Porous media packed beds

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

The fuel bed (packed bed) is a two-phase system, also referred to as a porous medium [20]. Thermochemical conversion processes, such as drying, pyrolysis, char combustion and char gasification, take place simultaneously in the conversion zone of the fuel bed (Figure 16). They are extremely complex, and are reviewed more in detail in section B. 4. Review of thermochemical conversion processes. 

A porous medium basically consists of a bed of many relatively closely packed particles or some other form of solid matrix which remains at rest and through which a fluid flows. If the fluid Alls all the gaps between the particles, the porous medium is said to be saturated with the fluid, i.e., with a saturated porous medium it is not possible to add more fluid to the porous medium without changing the conditions at which the fluid exists, e.g., its density if the fluid is a gas. A porous medium is shown schematically in Fig. 10.1. 

Soo and Radke (11) confirmed that the transient permeability reduction observed by McAuliffe (9) mainly arises from the retention of drops in pores, which they termed as straining capture of the oil droplets. They also observed that droplets smaller than pore throats were captured in crevices or pockets and sometimes on the surface of the porous medium. They concluded, on the basis of their experiments in sand packs and visual glass micromodel observations, that stable OAV emulsions do not flow in the porous medium as a continuum viscous liquid, nor do they flow by squeezing through pore constrictions, but rather by the capture of the oil droplets with subsequent permeability reduction. They used deep-bed filtration principles (i2, 13) to model this phenomenon, which is discussed in detail later in this chapter. 

Figure 5.9.3 Schematic diagram of fixed-bed porous-glass-sphere culture system (1) packed bed of porous glass spheres (2) sampling port (3) medium fed in (4) air/oxygen sparge (5) off-gas filters (6) pH probe (7) dissolved oxygen probe (8) peristaltic pump (9) inoculation port (10) harvest.

In gas-solid reactors when solid particles are held stationary (so-called fixed bed reactor), gas flows through a porous medium comprising macropores existing between pellets or packed solid particles and micropores within the catalyst pellets (or other porous solids). Issues such as isotropy of the porous medium, initial distribution of gases, characteristics of solid particles, ratio of characteristic length scale of solid particles and that of the reactor and so on, influence the flow within fixed bed reactors. Support screens are often used to cover the bed of solid particles to avoid fluidization and carry-over of bed particles. These reactors are extensively used in process industries. Some examples and illustrative flow simulations are discussed in Chapter 13. 

Porosity is one of the most important continuum-scale parameters. It is defined as the fraction of the total volume that comprises void space e = Woid/ ktotai-Equivalently, the solid volume fraction ((f) = 1 - e) is generally used for fibrous materials or other open structures. The term microporosity implies that the particles in a porous medium are themselves porous, usually at a much smaller scale. A common example is porous catalyst in a packed-bed reactor. 

Problem 9-22. Flow in a Brinkman Medium. Fluid flow in a packed bed or porous medium can be modeled as flow in a Brinkman medium, which we may envision as a bed of spherical particles. Each particle in the bed (there are n particles per unit volume) exerts a drag force on the fluid proportional to fluid velocity relative to the particle given by Stokes law, i.e., ( —Gtt/hiu, where a is the characteristic size of a bed particle). Thus the equations describing the fluid motion on an averaged scale (averaged over many bed particles, for example) are... 

A porous medium consists of a packed bed of solid particles in which the fluid in the pores between particles is free to move. The superficial fluid velocity V is defined as the volumetric flow rate of the fluid per unit of cross-sectional area normal to the motion. It is the imbalance between the pressure gradient (VP) and the hydrostatic pressure gradient (pg) that drives the fluid motion. The relation that includes both viscous and inertial effects is the Forscheimer equation [47]... 

When the porous medium is contained in a cylindrical bed, the multidimensional effect is given by equation 127. When the pressure drop is needed, there is no need to solve the governing equation numerically. Equations 106 and 128 can be used for the estimation of pressure drop as well as scaling-up of a packed bed. 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

Equipment and technique for HDC of silica sols are essentially the same as for SEC. The difference is that the packed bed of the separating column is composed of nonporous, rather than porous, particles. Typically, these particles are polystyrene-based beads, but glass or dense silica beads also are effective. Alternatively, a long, narrow capillary can be used as the separating medium [30]. 

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. 

The fluid velocity distribution given by Eqs. (93)-(96) are only valid for an isolated particle. However, there are a number of practically important situations, like the deep-bed filtration process, when the flow past an assembly of spheres (forming a porous mediiun) takes place. In this case, the flow field around a single sphere is influenced by the presence of other spheres. Various models that describe the flow field in the packed bed consisting of spheres are available. The sphere in cell models [81-83] assume that each sphere in the packed bed is surrounded by the spherical cavity filled with fluid. The size of the cavity is determined by the overall average porosity of the medium. The general solution of the Navier-Stokes equation for the stream function inside the cavity may be written as [7]... 

Abstract The theoretical background for the mechanistic description of flow phenomena in open channels and porous media is elucidated. Relevant works are described and the equations governing flow are explained. Fundamental concepts of dispersion, convection and diffusion are clarified and models that describe these processes are evaluated. The role of bulk and dispersive flow in dye transfer within a packed bed medium and the effect of including flow parameters on modelling dye dispersion and diffusion are then evaluated, and various models incorporating flow properties are examined. 

There are various conceptual ways of describing a porous medium. One concept is a continuous solid with holes in it. Such a medium is referred to as consolidated, and the holes may be unconnected (impermeable) or connected (permeable). Another concept is a collection of solid particles in a packed bed, where the fluid can pass through the voids between the particles, which is referred to as unconsolidated. Both of these concepts have been used as the basis for developing the equations which describe fluid flow behaviour. ... 

The term (-ur]lk) in Eq. 3.23 is the Darcy resistance term, and the term (rjW u) is the viscous resistance term the driving force is still considered to be the pressure gradient. When the permeability k is low, the Darcy resistance dominates the Navier-Stokes resistance, andEq. 3.23 reduces to Darcy s law. Therefore, the Brinkman equation has the advantage of considering both viscous drag along the walls and Darcy effects within the porous medium itself. In addition, because Brinkman s equation has second-order derivatives of u, it can satisfy no-slip conditions at solid surfaces bounding the porous material (e.g. the walls of a packed bed reactor), whereas Darcy s law cannot. In that sense, Brinkman s equation is more exact than Darcy s law. 

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... 

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