ME 423 Chapter 5 Axial Flow Compressors

ME 423ME 423

Chapter 5Chapter 5Axial Flow CompressorsAxial Flow Compressors

Prof. Dr. O. Cahit ERALPProf. Dr. O. Cahit ERALP

Me 423 Spring 2006Me 423 Spring 2006 Prof. Dr. O. Cahit ERALPProf. Dr. O. Cahit ERALP

Axial Flow CompressorsAxial Flow Compressors

Subsonic compressorsSubsonic compressors will be considered here as will be considered here as supersonic compressors have not proceeded beyond supersonic compressors have not proceeded beyond experimental stage.experimental stage.

A Comparison ofA Comparison of

Axial Flow Axial Flow CCompressors and Turbinesompressors and Turbines

TurbineTurbine :-:- Accelerating flow - Successive pressure drops Accelerating flow - Successive pressure drops and consequent reductions in enthalpy being converted and consequent reductions in enthalpy being converted into kinetic energy into kinetic energy

A1>A2 A1>A2 ⇨⇨ converging passages converging passages

Chapter 5 Axial Flow Compressors



A Comparison ofA Comparison of Axial Flow Compressors and TurbinesAxial Flow Compressors and Turbines

Compressor :-Compressor :- Decelerating flow - Pressure rises are Decelerating flow - Pressure rises are obtained through successive stages of diffusing obtained through successive stages of diffusing passages with consequent reduction in velocity. passages with consequent reduction in velocity.

A1<A2 A1<A2 ⇨⇨ diverging passages diverging passages PProblems of compressors & turbines are differentroblems of compressors & turbines are different

in in compressorscompressors - aerodynamic problems - aerodynamic problems

in in turbines turbines - problems due to entry temperature and - problems due to entry temperature and heat-transfer.heat-transfer.

Boundary Layers -Boundary Layers - regions of low momentum air where regions of low momentum air where viscous effects dviscous effects doominate over inertial effects.minate over inertial effects.




Boundary layers are far less happy in a compressive flowBoundary layers are far less happy in a compressive flow.. BL in a compressor operate in an unfavourable pressure BL in a compressor operate in an unfavourable pressure gradient [(+) 've ; p increase ]gradient [(+) 've ; p increase ]

BL in a turbine operate in a favourable pressure gradientBL in a turbine operate in a favourable pressure gradient [ [ (-)'ve ; p decrease. ](-)'ve ; p decrease. ]

This is the reason why a single stage turbine can create This is the reason why a single stage turbine can create enough power to drive a number of stages of compressor.enough power to drive a number of stages of compressor.

Bend thin plates and stick them behind each other forming Bend thin plates and stick them behind each other forming a stationary a stationary cascade of bladescascade of blades..

Let the flow be directed towards the inletLet the flow be directed towards the inlet ofof this this cascade cascade of bladesof blades without any without any incidenceincidence..




As As AA22 > A > A11 in subsonic flow (incompressible) in subsonic flow (incompressible)

WW22 < W < W11 && PP22 > P > P11

This is no more than a subsonic diffuser This is no more than a subsonic diffuser

To carryTo carry a mechanical load a mechanical load,, somesome thickness is required. thickness is required.

If M < 0.3 If M < 0.3 incompressible incompressible

i.e. i.e. thus thus

1 2

P W P W2 22

1 121

212

P0 0 2

2

2

112 2

1WWPP ..




Clearly the outlet velocity WClearly the outlet velocity W22 cannot decrease beyond a cannot decrease beyond a certain level (cannot be zero)certain level (cannot be zero) (or W (or W22 ≠≠ 0) 0)

p Wp W1122 (since W (since W22 is fixed by the lower limit) is fixed by the lower limit)

One should design the compressor at the highest inlet One should design the compressor at the highest inlet velocity velocity

But the But the losses losses ⇨⇨ PPoo αα W W1122

αα 1/ 1/ pp

Stage pressure ratio is limited and the number of stages Stage pressure ratio is limited and the number of stages are determined accordinglyare determined accordingly (single stage or multistage)(single stage or multistage)




Due to the contractionDue to the contraction,, the flow initially accelerates the flow initially accelerates pressure drops (favourable to BL)pressure drops (favourable to BL) ( A( A11 > A > A11' )' )

then then

The amount of pressure rise between 1' to 2 is larger The amount of pressure rise between 1' to 2 is larger than that of 1 to 2than that of 1 to 2..

i.e more diffusion the limit of Wi.e more diffusion the limit of Wmaxmax is than of sonic limit. is than of sonic limit.

More diffusion means less efficiencyMore diffusion means less efficiency i.e why we prefer compressor bi.e why we prefer compressor bllades ades to be to be as thin as as thin as possible.possible.

W

W

W

Wmax

2

1

2




The more the (camber)The more the (camber),, the more is the adverse pressure the more is the adverse pressure gradientgradient,, then seperation occurs earlier. then seperation occurs earlier.

The sepThe sepeerraated flow leaves the blade at an unwanted ted flow leaves the blade at an unwanted angle and unsteady situation.angle and unsteady situation.

All these problems in compressor cascades are due to All these problems in compressor cascades are due to Boundary layers.Boundary layers.

TurbineTurbine problems are completely different since we want problems are completely different since we want the pressure to drop along the flow direction.the pressure to drop along the flow direction.

