Questions: Mechanical Energy Balance with Pump and Turbine Work

Question 1

A pump moves water from a lower tank to an elevated tank through a long pipe. After applying the mechanical energy equation, a student finds H_pump = 40 m but the elevation difference is only 30 m and velocity/pressure differences are negligible. What happened to the missing 10 m of head?

• A. It was stored as potential energy in the pipe walls
• B. It was dissipated as heat by viscous friction in the pipe (head loss)
• C. It was converted to turbine work downstream
• D. The pump under-performed; 10 m of head was never delivered to the fluid

Answer: (B)

The energy equation is a strict accounting statement: H_pump = Δz + H_turbine + H_loss. With no turbine and 30 m of elevation gain, the remaining 10 m must equal H_loss — energy permanently destroyed by viscous friction and converted to heat. Head loss always appears on the outlet side of the equation and can never be recovered.

Question 2

A pump delivers Q = 0.05 m³/s against a pump head of H = 30 m. The pump's efficiency is η = 0.75. What shaft power must be supplied to the pump?

• A. ρgQH = 14.7 kW — shaft power equals hydraulic power
• B. ρgQH / η = 19.6 kW — shaft power is higher because energy is lost in the pump itself
• C. η × ρgQH = 11.0 kW — shaft power is lower because the pump multiplies input power
• D. ρg Q H η² = 8.3 kW — two efficiency factors apply, one for suction and one for discharge

Answer: (B)

The hydraulic power delivered to the fluid is P_hydraulic = ρgQH ≈ 14.7 kW. But because the pump is only 75% efficient, not all shaft power becomes fluid power — some is lost to friction, heat, and mechanical losses inside the pump. Therefore P_shaft = ρgQH / η = 14.7 / 0.75 ≈ 19.6 kW. Option A is the common error of forgetting that shaft power must be greater than delivered hydraulic power when efficiency < 1.

Question 3

In the mechanical energy equation, head loss appears on the outlet side of the equation because it represents energy removed from the fluid, just like turbine head.

• T. True
• F. False

Answer: False

Head loss and turbine head are both on the outlet (right) side, but for fundamentally different reasons. Turbine head represents useful work extracted — energy transferred to rotating machinery that can do work elsewhere. Head loss represents energy permanently destroyed by viscous friction and converted to heat; it cannot be recovered or redirected. The two terms have opposite physical significance: one is useful extraction, the other is irreversible waste.

Question 4

The concept of 'head' expresses each energy term in the mechanical energy equation as an equivalent height in meters, obtained by dividing energy per unit weight (J/N) by the gravitational constant.

• T. True
• F. False

Answer: True

Dividing each energy term (J/kg) by g gives units of meters — a 'height equivalent' of energy. Pressure head P/ρg is the height a static column would reach; velocity head V²/2g is the kinetic energy expressed as height; elevation head z is the actual height. Expressing everything in meters of head makes all terms directly comparable and greatly simplifies piping system calculations.

Question 5

A hydroelectric turbine extracts H_turbine = 50 m of head from water flowing at Q = 2 m³/s, and the turbine's efficiency is 0.85. What is the shaft power output, and why is it less than ρgQH_turbine?

Model answer: P_shaft = η × ρgQH_turbine = 0.85 × 1000 × 9.81 × 2 × 50 ≈ 835 kW. It is less than the full hydraulic power (ρgQH_turbine ≈ 981 kW) because the turbine cannot convert all fluid energy to shaft work — internal friction, fluid leakage, and mechanical losses dissipate some energy within the turbine itself.

The hydraulic power available from the fluid equals ρgQH_turbine. A turbine with efficiency η < 1 converts only a fraction of that to useful shaft output; the rest is dissipated internally. This mirrors the pump case: shaft input > hydraulic output for pumps (η < 1 means you pay more than you get), and shaft output < hydraulic input for turbines (η < 1 means you get less than is available).