Flow Measurement: Pitot Tubes, Venturi Meters, Orifices and Weirs

NCEES FE Civil spec: Fluid Mechanics, A. Flow measurement. FE Reference Handbook: Fluid Mechanics chapter, fluid flow measurement.

Flow measurement devices turn a measured pressure or head difference into a velocity or a flow rate. Every one of them is an application of the Bernoulli equation with a correction coefficient for real losses. On the FE Civil exam you will see Pitot tubes, Venturi meters, orifices and weirs, often with a manometer reading as the input.

What the NCEES spec covers

Section A, "Flow measurement", covers devices for closed conduits (Pitot tube, Venturi meter, orifice meter) and for free-surface flow (orifices in tanks, sharp-crested weirs), plus reading differential manometers.

Key concepts

Pitot tube. Measures the difference between stagnation and static pressure at a point, giving the local velocity.
Venturi and orifice meters force the flow through a smaller area. Continuity says velocity rises, Bernoulli says pressure drops, and the pressure drop gives Q. A Venturi recovers most of the pressure; an orifice plate loses much more.
Coefficients. The velocity coefficient Cv (about 0.95 to 0.99 for a Venturi) and discharge coefficient Cd (about 0.6 for a sharp-edged orifice) account for friction and contraction.
Differential manometer. A gauge liquid of specific gravity SGm deflected h under a flowing fluid of SGf indicates a head difference of h(SGm/SGf − 1) in the flowing fluid.
Weirs relate flow to the head H above the crest. Rectangular weirs vary with H^1.5, triangular (V-notch) weirs with H^2.5.

Core formulas

Pitot: V = C 2 g Δ h Orifice: Q = C_{d} A 2 g h

Venturi: Q = C_{v} A_{2} \frac{2 g Δ h}{1 - ( A _{2} / A _{1} ) ^{2}}, Δ h = (\frac{p _{1}}{γ} + z_{1}) - (\frac{p _{2}}{γ} + z_{2})

Manometer: Δ h = h_{m} (\frac{S G _{m}}{S G _{f}} - 1)

Rectangular weir: Q = \frac{2}{3} C_{d} b 2 g H^{3/2} V-notch: Q = \frac{8}{15} C_{d} tan (\frac{θ}{2}) 2 g H^{5/2}

Where it is in the FE Reference Handbook

The Fluid Mechanics chapter has a fluid flow measurement section covering the Pitot tube, Venturi meters and orifices, along with manometers under fluid statics. Weir equations for open channels are in the water resources part of the Civil Engineering chapter, often in a form with the coefficient already combined into a constant.

Worked example

Water flows through a horizontal Venturi meter with a 300 mm inlet and a 150 mm throat. A mercury manometer (SG 13.6) connected between inlet and throat shows a 200 mm deflection. With Cv = 0.98, find the flow rate.

Step 1: head difference in water.

Δ h = 0.200 (13.6 - 1) = 2.52 m of water

Step 2: areas.

A_{2} = \frac{π ( 0.15 ) ^{2}}{4} = 0.01767 m^{2} (\frac{A _{2}}{A _{1}})^{2} = (\frac{0.15}{0.30})^{4} = 0.0625

Step 3: flow rate.

Q = 0.98 (0.01767) \frac{2 ( 9.81 ) ( 2.52 )}{1 - 0.0625} = 0.98 (0.01767) (7.262) = 0.126 m^{3} / s

The ideal (frictionless) flow would be 0.128 m³/s; Cv trims about 2%. Because the meter is horizontal, the elevation terms cancel and the manometer reading is the whole story.

Common mistakes

Using the manometer deflection directly as head, without the (SGm/SGf − 1) factor.
Squaring the diameter ratio instead of raising it to the fourth power for (A₂/A₁)².
Mixing up Cv and Cd, or applying a coefficient twice.
Using H measured at the crest instead of upstream of the drawdown for a weir.
Forgetting elevation differences when the meter is inclined and pressures are read by gauges.

See the Venturi meter example for another case.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A Pitot tube in water measures a head difference of 0.20 m. Neglecting losses, the flow velocity is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
Water discharges through a 100-mm-diameter orifice under a 3.0-m head. If Cd = 0.62, the discharge is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
A rectangular sharp-crested weir has b = 1.2 m, H = 0.40 m and Cd = 0.62. Using Q = (2/3)·Cd·b·√(2g)·H^(3/2), Q is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
An ideal horizontal Venturi meter has D1 = 0.20 m and D2 = 0.10 m. The pressure difference is 30 kPa for water. The discharge is most nearly: See the worked solution or practice the Fluids & Hydraulics deck

Related Calculators and Examples

Frequently Asked Questions

How does a Venturi meter measure flow?

It narrows the pipe to a throat. Continuity makes the velocity rise there and Bernoulli makes the pressure drop. The measured pressure difference, together with the inlet and throat areas, gives the flow rate.

How do I convert a mercury manometer reading to head of water?

Multiply the deflection by (SG of mercury / SG of water - 1), which is 13.6 - 1 = 12.6. A 100 mm mercury deflection corresponds to 1.26 m of water.

What is the discharge coefficient of an orifice?

It is the ratio of actual to ideal flow. For a sharp-edged orifice it is typically about 0.6, mostly because the jet contracts to a smaller area just downstream of the opening.

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