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In (7.22) we can now replace the appearance of e by T ,
@e
@t D c v @T
;
@t
but we still have two unknowns, T and q, and only one equation:
%c v @T
@t C @q
@x f D 0:
(7.23)
The solution to this problem is to use Fourier's law of heat conduction. This law
says that the flow of heat is from hot to cold regions, in the direction of the greatest
decrease of T , i.e., along the negative gradient of T . Mathematically we can express
Fourier's law as
q D k @T
@x
:
(7.24)
The mathematical similarity with Fick's law should be noticed. From a physics point
of view, Fourier's law, with the transport of heat from hot to cold areas, is closely
connected to The Second Law of Thermodynamics.
The coefficient k reflects the medium's ability to conduct heat. In many applica-
tions, one desires to study heat conduction from one medium to another. The domain
will then consist of several different materials, each material having a specific value
of k.Thismeansthatk is a function of x, or, more specifically, a piecewise con-
stant function of x. The example of cooling a can of beer in a refrigerator involves
many materials: air, beer, aluminum (in the can), other goods, their packages, and
so forth. Besides having different k values, the materials also have different den-
sities (%) and heat capacities (c v ). Hence, % and c v will also typically be piecewise
constant functions.
Inserting (7.24)in(7.23) eliminates q and gives us a PDE governing the temper-
ature T :
k.x/ @T
@x
C f.x;t/:
%.x/c v .x/ @T
@t D @
(7.25)
@x
Contrary to the diffusive transport PDE (7.17), where a constant k is a reasonable
assumption, we have in the heat conduction PDE (7.25) variable coefficients %, c v ,
and k. These are constant if we consider heat conduction in a homogeneous medium.
We note that there are other ways of replacing @e=@t by a temperature expres-
sion. One common model ends up with a slightly different heat capacity, called heat
capacity at constant pressure. When looking for values of c v in the literature, one
should know that there is a related type of heat capacity.
Physical Interpretation of the Parameters
Let us try to explain what the parameters %, c v ,andk mean physically. First we dis-
cuss the effect of changing % and c v in a heat problem. Assume that we supply heat
 
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