Question

The power generated by a turbine operated by the fall of water from a height of 100 m at the rate of $24\text{ kg s}^{-1}$ is 21 kW. The percentage loss in power is

• A. 10%
• B. 12.5%
• C. 15%
• D. 20%
• E. 5%

Hint:

Logic Tip: Always convert Watts to kiloWatts (divide by 1000) early in the calculation to keep the numbers small and easy to compare with the given output power.

Answer 1

The Correct Option is B

Concept:
Power is defined as the rate of doing work or transferring energy. The theoretical input power ($P_{in}$) generated by falling water is the rate of change of potential energy: $P = \frac{mgh}{t} = \left(\frac{m}{t}\right)gh$, where $\frac{m}{t}$ is the mass flow rate. The percentage loss is calculated by comparing the actual output power to the theoretical input power.
Step 1: Calculate the theoretical input power ($P_{in}$).
Mass flow rate, $\frac{m}{t} = 24\text{ kg s}^{-1}$ Height, $h = 100\text{ m}$ Gravity, $g = 10\text{ ms}^{-2}$ (standard approximation for such problems) $$P_{in} = \left(\frac{m}{t}\right) \cdot g \cdot h$$ $$P_{in} = 24 \times 10 \times 100$$ $$P_{in} = 24000\text{ W} = 24\text{ kW}$$
Step 2: Determine the power loss.
Given actual output power, $P_{out} = 21\text{ kW}$. $$\text{Power Loss} = P_{in} - P_{out}$$ $$\text{Power Loss} = 24\text{ kW} - 21\text{ kW} = 3\text{ kW}$$
Step 3: Calculate the percentage loss.
$$% \text{ Loss} = \left( \frac{\text{Power Loss}}{P_{in}} \right) \times 100$$ $$% \text{ Loss} = \left( \frac{3}{24} \right) \times 100$$ $$% \text{ Loss} = \left( \frac{1}{8} \right) \times 100 = 12.5%$$