Flow in Packed Beds

Separative flow often occurs in a packed bed—typically a tube filled with a granular material. Chromatography in packed columns is the most important example of packed-bed flow. Similar flow is found in porous membranes used for membrane separation. The fluid flowing through such media can be a gas, a liquid, or a supercritical fluid. 

The relationship between flow and pressure drop is expressed by Darcy s law [7,8], an empirical equation applied (starting in the last century) to numerous forms of porous media. This law may be thought of as the Ohm s law of fluid flow, another linear law of transport expressing, in this case, the flux density q of fluid (analogous to current density) in the form 

Unfortunately, Darcy s law says nothing about the microscopic flow profile in the packing. The most serious weakness of Darcy s law, however, is that K0 varies strongly with the kind of packing material with no hint on the nature of the dependence. This shortcoming is addressed below. 

This is the same expression as Eq. 4.12 except that capillary radius rc has been replaced by dcl2, where dc is the capillary diameter. 

One other factor influences flow. Even with the same particle size, a loose 

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Specchia, V., and Baldi, G., Pressure drop and liquid hold-up for two-phase concurrent flow in packed beds. Chem. Eng. Sci. 32, 515-523 (1977). 

So far, there have only been a few modeling studies to try to link local fluid flow to bed structure. Chu and Ng (1989) and later Bryant et al. (1993) and Thompson and Fogler (1997) used network models for flow in packed beds. Different beds were established using a computer simulation method for creating a random bed. The model beds were then reduced to a network of pores, and either flow/pressure drop relations or Stokes law was used to obtain a flow distribution. 

Membrane flux is denoted by J, the usual symbol in the literature on membranes. It corresponds with uc, as used in Chapters 4 and 7 for flow in packed beds and filtration. 

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. 

Experiments show that the dispersion model well represents flow in packed beds and in pipes. Thus theory and experiment give lyiud for these vessels. We summarize them in the next three charts. 

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 ... 

D.C. Dankworth and S. Sunderasan, Time dependent vertical gas-liquid flow in packed beds, Chem. Engng. Science, 47 (1992) 337-346. 

A.E. Saez and R.G. Carbonnel, Hydrodynamic parameters for gas-liquid cocurrent flow in packed beds, AIChE Journal, 31 (1985) 52-62. 

There are a number of pressure drop correlations for two-phase flow in packed beds originating from the Lockhart-Martinelli correlation for two-phase flow in pipes. These correlate the two-phase pressure drop to the single-phase pressure drops of the gas and the liquid obtained from the Ergun equation. See, for instance, the Larkins correlation [Larkins, White, and Jeffrey, AIChE J. 7 231 (1967)]... 

Different mathematical formulations of multidimensional fluid flow in packed beds like those developed by Jeschar (4) and Szeke-ley ( ) are likely to overcome the first difficulty. 

Whitaker, S., Forced Convection Heat-Transfer Correlations for Flow in Pipes, Past Flat Plated, Single Cylinders, Single Spheres, and Flow in Packed Beds and Tube Bundles , AIChE J.. Vol. 18, pp. 361-371.1972. 

Chapter 7, Reactor Design, discusses continuous and batch stirred-tank reactors and die packed-bed catalytic reactor, which are frequently used. Heat exchangers for stirred-tank reactors described are the simple jacket, simple jacket with a spiral baffle, simple jacket with agitation nozzles, partial pipe-coil jacket, dimple jacket, and the internal pipe coil. The amount of heat removed or added determines what jacket is selected. Other topics discussed are jacket pressure drop and mechanical considerations. Chapter 7 also describes methods for removing or adding heat in packed-bed catalytic reactors. Also considered are flow distribution methods to approach plug flow in packed beds. 

Figure 3. Pressure drop for brine-air flow in packed bed.

In almost all situations involving flow in packed-bed reactors, the amoimt of material transported by diffusion or dispersion in the axial direction is negligible compared witlh that transported by convection (i.e., bulk flow) ... 

Whitaker, S. Forced convection heat transfer correlations 26. for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles. 27. AIChE J. 1972, 18, 361-371. 

As explained above, cocurrent gas-liquid flow in packed beds, packing being either catalytic or inert, is advantageously employed in the petroleum and chemical industries. Successful modeling of mass transfer in packed-bed reactors requires careful study of the three-phase hydrodynamics— fluid flow patterns, pressure drops, and liquid holdup. 

Fig. 17. Flow pattern diagrams in gas-liquid downward flow in packed beds (C12). (a) Nonfoamable liquid, (b) foamable liquid.

Experimental Data Available in Literature for Cocurrent Gas-Liquid Downward Flow in Packed Bed... 

Fig. 18. Correlations for AtuO values for cocurrent downward flow in packed beds (CIO).

Jiang, Y, Khadilkar, M.R., Al-Dahhan, M.H. and Dudukovic, M.P. (2001), CFD modeling of multiphase flow in packed bed reactors fluid flow, bed structure and modeling issues, AIChE J., submitted for publication. 

Turpin, J.L. Hungtinton, R.L. Prediction of pressure drop for two-phase, two component co-current flow in packed beds. Am. Inst. Chem. Eng. J. 1967, 6, 1196. 

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