To determine the drag coefficient of the general aviation aircraft, we can use the following relationship:
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Determine the true airspeed in meters per second: \[ \text{TAS} = 116.6 \text{ knots} \times \frac{0.514444 \text{ m/s}}{1 \text{ knot}} = 60.18 \text{ m/s} \]
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Calculate the air density at 850 m altitude: Under ISA conditions, the temperature at 850m is approximately 15°C - 0.0065°C/m * 850m = 10.5°C. In Kelvin, this is: \[ T = 10.5 + 273.15 = 283.65 \text{ K} \] The air density at sea level is approximately 1.225 kg/m³ and decreases with altitude. The air density at 850m can be approximated using the following formula: \[ \rho = \rho_0 \times \left(1 - \frac{L \cdot h}{T_0}\right)^{\frac{g \cdot M}{R \cdot L}} \] where:
- \( \rho_0 \) = 1.225 kg/m³ (sea level density)
- \( L \) = 0.0065 K/m (temperature lapse rate)
- \( h \) = 850 m (altitude)
- \( T_0 \) = 288.15 K (sea level temp)
- \( g \) = 9.81 m/s² (acceleration due to gravity)
- \( M \) = 0.0289644 kg/mol (molar mass of Earth's air)
- \( R = 8.314 J/(mol \cdot K) \) (universal gas constant)
For simplification, a commonly used approximate at 850m is: \[ \rho \approx 1.1 \text{ kg/m}^3 \]
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Calculate the drag force: The drag force is calculated using the thrust from the propeller: The propeller accelerates 80 kg of air per second to 105.56 m/s. Therefore, the thrust produced can be calculated using: \[ F_t = \dot{m} \cdot V_e \] where \( \dot{m} \) is the mass flow rate of the air and \( V_e \) is the exit velocity of air from the propeller.
Substituting the values: \[ F_t = 80 \text{ kg/s} \cdot 105.56 \text{ m/s} = 8444.8 \text{ N} \]
In steady level flight, thrust (T) equals drag (D): \[ D = 8444.8 \text{ N} \]
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Calculate the drag force using the drag equation: The drag force can also be expressed as: \[ D = \frac{1}{2} \cdot \rho \cdot V^2 \cdot S \cdot C_d \] where:
- \( \rho \) = air density (1.1 kg/m³)
- \( V \) = true airspeed (60.18 m/s)
- \( S \) = wing surface area (22 m²)
- \( C_d \) = drag coefficient (to be determined)
Rearranging for \( C_d \): \[ C_d = \frac{2D}{\rho V^2 S} \]
Substitute the known values into the equation: \[ C_d = \frac{2 \cdot 8444.8}{1.1 \cdot (60.18)^2 \cdot 22} \]
Calculating \( V^2 \): \[ V^2 = (60.18)^2 \approx 3621.41 \]
Now plug the values: \[ C_d = \frac{2 \cdot 8444.8}{1.1 \cdot 3621.41 \cdot 22} = \frac{16889.6}{1.1 \cdot 3621.41 \cdot 22} \]
\[ C_d = \frac{16889.6}{1.1 \cdot 3621.41 \cdot 22} = \frac{16889.6}{1.1 \cdot 79763.02} \] \[ C_d = \frac{16889.6}{87739.322} \approx 0.192 \]
Therefore, the calculated drag coefficient \( C_d \) is approximately 0.192.
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