To determine the lift coefficient (Cl) at which the aircraft achieves the maximum climb angle, we need to understand the relationship between lift and drag in the context of the aircraft's performance.
The maximum climb angle occurs when the ratio of lift to drag is maximized. The drag polar equation describes the relationship between the lift coefficient (Cl) and the drag coefficient (Cd) of the aircraft:
\[ C_d = C_{d0} + K \cdot C_l^2 \]
where \( K \) is the effective induced drag factor. The total drag coefficient can be written as:
\[ C_d = C_{d0} + K_1 \cdot C_l^2 + K_2 \cdot C_l \]
For this specific problem, we have:
- \( C_{d0} = 0.0162 \)
- \( K_1 = 0 \)
- \( K_2 = 0.0440 \)
Since \( K_1 = 0 \), the equation for drag simplifies to:
\[ C_d = C_{d0} + K_2 \cdot C_l \] Substituting in the values we know:
\[ C_d = 0.0162 + 0.0440 \cdot C_l \]
To find the lift coefficient that maximizes the climb angle, we need the relationship between thrust, lift, weight, and drag.
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For maximum climb angle, we set:
\[ \frac{L}{D} = \frac{W}{D_{\text{max}}} \]
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Substituting for lift (L = W = mg):
\[ L = \frac{1}{2} \cdot \rho \cdot V^2 \cdot S \cdot C_l \] \[ D = \frac{1}{2} \cdot \rho \cdot V^2 \cdot S \cdot C_d \]
Simplifying the expressions:
- Lift \( L = mg \)
- Drag \( D \) can be expressed with the drag polar.
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The climb performance can be characterized using thrust \( T \) and drag \( D \):
The condition for maximum climb angle can be evaluated by setting thrust equal to the drag at the point of maximum climb angle. That point will occur at a specific \( C_l \).
To achieve maximum lift-to-drag ratio \( (L/D) \):
The lift coefficient where \( \frac{d}{dC_l} \left( \frac{L}{D} \right) = 0 \), can be obtained by differentiating \( \frac{L}{D}= \frac{W}{D} \) and applying the quotient rule.
For simplification, it can typically be state derived that:
- The maximum \( L/D \) occurs when: \[ C_{l_{\text{max}}} = \sqrt{\frac{C_{d0}}{K}} \]
Substituting \( K_2 \) for our \( K \):
In our case:
- Calculate \( K \):
- Since \( K_1 = 0 \), we substitute only \( K_2 \) into \( K \)
\[ K = K_2 = 0.0440 \]
- Plugging the values into the equation to find \( C_{l_{\text{max}}} \):
\[ C_{l_{\text{max}}} = \sqrt{\frac{0.0162}{0.0440}} \approx 0.607 \]
Thus, the lift coefficient at which the MD-11 needs to fly to achieve the maximum climb angle is:
\[ C_l \approx 0.607 \]
This \( C_l \) should provide a suitable lift-to-drag ratio to maximize the climb angle.
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