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Math Help - Heat Transfer through a lagged sphere

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    62

    Heat Transfer through a lagged sphere

    i have the following question:

    A storage vessel maybe modeled as a hollow sphere of uniform material. The internal and external radii of r1 and r2 respectively. The temperature of the contents of the sphere is 0in and 0out is the ambient room temperature. The thermal coefficient of the material is ksph. The outside of the sphere is fitted with a uniform layer of lagging, of thickness x, which has a coefficient of thermal conductivity klag. The convective heat transfer coefficient at the inside of the sphere is hin and at the the outside is hout.

    I need to show that the rate of heat transfer q through convection and conduction through the lagged sphere can be modeled in the steady state by

    q = 4
    *pi*a^-1(0in - 0out)

    where

    a = 1/
    ksph(1/r1 - 1/r2) + 1/klag(1/r1 - 1/r2 + x) + 1/hin*r1^2 + 1/hout(r2 + x)^2



    Im not too sure where to start. Im pretty sure i have to use Fourier's law (d0/dr). I also understand that the
    1/r1 - 1/r2 parts relate to the heat loss through the different radii of the sphere itself.

    - Do 1/ksph etc come from the differentiation of a logarithm - dx/dy log x
    = 1/x?
    - Also where do the square parts come into it - those relating to
    1/hin*r1^2 and 1/hout(r2 + x)^2?

    anything to start me off would be good!

    cheers
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  2. #2
    Newbie
    Joined
    Jun 2008
    Posts
    1

    hint

    In equation q note they have a^-1.

    In equation a, for each term look at what a^-1 is, and have the standard conduction or convection equations in front of you, bearing in mind that in equation q, a^-1 is multiplied by 4*pi*(0in - 0out)

    It should start to become clear what they're getting at!

    Good luck

    p.s. remember surface area of sphere with radius r is 4*pi*r^2 but think about what the radius is for each part you're looking at and that'll help
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  3. #3
    Newbie
    Joined
    Apr 2008
    Posts
    6
    No need to go back as far as Fourier. Answer this question in stages:

    First stage is by convection from the gas inside to the inner surface of the sphere. You have the equation for heat transfer by convection:

    q=hA(0in -01)

    rearrange for ((0in - 01) and sub in values for A.

    then do the same thing for the following stages:

    By conduction through the sphere.
    By conduction through the lagging.
    By convection to the outside.

    Add together all the equations you're left with. The lh side all cancle down to (0in - 0out) add together and simplify the rh side.
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