Derive the theoretical flow equation for a pitot static tube.
Question
Derive the theoretical flow equation for a pitot static tube.
Solution
The Pitot-Static tube, also known as Pitot tube, is used to measure the fluid flow velocity. The principle of the Pitot tube is based on the Bernoulli Equation, which states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure.
Here is the step-by-step derivation of the theoretical flow equation for a Pitot tube:
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Bernoulli's Equation:
The Bernoulli's equation for fluid flow is given by:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Here, P is the pressure, ρ is the fluid density, v is the fluid velocity, g is the acceleration due to gravity, and h is the height.
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Applying Bernoulli's Equation to Pitot Tube:
In the case of a Pitot tube, the fluid is brought to rest at the tube's opening. This means that the velocity v₂ is zero. Also, the height difference h₁ - h₂ is negligible. So, the Bernoulli's equation simplifies to:
P₁ + ½ρv₁² = P₂
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Deriving the Flow Velocity:
We can rearrange this equation to solve for the fluid velocity v₁:
v₁ = sqrt((2(P₂ - P₁))/ρ)
This is the theoretical flow equation for a Pitot tube. It shows that the fluid velocity is proportional to the square root of the difference in pressure between the two points.
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