Flow in a packed bed

Neglecting radial effects, the dispersion equation for dispersed tracer with bulk flow in a packed bed is ... 

The physical understanding of gas flows through a packed bed is important for the study of the gas-solid transport system as it represents the limiting case of the gas-solid transport. A typical example is the minimum fluidization condition of gas-solid fluidization. In a gas-solid flow system, the presence of unsuspended particles is also common, and, thus, understanding the gas flow in a packed bed is essential. 

To illustrate this methodology, we show the case of pressure drop per unit length for one-phase flow in a packed bed. In laboratories, the pressure drop is measured over a 0.1 m length of packed bed using an apparatus as shown in Fig. 6.11. The fluid used is water at 20 °C (p = 1000 kg/m, t) = 10 kg/ms). While the tests are carried out the velocity is varied and the corresponding pressure drop is measured. Table 6.5 shows the results of these tests. 

The reduced interaction between gas and liquid during annular flow in the internally finned channel as compared with two-phase flow in a packed bed not only affects transfer... 

Bancroft coordinates) mass ratio mass ratio ic for flow in a packed-bed ... 

Fig. 1 Numerical simulation of flow in a packed-bed reactor. Velocity quivers illustrate near-wall channeling, which affects residence time distributions, heat transfer, etc. (From Ref. l)...

Usually, flow in a packed-bed absorber is countercurrent. Gravity causes the liquid to flow down through the packing, whereas a small pressure drop drives the gas flow upward. This pressure drop plays an important role in packed beds. Without liquid present, the pressure-drop across dry packing increases approximately with the square of the gas flowrate (see Fig. 8). 

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... 

Carberry and Bretton investigated axial dispersion of water flowing in a packed bed by pulse input-output analysis 10). 

To account for the nonlinear behavior of the flow in porous media, Forchheimer (10) hypothesized that the pressure drop for flow in a packed bed is a direct result from the viscous (linear in origin) and the inertial (quadratic) effects. Forchheimer s hypothesis in a multidimensional form can be expressed as... 

Figure 8.7, Schematic representation of a capillary electrochromatography column consisting of a packed bed and open tubular segment separated by a bed retaining frit (top) and the generation of electroosmotic flow in a packed bed (bottom).

This filter model considers that particles are transported from the flowing fluid to the filter media by Brownian diflFusion, fluid flow (interception), and settling. The eflFects of each of these mechanisms are assumed to be additive. Happel s (26) equations for flow in a packed bed as used by PfeflFer (27) are assumed in calculating the diflFusion and interception transport mechanisms. 

Figure 14.15 Material balance of steady-state flow in a packed bed of nanomaterial for chemical fixation.

Driving Force to Use in Mass Transfer. Derive Eq. (7.3-42) for the log mean driving force to use for a fluid flowing in a packed bed or in a tube. Hint Start by making a mass balance and a diffusion rate balance over a differential area dA as follows ... 

For laminar flow in a packed bed of particles, the Cnrman-Kozeny re/atio/i is similar to Eq. (14.2-1) and to the Biake-Kozeny equation (3.1-17) and has been shown to apply to filtration. 

The second type of reactor is the reverse-flow reactor (RFR), which has attracted signiflcant interest in recent years. In that operation, the direction of the flow in a packed-bed reactor is periodically reversed to trap a hot zone within the reactor. The cold feed is regeneratively heated up as the high-temperature zone moves downstream. Before the hot zone exits the bed, the feed-flow direction is reversed. A detailed review of the various applications of the RFR was presented by Matros and Bunimovich [12]. Experimental and theoretical studies have shown that many (usually more than 100) flow-reversals are ne ed before the reactor converges to the periodic state. There is still a need to determine the best control policy when the reactor is subjected to changes in the feed concentration and/or load. 

In fixed-bed adsorption, diffusion and mixing of metal ions in fluids occur due to adsorbate concentration gradients and non-uniformity of fluid flow. This gives rise to axial dispersion in the main direction of fluid flow and radial dispersion in the direction transverse to the main flow. The former is usually undesirable since it reduces separation efficiency, whereas the latter is desirable as it equalizes concentrations at the same axial position and reduces axial dispersion. For the simple case of singlephase flow in a packed bed of cylindrical configuration, the conservation equation for a metal ion in the solution is... 

Re Reynolds number for the flow in a packed bed Rep Particle Reynolds number... 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

Note that if there would be perfect plug flow in a packed bed reactor, the second term between the brackets is zero, and eq. (7.29) simplifies to a form identical to eq. (3.29). The order of magnitude of the effect of axial mixing on the conversion of a first order chemical reaction can be readily estimated. 

The flow in a packed bed, where the packing may be spherical, c indrical, etc., is quite complex (Figure 6.1.1(b)). However, it is often modeled as a collection of cylindrical capillaries of hydraulic radius R/, and length L, which is the packed bed length. Let e be the fractional void volume of the bed, Op be the total particle surface area per particle volume, v be the actual interstitial velocity in the void volume between particles and dp (= 2rp) be the mean particle diameter. Then the superficial velocity Vo based on the empty cross section of the packed bed is defined as... 

As already mentioned, the gas phase flow in a packed bed column is closer to the piston flow conditions than die liquid flow. That is why the effect of the axial mixing in the gas is many times smdler and taking it into account is not so important. Neverdieless, the information about this phenomcmon is substantial. 

Some attention has also been paid to the effect of bulk and dispersive flow. When fluid flows in a packed bed, two flow phenomena come into play these are known as bulk and dispersive flow. These flows create rate-determining steps in the sequence that determines the overall rate of transfer of dye to a given point on the fibre surface at a particular time. Burleydeveloped a model of the (iyeing process which included these effects. The nature of these flows is explained in detail below. 

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