IX. Energy Exchange with Moving Blades

A.   Introduction

So far we have only looked at the thermodynamic results of compressors and turbines (p’s and t’s).  Now we will look in more detail at how the components of a gas turbine engine produce these effects.  You will learn later (in 16.50 for example) that without heat transfer, it is only possible to change the total enthalpy of a fluid with an unsteady process (e.g. moving blades).  Still we will use many of the steady flow tools that we have discussed in thermodynamics and propulsion by considering the steady flow in and out of a component as shown in Figure 8.1.

Figure 9.1 Control volume around compressor or turbine.

B.   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 following model of the blade row:

Figure 9.2 Control volume for Euler Turbine Equation.

Applying conservation of angular momentum, we note that the torque, T, must be equal to the time rate of change of angular momentum in a streamtube that flows through the device (In the text: For more information about angular momentum and rotational energy, see pages 246 and 558 in Hibbler). 

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_{tangential out} > V_{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, w

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:

From the steady flow energy equation

                                           with                              

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

Which 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).

C.   Multistage Axial Compressors

An axial compressor is typically made up of many alternating rows of rotating and stationary blades called rotors and stators, respectively, as shown in Figures 9.3 and 9.4.  The first stationary row (which comes in front of the rotor) is typically called the inlet guide vanes or IGV Each successive rotor-stator pair is called a compressor stage.  Hence compressors with many blade rows are termed multistage compressors.

Figure 9.3 A typical multistage axial flow compressor (Rolls-Royce, 1992).

Figure 9.4 Schematic representation of an axial flow compressor.

One way to understand the workings of a compressor is to consider energy exchanges.  We can get an approximate picture of this using the Bernoulli Equation, where P_T is the stagnation pressure, a measure of the total energy carried in the flow, p is the static pressure a measure of the internal energy, and the velocity terms are a measure of the kinetic energy associated with each component of velocity (u is radial, v is tangential, w is axial).

The rotor adds swirl to the flow, thus increasing the total energy carried in the flow by increasing the angular momentum (adding to the kinetic energy associated with the tangential or swirl velocity, 1/2rv²).

The stator removes swirl from the flow, but it is not a moving blade row and thus cannot add any net energy to the flow.  Rather, the stator rather converts the kinetic energy associated with swirl to internal energy (raising the static pressure of the flow).  Thus typical velocity and pressure profiles through a multistage axial compressor look like those shown in Figure 9.5.

Figure 9.5 Pressure and velocity profiles through a multi-stage axial compressor (Rolls-Royce, 1992).

Note that the IGV also adds no energy to the flow.  It is designed to add swirl in the direction of rotor motion to lower the Mach number of the flow relative to the rotor blades, and thus improve the aerodynamic performance of the rotor.

D.  Velocity Triangles for an Axial Compressor Stage

Velocity triangles are typically used to relate the flow properties and blade design parameters in the relative frame (rotating with the moving blades), to the properties in the stationary or absolute frame.

We begin by “unwrapping” the compressor.  That is, we take a cutting plane at a particular radius (e.g. as shown in Figure 9.3) and unwrap it azimuthally to arrive at the diagrams shown in Figure 9.6.  Here we have assumed that the area of the annulus through which the flow passes is nearly constant and the density changes are small so that the axial velocity is approximately constant.

Figure 9.6 Velocity triangles for an axial compressor stage.  Primed quantities are in the relative frame, unprimed quantities are in the absolute frame.

In drawing these velocity diagrams it is important to note that the flow typically leaves the trailing edges of the blades at approximately the trailing edge angle in the coordinate frame attached to the blade (i.e. relative frame for the rotor, absolute frame for the stator).

We will now write the Euler Turbine Equation in terms of stage design parameters: w, the rotational speed, and b_b and b_c’ the leaving angles of the blades.

From geometry,

v_b = w_b tan b_b               and                   v_c = w_c tan b_c = wr_c - w_c tan

so

or

So we see that the total or stagnation temperature rise across the stage increases with the tip Mach number squared, and for fixed positive blade angles, decreases with increasing mass flow.  This behavior is represented schematically in Figure 9.7.

Figure 9.7 Compressor behavior

E.   Velocity Triangles for an Axial Flow Turbine Stage

We can apply the same analysis techniques to a turbine.  Again, the stator does no work.  It adds swirl to the flow, converting internal energy into kinetic energy.  The turbine rotor then extracts work from the flow by removing the kinetic energy associated with the swirl velocity.

Figure 9.8 Schematic of an axial flow turbine.

The appropriate velocity triangles are shown in Figure 9.9, where again the axial velocity was assumed to be constant for purposes of illustration.

As we did for the compressor, we can write the Euler Turbine Equation in terms of useful design variables:

Figure 9.9 Velocity triangles for an axial flow turbine stage.