A twin-engined propeller aircraft is equipped with two engines, each rated at 98 kW shaft power. During cruise flight at 72 m/s at an altitude of 3400 metres, the aircraft experiences a lift coefficient of 0.555. Its wing surface area is 28 square metres, with an average chord length of 2 metres. Its zero-lift drag coefficient is 0.01 and its span efficiency (or Oswald) factor is 0.7.
To determine the aircraft's total drag in cruise flight, we need to consider the various components of drag: induced drag, parasite drag, and zero-lift drag.
1. Induced Drag:
The induced drag is caused by the production of lift on the wings. It can be calculated using the following formula:
Di = (Cl^2 * p * S * k) / (2 * AR)
Where:
Di = Induced drag
Cl = Lift coefficient (0.555)
p = Air density (at 3400m altitude, approximately 0.845 kg/m^3)
S = Wing surface area (28 m^2)
k = Span efficiency or Oswald factor (0.7)
AR = Aspect ratio (calculated later)
2. Parasite Drag:
The parasite drag is the sum of the skin friction drag and the form drag. We need to calculate each component separately and then add them up.
a) Skin Friction Drag:
The skin friction drag can be estimated using the following equation:
Dsf = (0.455 * Cf * p * V^2 * Sf)
Where:
Dsf = Skin friction drag
Cf = Skin friction coefficient (approximated as 0.003, assuming smooth surface)
p = Air density (at 3400m altitude, approximately 0.845 kg/m^3)
V = Velocity (72 m/s)
Sf = Wetted area of the aircraft (approximated as 2 * S, assuming two sides of the aircraft are wetted)
b) Form Drag:
The form drag can be calculated using the following formula:
Df = (Cd * p * V^2 * Sf) / 2
Where:
Df = Form drag
Cd = Zero-lift drag coefficient (0.01)
p = Air density (at 3400m altitude, approximately 0.845 kg/m^3)
V = Velocity (72 m/s)
Sf = Wetted area of the aircraft (approximated as 2 * S, assuming two sides of the aircraft are wetted)
c) Total Parasite Drag:
The total parasite drag is the sum of skin friction drag and form drag:
Dp = Dsf + Df
3. Aspect Ratio (AR):
The aspect ratio of the wing can be calculated using the following formula:
AR = Wingspan^2 / Wing area
However, the wingspan is not provided in the given information. Without it, we cannot calculate the exact value of the aspect ratio.
Given the information provided, we can calculate the total drag equation using the formulas above. However, for a precise result, the wingspan information is needed.
To calculate the total drag on the aircraft during cruise flight, we need to consider the different components of drag.
1. Induced Drag: The induced drag is caused by the generation of lift. It can be calculated using the following equation:
Induced Drag = (L^2 / (0.5 * ρ * V^2 * S * e))
Where:
L is the lift force,
ρ is the air density,
V is the velocity,
S is the wing surface area, and
e is the span efficiency (Oswald's factor).
Given:
Lift coefficient (CL) = 0.555
ρ = air density at 3400m altitude (provided by relevant atmospheric model)
V = 72 m/s
S = 28 square meters
e = 0.7
First, we need to find the lift force (L):
L = CL * 0.5 * ρ * V^2 * S
L = 0.555 * 0.5 * ρ * (72^2) * 28
Now, we can calculate the induced drag:
Induced Drag = (L^2 / (0.5 * ρ * V^2 * S * e))
Induced Drag = (L^2 / (0.5 * ρ * (72^2) * 28 * 0.7))
2. Parasite Drag: The parasite drag consists of two components: the skin friction drag and the form drag.
Skin Friction Drag: It can be calculated using the following equation:
Skin Friction Drag = (0.5 * ρ * V^2 * S * Cf)
Where:
Cf is the skin friction coefficient, and it can be estimated using the Reynolds number (Re).
