# Step-by-Step Solution: Angle of Attack for Maximum \( L/D \) of a NACA 4416 Airfoil ## Given Data - **Airfoil:** NACA 4416 - **Aspect Ratio (\( AR \)):** \( 9.375 \) - **Oswald Efficiency (\( e \)):** \( .92 \) - **Zero-lift angle (\( \alpha_{L=} \)):** \( -4^\circ \) - **No drag due to air separation** (assume only induced and profile drag) ## Step 1: Find the Lift and Drag Coefficients ### Lift Coefficient (\( C_L \)) For a cambered airfoil: \[ C_L = a_ (\alpha - \alpha_{L=}) \] Where: - \( a_ \) = 2D lift slope (\( \approx 2\pi \) per radian for thin airfoils, NACA 4416 is close) - \( \alpha \) = angle of attack (in radians) - \( \alpha_{L=} \) = zero-lift angle (in radians) ### Drag Coefficient (\( C_D \)) The drag coefficient for a finite wing: \[ C_D = C_{D,} + \frac{C_L^2}{\pi e AR} \] Where: - \( C_{D,} \) = profile drag coefficient (parasitic drag, typically \( \approx .008 \) for NACA 4416) - \( e \) = Oswald efficiency number - \( AR \) = aspect ratio ## Step 2: Find Condition for Maximum \( L/D \) \[ \frac{L}{D} = \frac{C_L}{C_D} \] To maximize \( L/D \), set the derivative of \( \frac{C_L}{C_D} \) with respect to \( C_L \) to zero, or recall that: \[ \left( \frac{L}{D} \right)_{max} \text{ occurs when } C_{D,} = \frac{C_L^2}{\pi e AR} \] So, \[ C_L = \sqrt{C_{D,} \cdot \pi e AR} \] ## Step 3: Plug in the Numbers Assume \( C_{D,} = .008 \) (typical for NACA 4416). So, \[ C_L = \sqrt{.008 \cdot \pi \cdot .92 \cdot 9.375} \] Calculate step by step: 1. \( \pi \cdot .92 = 2.890 \) 2. \( 2.890 \cdot 9.375 = 27.093 \) 3. \( .008 \cdot 27.093 = .2167 \) 4. \( \sqrt{.2167} \approx .465 \) So, \[ C_L \approx .465 \] ## Step 4: Find the Angle of Attack Corresponding to \( C_L \) Recall: \[ C_L = a_ (\alpha - \alpha_{L=}) \] Where \( a_ = 2\pi \approx .11 \) per degree (since \( 2\pi \) per radian \( = .11 \) per degree). So, \[ .465 = .11 \cdot (\alpha - (-4)) \] \[ .465 = .11 \cdot (\alpha + 4) \] \[ \alpha + 4 = \frac{.465}{.11} = 4.23 \] \[ \alpha = 4.23 - 4 = .23^\circ \] ## Step 5: Final Answer ### The angle of attack for maximum \( L/D \) is: \[ \boxed{.2^\circ} \] ## Summary Table | Parameter | Value | |----------------------|----------| | \( AR \) | 9.375 | | \( e \) | .92 | | \( C_{D,} \) | .008 | | \( C_L \) at \( L/D_{max} \) | .465 | | \( \alpha_{L=} \) | \(-4^\circ\) | | \( \alpha_{L/D_{max}} \) | \(.2^\circ\) | --- **In words:** The NACA 4416 wing at \( AR = 9.375 \), Oswald \( e = .92 \), and zero-lift angle of \(-4^\circ\), will have its maximum efficiency (\( L/D_{max} \)) at an angle of attack of about **\( .2^\circ \)**. --- **Note:** - This calculation assumes typical values for \( C_{D,} \). If you have a more precise value from wind tunnel data, use that. - The result is sensitive to \( C_{D,} \) and the actual airfoil data. - Angle is **relative to the chord line**. --- **No image provided since the calculation is algebraic.**