Question

For compressible flow, which one of the following is true?

• A. \(\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{average}}>\ln\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{local}}\)
• B. \(\ln\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{average}}>\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{local}}\)
• C. \(\ln\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{average}}<\left( \dfrac{\rho_2}{\rho_1} \right)_{\text{local}}\)
• D. None of the above

Hint:

Remember: for positive numbers, \(\ln(x)<x - 1\) when \(x>1\).

Answer 1

The Correct Option is C

This follows from the logarithmic mean being less than the arithmetic mean, applicable in compressible flow calculations involving density changes.