Pitot Tubes

Overview

Engineering reference data for Pitot Tubes 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	—

Key Formulas

The following equations are fundamental to pitot tube operation and flow measurement:

$$p + \frac{1}{2} \rho v^2 + \rho g h = \text{constant along a streamline} \quad (1)$$ $$p_1 + \frac{1}{2} \rho v_1^2 = p_2 \quad (3)$$ $$v_1 = \sqrt{\frac{2 (p_2 - p_1)}{\rho}} = \sqrt{\frac{2 \Delta p}{\rho}} \quad (4)$$ $$v_1 = c \sqrt{2 g \Delta h} \quad (5)$$ $$v_a = \frac{\sum v_n}{n} = \sqrt{\frac{2 g \sum h_n}{n}} \quad (6)$$ $$q = v_a A \quad (7)$$

Pressure Definitions

Each term in the Bernoulli equation represents a specific type of pressure:

• Static Pressure ($p$): The pressure relative to the moving fluid, measurable through a flat opening parallel to the flow.
• Dynamic Pressure ($\frac{1}{2} \rho v^2$): The pressure component due to the fluid's kinetic energy.
• Hydrostatic Pressure ($\rho g h$ or $\gamma h$): The pressure component due to the fluid's elevation, where $\gamma = \rho g$ is the specific weight.

Measurement & Application

The pitot tube measures the difference between total (stagnation) pressure at the tip and static pressure from the side ports. This pressure difference ($\Delta p$ or $\Delta h$) is used with equations (4) or (5) to calculate the point velocity in the free stream.

For flow rate determination in a conduit, the point velocity-area method is used. Point velocities are measured across a traverse, and the average velocity ($v_a$) is calculated using equation (6). The volume flow ($q$) is then found from equation (7).

Practical Guidance:

• For round ducts larger than 10 inches (254 mm), a 10-point traverse is recommended at specific radial positions.
• For smaller ducts, the average velocity can be estimated as 81% of the centerline velocity.
• Pitot tubes are not suited for low-velocity flows due to the small dynamic pressure differential, which leads to inaccurate readings.

References

• Original Source