Packed beds flow through

How through packed beds under laminar conditions can be described by a model in which the flow is assumed to be through capillaries whose surface equals that of the solids comprising the bed. The capillary volume is set equal to the void volume of the bed. The model leads to the well-known Carman-Kozeny equation as follows  

The constant 5 is valid for lower porosity ranges and particles not too far from spherical in shape it generally depends on particle size, shape and porosity. Unfortunately, although the above equation has been found to work reasonably well for incompressible cakes over narrow porosity ranges, its importance is limited in cake filtration because it cannot be used for most practical compressible cakes. Its value is mostly in illustrating the high sensitivity of the pressure drop to the cake porosity and to the specific surface of the solids. A major criticism of the Carman-Kozeny equation is that it incorrectly tends to infinity when the voidage approaches 1. 

Darcy s law combines the constants in the last term of equation 18.20 into one factor, K, known as the permeability of the bed, which is a constant 

The modem filtration theory tends to prefer the Ruth form of Darcy s law as  

There is a hidden assumption in the above relafionships that the volume of the solids and the liquid retained in the cake is negligible. This is reasonable at low concentrations but can lead to errors at higher solids concentrations and moisture contents of cakes. The usual way to correct for this is by using a corrected value of the concentration, c, in equation 18.22b. 

We now examine the case in which the particles are stationary and in direct contact with their neighbours, which is known as a packed bed. 

In a packed bed, the fluid-particle interaction force is insufficient to support the weight of the particles. Hence, the fluid that percolates through the particles loses energy due to frictional dissipation. This results in a loss of pressure that is greater than can be accounted for by 

The total drop in fluid pressure across a length L of bed is given by  

When a fluid passes through a bed of porous material the pressure drop per unit length of bed is given by Darcy s law as  

It was found experimentally that 3/ = 1.75. With this value, Equation (23) becomes known as the Burke Plummer equation  

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Ca.rma.n-KozenyEfjua.tion, Flow through packed beds under laminar conditions can be described by the Carman-Kozeny equation in the... 

Equation 9.32 does not hold for flow through packed beds and should be replaced by the equation... 

Other correlations for flow through packed beds involve the porosity, e, e.g. 

In a straight pipe, the transition from streamline to turbulent flow occurs at Re 2300. For flow through packed beds, the transition occurs at a value of Rep of approximately 40. 

Figure 27.9 Flow-through packed-bed coulometric reactor.

We see that Apjl, the frictional pressure drop per unit depth of bed, is made up of two components. The first term on the right-hand-side accounts for viscous (laminar) frictional losses, cc pu. and dominates at low Reynolds numbers. The second term on the right-hand-side accounts for the inertial (turbulent) frictional losses, oc pu2, and dominates at high Reynolds numbers. For further information about flow through packed beds, see Chapter 7 An Introduction to Particle Systems . 

Flow through packed beds (eddy or multipath diffusion). In chromatography, component zones are carried through a bed of randomly packed particles. The streamlines in such flow veer back and forth to find passage between the particles (see Figure 5.4) and fluctuate in velocity... 

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. 

The effect of liquid flow rate on (AP/AZ)LG is taken into consideration by its effect on the liquid holdup. The form of Eq. (6-19) is taken as the same as the Ergun. equations25-60 for flow through packed beds using an effective voidage available fox gas flow defined as (1 — c — fi0L — k), where c is. the vo.lume fraction occupied by solid. /i0L is the liquid holdup per unit volume of column, and k is the effective deadspace volume per unit volume of column (taken here to be 0). 

The Peclet (or Bodenstein) numbers are significantly lower for trickle-flow conditions than in single-phase flow through packed beds. For example, as shown... 

Many empirical equations for predicting pressure gradients in countercurrent flow of gas and liquid are available in the literature.17,31,36 The pressure drop in countercurrent flow can be represented by an equation of the Carman-Kozeny type for flow through packed beds, Below the flooding point, the following equation is suggested36 and has been shown to agree well with experimental data ... 

Fig. 7. Radial velocity profiles for flow through packed beds at different ratios of bed diameter to particle diameter. After Fahien and Stankovic (5).

A correlation for flow through packed beds in terms of the Colburn J factor... 

Flow through packed beds of spherical particles or other packing... 

Iliuta, L Thyrion, F.C. Bolle, L. Giot, M. Comparison of hydrodynamic parameters for countercurrent and cocurrent flow through packed beds. Chem. Eng. Technol. 1997, 20, 171. 

These two frictional loss terms can be used in the mechanical energy balance, leading to the Blake-Kozeny equation (for low Re) and the Burke-Plummer equation (for higher Re). Ergun combined the two to give the most well-known equation for flow through packed beds ... 

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. 

If a population of particles is to be represented by a single number, there are many different measures of central tendency or mean sizes. Those include the median, the mode and many different means arithmetic, geometric, quadratic, cubic, bi-quadratic, harmonic (ref. 1) to name just a few. As to which is to be chosen to represent the population, once again this depends on what property is of importance the real system is in effect to be represented by an artificial mono-sized system of particle size equal to the mean. Thus, for example, in precipitation of fine particles due to turbulence or in total recovery predictions in gas cleaning, a simple analysis may be used to show that the most relevant mean size is the arithmetic mean of the mass distribution (this is the same as the bi-quadratic mean of the number distribution). In flow through packed beds (relevant to powder aeration or de-aeration), it is the arithmetic mean of the surface distribution, which is identical to the harmonic mean of the mass distribution. 

EFFECTIVE (OR AERODYNAMIC) PARTICLE DENSITY is when the measured volume includes both the closed and the open pores. This volume is within an aerodynamic envelope as seen by the gas flowing past the particle the value of density measured is therefore a weighted average of the solid and immobilised gas (or liquid) densities present within the envelope volume. The effective density is clearly of primary importance in applications involving flow round particles like in fluidization, sedimentation or flow through packed beds. 

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