Short Answer:

The conditions for maximum efficiency in turbines are achieved when the energy transfer between water and the turbine runner is at its highest level. For an impulse turbine like the Pelton wheel, maximum efficiency occurs when the velocity of the bucket is exactly half of the jet velocity. This condition ensures that the kinetic energy of the water is fully utilized with minimum energy loss.

In general, maximum efficiency is obtained when fluid flow is smooth, frictional losses are minimum, and the mechanical parts work without vibration. The correct relationship between jet velocity, bucket velocity, and flow direction ensures the turbine performs at its highest efficiency.

Detailed Explanation:

Conditions for Maximum Efficiency

Efficiency in a hydraulic turbine refers to how effectively the turbine converts the energy of water into useful mechanical energy. Maximum efficiency means the turbine is performing at its best condition, with minimum energy losses due to friction, turbulence, or mechanical resistance.

In turbines like the Pelton wheel, the condition for maximum efficiency is based on the relationship between the jet velocity (the speed of water striking the bucket) and the bucket velocity (the speed at which the bucket moves). When this relationship is ideal, the turbine extracts the highest possible amount of energy from the water jet.

Derivation of the Condition for Maximum Efficiency

Consider a jet of water striking the buckets of a Pelton wheel tangentially with a velocity . The bucket moves with a tangential velocity . The velocity of the water jet relative to the bucket is .

When the water jet hits the bucket, it gets deflected through an angle  (usually between 160° to 170°). The force exerted by the jet on the bucket can be expressed as:

Where,

•  = Density of water
•  = Cross-sectional area of the jet
•  = Jet velocity
•  = Bucket velocity
•  = Angle of deflection of the jet

The work done per second on the runner (or power developed) is:

The energy supplied by the water jet per second is:

Therefore, the hydraulic efficiency of the turbine can be written as:

Finding Condition for Maximum Efficiency

To find the condition for maximum efficiency, differentiate the efficiency equation with respect to  and equate it to zero.

Solving this gives:

This shows that maximum efficiency occurs when the bucket velocity is half of the jet velocity.

Substituting  into the efficiency equation:

For an ideal deflection angle ,

However, in practical situations, the deflection angle is between 160° to 170°, giving a hydraulic efficiency of about 90%.

Practical Conditions for Maximum Efficiency

Besides the velocity ratio, there are other practical conditions that help achieve maximum efficiency in a turbine system:

1. Proper Jet Alignment:
   The water jet should strike the bucket at the correct angle on its splitter to ensure smooth deflection and avoid splashing losses.
2. Smooth Flow of Water:
   The water should flow through the nozzle, buckets, and runner with minimum turbulence to avoid energy loss due to friction.
3. Optimum Head Utilization:
   The net head should be efficiently converted into kinetic energy without excessive frictional loss in the penstock or nozzle.
4. Efficient Mechanical System:
   The shaft, bearings, and other moving parts should have minimum friction to reduce mechanical losses.
5. Proper Maintenance of Equipment:
   Clean and smooth surfaces of buckets and nozzles ensure that no energy is lost due to surface roughness or scaling.
6. Correct Speed Regulation:
   The turbine should operate at its design speed where the energy transfer between water and runner is most effective.

Example for Condition of Maximum Efficiency

Let the net head .
Then, jet velocity .
For maximum efficiency,

Thus, the turbine will operate most efficiently when the bucket velocity is about 44.3 m/s for the given head.

Importance of Maximum Efficiency Condition

• Ensures maximum energy extraction from the water jet.
• Reduces energy losses due to improper flow or friction.
• Maintains steady operation with less vibration and wear.
• Increases the life span of turbine components.
• Provides economic operation by maximizing power output per unit of water used.

Conclusion:

The condition for maximum efficiency in a Pelton wheel or any impulse turbine occurs when the velocity of the moving buckets is half of the velocity of the water jet striking them. This allows the jet to transfer almost all its kinetic energy to the turbine without wasting energy in splashing or turbulence. Along with proper mechanical design, smooth flow, and correct deflection angle, this velocity ratio ensures the turbine operates at its best performance, achieving maximum overall efficiency.