1. Axial compressors 1 + compEDU tutorial
Elementary axial compressor theory
Velocity triangles
Limitations for compressor performance
relative Mach number limitations
deflection limitation
Degree of reaction
compEDU
Axial compressor tutorial
Overhaul and maintenance lab specification

2. Axial flow compressors
Working fluid is accelerated by the rotor and decelerated by the stator
Boundary layer growth and separation (stall) limits the rate of allowable diffusion
Diffusion (decrease of velocity and increase of static pressure) occurs in stator and in relative frame of rotor
Pictured here are the aerodynamic pressure contours from a computational simulation of a 21 blade row high performance compressor. The compressor is part of General Electric's GE90 turbofan engine that powers the new Boeing 777 airplane. The Average Passage NASA (APNASA) code was utilized for the 3- dimensional Navier-Stokes flow simulation. The computed pressure contours enable engineers to understand compressor performance before building and testing expensive hardware.

3. Elementary theory Energy equation for control volumes (again):
Adiabatic compression process (work added to system - sign convention added work = -w)
Rotor => -(-w) = cp(T02-T01) <=> w = cp(T02-T01)
Stator => 0 = cp(T03-T02) => T03= T02

4. How is the temperature rise related to the blade angles ?
We study change of angular momentum at mid of blade (as approximation)

5. Theory 9.1 – Stage temperature rise
Relative and absolute refererence frames are related by:
C = V + U
Many compressors have been designed assuming Ca=Ca1=Ca2. We will assume this for the following derivation
We repeat the derivation of theoretical work used for radial compressors and axial turbines:

6. Theory 9.1 – Stage temperature rise
Combining that flow occurs at a constant radius (=>U2=U1) and that the axial velocity is assumed to be constant (design assumption) we get:

7. Theory 9.1 – Stage temperature rise
Using the trigonometric relation from above together with the work relation we get:
Introducing the derived relation for work into the energy equation finally yields relation between air angles and temperature rise:

8. Conclusions To obtain a high temperature rise we should:
High blade speed (U)
High axial speed (Ca)
High fluid deflection (β1- β2)
Blade stresses and aerodynamic considerations limit these design selections
The isentropic efficiency then relates the blade angles to the pressure rise of the stage

9. Axial velocity and Mach numbers
Relative Mach number greatest at blade tip. Assuming axial and constant velocity over rotor entry:
The static temperature is:
Speed of sound is:
V = sqrt(450^ ^2)=492 m/s
T1 = (200^2)/(2*1005)=268.2 m/s
a = sqrt(1.4*287.0*T1)= m/s
Mt,relative=1.50
Typical (high) values : C=200 m/s, U = 450 m/s => Mrel,tip= 1.5 (check this yourself)

10. Fluid deflections and limitations
The relative velocity decreases in the rotor.
Too rapid retardation => separation and excessive losses.
A design criteria to limit the retardation is set by the de Haller number:
High fluid deflection => high rate of diffusion.

11. Fluid deflections and limitations
A better estimate of diffusion can be derived if pitch and chord is taken into account.
Greatest diffusion from Vmax to V2 on suction side. Here, boundary layer growth will be most severe => largest part of losses created in this region
We approximate the diffusion, D, by this velocity change according to:

12. Blockage
Boundary layer growth at annulus walls creates a peaky flow profile
For a fixed design (α 1 and β 2) can no longer be varied within the diffusion constraints), increasing Ca leads to a decrease in work output:
This is approximated by the use an empirical factor - the work done factor λ according to:

13. Degree of reaction Diffusion takes place in both rotor and stator.
The division characterizes the design
The quantity measuring this division is the degree of reaction - Λ :
We will derive Λ assuming:
variation in cp over temperature ranges is neglible => we can use temperatures
Ca constant
C3 = C1 => Δ TStage= Δ T0,Stage

14. Degree of reaction
Let Δ TA and Δ TB denote the static temperature rise in the rotor and stator respectively. Then:
Use that all work input occurs in the rotor, i.e.
Combining the two relations yields:

15. Degree of reaction
From the definition of Λ we then have:

16. Basic velocity triangles again:
Degree of reaction
Basic velocity triangles again:
Thus, we get:

17. Learning goals
Know how to relate blade angles to stage temperature rise
Understand how fluid mechanics limits the performance of axial compressor design:
Mach number limitations
Blockage
Have a basic insight of gas turbine overhaul and maintenance as given by compEDU tutorial