---
title: "Derivation of Euler's turbine equation"
subject: "Fluid Machines"
chapter: 1
key: "semester_5/fluid_machines:euler-turbine-equation-derivation"
source: https://www.bctengineer.com/q/semester_5/fluid_machines/euler-turbine-equation-derivation
site: BCT Engineering
---

# Derivation of Euler's turbine equation

Fluid Machines · Chapter 1: Introduction of Turbomachime and Dynamic Action of Fluid

- 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 Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/fluid_machines/euler-turbine-equation-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_5/fluid_machines/chapter-1

## Question

Derive Euler's equation for work done per unit weight in a turbomachine. Draw the velocity triangles for a radial-flow turbine runner.

## Short answer

Euler's equation from angular momentum: torque T = ρQ(Vw1 r1 - Vw2 r2), power P = Tω = ρQ(Vw1 u1 - Vw2 u2). Euler head He = (Vw1 u1 - Vw2 u2)/g. Velocity triangles show absolute velocity V, blade velocity u, relative velocity Vr, whirl component Vw, and flow component Vf.

## Answer

Euler's equation is the basic equation of all turbomachines. It follows from the **angular momentum principle**: the torque on the rotor equals the rate of change of angular momentum of the fluid passing through it.

#### Velocity triangles

At inlet (1) and outlet (2): $V$ = absolute velocity, $u$ = blade (peripheral) velocity, $V_r$ = relative velocity, $V_w$ = whirl component (along $u$), $V_f$ = flow component (radial for a radial machine), $\alpha$ = angle of $V$ with $u$, $\beta$ = angle of $V_r$ with $u$ (blade angle at inlet is $\theta$, at outlet $\phi$).

```text
  Triangle (same form at inlet 1 and outlet 2)

              apex
             /|\
        V   / | \  Vr
           /  |  \
          /   |Vf \
         / a  |  b \
        *-----+-----*------> blade motion
        |<-Vw->|
        |<----- u ----->|

  V  = u + Vr (vector sum),  Vw = V cos a,  Vf = V sin a
```

#### Derivation

Let a mass flow $m=\rho Q$ pass through the runner per second. Inlet radius $r_1$, outlet radius $r_2$.

Angular momentum per second at inlet: $m V_{w1} r_1$. At outlet: $m V_{w2} r_2$.

Torque on the runner:

$$
T = \rho Q\,(V_{w1} r_1 - V_{w2} r_2)
$$

Angular velocity $\omega = u_1/r_1 = u_2/r_2$, so the power given to the runner is

$$
P = T\omega = \rho Q\,(V_{w1} u_1 - V_{w2} u_2)
$$

Work done per second per unit weight of water (weight flow $= \rho g Q$) is the **Euler head**:

$$
\boxed{H_e = \frac{1}{g}\,(V_{w1} u_1 - V_{w2} u_2)}
$$

#### Notes

- If the water leaves without whirl ($V_{w2}=0$, radial discharge), $H_e = V_{w1}u_1/g$.
- For an axial machine $u_1=u_2=u$, so $H_e = u(V_{w1}-V_{w2})/g$.
- $V_{w2}$ is taken negative if it is opposite to the blade motion.
- For a pump the sign is reversed: $H_e = (V_{w2}u_2 - V_{w1}u_1)/g$.
- Hydraulic efficiency of a turbine: $\eta_h = H_e/H$, where $H$ is the net head.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Euler's equation follows from:
   A. Conservation of energy only
   B. Continuity only
   C. Bernoulli's equation
   D. The angular momentum principle: torque = rate of change of angular momentum
   Answer: D. Torque on the rotor equals rate of change of angular momentum of the fluid. (Key point: Principle)

2. The torque on the runner is:
   A. rho g Q H
   B. rho Q (Vw1 r1 - Vw2 r2)
   C. rho Q (Vf1 r1 - Vf2 r2)
   D. rho Q (Vw1 u1 - Vw2 u2)
   Answer: B. Angular momentum in minus out: T = rho Q (Vw1 r1 - Vw2 r2). (Key point: Torque)

3. The work done per unit weight is:
   A. (V1^2 - V2^2)/2g
   B. (Vw1 + Vw2)/g
   C. (Vw1 u1 - Vw2 u2)/g
   D. (Vf1 u1 - Vf2 u2)/g
   Answer: C. Power rho Q (Vw1 u1 - Vw2 u2) divided by weight flow rho g Q gives He. (Key point: Euler head)

4. If water leaves without whirl (Vw2 = 0), the Euler head is:
   A. Vw1 u1 / g
   B. 0
   C. Vw1^2 / g
   D. u1^2 / g
   Answer: A. With Vw2 = 0, He = Vw1 u1 / g. (Key point: Special case)

## Related in Fluid Machines

- [Francis turbine velocity triangles and power](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Afrancis-turbine-numerical) — Fluid Machines, chapter 3
- [Single-stage impulse turbine blade work and efficiency](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Ade-laval-turbine-velocity-diagram-numerical) — Fluid Machines, chapter 5
- [Parsons turbine stage work and efficiency](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Aparsons-reaction-turbine-numerical) — Fluid Machines, chapter 5
- [Cross-flow (Banki-Mitchell) turbine](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Across-flow-turbine-notes) — Fluid Machines, chapter 3
- [Kaplan turbine discharge and blade angles](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Akaplan-turbine-numerical) — Fluid Machines, chapter 3

## Related in other subjects

- [Euler work equation for turbomachines](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Aeuler-turbine-equation-derivation) — Turbo Machines, chapter 2
- [Steady state work vs steady flow; turbine, nozzle](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteady-state-work-vs-flow) — Fundamental of Thermodynamics & Heat Transfer, chapter 4
- [Impulse turbine blade work and diagram efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Aimpulse-turbine-diagram-efficiency) — Turbo Machines, chapter 2
- [Derive continuity equation for unsteady pipe flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontinuity-equation-unsteady-pipe-flow) — Hydraulics, chapter 4