Question 14.9: A fully loaded aircraft weighing 900 kN has a wing area of 2...
A fully loaded aircraft weighing 900 kN has a wing area of 230 m². If the wing has the characteristics of a NACA 23012 airfoil and during takeoff operates at a 6° angle of attack, what is the required takeoff speed at sea level? What is the takeoff speed at an elevation of 2000 m?
To take off, the lift force must overcome the weight, so FL = W,
\frac{1}{2}ρU^2AC_L = W
U=\sqrt{\frac{2W}{ρAC_L}}
From Appendix B for the U.S. Standard Atmosphere ρ(0 m) = 1.225 kg/m3 and ρ(2000 m) = 1.007 kg/m3. From Figure 14.36 CL = 0.79. Substituting the data into the preceding expression yields:
U=\sqrt{\frac{ 2(900×10^3\ N) }{(1.225\ kg/m^3)(230\ m^2)(0.79)}}=90\ m/s=324\ km/h
at sea level and at 2000 m
U=\sqrt{\frac{ 2(900×10^3\ N) }{(1.007\ kg/m^3)(230\ m^2)(0.79)}}=99\ m/s=356\ km/h
Thus an increase of almost 10% in takeoff speed is required at the higher elevation.
TABLE B.2 Properties of the U.S. Standard Atmosphere in SI Units
| Altitude (m) | Temperature (°C) | Acceleration of Gravity g (m/s²) | Pressure p [N/m²(abs)] | Density ρ (kg/m³) | Dynamic Viscosity \mu [(N•s)/m²] |
| −1,000 | 21.50 | 9.810 | 1.139 E + 5 | 1.347 E+0 | 1.821 E − 5 |
| 0 | 15.00 | 9.807 | 1.013 E + 5 | 1.225 E+0 | 1.789 E − 5 |
| 1,000 | 8.50 | 9.804 | 8.988 E + 4 | 1.112 E +0 | 1.758 E − 5 |
| 2,000 | 2.00 | 9.801 | 7.950 E + 4 | 1.007 E+0 | 1.726 E − 5 |
| 3,000 | −4.49 | 9.797 | 7.012 E + 4 | 9.093 E−1 | 1.694 E − 5 |
| 4,000 | −10.98 | 9.794 | 6.166 E + 4 | 8.194 E−1 | 1.661 E − 5 |
| 5,000 | −17.47 | 9.791 | 5.405 E + 4 | 7.364 E−1 | 1.628 E − 5 |
| 6,000 | −23.96 | 9.788 | 4.722 E + 4 | 6.601 E−1 | 1.595 E − 5 |
| 7,000 | −30.45 | 9.785 | 4.111 E + 4 | 5.900 E−1 | 1.561 E − 5 |
| 8,000 | −36.94 | 9.782 | 3.565 E + 4 | 5.258 E−1 | 1.527 E − 5 |
| 9,000 | −43.42 | 9.779 | 3.080 E + 4 | 4.671 E −1 | 1.493 E − 5 |
| 10,000 | −49.90 | 9.776 | 2.650 E + 4 | 4.135 E −1 | 1.458 E − 5 |
| 15,000 | −56.50 | 9.761 | 1.211 E + 4 | 1.948 E−1 | 1.422 E − 5 |
| 20,000 | −56.50 | 9.745 | 5.529 E + 3 | 8.891 E−2 | 1.422 E − 5 |
| 25,000 | −51.60 | 9.730 | 2.549 E + 3 | 4.008 E−2 | 1.448 E − 5 |
| 30,000 | −46.64 | 9.715 | 1.197 E + 3 | 1.841 E−2 | 1.475 E − 5 |
| 40,000 | −22.80 | 9.684 | 2.871 E + 2 | 3.996 E−3 | 1.601 E − 5 |
| 50,000 | −2.50 | 9.654 | 7.978 E + 1 | 1.027 E−3 | 1.704 E − 5 |
| 60,000 | −26.13 | 9.624 | 2.196 E + 1 | 3.097 E−4 | 1.584 E − 5 |
| 70,000 | −53.57 | 9.594 | 5.221 E + 0 | 8.283 E−5 | 1.438 E − 5 |
| 80,000 | −74.51 | 9.564 | 1.052 E + 0 | 1.846 E−5 | 1.321 E − 5 |
