#### Transcript ME 423 - Middle East Technical University

**ME 423 Chapter 5 Axial Flow Compressors**

# Prof. Dr. O. Cahit ERALP

*Axial Flow Compressors*

**Chapter 5**

**Axial Flow Compressors**

**Subsonic compressors**

supersonic compressors have not proceeded beyond experimental stage.

will be considered here as

**A Comparison of Axial Flow Compressors and Turbines**

**Turbine :-**

Accelerating flow - Successive pressure drops and consequent reductions in enthalpy being converted into kinetic energy A1>A2 ⇨ converging passages

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

**Compressor :-**

Decelerating flow - Pressure rises are obtained through successive stages of diffusing passages with consequent reduction in velocity. A1

**compressors**

- aerodynamic problems in

**turbines **

- problems due to entry temperature and heat-transfer.

**Boundary Layers -**

regions of low momentum air where viscous effects dominate over inertial effects.

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

Boundary layers are far less happy in a compressive flow. BL in a compressor operate in an unfavourable pressure gradient [(+) 've ; p increase ] BL in a turbine operate in a favourable pressure gradient [ (-)'ve ; p decrease. ] This is the reason why a single stage turbine can create enough power to drive a number of stages of compressor.

Bend thin plates and stick them behind each other forming a stationary

**cascade of blades**

.

Let the flow be directed towards the inlet of this

**cascade of blades**

without any

**incidence**

.

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

As

**A**

**2**

**> A**

**1**

in subsonic flow (incompressible)

**W**

**2**

**< W**

**1**

**& P**

**2**

**> P**

**1**

This is no more than a subsonic diffuser To carry a mechanical load, some thickness is required.

If M < 0.3 1 incompressible 2 i.e.

*P*

2 1

*W*

2 2 2 thus

*P*

0 0

*P*

1 1 2

*W*

1 2

*P*

**2**

*P*

**1**

**1 2**

..

*W*

**1 2**

*W*

**2 2**

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

Clearly the outlet velocity W 2 certain level (cannot be zero) cannot decrease beyond a (or W 2 ≠ 0)

**p W**

**1 2**

(since W 2 is fixed by the lower limit) One should design the compressor at the highest inlet velocity But the

**losses **

⇨ P o α α W 1 2

**1/ **

**p**

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

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

Due to the contraction, the flow initially accelerates pressure drops (favourable to BL) ( A 1 > A 1 ' ) then

*W*

max

*W*

2

*W*

1

*W*

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

i.e more diffusion the limit of W max is than of sonic limit.

More diffusion means less efficiency i.e why we prefer compressor blades to be as thin as possible.

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

The more the (camber), the more is the adverse pressure gradient, then seperation occurs earlier. The seperated flow leaves the blade at an unwanted angle and unsteady situation.

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

**Turbine**

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

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

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

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

The flow area should contract continuously all the way along the blades in order not to have an adverse pressure gradient BL along the row. Even an instantaneous discontinuity in the contraction of the passage results in a locally seperated BL, thus increased turbulence. This might happen due to simplified manufacture for curvatures such as two circles.

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

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

For the

**Axial Compressors and turbines **

the basic components are

**rotors and stators**

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

A

**compressor stage by a stator**

is composed of a

**rotor followed **

, where as a

**turbine stage stator followed by a rotor .**

is composed of

**a **

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

**In Compressors**

It is usual to provide a row of stator blades –

**Inlet Guide Vanes (IGV's)**

blades.

at the upstream of the first stage. These direct the axially approaching flow correctly into the first row of rotor blades. Thus deflect the flow from axial direction to off-axial direction. IGV's are turbine type of Two forms of

**rotor construction**

is used

**Drum type**

-suitable for industrial applications

**Disc type**

high cost) - suitable for aircraft applications low weight,

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

*Axial Flow Compressors*

**A Comparison of Axial Flow Compressors and Turbines**

Another important constructional detail is the contraction of the flow annulus from the low the high pressure end of the compressor. This is necessary to maintain a reasonably constant axial velocity along. most compressors are designed on the basis of constant axial velocity because of the simplification in design procedure. One could have a rising hub or a falling shroud in compressors.

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

Basic principle : Acceleration of the working fluid followed by diffusion to convert the acquired kinetic energy into a pressure rise. The flow is considered as occuring in the tangential plane at the mean blade height where the blade peripheral velocity is u.

