Problem Statement
A jet of water, having a velocity of 15 m/s, strikes a curved vane which is moving with a velocity of 6 m/s in the same direction as that of the jet at inlet. The vane is so shaped that the jet is deflected through 135°. The diameter of the jet is 150 mm. Assuming the vane to be smooth, find: (i) the force exerted by the jet on the vane in the direction of motion, (ii) power of the vane, and (iii) efficiency of the vane.
Given Data & Constants
- Velocity of jet, \(V = 15 \, \text{m/s}\)
- Velocity of vane, \(u = 6 \, \text{m/s}\)
- Diameter of jet, \(d = 150 \, \text{mm} = 0.15 \, \text{m}\)
- Deflection angle = 135°
- Angle of the jet at outlet, \(\theta = 180^\circ - 135^\circ = 45^\circ\)
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
Solution
(i) Force Exerted on the Vane (\(F_x\))
The force on a moving curved vane is determined by the mass of water striking the plate per second (based on relative velocity) and the change in velocity of the jet.
(ii) Power of the Vane (Work Done per Second)
The power delivered to the vane is the work done per second, which is the force in the direction of motion multiplied by the vane's velocity.
(iii) Efficiency of the Vane
Efficiency is the ratio of the useful power delivered to the vane to the initial kinetic energy of the jet.
(i) Force exerted on the vane: \( \approx 2442.6 \, \text{N} \)
(ii) Power of the vane: \( \approx 14.66 \, \text{kW} \)
(iii) Efficiency of the vane: \( \approx 49.2\% \)
Explanation of Key Concepts
- Force on a Moving Curved Vane: The force calculation is based on the impulse-momentum principle. The mass of water striking the vane is determined by the relative velocity (\(V-u\)). The change in velocity is also based on this relative velocity being deflected through the angle \(\theta\).
- Power: This is the useful work done by the jet on the vane. It's the component of force acting in the direction the vane is moving, multiplied by the vane's speed.
- Efficiency: This measures how effectively the initial kinetic energy of the water jet is converted into useful mechanical work on the vane. The efficiency is less than 100% because some kinetic energy is always lost; the water still has velocity when it leaves the vane.
