Kozeny-Carman

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

Thus the Carman-Kozeny equation appiicabie for the increment dx is given by... 

The specific cake resistance r(nr ) depends on particulate bed characteristics e and According to the Carman-Kozeny equation for packed beds (Chapter 2)... 

In principle, filter bed permeabilities can be calculated using the Carman-Kozeny equation 2.53. For slurries containing irregular particles, however, cake filtrabilities together with filter medium resistance are determined using the Leaf Test (Figure 4.13). In this technique, a sample of suspended slurry is drawn through a sample test filter leaf at a fixed pressure drop and the transient volumetric flowrate of clear filtrate collected determined. 

It was shown in Chapter 2 that the simplest models of solid-liquid separation are those based of the Carman-Kozeny equation for filtration in whieh the bed permeability (filtrability), F, may be expressed by... 

The permeability, K, is characteristic of the medium and can be related to measurable properties by the Carman-Kozeny equation ... 

Strictly, the value 5 in the Carman-Kozeny equation should be treated as an empirical constant, which has to be determined experimentally. However, for many systems of interest, the value is very close to 5. 

The specific resistances obtained are independent of applied load, suspension concentration and membrane type, as expected for non-compressible filter cakes. Tests with uniform latex particles have given permeabilities in very good agreement with Equation 2, using a value of 5 for the Carman-Kozeny constant. 

Calculate the minimum velocity at which spherical particles of density 1600 kg/m3 and of diameter 1.5 mm will be fluidised by water in a tube of diameter 10 mm on the assumption that the Carman-Kozeny equation is applicable. Discuss the uncertainties in this calculation. Repeat the calculation using the Ergun equation and explain the differences in the results obtained. 

As noted in Section 6.1.3 of Volume 2, the Carman-Kozeny equation applies only to conditions of laminar flow and hence to low values of the Reynolds number for flow in the bed. In practice, this restricts its application to fine particles. Approaches based on both the Carman-Kozeny and the Ergun equations are very sensitive to the value of the voidage and it seems likely that both equations overpredict the pressure drop for fluidised systems. 

Obtain a relationship for the ratio of the terminal falling velocity of a particle to the minimum fluidising velocity for a bed of similar particles. It may be assumed that Stokes Law and the Carman-Kozeny equation are applicable. What is the value of the ratio if the bed voidage at the minimum fluidising velocity is 0.4 ... 

The use of the Carman-Kozeny equation is discussed in Section 4.2.3 of Chapter 4. It is interesting to note that if e = 0.48, which was the value taken in Problem 6.1, then up/uf = 47.5, which agrees with the solution to that problem. 

Equation 9.33 is the Carman-Kozeny equation [Carman (1937)]. The parameter Kc has a value which depends on the panicle shape, the porosity and particle size range. The value lies in the range 3.5 to 5.5 but the value most commonly used is 5. 

In cake filtration, the filter medium acts as a strainer and collects the solid particles on top of the initial layer. A filter cake is formed and the flow obeys the Carman-Kozeny equation for packed beds. 

General expressions for flow through beds in terms of Carman-Kozeny equations... 

Deviations from the Carman-Kozeny equation (4.9) become more pronounced in these beds of fibres as the voidage increases, because the nature of the flow changes from one of channel flow to one in which the fibres behave as a series of obstacles in an otherwise unobstructed passage. The flow pattern is also different in expanded fluidised beds and the Carman-Kozeny equation does not apply there either. As fine spherical particles move far apart in a fluidised bed, Stokes law can be applied, whereas the Carman-Kozeny equation leads to no such limiting resistance. This problem is further discussed by Carman 141. 

Use of Carman-Kozeny equation for measurement of particle surface... 

The Carman-Kozeny equation relates the drop in pressure through a bed to the specific surface of the material and can therefore be used as a means of calculating S from measurements of the drop in pressure. This method is strictly only suitable for beds of uniformly packed particles and it is not a suitable method for measuring the size distribution of particles in the subsieve range. A convenient form of apparatus developed by Lea and Nurse 22 1 is shown diagrammatically in Figure 4.4. In this apparatus, air or another suitable gas flows through the bed contained in a cell (25 mm diameter, 87 mm deep), and the pressure drop is obtained from hi and the gas flowrate from h2. 

If flow conditions within the bed are streamline, the relation between fluid velocity uc, pressure drop (—A P) and voidage e is given, for a fixed bed of spherical particles of diameter d, by the Carman-Kozeny equation (4.12a) which takes the form ... 

There is evidence in the work reported in Chapter 5 on sedimentation 5) to suggest that where the particles are free to adjust their orientations with respect to one another and to the fluid, as in sedimentation and fluidisation, the equations for pressure drop in fixed beds overestimate the values where the particles can choose their orientation. A value of 3.36 rather than 5 for the Carman-Kozeny constant is in closer accord with experimental data. The coefficient in equation 6.3 then takes on the higher value of 0.0089. The experimental evidence is limited to a few measurements however and equation 6.3, with its possible inaccuracies, is used here. 

When the flow regime at the point of incipient fluidisation is outside the range over which the Carman-Kozeny equation is applicable, it is necessary to use one of the more general equations for the pressure gradient in the bed, such as the Ergun equation given in equation 4.20 as ... 

It is probable that the Ergun equation, like the Carman-Kozeny equation, also overpredicts pressure drop for fluidised systems, although no experimental evidence is available on the basis of which the values of the coefficients may be amended. 

Substituting each of fhese assumpfions into equation 1.35, and further assuming that the equivalent pore space length is proportional to the bed depth H, results in the Carman-Kozeny equation... 

The Carman-Kozeny expression for minimum fluidizing velocify is obtained by substituting for the pressure drop at minimum fluidization from equation 1.34 and therefore... 

The first term represents the pressure loss due to viscous drag (this is essentially the Carman-Kozeny equation) whilst the second term represents kinetic energy losses, which are significant at higher velocities (kinetic energy being proportional to velocity squared). Equation 1.43 is valid in the range 1 < Re < 2000 where the Reynolds number is defined by... 

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