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

These are called a

**2-D cascade of aerofoils**

.

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

A semi cascade can be produced if the cascade end boundary effects are eliminated (The flow in the channels are not aware of what happens at the ends) The aerodynamics of a cascade repeats itself with a periodicity of

**s**

(pitch).

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

To take care of this, BL is sucked; or a large "

**Aspect Ratio**

" cascade where the effect of end wall BL is less observed, is used.

**v-absolute velocity w-relative velocity u- peripheral blade velocity**

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

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

**w**

Connect the absolute velocity vectors (

**u and v**

) together arrow-head to arrow-head, the tails became the relative velocity vector (

**w**

) W 2 < W

*V*

1

*V*

3

*V*

2 1 P 2 > P 1

*P*

3

*P*

2 across the rotor across the stator

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

*Axial Flow Compressors Me 423 Spring 2006 Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

From the velocity triangles

**U/V**

**a**

**= tan **

**1**

**U/V**

**a**

**= tan **

**2**

**+ tan **

**1**

**+ tan **

**2**

(5.1) (5.2) The axial velocity V a the stage.

is assumed to be constant throughout The work absorbed by the stage, from the consideration of the"

**change of angular momentum**

", in terms of work done per unit mass flow rate or specific work input is: .

*W*

*U V*

2

*V*

1 )

*UV a*

(tan 2 tan 1 ) (5.3 , 5.4) or

*V*

2 .

*W W*

2

*V a*

tan 2

*UV a*

(tan 1 tan 2 )

*Me 423 Spring 2006*

(5.5)

*Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

V θ 2 – V θ 1 exit β 2 α 2 V 2 β 1 α 1 inlet V 1 W 2 V a U V θ1 V θ 2

**Combined Velocity Triangle for Axial Compressor Stage**

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

The

**input energy and v**

and is absorbed

**usefully to increase p waste fully to increase T**

(frictional losses) regardless of losses (efficiency)

**the whole input = **

**T**

**os**

If V 1 = V 3

*T os*

*T s*

*UV a*

(tan 1 tan 2 ) (5.6)

*C p*

In actual fact the stage temperature rise will be

**less**

this owing to 3D effects in the compressor annulus (growing end wall B/L) than

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

Analysis of experimental results has shown that it is necessary to multiply the results given by equation 5.6 by the so called

**work done factor**

which is a number < 1 λ = Actual work absorbing capacity / Ideal work absorbing capacity The explanation of this is based on the fact that the radial distribution of axial velocity is not constant across the annulus but becomes increasingly peaky as the flow proceeds as shown in the figure.

From eqn. 5.1 : Substitute into 5.5 : V a .

*W*

tan 1

*U*

*U*

= U- V a tan 1

*V a*

tan

**1**

*V a*

tan

**2**

since 1 & 1 are fixed while V a increase then

*w*

decrease

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

*V a V a mean V a V a mean*

From eqn. 5.1 : Substitute into 5.5 : V a .

*W*

tan 1

*U*

*U*

= U- V a tan 1

*V a*

tan

**1**

*V a*

tan

**2**

since 1 & 1 are fixed while V a increase then

*w*

decrease

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

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

a , the effect of an increase in V a in This reduction however should be compensated by increases in the regions of the root and tip of the blading because of the reductions in V a at these parts of the annulus. Unfortunately this is not the case since; Influence of BL's on the annulus walls Blade tip clearances has an adverse effect on this compensation and the net result is a loss in total work capacity) .W = Actual amount of work which can be supplied to the stage.

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

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

*Axial Flow Compressors*

**Elementary Theory For Axial Flow Compressors**

Actual stage temperature rise :

*T os*

*C p UV a*

(tan 1 tan 2 ) The pressure ratio:

*R s*

**1**

*s*

*T os T o*

**1**

**1**

s = stage isentropic efficiency

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

*Axial Flow Compressors*

**Degree of Reaction**

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

Since C p doesn't vary much across a stage, equal to the corresponding temperature rises.

will be

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

*Axial Flow Compressors*

**Degree of Reaction**

T R T ST = Temperature rise across the rotor = Temperature rise across the stator T S = Stage temp. Rise Assuming =1.0

*W*

.

*C p*

(

*T R*

*UV a*

(tan

**1**

*T ST*

) tan

**2**

)

*C p*

*T s*

*UV a*

(tan

**2**

tan

**1**

)

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

*Axial Flow Compressors*

**Degree of Reaction**

The steady flow energy eqn : with eqn (5.8) : .

