Flow Measurement · Venturi Tube · Differential Pressure
Venturi Tube Flow Meter: Working Principle, Formula Derivation, Calculator and Applications
A complete guide to the Venturi tube flow meter: how Bernoulli's principle creates the differential pressure, full derivation of the Venturi flow formula, beta ratio and discharge coefficient explained, an interactive three-mode calculator, worked examples and ISO 5167-4 standard requirements.
The Venturi tube is one of the oldest and most reliable flow measurement devices in process engineering. Invented by Giovanni Battista Venturi in 1797 and refined for industrial flow measurement by Clemens Herschel in 1887, it uses a simple and elegant physical principle: when a fluid is forced through a constriction, its velocity increases and its pressure falls. By measuring this pressure difference, you can calculate the flow rate precisely.
Despite its age, the Venturi tube remains widely specified in modern process plants for one compelling reason: it has the lowest permanent pressure loss of all differential pressure flow meters. An orifice plate wastes 40-60% of the measured differential pressure as permanent loss. A Venturi tube recovers 85-95% of the pressure in the diverging section. Over a year of operation on a large gas line, this difference translates directly into energy savings that can easily exceed the entire cost of the instrument.
This guide explains the Venturi tube completely: the geometry, the Bernoulli derivation, the continuity equation, the discharge coefficient, the beta ratio, the three key calculation modes, and a worked example covering a water pipeline application. An interactive calculator handles all three calculation modes instantly. For context on where Venturi tubes fit in the broader landscape of flow measurement, see our guide on types of flow meters and how to choose the right one.
Venturi Tube Geometry: The Four Sections
A standard Venturi tube according to ISO 5167-4 consists of four distinct sections. Understanding the geometry is the foundation for understanding why the pressure changes the way it does.
Figure 1: Venturi tube cross-section. Fluid accelerates from the wide inlet (D1) through the converging cone into the narrow throat (D2). Pressure drops to a minimum at the throat (P2). The differential pressure ΔP = P1 - P2 is measured and used to calculate flow rate. In the diverging section, velocity decreases and most pressure is recovered.
| Section | Function | ISO 5167-4 angle | What happens to velocity | What happens to pressure |
|---|---|---|---|---|
| Inlet cylinder | Establishes the upstream pressure tap and provides a uniform flow profile | Cylinder, length = D1 | V1 (baseline velocity) | P1 (upstream pressure, HIGH) |
| Converging cone | Accelerates the fluid smoothly into the throat | 21° half-angle (standard) | Increasing rapidly | Falling rapidly |
| Throat | The minimum diameter section where pressure tap P2 is located | Short cylinder | V2 (maximum velocity) | P2 (minimum pressure, LOW) |
| Diverging cone | Decelerates the fluid and recovers most of the differential pressure | 7° to 15° half-angle | Decreasing back towards V1 | Rising back towards P1 |
The Venturi Flow Formula: Step-by-Step Derivation
The Venturi flow formula comes directly from two fundamental fluid mechanics principles applied together: Bernoulli's equation (energy conservation) and the continuity equation (mass conservation). Understanding this derivation removes all mystery from the formula and makes it possible to use it correctly in any situation.
Step 1: Bernoulli's Equation
Bernoulli's equation states that for an incompressible, frictionless, steady-state flow along a streamline, the total mechanical energy per unit volume is constant:
Where:
P1, P2 = static pressure at points 1 and 2 (Pa)
ρ = fluid density (kg/m³)
V1, V2 = fluid velocity at points 1 and 2 (m/s)
g = acceleration due to gravity (9.81 m/s²)
z1, z2 = elevation at points 1 and 2 (m)
For a horizontal Venturi tube: z1 = z2, so the elevation terms cancel.
Simplified: P1 + ½ρV1² = P2 + ½ρV2²
Rearranging: P1 - P2 = ½ρ(V2² - V1²) So: ΔP = ½ρ(V2² - V1²)
Step 2: Continuity Equation
The continuity equation states that mass flow is constant throughout the Venturi tube. For an incompressible fluid, this means volumetric flow is also constant:
Where
A1, A2 are the cross-sectional areas at points 1 and 2.
