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example 1 A clay-water slurry of phosphate slimes is to be pumped over a horizontal distance of 2300 ft. (700 m), using a pipe of internal diameter 12 in. (305 mm). The slurry will be taken from a pond in which the relative density of the mixture is 1 • 13. Tests have been carried out with this material using a pipe of internal diameter 8 in. (203 mm). Test data for im and Vm appear in Table 1 together with values of 8Vm/D and t0 (the wall shear stress equals pwgimD/4). Figure 5 shows the values of im plotted versus Vm for the test pipe.

The preliminary design of the pumping system calls for a discharge Qm of 4770 gpm (0.30 m3/s) in the prototype pipe. However, this value is not yet definite, and Qm values of 3180 gpm (0.20 m3/s) and 6360 gpm (0.40 m3/s) are also to be considered.

a. Find the values of im for the three values of Qm just noted. First, the pipe area (0.785 ft2 or 0.0731 m2) is used to obtain the required velocities, i.e. 9.0, 13.5 and 18.0 ft/s (2.74, 4.11 and 5.48 m/s).

The next step is to scale the data points from the test pipe to the prototype pipe. For the laminar-flow points the appropriate scaling laws are given by Eqs. 15 and 16, which show that im scales inversely with the diameter ratio while Vm scales directly with this ratio. points from test runs 1 to 7 have been scaled on this basis and the first six are shown on Figure 5. For test runs 8 and 9 the flow is turbulent. The hydraulic gradient can still be scaled by Eq. 15, but Vm must now be scaled by Eq. 17. This requires evaluation of U«, i.e. Vr0/p. For example, run 8 with t0 = 0.01 lb/ft2 (67.0 Pa) has U* = 0.8 ft/s (0.243 m/s), for which Eq. 17 gives a scaled-up value of Vm equal to 17.6 ft/s (5.37 m/s).

The laminar-turbulent transition point is obtained by projecting the turbulent line back to intercept the laminar line. For the data in the test pipe this intercept occurred to the right of point 7, at a velocity of about 16 ft/s (say 5 m/s). For the prototype pipe the intercept lies between points 4 and 5, with a slightly lower velocity. Note that the scaled values for points 5 to 7 are not physically meaningful, because the equivalent flows in the prototype pipe are turbulent, not laminar.

Figure 5 shows that conditions will be laminar for the two lower flows to be investigated, and turbulent for the highest flow. The values of im can be taken directly from the figure, and are listed in Table 2.

The flat curve for laminar flow, plotted on Figure 5, shows a very small rise in im as Qm goes from 7.1 to 10.6 ft3/s (0.20 to 0.30 m3/s). On the other hand, the equal increase of Qm from 10.6 to 14.1 ft3/s (0.30 to 0.40 m3/s) requires a much larger increment in im, as a result of the shift from laminar to turbulent flow.

b. Suppose now that testing had stopped after run 6, so that no turbulent-data flow points were available.

FIGURE 5 Test data and scale-up for Example 1. [m/s X 3.28 = ft/s]