Fundamentals of turbomachines -The Euler’s equation

Fundamentals of turbomachines -The Euler’s equation:

Synopsis

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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