*W*

*C p*

*T R*

1 (

*V*

2 2 2

*V*

1 2 )

*C p*

*T R*

*UV a*

(tan 2 tan 1 ) 1 2 (

*V*

2 2

*V*

1 2 ) But

*V*

2

*V a*

sec 2

*V*

1

*V a*

sec 1

*C p*

*T R*

*UV a*

(tan 2 tan 1 ) 1

*V a*

2 2 (sec 2 2 sec 2 1 ) since

*C p*

*T R*

sec 2

*UV a*

(tan 2 tan 2 tan 1 ) 1

*V a*

2 2 (tan 2 2 1 tan 2 1 )

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

*Axial Flow Compressors*

**Degree of Reaction**

*UV a*

(tan 2 tan

*UV a*

1 ) (tan 1 2 2

*V a*

2 (tan 2 tan 1 ) 2 tan 2 1 )

*V a*

2

*U*

(tan 2 tan 1 ) 2

*U*

tan 1 tan 2 tan 1 tan 2

*V a V a*

2

*U*

( 2

*U V a*

tan 1 tan 2 )

*V a*

2

*U*

(tan 1 tan 2 ) (5.9)

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

*Axial Flow Compressors*

**Degree of Reaction**

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

*V a*

2

*U*

(tan 1 tan 2 ) from equations 5.1 & 5.2

⇨ 1 = 2 , 2 = 1 tan 1 tan 2 tan 1 tan 2

*U V a*

tan 1 tan 2

*V a*

*V*

1 cos 1

*V*

3 cos 3 since V 1 =V 3 1 = 3 (for repeating stages) For ⇨ symmetrical blading 1 = 2 = 3 , 1 = 2

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

*Axial Flow Compressors*

**Degree of Reaction**

Eqn 5.9 is derived for =1 Actually will differ from 50% slightly because of the influence of ; but still will be called

**symmetrical**

blading.

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

*Axial Flow Compressors*

**3D Flow**

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

For high H/T ratio 2D assumption is reasonable Low H/T ratio considered.

Radial flow components should be

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

*Axial Flow Compressors*

**3D Flow**

**Assumption **

Any radial flow within the annulus occurs only while the fluid is passing through the blade rows. The flow in the gaps between successive blade rows will be in

**Radial Equilibrium**

.

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

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

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

*Axial Flow Compressors*

**3D Flow**

dr p+dp p+dp/2 r p d θ V θ p+dp/2

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

*Axial Flow Compressors*

**3D Flow**

From the figure the force balance in radial direction i.e pressure forces = centrifugal forces V r =0 (

*p*

*prd*

2 (

*p*

*dp*

)

*dr*

2

*d*

2

*V*

2

*r*

Here sin( ) 2 2 for small Cancelling dq through the eqn and neglecting 2 nd orderterms such as dpdr.

1

*dp dr*

*V*

2

*r*

*Me 423 Spring 2006*

(Radial Equilibrium Condition)

*Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Radial Equilibrium Condition**

The Radial equilibrium equation may be used: to determine V a (r) once V (r) is chosen (design or indirect problem) to determine V a (r), V (r) produced by a selected blade shape i.e. a (r) (Direct problem) The stagnation enthalpy "h 0 " at any radius r 2

*h*

0

*V*

2 1 2 (

*V a*

2

*V*

2 ) since 1

*P*

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

*Axial Flow Compressors*

**Radial Equilibrium Condition**

*h*

0 1

*P*

1 2 (

*V a*

2

*V*

2 ) Differentiating wrt. r we have

*dh*

0

*dr*

1 1

*dP dr*

*P d*

2

*dr*

*V a dV a dr*

*V*

*dV*

*dr*

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

*P*

=const. is valid with little error.

In differential form

*d dr P*

0

**OR**

*d dr P*

0 substituting into the previous relation;

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

*Axial Flow Compressors*

**Radial Equilibrium Condition**

Introducing the

**Radial Equilibrium**

condition

*dh*

0

*dr*

*V a dV a dr*

*V*

*dV*

*dr*

*V*

2

*r*

Apart from the regions near the walls of the annulus the stagnation enthalpy (and T o ) is uniform across the annulus at the entry to the blade rows.

Thus

*dh*

0

*dr*

0 in any plane between a pair of blade rows.

