Problem IV

1) Based on what you have learned in Problem II, How would you expect k to vary with R?

2) Set T = 300, N = 300, W = 300, M = 1, find out how k varies with R. Explain your result.

3). Use larger disk diameter D and larger number of hard disks N and see how k changes with N in a dense hard disk system.

4). In a dense hard disk system (the mean free path no longer much greater than the particle radius), heat transfer occurs almost exclusively by collision, since an atom can hardly travel freely for a long time before it bumps into another atom. The expression for the conductivity k of a dense hard disk system is given by:

                                                             (1)
where              

While the expression for the conductivity k of a dilute hard disk system is given by:

                                                                                                    (2)

The difference between (1) and (2) is due to the different contribution of collisional heat transfer in the two systems.

Assume  and D are unity, plot k vs. ρ in a dense hard disk system, compare the graph with the one you obtained from 3), and explain the difference using what you have learned in Problem II. Remember ρ is proportional to N.

Problem V