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.

 

[Blueshift5]

Hello,

Thank you for reaching out to me with your question. I am always happy to help others understand complex concepts.

To reproduce Lord Kelvin's logic, we need to derive the temperature gradient at the surface of the Earth for a planet that is cooling by conduction. This can be done by modeling the planet as an infinite half-space.

First, let's define some variables. T0 represents the initial temperature at the surface of the Earth, T1 represents the temperature at the center of the Earth, and k is the thermal diffusivity of the Earth. We will also use z as the distance from the surface of the Earth, with z=0 representing the surface and z approaching infinity representing the center of the Earth.

Using the given boundary conditions, we can set up the following equation:

T(z,t) = T0 + (T1-T0)erf(z/(2*sqrt(kt)))

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

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

To find the temperature gradient at the surface, we can take the derivative of this equation with respect to z:

dT/dz = (T1-T0) * (2/sqrt(pi)) * e^(-(z/(2*sqrt(kt)))^2) / (2*sqrt(kt))

Since we are interested in the temperature gradient at the surface (z=0), we can simplify this equation to:

dT/dz = (T1-T0) * (2/sqrt(pi)) / (2*sqrt(kt)) = (T1-T0) / (sqrt(pi*kt))

Now, we can use this equation to calculate the temperature gradient at the surface of the Earth. This can then be used to determine the age of the Earth using Lord Kelvin's measurements.

I hope this helps and good luck with your homework! Remember, don't hesitate to ask for help if you get stuck again.