Question

Water flows from an elevated Dam through a pipeline to a turbine at a lower level and out of the turbine through a similar pipeline. At a point 150 m above the turbine the pressure is 260 kPa, and at a point 15 m below the turbine the pressure is 144 kPa. What must be the water flow rate, if the turbine output is 2.5 MW?

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Answer to a math question Water flows from an elevated Dam through a pipeline to a turbine at a lower level and out of the turbine through a similar pipeline. At a point 150 m above the turbine the pressure is 260 kPa, and at a point 15 m below the turbine the pressure is 144 kPa. What must be the water flow rate, if the turbine output is 2.5 MW?

Expert avatar
Fred
4.4
115 Answers
1. **Write Bernoulli's equation accounting for turbine work:**

\frac{P_1}{\rho g} + z_1 = \frac{P_2}{\rho g} + z_2 + \frac{\dot{W}}{\dot{m} g}

2. **Rewrite using \( \dot{m} = \rho Q \):**

\frac{\dot{W}}{\rho Q g} = \left( \frac{P_1 - P_2}{\rho g} \right) + (z_1 - z_2)

3. **Substitute known values:**

\frac{2.5 \times 10^6}{1000 \cdot Q \cdot 9.81} = \left( \frac{260000 - 144000}{1000 \cdot 9.81} \right) + 165

4. **Simplify:**

\frac{2.5 \times 10^6}{9810 \cdot Q} = 11.8267 + 165 = 176.8267

5. **Solve for \( Q \):**

Q = \frac{2.5 \times 10^6}{9810 \cdot 176.8267} \approx \frac{2.5 \times 10^6}{1,734,636.7} \approx 1.441 \, \text{m}^3/\text{s}

6. **Final answer:**

Q \approx 1.44 \, \text{m}^3/\text{s}

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