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1

Three Dimensional

Flow Analysis in

Axial Compressor

It may be recalled that this element is also executing a path through the curved diffusing passage between the rotor blades.

w

Simple three dimensional flow analysis : Initial assumptions

1)Radial movement of the flow is governed by the radial equilibrium of forces

2. Radial movements occur within the blade passage only and not outside it

3. Flow analysis involves balancing the radial force exerted by the blade rotation

4. Gravitational forces can be neglected

Resolving all the aerodynamic forces, acting on this element, in the radial direction,

we get ,

(p+dp)(r+dr).d θ .1 – p.r.1.d θ –

2(p+dp/2).dr.(d θ /2).

= ρ. dr. r. Cw^2 /r

Neglecting the second order terms (products of small terms e.g. dp.dr etc) the equation reduces to

w

1 dp 1

= .C

ρ dr r

This is called the

S imple Radial Equilibrium Equation

9

γ γ

2

dH dCa^ dCw 1 dp p dp = Ca +Cw +. - dr ρ dr dr dr dr -1 ρ

substituting for c p from Eqn(2) and then

differentiating the eqn (1) w.r.t. r ,

we get

Differentiating the eqn 3 (isentropic law) we get

γ

d ρ ρ dp

dr .p dr Substituting this in the new energy equation we get

dH dCa^ dCw 1 dp = C (^) a +Cw + dr ρ dr dr dr

• At entry to the compressor, except near the hub and the casing, enthalpy H (r) = constant.
• Using the condition of uniform work distribution along the blade length ( i.e. radially constant) we can say

dH = 0 dr

Thus, the energy equation would be written as, 2 a w w w

dC (^) dC C +C + = 0 dr dr r

Now, if Ca = constant at all radii, then the

first term is zero and the above equation

reduces to

dC C^2 C w^ = - w w (^) dr r

• The term Free Vortex essentially denotes that the strength of the vortex ( or lift per unit length ) created by each airfoil section used from the root to the tip of the blade remains constant

Lift , L = ρ.V.Γ

where, ρ is the density, V is the inlet velocity, and Γ is the strength of circulation

• It, therefore, means that at the trailing edge of the blade the trailing vortex sheet has constant strength from the root to the tip of the blade

Next Class ---

Free Vortex Design Law and Other Blade design laws