of Compressors and Turbines (AE 651) Autumn Semester 2009 - - PowerPoint PPT Presentation

Aerodynamics

• f

Compressors and Turbines

(AE 651)

Autumn Semester 2009

Instructor : Bhaskar Roy Professor, Aerospace Engineering Department I.I.T., Bombay e-mail : aeroyia@aero.iitb.ac.in

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AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Three – dimensional flow analysis

The assumption that the flow inside the axial flow compressor or turbine annulus is two– dimensional means that any effect of radial movement of the fluid through blade passage is ignored.

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Radial flow can appear due to the following reasons-- 1. Centrifugal action in the process of transfer

• f rotational motion is passed on to the

fluid 2. Convergence of the annulus passage is substantial for highly loaded compressor stages. 3. Twist and taper (chord-wise and thickness- wise) of the blade from hub to tip; 4. Tip clearance effects; (i.e. effect of tip flow around the open tip of the blades)

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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5 Double vortex formation in the blade passages; 6. Temperature/ Enthalpy / Entropy gradient in the radial direction (due to above 1 to 4); 7. Blade thickness blockage (including the effect of camber and stagger) 8. End wall (casing and hub) boundary layer blockage effects

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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The radial equilibrium theory is based on the assumption that the forces exerted by all these radial gradients contributing to the overall radial movement of the flow must be balanced by forces exerted by the pressure gradient existing in the flow, so that at any instant of time the fluid system is in radial equilibrium.

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Motion of a particle w.r.t two co–ordinate systems

 The two reference systems

have relative motion represented by R (motion

• f position vector R), ω

(rotation of the particle with respect to x,y,z) and vxyz are the rotational and translational motion of the particle with respect to xyz.

 Velocity of particle P with

respect to XYZ from fig.

r’

     

XYZ XYZ

= dr

V dt

r

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Velocity of P w.r.to small xyz,

     

xyz xyz

=

dρ' dt

v

Vectorially

       '

r = R+ ρ

                     

XYZ XYZ XyZ

= +

dr dR dρ' dt dt dt

Or,



XYZ xyz

= + + ω.ρ'

V R V

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Again, acceleration of P w.r.t. space coordinates XYZ,

     

XYZ

XYZ

= XYZ

dV a dt

Again, acceleration of P w.r.t. body-fixed coordinates, xyz,

     

xyz xyz =

dV a dt xyz

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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     

XYZ

XYZ

XYZ

V =

d dt

a

 

           

**

XYZ xyz

+ +

d d ωρ ' V xyz R dt dt

=

      XYZ

xyz =

dV dt

     

xyz xyz

xyz +

dV ω.V dt

 

                 

XYZ XYZ XYZ

= + '

d ω.ρ' dρ' dω ρ ω dt dt dt

           

XYZ xyz

= + '

dρ' dρ' ρ ω dt dt

Thus, acceleration of P w.r.t. space coordinates XYZ,

Also and Using

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Hence,

 

 

ρ' ' XYZ

= + + + + + ρ a a ω ω ω ω ω V R v xyz xyz xyz

. .   

     

**

xyz xyz+

+ + +

ρ' ρ' a ω ω ω ω R 2 v

=

For a frame of reference rotating with constant angular velocity ω, about the z-axis (axis of the rotating machine)

      XYZ =

+ + ρ' a a ω ω ω 2 v xyz xyz

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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For a compressor blade passage, the flow velocities are,

 

xyz

relative velocity

= V V

 

XYZ =

absolute velocity

C V

,

Differentiating,

 

 

       

     

XYZ

XYZ XYZ XYZ

=

• d V

d ω ρ' dt dt

a

Finally we get,

 

XYZ

xyz xyz

= + +

ρ' a a ω ω ω 2 v

XYZ xyz

= + =

= ω ρ' V + ω r V + u V V

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Considering the equilibrium of forces along the arbitrary flow direction, s – direction, we get, between any two axial stations, separated by a small distances ∆s, where area,

Ai is constant. ∆ p.Ai = Ai.ρ.∆s. aXYZ ,

XYZ

1 Δp = ρ Δs

a



2

1 Dv . p = + ω r + 2ωV ρ Dt

The flow in compressor blade is diffusing, DV

Dt

is negative,,

 

2

1 Dv . p =

• ω r + 2 ω V

ρ Dt

x x

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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where r, t and a are the radial, tangential (peripheral) and axial directions respectively.

Generalized flow path direction

Change of axis notations

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Assumptions made are:-

• The fluid is frictionless
• The rotor is rigid and rotates with constant

angular velocity ω

• The flow is steady relative to the rotor
• The radial variation of density is neglected
• This still leaves enough scope for formation
• f i) vorticity,

ii) entropy gradients, and iii) stagnation enthalpy gradients in the flow field.