The flow is a high "h" enthalpy or high temperature, high The flow is a high "h" enthalpy or high temperature, high pressure (to low T low P) flow. pressure (to low T low P) flow.




The blades are such that minimum c/s area occurs at the The blades are such that minimum c/s area occurs at the trailing edge of the blades trailing edge of the blades which is which is called the throat.called the throat.

The flow area should contract continuously all the way The flow area should contract continuously all the way along the balong the bllades in order not to have an adverse ades in order not to have an adverse pressure gradient BL along the row. pressure gradient BL along the row.

Even an instantaneous discontinuity in the contraction of Even an instantaneous discontinuity in the contraction of the passage results in a locally seperated BL, thus the passage results in a locally seperated BL, thus increased turbulence. increased turbulence.

This might happen due to simplified manufacture for This might happen due to simplified manufacture for curvatures such as two circles.curvatures such as two circles.

This results in extremely high heat transfer coefficientThis results in extremely high heat transfer coefficient, , thus the blade will not last 10 minutes. thus the blade will not last 10 minutes.




For the For the Axial Compressors and turbines Axial Compressors and turbines the basic the basic components are components are rotors and statorsrotors and stators, the former carrying , the former carrying the rotating blades and the latter the stationary rows the rotating blades and the latter the stationary rows which serve to recover the pressure rise from the kinetic which serve to recover the pressure rise from the kinetic energy imparted to the fluid by the rotor blades as in energy imparted to the fluid by the rotor blades as in compressors and/or to redirect the flow into an angle compressors and/or to redirect the flow into an angle suitable for entry to the next row of moving blades.suitable for entry to the next row of moving blades.

A A compressor stagecompressor stage is composed of a is composed of a rotor followed rotor followed by a statorby a stator, where as a , where as a turbine stageturbine stage is composed of is composed of a a stator followed by a rotor .stator followed by a rotor .




In CompressorsIn Compressors

It is usual to provide a row of stator blades –It is usual to provide a row of stator blades –Inlet Guide Inlet Guide Vanes (IGV's)Vanes (IGV's) at the upstream of the first stage. These at the upstream of the first stage. These direct the axially approaching flow correctly into the first direct the axially approaching flow correctly into the first row of rotor blades. Thus deflect the flow from axial row of rotor blades. Thus deflect the flow from axial direction to off-axial direction. IGV's are turbine type of direction to off-axial direction. IGV's are turbine type of blades.blades.

Two forms of Two forms of rotor constructionrotor construction is used is used

Drum typeDrum type-suitable for industrial applications-suitable for industrial applications

Disc typeDisc type - suitable for aircraft applications low weight, - suitable for aircraft applications low weight, high cost)high cost)




Another important constructional detail is the contraction Another important constructional detail is the contraction of the flow annulus from the low the high pressure end of of the flow annulus from the low the high pressure end of the compressor. the compressor.

This is necessary to maintain a reasonably constant axial This is necessary to maintain a reasonably constant axial velocity along. most compressors are designed on the velocity along. most compressors are designed on the basis of constbasis of constantant axial velocity because of the axial velocity because of the simplsimpliification in design procedure. One could have a fication in design procedure. One could have a rising hub or rising hub or a a falling shroud in compressors.falling shroud in compressors.



Elementary Theory For Axial Flow CompressorsElementary Theory For Axial Flow Compressors

Basic principle : Acceleration of the working fluid followed Basic principle : Acceleration of the working fluid followed by diffusion to convert the acquired kinetic energy into a by diffusion to convert the acquired kinetic energy into a pressure rise. pressure rise.

The flow is considered as occuring in the tangential plane The flow is considered as occuring in the tangential plane at the mean blade height where the blade peripheral at the mean blade height where the blade peripheral velocity is u.velocity is u.

When the annulus is unrolled, since the blade C/S When the annulus is unrolled, since the blade C/S changes from Hub to Tip, one C/S is chosen changes from Hub to Tip, one C/S is chosen (e(e..gg.. at mid blade height) at mid blade height) and a series of constant C/S aerofoils result.and a series of constant C/S aerofoils result.

These are called a These are called a 2-D cascade of aerofoils2-D cascade of aerofoils..




A semi-A semi- cascade can be produced if the cascade end cascade can be produced if the cascade end boundary effects are eliminated (The flow in the channels boundary effects are eliminated (The flow in the channels are not aware of what happens at the ends)are not aware of what happens at the ends)

TheThe aerodynamics of a cascade repeat aerodynamics of a cascade repeatss itself with a itself with a periodicity of periodicity of ss (pitch). (pitch).

As the flow is going through the cascade, the end wall BL As the flow is going through the cascade, the end wall BL grows in thickness, thus the axial velocity grows.grows in thickness, thus the axial velocity grows.

