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Euler work equation for turbomachines

Short answer

For a turbine, work per unit mass is w = U1Cw1 - U2Cw2. From velocity triangles and cosine rule, it equals (C1^2 - C2^2)/2 + (U1^2 - U2^2)/2 + (Vr2^2 - Vr1^2)/2, representing change in absolute kinetic energy, centrifugal effect, and change in relative kinetic energy.

Derive the Euler turbine equation for the work done per unit mass of fluid in a turbomachine using the velocity triangles at inlet and outlet. Express it in terms of the components of velocity and show that it can be written in the form of three energy components.

Derivation

Consider a rotor with fluid entering at radius r1r_1 with absolute velocity C1C_1 and leaving at r2r_2 with C2C_2. The whirl (tangential) components are Cw1C_{w1} and Cw2C_{w2}. For a steady mass flow m˙\dot m, the moment of momentum equation gives the torque on the rotor:

T=m˙(r1Cw1−r2Cw2)T = \dot m (r_1 C_{w1} - r_2 C_{w2})

Power =Tω= T\omega and the blade speed is U=ωrU = \omega r:

P=m˙(U1Cw1−U2Cw2)P = \dot m (U_1 C_{w1} - U_2 C_{w2})

Work per unit mass (Euler equation for a turbine):

w=U1Cw1−U2Cw2\boxed{w = U_1 C_{w1} - U_2 C_{w2}}

For an axial machine U1=U2=UU_1 = U_2 = U, so w=U(Cw1−Cw2)w = U(C_{w1} - C_{w2}). For a compressor or pump the sign is reversed: w=U2Cw2−U1Cw1w = U_2 C_{w2} - U_1 C_{w1}.

Three energy components

From the velocity triangle, the cosine rule gives UCw=12(C2+U2−Vr2)U C_w = \tfrac12 (C^2 + U^2 - V_r^2), where VrV_r is the relative velocity. Substituting at inlet and outlet:

w=C12−C222+U12−U222+Vr22−Vr122w = \frac{C_1^2 - C_2^2}{2} + \frac{U_1^2 - U_2^2}{2} + \frac{V_{r2}^2 - V_{r1}^2}{2} 
  inlet triangle           outlet triangle
       C1  Vr1                 C2  Vr2
        \  /|                   \  /|
         \/ | Ca1                \/ | Ca2
   ------U1--                ----U2--
  1. C12−C222\dfrac{C_1^2 - C_2^2}{2} - change in absolute kinetic energy (impulse effect).
  2. U12−U222\dfrac{U_1^2 - U_2^2}{2} - centrifugal effect, work from change of radius (zero in axial machines).
  3. Vr22−Vr122\dfrac{V_{r2}^2 - V_{r1}^2}{2} - change of relative kinetic energy, the acceleration inside the rotor passage (reaction effect).

For an axial impulse turbine, terms 2 and 3 are (nearly) zero, and the work comes only from the first term. The equation holds whatever the losses, since it depends only on the velocity triangles.

✓ Answer checked by Claude Opus 5.5 (high effort)

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