Lab 2: Introduction to Thermal Modeling

Due Date:

The objective of this lab is to apply basic equations of conductive heat transfer

to geologic problems. This is practice for next week, when we will use this

technique on real data.

Background:

Temperature diffuses in the same way chemicals diffuse or landscapes diffuse; these are all

described by a simple differential equation:

6T 2 K 82T

70? (922

There are 2D and 3D versions of this equation, but we will only consider temperature in

1D (depth). The constant K: is called the thermal diffusivity, which describes how rapidly

temperature diffuses in a material. It is a function of a material’s thermal conductivity,

density, and heat capacity. Notice that if the second derivative with respect to 2 on the right

hand side is large (i.e. if there is a sharp peak or trough in temperature), the left hand time

derivative will also be large, so the temperature will change rapidly at that location. The

effect of this is that temperature spikes will tend to be smoothed over time, and the larger

the spike, the faster the temperature change.

Exact solutions to the thermal diffusion equation involve the error function, but we will

use a numerical approximation scheme commonly employed for differential equations, called

the finite difference method. The finite difference equation we will use for this lab exercise

is:

T2105) 2 T2¢(t _ 1) + A[T2i+1(t _ 1) + TZi-1(t_ 1) _ 2TZ1 (t _ 1”

where A = [565-532

Although this looks imposing, it is quite simple. It says that the temperature at time step t

and spot 2,- (T z, (t)) is the temperature in the same spot at the previous time step (T z, (t – 1))

plus an adjustment factor that is an approximation to the second derivative.

To solve for the entire behavior of the system, we need initial and boundary conditions,

such as the temperature distribution at time zero.

We could optimize the speed of the finite difference method by programming it in Matlab

or another programming language, but it is more illuminating (and requires no programming

knowledge) to do the calculations in Microsoft Excel. In the Excel template you can download

from ANGEL, the rows (from top to bottom) are the times, and the columns (from left to

right) are the depths.