## Solution

1. Head versus flow curves A, B, and C of Figure 11 are added together in parallel, giving curve (A + B + C). Curves (A + B + C), D, and E are added together in series, resulting in curve (A + B + C) + E + E. This latter curve indicates that 450 ft (137 m) total pump head is required at 3600 gpm (817 m3/h), point X. Curve (A + B + C) crosses point Y at 3600 gpm (817 m3/h) flow through the branches, and this condition requires 310 ft (94.5 m) total head, which is the head across branch points 1 and 2. From each individual component curve, the flow through branches A, B, and C can be read as 1250, 900, and 1450 gpm (284, 204, and 329 m3/h), respectively.

2. Because the total flow through components A, B, and C need be only 2500 gpm (800 + 700 + 1000) [568 m3/h (182 + 159 + 227)], the bypass should be designed to pass 1100 gpm (249 m3/h). Component B requires the maximum head, 200 ft (61 m) differential (point b) across points 1 and 2. Throttling valves are needed in components A and C and bypass F to increase the head in each branch to 200 ft (61 m) at the required flows. Branch A (point a) requires only 140 ft (42.7 m) total head to pass 800 gpm (182 m3/h); therefore the throttling valve must be designed for a 60-ft (42.7-m) head loss. Branch C (point c) requires only 150 ft (45.7 m) total head to pass 1000 gpm (227 m3/h), requiring a throttling valve for a 50-ft (15.3-m) head loss. The bypass control valve and piping should be designed to produce a 200-ft (61-m) head drop at 1100 gpm (249 m3/h).

At 3600 gpm (817 m3/h), the pump is now required to overcome 200 ft (61 m) total head across points 1 and 2, 40 ft (12.2 m) total head through component D, and 100 ft (30.1 m) total head through the system pipe and fittings, component E, for a total of 340 ft (103.3 m). The reduction in pumping head from 450 to 340 ft (137.1 to 103.6 m), a saving of 110 ft (33.5 m), or 24.4% water power, is the result of decreasing the branch head from 310 to 200 ft (94.5 to 61 m) by bypassing the excess flow.

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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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