Chalmers University of Technology Axial compressors 1 comp

Chalmers University of Technology Axial compressors 1 + comp. EDU tutorial • Elementary axial compressor theory – Velocity triangles – Limitations for compressor performance • relative Mach number limitations • deflection limitation – Degree of reaction • comp. EDU – Axial compressor tutorial – Overhaul and maintenance lab specification

Chalmers University of Technology 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

Chalmers University of Technology Elementary theory • Energy equation for control volumes (again): • Adiabatic compression process (work added to system sign convention added work = -w) – Rotor => -(-w) = cp(T 02 -T 01) <=> w = cp(T 02 -T 01) – Stator => 0 = cp(T 03 -T 02) => T 03= T 02

Chalmers University of Technology How is the temperature rise related to the blade angles ? • We study change of angular momentum at mid of blade (as approximation)

Chalmers University of Technology Theory 9. 1 – Stage temperature rise • Relative and absolute refererence frames are related by: C=V+U • Many compressors have been designed assuming Ca=Ca 1=Ca 2. We will assume this for the following derivation • We repeat the derivation of theoretical work used for radial compressors and axial turbines:

Chalmers University of Technology Theory 9. 1 – Stage temperature rise • Combining that flow occurs at a constant radius (=>U 2=U 1) and that the axial velocity is assumed to be constant (design assumption) we get:

Chalmers University of Technology 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:

Chalmers University of Technology Conclusions To obtain a high temperature rise we should: • High blade speed (U) • High axial speed (Ca) • High fluid deflection (β 1 - β 2) Blade stresses and aerodynamic considerations limit these design selections • The isentropic efficiency then relates the blade angles to the pressure rise of the stage

Chalmers University of Technology 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: – • Typical (high) values : C=200 m/s, U = 450 m/s => Mrel, tip= 1. 5 (check this yourself)

Chalmers University of Technology 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.

Chalmers University of Technology 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 V 2 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:

Chalmers University of Technology 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:

Chalmers University of Technology 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 – C 3 = C 1 => Δ TStage= Δ T 0, Stage

Chalmers University of Technology 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:

Chalmers University of Technology Degree of reaction • From the definition of Λ we then have:

Chalmers University of Technology Degree of reaction Basic velocity triangles again: Thus, we get:

Chalmers University of Technology 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 comp. EDU tutorial