## Other Factors

Vertical Flows For industrial application of vertical transportation of a solid-liquid mixture in a pipe, the operating velocity must be sufficient to maintain a continuous flow of solids at the discharge end. However, unnecessarily high velocity causes excessive pipe wear and energy losses. The appropriate operating velocity depends on the settling conditions of the solids, indicating that size, density, and concentration of particles are key parameters in the hydraulic design of a vertical particle-fluid transportation system.

Many experimental studies have been made of vertical slurry transport. For example, Sellgren (1979) used a pilot-scale facility with a centrifugal pump to investigate important design parameters for ores and industrial minerals taken from in-plant crushing and milling. The results of these experiments are summarized in the following paragraphs.

It is suggested that the allowable minimum mixture velocity be based on the settling velocity of the largest particles in still water multiplied by a factor of 4 or 5. Provided the velocity exceeds this value, then in most industrial applications, with volumetric concentrations of 15-30%, the corresponding pressure requirement in the vertical system can be determined by the equivalent-fluid model. As noted in connecetion with Eq. 5, this model is based on the density of the slurry and the friction factor for water. Applied to vertical flow, it gives

Here p is the pressure required, Sm is the relative density of the slurry, z is the length of vertical pipe, and Vall is the allowable minimum mixture velocity, approximately four times the settling velocity of the largest particle.

The settling velocity in still water of industrially-crushed mineral particles is normally reduced significantly compared to smooth spheres of corresponding size. Tests have shown that, on average the settling velocity for particles in the range of 0.04 in. to 1.2 in. (1 mm to 30 mm) is reduced approximately 50%. Therefore, the criterion previously given for the minimum allowable velocity could alternatively be formulated as: Vall is twice the settling velocity of a smooth sphere of the same size as the largest particles.

Within the constraints previously discussed, systems operate under conditions where the effect of relative velocity between the components appears to be negligible. The maximum particle sizes considered are in the range of 0.04 in. to 1.2 in. (1 mm to 30 mm) in pipe diameters of 4 to 12 in. (0.1 m to 0.3 m). With larger particle sizes (up to 4 to 6 in./100 to 150 mm) and low concentrations, the relative velocity between the components becomes significant. Boundary-layer transitional effects may also introduce certain instabilities that must be carefully evaluated in long vertical risers. Particles larger than one-fifth the pipe diameter can promote slugging instability in vertical hoisting, and particles larger than one-third the pipe diameter may jam the pipe and should be avoided.

example 4 Centrifual slurry pumps are used to pump a sand slurry (d50 = 0.06 in./1.5 mm) out of a quarry. The pipe is vertical with a length of 328 ft. (100 m) and a diameter of 4 in. (0.10 m). Tests have shown that the settling velocity of the largest particles is approximately 1.5 ft/s (0.45 m/s). Select the operating velocity and calculate the head requirement in meters of slurry.

Solution Following the guidelines previously given, the velocity V^ is four times 0.45, i.e. 6 ft/s (1.8 m/s). At this velocity, Vj2g is 0.54 ft (0.165 m). With the friction factor fw taken as 0.016 for smooth-pipe conditions (see Figure 2), the head is obtained from Eq. 30 as

336.5 ft of slurry

3.94/12

in USCS units or 336.5 Sm in ft of water.

In SI units the calculation becomes

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