TTo take care of thiso take care of this,, BL BL is is sucksucked;ed; or a large " or a large "Aspect RatioAspect Ratio" " cascade where the effect of end wall BL is less observedcascade where the effect of end wall BL is less observed, is used, is used.. v-absolute velocity v-absolute velocity w-relative velocity w-relative velocity u- peripheral blade velocityu- peripheral blade velocity




On the rotor, turn your head into the wind, and the On the rotor, turn your head into the wind, and the drought you feel is the relative velocity drought you feel is the relative velocity ww

Connect the absolute velocity vectors (Connect the absolute velocity vectors (u and vu and v) together ) together arrow-head to arrow-head, the tails became the relative arrow-head to arrow-head, the tails became the relative velocity vector (velocity vector (ww) )

WW22< W< W11 PP22 > P > P11 across the rotor across the rotor

across the statoracross the statorV V V1 3 2 P P3 2




From the velocity trianglesFrom the velocity trianglesU/VU/Vaa= tan = tan 11 + tan + tan 11 (5.1)(5.1)U/VU/Vaa= tan = tan 22 + tan + tan 22 (5.2)(5.2)

The axial velocity VThe axial velocity Vaa is assumed to be constant throughout is assumed to be constant throughout the stage.the stage.

The work absorbed by the stage, from the consideration of The work absorbed by the stage, from the consideration of the"the"change of angular momentumchange of angular momentum", in terms of work done ", in terms of work done per unit mass flow rate or specific work input isper unit mass flow rate or specific work input is::

(5.3 , 5.4)(5.3 , 5.4)

oror (5.5)(5.5)

W U V V UVa.

( ) (tan tan ) 2 1 2 1

V U W U Va 2 2 2 tan

W UVa.

(tan tan ) 1 2




Vθ2 – Vθ1

Vθ2

Vθ1U

Va

exit

W2

V1

inletβ1 α1

V2

α2

β2

Combined Velocity Triangle for Axial Compressor Stage




The The input energyinput energy is absorbed is absorbed usefully tousefully to increase p increase p and vand v and and waste fully to increase Twaste fully to increase T (frictional losses) (frictional losses)

regardless of losses (efficiency)regardless of losses (efficiency) the whole input = the whole input = TTosos

If VIf V11 = V = V33

(5.6)(5.6)

In actual fact the stage temperature rise will be In actual fact the stage temperature rise will be lessless than than this owing to 3D effects in the compressor annulus this owing to 3D effects in the compressor annulus (growing end wall B/L)(growing end wall B/L)

T TUV

Cos sa

p (tan tan ) 1 2




Analysis of experimental results has shown that it is Analysis of experimental results has shown that it is necessary to multiply the results given by equation 5.6 by necessary to multiply the results given by equation 5.6 by the so called the so called work done factorwork done factor which is a number < 1 which is a number < 1

λλ = Actual work absorbing capacity /= Actual work absorbing capacity / Ideal work absorbing capacityIdeal work absorbing capacity

The explanation of this is based on the fact that the radial The explanation of this is based on the fact that the radial distribution of axial velocity is not constant across the distribution of axial velocity is not constant across the annulus but becomes increasingly peaky as the flow annulus but becomes increasingly peaky as the flow proceeds as shown in the figure.proceeds as shown in the figure.

From eFrom eqqn. 5.1 :n. 5.1 : VVaa tan tan 11 = U- V = U- Vaatan tan 11

SubstSubstituteitute into 5.5 : into 5.5 :

sincesince 11 & & 11 are fixed are fixed while Vwhile Vaa increase then increase then ww decrease decrease

21 tantan

.

aaVVUUW




From eFrom eqqn. 5.1 :n. 5.1 : VVaa tan tan 11 = U- V = U- Vaatan tan 11

SubstSubstituteitute into 5.5 : into 5.5 :

sincesince 11 & & 11 are fixed are fixed while Vwhile Vaa increase then increase then ww decrease decrease

21 tantan

.

aaVVUUW

Va Va

Va meanVa mean




If the compressor has been designed for a constant radial If the compressor has been designed for a constant radial distribution of Vdistribution of Vaa, the effect of an increase in V, the effect of an increase in Vaa in the in the central region will be to reduce the work capacity of central region will be to reduce the work capacity of blading in that area.blading in that area.

This reduction however should be compensated by This reduction however should be compensated by increases in the regions of the root and tip of the blading increases in the regions of the root and tip of the blading because of the reductions in Vbecause of the reductions in Vaa at these parts of the at these parts of the annulus. Unfortunately this is not the case sinceannulus. Unfortunately this is not the case since;; Influence of BL's on the annulus wallsInfluence of BL's on the annulus walls Blade tip clearancesBlade tip clearanceshas an adverse effect on this compensation and the net has an adverse effect on this compensation and the net

result is a loss in total work capresult is a loss in total work capaacity)city)

.W .W = Actual amount of work which can be supplied to = Actual amount of work which can be supplied to the the stage.stage.




Actual stage temperature rise :Actual stage temperature rise :

The pressure ratio: The pressure ratio:

ss= stage isentropic efficiency= stage isentropic efficiency

TCUVosp

a (tan tan )1 2

1

1

1

o

oss

s T

TR



Degree of ReactionDegree of Reaction

== static pressure rise across the rotorstatic pressure rise across the rotor //

/ / static pressurestatic pressure rise across the whole stagerise across the whole stage

It is also a measure of how much of the total pressure It is also a measure of how much of the total pressure rise across the stage occurs in the rotor.rise across the stage occurs in the rotor.

Since CSince Cpp doesn't vary much across a stage, doesn't vary much across a stage, will be will be

equal to the corresponding temperature rises.equal to the corresponding temperature rises.




TTRR = Temperature rise across the rotor = Temperature rise across the rotor

TTSTST = Temperature rise across the stator = Temperature rise across the stator

TTSS = Stage temp = Stage temp.. Rise Rise

Assuming Assuming =1.0 =1.0

)tan(tan)tan(tan

)(.

