|Flywheel Torque Fluctuation.
Aflywheel is a mechanical device that uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. Flywheels are often used to provide continuous power output in systems where the energy source is not continuous. For example, a flywheel is used to smooth the fast angular velocity fluctuations of the crankshaft in a reciprocating engine. A flywheel is a heavy wheel for opposing and moderating by its inertia any fluctuation of speed in the machinery with which it revolves.
In the case of steam engines, internal combustion engines, reciprocating compressors, and pumps, the energy is developed during one stroke and the engine is to run for the whole cycle on the energy produced during this one stroke. For example, in I.C. engines, the energy is developed only during power stroke which is much more than the engine load, and no energy is being developed during suction, compression, and exhaust strokes in the case of four-stroke engines and during compression in the case of two-stroke engines. The excess energy developed during a power stroke is absorbed by the flywheel and released to the crankshaft during other strokes in which no energy is developed, thus rotating the crankshaft at a uniform speed. A little consideration will show that when the flywheel absorbs energy, its speed increases, and when it releases, the speed decreases. Hence a flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed.
Basic Angular Relationship
A flywheel is essentially a device for storing angular kinetic energy for which the formula is
When a rotating body changes speed, the angular acceleration is related to the moment of inertia and the applied torque by the formula,
When an increase in torque occurs, the flywheel will speed up and absorb energy. The greater the moment of inertia, the more energy it will absorb. The result is that it speeds up less than it would do with a smaller moment of inertia.
When the torque decreases, the flywheel will slow down but the inertia of the system will limit the amount it slows.
When a torque is applied to a body and it rotates, the work done is the product of torque T.Nm and angle ϴ radian.
Work = Tϴ
This leads us to torque — angle diagrams otherwise known as turning moment diagrams.
Torque Analysis
When the flywheel absorbs energy its speed increases and vice versa. This reduces speed fluctuations.
Considering the equilibrium of torques,
I (dω/DT) = Ti — To
where;
When (Ti — To ) is positive then the flywheel is accelerated.
When (Ti — To ) is negative then the flywheel is retarded.
Turning Moment Diagram
A curve showing the variation of the engine output torque or turning moment with the crank rotation is called the turning moment diagram. In a four-stroke engine, the four strokes are identified as suction, compression, power, and exhaust. Power due to combustion is generated only in one of the four strokes and hence the resulting torque on the crankshaft will also fluctuate similarly. In the turning moment diagram, there is one power peak for every two revolutions, i.e. 720°, of the crankshaft. The mean height of the torque curve gives the mean torque on the crankshaft.
For one complete thermodynamic cycle, the turning moment diagram for a single-cylinder double-acting steam engine is shown inFigure 2. The torque is zero when the crank angle (θ) is zero and maximum when the crank angle is 90° and it is again zero when the crank angle is 180°. The area of the turning moment diagram represents the work done per revolution. The line AF is the mean resisting torque curve. In actual practice, the engine is assumed to work against the mean resisting torque, as shown by a horizontal line AF. When the turning moment is positive, between points B and C (or D and E), the crankshaft accelerates. When the turning moment is negative (between points C and D), the crankshaft retards. The work done per cycle is,
P = Tmean × θ
where, Tmean = Mean torque,
θ = Angle turned (in radians) in one revolution
Coefficient of fluctuation of speeds
In most applications, particularly engines and machines, the flywheel’s job is to reduce fluctuations in speed by storing and releasing of kinetic energy. A measure of this ability of the flywheel is called the coefficient of speed fluctuation. The difference between the maximum and minimum speeds during a cycle is called the maximum fluctuation of speed. The ratio of the maximum fluctuation of speed to the mean speed is called the coefficient of fluctuation of speed.
Let, N1 and N2 = Maximum and minimum speeds in r.p.m. during the cycle, then the Coefficient of Fluctuation of Speed will be,
Table 1: Permissible values for the coefficient of fluctuation of speed
The coefficient of fluctuation of speed is a limiting factor in the design of the flywheel. It varies depending on the nature of the service to which the flywheel is employed. The reciprocal of the coefficient of fluctuation of speed is known as the coefficient of steadiness and is denoted by m.
Fluctuation of Energy
The fluctuation of energy may be determined by the turning moment diagram for one complete cycle of operation. The variations of energy above and below the mean resisting torque line are called fluctuation of energy. The areas BbC, CcD, DdE etc. represent fluctuations of energy.
