Externally imposed temperature gradient thermal diffusion

5 Externally imposed temperature gradient thermal diffusion 

Use the one-dimensional form of this relation since the species movement is only in one coordinate direction  

kr is expressed as -fc j.Xi(l-Xi), where fc j. is a thermal diffusion constant this leads to 

Integrate between region 1 at T, and region 2 at Tj. The extent of separation generated by thermal diffusion is rather low therefore the product Xi (1 - xi) is considered to he a constant at an averaged value Xi(l-Xi). Integrating equation (4.2.64) now yields 

This relation provides an expression for the enrichment factor 12 for two species 1 and 2 between the two regions in terms of fc). and the two temperatures. Consider now an equimolar mixture of H2 and N2 subjected to a temperature difference of 260 °C in the hot bulb and 15 °C in the cold bulb. In the above result, (xu - x ) may be estimated if we can estimate fc j-(xi2)(l-Xn), which is equal to -kr, a rough estimate of this value for illustration is 0.066 (Ibbs et a/., 1939). The value of (x - X12) we obtain is 0.0406, indicating about a 4% difference in mole fraction for the lighter species H2 between the hot region (rj and the cold region (72). 

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The thermophoretic velocity u is defined as u = — DtVT, the coefficient Dj- is the thermal diffusion coefficient, and VT is the temperature gradient either imposed externally or induced internally by some means. The determinatiOTi of thermal diffusion coefficient is the starting point of any thermophoresis study. 

Here is the mobility tensor satisfying f = X- Again, this diffusion coefficient contains two contributions, one depending on the temperature T of the flmd and another one independent from it and proportional to the velocity gradient of the imposed external flow. Therefore, the first contribution is related to the usual thermal Brownian motion whereas the last one is related to non-thermal effects. Notice also that the Smoluchowski equation (26) also contains the usual convective term V (pvo). 

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