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to a system by the function f (think of an oven in a room or a heat generation device
embedded in a solid material). If we neglect the spread of heat due to conduction,
we can omit the term with spatial derivatives, such that the PDE reduces to
%c v @T
@t D f:
With a constant heat supply in time and space, f D C D const, and constant %c v ,
we obtain
T D C
%c v
t C integration constant :
That is, the density and heat capacity of the material determine how fast the tem-
perature rises due to the heat supply. When the sun shines and thereby heats up
water and land, the land is heated much more quickly than the water, because the
heat capacity is much smaller for land compared to water. At night, land loses heat
much faster than water, for the same reason. The air above land gets warmer than
the air above the ocean during the day. Since warm air flows upward, wind from the
ocean toward land often arises in the afternoon. A wind in the opposite direction is
frequently experienced in the morning during periods of good weather.
The parameter k is connected to the flow of heat. If k is large, heat is effi-
ciently transported through the material. Think of the application of bringing two
metal pieces, at different temperatures, into contact with each other, as depicted in
Figs. 7.4 and 7.5 .Alargek in the two materials makes heat flow quickly, and the
temperature difference is rapidly smoothed out. The smoothing process runs more
slowly when k is small. In Sect. 7.3.5 we argue that a characteristic time for changes
in the temperature behaves like %c v =k.Alargek or small c v yields quick time-
dependent responses in temperature. While c v is related to temperature changes in
time at a point in space, k is related to heat transport in space.
Deriving a Three-Dimensional PDE
The derivation of a three-dimensional version of (7.25) follows the same mathemat-
ical steps as the derivation of the three-dimensional PDE for diffusive transport. The
energy balance in an arbitrarily chosen volume V can be written
Z
q n t dS C Z
ft dV D Z
% @e
@t t d V :
@V
V
V
Here, q is the heat flow vector field, e and f are internal energy and heat generation,
respectively, while the rest of the symbols are as defined in Sect. 7.3.1 . The integral
on the left-hand side is transformed, using the divergence theorem, to a volume
integral R V
rq dV . Writing the equation as one integral that equals zero reveals
that the integrand must vanish, resulting in the PDE
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