You are probably thinking that if heat is conducted into the element from
both sides, as assumed in the formulation, the temperature of the medium will
have to rise and thus heat conduction cannot be steady. Perhaps a more realistic
approach would be to assume the heat conduction to be into the element on
the left side and out of the element on the right side. If you repeat the formulation
using this assumption, you will again obtain the same result since the
heat conduction term on the right side in this case will involve Tm Tm 1 instead
of Tm 1 Tm, which is subtracted instead of being added. Therefore,
the assumed direction of heat conduction at the surfaces of the volume elements
has no effect on the formulation, as shown in Figure 5–12. (Besides, the
actual direction of heat transfer is usually not known.) However, it is convenient
to assume heat conduction to be into the element at all surfaces and not
worry about the sign of the conduction terms. Then all temperature differences
in conduction relations are expressed as the temperature of the neighboring
node minus the temperature of the node under consideration, and all conduction
terms are added.