Worked problem

Pump Head, Shaft Power and Motor Size

Total dynamic head, hydraulic and shaft power, and a standard motor size for water at 20 L/s through 100 m of 150 mm pipe.

Mechanical, HVAC & Fluids · Last reviewed

Given

Water at 20 L/s, static lift 25 m, 150 mm pipe 100 m long, Darcy friction factor f = 0.02, minor-loss coefficients totaling K = 4.0, pump efficiency 72%, motor efficiency 90%.

Find

Total head, shaft power, motor input power, and a standard motor size.

Solution

  1. Pipe area = π/4 × 0.15² = 0.01767 m². Velocity = 0.020 ÷ 0.01767 = 1.132 m/s. Velocity head = V² ÷ 2g = 0.0653 m.
  2. Friction head = f × (L/D) × V²/2g = 0.02 × (100 ÷ 0.15) × 0.0653 = 0.87 m.
  3. Minor losses = K × V²/2g = 4.0 × 0.0653 = 0.26 m.
  4. Total head = 25 + 0.87 + 0.26 = 26.13 m.
  5. Hydraulic power = ρ g Q H = 1000 × 9.81 × 0.020 × 26.13 = 5.13 kW.
  6. Shaft power = 5.13 ÷ 0.72 = 7.12 kW. Motor input = 7.12 ÷ 0.90 = 7.91 kW.
  7. A 7.5 kW motor leaves only about 5% margin over 7.12 kW. A common margin is 10 to 15%: 7.12 × 1.15 = 8.2 kW, so choose the next standard size, 11 kW (or 10 hp).

Answer

26.1 m total head | 7.1 kW at the shaft | 11 kW motor

Common mistakes

  • Multiplying by efficiency instead of dividing.
  • Mixing L/s and m³/s. Q = 20 L/s is 0.020 m³/s.
  • Leaving out minor losses from fittings and valves.

Try a variation: half the flow

  1. At 10 L/s the velocity is 0.566 m/s and the velocity head is 0.0163 m.
  2. Friction head = 0.02 × (100 ÷ 0.15) × 0.0163 = 0.22 m. Minor losses = 4.0 × 0.0163 = 0.07 m.
  3. Total head = 25 + 0.22 + 0.07 = 25.28 m, barely below the 26.13 m at full flow.
  4. Hydraulic power = 9,810 × 0.010 × 25.28 = 2.48 kW. Shaft power at 72% = 3.44 kW.

The flow halved, but the head dropped only 3% because the 25 m of static lift dominates the system. Shaft power fell from 7.12 kW to 3.44 kW, to about 48% of the original, because power follows flow when the head is nearly constant.

Sanity check

A quick check for water: hydraulic power in kW ≈ flow in L/s × head in m ÷ 102. For the original problem, 20 × 26.13 ÷ 102 = 5.12 kW, which matches the full calculation.

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Questions

Why add a margin to the motor size?

The operating point can move with wear, fluid changes or system changes, and a motor running at its limit has little room. The margin depends on the application and the standard you follow.

Does this include NPSH?

No. NPSH available against NPSH required is a separate check, and it matters most for suction lift and hot liquids.

What is total dynamic head?

The total energy the pump must add per unit weight of liquid: the static lift plus the pressure difference between the two ends of the system plus the friction and fitting losses, all expressed as a height of liquid.

Why is the discharge velocity head sometimes added?

If the liquid leaves the system at speed, for example from an open pipe end, the pump must supply that kinetic energy too. In this example it is 0.065 m, small enough to ignore, but it is added in careful calculations.

How does the system curve relate to the pump curve?

The system curve shows the head the piping needs at each flow. The pump curve shows the head the pump gives at each flow. The pump runs where the two curves cross.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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