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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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