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Diff. Eq. with heat flow

  1. Apr 10, 2013 #1
    1. The problem statement, all variables and given/known data

    The fluid inside the pipe shown has a temperature of
    350 K, but the temperature of the air in the room is only 306 K.
    Therefore, heat flows at a constant rate from the fluid, through
    the pipe walls, and into the room. The inner pipe radius is 4
    cm, and the outer radius is 8 cm. The heat equation is:

    dQ/dt = kA(dT/dx)

    where x is the direction of heat flow, A is the area through
    which the heat flows (i.e., perpendicular to x), and k is the
    conductivity of the material through which the heat is flowing.
    Determine the temperature of the pipe metal at r = 5.92 cm.

    2. Relevant equations

    T(r<4)=350
    T(r>8) = 306
    3. The attempt at a solution

    I've tried this a bunch of times, but can't see to get it. I have done:

    Q'=kA(dT/dr)
    where Q' is a constant

    A=pi*r^2

    dT=Q'/(k*pi*r^2) *dr

    T=(Q'/k*pi)(-1/r)+c1

    I let Q'/k = c2

    so

    T=c1-c2/r*pi

    After imposing the initial conditions:

    350 = c1 - c2/4pi

    306 = c1 - c2/8pi

    from this
    c1 = 262
    c2= 1105.84

    and got the temperature at r=5.92 to be 321.46 K, but this wasn't right.

    Any ideas?

    I think I went wrong with the initial conditions somewhere
     
  2. jcsd
  3. Apr 11, 2013 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    There's a mistake here, since the area through which the heat flows must depend on the length of the pipe.
     
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