Calculate the power output, in kW, for a hydropower site with an effective head of 15 m, a turbo-generator efficiency of 80% and a flow rate of 0.04 m3/s.

Do not round your answer.

To calculate the power output for a hydropower site, you can use the following formula:

\[
P = \eta \cdot \rho \cdot g \cdot H \cdot Q
\]

Where:
- \(P\) is the power output in watts (W).
- \(\eta\) is the efficiency of the turbo-generator (in decimal form).
- \(\rho\) is the density of water (approximately \(1000 \, \text{kg/m}^3\)).
- \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)).
- \(H\) is the effective head (in meters).
- \(Q\) is the flow rate (in cubic meters per second, m³/s).

Now, let's substitute the given values:

- \(\eta = 0.8\)
- \(\rho = 1000 \, \text{kg/m}^3\)
- \(g = 9.81 \, \text{m/s}^2\)
- \(H = 15 \, \text{m}\)
- \(Q = 0.04 \, \text{m}^3/\text{s}\)

Substituting these into the formula:

\[
P = 0.8 \times 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 15 \, \text{m} \times 0.04 \, \text{m}^3/\text{s}
\]

\[
P = 0.8 \times 1000 \times 9.81 \times 15 \times 0.04
\]

Now, calculating the value:

\[
P = 0.8 \times 1000 \times 9.81 \times 15 \times 0.04 = 4708.8 \, \text{W}
\]

Since 1 kW = 1000 W, the power output in kilowatts is:

\[
P = \frac{4708.8}{1000} \, \text{kW} = 4.7088 \, \text{kW}
\]

So, the power output is **4.7088 kW**.

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. How would the power output change if the effective head increased to 20 m?
2. What happens to the power output if the efficiency of the turbine decreases to 70%?
3. How would the power output differ if the flow rate was doubled to 0.08 m³/s?
4. What is the impact of water density variations on the power output calculation?
5. How would seasonal changes in water flow affect the annual power production?

**Tip:** Always convert the final power output to kilowatts (kW) for practical applications, as this is the standard unit used in power generation discussions.

Learn how to calculate the power output for a hydropower site using the efficiency of the turbo-generator, water density, acceleration due to gravity, effective head, and flow rate. Explore a detailed solution with calculations. Power output is determined to be 4.7088 kW.

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