1221

aa

spSTRp

UVUV

TCTTCW




The steady flow energy eqn :The steady flow energy eqn :

with eqn (5.8) :with eqn (5.8) :

ButBut

sincesince

W C T V Vp R

.( )

1

2 22

12

C T UV V Vp R a (tan tan ) ( ) 2 1 22

121

2

V Va2 2 sec V Va1 1 sec

C T UV Vp R a a (tan tan ) (sec sec ) 2 12 2

22

11

2

C T UV Vp R a a (tan tan ) (tan tan ) 2 12 2

22

11

2

sec tan2 2 1




(5.9)(5.9)

UV V

UV

V

U

a a

a

a(tan tan ) (tan tan )

(tan tan )(tan tan )

2 12 2

22

1

2 12 1

12 1

2

21 2 1 2

U

Va tan tan tan tan

12

21 2

VU

UV

a

a( tan tan )

V

Ua

2 1 2(tan tan )




For 50% reaction which is a wide practice (For 50% reaction which is a wide practice (=0.50)=0.50)

from eqfrom equatiouations 5.1ns 5.1 & 5.2& 5.2

⇨⇨ 11 == 22 , , 22 == 11

since Vsince V11=V=V33 11 == 33 (for repe (for repeatingating stages) stages)

For For ⇨⇨ symmetrical blading symmetrical blading

11 == 22 == 33 , , 11 == 22

0 502 1 2. (tan tan ) V

Ua tan tan 1 2

U

Vatan tan 1 2 tan tan 1 2

V V Va 1 1 3 3cos cos




EEqn 5.9 is derived for qn 5.9 is derived for =1=1

Actually Actually will differ from 50% slightly because of the will differ from 50% slightly because of the influence of influence of ;;

but still will be called but still will be called symmetricalsymmetrical blading. blading.



33DD Flow Flow

Up to here the analysis has been confined to a 2Up to here the analysis has been confined to a 2DD flow flow basis at one particular radial position in the annulus ; basis at one particular radial position in the annulus ; which is usually chosen to be "at the mean blade height" which is usually chosen to be "at the mean blade height" Before considering its extension to cover the whole blade Before considering its extension to cover the whole blade height , attention must be given to some basic principles height , attention must be given to some basic principles of 3of 3DD flow. flow.

For high H/T ratio For high H/T ratio 2 2DD assumption is reasonable assumption is reasonable

Low H/T ratio Low H/T ratio Radial flow components should be Radial flow components should be considered.considered.



33DD Flow Flow

Assumption Assumption Any radial flow within the annulus occurs only while Any radial flow within the annulus occurs only while

the fluid is passing through the blade rows. The flow in the fluid is passing through the blade rows. The flow in the gaps between successive blade rows will be in the gaps between successive blade rows will be in Radial EquilibriumRadial Equilibrium..

Basic Assumption VBasic Assumption V r =0 at the entry and exit of a r =0 at the entry and exit of a blade row.blade row.

A commonly used design method is based on this A commonly used design method is based on this principle and an equation is set up to fulfill the principle and an equation is set up to fulfill the requirement that radial pressure forces must act on requirement that radial pressure forces must act on the air elements in order to provide the necessary the air elements in order to provide the necessary radial acceleration associated with the peripheral radial acceleration associated with the peripheral velocity component Vvelocity component V..



33DD Flow Flow

p+dp

p

dθ

r

dr

Vθ

p+dp/2p+dp/2



33DD Flow Flow

From the figure the force balance in radial direction i.e From the figure the force balance in radial direction i.e pressure forces = centrifugal forces Vpressure forces = centrifugal forces Vrr =0 =0

Here Here for for small small

Cancelling dq through the eqn and neglecting 2Cancelling dq through the eqn and neglecting 2ndnd orderterms such as dpdr.orderterms such as dpdr.

(Radial Equilibrium Condition)(Radial Equilibrium Condition)

( )( ) ( )p dp r dr d prd pdpdrd

rdrdV

r 2

2 2

2

sin( ) 2 2

1 2

dp

dr

V

r



Radial Equilibrium ConditionRadial Equilibrium Condition

The Radial equilibrium equation may be used:The Radial equilibrium equation may be used:

to determine Vto determine Vaa (r) once V (r) once V(r) is chosen(r) is chosen (design or(design or

indirect problem)indirect problem)

to determine Vto determine Vaa (r), V (r), V (r) produced by a selected (r) produced by a selected

blade shape i.e. a (r) (Direct problem)blade shape i.e. a (r) (Direct problem)

The stagnation enthalpy "hThe stagnation enthalpy "h00" at any radius r" at any radius r

sincesince

h hV

C T V Vp a0

22 2

2

1

2 ( )

C TP

p 1




Differentiating wrt. r we haveDifferentiating wrt. r we have

Lets assume that the change in pressure across the Lets assume that the change in pressure across the annulus is small and the isentropic relation can be used. annulus is small and the isentropic relation can be used.

ii.e .e =const=const.. is valid with little error. is valid with little error.

