Figure 10.31a illustrates the variation of the lift coefficient, C L as a function of angle of attack, α and the aspect ratio, AR, while Figure 10.31b illustrates the variation of the drag coefficient, CD as a function of angle of attack, α and the aspectratio, AR. Consider an airplane with wings

Question

Figure 10.31 a illustrates the variation of the lift coefficient, [latex] C_{L} [/latex] as a function of angle of attack, α and the aspect ratio, AR, while Figure 10.31b illustrates the variation of the drag coefficient, [latex] C_{D} [/latex] as a function of angle of attack, α and the aspectratio,AR. Consider an airplane with wings that have aplanform areaof [latex] 1600 ft^{2} [/latex],as illustrated in Figure EP 10.25. Assume that the airplane cruises at an altitude of 45,000ft (air at standard atmosphere at an altitude of 45,000ft, [latex] \rho = 0.00046227 slugs/ft^{3} [/latex]) at a steady cruising speed of 350 mph, with a cruising angle of attack of [latex] 5^{\circ } [/latex]. (a) Determine the lift force and the drag force for an aspect ratio of 7. (b) Determine the lift force and the drag force for an aspect ratio of 3.

Accepted Answer

(a)–(b) The lift force is determined by applying Equation 10.21 [latex] F_{L} = C_{L} \frac{1}{2} ho.V^{2} . A[/latex]. Note, the lift force must equal the weight of the aircraft at steady cruising speed. The drag force acting on the aircraft wings during steady cruising speed is determined by applying Equation 10.12 [latex] F_{D} = C_{D} \frac{1}{2} ho v^{2}A [/latex]. Furthermore, the lift and drag coefficients are determined from Figures 10.31a and b, respectively, for the assumed aspect ratio, for an angle of attack, [latex] \alpha = 5^{\circ } [/latex].

(a) The lift force and the drag force for an aspect ratio of 7 are determined as follows:

[latex] slug: = 1 lb \frac{sec^{2}}{ft} [/latex] [latex] A_{plan}: = 1600 ft^{2} [/latex] [latex] \alpha : = 5 deg [/latex]

[latex] ho _{cruise}: = 0.00046227 \frac{slug}{ft^{3}} [/latex] [latex] V_{cruise}: = 350 mph = 513.333 \frac{ft}{s} [/latex]

[latex] AR: = 7[/latex] [latex] C_{Lcruise}: = 0.9 [/latex] [latex] C_{Dcruise}: = 0.011 [/latex]

Guess value: [latex] b: = 50 ft [/latex] [latex] c: = 20 ft [/latex] [latex] F_{Dcruise}: = 100 lb [/latex] [latex] F_{Lcruise}: = 100 lb [/latex]

Given

[latex] AR = \frac{b}{c} [/latex] [latex] A_{plan} = 2.b.c [/latex]

[latex] F_{Lcruise} = C_{Lcruise} \frac{1}{2} ho _{cruise} .V_{cruise}^{2} . A_{plan} [/latex]
[latex] F_{Dcruise} = C_{Dcruise} \frac{1}{2} ho _{cruise} .V_{cruise}^{2} . A_{plan} [/latex]

[latex] \left ( \begin{matrix} b \ c \ F_{Dcruise} \ F_{Lcruise} \end{matrix} ight ) : =Find (b, c, F_{Dcruise}, F_{Lcruise} ) [/latex]

[latex] b = 74.833 ft [/latex] [latex] c = 10.69 ft [/latex]
[latex] F_{Lcruise} = 8.771 imes 10^{4} lb [/latex] [latex] F_{Dcruise} = 1.072 imes 10^{3} lb [/latex]

(b) The lift force and the drag force for an aspect ratio of 3 are determined as follows:

[latex] AR:= 3 [/latex] [latex] C_{Lcruise}: = 0.7 [/latex] [latex] C_{Dcruise}: = 0.012 [/latex]

Guess value: [latex] b: = 50 ft [/latex] [latex] c: = 20 ft [/latex] [latex] F_{Dcruise}: = 100 lb [/latex] [latex] F_{Lcruise}: = 100 lb [/latex]

Given

[latex] AR = \frac{b}{c} [/latex] [latex] A_{plan} = 2.b.c [/latex]

[latex] F_{Lcruise} = C_{Lcruise} \frac{1}{2} ho _{cruise} .V_{cruise}^{2} . A_{plan} [/latex]
[latex] F_{Dcruise} = C_{Dcruise} \frac{1}{2} ho _{cruise} .V_{cruise}^{2} . A_{plan} [/latex]

[latex] \left ( \begin{matrix} b \ c \ F_{Dcruise} \ F_{Lcruise} \end{matrix} ight ) : =Find (b, c, F_{Dcruise}, F_{Lcruise} ) [/latex]

[latex] b = 48.99 ft [/latex] [latex] c = 16.33 ft [/latex]
[latex] F_{Lcruise} = 6.822 imes 10^{4} lb [/latex] [latex] F_{Dcruise} = 1.169 imes 10^{3} lb [/latex]

Therefore, for the given angle of attack, α of [latex] 5^{\circ } [/latex], as the aspect ratio, AR decreases from 7 to 3, the lift coefficient, [latex] C_{L} [/latex] decreases from 0.9 to 0.7, while the drag coefficient, [latex] C_{D} [/latex] increases from 0.011 to 0.012. Furthermore, the corresponding lift force decreases from 8.771 [latex] imes 10^{4} [/latex] lb to 6.822 [latex] imes 10^{4} [/latex] lb, while the drag force increases from 1.072 [latex] imes 10^{3} [/latex] lb to 1.169[latex] imes 10^{3} [/latex] lb, as the lift force and the drag force are directly proportional to the lift coefficient and the drag coefficient, respectively.

Question 10.25: Figure 10.31a illustrates the variation of the lift coeffici...

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