Particle friction factor

Both of these functions f and g are specific for the solid material in question and to some extent also for the diameter of the tube and the mixture ratio p. The great advantage of Eq. (14.113) is that no material or particle friction factors are needed. 

The dynamic filtration theory of Outmans (127) requires experimental terms such as particle-particle stresses, particle friction factors, and thickness of a shear zone within the filter cake that would be difficult to determine. However, the qualitative picture of dynamic filtration presented by Outmans, namely, irreversible adhesion of solid particles up to a certain thickness that is determined by the shear stress (or shear rate) at the surface of the cake, accords with the experiments of Fordham and co-workers (129,135). Once a filter cake has formed under dynamic conditions, it is difficult to remove it by subsequent changes in yc or vm. Figure 44 shows the effect of changes in the flow rate on cumulative filtrate volume. The limiting filtration rate obtained when the initial flow rate of the drilling fluid was 1.8 m3/h remained unaltered when the flow rate of the drilling fluid was increased to 7.0 m3/h in a step-... 

Particle friction factor as a function of system Reynolds number. Re. (Adapted from Rose, H. E. 

The frictional energy lost due to the particle flow can be estimated using Equation 3.40. In such equation, the gas flux may be calculated as the product of the gas velocity by its density. Since the air density equals unity, the gas flux is C = 117 kg/ m s. Also, the particle friction factor /j, to be substituted into Equation 3.40 can be determined from the system Reynolds number Re as in Figure 3.34. In such figure, for Re = 380,250, /j, = 0.00007. Substituting, thus, all the values in Equation 3.40 ... 

Frequency (dimensionless) frequency (1 /T), fanning friction factor, ratio of filtration to cycle time in Equation 10.125 Size fraction of one component of average weight rUa Bend friction factor Gas friction factor Particle friction factor Flow factor... 

In Chap. 9 we shall discuss in considerable detail a parameter called the molecular friction factor f. For velocities that are not too great, the friction factor expresses the proportionality between the frictional force a particle experiences and its velocity ... 

For spherical particles of radius R moving through a medium of viscosity 17, Stokes showed that the friction factor is given by... 

This concludes our discussion of the viscosity of polymer solutions per se, although various aspects of the viscous resistance to particle motion continue to appear in the remainder of the chapter. We began this chapter by discussing the intrinsic viscosity and the friction factor for rigid spheres. Now that we have developed the intrinsic viscosity well beyond that first introduction, we shall do the same (more or less) for the friction factor. We turn to this in the next section, considering the relationship between the friction factor and diffusion. 

We shall see in Sec. 9.9 that D is a measurable quantity hence Eq. (9.79) provides a method for the determination of an experimental friction factor as well. Note that no assumptions are made regarding the shape of the solute particles in deriving Eq. (9.79), and the assumption of ideality can be satisfied by extrapolating experimental results to c = 0, where 7=1. 

In contrast with Eq. (9.79), prior theoretical discussions of the friction factor for various particles were based on some assumed structure or geometry for the molecule ... 

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

At high Reynolds numbers the friction factor becomes nearly constant, approaching a value of the order of unity for most packed beds. In terms of S, particle surface area per unit volume of bed,... 

To allow for the effect of roughness one can use the results of empirical tests in ducts that have been artificially roughened with particles glued on the surface. This approach allows roughness levels to be determined as a function of the particle diameter k. The following friction factor equation has been derived for large Reynolds numbers ... 

In practice the friction factors are calculated either by integration of Eq. (4.51) or by reference to a Moody chart. This is based on Eq. (4.51) by using equivalent roughness values representing the sand particle roughness (see Table 4.3). 

If the calculated value of the Reynolds number is below 2,000, the flow will generally be laminar, that is, the fluid particles will follow parallel flow paths. For laminar flow the friction factor is... 

The solids contribution to the pressure drop, APls, is a consequence of both the particle-wall and particle-particle interactions. The latter is reflected in the dependence of the friction factor fs on the particle diameter, drag coefficient, density, and relative (slip) velocity by (Hinkel, 1953) ... 

Using the Ergun (1952) equation for the interfacial friction factor, Wen and Yu (1966) derived the following general equation to estimate the minimum fluidization superficial velocity Umf for spherical particles ... 

Use trial and error between Eqs. (12)—(14) to evaluate U (the equilibrium particle velocity in the draft tube after acceleration), er (the equilibrium voidage in the draft tube after acceleration), and f (the solid friction factor). 

When evaluating a material for the purpose of establishing dense-phase and long-distance suitability, it is important to undertake all the necessary tests (e.g., particle sizing, particle and bulk densities, fluidization and deaeration). Also, if possible, it is useful to compare such results with those obtained on previously conveyed similar materials (e.g., fly ash). However, it should be noted that such an evaluation only is a qualitative one and it is not possible to predict say, minimum air flows or pipeline pressure drop based on such data (i.e., pilot-scale tests normally are required to confirm minimum velocities, friction factors, etc., especially over long distances and for large-diameter pipes). 

Xb Particle-wall friction factor in bend As Particle-wall friction factor in straight pipe pu Loose-poured bulk density, kg nr3 Py Air density, kg mr3 ps Particle density, kg nr3... 

It may be assumed that the particles are spherical and that, in both cases, the friction factor, R /pu2 is constant at 0.22, where R is the force on the particle per unit of projected area of the particle, p is the fluid density and u the velocity of the particle relative to the fluid. 

The results were expressed as the friction factor (R /pu2), which was found to have a constant value of 0.40 for particle Reynolds numbers (Re ) over the range from 3000 to 9000, and for tube Reynolds numbers (Re) from 12,000 to 26,000. Thus the value of R /pu2 has been approximately doubled as a result of turbulence in the fluid. 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

A particle drag coefficient Cd can now be defined as the drag force divided by the product of the dynamic pressure acting on the particle (i.e. the velocity head expressed as an absolute pressure) and the cross-sectional area of the particle. This definition is analogous to that of a friction factor in conventional fluid flow. Hence... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

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