Problem Statement
Calculate the numerical factor by which the actual pressure difference shown by the gauge of a pitot-tube should be multiplied to allow for compressibility when the value of the Mach number is 0.7. Take k = 1.4.
Given Data & Constants
- Mach number, \(M = 0.7\)
- Adiabatic index, \(k = 1.4\)
Solution
1. Formula for Compressibility Correction Factor
The correction factor (F) is the ratio of the compressible dynamic pressure to the incompressible dynamic pressure. It is derived from the isentropic flow equations.
2. Substitute Values and Calculate
We substitute the given values of M = 0.7 and k = 1.4 into the formula.
The numerical factor is approximately 1.136.
Explanation of the Factor
When a fluid is moving at high speeds (typically when the Mach number is greater than 0.3), its density changes as it is brought to a stop at the tip of a pitot tube. The simple Bernoulli equation, which assumes constant density, becomes inaccurate.
The calculated factor of **1.136** is a correction that accounts for this compression. It means that at Mach 0.7, the true dynamic pressure is about **13.6% higher** than the value that would be calculated assuming the air is incompressible. To get the correct dynamic pressure, the pressure difference measured by the pitot tube gauge must be multiplied by this factor.