In differential formIn differential form OROR substituting into the previous relationsubstituting into the previous relation;;

hP

V Va02 2

1

1

2

( )

dhdr

dPdr

P ddr

VdVdr

VdVdra

a021

1

P

ddr

P

0ddr

P

0




Introducing the Introducing the Radial EquilibriumRadial Equilibrium condition condition

Apart from the regions near the walls of the annulus the Apart from the regions near the walls of the annulus the stagnation enthalpy (and Tstagnation enthalpy (and Too) is uniform across the ) is uniform across the

annulus at the entry to the blade rows.annulus at the entry to the blade rows.

Thus Thus

in any plane between a pair of blade rows.in any plane between a pair of blade rows.

dh

drVdV

drVdV

dr

V

raa0

2

dh

dr0 0

VdV

drVdV

dr

V

raa

2

0




A special case may now be considered in whichA special case may now be considered in which VVaa=const=const.. is maintained across the annulus, so that is maintained across the annulus, so that

OR OR

Integrating this gives:Integrating this gives: OROR

Thus the whirl velocity component of the flow varies Thus the whirl velocity component of the flow varies inversely with the radius.inversely with the radius.

This is the This is the Free Vortex Free Vortex condition.condition.

dV

dr

V

r

dV

Vdrr

ln lnV r const V r const




The The Free Vortex Radial Equilibrium is Satisfied by:Free Vortex Radial Equilibrium is Satisfied by:

Constant specific work input dhConstant specific work input dhoo/dr = 0/dr = 0

Constant axial velocity at all radii i.e. dVConstant axial velocity at all radii i.e. dVaa/dr =0/dr =0

Free Vortex variation of whirl velocity (VFree Vortex variation of whirl velocity (Vr =const)r =const)




There is no reason why the specific work input should notThere is no reason why the specific work input should not

be varied with radius i.e.be varied with radius i.e. . . It would then be It would then be

necessarynecessary to choose a radial variation of one of the other to choose a radial variation of one of the other variables say Vvariables say Vaa (r) and determine the variation of V (r) and determine the variation of V with with r to satisfy the radial equilibrium. Thus in general a r to satisfy the radial equilibrium. Thus in general a design can be based on arbitrarily choosen radial design can be based on arbitrarily choosen radial distributions of any two variables and the appropriate distributions of any two variables and the appropriate variation of the third can be determined by using the variation of the third can be determined by using the equationequation

Note:Note: or any other variable may be used instead ofor any other variable may be used instead of

dh

dr0 0

dh

drVdV

drVdV

dr

V

raa0

2

d

dr

dV

dr


Axial Flow CompressorsAxial Flow CompressorsMethod of

DesignWork

Variationswith radius

ho(r)

Tangentialvelocity

Distibution Vθ(r)

Axial velocitydistribution

withRadius Va(r)

Reactiondistribution with

Radius Λ(r)

RadialEquilib.

Remarks

ATwo-Dimensional

Supposedconstant

Supposed constant Supposed constant

Supposed constant

Ignored All variations of flow with radius are ignoredMethod for: high H/T stages

BFree Vortex

Constant Vθr = constant Constant Incresed with radius

Yes Limited by high rotor root deflection(approx. const. stator defl.)

C Constant Reaction (without equilibrium)

Supposedconstant

Vθ = ar ± b/r Supposed constant

Supposed constant

Ignored Λ and work distr. will NOT be const. since true variation in Va is not considered

D Constant Reaction

Constant Vθ = ar ± b/r From radial equilib.

Constant Yes Logical design methodHighly twisted blades

EHalf Vortex

Supposedconstant

Arithmetic meanof free vortex andconst. reaction dist.

Supposed constant

Not far from const.

Ignored Λ and work distr. will NOTbe const. since true variationin Va is not considered

FConstant α2

Supposedconstant

Fixed by conditionVθ2 = cost.

[stator entry]Vθ1 = a – b/r

[rotor entry]

Supposed constant

Not far from const.

Ignored Widely used but itsperformance and advantagesnot widely understood

G Forced Vortex

Increaseswith r2

Vθ α r From radial equilib.

Varies with radius Yes Rarely used

HExponential

Constant Vθ = a ± b/r From radial equilib.

Varies with radius Yes A logicl design method



Blade DesignBlade Design

Having determined the air angle distributions to give the Having determined the air angle distributions to give the required stage work it is now necessary to convert these required stage work it is now necessary to convert these into blade angle distributions from which the correct into blade angle distributions from which the correct geometry of the blade forms may be determined.geometry of the blade forms may be determined.

Air Angles Air Angles Blade angles Blade angles Blade Geometry Blade Geometry

The common practice is to use the results of the wind The common practice is to use the results of the wind tunnel tests to determine the blade shapes to give the tunnel tests to determine the blade shapes to give the required air angles. The aim of the cascade testing is to required air angles. The aim of the cascade testing is to determine the required angles fordetermine the required angles for

Maximum mean deflectionMaximum mean deflection

Minimum mean total head loss.Minimum mean total head loss.

