Flow Measurement - Theory & Concepts

Flow Measurement

Devices and methods for measuring flow rate, including orifices, venturi meters, and weirs.

Overview

Accurate measurement of fluid flow is essential in engineering for billing, process control, and environmental monitoring. Devices are categorized by whether they measure flow rate (

Q

) or velocity (

V

Orifices

An orifice is an opening in a tank or pipe through which fluid flows.

Orifice Discharge Equation

Based on Torricelli's Theorem, but modified for real fluid effects.

Orifice Discharge Equation

An orifice is an opening in a tank or pipe through which fluid flows.

Q = C_d A \sqrt{2gh}

Variables

Symbol	Description	Unit
$C_d$	Coefficient of discharge	$C_d = C_c C_v$
$C_c$	Coefficient of contraction	$A_{jet}/A_{orifice}$
$C_v$	Coefficient of velocity	$V_{actual}/V_{theoretical}$
$h$	Head acting on the orifice center.	-

Venturi Meters

A converging-diverging pipe section used to measure flow. High accuracy, low head loss.

Venturi Meters

A converging-diverging pipe section used to measure flow. High accuracy, low head loss.

Q = C_d A_1 A_2 \sqrt{\frac{2g(h_1-h_2)}{A_1^2 - A_2^2}}

Variables

Symbol	Description	Unit
$A_1$	Inlet Area	-
$A_2$	Throat Area	-
$h_1, h_2$	Piezometric heads at inlet and throat.	-

Venturi Meter Simulation

Inlet Diameter (D1)0.20 m

Throat Diameter (D2)0.10 m

Head Difference (h)0.50 m

Discharge Coefficient (C_d)0.98

Results

Inlet Area (A1):0.0314 m²

Throat Area (A2):0.0079 m²

Discharge (Q):24.9 L/s (0.0249 m³/s)

Nozzles and Tubes

Other devices to control and measure flow based on orifices.

Nozzles and Short Tubes

These are essentially extended orifices used to direct jets or measure discharge.

Standard Short Tube: A pipe whose length is 2.5 to 3 times its diameter, attached to an orifice. It flows full at the exit. Compared to a standard sharp-edged orifice, a short tube has a higher coefficient of discharge ( $C_d \approx 0.82$ ) but a lower coefficient of velocity.
Nozzles: Converging tubes attached to the end of pipes. They gradually reduce the cross-sectional area to increase the velocity of the exiting jet ( $C_d \approx 0.95$ to $0.99$ ).

The discharge equation is the same form as the orifice equation:

Q = C_d A \sqrt{2gh}

Weirs

Overview of Weirs

Weirs are overflow structures built across open channels to measure discharge. The flow rate is related to the head (

H

) above the weir crest.

Rectangular Weir

Rectangular Weir

Other devices to control and measure flow based on orifices.

Q = \frac{2}{3} C_d \sqrt{2g} L H^{3/2}

Francis Formula (Suppressed Weir - Full width):
$Q = 1.84 L H^{3/2}$
Contracted Weir (End contractions):
$Q = 1.84 (L - 0.2H) H^{3/2}$

Triangular (V-Notch) Weir

Best for measuring small flows with high accuracy.

Triangular (V-Notch) Weir

Best for measuring small flows with high accuracy.

Q = \frac{8}{15} C_d \sqrt{2g} \tan\left(\frac{\theta}{2}\right) H^{5/2}

For 90° V-Notch:
$Q = 1.4 H^{5/2} \quad (\text{SI Units})$
$Q = 2.5 H^{5/2} \quad (\text{English Units})$

Trapezoidal (Cipolletti) Weir

Designed to compensate for end contractions automatically. Side slope is 1H:4V.

Trapezoidal (Cipolletti) Weir

Designed to compensate for end contractions automatically. Side slope is 1H:4V.

Q = 1.86 L H^{3/2}

Key Takeaways

An orifice provides a simple way to measure flow based on the head driving the fluid.
Real fluid behavior requires empirical coefficients to correct for the contraction of the jet ( $C_c$ ) and friction losses ( $C_v$ ).
Orifices are easy to install but cause significant permanent energy (head) loss in a pipe system.
Nozzles and Short Tubes: Act as extended orifices with different empirical coefficients, generally providing a higher coefficient of discharge than a simple sharp-edged orifice.
A Venturi meter measures discharge in pipes by deliberately restricting the flow area to increase velocity and decrease pressure.
By measuring the pressure difference between the normal pipe and the throat, discharge can be calculated with high precision.
Because of its gradual expansion section, a Venturi meter recovers most of the pressure head, making it highly efficient compared to an orifice meter.
Weirs are the primary structures used to measure discharge in open channels. Flow rate is a function of the head ( $H$ ) over the crest.
For rectangular weirs, discharge is proportional to $H^{3/2}$ . The Francis formula is commonly used, with adjustments made if the weir has end contractions.
For V-notch (triangular) weirs, discharge is proportional to $H^{5/2}$ . They are highly sensitive to small changes in head, making them ideal for measuring low flows accurately.

PreviousFlow in Pipes: Systems & Networks - Examples & Applications

Quiz Me

NextFlow Measurement - Examples & Applications