## Im fD Vm 022Ss 1VM0CvdVmM23

It is typical to find a moderate decrease in fw with increasing pipe size, which produces a small increase in V50. Increases in D have the opposite effect, but for small ratios of d/D, as is the case for the data of Figure 7, this effect is not significant.

Other problems arise when pipeline experiments have not been carried out with the particular slurry of interest. In many cases of practical importance, information on the size and grading of the material to be pumped is limited, but estimates of the solids effect must be made. For example, consider the case of an ore that is to be crushed and then transported by pipeline. At the initial stage of the design there may be no adequate sample of the crushed ore, but estimates of the effect of the solids on the head loss must be made for feasibility studies, preliminary designs and cost estimates, and even for justifying the expenses of laboratory or pilot-plant testing.

To estimate the solids effect, two parameters are required: the power M and the velocity V50. The value of M has a lower limit of 0.25 (for fully-stratified flow) and approaches 1.7 for slurries with narrow particle grading. If only a rough idea of the grading is available, it may be adequate to use the following approximation, which requires only an estimate of the particle diameter ratio d85/d50 (d50 is the mass-median particle diameter and d85 is the diameter for which 85% by mass of the particles are smaller). Using this evaluation, the approximation for M is written

Here /n is the natural logarithm. Also, M should not be allowed to exceed 1.7 or fall below 0.25.

The next step is to obtain a commensurate approximation for V50 (Wilson et al., 1997). This formula reads

Here d50 is in mm. The coefficient 3.93 applies for velocities in m/s; for velocities in ft/s this coefficient becomes 12.9. With sand-weight solids (Ss — 1) equals 1.65 and the bracketed portion of Eq. 25 equals 1.00. The value of V50 obtained from Eq. 25 is substituted into Eq. 22 to obtain the solids effect (im — if), where if is the gradient for an equal flow of water. Note, Eq. 25 is only applicable for 0.006 in < d50 < 0.055 in (0.15 mm < d50 < 1.4 mm). For larger particles the value of V50 given by Eq. 25 should be multiplied by cosh(60d50/D).

Remarks on Complex Slurry Flows In the introduction to this section, it was mentioned that slurry flows can be divided into three types on the basis of mechanisms of particle support. These types are homogeneous, partially-stratified and fully stratified, and the applicable methods of analysis for these flows have been outlined in the appropriate preceding sections. Quite often, the particle grading curve is sufficiently broad to span two of the flow types, or even all three. This gives rise to complex slurry flows. The larger particles, which would settle readily in water, often receive considerable support from the smaller particles and the carrier fluid, promoting efficient transport. A complete analysis of such flows is not yet available, but it is hoped that the following remarks will aid the design engineer.

Whenever some coarse particles settle, they form contact load. As shown in earlier sections of this chapter, this has an effect on pressure drop which is quite different from that of particles suspended by the fluid. The contact-load effect, analyzed previously for the case of a Newtonian carrier fluid, must eventually be combined with the scaling laws for non-Newtonian fluids presented in an earlier part of this chapter. As laminar flows which have significant particle settling are usually avoided in design, only turbulent flows will be considered here.

Maciejewski et al. (1993) compared large-diameter transportation of coarse particles of about 4 in. (100 mm) in clay suspensions and in oil-sand tailings slurries (particle size below 0.03 in./0.8 mm). They found that the sand slurry was more effective as a transport medium than a viscous, homogeneous clay slurry. The important role of particles with sizes of 0.004 to 0.020 in. (0.1 to 0.5 mm) in reducing friction was further shown in studies by Sundqvist et al. (1996a, 1996b) for products with d50 of 0.024 to 0.027 in. (0.6 to 0.7 mm) and various size distributions, with maximum sizes of up to 6 in. (150 mm).

In studying the behavior of complex slurries like these, it is logical to begin by dividing the total concentration of solids Cv into three components, each associated with a support mechanism. Thus Ch stands for homogeneous, Cmi for partly stratified (the "middlings") and Cd for the coarse fully stratified particles (the "clunkers"). On the basis outlined previously the particle size of 200 mesh (75 mm) separates Ch and Cmi and the size 0.018D separates Cmi and Cd. This point is best illustrated by an example. Take Ss = 2.65; and a concentration of 30% by volume (Cv = 0.30). From the solids grading curve, suppose that 20% of the total is slimes, 50% middlings and 30% clunkers. Thus, the concentration of slimes in the slurry is (0.30)(0.20) = 0.06, and similarly 0.15 and 0.09 for middlings and clunkers, respectively. The equivalent fluid based on the slimes has specific gravity Sh = 1 + (Ss - 1)Ch = 1 + (1.65)(0.06) = 1.099 and that for the combined slimes and middlings is Shmi = 1 + (1.65)(.21) = 1.347. Thus, for the middlings the specific gravity difference between solids and carrier fluid is (2.650 - 1.099) = 1.551 (rather than 1.650). For the clunkers, the equivalent difference is (2.650 - 1.347) = 1.303.

The homogeneous fraction now forms the carrier fluid for the rest of the slurry, and its hydraulic gradient ih replaces iw in equations like Eq. 20 and Eq. 22. These are used to determine the solids effect for middlings and clunkers, which may be written Aimi and Aid. The gradient for the mixture im represents the sum of ih and the solids effects for the middlings and the clunkers, i.e.

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