Euler’s turbine equation (1754)

Leonhard Euler, studying Segner’s reaction water wheel at the Berlin academy, wrote the torque of a rotor as the change in the angular momentum of the water passing through it. Multiplied by the speed, it gives the head a pump adds or a turbine takes, \(H = (u_2V_{t2} - u_1V_{t1})/g\) (euler_turbomachine_head()). This example draws the ideal head of a centrifugal impeller for backward, radial and forward blades, and shows that the head scales with the square of the speed, the affinity law (affinity_laws()).

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.fluid_engineering import affinity_laws, euler_turbomachine_head

The outlet velocity triangle

Water leaves an impeller of radius \(r_2\) and width \(b_2\) with radial speed \(V_{r2} = Q/(2\pi r_2 b_2)\). Relative to the blade it leaves at the blade angle \(\beta_2\), so its swirl is \(V_{t2} = u_2 - V_{r2}\cot\beta_2\). With no swirl at the inlet, the ideal head falls linearly with flow for backward-curved blades.

r2, b2, rpm = 0.15, 0.02, 1750
omega = rpm * 2 * np.pi / 60
u2 = omega * r2
Q = np.linspace(0, 0.08, 50)
V_r2 = Q / (2 * np.pi * r2 * b2)

fig, ax = plt.subplots(figsize=(6, 4))
heads = {}
for label, beta in [("backward, β₂ = 25°", 25), ("radial, β₂ = 90°", 90), ("forward, β₂ = 140°", 140)]:
    heads[beta] = euler_turbomachine_head(u2, u2 - V_r2 / np.tan(np.radians(beta)))
    ax.plot(Q * 1000, heads[beta], label=label)
ax.set(xlabel="flow [L/s]", ylabel="ideal head [m]", title=f"Euler head of an impeller at {rpm} rpm")
ax.legend()
fig.tight_layout()
print(f"shut-off head u2^2/g = {heads[25][0]:.1f} m for every blade angle")

shut-off head u2^2/g = 77.1 m for every blade angle

Backward blades are used in practice: their head falls with flow, so the pump is stable on its system curve and its power does not run away.

The head at twice the speed

At the same flow coefficient, the velocity triangle keeps its shape and every speed doubles, so the head goes up four times.

Q_bep, H_bep = 0.04, float(np.interp(0.04, Q, heads[25]))
Q2, H2, _ = affinity_laws(Q_bep, H_bep, 0.0, 2.0)
V_r2_fast = Q2 / (2 * np.pi * r2 * b2)
H2_euler = euler_turbomachine_head(2 * u2, 2 * u2 - V_r2_fast / np.tan(np.radians(25)))
print(f"at 2x speed: affinity law {H2:.1f} m, Euler {H2_euler:.1f} m")

assert np.allclose([heads[25][0], heads[90][0], heads[140][0]], u2**2 / 9.80665)
assert np.allclose(heads[90], heads[90][0]) and np.all(np.diff(heads[25]) < 0) and np.all(np.diff(heads[140]) > 0)
assert np.isclose(H2, H2_euler)

plt.show()

at 2x speed: affinity law 257.2 m, Euler 257.2 m