Short Answer:

The coefficient of fluctuation of energy is a measure that shows how much the energy stored in a flywheel changes during one complete cycle of operation. It represents the ratio of the difference between the maximum and minimum energy in the flywheel to the mean energy of rotation.

In simple terms, it tells us how much the energy in the flywheel fluctuates while maintaining a constant speed in the machine. A smaller coefficient means less energy variation and smoother running of the engine or machine, while a larger coefficient indicates more fluctuation in stored energy.

Detailed Explanation:

Coefficient of Fluctuation of Energy

The coefficient of fluctuation of energy is an important term used in the study of flywheels and rotating machinery. It is used to express how much the energy stored in a flywheel varies during one complete revolution or working cycle. Since the torque and speed in mechanical systems are not uniform throughout their cycle, the energy stored in the flywheel continuously increases and decreases depending on the torque output.

This variation in stored energy is measured by the coefficient of fluctuation of energy. It helps in designing a flywheel with sufficient capacity to store and release energy as required, ensuring smooth and steady motion in the machine.

Mathematically, the coefficient of fluctuation of energy is expressed as:

Where,
= Coefficient of fluctuation of energy
= Maximum energy stored in the flywheel during a cycle
= Minimum energy stored in the flywheel during a cycle
= Mean energy of rotation of the flywheel

The term represents the total fluctuation of energy, which is the difference between the maximum and minimum energy levels of the flywheel during one cycle.

Explanation of Energy Fluctuation

During the operation of a machine, especially in reciprocating engines or compressors, the energy input and output are not constant. The energy supplied by the engine varies with each stroke, leading to variations in speed and torque. When the energy supplied is more than the energy required for load, the excess energy is stored in the flywheel in the form of rotational kinetic energy.

On the other hand, when the energy required by the load is more than the energy supplied, the flywheel releases the previously stored energy to maintain a constant speed. This alternate storing and releasing of energy causes fluctuation in the amount of energy stored in the flywheel during each revolution.

The coefficient of fluctuation of energy helps engineers to understand and control these changes. It is an essential factor in determining the size, weight, and moment of inertia of the flywheel required for a particular machine.

Derivation and Relation with Flywheel Energy

The rotational kinetic energy of a flywheel is given by the equation:

Where,
= Energy stored in the flywheel
= Moment of inertia of the flywheel
= Angular velocity (rad/s)

During one revolution, the speed of the flywheel is not constant and varies between a maximum value and a minimum value . Therefore, the maximum and minimum energies can be written as:

The fluctuation of energy (ΔE) is the difference between these two:

The mean energy of rotation can be taken as:

Hence, the coefficient of fluctuation of energy becomes:

This equation shows that the coefficient of fluctuation of energy depends directly on the variation in the angular velocity of the flywheel. The smaller the difference between maximum and minimum speeds, the smaller the fluctuation of energy.

Significance of Coefficient of Fluctuation of Energy

Design of Flywheel:
The coefficient of fluctuation of energy is used in determining the required mass and moment of inertia of a flywheel. A higher energy fluctuation requires a larger flywheel to store and release sufficient energy to maintain steady motion.
Machine Performance:
A small coefficient indicates that the machine runs smoothly with less variation in energy and speed. This is desirable for engines, generators, and precision machines where constant speed is important.
Energy Stability:
It ensures that the energy supply remains constant even when there are variations in torque or load. The flywheel compensates for these variations by balancing the energy flow.
Control of Speed Variation:
Since the energy fluctuation is directly related to speed fluctuation, controlling the coefficient of energy fluctuation also helps in reducing speed variation and mechanical stress.
Efficiency and Longevity:
Machines with lower coefficients operate more efficiently, with reduced vibrations and wear, leading to longer operational life and better performance.

Typical Values

The coefficient of fluctuation of energy varies depending on the type of machine:

For engines: = 0.01 to 0.04
For punching or rolling machines: = 0.05 to 0.1
For electric generators: = 0.005 or less

These values indicate how much energy variation is tolerable for different types of mechanical systems.

Example

If a flywheel has a maximum energy of 2100 J and a minimum energy of 1900 J, and its mean energy of rotation is 2000 J, then:

Thus, the coefficient of fluctuation of energy is 0.1, or 10%. This means the energy in the flywheel varies by 10% of the mean energy during operation.

Conclusion:

The coefficient of fluctuation of energy measures the relative variation of energy stored in a flywheel during operation. It is a key factor in flywheel design and helps to ensure smooth, steady, and efficient machine performance. A smaller value of this coefficient indicates better stability and less variation in motion. Therefore, by controlling this factor, engineers can design machines that operate with minimum speed fluctuation and maximum efficiency.