= 1

2



Blade DesignBlade Designββ1v1v,,αα1v1v = Blade inlet angle = Blade inlet angle

ββ2v2v,,αα2v2v = Blade outlet angle = Blade outlet angle

ββ11, , αα11 = Air inlet angle = Air inlet angle

ββ22, , αα2 2 = Air outlet angle = Air outlet angle

WW11,V,V11 = Air inlet velocity = Air inlet velocity

WW22,V,V22 = Air outlet velocity = Air outlet velocity

s = pitchs = pitchc = chordc = chord

θθ = camber = = camber = αα1v1v – – αα2v2v

ξξ = stagger = 0.5( = stagger = 0.5(αα1v1v + + αα2v2v) )

єє = deflection = = deflection = αα11 – – αα22

i = incidence = i = incidence = αα11 - - αα1v1v

δδ = deviation = = deviation = αα22 – – αα2v2v




The loss in non-dimensional form The loss in non-dimensional form ww = =

It is desirable to avoid numbers with common multiples It is desirable to avoid numbers with common multiples for the bfor the bllades in successive rows to reduce the likelihood ades in successive rows to reduce the likelihood of introducing resonant frequencies.of introducing resonant frequencies.

The common practice is to choose an The common practice is to choose an even number for even number for the stator blades and a prime number for the rotor the stator blades and a prime number for the rotor blades.blades.

The bladeThe blade outlet angle outlet angle ““2v2v”” can not be determined from can not be determined from the air outlet angle the air outlet angle ““22”” until the deviation angle until the deviation angle ““”” has been determined.has been determined.

P P

V

01 02

121

2

2 2v



β2 α2

nr prime

ns even

β1

ε = β1 – β2

β2

ε*

s/c

s

c

h h/c

know how ≈ 3

rm

nn = 2πrm/snumber of

blades

Des. Defl.Curve

recalculate s/c , h/c

Design Procedure





Ideally the mean direction of the air leaving the cascade Ideally the mean direction of the air leaving the cascade would be that of the outlet angle is of the bladeswould be that of the outlet angle is of the blades..

BBut in practice it is found that there is a deviation which is ut in practice it is found that there is a deviation which is due to the reluctance of the air to turn through the full due to the reluctance of the air to turn through the full angle required by the shape of the blade.angle required by the shape of the blade.

EmpEmpiirical equations are employed to estimate rical equations are employed to estimate ..

wherewhere : :

where "a" = the distance to the point of maximum camber where "a" = the distance to the point of maximum camber from the leading edge.from the leading edge.

If the camber arc is circular If the camber arc is circular ((22 a/ca/c)) = = 11

m sc

ma

c 0 23

20 1

502 2. ( ) . ( )




Using the values of Using the values of ““c, c, 1v1v, , 2v2v, , ““ ; ; it is possible to it is possible to

construct the circular arc camberconstruct the circular arc camber line of the blade around line of the blade around which an aerofoil section can be built up.which an aerofoil section can be built up.

This method can now be applied to a selected number of This method can now be applied to a selected number of points along the blade length to get a complete picture of points along the blade length to get a complete picture of the blade form.the blade form.



Calculation Calculation ooff Stage PerformanceStage Performance

After the completion of stage design it will now be After the completion of stage design it will now be necessary to check over the performance, particularly in necessary to check over the performance, particularly in regard to the efficiency which for a given work input will regard to the efficiency which for a given work input will completely govern the final pressure ratio. completely govern the final pressure ratio.

This efficiency is dependent of the total This efficiency is dependent of the total pressure dpressure drop for rop for each of the blade rows comprising the stage and in order each of the blade rows comprising the stage and in order to evaluate these quantities it will be necessary to revert to evaluate these quantities it will be necessary to revert the loss the loss mmeasurements in cascade tests.easurements in cascade tests.

Lift and profile drag coefficients Lift and profile drag coefficients CCLL and and CCDPDP can be can be obtained from measured values of mean loss obtained from measured values of mean loss ww..




The static pressure rise across the blades is given by The static pressure rise across the blades is given by (incompressible assumption)(incompressible assumption)

P P P P V P V 2 1 02 22

01 121

212

( ) ( )

P V V P P 1

2 12

22

02 01( ) ( ) P P02 01

P Va 1

22 2

12

2 (tan tan )




Assuming Assuming VVaa == VVa1a1 == VVa2a2;;

The axial force per unit length of each blade is = s The axial force per unit length of each blade is = s PP

From the consideration of momentum changes the forces From the consideration of momentum changes the forces acting along ethe cascade is given byacting along ethe cascade is given by

FF = s = s VVaa

change in tangent velocity component along the cascadechange in tangent velocity component along the cascade

FF = s= sVVaa *V *Vaa (tan (tan11 -- tantan2)2)




The coefficients CThe coefficients CLL and C and CDPDP are based on arbitrarily are based on arbitrarily

defined vector mean velocity Vdefined vector mean velocity Vmm,,

where where

D =D = Drag force along vector mean velocity Drag force along vector mean velocity

L =L = Lift force perpendicular to vector mean velocity Lift force perpendicular to vector mean velocity

V Vm a m sec

tan (tan tan ) m 1

2 1 2

D V cCm Dp1

22 L V cCm L

1

22




After some manipulationsAfter some manipulations

CCDPDP and C and CLL can be evaluated if a can be evaluated if a HowellHowell like curve is like curve is

known from cascade test results and known from cascade test results and

Csc V

Dpm

( )cos

cos

1

2 12

3

21

Csc

CL m Dp m 2 1 2tan tan cos tan

1 1

2 1

1 1 2

1

2

v

m

i *

tan (tan tan )



Calculation Calculation of of Stage PerformanceStage Performance

CDP

CL

İncidence i degrees

Dra

g co

effic

ient

CD

P Lift

coe

ffic

ient

CL

0.075

-10 -5 0 5 10

0

0.5

1.0

1.5

0.025

0.050

-15-200




Using the values of Using the values of from cascade test from cascade test

results for known values of (s/c); Cresults for known values of (s/c); CDPDP and C and CLL can be plotted can be plotted

against incidence.against incidence.

