Impact of jets and velocity triangles

Why it matters

Every turbine, pump, compressor and turbocharger transfers energy by turning a stream of fluid on moving blades. Jet impact on plates and vanes is the simplest version of that exchange, and the velocity triangles you draw here are exactly the ones used for Pelton wheels, Francis runners, pump impellers and steam-turbine stages. Master the triangles once and every turbomachine numerical becomes routine.

Key ideas

• Force from momentum change. The force on a plate or vane equals the rate of change of momentum of the jet, in the chosen direction, with the sign reversed (force of fluid on solid = −force of solid on fluid). Friction and gravity are usually neglected, so the jet speed relative to a stationary vane is unchanged.
• Stationary surfaces.
  • Flat plate normal to the jet: the jet leaves along the plate, so all of its normal momentum is destroyed.
  • Inclined flat plate: only the velocity component normal to the plate is destroyed. The force acts normal to the plate (no friction).
  • Curved vane: turning the jet back on itself gives a larger momentum change. A symmetrical vane that deflects the jet through 180° gives twice the force of a flat plate.
• Moving surfaces. What matters is the velocity relative to the vane, V − u.
  • Single moving plate or vane: the mass striking it per second is ρA(V − u), because the vane runs away from the jet.
  • Series of plates or vanes on a wheel: some vane is always in the jet, so the whole jet mass flow ρAV is used.
  • Work is done only by the force component in the direction of vane motion (force × u).
• Velocity triangles (for a vane or blade moving at speed u):
  • Absolute velocity V = blade velocity u + relative velocity V_r (a vector sum).
  • Resolve V into the whirl (tangential) component V_w, along u, and the flow component V_f, perpendicular to u.
  • α = angle of V with the direction of motion (the nozzle or guide-vane angle); β = angle of V_r with the direction of motion (the blade angle).
  • For shock-free entry, the blade inlet angle must equal the β of the relative velocity, so V_r slides smoothly onto the vane.
  • On a frictionless vane, |V_r2| = |V_r1|; friction reduces it by a blade-velocity coefficient k.
• Work and efficiency. Work per unit mass equals u × (change of whirl velocity), with the whirl components added if they point in opposite directions. Efficiency is work done divided by the jet's kinetic energy V²/2.
• Best blade speed.
  • Series of flat plates: maximum efficiency 50 % at u = V/2.
  • Single moving flat plate: maximum 29.6 % (8/27) at u = V/3.
  • Series of curved vanes deflecting through 180° at u = V/2: the jet leaves with zero absolute velocity and efficiency reaches 100 % in theory. The Pelton wheel approaches this with buckets turning the jet through about 165°, so the water clears the next bucket.
• These results generalise to the Euler turbomachine equation in the next topics: work per unit mass = V_w1·u₁ ∓ V_w2·u₂.

Formulas

F = ρ·A·V² Force (N) of a jet of area A (m²) and speed V (m/s) on a fixed flat plate normal to it; ρ in kg/m³.

F_n = ρ·A·V²·sin θ Fixed inclined plate; θ angle between jet and plate; F_n normal to the plate.

F = ρ·A·V²·(1 + cos φ) Fixed symmetrical curved vane; φ angle between the leaving jet and the reversed jet direction (φ = 0 for full 180° turn, giving 2ρAV²).

F = ρ·A·(V − u)² single moving flat plate; u plate speed (m/s). F = ρ·A·V·(V − u) series of flat plates on a wheel; work per second = F·u, η = 2u(V − u)/V².

V⃗₁ = u⃗ + V⃗_r1 V_w1 = V₁·cos α₁ V_f1 = V₁·sin α₁ tan β₁ = V_f1 / (V_w1 − u) Inlet velocity triangle; angles from the direction of blade motion.

W = (V_w1 ± V_w2)·u (J/kg) Work per unit mass on a moving vane (u₁ = u₂ = u); use + when V_w2 points opposite to u.

η = 2·(V_w1 ± V_w2)·u / V₁² Hydraulic (blade) efficiency.

Worked examples

Example 1 (standard) — fixed and moving plates. Given: water jet of diameter 50 mm and speed V = 25 m/s (ρ = 1000 kg/m³). Find the force on (a) a fixed plate normal to the jet, (b) a fixed plate inclined at 30° to the jet, (c) a single plate moving at u = 10 m/s away from the jet, and (d) the power developed by a series of such plates at u = 10 m/s.

