## 01

102.6 m of slurry or 102.6 Sm in meters of water.

Inclined Flows Lengths of pipe with an adverse slope often form part of pipelines transporting solids. Compared to the horizontal case, flow up an incline tends to require higher throughput velocities in order to avoid deposition. This is of greatest significance for coarse-particle flow.

In an experimental investigation carried out by Wilson & Tse (1984), four particle sizes between 0.04 and 0.24 in. (1 and 6 mm) were tested in a pipe at angles of inclination up to 40 degrees from the horizontal. It was found that the velocity at the limit of deposition initially increases with the angle of upward inclination, Q, reaching a maximum when this angle is about 30 degrees. For the materials tested this maximum velocity was approximately 50 percent larger than that required to move a deposit in a horizontal pipe. This large difference is clearly a matter of importance for both design and operation of pipelines with inclined sections.

As indicated schematically on Figure 1, Yam marks the lower end of the range of desirable operating velocities for a pipeline. For purposes of comparison, it is appropriate to represent the deposition limit for inclined flow in terms of the dimensionless velocity, or Durand number, Vsm/[2,g'(Ss — 1)D]m . The difference between the Durand number for inclined flow and that for horizontal flow, AD, is plotted against Q on Figure 9.

For the effect of pipe inclination on friction loss, the following widely-used formula by Worster & Denny (1955) may be employed for heterogeneous flows and homogeneous flows in which the hold-up effects are small. Their approach, based on water as the carrier fluid, deals with the extra pressure gradient (Ai(Q), expressed in height of water per length of pipe) beyond that for pumping water alone. For horizontal flow, this extra gradient is simply the solids effect (im — iw), which may be written Ai(0). Worster and Denny's formula states that

For highly stratified flows, Eq. 31 underestimates the losses. For further information see, for example, Wilson et al. (1997).

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