Thermodynamics and Propulsion

12.3 The Euler Turbine Equation

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.

Figure 12.2: Control volume for Euler Turbine Equation.

Applying conservation of angular momentum, we note that the torque, $\mathcal{T}$ , must be equal to the time rate of change of angular momentum in a streamtube that flows through the device

$$\mathcal{T} = \dot{m}(v_c r_c - v_b r_b).$$

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 $V_{\textrm{tangential out}} > V_{\textrm{tangential in}}$ -- 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, $P$ , is the torque multiplied by the angular velocity, $\omega$ :

$$P = \mathcal{T} \omega = \omega \dot{m}(v_c r_c - v_b r_b).$$

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:

• If the tangential velocity increases across a blade row (where positive tangential velocity is defined in the same direction as the rotor motion) then work is added to the flow (this happens in a compressor).
• If the tangential velocity decreases across a blade row (where positive tangential velocity is defined in the same direction as the rotor motion) then work is removed from the flow (this happens in a turbine).

From the steady flow energy equation,

$$\dot{q} -\dot{w}_s = \dot{m} \Delta h_T$$

with

$$\dot{q} = 0 \quad \textrm{and} \quad -\dot{w}_s = P,$$

$$P = \dot{m}(h_{Tc}-h_{Tb}).$$

Then equating this expression of conservation of energy with our expression from conservation of angular momentum, we arrive at:

$$h_{Tc} - h_{Tb} = \omega (r_c v_c - r_b v_b)$$

or for a perfect gas with $c_p=\textrm{constant}$ ,

$$c_p (T_{Tc} - T_{Tb}) = \omega (r_c v_c - r_b v_b).$$ (12..3)

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).

• If angular momentum increases across a blade row, then $T_{Tc} > T_{Tb}$ and work was done on the fluid (a compressor).
• If angular momentum decreases across a blade row, then $T_{Tc} < T_{Tb}$ and work was done by the fluid (a turbine).