Fluid Flow Through a Packed Bed of Particles

The pressure drop experienced by a fluid as it flows through a packed bed of particles of diameter dp may be described by the Blake-Kozeny equation (6.1.4f), among others ... 

Permeametry, the measurement of the rate of flow of a fluid through a porous medium under a known pressure gradient, is a technique by means of which a mean particle size (but not a particle size distribution) can be determined. The equation for the rate of fluid flow through a packed bed of uniform spheres is the semiempirical Ergun equation... 

A certain amount of information on particle-to-liquid and bed-to-wall heat transfer is available for single-phase fluid flow through a packed bed. It is not clear, however, to what extent this information can be applied to... 

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 ... 

When the gas stream flows upwards through a packed bed of particles, there is a pressure drop due to the flow resistance from these particles. This pressure drop increases with the gas velocity based on the theory of fluid mechanics [64], A fluidised bed of particles is formed if the pressure drop across the bed is equal to the weight of the bed particles per unit area, i.e. the pressure (or the force in a unit area) for pushing the particle upwards to form the fluidised bed equals the weight of the particles of the unit area, as shown in Figure 3.36. 

J. Minimum velocity and porosity for fluidization. When a fluid flows upward through a packed bed of particles at low velocities, the particles remain stationary. As the fluid velocity is increased, the pressure drop increases according to the Ergun equation (3.1-20). Upon further increases in velocity, conditions finally occur where the force of the pressure drop times the cross-sectional area just equals the gravitational force on the mass of particles. Then the particles just begin to move, and this is the onset of fluidization or minimum fluidization. The fluid velocity at which fluidization begins is the minimum fluidization velocity v f in m/s based on the empty cross section of the tower (superficial velocity). 

What mean particle size do we use in calculating the pressure gradient for flow of a fluid through a packed bed of particles using the Carman-Kozeny equation (see Chapter 6) ... 

As an example of the application of the above analysis for flow through packed beds of particles, we will briefly consider cake filtration. Cake filtration is widely used in industry to separate solid particles from suspension in liquid. It involves the build up of a bed or cake of particles on a porous surface known as the filter medium, which commonly takes the form of a woven fabric. In cake filtration the pore size of the medium is less than the size of the particles to be filtered. It will be appreciated that this filtration process can be analysed in terms of the flow of fluid through a packed bed of particles, the depth of which is increasing with time. In practice the voidage of the cake may also change with time. However, we will first consider the case where the cake voidage is constant, i.e. an incompressible cake. 

There is a tendency for both axial and radial dispersion of mass to occur when a fluid flows through a packed bed. Since the bed diameter is normally far greater than the particle diameter in an adsorption bed, it is common not to have to consider the effects of radial dispersion. Hence the prevalence of plug and axially dispersed plug flow models in rigorous design methods. 

As the upward velocity of flow of fluid through a packed bed of uniform spheres is increased, the point of incipient fluidisation is reached when the particles are just supported in the fluid. The corresponding value of the minimum fluidising velocity (umf) is then obtained by substituting emf into equation 6.3 to give ... 

The generalized relation for the pressure drop for flows through a packed bed was formulated by Ergun (1952). The pressure loss was considered to be caused by simultaneous kinetic and viscous energy losses. In Ergun s formulation, four factors contribute to the pressure drop. They are (1) fluid flow rate, (2) properties of the fluid (such as viscosity and density), (3) closeness (such as porosity) and orientation of packing, and (4) size, shape, and surface of the solid particles. 

We have studied in earlier subsections how separation takes place as a fluid containing various species to be separated moves through a packed bed of adsorbents or chromatographic medium. As the fluid flows, there is a significant pressure drop as a certain level of separation is achieved. The question of interest here is what happens if the particle size is reduced This is part of a broader question how can one improve separation in a packed column based process ... 

Many events occur in the MTZ during adsorption which render the analysis complex. First, one or more adsorbates transfer from the fluid bulk by convection or diffusion across the fluid film which is external to the solid surface. Secondly, these adsorbates penetrate the particle by Maxwell, Knudsen and surface diffusion mechanisms (see Chapter 4), and adsorb onto the internal surface where the heat of adsorption is released. Heat may then be transferred to the adsorbent, to the flowing process fluid, and, via the vessel wall, to the surrounding environment. Heat and mass transfer may occur in the MTZ by bulk and diffusive flows in both the radial and axial directions. An additional complexity is that the flow through a packed bed may not be uniform across its entire cross-sectional area. This may be because of channelling of fluid at the wall or because of temperature gradients created when the heat of adsorption is released. 

To illustrate this idea, consider a reactor that is packed with catalyst particles. A fluid containing the reactant(s) flows through the fixed bed of particles. The reactor could be a differential reactor, as described in Chapter 6, or it could be an integral, ideal plug-flow reactor. Let s consider the PFR. The design equation is... 

Figure 1 Schematic examples of multiphase systems in which a discrete dispersed phase is moving through, or moved by, a continuous fluid phase. The discrete phase can be a solid (left), a gas (center), or a liquid (right). In many cases, inhomogeneous mesoscale structures appear in the spatial distribution of the discrete phase, caused by interplay of hydrodynamic flow and local energy dissipation. More complicated cases with three or more phases are also possible, such as encountered in slurry reactors (where solid particles are also present in the continuous liquid phase) or trickle bed reactors (where the droplets are sprayed on a packed bed of particles). To focus on the essentials, the topical sections will focus mostly on the two-phase examples depicted here.

The column length, as well as providing the required efficiency, is also defined by the D Arcy equation. The D Arcy equation describes the flow of a liquid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D Arcy equation for an incompressible fluid is given as follows. 

Filtration is the concentration of solids (or clarification of liquor) from slurry by fluid flow through a permeable medium. This normally takes the form of a membrane, filter leaf or packed bed, which restricts the particles, more than the fluid (Figure 4.4). 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

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