Understanding Radial Equilibrium in Axial Flow Compressors: 3D Flow Analysis, Slides of Turbomachinery

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1

Three dimensional flow analysis

in

Axial Flow Compressors

Radial flow can appear due to following reasons—

1. Centrifugal action on being imparted rotational

motion is experienced by the fluid

2. Convergence of the annulus flow track introduces

radiality in highly loaded compressor stages.

3. Twist and taper (chord and thickness-wise) of the

blade introduces radial component to the fluid;

4. Tip clearance effects - effect of tip flow around

the open tip of the blade rotor

5 Passage vortex formation inside the blade

passages;

6. Temperature/ Enthalpy / Entropy gradient in

the radial direction (due to 1 to 4 above);

7. Blade solid body thickness blockage (including

the effect of camber and stagger)

8. End wall (casing and hub) boundary layer

blockage effects, that deflect the flow inward,

in addition to reducing the main flow rate

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

In axial flow compressors rotors need a rotating co-ordinate system, whereas the stator may use a static co-ordinate system

Assume that a fluid particle, p is moving in an arbitrary path within the two coordinate systems

o

O’

 The two reference systems have relative motion represented by Ṙ (motion of vector R w.r.t. fixed origin, o).  ω is rotation of the particle with respect to moving axes system xyz, and vxyz is the translational motion of the particle with respect to moving axes xyz.

     

XYZ XYZ

=

dr V dt

• Velocity of the particle P with respect to fixed axes system XYZ from the figure is :

o

O’

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Acceleration of P w.r.t. Fixed coordinates XYZ ,

     

XYZ XYZ

=

XYZ

dV a dt

And, acceleration of P w.r.t. Rotating coordinates xyz,

     

xyz

xyz

=

dV a dt (^) xyz

Definitions of the accelerations

Thus, total acceleration of P w.r.t the fixed coordinate system XYZ,

     

XYZ XYZ

= dV XYZ dt

a

  (^)  (^) ( )   (^)     ^ 

__**

xyz XYZ

+ +

d (^) V xyz dω ρ' R dt dt

=

..

 •^  • •

_** = aωxyz + (^) R 2+ (^) **ω ω vxyz +** (^) **ω ρ' + ρ'_**

Acceleration of the particle p w.r.t the fixed origin O may be written as

.. a (^) XYZ

    XYZ (^)  

a ω =^ axyzω + 2 ω v +^ ρ' xyz

For motion with constant angular velocity ω

a

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For a compressor blade passage, the flow velocities are

( ) xyz V =V relative velocity

V XYZ =^ C^ (^ absolute velocity )

Differentiating, ( ) ^ ( )

     

XYZ XYZ XYZ XYZ

d V (^) dω ρ' a dt dt

Final accn. (^) ( ) a (^) XYZω = axyz+ω^2 ωv xyz+^ ρ'

V (^) XYZ =^ V xyz+^ ω ρ'^ =^ V + ω r =V + u

XYZ

Translational motion Rotation

As the flow in compressor blade is diffusing

DV Dt

is negative

− ∇

1 Dv 2

. p = ω r + 2 ω V- ρ Dt

x x

17

where r , w and a are the radial, whirl(peripheral) and axial directions respectively.

New of axis notations

W

19

Then from the definition of unit vectors ∧ ∧ ∧ r t t t

D = = Dt r

i ω V i i

∧ ∧ ∧ t t r r

D = - = - Dt r

i ω V and i i

∧ ∧ ∧ V = V i +V i +V r r t t a ia using,

( ) ( )      

D D

= V +

Dt Ds

ds

dt

ds = 0 dt

as For Steady State flow

By definition

0

r r r w

w (^) w

r w w

s is length in any direction

Now, the equation for flow inside the compressor blade passage may be recast using r, θ and z coordinate system and modified to :

 ^ ^    (^)     (^)  

∧ ∧ ∧ t

r t a r w r a

DV DV d DV DV dDt^ θ^ θ^ = i Dt - V dt + i Dt +V dt + i Dt

w w

Using the coordinate systems the flow velocity in relative frame is V and its components may be shown as Va , Vr , Vw