Venturimeter

A Venturimeter is a differential-pressure flow meter that measures volumetric or mass flow rate in a pipe using a converging section, a short throat, and a diverging diffuser. It relies on continuity and Bernoulli’s equation: the fluid accelerates in the throat, pressure drops, and the pressure difference relates to the flow rate.

Geometry and Notation

Sections: 1 (upstream), 2 (throat). Pipe is assumed circular and full.
Diameters: \(d_1\) (upstream), \(d_2\) (throat), with \(\beta = d_2/d_1\).
Areas: \(A_1 = \dfrac{\pi d_1^2}{4}\), \(A_2 = \dfrac{\pi d_2^2}{4}\), so \(A_2/A_1 = \beta^2\).
Velocities: \(V_1, V_2\); pressures: \(p_1, p_2\); elevations: \(z_1, z_2\).
Fluid density: \(\rho_f\); gravitational acceleration: \(g\).
Flow rate: volumetric \(Q\), mass \(\dot{m} = \rho_f Q\).

Governing Equations

Continuity

\[ Q \;=\; A_1 V_1 \;=\; A_2 V_2 \quad\Rightarrow\quad V_1 \;=\; \beta^2 V_2. \]

Bernoulli Between 1 and 2 (Incompressible, No Shaft Work)

\[ \frac{p_1}{\rho_f g} + \frac{V_1^2}{2g} + z_1 \;=\; \frac{p_2}{\rho_f g} + \frac{V_2^2}{2g} + z_2 + h_L, \] where \(h_L\) is the head loss between taps (small for a well-designed Venturi). For a horizontal installation \((z_1 = z_2)\) and neglecting \(h_L\) (ideal), \[ \frac{p_1 - p_2}{\rho_f g} \;=\; \frac{V_2^2 - V_1^2}{2g} \;=\; \frac{V_2^2}{2g}\,(1 - \beta^4). \] Define the differential head in the flowing fluid: \[ h \;\equiv\; \frac{p_1 - p_2}{\rho_f g}. \] Then \[ V_2 \;=\; \sqrt{\frac{2 g\, h}{1 - \beta^4}} \quad\Rightarrow\quad Q_{\text{ideal}} \;=\; A_2 \sqrt{\frac{2 g\, h}{1 - \beta^4}}. \]

Real Flow: Discharge Coefficient

Accounting for viscous effects and non-idealities via \(C_d\) (typically \(0.97\!-\!0.99\) for Venturis), \[ Q \;=\; C_d\, A_2 \sqrt{\frac{2 g\, h}{1 - \beta^4}}. \]

Manometer Relationships

Differential Manometer with a Different Fluid

If a U-tube manometer uses a fluid of density \(\rho_m\) different from the line fluid (\(\rho_f\)), the manometer reading \(h_m\) (height difference) corresponds to \[ \Delta p \;=\; p_1 - p_2 \;=\; g\, h_m \,(\rho_m - \rho_f). \] Hence the equivalent head in the flowing fluid: \[ h \;=\; \frac{\Delta p}{\rho_f g} \;=\; h_m\!\left(\frac{\rho_m}{\rho_f} - 1\right). \] Combine with the discharge equation: \[ Q \;=\; C_d\, A_2 \sqrt{\frac{2 g\, h_m\!\left(\frac{\rho_m}{\rho_f} - 1\right)}{1 - \beta^4}}. \]

Manometer Filled with the Same Fluid

If \(\rho_m = \rho_f\) (e.g., piezometric taps connected to a gauge filled with the same liquid), then \[ \Delta p \;=\; \rho_f g\, h_m \quad\Rightarrow\quad h \;=\; h_m. \]

Elevation Difference Between Taps

For non-horizontal installations, include elevation: \[ \frac{p_1 - p_2}{\rho_f g} \;=\; h \;=\; h_{\text{read}} + (z_2 - z_1), \] where \(h_{\text{read}}\) is the manometer head referenced to the same datum. Many setups are designed so taps are at the same elevation.

Head Loss and Energy Considerations

Venturi head loss \(h_L\) is typically small compared to the measured differential head \(h\) due to the smooth diffuser recovery.
Power extracted is zero (no shaft work); the DP is a measuring signal, not a consumed energy like across a throttling valve.
Mass flow rate: \(\dot{m} = \rho_f Q\).