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Then from the definition of unit vectors,

∧ ∧ ∧

t

r t t t t

D = = = = Dt r

V i ω i i i i

∧ ∧ ∧ t t t t r r r r

D = - = - Dt r

i V i V ω i i i i

And

∧ ∧ ∧ r r t t a a

V = V i +V i +V i

Again,

    

    

Ο

D D = V + Dt Ds

ds dt

ds = 0 dt

           

∧ ∧ ∧ t

t a t a r r w r a

DV DV DV DV DV DV DV dθ dθ =

• +

+ + Dt Dt dt Dt dt Dt

V V V V

i i i

And Steady State As

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Now, the equation for flow inside the compressor blade passage may be resolved in its three components, using r, q and z coordinate system and modified the above,

 

     

2

t r

ω r V D V 1 ∂ p

• =
• ρ ∂ r

D s r

+ + V

     

r r

a

t

D V 1 ∂ p V V V V

• .

= + + 2.ω.V V ρ r.∂ θ D s

. r

     

a

D V 1 ∂ p

• =

ρ ∂ z D s

V

• --------- (a)
• ------- (b)
• ------------- (c)

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Vt = Vw

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Assuming some arbitrary velocity triangle for the flow at the station under consideration,

t t =

+ = +

C ω r V

Then equation (a) and (b) can be rewritten as

 

       

r

2 t

C DV 1 p

• =
• ρ

r Ds

V r

 

     

t

D r.C 1 p 1 p V

• =

ρ r. θ D s r ∂ ∂

• --------- (d)
• ---------- (e)

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Now, we can write the kinematic relation as,

   

a

D D D D = Ds Da

V V V V

Where Va and a are axial components

• f V and s respectively Now, if we

define a new meridional direction by ∧ ∧ ∧ m m r r a a

D i = D i + D i

Hence from equation (d) we can write,

 

     

2 m

t r

C DV DV 1 ∂p

• =
• ρ ∂r

Dm

V r

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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By the definition of meridional direction

r Z

tan.φ = V

V

and V = V sin

Hence the flow equation in radial direction can be written down as

2 2 t m m m m m

1 ∂p Dsin.φ =

• si

sin.φ ρ ∂ρ r Dm Dm

D C V V V V V

   

m

D D D D = Ds Dm

V V V V

Now, by definition

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Now,

   D sin. D = cos. Dm D m



m

D 1 D 1 = - = - Dm Dm

r

rm is the radius of curvature of the meridional plane,

The negative sign is arbitrary. But, for axial flow compressor the flow track inside generally moves towards lesser φ or higher rm . Hence,

r

2 2 m m m m m

t

D 1 p 1 p = + cos.φ

• ρ

ρ r Dm

C V V V V V r

∂ ∂

This is the full radial – equilibrium Equation for circumferentially averaged (blade to blade) flow properties inside of a turbo machine blade row Flow Models for Turbomachinery

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

and

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AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery

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Flow Models for Turbomachinery

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

• For old fashioned compressor designs Vm (instead of

axial velocity Va or Ca) may be considered constant and the last term consequently drops. In the very early design of compressor the flow path was considered linear and hence even the 2nd term was not considered, giving the simplest form of radial equilibrium equation

.

2 2

t w

1 p 1 p = = = = ρ ρ r r

C C

∂ ∂

For blades, as the hub/ tip ratio decreases, giving longer blades and increasing blade loading, as in the case of modern compressor, this simple relationship is inadequate.

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Flow Models for Turbomachinery

AE 651 - Prof Bhaskar Roy, IITB Lect - 19



Wherever the blading is not exerting the centrifugal action, the radial equilibrium relation can not be applied. Experiment has shown that even in between the blades, in the axial gaps, there would be radial shift of the flow. Hence it is necessary that for accurate analysis the full radial–equation be used.

 For using the full equation the following further steps

need to be taken. i) The R.E.E is to be transformed into a form that contains partial derivatives with respect to r and θ ii) Next, the circumferential average is taken by integrating over θ from pressure side of one blade to the suction side of the other blade. iii) The flow is analysed at various axial stations with a) Energy equation, b) Continuity condition and c) R.E.E.

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Flow Models for Turbomachinery

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

• It is necessary that flow properties obtained in this

manner at various axial stations be consistent with

• ne another as the flow properties are evaluated

from hub to tip at each station. That means radial acceleration that the fluid particle undergoes from station to station be accounted for in the R.E.E.; this can be achieved by assuming shapes for the meridional streamlines consistent with the continuity condition expressing the radial acceleration in terms

• f the streamline slope and curvature. This implies

an iterative method of solution.

• This method, in which types of surfaces are used to

build up axisymmetric flow in a turbomachinery, has been used widely used. The equations on the blade- to-blade surface and those on the meridional plane need to be solved separately.

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Flow Models for Turbomachinery

AE 651 - Prof Bhaskar Roy, IITB Lect - 19

The 3-D flow computations has now been in existence for nearly 25 years and has provided immense assistance to engine designers. It has helped cut down on design time and has reduced dependence on costly experimental analysis. The 3-D methods have helped understand various flow phenomena e.g. secondary flow development, chocking in the stages, effects of end-wall flows etc. However, the designer uses these solutions in conjunction with many empirical relations (specially for losses, blockage, tip flow models etc.) and experimental data to make the design as best as the situation permits. There is still a large scope for improvement on these methods and for reducing dependence on empirical relations