Impact of jets and velocity triangles

Why it matters

A jet striking a plate or a curved vane is the simplest turbomachine, and it is exactly how a Pelton wheel produces power. The same momentum reasoning sizes water-jet cleaners and cutters, the thrust on a nozzle or deflector, and the forces on valve spools. Velocity triangles, introduced here, are the language used for every turbine, pump and compressor that follows.

Key ideas

Force from a jet. A jet of area a and speed V carries momentum flux ρ·a·V². When a surface changes the jet's velocity (in magnitude or direction), the surface must push on the fluid, and the fluid pushes back with an equal and opposite force. Apply the steady momentum equation to a control volume around the jet and surface. Assumptions usually made: the jet is at atmospheric pressure throughout (so no pressure forces on the CV), gravity is negligible over the small region, and friction on the surface is neglected unless a friction factor is given (relative speed then stays constant along the vane).

Fixed surfaces.

• Flat plate normal to the jet: the jet is turned through 90° and leaves sideways, losing all its momentum in the jet direction. F = ρaV².
• Flat plate inclined at θ to the jet: with no friction, the force is normal to the plate, F_n = ρaV²·sinθ. The jet splits unequally along the plate.
• Curved vane: turning the jet back on itself increases the force. A symmetrical vane that turns the jet through (180° − φ) gives F = ρaV²(1 + cosφ); a semicircular cup (φ = 0) gives the maximum, 2ρaV².

Moving surfaces. Work is done only if the surface moves. Work the problem in the frame of the vane, using relative velocity:

• Single plate or vane moving away at u: the jet catches up with it at V − u, so the mass striking per second is ρa(V − u) and F = ρa(V − u)². This single-vane case is a textbook idealisation; it cannot be used for a wheel.
• Series of vanes (a wheel): there is always a vane in the jet's path, so all the water issuing from the nozzle, ρaV, is used, and F = ρaV(V − u) for flat plates. Efficiency η = 2u(V − u)/V², maximum 50 % at u = V/2. With curved buckets that turn the jet through nearly 180°, η = 2u(V − u)(1 + cosφ)/V², which approaches 100 % at u = V/2 — the basis of the Pelton wheel.

Velocity triangles. At the inlet and outlet of a moving blade the absolute velocity V equals the vector sum of the blade velocity u and the relative velocity V_r: V = u + V_r. Standard notation:

• V_w (whirl) = component of V along the blade motion; V_f (flow) = component perpendicular to it.
• α = angle of the absolute velocity (guide or nozzle angle), β (or θ, φ) = blade angle, the angle of the relative velocity, measured from the direction of blade motion.
• Shockless entry: the blade inlet angle must match the direction of V_r at the inlet, so that fluid enters smoothly without impact loss.
• Without friction, the relative speed is unchanged over the blade (V_r1 = V_r2); friction reduces V_r2 by a factor k.

Work and Euler's equation. The torque on a rotor equals the rate of change of angular momentum of the fluid. For a blade moving at u, the work per unit weight of fluid is (V_w1·u₁ ± V_w2·u₂)/g. Use + when the outlet whirl is opposite to the blade motion and − when it is in the same direction (with vectors, it is simply V_w1·u₁ − V_w2·u₂ with signed V_w2). Hydraulic efficiency of a jet-driven wheel is work done divided by the jet kinetic energy, and with no friction it equals 1 − V₂²/V₁², where V₂ is the absolute exit velocity: the leaving kinetic energy is the loss.

Formulas

F = ρ·a·V² (fixed flat plate, normal)

• F = force (N), ρ = density (kg/m³), a = jet area (m²), V = jet velocity (m/s).

F_n = ρ·a·V²·sinθ (fixed flat plate inclined at θ to the jet)

F = ρ·a·V²·(1 + cosφ) (fixed symmetrical curved vane; φ = angle of the leaving jet measured from the direction exactly opposite to the incoming jet)

F = ρ·a·(V − u)² (single vane moving at u, jet direction)

F = ρ·a·V·(V − u), η = 2u·(V − u) / V² (series of flat plates; η_max = 0.5 at u = V/2)

η = 2u·(V − u)·(1 + k·cosφ) / V² (series of curved buckets; φ = bucket outlet angle measured from the reversed jet direction, k = V_r2/V_r1, 1 if frictionless)

W/g = (V_w1·u₁ ± V_w2·u₂) / g

• Work per unit weight (m, i.e. J/N); u = blade speed (m/s), V_w = whirl component (m/s).

