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An in-depth analysis of three-dimensional flow in axial compressors, focusing on the concepts of radial equilibrium and free vortex law. The derivation of the simple radial equilibrium equation and its application in the energy equation. It also explains the significance of the free vortex law in blade design.
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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
Radial movements occur within the blade passage only and not outside it
Flow analysis involves balancing the radial force exerted by the blade rotation
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).
Neglecting the second order terms (products of small terms e.g. dp.dr etc) the equation reduces to
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
Differentiating the eqn 3 (isentropic law) we get
γ
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
dH = 0 dr
Thus, the energy equation would be written as, 2 a w w w
dC (^) dC C +C + = 0 dr dr r
dC C^2 C w^ = - w w (^) dr r
where, ρ is the density, V is the inlet velocity, and Γ is the strength of circulation
Next Class ---
Free Vortex Design Law and Other Blade design laws