which apply only to a given impeller with altered D and constant efficiency but not to a geometrically similar series of impellers. The assumptions on which Eqs. 12 were based are rarely if ever fulfilled in practice, so exact predictions by the equations should not be expected. A common example is the low-ns radial discharge impeller with parallel radial hub and shroud profiles over most of the path from inlet to exit (Figure 6). Here, Am2 decreases with cutting, and H falls more than would be predicted by Eq. 12b. [This type of impeller is often found in multistage pumps, particularly those in which the designer, driven by cost reduction goals, has minimized a) the axial length occupied by each stage and b) the number of stages, thereby pushing down the ns of the individual stage to the point that a tolerable sacrifice in efficiency results.]

radial discharge impellers Impellers of low specific speed may be cut successfully provided the following items are kept in mind:

1. The angle b2 may change as D is reduced, but this usually can be corrected by filing the blade tips. (see the discussion on blade-tip filing that follows.)

2. Tapered blade tips will be thickened by cutting and should be filed to restore the original shape. (see the discussion below on blade-tip filing.)

3. Bearing and stuffing box friction remain constant, but disk friction should decrease with decreasing D.

4. The length of flow path in the pump casing is increased by decreasing D.

5. Because cm1 is smaller at the reduced capacity, the inlet triangles no longer remain similar before and after cutting, and local flow separation may take place near the blade entrance tips.

6. The second right-hand term in Eq. 5 was neglected in arriving at Eqs. 12, but it may represent a significant decrease in head as D is reduced.

7. Some blade overlap should be maintained after cutting. Usually the initial blade overlap decreases with increasing specific speed, so the higher the specific speed, the less the allowable diameter reduction.

8. Diameter reductions greater than from 10 to 20% of the original full diameter of the impeller are rarely made.

Most of the losses are approximately proportional to Q2, and hence to D2 by Eq. 12. Because the power output decreases approximately as D3, it is reasonable to expect the maximum efficiency to decrease as the wheel is cut, and this often is the case. By Eq. 12 and the ns-definition (Eq. 38a of Section 2.1), the product nfl should remain constant so the specific speed at best efficiency increases as the wheel diameter is reduced (Table 4).

The characteristics of the pump shown in Figure 13 may be used to illustrate reduction of diameter at constant speed. Starting with the best efficiency point and D = 1616 in (41.4 cm), let it be required to reduce the head from H = 224.4 to H = 192.9 ft (68.4 m to 58.8 m) and to determine the wheel diameter, capacity, and power for the new conditions.

Because the speed is constant, Eqs. 12 may be written

where kH and kp may be obtained from the known operating conditions at D = l616 in (41.4 cm). Plot a few points for assumed capacities and draw the curve segments as shown by the solid lines in Figures 13b and 13c. Then, from Eqs. 12

from which D = 151 in (38.4 cm), Q = 3709 gpm (234 l/s), and P = 215.5 hp (160.7 kW). In Figure 13, the initial conditions were at points A and the computed conditions after cutting at points B. The test curve for D = 15g in (38.4 cm) shows the best efficiency point a at a lower flow rate than predicted by Eqs. 14, but the head curve satisfies the predicted values very closely. The power prediction was not quite as good. Table 4 and Figure 13 give actual and predicted performance for three impeller diameters. The error in predicting the best efficiency point was computed by (predicted value minus test value) (100)/(test value). As the wheel diameter was reduced, the best efficiency point moved to a lower flow rate than predicted by Eq. 14 and the specific speed increased, showing that the conditions for Eqs. 12 to hold were not maintained.

Wheel cutting should be done in two or more steps with a test after each cut to avoid too large a reduction in diameter. Figure 14 shows an approximate correction, given by Stepanoff,12 that may be applied to the ratio D'/D as computed by Eqs. 12 or 14. The accuracy of the correction decreases with increasing specific speed. Figure 15 shows a correction proposed by Rutschi13 on the basis of extensive tests on low-specific-speed pumps. The corrected diameter reduction AD is the diameter reduction D — D given by Eqs. 14 and multiplied by k from Figure 15. The shaded area in Figure 15 indicates the range of scatter of the test points operating at or near maximum efficiency. Near shutoff the values of k were smaller and at maximum flow rate the values of k were larger than shown in Figure 15. Table 5 shows the results of applying Figures 14 and 15 to the pump of the preceding example.

There is no independent control of Q and H in impeller cutting, although Q may be increased somewhat by underfiling the blade tips as described later. The flow rate and power will automatically adjust to the values at which the pump head satisfies the system head-flow curve.

mixed-flow impellers Diameter reduction of mixed-flow impellers is usually done by cutting a maximum at the outside diameter D0 and little or nothing at the inside diameter Di, as shown in Figure 16. Stepanoff14 recommends that the calculations be based on the average diameter Dav = (Di + D0)/2 or estimated from the blade-length ratio FK/EK or GK/EK in Figure 16d. Figure 16 shows a portion of the characteristics of a mixed-flow impeller on which two cuts were made as in Figure 16b. The calculations were made by Eqs. 14 using the mean diameter

instead of the outside diameter in each case. The predictions and test results are shown in Figure 16 and Table 6. It is clear that the actual change in the characteristics far exceeded

TABLE 4 Predicted characteristics at different impeller diameters on a radial-flow pump

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