Flow Velocity Calculation for Kaplan Turbine

Given:
Discharge through the runner = 52 m3/s
Outer diameter of the blade = 3 m
Hub diameter = 1 m

To find: Flow velocity at the inlet

Approach:
- Use the continuity equation to find the velocity at the inlet
- The continuity equation states that the mass flow rate is constant at all points in the turbine
- The mass flow rate is given by the product of density, area, and velocity
- Since the density is constant, we can equate the mass flow rates at the inlet and outlet to get the velocity at the inlet

Calculation:
- The cross-sectional area of the runner can be approximated as a ring with inner diameter = hub diameter and outer diameter = blade diameter
- Cross-sectional area = π/4 * (3^2 - 1^2) = 7π m2
- Mass flow rate = density * discharge = 1000 * 52 = 52,000 kg/s
- Velocity at the inlet = mass flow rate / (density * area) = 52,000 / (1000 * 7π) = 2.34 m/s
- However, this is the average velocity across the runner. The actual velocity at the inlet may be higher due to the converging shape of the inlet duct
- The velocity increase can be approximated using the Bernoulli's principle, which states that the sum of pressure, kinetic energy, and potential energy is constant along a streamline
- Assuming negligible changes in potential energy, we can equate the pressure at the inlet and outlet to get the increase in kinetic energy
- Using the ideal gas law, the pressure at the inlet can be calculated as P = ρRT, where R is the gas constant and T is the temperature. Assuming standard conditions, P = 101,325 Pa
- Using the same equation for the outlet pressure, we get ΔP = 0, since the pressure remains constant
- Therefore, the increase in kinetic energy is ΔKE = ΔP / ρ = 0 / 1000 = 0
- Thus, the actual velocity at the inlet is approximately the same as the average velocity across the runner

Answer:
The flow velocity at the inlet of the Kaplan turbine is approximately 2.34 m/s. Therefore, option (1) 7.47m/s, option (2) 8.28m/s, option (3) 10.42m/s, and option (4) 12.36m/s are all incorrect.