Since the value of CSince the value of Cpp tan tan mm in C in CLL equation is negligibly small, it equation is negligibly small, it

is usual to use a more convenient theoretical value of Cis usual to use a more convenient theoretical value of CLL given given

byby

In which the effect of profile drag is ignored. Using this formula, In which the effect of profile drag is ignored. Using this formula, curves of Ccurves of CLL can be plotted for nominal (or design) conditions to can be plotted for nominal (or design) conditions to

correspond with the curves of deflection. These curves are again correspond with the curves of deflection. These curves are again plotted against plotted against 22 for fixed values of s/c for fixed values of s/c

12 1

2V

CscL m

2 1 2tan tan cos




Before applying these coefficients to the blade rows of Before applying these coefficients to the blade rows of the compressor stage two additional factors must be the compressor stage two additional factors must be taken into account.taken into account.

Annulus DragAnnulus Drag: Drag effects due to the walls of the : Drag effects due to the walls of the Compressor annulus = Compressor annulus = CCDADA

CCDADA = 0.02 (s/h) = 0.02 (s/h)

Secondry LossesSecondry Losses: Due to the trailing vortices and tip : Due to the trailing vortices and tip ccllearances earances = C= CDSDS The following emprical relations can be The following emprical relations can be usedused

CCDSDS = 0.018 = 0.018 CCLL22




The overall Drag Coefficient is given byThe overall Drag Coefficient is given by

ProfileProfile + annulus + secondary+ annulus + secondary

(the annular cascade C(the annular cascade CDPDP is replaced by C is replaced by CDD) thus) thus

This enables the loss coefficientThis enables the loss coefficient for the blade for the blade row to be determined.row to be determined.

C C C CD DP DA DS

Csc V

Dm

1

2 12

3

21

cos

cos

12 1

2V




The theoretical pressure rise (i.e The theoretical pressure rise (i.e ww =0) =0)

Efficiency of the blade rowEfficiency of the blade row

P V Vth a a 1

2

1

22 2

12

22 2

12

2 (tan tan ) (sec sec )

P

V

th

a12

12 2

1

22

21

sec

sec

sec

P

V

th12

1

12

21

22

cos

cos

thth

th th

PP

V

P

V

1

12

12

12

12




For a case where For a case where = 50 % Rotor and stator rows are = 50 % Rotor and stator rows are similar thus this calculation carried at design diameter similar thus this calculation carried at design diameter can be applied to the whole stagecan be applied to the whole stage

forfor = = 1/21/2

stage efficiencystage efficiency

bl ‘

ssP P

P P

P

P

T

T

2 1

2 1

2

1 1

1

1 2( )

s

is‘

act

TT




thenthen

expanding and neglecting 2expanding and neglecting 2ndnd order terms; order terms;

forforT K

T K

so

o

20

4001

PP

TT

‘s2

1 1

11

2

bl ‘

ss

s

PP

PP

TT

TT

21

21

1

1

1

1

1

1

12

1

12

1

bl s

ss

TT

11

1 41

1

( )

bl s




For cases other than 50 % reaction at the design For cases other than 50 % reaction at the design diameter an approximate stage efficiency is given bydiameter an approximate stage efficiency is given by

If If far removed from 50% far removed from 50%

s blR bl-ST 1

2

s bl-R bl-ST 1



Summary of the Design ProcedureSummary of the Design Procedure

Assume Assume Ts and at the design radius Ts and at the design radius Calculate the air Calculate the air anglesangles

Applying chosen design condition (Free Vortex, Constant Applying chosen design condition (Free Vortex, Constant Reaction etc) Calculate air angles at all radiiReaction etc) Calculate air angles at all radii

Results of Cascade Tests Results of Cascade Tests Blade shapes (Blade Blade shapes (Blade angles)angles) C CDD and C and CLL

Calculate Calculate ηηss and and RRss



Overall PerformanceOverall Performance

Assuming thatAssuming that ηηs s = = ηη∞∞

((ηη constant through all compressor stages), constant through all compressor stages), for a compressor consisting of N similar stages, each with for a compressor consisting of N similar stages, each with ηηs s = = ηη∞∞

R is the “R is the “Overall Pressure RatioOverall Pressure Ratio”” where where ;;

RN TTos

o

nn

11

1

nn s

1 1 1



Overall PerformanceOverall Performance

Although the use of polytropic law gives a rapid means of Although the use of polytropic law gives a rapid means of estimating the overall performance of a multistage estimating the overall performance of a multistage compressor, it is necessary in practice to make a step by compressor, it is necessary in practice to make a step by step final performance calculation.step final performance calculation.

Latest blade manufacturing technology allows different Latest blade manufacturing technology allows different blade shapes for different rows.blade shapes for different rows.



Compressibility EffectsCompressibility Effects

High air velocities between the blades effects the High air velocities between the blades effects the compressor performancecompressor performance..

Critical MachCritical Mach numbernumber MMcc is defined such that at entry is defined such that at entry velocities lower than thisvelocities lower than this;; the performance of the c the performance of the caascade scade differs very little from that at low speeds. Above this differs very little from that at low speeds. Above this losses begin to show a marked increase.losses begin to show a marked increase.

