Question

Consider a conventional subsonic fixed-wing airplane. \(e\) is the Oswald efficiency factor and \(AR\) is the aspect ratio. Corresponding to the minimum \(\left(\frac{C_D}{C_L^{3/2}}\right)\), which of the following relations is true?

• A. \(\frac{C_D}{C_L^2} = \frac{1}{\pi e AR}\)
• B. \(\frac{C_D}{C_L^2} = \frac{4}{3 \pi e AR}\)
• C. \(\frac{C_D}{C_L} = \frac{1}{\pi e AR}\)
• D. \(\frac{C_D}{\sqrt{C_L}} = \frac{1}{\sqrt{\pi e AR}}\)

Hint:

To minimize drag-to-lift ratios in subsonic flight, optimize the aspect ratio and the Oswald efficiency factor to achieve better aerodynamic performance.

Answer 1

The Correct Option is B

Step 1: Understanding the given parameters.
In the context of subsonic flight, the drag coefficient \(C_D\) and lift coefficient \(C_L\) are related to the aerodynamic efficiency of the aircraft. We are asked to minimize the ratio \(\frac{C_D}{C_L^{3/2}}\), which involves considering the effects of the Oswald efficiency factor \(e\) and the aspect ratio \(AR\).
Step 2: Equation for drag and lift coefficients.
For a conventional subsonic fixed-wing airplane, the drag coefficient \(C_D\) at the minimum \(\frac{C_D}{C_L^{3/2}}\) is given by the relationship: \[ \frac{C_D}{C_L^2} = \frac{4}{3 \pi e AR}. \] This equation is derived from aerodynamic theory and takes into account the effects of the aircraft’s geometry and the Oswald efficiency factor.
Step 3: Conclusion.
The correct option is (B), where the relationship \(\frac{C_D}{C_L^2} = \frac{4}{3 \pi e AR}\) holds true at the minimum value of \(\frac{C_D}{C_L^{3/2}}\).