A = π × (D/2)² = π × D² / 4
Solving for V1: V1 = V2 × (A2/A1) = V2 × (D2/D1)²
Define beta ratio β = D2/D1 (throat diameter / inlet diameter)
Therefore: V1 = V2 × β²
Step 3: Substituting continuity into Bernoulli
Solving for V2: V2² = 2ΔP / (ρ(1 - β⁴)) V2 = sqrt(2ΔP / (ρ(1 - β⁴)))Volumetric flow rate (theoretical): Q_theoretical = A2 × V2 = (π/4 × D2²) × sqrt(2ΔP / (ρ(1 - β⁴)))
Step 4: Adding the discharge coefficient
The theoretical formula assumes perfectly frictionless, uniform flow. In reality, viscous effects, boundary layer separation and non-uniform velocity profiles reduce the actual flow slightly below the theoretical value. The discharge coefficient Cd corrects for this:
Where:
Q = volumetric flow rate (m³/s)
Cd = discharge coefficient (dimensionless, typically 0.984 to 0.995)
D2 = throat diameter (m)
ΔP = P1 - P2 = differential pressure (Pa)
ρ = fluid density (kg/m³)
β = D2/D1 = beta ratio (dimensionless)
Mass flow rate: qm = ρ × Q = Cd × (π/4 × D2²) × sqrt(2ρΔP / (1 - β⁴))
This is the complete formula used in all industrial Venturi tube flow calculations.
Beta Ratio and Discharge Coefficient: The Two Most Important Parameters
Beta Ratio (β)
The beta ratio is the ratio of the throat diameter to the inlet pipe diameter: β = D2/D1. It is one of the most important design parameters for a Venturi tube. ISO 5167-4 specifies that the beta ratio must be within the range 0.3 to 0.75 for the standard discharge coefficient values to apply.
| Beta ratio (β) | Throat diameter as % of pipe | Effect on ΔP | Effect on velocity | Practical consideration |
|---|---|---|---|---|
| 0.3 | 30% of D1 | Very large ΔP (high sensitivity) | Very high throat velocity | Risk of erosion, cavitation and high permanent pressure loss at small beta values. |
| 0.4 to 0.5 | 40-50% of D1 | Large ΔP (good signal) | High throat velocity | Good sensitivity with acceptable erosion and pressure loss. Preferred for high-accuracy applications. |
| 0.5 to 0.65 | 50-65% of D1 | Moderate ΔP | Moderate throat velocity | Most common design range. Good balance of signal strength, pressure loss and velocity. |
| 0.65 to 0.75 | 65-75% of D1 | Small ΔP (lower sensitivity) | Lower throat velocity | Large pipe applications, slurries, dirty fluids. Lower ΔP means more sensitive DP transmitter needed. |
Discharge Coefficient (Cd)
The discharge coefficient Cd accounts for the difference between ideal (frictionless, uniform) flow and real flow. For a machined classical Venturi tube (rough-cast, machined or rough-welded convergent section), ISO 5167-4 specifies:
- Classical Venturi with machined convergent section: Cd = 0.995
- Classical Venturi with rough-cast convergent section: Cd = 0.984
- Classical Venturi with rough-welded sheet-iron convergent section: Cd = 0.985
- Uncertainty on Cd: ±0.5% to ±1.5% depending on type and Reynolds number
For most industrial calculations, Cd = 0.98 is a safe and commonly used default value. For custody transfer metering, the Cd must be determined by in-situ calibration against a traceable standard.
Venturi Tube Flow Calculator: Three Calculation Modes
The calculator below handles all three standard calculation modes for Venturi tube applications. Select the mode you need using the tabs.
Worked Example: Water Pipeline Venturi Tube
A 200 mm diameter water pipeline needs a Venturi tube to measure flow up to 150 m³/h. The throat diameter is 100 mm. Fluid density is 998 kg/m³ (water at 20°C). Cd = 0.98. Find the differential pressure at maximum flow and check the beta ratio.