1. A = π × 0.05²/4 = 1.9635 × 10⁻³ m².
2. (a) F = ρAV² = 1000 × 1.9635 × 10⁻³ × 625 = 1227 N.
3. (b) F_n = 1227 × sin 30° = 613.6 N (normal to the plate).
4. (c) F = ρA(V − u)² = 1000 × 1.9635 × 10⁻³ × 15² = 441.8 N; power = 441.8 × 10 = 4418 W.
5. (d) Series: F = ρAV(V − u) = 1000 × 1.9635 × 10⁻³ × 25 × 15 = 736.3 N; power = 7363 W; η = 2 × 10 × 15/625 = 48 %. Answer: (a) 1227 N, (b) 614 N, (c) 442 N, (d) 7.36 kW at 48 % efficiency.

Example 2 (GATE level) — velocity triangles on a moving curved vane. Given: a jet at V₁ = 30 m/s, inclined at α₁ = 30° to the direction of motion, enters a vane moving at u = 12 m/s. The vane outlet angle is β₂ = 20° (measured from the direction of motion), and there is no friction. Find the inlet blade angle, the work done per kg, the absolute exit velocity and the efficiency.

1. V_w1 = 30 cos 30° = 25.98 m/s; V_f1 = 30 sin 30° = 15.00 m/s.
2. Relative whirl at inlet: V_w1 − u = 25.98 − 12 = 13.98 m/s → tan β₁ = 15.00/13.98 → β₁ = 47.0°.
3. V_r1 = √(13.98² + 15.00²) = 20.51 m/s; frictionless → V_r2 = 20.51 m/s.
4. Outlet: V_r2 cos β₂ = 20.51 × cos 20° = 19.27 m/s, directed backwards. Since 19.27 > u, V_w2 = 19.27 − 12 = 7.27 m/s, opposite to u.
5. V_f2 = 20.51 × sin 20° = 7.01 m/s; V₂ = √(7.27² + 7.01²) = 10.10 m/s.
6. Work: W = (V_w1 + V_w2)·u = (25.98 + 7.27) × 12 = 399.0 J/kg.
7. Efficiency: η = 2W/V₁² = 798.0/900 = 0.887.
8. Check by energy: (V₁² − V₂²)/2 = (900 − 102.0)/2 = 399 J/kg ✓. Answer: β₁ ≈ 47.0°, W ≈ 399 J/kg, V₂ ≈ 10.1 m/s, η ≈ 88.7 %.

Common mistakes

• Using ρAV instead of ρA(V − u) for the mass flow striking a single moving plate (or the reverse for a series of plates).
• Subtracting u from V_w — whirl velocity is a component of the absolute velocity; V_w − u is the whirl component of the relative velocity.
• Adding whirl components when they point in the same direction (or subtracting when opposite).
• Measuring blade angles from the wrong reference; state whether angles are from the direction of motion or from the axial direction.
• Assuming the absolute velocity is unchanged across a moving vane; on a frictionless vane, it is the relative speed that stays constant.

For GATE ME

Expect: force on fixed or moving plates and vanes, the blade speed for maximum efficiency and its value (50 % for flat plates, up to 100 % for 180° vanes), inlet and outlet blade angles from velocity triangles, and work or power per unit mass flow. Practise drawing both triangles to scale and labelling V, u, V_r, V_w, V_f, α and β every time.

Quick check

1. A jet of 0.03 m³/s at 20 m/s hits a fixed normal plate. What is the force?
2. For a series of flat plates, what blade speed gives maximum efficiency, and what is that efficiency?
3. A jet at 25 m/s enters at 30° to the direction of motion. What is its whirl velocity?
4. On a frictionless moving vane, which velocity has the same magnitude at inlet and outlet?
5. What deflection angle would ideally give 100 % efficiency on a series of vanes? Answers: 1. 1000 × 0.03 × 20 = 600 N. 2. u = V/2; 50 %. 3. 25 cos 30° = 21.65 m/s. 4. The relative velocity. 5. 180° (with u = V/2).