Opposite ends of an $Fe-0\mathrm{.}8wt\%C$ bar are placed in contact with thermal

reservoirs at $1000C$ and $1200C.$ In this temperature range, the $C$ atoms

occupy interstitial positions in f$.$c$.$c$.$ austenite, forming an interstitial solid solution.

As a result of the heat transport along the bar, it is found that $C$ atoms migrate

towards the hot temperature end. Assume that only $C$ atoms are mobile. You

can also assume that, initially, the concentration of $C$ in the bar is uniform.

$10$

a$.(2$ points$)$ Draw arrows on the schematic above to indicate directions

corresponding to the flow of heat and the flow of mass.

b$.(4$ points$)$ Write out a system of linear flux equations that describe the

flow of heat and matter due to the operant driving forces.

c$.(10$ points$)$ What is the sign of the Onsager indirect coefficient?

Hint: assume that, initially, the $C$ concentration in the bar is

uniform such that vec$(grad){}_C=0.$ Thus, the only operant driving

force is that of the imposed temperature gradient.

d$.(4$ points$)$

How might you measure this Onsager coupling coefficient

experimentally? Explain in a few sentences.

Hint: consider what happens to the system at long times, i$.$e$.,$

when steady$-$state behavior has been reached $).$