If I wanted to calculate, say, the heat released during space shuttle re-entry, how should I do it? I know they provide the heat on their website, but if I wanted to arrive at the result just through calculation, would that be easy?
You can get an estimate for the maximum amount of energy converted to heat in the shuttle and atmosphere during reentry by taking the difference in the amount of mechanical energy (kinetic energy due to the speed of the shuttles relative to the atmosphere or ground plus potential energy from being at a certain height in the Earth's gravity field) the shuttle has just before reentry compared to the mechanical energy at some other point of interest, for instance when it stops at the runway.
To calculate the rate of heat generation at any particular point in time during re-entry is a very difficult problem . Complex computer simulations and data from many years of experimental work are needed .
However working out the total heat generation for the complete re-entry from first entering the atmosphere to touch down is relatively simple . Can you see how this could be done ?