*V a dV a dr*

*V*

*dV*

*dr*

*V*

2

*r*

0

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

*Axial Flow Compressors*

**Radial Equilibrium Condition**

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

*dV*

*dr*

*V*

*r*

OR

*dV*

*V*

*dr r*

Integrating this gives: ln

*V*

ln

*r*

*const*

OR

*const*

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

This is the

**Free Vortex **

condition.

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

*Axial Flow Compressors*

**Radial Equilibrium Condition**

The

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

Constant specific work input dh o /dr = 0 Constant axial velocity at all radii i.e. dV a /dr =0 Free Vortex variation of whirl velocity (V r =const)

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

*Axial Flow Compressors*

**Radial Equilibrium Condition**

There is no reason why the specific work input should not be varied with radius i.e.

Note:

*d*

*dr*

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

*dh*

0

*dV a*

*dV*

*V*

2

*dr V a dr V*

*dr r*

or any other variable may be used instead of

*dV*

*dr dh*

0 0 . It would then be

*dr*

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

*Axial Flow Compressors*

**Method of Design A**

Two Dimensional

**B**

Free Vortex

**Work Variations with radius h o (r)**

Supposed constant Constant

**Tangential velocity Distibution V θ (r)**

Supposed constant V θ r = constant

**Axial velocity distribution with Radius V a (r)**

Supposed constant Constant

**C**

Constant Reaction (without equilibrium) Supposed constant V θ = ar ± b/r

**D**

Constant Reaction

**E**

Half Vortex

**F**

Constant α 2 Constant Supposed constant Supposed constant V θ = ar ± b/r Arithmetic mean of free vortex and const. reaction dist.

Fixed by condition V θ2 = cost.

[stator entry] V θ1 = a – b/r [rotor entry] V θ α r

**G**

Forced Vortex Increases with r 2

**H**

Exponential Constant V θ = a ± b/r

*Me 423 Spring 2006*

Supposed constant From radial equilib.

Supposed constant Supposed constant From radial equilib.

From radial equilib.

**Reaction distribution with Radius Λ(r) Radial Equilib.**

Supposed constant Incresed with radius Supposed constant Constant Not far from const.

Not far from const.

**Remarks**

Ignored Yes Ignored Yes Ignored Ignored All variations of flow with radius are ignored Method for: high H/T stages Limited by high rotor root deflection (approx. const. stator defl.) Λ and work distr. will NOT be const. since true variation in V a is not considered Logical design method Highly twisted blades Λ and work distr. will NOT be const. since true variation in V a is not considered Widely used but its performance and advantages not widely understood Varies with radius Yes Varies with radius Yes Rarely used A logicl design method

*Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Blade Design**

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

**Air Angles **

**Blade angles **

**Blade Geometry**

The common practice is to use the results of the wind tunnel tests to determine the blade shapes to give the required air angles. The aim of the cascade testing is to determine the required angles for Maximum mean deflection 1 2 Minimum mean total head loss.

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

*Axial Flow Compressors*

**Blade Design**

*Me 423 Spring 2006*

β 1v ,α 1v = Blade inlet angle β 2v ,α 2v β 1 , α 1 β 2 , α 2 = Blade outlet angle = Air inlet angle = Air outlet angle W 1 ,V W 2 ,V 2 1 = Air inlet velocity = Air outlet velocity s = pitch c = chord θ = camber = α 1v – α 2v ξ = stagger = 0.5(α 1v + α 2v ) є = deflection = α 1 – α 2 i = incidence = α 1 δ = deviation = α 2 - α 1v – α 2v

*Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Blade Design**

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

*Axial Flow Compressors*

**Blade Design**

The loss in non-dimensional form

**w**

=

*P*

01 1

*P*

02

*V*

1 2 2 It is desirable to avoid numbers with common multiples for the blades in successive rows to reduce the likelihood of introducing resonant frequencies.

The common practice is to choose an

**blades.**

**even number for the stator blades and a prime number for the rotor **

The blade outlet angle the air outlet angle “

**2**

**”**

“

**2v**

**”**

has been determined.

can not be determined from until the deviation angle “ ” 2 2

*v*

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

*Axial Flow Compressors*

**Blade Design**

Design Procedure ε = β 1 – β 2 β Des. Defl.

