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?
Fred
4.4115 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}
2. **Rewrite using \( \dot{m} = \rho Q \):**
3. **Substitute known values:**
4. **Simplify:**
5. **Solve for \( Q \):**
6. **Final answer:**
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