Heat Flow
The results of exercises 1 and 2 suggest that the direction of
the heat flow at a point is perpendicular to the
isothermal curve,
,
through that point. In fact, this is easily proved (see
).
The gradient of the temperature,
, at a point is also normal (or perpendicular) to the
isotherm through that point. But the heat flow and
are not in the same direction.
The gradient of
and the heat flow are, in fact, oppositely directed, for the
gradient (by definition) is in the direction of increasing
temperature, while heat flows from higher to lower
temperatures.
Until now we have used the phrase "heat flow" in an intuitive
sense; but to formulate the basic law of heat conduction,
equation (2.1) below, we must be more precise. To this end we
introduce the heat flux vector,
, as defined following equation (2.1).
It has been established by experiment that heat flows at a
maximum rate in the direction of the negative gradient of the
temperature function
,
and that the heat flux is proportional to the magnitude of the
gradient. That is,
Equation
3
where the vector
represents the heat flux in units of watts per square meter
.
is in degrees
Kelvin and the (positive) proportionality factor
is called the thermal conductivity and has units of watts
per meter per degree Kelvin
. The scalar
is a property of the conducting medium.
Equation (2.1), the fundamental relation in heat
conduction, is called Fourier's law.
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