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

  • Thread starter mkruzic
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  • #1
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Homework Statement



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.)

Homework 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


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

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Redbelly98
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Science Advisor
Homework Helper
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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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