Calculate Drag Force & Power for Long Cylindrical Antenna

[Nurah]

Homework Statement

A long cylindrical antenna of 9 mm (0.009 m) diameter is mounted vertically on a car moving at 100 km/h (27.778 m/s). Antenna is 920 mm long (0.92 m). Calculate the power required to move antena through the air with standard kinematic viscositiy ν = 1.46 ⋅ 10^-5 m²/s.

The given result is P = 104 W.


2. Homework Equations
F = 0.5 ⋅ ρ ⋅ v² ⋅ C_D ⋅ A

where:
F - drag force
ρ = 1.2 kg/m³ - density of the air
v = 27.778 m/s - speed
C_D - drag coefficient
A - relevant area

P = F ⋅ v

where:
P - reguired power


3. The Attempt at a Solution

First, I tried to find drag coefficient. To do that, I must find Reynolds number:

R_e = (v ⋅ d) / ν
R_e = (27.778 ⋅ 0.009) / (1.46 ⋅ 10^-5)
R_e = 17123.4

According to the diagram, for infinitely long cylinders, C_D ≈1

Also, relevant area is:
A = diameter ⋅ height
A = 0.009 ⋅ 0.92
A = 8.28 ⋅ 10^-3 m²

Now, the drag force is:
F = 0.5 ⋅ 1.2 ⋅ 27.778² ⋅ 1 ⋅ 8.28 ⋅ 10^-3
F = 3.83 N

Required power is equal to:
P = F ⋅ v
P = 4.37 ⋅ 27.778
P = 106.4 W

Is this the correct way to calculate this? Am I getting 2 watts more just because of "not so accurate" reading from the diagram above?

Thanks.

 

[mfb]

Why did the force change from 3.83 N to 4.37 N?
Where does the value for the density of air come from?
2% deviation is certainly fine if you take C_D from the diagram.

 

[Nurah]

Thanks for reply.

When I calculated the force for the first time, I read C_D = 1.14 from the diagram and I got F = 4.37 N and P = 121.4 W, which was too much.
Then I realized that the C_D might be a little lower, so in second reading from the diagram, I read C_D = 1. Then I got F = 3.83 N and P = 106 W.
Since I typed that in a hurry, I missed to correct the value for the force in all equations. Sorry for that.

In the book where I found this example under the text it is said that the air has standard characteristics, so the density is 1.2 kg/m³.

So, this procedure is fine?

Is there any other way to calculate C_D in this case, because the "reading from the diagram" method is not so accurate?

Thanks.

 

[mfb]

The assumption that the antenna is in a free air flow is not so accurate either. Usually car antennas are attached to cars, and those influence the air flow. Taking a value of 1 should be fine, you won't get a 1% accuracy anyway.