Knudsen effect, effective thermal conductivity

The effective thermal conductivity k in the dried material has been found to vary significantly with the total pressure and with the type of gas present. Also, the type of material affects the value of k (SI, Kl). The effective diffusivity D of the dried material is a function of the structure of the material, Knudsen diffusivity, and molecular diffusivity (Kl). 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

This inaccuracy stems from their calculation of molecular transport effects, such as viscous dissipation and thermal conduction, from bulk flow quantities, such as mean flow velocity and temperature. This approximation of microscale phenomena with macroscale information fails as the characteristic length of the (gaseous) flow gradients approaches the average distance travelled by molecules between collisions - the mean path. The ratio of these quantities is referred to as Knudsen number. 

Figure 9.37 Effect of the pore (cell) size on the gas thermal conductivity in open-pore polymeric foams together with the theoretical predictions obtained from the Knudsen equation and some of the previous results in aerogels. PMMA, poly(methyl methacrylate).

The low gas thermal conductivity Agas of aerogels in general can be explained by the Knudsen effect, expressing the gas conduction in a porous media as function of the air pressure and the characteristic pore size and written in a simplified way as follows (see also Figure 45.3) [31,69-71] ... 

Research is also ongoing to develop new high-performance thermal insulation materials, for example, nano insulation materials (NIM) [31,69,70,103-114] that utilize the Knudsen effect to achieve their low thermal conductivity without applying a vacuum in the pores. One possible pathway toward NlMs is to... 

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