Step 2: Areas A1 = π/4 × (0.200)² = 0.031416 m² A2 = π/4 × (0.100)² = 0.007854 m²
Step 3: Throat velocity from continuity V2 = Q / A2 = 0.04167 / 0.007854 = 5.305 m/s
Step 4: Differential pressure from Bernoulli DP = (Q / (Cd × A2))² × ρ × (1 - β⁴) / 2 = (0.04167 / (0.98 × 0.007854))² × 998 × (1 - 0.5⁴) / 2 = (0.04167 / 0.007697)² × 998 × 0.9375 / 2 = (5.414)² × 468.6 = 29.31 × 468.6
ΔP = 13,733 Pa ≈ 13.7 kPa ≈ 1,400 mmWC at 150 m³/h
Select a DP transmitter with range 0-16 kPa (0-1,600 mmWC) to give 85% utilisation at maximum flow.
Venturi Tube vs Orifice Plate: When to Choose Each
| Parameter | Venturi Tube | Orifice Plate |
|---|---|---|
| Permanent pressure loss | 5 to 15% of measured ΔP (excellent) | 40 to 60% of measured ΔP (significant) |
| Turndown ratio | 4:1 to 5:1 | 3:1 to 4:1 |
| Installation cost | Higher (longer body, heavier) | Lower (simple plate, flanges only) |
| Maintenance | Very low (no sharp edges to erode) | Moderate (sharp edge degrades over time) |
| Dirty/slurry fluids | Better suited (no sharp orifice to block) | Poor (sharp edge traps debris, erodes rapidly) |
| Accuracy | ±0.5% to ±1% (Cd very stable) | ±0.5% to ±2% (Cd sensitive to edge sharpness) |
| Large diameter pipes | Preferred (lower energy loss matters most) | Available but high energy loss is costly |
| ISO standard | ISO 5167-4:2022 | ISO 5167-2:2022 |
| Best application | Large pipes, high flow, dirty fluids, long-term installation where energy savings justify higher initial cost | Clean fluids, shorter pipes, applications requiring easy replacement, lower capital budget |
ISO 5167-4 Standard Requirements for Venturi Tubes
ISO 5167-4:2022 is the international standard for measurement of fluid flow using Venturi tubes. Key requirements include:
- Beta ratio: 0.3 to 0.75
- Reynolds number range: Minimum Re = 2 × 10⁵ to 1 × 10⁶ depending on Venturi type and beta ratio
- Throat diameter D2: Minimum 50 mm (standard Venturi)
- Upstream straight pipe: 10 to 30D depending on upstream fittings
- Downstream straight pipe: Minimum 5D
- Surface finish: Machined convergent section for Cd = 0.995
- Pressure taps: Located at specific positions on the inlet cylinder and at the throat
Further Reading and External Resources
- ISO 5167-4:2022: Measurement of Fluid Flow by Means of Pressure Differential Devices: Venturi Tubes. The international standard governing Venturi tube design, installation, and discharge coefficients.
- Emerson: Differential Pressure Flow Measurement Technology. Technical resource on DP flow measurement including Venturi tubes, orifice plates and flow nozzles from a leading instrumentation supplier.
- Omega Engineering: Venturi Flow Meter Reference. Practical engineering reference on Venturi tube principles, sizing and industrial applications from a leading instrumentation manufacturer.
- AutomationForum: Venturi Tube Flow Calculator Reference. The original reference article that inspired this guide, with additional calculator tools for flow measurement applications.
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What we learn today
- A Venturi tube measures flow by accelerating fluid through a throat constriction, creating a pressure drop. The flow formula Q = Cd × A2 × sqrt(2ΔP / (ρ(1-β⁴))) comes directly from combining Bernoulli's equation (energy conservation) and the continuity equation (mass conservation). The (1-β⁴) term arises because V1 = V2 × β² from continuity, and velocity appears squared in Bernoulli.
- The beta ratio β = D2/D1 must be between 0.3 and 0.75 per ISO 5167-4. Lower beta gives higher ΔP (better signal) but higher velocity at the throat. Higher beta gives lower ΔP (needs more sensitive DP transmitter) but lower throat velocity. Most applications use β between 0.4 and 0.65.
- The discharge coefficient Cd corrects for real-world viscous effects. For a machined classical Venturi tube, ISO 5167-4 specifies Cd = 0.995. For rough-cast, Cd = 0.984. Use Cd = 0.98 as a safe default for preliminary calculations.
- The Venturi tube recovers 85-95% of the measured differential pressure in the diverging section. An orifice plate permanently loses 40-60% of ΔP. For large pipes and high flow rates where energy cost over time matters, this difference makes the Venturi tube the economically correct choice despite its higher initial cost.