|Thermodynamics and Propulsion|
The Euler turbine equation relates the power added to or removed from the flow, to characteristics of a rotating blade row. The equation is based on the concepts of conservation of angular momentum and conservation of energy. We will work with the model of the blade row shown in Figure 12.2.
Applying conservation of angular momentum, we note that the torque, , must be equal to the time rate of change of angular momentum in a streamtube that flows through the device
This is true whether the blade row is rotating or not. The sign matters (i.e. angular momentum is a vector - positive means it is spinning in one direction, negative means it is spinning in the other direction). So depending on how things are defined, there can be positive and negative torques, and positive and negative angular momentum. In Figure 9.2, torque is positive when -- the same sense as the angular velocity.
If the blade row is moving, then work is done on/by the fluid. The work per unit time, or power, , is the torque multiplied by the angular velocity, :
If torque and angular velocity are of like sign, work is being done on the fluid (a compressor). If torque and angular velocity are of opposite sign work is being extracted from the fluid (a turbine). Here is another approach to the same idea:
Then equating this expression of conservation of energy with our expression from conservation of angular momentum, we arrive at:
or for a perfect gas with ,
Equation (12.3) is called the Euler Turbine Equation. It relates the temperature ratio (and hence the pressure ratio) across a turbine or compressor to the rotational speed and the change in momentum per unit mass. Note that the velocities used in this equation are what we will later call absolute frame velocities (as opposed to relative frame velocities).