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.
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.
To determine the plane's drag coefficient (C_D), we need to use the equation:
Drag force (D) = 0.5 * ρ * V^2 * C_D * A
Where:
- Drag force (D) is the force resisting the motion of the plane.
- ρ is the air density.
- V is the true airspeed of the plane.
- C_D is the drag coefficient.
- A is the wing surface area.
In this case, we need to solve for C_D.
First, we need to calculate the air density (ρ) at 850 meters altitude. The air density at a given altitude can be calculated using the following equation:
ρ = ρ_0 * e^(-h / H)
Where:
- ρ_0 is the air density at sea level under ISA conditions (1.225 kg/m^3).
- h is the altitude (in this case, 850 meters).
- H is the scale height of the atmosphere (approximately 7,400 meters).
Using this equation, we can calculate ρ:
ρ = 1.225 * e^(-850 / 7,400)
Next, we need to convert the true airspeed from knots to meters per second:
V = 116.6 knots * (0.5144 m/s per knot)
Now, we can substitute the values into the drag equation:
D = 0.5 * ρ * V^2 * C_D * A
Rearranging the equation to solve for C_D, we get:
C_D = D / (0.5 * ρ * V^2 * A)
We know the propeller is accelerating 80 kilograms of air to a velocity of 105.56 m/s every second. The force required to accelerate this air is the drag force (D) in our equation:
D = m * a
Where
- m is the mass of air accelerated (80 kg).
- a is the acceleration (105.56 m/s^2).
Substituting these values into the equation, we get:
D = 80 kg * 105.56 m/s^2
Now, we can calculate C_D:
C_D = (80 kg * 105.56 m/s^2) / (0.5 * ρ * V^2 * A)
Plug in the values for ρ, V, and A that we've calculated, and solve for C_D.
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.
To determine the plane's drag coefficient, we can use the formula:
C_D = (2 * D) / (ρ * v^2 * A)
Where:
- C_D is the drag coefficient
- D is the drag force
- ρ is the air density
- v is the true airspeed
- A is the wing surface area
First, we need to find the value for the drag force, D. We know that the propeller is able to accelerate 80 kilograms of air to a velocity of 105.56 m/s every second. The acceleration of this air is given by:
F = m * a
Where:
- F is the force (80 kg * 105.56 m/s)
- m is the mass of the air accelerated (80 kg)
- a is the acceleration
Since we know the force, we can write the equation for drag force:
D = F - Thrust
But since the plane is flying at a constant velocity, the thrust is equal to the drag force. Therefore, we have:
D = F - D
Rearranging the equation, we get:
2D = F
Now we can substitute the respective values into the equation:
2D = 80 kg * 105.56 m/s
Simplifying:
2D ≈ 8444.8 N
D ≈ 8444.8 N / 2
D ≈ 4222.4 N
Next, we need to find the air density, ρ. We can use the International Standard Atmosphere (ISA) model to calculate ρ at the given altitude of 850 meters:
ρ = ρ_0 * (1 - (L * h) / T_0)^(g0 / (R * L))
Where:
- ρ_0 is the standard sea-level density (1.225 kg/m^3)
- L is the temperature lapse rate (0.0065 K/m)
- h is the altitude (850 m)
- T_0 is the standard sea-level temperature (288.15 K)
- g0 is the standard acceleration due to gravity at sea level (9.80665 m/s^2)
- R is the specific gas constant for dry air (287.05287 J/(kg·K))
Plugging in the values:
ρ = 1.225 kg/m^3 * (1 - (0.0065 K/m * 850 m) / 288.15 K)^(9.80665 m/s^2 / (287.05287 J/(kg·K) * 0.0065 K/m))
Simplifying:
ρ ≈ 1.148 kg/m^3
Now we can substitute the values into the equation for the drag coefficient:
C_D = (2 * D) / (ρ * v^2 * A)
C_D = (2 * 4222.4 N) / (1.148 kg/m^3 * (116.6 knots * 0.514444 m/s)^2 * 22 m^2)
Simplifying and converting knots to m/s:
C_D = 2 * 4222.4 N / (1.148 kg/m^3 * (116.6 * 0.514444)^2 m/s^2 * 22 m^2)
C_D ≈ 0.0259
Therefore, the plane's drag coefficient (C_D) is approximately 0.0259.