A horizontal-axis tidal turbine extracts kinetic energy from a moving current the same way a wind turbine extracts it from moving air — only water is about 800× denser, so a tidal rotor a fraction of the diameter of a wind turbine can capture comparable power.
Available power: P_avail = ½ ρ A U³
Extracted power: P = ½ ρ A U³ · C_p(λ, θ)
Tip-speed ratio: λ = ω R / U
The rotor never reaches an arbitrary speed — its angular momentum obeys a real torque balance each frame:
I dω/dt = T_hydro − T_gen
T_hydro = P(ω) / ω, T_gen = k · ω
where I is the rotor's moment of inertia and k is the generator's load coefficient. The rotor spins up until the hydrodynamic torque and the generator's resistive torque balance, settling on a tip-speed ratio λ. The power coefficient itself follows the standard empirical rotor curve (the same functional form used for wind-turbine blade-element analysis, equally valid for a tidal rotor's aerofoil sections):
1/λi = 1/(λ + 0.08θ) − 0.035/(θ³ + 1)
C_p(λ,θ) = 0.5176(116/λi − 0.4θ − 5)e^(−21/λi) + 0.0068λ
- Current speed U — the free-stream tidal current; power scales with the cube of U.
- Blade pitch θ — coarsening the pitch away from 0° trades peak efficiency for a flatter, lower curve — exactly how real turbines shed load in fast spring tides.
- Generator load k — higher electrical load holds the rotor at a lower λ; too little load and it overspins past its efficient operating point, too much and it stalls.
- Seawater vs air — swaps ρ only, holding every other control fixed, to show why tidal turbines are so much more compact than wind turbines for the same power.
The Cp(λ) curve peaks below the theoretical open-flow Betz limit of 16/27 ≈ 59.3%, matching real rotor efficiencies (~45–48%) once real blade aerodynamics are accounted for.