Conductors in Series—Analytical Approach

Recall that Fourier’s Law only describes heat flow through a single medium. Extending Fourier’s Law to multiple media can be achieved by minimizing thermal resistance. If, proportionally, the most heat flows though the path of least thermal resistance, then the flow of heat is maximized.

A series of two media of differing thermal conductivities is the simplest to model. The path of minimum thermal resistance will take a longer path through the material with greater thermal conductivity and a shorter path through the material with lesser conductivity. Hence, the flow of heat is refracted. This is similar to the refraction of light in Snell’s Law, where the proportional velocity of photons through each medium determines the dependent path. Both the refraction of light and heat flow are a consequence of the Principle of Least Time. When thermal resistance is minimized, a particular quantity of heat shall flow though the media in the least amount of time.

This simulation concerns the heat flow through two materials in series with differing thermal conductivities (see Figure 1). The materials are inexpensive bars of metal that are 5.0 cm long and 1.3 cm wide and high that are available though science education supply firms. The bars are arranged parallel lengthwise and firmly touch each other along one side of their length. Heat is allowed to flow into one extreme corner of one conductor and exit through the opposite corner of the other conductor. This is managed by placing a hot object at one corner and a cold object at the other corner. Except for these corners, the conductors are completely surrounded by a good insulator.

For more information, read A Colorful Demonstration of Thermal Refraction (2014).


Thermal refraction demonstration diagram

Thermal refraction demonstration diagram




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