Water falls from a height of 30 m at the rate of 20 kg/s to operate a turbine. The losses due to frictional forces are 20% of energy. How much power is generated by the turbine ? (g = 10 m/`s^(2)`)

Options

A. 12.3 kW
B. 7.0 kW
C. 4.8 kW
D. 10.2 kW

Explanation

To solve the problem of how much power is generated by the turbine when water falls from a height of 30 m at a rate of 20 kg/s, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Height (h) = 30 m
- Mass flow rate (dm/dt) = 20 kg/s
- Gravitational acceleration (g) = 10 m/s²
- Loss due to friction = 20%

2. **Calculate the Potential Energy per Second:**
The potential energy (PE) of the water falling can be calculated using the formula:
\[ PE = m \cdot g \cdot h \]
Since water is falling at a rate of 20 kg/s, we can substitute:
\[ PE = (20 \, \text{kg/s}) \cdot (10 \, \text{m/s}^2) \cdot (30 \, \text{m}) \]

3. **Calculate the Potential Energy:**
\[ PE = 20 \cdot 10 \cdot 30 = 6000 \, \text{J/s} = 6000 \, \text{W} \]

4. **Account for Energy Losses:**
Since 20% of the energy is lost due to friction, only 80% of the potential energy is converted into useful power:
\[ \text{Useful Power} = PE \cdot (1 - \text{Loss Fraction}) = 6000 \cdot 0.8 \]

5. **Calculate the Useful Power:**
\[ \text{Useful Power} = 6000 \cdot 0.8 = 4800 \, \text{W} \]

6. **Convert Watts to Kilowatts:**
\[ \text{Power in kilowatts} = \frac{4800 \, \text{W}}{1000} = 4.8 \, \text{kW} \]

### Final Answer:
The power generated by the turbine is **4.8 kW**.

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