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Drag force and power

  1. Jan 3, 2016 #1
    1. The problem statement, all variables and given/known data
    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. Relevant equations

    F = 0.5 ⋅ ρ ⋅ v2 ⋅ CD ⋅ A

    where:
    F - drag force
    ρ = 1.2 kg/m3 - density of the air
    v = 27.778 m/s - speed
    CD - 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:

    Re = (v ⋅ d) / ν
    Re = (27.778 ⋅ 0.009) / (1.46 ⋅ 10-5)

    Re = 17123.4

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

    b010a828f330f2c43cc493c60e7c70b5.jpg

    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.




     
  2. jcsd
  3. Jan 3, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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 CD from the diagram.
     
  4. Jan 3, 2016 #3
    Thanks for reply.

    When I calculated the force for the first time, I read CD = 1.14 from the diagram and I got F = 4.37 N and P = 121.4 W, which was too much.
    Then I realised that the CD might be a little lower, so in second reading from the diagram, I read CD = 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 CD in this case, because the "reading from the diagram" method is not so accurate?

    Thanks.
     
    Last edited: Jan 3, 2016
  5. Jan 4, 2016 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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.
     
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