Interdisciplinary Applied Mathematics

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Ti >T2.


If these two containers are connected with a relatively thick channel (A ^ h), the equilibrium condition requires no-flow in the channel. If the channel thickness (h) becomes comparable to the mean free path (A), rarefied gas effects    have    to be taken    into    account.    In    such    a case    the    local    equi


librium mechanism is very complex, and interaction of the gas molecules with the walls must also be considered. Here, we consider free-molecular flow conditions (i.e., A ^ h) to simplify the discussion. In this flow regime, the intermolecular collisions are negligible compared to the interaction of molecules with the surfaces. If we assume that molecule-wall interactions are specular (i.e., av = 0), then the following analysis is valid. We assume that the density of the fluid is proportional to the number density (number of molecules per unit volume),


p ж n,


and the temperature of the fluid is proportional to the square of average molecular speed,


T ж c2.


The mass fluxes at the hot and the cold ends of the channel are


mnici and mn2C2,


respectively; here m is the mass of the gas molecules. Then


mnici pi


mn2c2 p2



T2



0.5



Pi


P2



Ti


Ti



0.5



Ti


Ti



0.5


< i,


where we have used the equations of state


P



pRT and



Pi


P2



The above analysis indicates a flow creeping from cold to hot. Osborne Reynolds was the first to realize this phenomenon, and he coined the term thermal transpiration (Reynolds, 1879). At about the same time, Maxwell developed independently a theory for thermal creep. In the early 1900s, Knudsen built a molecular compressor based on the thermal transpiration idea by connecting a series of tubes with constrictions arranged between each tube (Knudsen, 1910a; Knudsen, 1910b). The constrictions were very small, so that the rarefaction effects became important in the constrictions. By heating the same side of these constrictions to very high temperatures (773 K), Knudsen was able to maintain considerable pressure gradients (Loeb, 1961). According to (Ebert and Albrand, 1963), Gaede also described a thermal pump in his unpublished notes. Other developments in thermal creep driven vacuum pumps can be found in (Vargo et al., 1998).

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