Maximum Mach Number is Maximum Mach Number is defined as thedefined as the air speed at air speed at which losses cancel the pressure risewhich losses cancel the pressure rise..

FFor a typical low speed cascade Mor a typical low speed cascade Mcc = 0.7 M = 0.7 Mmm =0.85 =0.85

Increased Mach number also narrows the operating Increased Mach number also narrows the operating range of incidence leading to poor performance at off range of incidence leading to poor performance at off design conditions.design conditions.




In the sketch the variations in Mach # across the annulus In the sketch the variations in Mach # across the annulus is shown for Free Vortex and constant reaction blading. is shown for Free Vortex and constant reaction blading. Free Vortex blading shown large Mach number variations Free Vortex blading shown large Mach number variations which extreme care should be taken.which extreme care should be taken.

Since the velocity of sound in air increases with Since the velocity of sound in air increases with increasing temperature the Mach numbers will decrease increasing temperature the Mach numbers will decrease through the compressor due to the progressively through the compressor due to the progressively increasing temperature.increasing temperature.




Not to suffer from compressibNot to suffer from compressibiility effects in early stages lity effects in early stages one mighone mightt use constant reaction design if no other use constant reaction design if no other precaprecauution can be taken.tion can be taken.

Transonic stages where the flow is actually supersonic Transonic stages where the flow is actually supersonic over a part of the blade height can now be designed over a part of the blade height can now be designed utilizing very thin and special shaped blades. One utilizing very thin and special shaped blades. One advantage is eliminating IGVadvantage is eliminating IGV’’s less noise. s less noise.



Some Deductions from the Compressor Some Deductions from the Compressor CharacteristicsCharacteristics The overall compressor characteristic is composed of the The overall compressor characteristic is composed of the

stage characteristics stacked.stage characteristics stacked.

The mass flow through the compressor is controlled by The mass flow through the compressor is controlled by the choking of various stages in some cases early the choking of various stages in some cases early stages, in the others the rear stagesstages, in the others the rear stages..

If the axial flow compressor is designed for constant axial If the axial flow compressor is designed for constant axial velocity throughoutvelocity throughout ; ; the annulus area must decrease the annulus area must decrease along due to the increasing density.along due to the increasing density.

The annulus area for each stage is determined for the The annulus area for each stage is determined for the design condition. At any other operating conditon the design condition. At any other operating conditon the design point calculated area will result in a variation of design point calculated area will result in a variation of axial velocityaxial velocity..



Some Deductions from the Compressor Some Deductions from the Compressor CharacteristicsCharacteristics

When the compressor is run at When the compressor is run at a speed lower than a speed lower than designdesign, , T and RT and Rcc are reduced than the are reduced than the densitydensity at the at the rear stages will be lower than the design value.rear stages will be lower than the design value.

As a result the axial velocity at the rear stages will As a result the axial velocity at the rear stages will increase, eventually choking will occurincrease, eventually choking will occur..

Thus Thus at low speedsat low speeds mm is determined by the is determined by the choking of choking of the rear stages.the rear stages. As the speed is increased density of the As the speed is increased density of the rear stages increases (V decrease) thus gets unchoked.rear stages increases (V decrease) thus gets unchoked.

At veryAt very high speeds choking will occur at the inlethigh speeds choking will occur at the inlet. . The vertical line of constant speed is due to choking at The vertical line of constant speed is due to choking at the inlet of the compressor.the inlet of the compressor.




A A B B

At the design speed if we consider the moving of At the design speed if we consider the moving of operating point from A to B. operating point from A to B.

At point B (on the surge line), the density at the At point B (on the surge line), the density at the compressor exit will be increased due to the compressor compressor exit will be increased due to the compressor exit will be increased due to the increase in delivery exit will be increased due to the increase in delivery pressure; also pressure; also ṁṁ is slightly reduced. is slightly reduced.

Axial velocity in the last stage is reduced incidence in the Axial velocity in the last stage is reduced incidence in the last stage is increased. Rotor blades are expected to last stage is increased. Rotor blades are expected to stall from the last stagesstall from the last stages..




AA C C

ṁṁ falls rapidly falls rapidly ; ; VVaa at the inlet decrease at the inlet decreases,s, incidence of incidence of the first stage increasethe first stage increases.s. But the incidence of the later But the incidence of the later stages decrease due to the increase of Vstages decrease due to the increase of Vaa (due to lower (due to lower pressure and density). At low speeds surging is probably pressure and density). At low speeds surging is probably due to first stages stalling. due to first stages stalling.

At conditions far removed from surge At conditions far removed from surge RR is very low is very low high high VVaa large decrease in incidence large decrease in incidence result in stall in result in stall in negative incidence negative incidence very low very low ηηcc




At high pressure operation At high pressure operation Blow-offBlow-off at an intermediate at an intermediate stage (wasteful). Incidence can be maintained at design stage (wasteful). Incidence can be maintained at design value by increasing the speed of last stage (HPC) and value by increasing the speed of last stage (HPC) and decreasing the speed of first stage (LPC)decreasing the speed of first stage (LPC);;

Two spools are mechanically independent but Two spools are mechanically independent but aerodynamically coaerodynamically couupled.pled.

ME 423 Chapter 5 Axial Flow Compressors

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Transcript of ME 423 Chapter 5 Axial Flow Compressors