wide range of values. It should be noted that these values of B apply only to the laminar flow regime. 

For laminar flow of a power-law fluid through a packed bed, Kemblowski et alS25) have developed an analogous Reynolds number (Rei) , which they have used as the basis for the calculation of the pressure drop for the flow of power-law fluids ... 

This relationship is based on an assumption of laminar flow between particles in a packed bed. Consequently the particle Reynolds number is limited to Re j < 20 (Kunii and Levenspiel, 1991) where... 

In a packed bed, the transition between laminar and turbulent flow occurs in the region of Rep 40 Rep is the Reynolds number based on the equivalent particle... 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Checks on the relationships between the axial coefficients were provided in empty tubes with laminar flow by Taylor (T2), Blackwell (B15), Bournia et al. (B19), and van Deemter, Breeder and Lauwerier (V3), and for turbulent flow by Taylor (T4) and Tichacek et al. (T8). The agreement of experiment and theory in all of these cases was satisfactory, except for the data of Boumia et al. as discussed previously, their data indicated that the simple axial-dispersed plug-flow treatment may not be valid for laminar flow of gases. Tichacek et al. found that the theoretical calculations were extremely sensitive to the velocity profile. Converse (C20), and Bischoff and Levenspiel (B14) showed that rough agreement was also obtained in packed beds. Here, of course, the theoretical calculation was very approximate because of the scatter in packed-bed velocity-profile data. 

The use of the tanks-in-series model for packed beds can be more strongly justified. The fluid can be visualized as moving from one void space to another through the restrictions between particles. If the fluid in each void space were perfectly mixed, the mixing could be represented by a series of stirred tanks each with a size the order of magnitude of the particle. This has been discussed in detail by Aris and Amundson (A14). The fluid in the void spaces is not perfectly mixed, and so an efficiency of mixing in the void spaces has to be introduced (C6). This means that the analogy is somewhat spoiled and the model loses some of its attractiveness. In laminar flow the tanks-in-series model may be still less applicable. 

Laminar flow of a fluid in a packed bed may often be described by the Carman-Kozeny Equation (6) ... 

The up-scaling from microreactor to small monoliths principally deals with the change of geometry (from powdered to honeycomb catalyst) and fluid dynamics (from turbulent flow in packed-bed to laminar flow in monolith channels). In this respect, it involves therefore moving closer to the conditions prevailing in the real full-scale monolithic converter, while still operating, however, under well controlled laboratory conditions, involving, e.g. the use of synthetic gas mixtures. 

For packed beds of naphthalene and caffeine, Lim et al. [28] took into consideration mass--transfer by both types of convection in opposite flows. Their equation for laminar flow in packed beds (10 to 203 bar and 35 to 45°C) is as follows ... 

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. 

For the fluidized bed process the bed expansion as a consequence of an increase in linear flow rate has to be considered. In a simplified picture diffusive transport takes place in a boundary layer around the matrix particle which is frequently renewed, this frequency being dependent on velocity and voidage, as long as convective effects, e.g. the movement of particles are neglected. Rowe [74] has included these considerations into his correlation for kf in fluidized beds, which is applicable for a wide range of Reynolds numbers, including the laminar flow regime where fluidized bed adsorption of proteins takes place (Eq. 19). The exponent m is set to 1 for a liquid fluidized bed, a represents the proportionality factor in the correlation for packed beds (Eq. 18) and is assumed as 1.45. 

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