Surge Tank Oscillation

What is modelled

A hydropower station is fed through a long tunnel. Stopping the turbines quickly would need to stop a huge mass of moving water, so a surge tank is placed near the turbines: the water goes into the tank and the level swings, like a mass on a spring. With a turbine that holds constant power the swing can even grow if the tank is too small, which is the Thoma stability limit.

Equations

• Tank As dz/dt = AF v − Qt; tunnel (L/g) dv/dt = −(z + α v|v|) with α = f L / (2 g D).
• Undamped period T = 2π √(L As / (g AF)) and peak rise zmax = v0 √(L AF / (g As)).
• Thoma area ATh = AF L / (2 g α (H0 − α v02)).

Checks run in the page

• The frictionless period against the closed form.
• The frictionless peak rise against the closed form.
• Conservation of the energy-like quantity without friction.
• The sign of the small-swing growth rate on either side of the Thoma area, confirmed by a nonlinear run.

Honest simplifications

The tunnel is rigid and the water incompressible, so there is no water hammer; the tank has a constant cross-section with no orifice or overflow; the turbine head is the gross head plus the tank level; the friction factor is constant. A real design uses the full transient with the penstock, the governor and the worst cases.

• Press Full load rejection, then Wide tank and compare the period and the peak.
• Press Governed, too small tank and watch the swing grow, then Governed, stable tank.
• Read the growth-rate chart: it crosses zero at the Thoma diameter.