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Fundamentals of turbomachines -The Euler’s equation

Fundamentals of turbomachines -The Euler’s equation:
Rotating machines are called as turbomachines.
Turbomachines are a device, in which the rotating fluid interacts with the rotor which rotates about an axis.
When a fluid interacts with a rotor, two things take place
  1. The energy transfer and
  2. The energy transformation
The energy transfer takes place in the rotating parts( i.e) it may be transferred from the fluid to rotor or vice versa
In case if the energy is transferred from fluid to rotor it is called as a turbine
If the energy is transferred from rotor to fluid it is called as a compressor.
Energy transfer takes place only in the rotating parts
Whereas the energy transformation takes place both in the rotating and stationary parts.
Energy transformation means the change of kinetic energy to the pressure energy.
The forces developed in the turbomachines is because of Newton’s second law of motion or the combined action of both newtons second and third law.
Newton’s second law states that the rate of change of momentum is directly proportional to the force applied.
In case of turbomachines, the angular momentum takes place,
Therefore the rate of change of angular momentum is directly proportional to the applied torque.
fundamentals of turbo machines - the Eulers energy equation
Consider a fluid of mass m and interacts with the rotor with a tangential velocity C1 at a distance r1 from the axis of the rotor and leaves the rotor with a tangential velocity c2 at a distance r2 from the rotor. as this fluid (gas) interacts with the rotor which rotates about the axis A-A, there comes an angular momentum which is directly proportional to the torque applied.
The angular momentum at the inlet is mc1r1
The angular momentum at the exit is mc2r2
Applying Newton’s second law,
The applied torque = mc2r2 -mc1r1
The energy transfer E = T×ω(energy = torque * angular momentum)
E=(mc2r2 -mc1r1ω
E=m(c2r2 -c1r1ω
E=m(c2r2ω -c1r1ω)
E=m(c2u2 -c1u1) (u=rω )
If m=1
 E=(c2u2 -c1u1)
this is the Euler’s energy equation.
where u2 and u1 is the  tangential velocity of the rotoe and c2 and c1 is the tangential velocity of the gas.
c2u2 >c1u1 the energy transfer is positive and is a Turbine
If c2u2 < c1u1 the energy transfer is negative and is a compressor
Energy transfer equation of a Turbine is ET=(c2u2 -c1u1)
Energy transfer equation of a compressor is Ec=(c1u1 -c2u2)
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Quick Reference Corner

4 basics every engineer must know

4 basics every engineer must know:

Pressure:

What pressure is and how it works is so fundamental to the understanding of aerodynamics.

There are two ways to look at pressure:

Action of individual air molecules(using kinetic theory of gases) .

The action of a large number of molecules

.

By definition, Pressure is defined as the normal force per unit area.

P=FA

Unit of pressure is N/m2 , Bar, Pascal.

Pressure is a Scalar quantity (i.e. it has only magnitude, not the direction)

A gas is composed of a large number of molecules that are very small relative to the distance between molecules.The molecules of a gas are in constant, random motion and frequently collide with each other and with the walls of any container.The molecules possess the physical properties of mass, momentum, and energy.

The momentum of a single molecule is the product of its mass and velocity,

Momentum P=m×v

As the gas molecules collide with the walls of a container, the molecules impart momentum to the walls, producing a force perpendicular to the wall.molecule striking wall

The sum of the forces of all the molecules striking the wall divided by the area of the wall is defined to be the pressure.

The pressure of a gas is then a measure of the average linear momentum of the moving molecules of a gas. The pressure acts perpendicular (normal) to the wall.

The tangential (shear) component of the force is the shear stress.

If the gas as a whole is moving, the measured pressure is different in the direction of the motion. The ordered motion of the gas produces an ordered component of the momentum in the direction of the motion. We associate an additional pressure component, called dynamic pressure, with this fluid momentum.

Dynamic pressure

q=12ρν2

The pressure measured in the direction of the motion is called the total pressure and is equal to the sum of the static and dynamic pressures described by Bernoulli’s equation.

Density:

An important property of any gas is its density. Understanding density and how it works is fundamental to the understanding of rocket aerodynamics.

Density is defined as the mass of an object per unit volume. We know that some objects are heavier than other objects, even though they are the same size.

Density is a scalar quantity. Its unit is kg/m3

Different materials have different density. For example aluminum is less dense than iron. That is why airplanes, rockets, and some automobile parts are made from aluminum. For the same volume of material, one metal weighs less than another does if it has a lower density.

For solids, the density remains constant because the molecules are tightly bound. For example, a pure silver coin on the earth weighs same as in the moon.

However, for gases, the density can vary over a wide range because the molecules are free to move. Air at the sea level is different from the air at the stratosphere.

molecules are in constant random motionA gas is composed of a large number of molecules that are very small relative to the distance between molecules. The molecules are in constant, random motion and frequently collide with each other and with the walls of a container. Because the molecules are in motion, a gas will expand to fill the container. Density depends directly on the size of the container in which a fixed mass of gas is confined.

As a simple example, consider the figure. We have 10 molecules of a mythical gas. Each molecule has a mass of 50 grams (.05 kilograms), so the mass of this gas is .05 kg. We have confined this gas in a rectangular tube that is 1 meter on each side and 3 meters high. We are viewing the tube from the front, so the dimension into the slide is 1 meter for all the cases considered. The volume of the tube is 3 cubic meters, so the density is .16 kg/cubic meter.

Temperature:

An important property of any gas is temperature.  An entire branch of physics, called thermodynamics, is devoted to studying the temperature of objects and the transfer of heat between objects of different temperatures.

The molecules are in constant, random motion and frequently collide with each other and with the walls of any container. The molecules possess the physical properties of mass, momentum, and energy. The momentum of a single molecule is the product of its mass and velocity, while the kinetic energy is one half the mass times the square of the velocity.

Unit is K, degree Celsius and degree farenheit.

The speed of sound:

The speed of “sound” is actually the speed of transmission of a small disturbance through a medium. The sound itself is a sensation created in the human brain in response to sensory inputs from the inner ear.

Disturbances are transmitted through a gas as a result of collisions between the randomly moving molecules in the gas.

The conditions in the gas are the same before and after the disturbance passes through. Because the speed of transmission depends on molecular collisions, the speed of sound depends on the state of the gas.

The speed of sound is a constant within a given gas and the value of the constant depends on the type of gas (air, pure oxygen, carbon dioxide, etc.) and the temperature of the gas.

Speed of sound =

γRT

These 4 basic definitions play a strong role in every sort of science. there is no need of memorising these definitions, but understanding provides good foundation for learning. These values vary from region to region in the atmosphere.

 

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