Bernoulli's equation and flow measurement

Why it matters

Bernoulli's equation is the working tool for carburettor venturis, intake-manifold flow, fuel injection pressure, siphons and every differential-pressure flow meter on a test rig. Used correctly, with its assumptions respected and losses added where needed, it turns a pressure reading into a velocity or flow rate in two lines.

Key ideas

• Origin. Integrating Euler's equation of motion along a streamline gives Bernoulli's equation. Each term is an energy per unit weight (a "head", in metres): pressure head p/(ρg), velocity head V²/(2g) and datum head z.
• Assumptions. Steady flow, incompressible fluid, negligible viscosity (no friction loss), along a single streamline (or between any two points if the flow is irrotational), and no shaft work or heat transfer between the two points.
• Real flows. Add a head-loss term h_L for friction and fittings, a pump head H_p and a turbine head H_t, giving the energy equation. The kinetic-energy correction factor α (1 for uniform, 2 for laminar pipe flow, about 1.05 for turbulent) corrects V²/2g when the profile is non-uniform.
• Energy and hydraulic grade lines. The total-energy line (TEL) is at p/(ρg) + V²/(2g) + z; the hydraulic grade line (HGL) is at p/(ρg) + z, one velocity head lower. The TEL can only fall in the flow direction unless a pump adds energy. Where the HGL drops below the pipe, the pressure is below atmospheric.
• Static, dynamic and stagnation pressure. Static p is measured by a wall tap. Dynamic pressure is ½ρV². Stagnation pressure p₀ = p + ½ρV² is reached where the flow is brought to rest without loss, at the nose of a Pitot tube.
• Flow-measurement devices.
  • Venturi meter: a converging cone, throat and gentle diverging cone. Low permanent loss; discharge coefficient C_d ≈ 0.95–0.99.
  • Orifice meter: a sharp-edged plate. Cheap and compact but has a large permanent pressure loss. The jet contracts to a vena contracta downstream, so C_d is much lower (about 0.6–0.65).
  • Flow nozzle: between the two in cost and loss.
  • Pitot-static tube: measures stagnation minus static pressure at a point, so it gives local velocity, not flow rate directly.
  • Values of C_d depend on geometry and Reynolds number — take them from the meter's data or the standard.
• Orifice coefficients. C_c = area of the vena contracta / orifice area; C_v = actual / ideal jet velocity; C_d = C_c·C_v.
• Torricelli's theorem. A jet from a small orifice at depth H below the free surface of a large tank leaves at V = √(2gH).
• Manometer reading to head. For a differential U-tube with gauge liquid relative density S_m under a flowing liquid of relative density S, the piezometric head difference is h = x·(S_m/S − 1).
• Cavitation limit. Bernoulli can predict an absolute pressure below the vapour pressure, for example at a siphon summit or venturi throat. In reality the liquid then vaporises and the flow is limited.

Formulas

p₁/(ρg) + V₁²/(2g) + z₁ = p₂/(ρg) + V₂²/(2g) + z₂ Bernoulli, ideal flow; p (Pa), ρ (kg/m³), V (m/s), z elevation (m), g = 9.81 m/s². Each term in metres.

p₁/(ρg) + α₁·V₁²/(2g) + z₁ + H_p = p₂/(ρg) + α₂·V₂²/(2g) + z₂ + H_t + h_L Energy equation; H_p pump head, H_t turbine head, h_L head loss (all m).

Q = C_d·A₁·A₂·√(2gh) / √(A₁² − A₂²) Venturi or orifice meter; Q (m³/s), A₁ inlet area, A₂ throat or orifice area (m²), h piezometric head difference (m of flowing fluid), C_d discharge coefficient.

h = x·(S_m/S − 1) x manometer deflection (m); S_m, S relative densities of gauge and flowing liquids.

V = C_v·√(2·Δp/ρ) Pitot-static tube; Δp = p₀ − p (Pa); C_v ≈ 1 for a good probe.

V = √(2gH), Q = C_d·a·√(2gH) Torricelli: jet from an orifice of area a at head H (m) in a large open tank.

