Question

If H is the available head for a hydraulic turbine, the speed (N), the discharge (Q), and the power (P), respectively are proportional to

• A. \( N \propto \sqrt{H}; Q \propto \sqrt{H}; P \propto H^{3/2} \)
• B. \( N \propto \sqrt{H}; Q \propto H; P \propto H^{3/2} \)
• C. \( N \propto \sqrt{H}; Q \propto H^{3/2}; P \propto H^{5/2} \)
• D. \( N \propto H; Q \propto \sqrt{H^{3}}; P \propto H^{5/2} \)

Hint:

These proportionality relationships are fundamental in understanding the performance characteristics of hydraulic turbines under varying head conditions. They are often used in preliminary design and analysis. Remember that these are ideal proportionalities, and actual performance might deviate due to changes in efficiency and other factors.

Answer 1

The Correct Option is A

Step 1: Understand the relationships between turbine parameters and available head.
The speed, discharge, and power developed by a hydraulic turbine are related to the available head \( H \), among other factors like the turbine design and efficiency. For a given turbine operating under different heads, certain proportionality relationships hold. Step 2: Recall the general proportionality relationships derived from similarity laws or basic principles.
Speed (N): The speed of a turbine is primarily related to the velocity of the water striking the runner. This velocity is proportional to \( \sqrt{2gH} \), where \( g \) is the acceleration due to gravity. Therefore, for a given turbine geometry, the speed \( N \) is proportional to \( \sqrt{H} \). $$N \propto \sqrt{H}$$ Discharge (Q): The discharge through a turbine is related to the velocity of flow and the area of flow. Since the velocity is proportional to \( \sqrt{H} \) and the flow area is generally considered constant for a given operating condition relative to the turbine size, the discharge \( Q \) is also proportional to \( \sqrt{H} \). $$Q \propto \sqrt{H}$$ Power (P): The power developed by a hydraulic turbine is the product of the head, discharge, and efficiency (\( \eta \)). The weight flow rate of water is proportional to \( \rho g Q \), and the energy per unit weight is proportional to \( H \). Thus, the power \( P \propto \rho g Q H \eta \). Assuming efficiency is relatively constant over a range of operating heads for a given turbine, and substituting \( Q \propto \sqrt{H} \), we get: $$P \propto H \cdot \sqrt{H} = H^{3/2}$$ So, \( P \propto H^{3/2} \). Step 3: Combine the proportionality relationships for N, Q, and P.
Based on the above derivations:
\( N \propto \sqrt{H} \)
\( Q \propto \sqrt{H} \)
\( P \propto H^{3/2} \)
Step 4: Identify the option that matches these proportionalities.
Looking at the given options: Option (1) \( N \propto \sqrt{H}; Q \propto \sqrt{H}; P \propto H^{3/2} \) matches our derived relationships.
Options (2), (3), and (4) have different proportionalities for \( Q \) and \( P \), making them incorrect.
Therefore, the correct answer is (1).