Rate Constants for Heat Transfer

We can show that a is zero by use of the second condition, equivalent to the assumption of a long bar. From the first condition, we find that 

Thus the temperature drops off exponentially as we get farther and farther away from the bar s base. This is what Biot observed experimentally. 

This problem is difficult because of the boundary condition that the bar is very long and because of the term in the energy balance for heat loss into the air. This second aspect gave Fourier himself a lot of trouble, so if it is not clear the first time, try again. 

Up to this point, we have treated the thermal conductivity kr, the thermal diffusivity a, and the heat transfer coefficient h as unknowns, adjustable parameters in any calculation. In fact, we often want to use previously measured values of these quantities to make predictions about new situations. Values for gases can be predicted from kinetic theory, and values for liquids and solids are best found by experiment. In this section, we report a few selected values of these quantities. 

Estimates of thermal conductivities of gases depend on the following result of kinetic theory  

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