Question

Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional force are 10% of the input energy. How much power is generated by the turbine (g=10 m/s²)

• A. 7.0 kW
• B. 10.2 kW
• C. 8.1 kW
• D. 12.3 kW

Answer 1

The Correct Option is C

To solve the problem of calculating the power generated by the turbine, we will use the principle of conservation of energy. Here's a step-by-step breakdown:

1. First, determine the potential energy per second of the water falling from 60 m height. The potential energy (PE) gained by the water per second is given by the formula: \(PE = m \cdot g \cdot h\)where:
   • \(m = 15 \text{ kg/s}\)(mass flow rate) 
   • \(g = 10 \text{ m/s}^2\)(acceleration due to gravity)
   • \(h = 60 \text{ m}\)(height)
2. Substitute the values into the potential energy formula: \(PE = 15 \cdot 10 \cdot 60 = 9000 \text{ J/s} \text{ or } 9 \text{ kW}\)
3. Next, account for the energy losses due to friction. The problem states that 10% of the input energy is lost. Therefore, 90% of the potential energy is converted into useful energy: \(\text{Useful energy} = 0.9 \times 9000 = 8100 \text{ J/s} \text{ or } 8.1 \text{ kW}\)
4. Thus, the power generated by the turbine is \(8.1 \text{ kW}\).

This result matches the correct answer option: 8.1 kW.

In summary, despite the frictional losses, the system is capable of generating a significant amount of power, reaching 8.1 kW after accounting for efficiency losses. This solution involves understanding the conversion of gravitational potential energy to mechanical energy while incorporating efficiency factors that affect real-life applications.