Laminar flow through packed beds

Leva did extensive work on the pressure drop in packed beds of particles with various shapes [16]. He suggests the following equation for laminar flow through packed beds ... 

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

We see that Apjl, the frictional pressure drop per unit depth of bed, is made up of two components. The first term on the right-hand-side accounts for viscous (laminar) frictional losses, cc pu. and dominates at low Reynolds numbers. The second term on the right-hand-side accounts for the inertial (turbulent) frictional losses, oc pu2, and dominates at high Reynolds numbers. For further information about flow through packed beds, see Chapter 7 An Introduction to Particle Systems . 

The Sherwood number is also known as the Nusselt number for mass iransfer. Notice that the diameter of the catalyst pellet is used in the Reynolds and Sherwood numbers as the characteristic length dimension of the system. For flow through packed beds, the transition between laminar and turbulent flow occurs at a Reynolds number of approximately 40. 

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. 

Figure 9 shows the value of k as a function of the ratio of bed diameter over particle diameter, as determined from packed gas chromatographic columns (d). It can be seen that k tends to decrease as the diameter ratio increases, which implies that flow becomes more uniform. Whereas at low ratios k is inevitably high (of the same order of magnitude as in laminar flow through empty tubes) due to the wall effect, at higher diameter ratios k can vary more widely since its value depends upon whether the colunm is well or badly packed. 

Fig. 7.9 shows the effect ofpH on the measured haematite deposit mass per unit area with time. In these experiments the particulate concentration was 100 mg/kg and the Reynolds number 11,000. The high levels of deposit mass per unit area at a pH of around 6 may also be compared to the data on Fig. 7.10. The pH controls the magnitude and sign of charges on particles and substrate, as shown in the work of Matijevic [1982]. It may be possible with systems having these characteristics that particle deposition could be limited by suitable pH control. It is also interesting that the work of Newson et al [1988] with turbulent flow conditions gave similar results to Kuo and Matijevic [1980] for laminar flow through a packed bed.

High-performance liquid chromatography requires that liquid be pumped across a packed bed within a tubular configuration. Snyder and Kirkland have used the Hagen-Poiseuille equation for laminar flow through tubes... 

If there is a difference in total pressure across a particle, then there will be a direct contribution to the adsorption flux from forced laminar flow through the macropores. This effect is generally negligible in a packed bed since the pressure drop over an individual particle is very small. The effect may be of greater significance in the direct laboratory measurement of uptake rates in a vacuum system. From Poiseuille s equation it may be shown that the equivalent diffusivity is given by... 

How through packed beds under laminar conditions can be described by a model in which the flow is assumed to be through capillaries whose surface equals that of the solids comprising the bed. The capillary volume is set equal to the void volume of the bed. The model leads to the well-known Carman-Kozeny equation as follows ... 

This is a method of size analysis based on fluid flow through a packed bed (see Chapter 6). The Carman-Kozeny equation for laminar flow through a randomly packed bed of uniformly sized spheres of diameter x is [Equation 6.9] ... 

The Carman-Kozeny equation for laminar flow through a randomly packed bed of particles is ... 

The flow of a fluid through a packed bed of solid particles may be analysed in terms of the fluid flow through tubes. The starting point is the Hagen-Poiseuille equation for laminar flow through a tube ... 

Y Cohen, AB Metzner. Wall effects in laminar flow of fluids through packed beds. AIChE J 27 705-715, 1981. 

T. Cohen, Y. and A. B. Metzner. Wall Effects for Laminar Flow of Fluids through Packed Beds. AIChE J. 27 (l98l) 705-715. 

At low fluid velocities through packed beds of powders the laminar flow term predominates, whereas at higher velocities both viscous and kinetic effects are important. Er n and Oming [14] found that in the transitional region between laminar and turbulent flow, the equation relating pressure gradient and superflcial fluid velocity uf was ... 

The analysis of elution chromatography is a close parallel to that for adsorption. The best theoretical development, due to Golay, assumes the process takes place not in a packed bed but in a thin cylindrical tube with adsorbent-coated walls. Solvent moves steadily in laminar flow through the tube. At time zero a pulse of solutes is injected at one end of the tube. Each solute elutes with a concentration profile given by Eq. 4.4-31... 

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 ... 

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. 

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