P = ρ·Q·(V_w1·u₁ ± V_w2·u₂)

• P = power (W), Q = flow rate striking the wheel (m³/s).

η_h = 2·(V_w1·u₁ ± V_w2·u₂) / V₁²

• Hydraulic efficiency of a jet-driven wheel (–).

Worked examples

Example 1 — jet on plates (standard). A water jet 50 mm in diameter has a speed of 30 m/s. Find the force (a) on a fixed plate normal to the jet, (b) on a single plate moving away at 10 m/s, and (c) on a series of plates moving at 10 m/s, with the power and efficiency in (c).

1. Jet area a = π × 0.05²/4 = 1.963 × 10⁻³ m².
2. (a) F = ρaV² = 1000 × 1.963 × 10⁻³ × 900 = 1767 N.
3. (b) F = ρa(V − u)² = 1000 × 1.963 × 10⁻³ × 20² = 785.4 N.
4. (c) F = ρaV(V − u) = 1000 × 1.963 × 10⁻³ × 30 × 20 = 1178 N; power = F·u = 11 781 W.
5. Efficiency η = 2u(V − u)/V² = 2 × 10 × 20/900 = 0.444. Answer: (a) 1767 N; (b) 785 N; (c) 1178 N, 11.8 kW, η = 44.4 %.

Example 2 — velocity triangles on a moving curved vane (GATE level). A water jet at 40 m/s meets a series of curved vanes moving at 15 m/s, at 20° to the direction of vane motion. The vane outlet angle is 30°, measured from the direction opposite to the vane motion. Neglecting friction, find the vane inlet angle for shockless entry, the work done per unit weight and the hydraulic efficiency.

1. Inlet components: V_w1 = 40·cos20° = 37.59 m/s; V_f1 = 40·sin20° = 13.68 m/s.
2. Relative velocity at inlet: along the motion 37.59 − 15 = 22.59 m/s; across 13.68 m/s. Inlet blade angle θ = tan⁻¹(13.68/22.59) = 31.2°; V_r1 = √(22.59² + 13.68²) = 26.41 m/s.
3. No friction: V_r2 = 26.41 m/s at 30° backward. Its component along the motion is −26.41·cos30° = −22.87 m/s.
4. Outlet whirl: V_w2 = u + (−22.87) = −7.87 m/s, i.e. 7.87 m/s opposite to the vane motion, so the + sign applies.
5. Work per unit weight: (V_w1 + |V_w2|)·u/g = (37.59 + 7.87) × 15/9.81 = 69.5 m (J/N).
6. Efficiency: η = 2(V_w1 + |V_w2|)u/V₁² = 2 × 45.46 × 15/1600 = 0.852. Check: V_f2 = 26.41·sin30° = 13.20 m/s, V₂ = √(7.87² + 13.20²) = 15.37 m/s, and 1 − (15.37/40)² = 0.852. Answer: inlet vane angle ≈ 31.2°; work ≈ 69.5 J/N; η ≈ 85.2 %.

Common mistakes

• Using ρa(V − u) as the mass flow for a series of vanes; the whole jet ρaV is used by a wheel.
• Using ρaV(V − u) for a single moving plate; there it is ρa(V − u)².
• Getting the sign of the outlet whirl wrong. Draw the triangle and decide whether V_w2 points with or against u.
• Measuring blade angles from the wrong reference line. State whether the angle is from the direction of motion or from the opposite direction.
• Forgetting that a frictionless vane keeps the relative speed constant, not the absolute speed.
• Dividing work by V²/2g of the wrong velocity when finding efficiency; use the jet (inlet absolute) velocity.

For GATE ME

Expect jet-force numericals on fixed and moving plates and vanes, the condition u = V/2 for maximum efficiency and the corresponding efficiency, Pelton-type bucket calculations with a friction factor, and velocity-triangle problems giving blade angles, whirl components, work and efficiency. These questions overlap heavily with the turbines topic, so practise drawing the inlet and outlet triangles neatly with clear angle conventions.

Quick check

1. What is the force of a jet on a fixed semicircular cup that turns it back through 180°?
2. For a series of flat plates, at what blade speed is the efficiency maximum, and what is that maximum?
3. What is the mass flow rate striking a single plate moving away from the jet at u?
4. What is meant by shockless entry?
5. If the jet velocity doubles on a fixed plate, how does the force change?

Answers: 1. 2ρaV²; 2. u = V/2, 50 %; 3. ρa(V − u); 4. the blade inlet angle matches the direction of the relative velocity, so the fluid enters without impact; 5. it becomes four times as large.