Water Flow Measurement

Overview

Engineering reference data for Water Flow Measurement in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Common Water Flow Measurement Devices

The USBR manual describes several standard devices for measuring irrigation water. Key types include:

• Weirs (sharp-crested, broad-crested): Structures over which water flows, with flow rate calculated from upstream head.
• Flumes (Parshall, Cutthroat, HSMB): Constricted channel sections that accelerate flow, creating a unique head-discharge relationship.
• Orifices (submerged, unsubmerged): Precisely sized openings in a bulkhead or pipe, with flow proportional to the square root of head.
• Meters (propeller, electromagnetic, ultrasonic): Inline instruments that measure velocity or volume directly.

Key Formulas for Primary Devices

The fundamental equations for calculating discharge ($Q$) from head ($H$) for common devices are:

For a rectangular weir (Thomson's formula): $Q = C_d \frac{2}{3} \sqrt{2g} \cdot b \cdot H^{3/2}$

For a Parshall flume (free-flow discharge): $Q = C \cdot H^n$ where $C$ and $n$ are constants specific to the flume size.

For a submerged circular orifice: $Q = C_d A \sqrt{2gH}$ where $A$ is the orifice area.

Selection Criteria

Choosing the appropriate device depends on several factors:

1. Flow Range: The device must handle expected minimum and maximum flows.
2. Accuracy Requirements: Applications range from ±2% (high-value water) to ±10% (general irrigation).
3. Head Loss: Available hydraulic gradient; flumes typically cause less head loss than weirs.
4. Sediment & Debris Load: Orifices are prone to clogging; flumes perform better in sediment-laden canals.
5. Maintenance & Inspection Access: Devices require periodic cleaning and calibration checks.

Definitions of Critical Terms

• Head ($H$): The vertical distance from the water surface upstream of the device to a specific measurement point (e.g., weir crest or flume throat).
• Discharge Coefficient ($C_d$): An empirical factor that accounts for energy losses and the contraction of the flow stream. It is device-specific and varies with Reynolds number.
• Free Flow: Flow condition where the downstream water level does not affect the discharge measurement.
• Submerged Flow: Flow condition where the downstream water level is high enough to reduce the discharge through the device for a given upstream head.

References

• Original Source