Chalmers University of Technology Axial compressors 1 + compEDU tutorial Elementary axial compressor theory –Velocity triangles –Limitations for compressor

Chalmers University of Technology

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


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


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

0103

0103

21

1

23

3

00103

22TTchh

Vh

Vhwq p

gasPerfect

hh


How is the temperature rise related to the blade angles ?

• We study change of angular momentum at mid of blade (as approximation)


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:

12

11221122

1122

radiusconstant at Flow

ww

wwww

ww

CCU

UCUCrCrCworklTheoretica

rCrCtorquelTheoretica

momentumangularofchangeofRate


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:

2112

2211

22

11

tantantantan

tantantantan

tantan

tantan

22

11

ww

ww

V

a

C

a

V

a

C

a

CCU

CCU


Theory 9.1 – Stage temperature rise• Using the trigonometric relation from above together with the work relation we get:

tantan

relation rictrigonomettantan

21

1212

a

aww

mUC

mUCCCmUW

• Introducing the derived relation for work into the energy equation finally yields relationbetween air angles and temperaturerise:

210102 tantan p

a

c

UCTT


Conclusions

210102 tantan p

a

c

UCTT

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


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: –

pc

CTT

2

21

011

• Typical (high) values : C=200 m/s, U = 450 m/s => Mrel,tip= 1.5 (check this yourself)

22tt UCV

1RTa


• The relative velocity decreasesin the rotor. – Too rapid retardation => separation and excessive losses. – A design criteria to limit the retardation is set by the de Haller number:

72.01

2 V

V

• High fluid deflection => high rate of diffusion.

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:

Fluid deflections and limitations

c

s

V

C

V

V

V

VcsC

V

VforresultsTestV

VVD

w

numberHallerde

w

11

2

1

21

max1

2max

212


Blockage

210102 tantan a

p

UCc

TT

21

21

tantan

tantan

a

aa

CUmU

CCUmUW

• 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:


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 :

stageinriseenthalpystatic

rotorinriseenthalpystatic

• 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


Degree of reaction• Let Δ TA and Δ TB denote the static temperature rise in the rotor

and stator respectively. Then: 1221 tantantantan aaSpBAp UCUCTcTTcW

• Use that all work input occurs in the rotor, i.e.

21

22

21

1

22

20102 2

1

22CCTc

c

CT

c

CTcTTcW Ap

pppp

• Combining the two relations yields:

2

1

2

212

11

22

21

2212

21

22

coscos2

1tantan

cos,

cos

2

1tantan

2

1

aaa

aa

aAp

CCUC

CC

CC

CCUCCCWTc


Degree of reaction

Apa

a

aa

aa

TcC

UC

CUC

CUC

12

22

2

12

12

12

12

22

22

222

12

22

12

22

2

12

tantan2

tantan

cos

sincos

cos

sincos

2tantan

1sincos that Usecos

1

cos

1

2tantan....

• From the definition of Λ we then have:

12

12

1212

2

12

12

12

22

2

12

tantan2

1

tantan

tantantantan2

tantan

tantan

tantan2

tantan

U

C

UC

CUC

UC

CUC

TT

T

a

a

aa

a

aa

BA

A


Degree of reaction

equationsSum

2121

2211

tantan2

tantan2

1

tantantantan2

U

C

U

C

C

U

aa

a

Basic velocity triangles again:

22

11

tantan

tantan

a

a

C

U

C

U

2112 tantan2

tantan2

1 U

C

U

C aa

Thus, we get:


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

Documents

Chalmers University of Technology Axial compressors 1 + compEDU tutorial Elementary axial compressor theory –Velocity triangles –Limitations for compressor