Unrecoverable pressure loss

The total drop in fluid pressure across a length L of bed is Ap. Of this a portion AP comes about as a result of frictional interaction between the fluid and the particles it represents energy irrevocably lost by the fluid, dissipated as heat. It is therefore convenient to refer to AP as the unrecoverable pressure loss. If the total pressure loss in the fluid is attributable solely to fluid particle frictional interaction and the gain in gravitational potential energy in the rising fluid (as may be assumed in the applications described in this and subsequent chapters), then we have ... 

Theoretical expressions for unrecoverable pressure loss in Newtonian fluids in laminar flow were first derived in the mid-nineteenth century. For a path length L in a cylindrical tube of diameter D, it becomes ... 

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. 

This form shows that for Rep equal to 85.7 (1 — e), which is approximately 50 for normal packed beds (e 0.4), the viscous and inertial contributions to the unrecoverable pressure loss are of equal magnitude. For much smaller Rcp the viscous effects clearly dominate, as do the inertial effects for much larger Rcp. 

A satisfactory interpolation between the packed bed and fully expanded limits can be achieved by first considering a typical particle in a bed of many others. The unrecoverable pressure loss AP comes about as a result of energy dissipation in the bed, and is therefore directly related to particle drag/ . We have expressions for particle drag for the unhindered case under low and high Reynolds number conditions eqns (2.5) 2.7). What are now required are counterpart expressions for a particle in a concentrated bed. These may be deduced from the above unrecoverable pressure loss expressions, eqns (3.21) and (3.23). 

Relation of particle drag to the unrecoverable pressure loss AP... 

We may now apply the relation between particle drag and unrecoverable pressure loss, eqn (3.26), to eqns (3.30) and (3.31) to yield the pressure loss equations, applicable over the full expansion range, 1 > e > 0.4 ... 

The results of these three investigations are shown in Figure 3.6, the first as broken lines representing the published correlations, the other two as raw data points. The range covered is enormous six orders of magnitude in both dimensionless unrecoverable pressure loss and particle Reynolds number from well inside the viscous to deep into the inertial flow regimes. 

The comparisons shown in Figure 3.6 are encouraging, supporting as they do the predictive ability of the derived, unrecoverable pressure loss expression, eqn (3.37), for fluid flow through beds of spheres over the full expansion range encountered with fluidized systems. 

Expressions were derived in the previous chapter for the unrecoverable pressure loss AP in a fluid flowing through a bed of particles. These were shown to apply to beds expanded by various mechanical means to void fractions normally encountered only in fluidized systems. In this chapter we make use of these relations in an analysis of the equilibrium state of homogeneous fluidization. 

The unrecoverable pressure loss, an indelible consequence of maintaining the particles in suspension, is thus ... 

An important property of fluidized beds follows immediately from this simple relation. If we apply it to the whole bed, of height Lb, rather than just a fixed slice of height L, then the product (1 -s)Lb represents the total volume Vb of particles per unit cross-section, which remains unchanged as the bed expands as the fluid flux is increased, Lb increases and (1 — e) decreases so as to maintain their product at a constant value. Thus the unrecoverable pressure loss APb for the whole bed becomes ... 

A general expression for the expansion characteristics will now be derived on the basis of the following constitutive relation for the unrecoverable pressure loss over the bed as a whole ... 

This form is identical to the Richardson-Zaki equation, eqn (4.4). It therefore relates the parameter n in that empirical relation to the ratio of the void fraction and fluid flux exponents in the expression for unrecoverable pressure loss, eqn (4.8) n = —b/a. Note that under both viscous and inertial flow conditions (a=l, = 4.8 and a = 2, =2.4, respectively), the void fraction exponent b assumes the value of —4.8. This unexpected coincidence will now be put to effective use. 

Two working hypotheses. The above interpretation of the empirical parameter n, together with the evidence for effectively identical void fraction dependencies in the unrecoverable pressure loss relations for... 

In Chapter 3, expressions for the drag force on a particle in a bed of particles were obtained from the relations for unrecoverable pressure loss AP. These apply quite generally, regardless of how the particles are supported. They will now be applied to particles in a fluidized bed. 

The rate of energy dissipation, per unit area of bed cross-section, in a gas flowing through a system that gives rise to a total unrecoverable pressure loss AP is, by definition, /qAP. For a fluidized bed, AP remains constant (equal to APb) regardless of the fluid flux 7q. For a slugging bed, however,... 

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