Reynolds number (Re) = (ρ * V * c) / μ
Given:
c = average chord length = 2 meters
We need to find the air viscosity (μ) at the given altitude:
(The viscosity decreases with altitude and can be estimated using relevant atmospheric models or tables.)
Once we have the air viscosity, we can calculate the Reynolds number (Re).
Then, using the Reynolds number and appropriate reference data, we can estimate the skin friction coefficient (Cf).
Form Drag: It can be calculated using the following equation:
Form Drag = (0.5 * ρ * V^2 * S * Cd)
Where:
Cd is the drag coefficient.
Given: Cd = zero-lift drag coefficient = 0.01
Now, we can calculate the parasite drag:
Parasite Drag = Skin Friction Drag + Form Drag
3. Total Drag: The total drag is the sum of the induced drag and the parasite drag.
Total Drag = Induced Drag + Parasite Drag
Once we have calculated the total drag, we can proceed to other calculations or analysis using the obtained value.
To determine the total engine power required by the aircraft during cruise flight, we need to consider the four main forces acting on the aircraft: lift, drag, weight, and thrust.
1. Lift force (L) can be calculated using the lift coefficient (CL), air density (ρ), velocity (V), and wing surface area (S):
L = 0.5 * CL * ρ * V^2 * S
2. To find the air density (ρ) at an altitude of 3400 meters, we can use the International Standard Atmosphere (ISA) model. At this altitude, the air density can be approximated as:
ρ = ρ0 * e^(-h/H)
where ρ0 is the air density at sea level (1.225 kg/m³), h is the altitude (3400 m), and H is the scale height (7000 m):
ρ = 1.225 * e^(-3400/7000)
3. Drag force (D) can be calculated considering two components: induced drag (Di) and parasite drag (Dp). The sum of these two components is the total drag (D).
D = Di + Dp
a. Induced drag (Di) is related to lift force (L) and the span efficiency factor (e):
Di = L^2 / (0.5 * ρ * V^2 * S * π * AR * e)
where AR is the aspect ratio of the wing. Since the aspect ratio is not given, let's assume it to be 8 (a common value for general aviation aircraft).
b. Parasite drag (Dp) is related to the zero-lift drag coefficient (CD0):
Dp = 0.5 * CD0 * ρ * V^2 * S
4. Weight force (W) can be calculated using the mass (m) and acceleration due to gravity (g):
W = m * g
5. Finally, the total power required (P) can be determined by multiplying the total drag force (D) with the velocity (V):
P = D * V
Let's calculate each of these quantities step by step:
1. Lift Force:
L = 0.5 * CL * ρ * V^2 * S
= 0.5 * 0.555 * (1.225 * e^(-3400/7000)) * (72)^2 * 28
≈ 38573 N
2. Air Density:
ρ = ρ0 * e^(-h/H)
= 1.225 * e^(-3400/7000)
≈ 0.785 kg/m³
3a. Induced Drag:
AR = 8
e = 0.7
Di = L^2 / (0.5 * ρ * V^2 * S * π * AR * e)
= (38573)^2 / (0.5 * (0.785) * (72)^2 * 28 * π * 8 * 0.7)
≈ 1147 N
3b. Parasite Drag:
CD0 = 0.01
Dp = 0.5 * CD0 * ρ * V^2 * S
= 0.5 * 0.01 * (0.785) * (72)^2 * 28
≈ 792 N
Total Drag:
D = Di + Dp
= 1147 + 792
≈ 1939 N
4. Weight:
Assuming the mass (m) of the aircraft is not given, we cannot directly calculate the weight force. However, we can find the power required to overcome the weight force by using the following formula:
P_w = m * g * V
The power required to overcome the weight force should be equal to the total power required (P) calculated in step 5.
5. Total Power Required:
P = D * V
= 1939 * 72
≈ 139,608 N·m/s (or Watts)
Please note that the mass (m) of the aircraft is not given in the provided information, so the specific weight forces or power required to overcome the weight force cannot be determined without this information.