Fluid flow through packed beds

In a packed bed of unit volume, the volumes occupied by the voids and the solid particles are e and (1 — e) respectively where e is the voidage fraction or porosity of the bed. Let S0 be the surface area per unit volume of the solid material in the bed. Thus the total surface area in a packed bed of unit volume is (1 — e)S0. 

For a spherical particle of diameter dp the value of S = 6ldp. For a non-spherical particle with an average particle diameter dp, the value Sa = 6/(dpip) where p = 1 for a spherical particle. Values of tp for other shapes are readily available [Perry (1984)]. 

An equivalent diameter de for flow through the bed can be defined as four times the cross-sectional flow area divided by the appropriate flow perimeter. For a random packing, this is equal to four times the volume occupied by the fluid divided by the surface area of particles in contact with the fluid. 

The velocity calculated by dividing the volumetric flow rate by the whole cross-sectional area of the bed is known as the superficial velocity u. The mean velocity within the interstices of the bed is then ub = ule. 

A Reynolds number for flow through a packed bed can be defined as 

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For further information on fluid flow through packed beds and on filtration the reader is referred to the following ... 

Ergun, S. (1952) Fluid flow through packed beds. Chem. Eng. Prog., 48, 89-94. 

These few examples of percolation problems evidence the influence that percolation structures may have cxi fluid flow hydrodynamics when fluid-solid interaction is the factor determining fluid flow through packed beds (i.e. at low gas-liquid interactions). 

Substitution of Eqs. (6.19) and (6.20) into Eqs. (6.15) and (6.16) leads to the following general equations for fluid flow through packed beds ... 

Rong LW, Dong KJ, Yu AB Lattice-Boltzmann simulation of fluid flow through packed beds of imiform spheres effect of porosity, Chem Eng Sci 99 44—58, 2013. 

We further mention that at low values of the Reynolds number (that is at very low fluid velocities or for very small particles) for flow through packed beds the Sherwood number for the mass transfer can become lower than Sh = 2, found for a single particle stagnant relative to the fluid [5]. We refer to the relevant papers. For the practice of catalytic reactors this is not of interest at too low velocities the danger of particle runaway (see Section 4.3) becomes too large and this should be avoided, for very small particles suspension or fluid bed reactors have to be applied instead of packed beds. For small particles in large packed beds the pressure drop become prohibitive. Only for fluid bed reactors, like in catalytic cracking, may Sh approach a value of 2. 

Flow of fluids through packed beds of granular particles occurs frequently in chemical processes. Examples are flow through a fixed-bed catalytic reactor, flow through a filter cake, and flow through an absorption or adsorption column. An understanding of flow through packed beds is also important in the study of sedimentation and fluidization. 

Uniform fluid flow through the bed is desirable for good utilization of the catalyst and control of the process. To avoid channeling, the bed is packed as evenly as possible. A rule of thumb dictates that the reactor to particle diameter ratio should be from five to ten, with the reactor length at least SO-100 times the particle diameter." This ensures that the flow is turbulent, uniform, and approximates plug flow. For most commercial reactors these criteria are met. Only in the narrow>tube reactors found in highly endo- or exothermic processes is there any concern. 

Packed beds, often composed of catalyst pellets, are used widely in the chemical industries. Fluid flows through the bed and exchanges heat with the bed material. Heat-transfer processes within the bed and between the bed and the container walls are also of concern. Gnielinski ([29], summarized in [1]) presents a comprehensive listing of available data and gives the following equations valid over a Reynolds number range from 100 to 2 x 1(1 and void fractions 0.26 < / < 1 ... 

Pressure drop and liquid holdup are very important parameters, indispensable for the design of trickle-bed reactor (6, 7, 13, 16, 17, 19, 23, 27, 29, 30, 32, 42). Their values influence directly the interfacial parameters between the fluid phases and between liquid and solid pliases too. Many workers have proposed different correlations for predicting the two-phase pressure drop in co-current downward flow through packed beds (6, 17, 19, 23,... 

In most adsorption processes the adsorbent is contacted by the fluid phase in a packed column. Such variables as the particle size, fluid velocity, and bed dimensions determine the pressure drop and have an important impact on the economics of the process since they determine the pumping cost as well as the extent of axial mixing and the heat transfer properties. The hydrodynamics of flow through packed beds have been extensively studied, and detailed accounts may be found in many chemical engineering textbooks. The present review is therefore limited to a brief summary of the principal features of the flow behavior which are important in the design of fixed bed absorbers. 

The flow of fluids outside immersed bodies appears in many chemical engineering applications and other processing applications. These occur, for example, in flow past spheres in settling, flow through packed beds in drying and filtration, flow past tubes in heat exchangers, and so on. It is useful to be able to predict the frictional losses and/or the force on the submerged objects in these various applications. 

As the use of non-Newtonian fluids in industry (such as in plastics and synthetic fibre manufacture, in polymer processing, in enhanced oil recovery, in biochemistry and biotechnology, and in petrochemicals) is increasing, the interest and research in filtration of such fluids are also growing. Such research is closely linked to further work on the fundamentals of flow through packed beds for non-Newtonian fluids, such as, for example, the recent work of Machac and co-workers on purely viscous and viscoelastic fluids. In future editions of this book or any other in this subject I expect more prominence given to the filtration of non-Newtonian liquids. 

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. 

In this chapter we will explore some other significant areas of fluid dynamics relevant to processes. The topics that will be dealt with include flow around objects, motion of particles, flow through packed beds, non-Newtonian fluids, and agitation and mixing. 

Gnielinski, V., 2010. Fluid-particle heat transfer in flow through packed beds of solids, in Heat atlas, 2nd edn. Springer, Berlin, Germany, Sect. G9, pp. 743-744. 

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 ... 

While plug flow is an idealization, fluid flow through long pipes, gas flow through packed beds under certain conditions, and some transfer line reactors may be regarded as plug flow systems as a reasonable approximation. 

The confrontation of this expression for the unrecoverable pressure loss with experimental measurements has led to the constant in eqn (3.10) being increased from 72 to 150, with which value the relation becomes known as the Blake-Kozeny equation. A major reason for the increase has been attributed to the fact that fluid flowing through packing follows a tortuous path, which is considerably greater than the bed length L (Carman, 1937). We consider this phenomenon in some detail in the following section, in particular in relation to its effect for expanded particle beds. 

In the CFD modelling of membrane filtration process, membranes are usually modelled as a porous wall while the flow within a membrane is usually solved using both Navier-Stokes and Darqr equations (Ghidossi et al, 2006). A porous media model is widely used for determining the pressure loss during flow through packed beds, filter papers, perforated plates, flow distributors and tube banks (ANSYS, 2010). A momentum source term is added to the governing momentum equations, which creates a pressure drop that is proportional to the fluid velocity ... 

The column may be packed or it may be an open tube but in this example, a packed column will be specifically considered. The column is considered to have a length (L) and inlet and outlet pressures and inlet and outlet velocities of (Pi), (Po) (ui) and (uo), respectively. The pressure and velocity at a distance (x) from the front of the column is (Px) and (ux), respectively. According to D Arcy s equation for fluid flow through a packed bed, at any point in the column. 

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