A hydroelectric power plant has the following parameters:

Flow rate (Q) = 50 m³/s

Electrical power output (Pout) = 8 MW

Water density (ρ) = 1000 kg/m³

Acceleration due to gravity (g) = 10 m/s²

To achieve an overall efficiency of 80%, what should be the required net head (H)?

Explanation:

Hydroelectric Power Plant Efficiency and Net Head Calculation

Definition: A hydroelectric power plant converts the potential energy of water into electrical energy. The efficiency of the system depends on various factors, including the flow rate of water, the net head (height difference between water source and turbine), water density, acceleration due to gravity, and the overall efficiency of the system.

Problem Statement:

We are given the following parameters:

• Flow rate (Q) = 50 m³/s
• Electrical power output (P_out) = 8 MW = 8 × 10⁶ W
• Water density (ρ) = 1000 kg/m³
• Acceleration due to gravity (g) = 10 m/s²
• Overall efficiency (η) = 80% = 0.8

Objective: To calculate the required net head (H) for the hydroelectric power plant to achieve an overall efficiency of 80%.

Formula:

The overall efficiency (η) of the hydroelectric power plant is given by:

η = (P_out) / (P_input)

Where P_input is the power input due to the potential energy of water, which can be expressed as:

P_input = ρ × g × Q × H

Rearranging the formula to find H:

H = (P_out) / (η × ρ × g × Q)

Substituting the given values:

• P_out = 8 × 10⁶ W
• η = 0.8
• ρ = 1000 kg/m³
• g = 10 m/s²
• Q = 50 m³/s

H = (8 × 10⁶) / (0.8 × 1000 × 10 × 50)

H = (8 × 10⁶) / (400,000)

H = 20 m

Conclusion:

The required net head (H) for the hydroelectric power plant to achieve an overall efficiency of 80% is 20 meters.

Correct Option: Option 4

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 19.50 m

This option is incorrect because the calculated net head (H) using the formula does not match this value. It may arise from a miscalculation or incorrect assumptions in the problem-solving process.

Option 2: 22.15 m

This option is also incorrect for the same reasons as option 1. The value does not align with the mathematical calculation based on the given parameters.

Option 3: 18.75 m

This option is incorrect as well. It does not correspond to the calculated value of H using the formula provided.

Option 4: 20 m

This is the correct option, as demonstrated by the step-by-step calculation above. The net head required for the given efficiency is 20 meters.

Conclusion:

The formula for calculating the net head (H) in a hydroelectric power plant is straightforward when the efficiency, flow rate, water density, and acceleration due to gravity are known. By carefully substituting the given values into the formula, the correct net head of 20 meters is determined, making option 4 the correct choice. This calculation is essential for designing efficient hydroelectric power plants and ensuring optimal energy output.