Computer Algebra Recipes

266 CHAPTER 6. LINEAR PDE MODELS. PART 2
6.2.3 Radioactive Contamination
The unexamined life is not worth living.
Socrates, Greek philosopher (470{399 BC)
When coauthor Richard was a student, he worked one summer as a chemistry
lab assistant at a uranium mine on Great Bear Lake, which straddles the
Arctic Circle in Northern Canada. Because it was a summer job, there was a
period of several weeks when the sun never set, so evening baseball games were
never called o® because of darkness. The baseball diamond was located on the
leveled mine tailings, the only relatively °at spot in the small northern min-
0
x
Figure 6.1: The radioactive disposal site.
ing town of Port Radium, which is perched on almost treeless primordial rock
formations. The author has always wondered what long-term health problems
eventually arose among the permanent workers because of working in the mine
and playing baseball on the radioactive mine tailings. Motivated by these reminiscences,
we could not resist including a related example involving radioactive
contamination.
A radioactive gas is di®using at a steady rate into the atmosphere from
a leveled contaminated disposal site. The ground and the atmosphere will
be taken to be semi-in¯nite media with X = 0 at the boundary as shown in
Figure 6.1. The concentration C(X; T) of radioactive gas in the atmosphere
obeys the concentration equation (a modi¯ed di®usion equation)
@C(X; T)
@T
= d @2 C(X; T)
@X 2 ¡ ¸C(X; T); (6.8)
where d is the di®usion constant and ¸ is the decay rate of the radioactive gas.
The boundary condition at X =0isgivenbyFick's law,
¡d @C(0;T)
= K; (6.9)
@X