Curve 1 number of blades ε* r m β 2 s/c β 2 α 2 n h c know how ≈ 3 h/c s n = 2πr m /s n s n r even prime recalculate s/c , h/c

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

*Axial Flow Compressors*

**Blade Design**

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

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

Empirical equations are employed to estimate .

*m*

*s c*

where :

*m*

0 23 ( 2

*c a*

) 2 2 50 ) where "a" = the distance to the point of maximum camber from the leading edge.

If the camber arc is circular (

**2 a/c) = 1**

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

*Axial Flow Compressors*

**Blade Design**

Using the values of “

**c, **

**1v**

**, **

**2v**

**, **

**“**

; it is possible to construct the circular arc camber line of the blade around which an aerofoil section can be built up.

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

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

After the completion of stage design it will now be necessary to check over the performance, particularly in regard to the efficiency which for a given work input will completely govern the final pressure ratio. This efficiency is dependent of the total pressure drop for each of the blade rows comprising the stage and in order to evaluate these quantities it will be necessary to revert the loss measurements in cascade tests.

Lift and profile drag coefficients

**C**

**L**

and

**C**

**DP**

can be obtained from measured values of mean loss

**w**

.

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

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

*P*

*P*

2

*P*

1 (

*P*

02 1 2

*V*

2 2 ) (

*P*

01 1 2

*V*

1 2 )

*P*

1 2 (

*V*

1 2

*V*

2 2 ) (

*P*

02

*P*

01 )

*P*

02

*P*

01

*P*

1

*V a*

2 2 (tan 2 1 tan 2 2 )

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

Assuming

**V**

**a**

**= V**

**a1**

**= V**

**a2**

; The axial force per unit length of each blade is = s P From the consideration of momentum changes the forces acting along ethe cascade is given by

**F = s **

**V**

**a**

change in tangent velocity component along the cascade

**F = s**

**V**

**a**

***V**

**a**

**(tan**

**1 - tan**

**2)**

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

The coefficients C L and C DP are based on arbitrarily defined vector mean velocity V m , where

*V m*

*V a*

sec

*m*

tan

*m*

1 2 (tan 1 tan 2 )

**D =**

Drag force along vector mean velocity

**L =**

Lift force perpendicular to vector mean velocity

*D*

1

*V cC Dp*

2

*L*

1

*V cC L*

2

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

After some manipulations

*C Dp*

*s*

( )

*c*

1 2

*V*

1 2 cos 3 cos 2

*m*

1

*C L*

2

*s c*

tan 1 tan 2 cos

*m*

*C Dp*

tan

*m*

C DP and C L can be evaluated if a

**Howell**

known from cascade test results and like curve is 1 1

*v*

*i*

2 1 *

*m*

tan 1 1 2 1

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

C L 1.5

1.0

0.5

0.075

0.050

0.025

0 -20 -15 C DP -10 -5 0 İncidence i degrees 5

*Me 423 Spring 2006*

10 0

*Prof. Dr. O. Cahit ERALP*

*Axial Flow Compressors*

**Calculation of Stage Performance**

Using the values of 1

*V*

1 2 from cascade test 2 results for known values of (s/c); C DP and C L can be plotted against incidence.

Since the value of C p tan m in C L equation is negligibly small, it is usual to use a more convenient theoretical value of C L given by

*C L*

2

*s*

tan 1 tan 2 cos

*m c*

In which the effect of profile drag is ignored. Using this formula, curves of C L can be plotted for nominal (or design) conditions to correspond with the curves of deflection. These curves are again plotted against 2 for fixed values of s/c

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

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

**Annulus Drag**

: Drag effects due to the walls of the Compressor annulus =

**C**

**DA**

**C**

**DA**

**= 0.02 (s/h) Secondry Losses**

: Due to the trailing vortices and tip clearances used

**= C**

**DS**

The following emprical relations can be

**C**

**DS**

**= 0.018**

C L 2

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

The overall Drag Coefficient is given by

*C D*

*C DP*

*C DA*

*C DS*

**Profile + annulus + secondary**

(the annular cascade C DP

*C D*

*s c*

1 2

*V*

1 2 cos 3 cos 2 is replaced by C D ) thus

*m*

1 This enables the loss coefficient row to be determined.

1 2

*V*

1 2 for the blade

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

The theoretical pressure rise (i.e

**w**

=0)

*P th*

1

*V a*

2 (tan 2 2 1 tan 2 2 ) 1 2

*V a*

2 (sec 2 1 sec 2 2 )

