Thermal Conductivity at Low Pressures

When the gas density between two plates at different temperatures is such that the mean free path of the gas is much greater than the distance between the plates, the transport of thermal energy is directly by molecular impacts upon the plates. This process can be analyzed by following the procedure used for momentum transport at low densities (Sec. VIII.5). 

If a is used to represent the accommodation coefficient of energy transfer between molecules and plates, then the gas between the two plates may be assumed to consist of two independent Maxwellian distributions of densities Ni and N2 and temperatures Ti and T y where the plate temperatures are and T2, respectively, and the total gas density is Nt = Ni + AT2. 

After a steady state has been reached, the total flow of molecules to the hot plate T2 must be equal to the total flow of molecules to the cold plate. By using the formula for the number of impacts per unit area [Eq. (VII.6.6)] we have  

The total net flow of energy carried by these two streams per unit area per second is then given by the product 

When AT is small compared to 1, this may be reduced to an approximate form 

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