Blake-Kozeny equation

This is known as the Blake-Kozeny equation and, as noted, applies for A/e.PM < 10-... 

This equation with a value of 150 instead of 180 is called the Ergun equation and is simply the sum of Eqs (13-15) and (13-16). (The more recent references favor the value of 180, which is also more conservative.) Obviously, for ARcRM < 10 the first term is small relative to the second, and the Ergun equation reduces to the Blake-Kozeny equation. Likewise, for A Re PM > 1000 the first term is much larger than the second, and the equation reduces to the Burke-Plummer equation. 

If the Blake-Kozeny equation for laminar flow is used to describe the friction loss, which is then equated to AP/p from the Bernoulli equation, the resulting expression for the flow rate is... 

It should be noted that the importance of the continuity equation is in evaluating actual velocities within the reactor bed as influenced by the mole, temperature, and pressure changes. Because of the use of mass velocities (pgug), the importance of the actual velocities is really restricted to cases where pressure relationships such as the Blake-Kozeny equation or velocity effects on heat transfer parameters are considered. As will be shown later, very little increased computational effort is introduced by retaining the continuity equation, since it is solved as a set of algebraic equations. 

For laminar flow, the second term of the Ergun equation may be dropped, resulting in the Blake-Kozeny equation ... 

Using the Blake-Kozeny equation, the screw conveyor reactor can be modeled as rotating equipment in which the fluid is moved by screw rotation. Although 10 small horizontal cylindrical baffles exist in the screw flights, the interaction between the screw and baffles can be ignored. For this case, the rotating reference frame was used instead of the inertial reference frame. 

In Equation 6.17, P is flie pressure at any point in the cake shown schematically in Figure 6.11, s, the specific surface (surface area per unit volume of particle), p, the hquid viscosity, vs, the superficial liquid velocity, and 8, the porosity of the cake. The Kozeny-Carmen equation is derived in a number of texts. See, for example. Bird et al. [26], who have called the equation the Blake-Kozeny equation. 

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 ... 

The Ergun equation was empirically obtained based on a more general form of the dimensionless pressure drop equation, namely, the Blake-Kozeny equation, which was also theoretically derived by Irmay (23). Irmay s model yields... 

Equation 12.13 is known as the Blake-Kozeny equation or the Kozeny-Carman equations it describes the experimental data for steady flow of newtonian fluids through beds of uniform-size spheres satisfactorily for less than about 10. 

Eq. (4.93) is the Blake-Kozeny equation, which is valid only for the laminar flow regime [10]. 

The Blake-Kozeny equation and the Burke—Plummer equation define the limiting flow regimes for a fluid passing through a mass of spherical particles. Ergun added these equations to obtain... 

The proposed equation compared well with published experimental data obtained from high-voidage fixed beds of spheres it represented a significant improvement over that of Ergun. The equation is a combination of two equations, one for the laminar regime and the other for fully turbulent flow. The laminar flow regime equation is derived to match the Blake-Kozeny equation at e equal to 0.4 and can be expressed... 

The equations for minimum fluidization are similar to those presented for a fixed bed, i.e., Equation (18.13). For laminar flow conditions (Re < 10), the Blake-Kozeny equation is used to express the pressure in terms of the superficial gas velocity at minimum fluidization, Vmf, and other fluid and bed properties. This equation is obtained from Equations (18.10) and (18.14)... 

ILLUSTRATIVE EXAMPLE 18.16 Comment on the relationship between the Ergun equation and the Burke-Plummer and Blake-Kozeny equations. 

Solution. Note that for high rates of flow, the first term on the right-hand side of the Ergun equation drops out and the equation reduces to the Burke-Plummer equation. At low rates of flow, the second term on the right-hand side drops out and the Blake-Kozeny equation is obtained. It should be emphasized that the Ergun equation is but one of many that have been proposed for describing the pressure drop across packed columns. For example, Theodore provide the following simple equation. 

In reality, the length of the flow path in the hypothetical capillaries in the bed is larger than L due to the path tortuosity. Experimental measurements indicate that the following equation (the Blake-Kozeny equation) is instead more accurate as long as fi < 0.5 and dpp Votp (1 - c)) <10 ... 

The pressure drop experienced by a fluid as it flows through a packed bed of particles of diameter dp may be described by the Blake-Kozeny equation (6.1.4f), among others ... 

For most conditions encountered in permeametry measurements, the second term on the r.h.s. of the Ergun equation will be negligibly small compared with the first. The simplified equation obtained by striking out this negligible term is the Blake-Kozeny equation. 

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. 

Inserting the relation for T of eqn (3.19) in eqn (3.18), and making an empirical adjustment to the constant (from 72 to 60), produces an expression for AP that is in exact agreement with the Blake-Kozeny equation at the normal packed bed void fraction of 0.4 but which, beyond that point, reflects the above tortnosity considerations for expanded beds, e > 0.4 ... 

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