*P th*

1 2

*V a*

2 sec 2 1 sec 2 sec 2 2 1

*P th*

1

*V*

1 2 2 cos 2 cos 2 1 2 Efficiency of the blade row

*th*

*P th*

*P th*

1

*P th*

2 1

*V*

1 2

*V*

1 2 2

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

For a case where = 50 % Rotor and stator rows are similar thus this calculation carried at design diameter

*bl*

can be applied to the whole stage

*P P*

2

*‘*

2

*P*

1

*P*

1

*P*

2

*P*

1

*s*

(

*T s*

2

*T*

1 ) 1 for = 1/2

*s*

*T ‘ is*

*T act*

stage efficiency

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

*P*

2

*‘ P*

1 1

*T s*

2

*T*

1 1 then

*bl*

*P*

2

*P*

1 1

*P*

2

*‘ P*

1 1 1

*s*

1 2

*T*

1

*T s*

1 1 2

*T*

1

*T s*

1 1 expanding and neglecting 2 nd

*bl*

*s*

1 order terms; 1

*T s*

1 4

*T*

1 ( 1

*s*

) for

*T s*

20

*o K T*

1 400

*o K*

*bl*

*s*

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

*Axial Flow Compressors*

**Calculation of Stage Performance**

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

*s*

1 2

*blR*

*bl-ST*

If far removed from 50%

*s*

*bl-R*

1

*bl-ST*

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

*Axial Flow Compressors*

**Summary of the Design Procedure**

Assume Ts and at the design radius angles Calculate the air Applying chosen design condition (Free Vortex, Constant Reaction etc) Calculate air angles at all radii Results of Cascade Tests angles) C D and C L Blade shapes (Blade Calculate

**η**

**s**

and

**R**

**s**

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

*Axial Flow Compressors*

**Overall Performance**

Assuming that η s = η ∞ ( η constant through all compressor stages), for a compressor consisting of N similar stages, each with η s = η ∞

*n R*

*os n*

1

*T o*

1 R is the “Overall Pressure Ratio” where ;

*n n*

1 1 1

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

*Axial Flow Compressors*

**Overall Performance**

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

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

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

*Axial Flow Compressors*

**Compressibility Effects**

High air velocities between the blades effects the compressor performance.

**Critical Mach number M**

**c**

is defined such that at entry velocities lower than this; the performance of the cascade differs very little from that at low speeds. Above this losses begin to show a marked increase.

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

For a typical low speed cascade M c = 0.7 M m =0.85

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

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

*Axial Flow Compressors*

**Compressibility Effects**

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

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

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

*Axial Flow Compressors*

**Compressibility Effects**

Not to suffer from compressibility effects in early stages one might use constant reaction design if no other precaution can be taken.

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

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

The overall compressor characteristic is composed of the stage characteristics stacked.

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

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

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

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

When the compressor is run at

**a speed lower than design**

,

**T and R**

**c**

are reduced than the

**density**

at the rear stages will be lower than the design value.

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

Thus

**at low speeds m**

is determined by the

**choking of the rear stages.**

As the speed is increased density of the rear stages increases (V decrease) thus gets unchoked.

**At very high speeds choking will occur at the inlet**

the inlet of the compressor.

. The vertical line of constant speed is due to choking at

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

**A **

**B**

At the design speed if we consider the moving of operating point from A to B. At point B (on the surge line), the density at the compressor exit will be increased due to the compressor exit will be increased due to the increase in delivery pressure; also

**ṁ**

is slightly reduced. Axial velocity in the last stage is reduced incidence in the last stage is increased. Rotor blades are expected to

**stall from the last stages**

.

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

**A**

**C**

**ṁ falls rapidly**

;

**V**

**a**

at the inlet decreases, incidence of the first stage increases. But the incidence of the later stages decrease due to the increase of V due to first stages stalling. a (due to lower pressure and density). At low speeds surging is probably At conditions far removed from surge

**R**

high

**V**

**a**

large decrease in incidence negative incidence

**very low η**

**c**

is very low result in stall in

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

*Axial Flow Compressors*

**Some Deductions from the Compressor Characteristics**

At high pressure operation

**Blow-off**

at an intermediate stage (wasteful). Incidence can be maintained at design value by increasing the speed of last stage (HPC) and decreasing the speed of first stage (LPC); Two spools are mechanically independent but aerodynamically coupled.

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