Design and Installation Notes

Straight runs: provide adequate upstream/downstream straight lengths (e.g., upstream \(10\text{–}20\,D\) after strong disturbances; downstream \(4\text{–}8\,D\)). Flow conditioners reduce requirements.
Tap locations: pressure taps should be at the plane sections of the upstream pipe and the throat, free from separation zones.
Cavitation check: ensure throat pressure stays above vapor pressure: \[ p_2 \;=\; p_1 - \rho_f g h \;>\; p_v \;\Rightarrow\; h \;<\; \frac{p_1 - p_v}{\rho_f g}.
Reynolds number: maintain sufficiently high \(Re\) (typically \(> 2\times10^4\)) to keep \(C_d\) stable; use calibrated \(C_d\) otherwise.
Compressible flows: for gases with \(M \lesssim 0.3\), incompressible treatment is acceptable. At higher compressibility, apply expansibility factor \(Y\) and standards-based correlations.

Worked Example (Water, Mercury Manometer)

Water at \(20^\circ\text{C}\) \((\rho_f \approx 1000\ \text{kg/m}^3)\) flows through a Venturi with \(d_1 = 0.10\ \text{m}\), \(d_2 = 0.050\ \text{m}\) (\(\beta = 0.5\)). A U-tube manometer filled with mercury \((\rho_m \approx 13{,}600\ \text{kg/m}^3)\) reads \(h_m = 0.25\ \text{m}\). Take \(C_d = 0.98\), \(g = 9.81\ \text{m/s}^2\). Find \(Q\).

Equivalent head in water:

\[ h \;=\; h_m\!\left(\frac{\rho_m}{\rho_f} - 1\right) \;=\; 0.25\,(13.6 - 1) \;=\; 3.15\ \text{m}. \]

Throat area:

\[ A_2 \;=\; \frac{\pi d_2^2}{4} \;=\; \frac{\pi (0.05)^2}{4} \;=\; 1.9635\times 10^{-3}\ \text{m}^2. \]

Denominator factor:

\[ 1 - \beta^4 \;=\; 1 - 0.5^4 \;=\; 0.9375. \]

Flow rate:

\[ Q \;=\; 0.98 \times 1.9635\times 10^{-3} \times \sqrt{\frac{2\times 9.81 \times 3.15}{0.9375}} \;\approx\; 1.56\times 10^{-2}\ \text{m}^3/\text{s} \;\approx\; 15.6\ \text{L/s}. \]

Common Pitfalls

Manometer conversion: using \(h = h_m\) even when the manometer fluid differs from the line fluid (must use \(h_m(\rho_m/\rho_f - 1)\)).
Ignoring elevation: if taps are at different heights, include \(z\)-terms.
Wrong \(\beta\): remember \(A_2/A_1 = \beta^2\) and the key factor is \(1 - \beta^4\).
Using uncalibrated \(C_d\): adopt a standard \(C_d\) only within its validity; otherwise calibrate.
Cavitation and air entrainment: throat pressure must not drop below vapor pressure; remove bubbles before manometer.

Quick Venturi Calculator

Enter pipe/throat diameters, manometer reading, densities, and \(C_d\). Assumes horizontal taps. For non-horizontal taps, adjust \(h\) accordingly.

Results

Key Formulas Summary

Continuity: \[ Q = A_1 V_1 = A_2 V_2,\quad V_1 = \beta^2 V_2,\quad \beta = \frac{d_2}{d_1}. \]
Bernoulli (ideal, horizontal): \[ h = \frac{p_1 - p_2}{\rho_f g} = \frac{V_2^2 - V_1^2}{2g} = \frac{V_2^2}{2g}(1 - \beta^4). \]
Discharge (real): \[ Q = C_d\, A_2 \sqrt{\frac{2 g\, h}{1 - \beta^4}}. \]
Manometer conversion (different fluid): \[ h = h_m\!\left(\frac{\rho_m}{\rho_f} - 1\right);\quad \Delta p = g h_m(\rho_m - \rho_f). \]
Mass flow: \[ \dot{m} = \rho_f\, Q. \]