A little consideration will show that the engine has a maximum speed of either at q or at s. This is because the flywheel absorbs energy while the crank moves from p to q and from r to s. On the other hand, the engine has a minimum speed of either p or r. The reason is that the flywheel gives out some of its energy when the crank moves from a to p and from q to r.
Coefficient of Fluctuation of Energy
It is defined as the ratio of the maximum fluctuation of energy to the work done per cycle. It is usually denoted by CE. Mathematically, the coefficient of fluctuation of energy,
Table 2: Coefficient of fluctuation of energy (Cf) for steam and internal combustion engines.
Determination of Maximum Fluctuation of Energy
The difference between the maximum and the minimum energies is known as the maximum fluctuation of energy. A turning moment diagram for a multi-cylinder engine is shown by a wavy curve in Fig. . The horizontal line AG represents the mean torque line. Let a1, a3, and a5 be the areas above the mean torque line and a2, a4 and a6 be the areas below the mean torque line. These areas represent some quantity of energy that is either added or subtracted from the energy of the moving parts of the engine.
Let the energy in the flywheel at A = E, then from Fig., we have
Energy at B = E + a1
Energy at C = E + a1 — a2
Energy at D = E + a1 — a2 + a3
Energy at E = E + a1 — a2 + a3 — a4
Energy at F = E + a1 — a2 + a3 — a4 + a5
Energy at G = E + a1 — a2 + a3 — a4 + a5 — a6 = Energy at A
Let us now suppose that the maximum of these energies is at B and the minimum at E.
∴ Maximum energy in the flywheel
= E + a1
and minimum energy in the flywheel
= E + a1 — a2 + a3 — a4
Maximum fluctuation of energy,
∆ E = Maximum energy — Minimum energy
= (E + a1) — (E + a1 — a2 + a3 — a4) = a2 — a3 + a4
Torque Variation and Energy
The required change in kinetic energy Ek is obtained from the known torque time relation or curve by integrating it for one cycle.
Computing the kinetic energy Ek needed is illustrated by below conditions:
- Torque Time Relation without Flywheel
- Torque Time Relation with Flywheel
A typical torque time relation for example of a mechanical punching press without a flywheel is shown in the figure. In the absence of a flywheel surplus or positive energy is available initially and intermediate and energy absorption or negative energy during punching and stripping operations. A large magnitude of speed fluctuation can be noted as shown below.
To smoothen out the speed fluctuation flywheel is to be added and the flywheel energy needed is computed as illustrated below
Example
Find deltaE for the torque — angle diagram is shown. The enclosed areas are A1 = 400 J A2 = 800 J A3 = 550 J A4 = 150 J. find the moment of inertia for a flywheel which will keep the speed within the range 410 to 416 rev/min. Find the mass of a suitable flywheel with a radius of gyration of 0.5 m
- First, find the greatest fluctuation in energy.
The energy at = EA
Energy at B = EB = EA –A1 = EA — 400
Energy at C = EC = EB +A2 = EA — 400 + 800 = EA + 400
Energy at D = ED = EC — A3 = EA + 400–550 = EA — 150
Energy at E = EE = ED +A4 = EA — 150 + 150 = EA
If the last figure is not equal to EA then there would be an error.
The largest energy value is EA + 400 and the smallest value is EA — 400 so the greatest fluctuation is from +400 to -400 giving 800 Joules.
Equate delatE = 800 Joules.
Now find the coefficient of fluctuation in speed.
Mean speed N = (410 + 416)/2 = 413 rev/min.
There is no need to convert this into radian to find
= (2–1)/ = (N2 — N1)/N = (416–410)/413 = 0.01452
We need the mean in radian/s so = 413 x (2/60) = 43.25 rad/s
Now find the moment of inertia of the flywheel using
I = 800/43.25² * 0.01452 = 29.44 kg m²
Now find the mass of the flywheel.
I = mk2
29.44 = m x 0.5 = 0.25m
m = 29.44/0.25 = 117.8 kg
Conclusion
1. The turning moment diagram (also known as the crank effort diagram) is the graphical representation of the turning moment or crank effort for various positions of the crank.
2. Flywheel absorbs energy, its speed increases, and when it releases energy, the speed decreases.
3. Hence a flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed.
4. Flywheel controls the speed variations caused by the fluctuation of the engine turning moment during each cycle of operation.