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Temp Gradient derivation at Surface, earth as infinite half-space

  1. May 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Lord Kelvin used the heat flow at the surface of the Earth to argue that the Earth was 100 million years old withing a factor of 4 error.

    a.) Reproduce his logic by deriving the temperature gradient at the surface of the Earth for a planet that is cooling by conduction. Model the planet as an infinite half space.

    b.) for this part we will infer the age of the earth using kelvin's measurements. I feel fine doing this but am stuck on part a.)

    2. Relevant equations

    for all eqns: T = temp, t = time, k = kappa (thermal diffusivity), x is a variable used to define erf function, and z = positive in downward direction towards T0 (i.e. towards center of earth)

    T = T0 erf(z/(kt).5), t > 0

    Where "erf" is the error function and defined as:

    Erf(x) = (2 / (pi)^.5) * integral( e(-(x^2)) dx) from 0 to x


    3. The attempt at a solution
    First I set these boundary conditions:

    At t = 0, T = T0 everywhere
    At z = 0, T = T1
    And as z goes to infinity, T = T0 everywhere

    Next, by substituting z / [(kt)^.5] into erf i got:

    T = T0 * (2 / (pi)^.5) * [integral of (e^ -(z/root(kt))2) d(-z/root(kt))] from 0 to [z/root(kt)]

    Then i attempted to take dT / dz and got very confused with the calculus.

    Any ideas on how to go about solving this problem?

    peace
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 31, 2009 #2

    Redbelly98

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    Staff Emeritus
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    Homework Helper

    Welcome to Physics Forums!

    It took me a while to get around to taking a careful look at this problem, or I would have responded sooner.

    You may treat t as a constant, since we are looking for ∂T/∂z at a fixed time.

    Hope that helps ... if not, post again.
     
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