Bernoulli's Equation and Pitot Tube Analysis
1. Bernoulli's Equation (General Form)
Bernoulli's equation is derived from the conservation of energy for incompressible, non-viscous, steady flow
of fluid:
P/(rhog) + v^2/(2g) + z = constant
Where:
- P = pressure (Pa)
- rho = fluid density (kg/m³)
- v = fluid velocity (m/s)
- g = acceleration due to gravity (9.81 m/s^2)
- z = elevation head (m)
2. Bernoulli's Equation Between Two Points
P1/(rhog) + v1^2/(2g) + z1 = P2/(rhog) + v2^2/(2g) + z2
If the flow is horizontal (z1 = z2):
P1/(rhog) + v1^2/(2g) = P2/(rhog) + v2^2/(2g)
3. Pitot Tube Analysis
A Pitot tube measures the velocity of fluid flow by comparing stagnation pressure and static pressure.
Key Concepts:
- Stagnation Point: Fluid is brought to rest (velocity = 0)
- Static Pressure: Measured from side port, unaffected by velocity
- Dynamic Pressure: Due to fluid motion
4. Pitot Tube Equation Derivation
Apply Bernoulli's equation between:
Bernoulli's Equation and Pitot Tube Analysis
- Point 1: Free-stream (fluid moving at velocity v), pressure P
- Point 2: Stagnation point (velocity = 0), pressure P0
P/(rhog) + v^2/(2g) = P0/(rhog)
Rearranged:
v = sqrt[2(P0 - P)/rho]
5. Head Form
In terms of pressure head:
v = sqrt(2g(p0 - p))
If pressure is measured in terms of head h:
v = sqrt(2gh)
Where h = (P0 - P)/(rhog) is the velocity head.
6. Practical Notes
- Ideal for measuring aircraft speed, wind tunnel testing, pipe flow velocity
- Sensitive to alignment with flow direction
- Can be affected by viscosity and compressibility at high speeds
7. Types of Pitot Tubes
- Simple Pitot Tube: Measures stagnation pressure only
- Pitot-Static Tube: Combines static and stagnation ports for velocity calculation
- Multi-hole Tubes: Used in 3D flow measurements (e.g., 5-hole, 7-hole)