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.
Determine the propulsive efficiency (in percent) in this flight condition.
To determine the propulsive efficiency of the aircraft, we need to calculate the total drag and the total power.
Step 1: Calculate the lift force (L):
Lift force (L) = 0.5 * air density * velocity^2 * wing surface area * lift coefficient
Given:
Air density = 1.225 kg/m^3 (standard air density at sea level and 15 degrees Celsius)
Velocity (v) = 72 m/s
Wing surface area (S) = 28 m^2
Lift coefficient (Cl) = 0.555
L = 0.5 * 1.225 * (72)^2 * 28 * 0.555
L ≈ 8,281.65 N
Step 2: Calculate the lift-induced drag (D_i):
Lift-induced drag (D_i) = (L^2) / (0.5 * air density * velocity^2 * wing surface area * span efficiency factor)
Given:
Span efficiency factor (e) = 0.7
D_i = (8,281.65^2) / (0.5 * 1.225 * (72)^2 * 28 * 0.7)
D_i ≈ 1,283.10 N
Step 3: Calculate the parasitic drag (D_p):
Parasitic drag (D_p) = zero-lift drag coefficient * (0.5 * air density * velocity^2) * wing surface area
Given:
Zero-lift drag coefficient (Cd0) = 0.01
D_p = 0.01 * (0.5 * 1.225 * (72)^2) * 28
D_p ≈ 6,772.32 N
Step 4: Calculate the total drag (D):
Total drag (D) = D_i + D_p
D = 1,283.10 + 6,772.32
D ≈ 8,055.42 N
Step 5: Calculate the total power (P):
Total power (P) = D * velocity
P = 8,055.42 * 72
P ≈ 579,269.42 W
Step 6: Calculate the propulsive power (P_p) used to overcome drag:
Propulsive power (P_p) = P - (2 * engine power)
Given:
Engine power = 98 kW
P_p = 579,269.42 - (2 * 98,000)
P_p ≈ 383,269.42 W
Step 7: Calculate the propulsive efficiency (E_p):
Propulsive efficiency (E_p) = P_p / P * 100
E_p = (383,269.42 / 579,269.42) * 100
E_p ≈ 66.21%
Therefore, the propulsive efficiency in this flight condition is approximately 66.21%.
The propulsive efficiency is a measure of how effectively an aircraft's engines convert fuel energy into thrust. It is expressed as a percentage and can be calculated using the following equation:
Propulsive Efficiency = (Propulsive Power / Shaft Power) x 100
To calculate the propulsive power, we need to determine the drag of the aircraft at the given flight condition. The drag can be divided into two components: parasitic drag and induced drag.
1. Parasitic drag:
Parasitic drag can be calculated using the zero-lift drag coefficient, wing surface area, and air density. The equation for parasitic drag is:
Parasitic Drag = 0.5 x Zero-Lift Drag Coefficient x Wing Surface Area x Air Density x Velocity^2
where:
Zero-Lift Drag Coefficient = 0.01
Wing Surface Area = 28 square meters
Air Density = ρ (which can be approximated as 1.225 kg/m^3 at sea level)
2. Induced drag:
Induced drag is directly related to the coefficient of lift, wing surface area, air density, velocity, and span efficiency factor. The equation for induced drag is:
Induced Drag = (Coefficient of Lift^2 x Wing Surface Area x Air Density x Velocity^2) / (2 x Span Efficiency Factor)
where:
Coefficient of Lift = 0.555
Wing Surface Area = 28 square meters
Air Density = ρ (which can be approximated as 1.225 kg/m^3 at sea level)
Velocity = 72 m/s
Span Efficiency Factor = 0.7
3. Total Drag:
Now, we can calculate the total drag by summing the parasitic drag and induced drag:
Total Drag = Parasitic Drag + Induced Drag
4. Propulsive Power:
Propulsive power is the product of total drag and velocity:
Propulsive Power = Total Drag x Velocity
Finally, we can calculate the propulsive efficiency using the propulsive power and the given shaft power of each engine:
Propulsive Efficiency = (Propulsive Power / (2 x Shaft Power)) x 100
By substituting the values into the equations and performing the calculations, you will be able to determine the propulsive efficiency of the aircraft in this flight condition.
To determine the propulsive efficiency, we first need to find the drag force acting on the aircraft.
The drag force can be calculated using the drag equation:
Drag Force = 0.5 * Air Density * Velocity^2 * Wing Surface Area * Drag Coefficient
First, let's calculate the air density at the given altitude using the International Standard Atmosphere model:
Air Density = 1.225 * e^(-0.000118 * altitude)
Air Density = 1.225 * e^(-0.000118 * 3400) ≈ 0.812 kg/m^3
Now, we can calculate the drag force:
Drag Force = 0.5 * 0.812 * (72)^2 * 28 * 0.01
Drag Force ≈ 6352 N
Next, we need to calculate the lift force acting on the aircraft.
The lift force can be calculated using the lift equation:
Lift Force = 0.5 * Air Density * Velocity^2 * Wing Surface Area * Lift Coefficient
Lift Force = 0.5 * 0.812 * (72)^2 * 28 * 0.555
Lift Force ≈ 17551 N
Now, we can calculate the induced drag force using the lift and drag forces:
Induced Drag Force = Lift Force^2 / (0.5 * Air Density * Velocity^2 * Wing Surface Area * Oswald Factor)
Induced Drag Force = 17551^2 / (0.5 * 0.812 * (72)^2 * 28 * 0.7)
Induced Drag Force ≈ 2882 N
The remaining drag force is the parasite drag:
Parasite Drag Force = Drag Force - Induced Drag Force
Parasite Drag Force ≈ 6352 - 2882
Parasite Drag Force ≈ 3470 N
Now, we can calculate the power required to overcome the parasite drag:
Power Required = Parasite Drag Force * Velocity
Power Required = 3470 * 72
Power Required ≈ 249840 W
The propulsive power is equal to the power required, so it is 249840 W.
To calculate the propulsive efficiency, we need to know the power output of the engines. Since each engine is rated at 98 kW, the total power output is 2 * 98 kW = 196 kW = 196000 W.
Propulsive Efficiency = (Power Output / Power Required) * 100
Propulsive Efficiency = (196000 / 249840) * 100
Propulsive Efficiency ≈ 78.45 %
Therefore, the propulsive efficiency in this flight condition is approximately 78.45%.