Worked examples

Example 1 (standard) — venturi meter with a mercury manometer. Given: horizontal venturi, inlet 300 mm, throat 150 mm, carrying water. A mercury differential manometer reads x = 200 mm. C_d = 0.98. Find Q.

1. Head difference: h = x·(S_m/S − 1) = 0.2 × (13.6 − 1) = 2.52 m of water.
2. Areas: A₁ = π × 0.3²/4 = 0.070686 m²; A₂ = π × 0.15²/4 = 0.017671 m².
3. √(2gh) = √(2 × 9.81 × 2.52) = 7.0315 m/s.
4. Ideal Q = A₁A₂√(2gh)/√(A₁² − A₂²) = 0.12833 m³/s.
5. Actual Q = 0.98 × 0.12833 = 0.12577 m³/s. Answer: Q ≈ 0.126 m³/s (126 L/s).

Example 2 (GATE level) — siphon and summit pressure. Given: a 50 mm siphon draws water from a large open tank. The outlet discharges to atmosphere 3 m below the tank's free surface, and the summit is 1.5 m above the free surface. Neglect losses; atmospheric pressure is 101.325 kPa. Find Q and the absolute pressure at the summit.

1. Bernoulli from the free surface (1) to the outlet (2), both atmospheric, V₁ ≈ 0: V₂ = √(2g × 3) = √58.86 = 7.672 m/s.
2. Q = (π × 0.05²/4) × 7.672 = 1.9635 × 10⁻³ × 7.672 = 0.01506 m³/s.
3. Summit S: the velocity there equals V₂ (same diameter), so V²/2g = 3 m.
4. Bernoulli from 1 to S: 0 + 0 + 0 = p_S/(ρg) + 3 + 1.5 → p_S/(ρg) = −4.5 m.
5. p_S = −4.5 × 9810 = −44 145 Pa gauge → absolute = 101 325 − 44 145 = 57 180 Pa. Answer: Q ≈ 15.1 L/s; summit pressure ≈ 57.2 kPa absolute (−44.1 kPa gauge). If the summit were raised until this fell below the vapour pressure, the siphon would stop working.

Example 3 (quick) — Pitot-static tube in air. A water manometer on a Pitot-static tube in an air stream (ρ_air = 1.2 kg/m³) reads 50 mm. Δp = (1000 − 1.2) × 9.81 × 0.05 = 489.9 Pa; V = √(2 × 489.9/1.2) = 28.6 m/s.

Common mistakes

• Writing h = x·S_m instead of h = x·(S_m/S − 1) for a differential manometer, which overestimates the head.
• Using diameters instead of areas in the venturi formula, or forgetting the √(A₁² − A₂²) term (approach velocity).
• Using air density in the manometer equation and water density in the velocity formula — or the reverse.
• Applying Bernoulli across a pump, a turbine or a sudden expansion without adding the work or loss terms.
• Mixing gauge and absolute pressure when checking for cavitation.
• Assuming an orifice meter's C_d is close to 1.

For GATE ME

Expect venturi or orifice meter discharge with a manometer, Pitot-tube velocity, tank draining through an orifice (including time to empty), siphon summit pressure, and energy-equation problems with a pump or turbine and a given head loss. Conceptual MCQs ask about assumptions, the TEL and HGL, and why the venturi has lower loss than the orifice meter.

Quick check

1. Name the three heads in Bernoulli's equation.
2. What does a Pitot-static tube measure directly?
3. Why does an orifice meter have a lower C_d than a venturi meter?
4. Water leaves a small orifice 5 m below the free surface. What is the ideal jet velocity?
5. A mercury–water manometer on a venturi reads 100 mm. What is the head difference in metres of water? Answers: 1. Pressure, velocity and datum (potential) head. 2. Stagnation minus static pressure, i.e. dynamic pressure. 3. Because of jet contraction at the vena contracta and the losses at the sharp edge. 4. √(2 × 9.81 × 5) = 9.90 m/s. 5. 0.1 × 12.6 = 1.26 m.