Presets
Playback
Plant design
Changing these rebuilds the runner and penstock.
Operation
Reservoir level, guide-vane target, servo speed and runner height.
Experiments
Layers
Model checks
The science
Velocity triangles and Euler work
Water leaves the guide vanes with absolute velocity c1 at angle α to the tangent. The runner blade at radius r1 moves at u1 = ω r1, so the water meets the blade with relative velocity w1 = c1 - u1. The runner turns the swirl cu1 down to cu2 at the outlet, and the specific work is the Euler equation He = (u1 cu1 - u2 cu2) / g (metres of water). The guide-vane opening sets the radial speed cm1 = Q / A1 and so the swirl cu1 = cm1 / tan α.
Head balance
The available head is shared as H = hf + He + h_shock + h_runner-friction + h_exit. hf is Darcy-Weisbach penstock friction (Haaland friction factor, plus 0.3 velocity heads for entrance and bends). Shock loss appears whenever the swirl differs from the shock-free value the blades were designed for, which is why efficiency falls away from the best opening and gives the "hill" shape.
Water column dynamics
Treating the penstock water as a rigid column, (L / g Ap) dQ/dt = H - hf(Q) - H_runner(Q, α). The water starting time Tw = L Q / (g Ap H) sets how fast the flow can follow the gates. Fast closure compresses the water: the Joukowsky limit is ΔH = a v / g, and for closure slower than the wave round trip 2L/a the Michaud estimate is ΔH = 2 L v / (g Tc). The rigid-column model itself does not contain the pressure wave, so these two are shown as formula estimates only.
Cavitation
The Thoma number is σ = (Hatm - Hv - Hs) / H. A plant is considered safe when σ exceeds the critical value; the empirical Francis correlation sigma_c = 0.0432 (ns/100)^1.64 used here is indicative only, not a design rule.
Simplifications
One-dimensional mean-line model, fixed loss coefficients (illustrative), no draft-tube vortex rope, no runner leakage detail, no elastic water hammer, generator at fixed synchronous speed.
What to try
- Click "Go to best opening" and read the efficiency, then drag the opening away in either direction and watch the shock loss grow in the head-budget graph.
- Pick "Low head 25 m" and compare the runner diameter, speed and specific speed against the 300 m preset.
- Press "Fast close" and compare the slow servo (60 s) with a fast one (2 s): the water-hammer estimate changes by about 30 times.
- Raise the runner height above the tailwater (Hs) until the cavitation margin turns red.