A propeller engine for a model airplane is measured to have a jet velocity of 50 kilometres per hour. When strapped to a fixed support it is measured to produce 10 Newtons of thrust. Determine the mass flow through this propeller (in kilograms per second)Again consider the model propeller of the previous problem. If we assume the jet velocity to be equal to the velocity of the air through the propeller, then we can say that , where is the air density, is the propeller area and is the speed of the flow through the propeller.
Given that the test was performed with an air density of 1.225 kilograms per cubic metre, determine the propeller diameter (in metres).
To determine the mass flow through the propeller, we need to use the equation:
Mass flow rate (ṁ) = Air density (ρ) * Propeller area (A) * Speed of flow (V)
We are given the following information:
Jet velocity = 50 kilometers per hour = 50,000 meters per hour = 50,000/3600 meters per second ≈ 13.89 meters per second
Thrust force = 10 Newtons
Air density = 1.225 kilograms per cubic meter
We know that thrust force is equal to the mass flow rate times the change in velocity:
Thrust force (F) = Mass flow rate (ṁ) * Change in velocity (ΔV)
Rearranging the equation, we get:
Mass flow rate (ṁ) = Thrust force (F) / Change in velocity (ΔV)
The change in velocity is the difference between the jet velocity and the velocity of the air through the propeller:
Change in velocity (ΔV) = Jet velocity - Velocity of air through propeller
Rearranging the equation again, we get:
Velocity of air through propeller = Jet velocity - Change in velocity (ΔV)
Now, let's substitute the given values into the equations:
Velocity of air through propeller = 13.89 m/s - ΔV
Since the jet velocity is measured when the propeller is strapped to a fixed support, the velocity of the air through the propeller is equal to the jet velocity. Therefore:
Velocity of air through propeller = 13.89 m/s
Now we can substitute all the values into the mass flow rate equation:
Mass flow rate (ṁ) = 10 N / (13.89 m/s)
Simplifying the equation, we find:
Mass flow rate (ṁ) ≈ 0.719 kg/s
To determine the propeller diameter, we need to use the equation:
Propeller area (A) = Mass flow rate (ṁ) / (Air density (ρ) * Speed of flow (V))
Substituting the given values into the equation:
Propeller area (A) = 0.719 kg/s / (1.225 kg/m^3 * 13.89 m/s)
Simplifying the equation, we find:
Propeller area (A) ≈ 0.0473 m^2
To calculate the propeller diameter (d), we can use the formula for the area of a circle:
A = π * (d/2)^2
Rearranging the equation, we get:
d/2 = √(A/π)
Now let's solve for the propeller diameter:
d/2 = √(0.0473 m^2 / π)
d/2 ≈ √(0.0151 m^2)
d/2 ≈ 0.1229 m
Finally, multiplying both sides of the equation by 2, we find:
d ≈ 2 * 0.1229 m
d ≈ 0.2458 m
Therefore, the propeller diameter is approximately 0.2458 meters.
To determine the mass flow through the propeller, we can use the equation:
Mass flow rate = Thrust / Jet velocity
Given that the thrust is 10 Newtons and the jet velocity is 50 kilometres per hour (which needs to be converted to meters per second), we can calculate the mass flow rate:
Jet velocity = 50 km/h = 50,000 m/3600 s = 13.89 m/s
Mass flow rate = 10 N / 13.89 m/s = 0.72 kg/s
Now, we can use the equation:
Mass flow rate = Air density * Propeller area * Speed of flow through the propeller
Given that the air density is 1.225 kg/m^3, the mass flow rate is 0.72 kg/s, and the speed of flow through the propeller is equal to the jet velocity (13.89 m/s), we can solve for the propeller area:
0.72 kg/s = 1.225 kg/m^3 * Propeller area * 13.89 m/s
Propeller area = 0.72 kg/s / (1.225 kg/m^3 * 13.89 m/s) = 0.0525 m^2
Finally, we can find the propeller diameter by assuming it is a circular shape:
Propeller area = π * (Propeller diameter/2)^2
0.0525 m^2 = π * (Propeller diameter/2)^2
Solving for the propeller diameter:
(Propeller diameter/2)^2 = 0.0525 m^2 / π
Propeller diameter/2 = sqrt(0.0525 m^2 / π)
Propeller diameter = 2 * sqrt(0.0525 m^2 / π) ≈ 0.321 m
Therefore, the propeller diameter is approximately 0.321 meters.
To determine the mass flow through the propeller, we can use the equation:
Mass Flow Rate = Density x Area x Velocity
Given that the jet velocity is 50 kilometers per hour, we need to convert it to meters per second:
Jet Velocity = 50 km/h = 50,000 m/3600 s = 13.89 m/s
The air density is given as 1.225 kilograms per cubic meter.
Let's assume the propeller's area is A and the speed of the flow through the propeller is V. Therefore, using the equation:
10 N = (1.225 kg/m^3) x A x 13.89 m/s
We can rearrange the equation to solve for the area:
A = 10 N / [(1.225 kg/m^3) x 13.89 m/s]
A ≈ 10 / (1.225 x 13.89)
Finally, we need to determine the propeller diameter from the area. The area of a circle is given by:
Area = π x (Diameter/2)^2
Rearranging this equation, we can solve for the diameter:
Diameter = √(4 x Area / π)
Substituting the calculated area into the equation, we get:
Diameter = √(4 x A / π)
Now we can substitute the calculated value of A to find the diameter.