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

[mkruzic]

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

 

[Redbelly98]

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.