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

Calculates flow rate through an orifice accounting for real fluid contraction and velocity losses.

Q = C_d A \sqrt{2gh}

Variables

Symbol	Description	Unit
$Q$	Actual flow rate (discharge)	$m^{3} / s$
$C_{d}$		-
$C_{c}$	Coefficient of contraction ( $A_{jet}/A_{orifice}$ )	-
$C_{v}$	Coefficient of velocity ( $V_{actual}/V_{theoretical}$ )	-
$A$	Area of the orifice opening	$m^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$
$h$	Head acting on the center of the orifice	m

Venturi Meters

A Venturi meter is a converging-diverging pipe section used to measure flow with high accuracy and low permanent head loss.

Venturi Meter Equation

Calculates flow rate through a Venturi meter based on pressure head differences.

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

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$C_{d}$	Coefficient of discharge (typically $0.95$ to $0.99$ )	-
$A_{1}$	Inlet area of the pipe	$m^{2}$
$A_{2}$	Throat area of the meter	$m^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$
$h_{1}$	Piezometric head at the inlet	m
$h_{2}$	Piezometric head at the throat	m

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: $Q = C_d A \sqrt{2gh}$ .

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 Weirs

Rectangular weirs are sharp-crested overflow structures whose discharge varies with the crest length and the head raised to the $3/2$ power.

Rectangular Weir Equation

Theoretical discharge equation for a sharp-crested rectangular weir.

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

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$C_{d}$	Coefficient of discharge	-
$g$	Acceleration due to gravity	$m / s^{2}$
$L$	Crest length of the weir	m
$H$	Head above the weir crest	m

Francis Suppressed Weir Formula

Empirical equation for a suppressed rectangular weir (no end contractions).

Q = 1.84 L H^{3/2}

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$L$	Crest length of the weir	m
$H$	Head above the weir crest	m

Francis Contracted Weir Formula

Empirical equation for a rectangular weir with two end contractions.

Q = 1.84 (L - 0.2H) H^{3/2}

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$L$	Crest length of the weir	m
$H$	Head above the weir crest	m

Triangular (V-Notch) Weirs

Triangular V-notch weirs are best for measuring small flows with high accuracy because discharge is highly sensitive to head.

Triangular (V-Notch) Weir Equation

Calculates flow rate through a V-notch weir with any notch angle.

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

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$C_{d}$	Coefficient of discharge	-
$g$	Acceleration due to gravity	$m / s^{2}$
	V-notch angle	degrees
$H$	Head above the notch vertex	m

90° V-Notch Weir Equation (SI Units)

Simplified empirical equation for a 90-degree V-notch weir in metric units.

Q = 1.4 H^{5/2}

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$H$	Head above the notch vertex	m

90° V-Notch Weir Equation (English Units)

Simplified empirical equation for a 90-degree V-notch weir in US customary units.

Q = 2.5 H^{5/2}

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$f t^{3} / s$
$H$	Head above the notch vertex	ft

Trapezoidal (Cipolletti) Weirs

Trapezoidal Cipolletti weirs are designed to compensate for end contractions automatically. The standard side slope is 1H:4V.

Trapezoidal (Cipolletti) Weir Equation

Empirical equation for a Cipolletti weir, compensating for end contractions.

Q = 1.86 L H^{3/2}

Variables

Symbol	Description	Unit
$Q$	Flow rate (discharge)	$m^{3} / s$
$L$	Crest length of the weir	m
$H$	Head above the weir crest	m

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.

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