--- title: "Euler work equation for turbomachines" subject: "Turbo Machines" chapter: 2 key: "semester_7/turbo_machines:euler-turbine-equation-derivation" source: https://www.bctengineer.com/q/semester_7/turbo_machines/euler-turbine-equation-derivation site: BCT Engineering --- # Euler work equation for turbomachines Turbo Machines · Chapter 2: Velocity Vector Diagram - Asked in: year not given, 6 marks - Video: [Euler's equation for Turbine - #TURBO_MACHINES](https://www.youtube.com/watch?v=SxwBSkot0W0) — QUICK MECH SHOTS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/turbo_machines/euler-turbine-equation-derivation - Whole chapter: https://www.bctengineer.com/md/semester_7/turbo_machines/chapter-2 ## Question 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. ## 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. ## Answer #### Derivation Consider a rotor with fluid entering at radius $r_1$ with absolute velocity $C_1$ and leaving at $r_2$ with $C_2$. The whirl (tangential) components are $C_{w1}$ and $C_{w2}$. For a steady mass flow $\dot m$, the moment of momentum equation gives the torque on the rotor: $$ T = \dot m (r_1 C_{w1} - r_2 C_{w2}) $$ Power $= T\omega$ and the blade speed is $U = \omega r$: $$ P = \dot m (U_1 C_{w1} - U_2 C_{w2}) $$ Work per unit mass (Euler equation for a turbine): $$ \boxed{w = U_1 C_{w1} - U_2 C_{w2}} $$ For an axial machine $U_1 = U_2 = U$, so $w = U(C_{w1} - C_{w2})$. For a compressor or pump the sign is reversed: $w = U_2 C_{w2} - U_1 C_{w1}$. #### Three energy components From the velocity triangle, the cosine rule gives $U C_w = \tfrac12 (C^2 + U^2 - V_r^2)$, where $V_r$ is the relative velocity. Substituting at inlet and outlet: $$ w = \frac{C_1^2 - C_2^2}{2} + \frac{U_1^2 - U_2^2}{2} + \frac{V_{r2}^2 - V_{r1}^2}{2} $$ ```text inlet triangle outlet triangle C1 Vr1 C2 Vr2 \ /| \ /| \/ | Ca1 \/ | Ca2 ------U1-- ----U2-- ``` 1. $\dfrac{C_1^2 - C_2^2}{2}$ - change in **absolute kinetic energy** (impulse effect). 2. $\dfrac{U_1^2 - U_2^2}{2}$ - **centrifugal effect**, work from change of radius (zero in axial machines). 3. $\dfrac{V_{r2}^2 - V_{r1}^2}{2}$ - **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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the Euler work per unit mass for a turbine? A. U2 Cw2 - U1 Cw1 B. U1 Cw1 - U2 Cw2 C. (U1 + U2)(Cw1 + Cw2) D. m (r1 + r2) Answer: B. Moment of momentum gives w = U1 Cw1 - U2 Cw2 for a turbine; the sign reverses for compressors and pumps. (Key point: Euler equation) 2. Which principle gives the torque T = m(r1 Cw1 - r2 Cw2)? A. Conservation of energy B. Moment of momentum equation C. Bernoulli's equation D. Continuity equation Answer: B. Torque equals the rate of change of angular momentum of the fluid; power is then T times omega. (Key point: Origin) 3. Which term in the Euler equation is the centrifugal effect? A. (C1^2 - C2^2)/2 B. (Vr2^2 - Vr1^2)/2 C. (U1^2 - U2^2)/2 D. U (Cw1 - Cw2) Answer: C. (U1^2 - U2^2)/2 is work from change of radius, zero in axial machines. (Key point: Three energy components) 4. In an axial impulse turbine, where does the work come from? A. Only from the change in relative kinetic energy B. Only from the centrifugal term C. Equally from all three terms D. Only from the change in absolute kinetic energy Answer: D. For axial impulse, U1 = U2 and Vr2 is about Vr1, so terms 2 and 3 are nearly zero. 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