Francis Turbine Runner Blade Hydraulics

Operating Point

Guide vane angle α₁ 18°

Runner speed N 300 rpm

Net head H 60 m

Velocity Triangle

U — blade (peripheral) speed

C — absolute velocity

W — relative velocity

fixed blade angle (design)

Live Readouts

Euler head E

— m

Hydraulic η

—%

Shaft power

— kW

Incidence Δβ

—°

Flow rate Q (continuity)

— m³/s

How it works

Water leaves the guide vanes at absolute velocity C₁, angled α₁ from the tangential direction. The runner rim moves at blade speed U₁ = πD₁N/60. Subtracting U₁ from C₁ (vector subtraction) gives the relative velocity W₁ — the direction the blade edge itself must be curved to, for the water to enter without a sudden change of direction ("shockless entry").

C₁ = Cv·√(2gH)         (guide-vane exit velocity)
Cw₁ = C₁cos α₁,  Cr₁ = C₁sin α₁
U₁  = πD₁N/60
Euler head:  E = U₁Cw₁ / g     (radial-exit design, Cw₂≈0)
η_h = E / H

This site's runner blade is fixed with a radial inlet angle (β₁=90°), so it accepts a shock-free flow only when Cw₁ = U₁ — one specific combination of α₁, N and H. Away from that point, W₁ no longer lines up with the blade edge; the gap between them, the incidence angle Δβ, is the classic source of runner entry-shock loss in off-design operation. The outlet triangle (toggle above) shows the same idea against the fixed exit blade angle β₂, using the meridional velocity Cr₂ recovered from continuity through the runner exit area.

Guide vane angle α₁ — steers the incoming jet; the vane ring in the 3D view visibly rotates with it.
Runner speed N — sets blade speed U₁/U₂; the runner and its 13 blades spin at this rate.
Net head H — sets the guide-vane exit speed C₁ and therefore the flow rate Q by continuity.

This is the same velocity-triangle method used to design and troubleshoot real Francis units — a mismatched incidence angle at part-load is a known cause of runner blade cavitation pitting and vibration in hydropower plants.

Diagram at inlet rim · vectors rescale live with the controls

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