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7.3.2
Heat Conduction
Energy Balance
Derivation of the diffusion equation for heat conduction is much more complicated,
from a physical point of view, than the derivation of the same equation for the dif-
fusive transport of a substance. This is due to the fact that the diffusion equation
for heat conduction is based on the
First Law of Thermodynamics
combined with
Fourier's law, which is closely connected to the
Second Law of Thermodynamics
,
plus other relations from thermodynamics. These models and equations look more
complicated than the simple mass conservation principle used in diffusive transport.
The First Law of Thermodynamics expresses an energy balance (some call it the
conservation of energy): The increase in total energy of a system equals the work on
the system plus the supplied heat. In a simplified version, adapted to the present heat
conduction application, one can state that the
increase in internal energy of a system
equals the supplied heat.
We will use this principle to derive a PDE, like (7.1), gov-
erning temperature distributions. The derivation is more compact than in Sect.
7.3.1
.
Therefore, you should study Sect.
7.3.1
carefully before proceeding, since many of
the ideas and mathematical details are the same, but exposed in a different physical
context.
Deriving a One-Dimensional PDE
Again we consider a one-dimensional heat conduction application and apply our
refined version of energy balance to an arbitrarily chosen interval ˝
D
.a; b/. Phys-
ically, we have a three-dimensional problem, like heat conduction in an insulated
tube, such that we can assume that all quantities depend on x only (if the x axis
coincides with the center line of the tube). Let e.x; t/ be the internal energy per unit
mass, let % be the mass density, and let q.x; t/ be the flow of heat (defined per unit
time). During a time interval t , the increase in total energy in ˝ is
Z
b
%
@e
@t
t
dx
:
a
The supplied heat consists in our case of heat conducted from the surroundings
plus heat generated in the medium, e.g., by chemical reactions or radioactivity. The
heat conducted from the surroundings is represented by the flow of heat, q. The net
inflow of heat into Œa; b in a time interval t is
t.q.a; ; t/
q.b; t// :
The generated heat per unit time and unit volume is supposed to be represented
by a function f.x;t/, such that the total amount of heat generated in ˝ in the
