A general aviation aircraft (m = 1200 kg) flies (under ISA conditions) at 850 metres altitude, with a constant velocity (true airspeed) of 116.6 knots. Its wing surface area is 22 square metres.

Question

Given that its propeller is able to accelerate 80 kilograms of air to a velocity of 105.56 m/s every second, determine the plane's drag coefficient C_D.

Answer 1

We can start by calculating the plane's lift coefficient C_L using the lift equation:

L = 0.5 * rho * V^2 * S * C_L

where L is the lift force, rho is the air density, V is the true airspeed, S is the wing surface area, and C_L is the lift coefficient.

Assuming standard atmospheric conditions (ISA) at 850 meters altitude, the air density can be calculated as:

rho = 1.225 * (288.15 - 0.0065 * 850) / 288.15 = 1.056 kg/m^3

Substituting the given values, we get:

L = 0.5 * 1.056 * (116.6 * 0.51444)^2 * 22 * C_L
L = 15320.5 * C_L

The weight of the plane (mg) is:

mg = 1200 * 9.81 = 11772 N

At constant velocity, the lift force equals the weight, so:

L = mg

Substituting, we get:

15320.5 * C_L = 11772
C_L = 0.768

Now we can use the drag equation to solve for the drag coefficient C_D:

D = 0.5 * rho * V^2 * S * C_D

where D is the drag force.

The only unknown in this equation is C_D, so we can rearrange it as:

C_D = 2 * D / (rho * V^2 * S)

We need to find the value of D, which can be obtained from the thrust equation:

T = D + ma

where T is the thrust force, m is the mass of the air accelerated by the propeller (80 kg per second), and a is the acceleration of that air (105.56 m/s every second).

Assuming that the propeller has a constant efficiency of 80%, the power output can be calculated as:

P = T * V / 0.8

where V is the true airspeed in meters per second (converted from knots).

Substituting the given values, we get:

P = 0.8 * 80 * 105.56 * 0.51444 = 2849.4 W

The power required for level flight can be calculated as:

P_req = D * V

where P_req is the power required, assuming no wind (thus no headwind or tailwind component).

Substituting the given values, we get:

P_req = 0.5 * 1.056 * (116.6 * 0.51444)^3 * 22 * C_D
P_req = 19810.4 * C_D

Since the plane is flying at constant velocity, the power output must equal the power required:

P = P_req

Substituting, we get:

2849.4 = 19810.4 * C_D

Solving for C_D, we get:

C_D = 0.144

Therefore, the drag coefficient of the plane is approximately 0.144.