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Homework Help: Sequence of insulating layers of different conductivities for pipe

  1. Apr 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Two insulating materials of thermal conductivity K and 2K respectively are to be used as two layers of insulation for lagging a pipe carrying hot fluid. If the radial thickness both the layers is to be same, then -

    i) the first material (thermal conductivity K) should be the inner layer, and the second material should be the outer layer

    ii) the second material should be the inner layer

    iii) the order of the materials is irrelevant

    iv) numerical values required to give a specific answer

    2. Relevant equations
    The heat transfer in this case would be given by the following equation, I suppose:

    q=[itex]\frac{2*pi*ΔT}{1/h1r1 +ln(r2/r1)/K1 + ln(r3/r2)/K2 +1/r3h3}[/itex]

    q= radial heat transfer per unit time per unit length of pipe
    ΔT= temperature drop across the pipe thickness i.e. hot fluid temperature inside the pipe - ambient temperature outside the pipe
    h1= heat transfer coefficient of the hot fluid
    h3=heat transfer coefficient of the ambient air outside the pipe
    r1=inside radius of first layer
    r2=outside radius of the first layer=inside radius of the second layer=r1+t, t=radial thickness of one layer of insulation
    r3=outside radius of the second layer=r2+t=r1+2t
    K1=thermal conductivity of the first layer
    K2=thermal conductivity of the second layer
    3. The attempt at a solution

    In the book where I found the question, (i) is the given answer, but I don't understand how that is. Looking at the equation, I guess ln(r2/r1)/K1+ln(r3/r2)/K2 should be maximum, but I don't know how to reach the conclusion that K1=K and K2=2K is the optimum option.
  2. jcsd
  3. Apr 20, 2013 #2


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    Staff: Mentor

    Just use r2=r1+t, r3=r2+t

    Then you can write those logarithms as ln(1+t/r1) and ln(1+t/r2), where r2>r1. Which one is larger? How can you maximize the sum with this knowledge?
  4. Apr 21, 2013 #3
    So the logarithms are ln(1 + x) and ln(1+x/(1+x)), where x = t/r1. Now, expand each log term in a taylor series in x, and retain terms only up to x2.
  5. Apr 21, 2013 #4


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    Staff: Mentor

    It is sufficient to know that ln() is monotonically increasing.
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