Kinetic theory of thermal transpiration and mechanocaloric effect. II

Abstract

The problem of thermal transpiration between parallel plates and in a cylindrical capillary is considered by using the Maxwellian diffuse-specular reflection at the surface. The relevant integral equations are solved by a simple numerical scheme and results for both the Poiseuille flow and the thermal creep flow rate are obtained. Some recently reported variational results for the Poiseuille flow appear to be inaccurate (off by about 10%–15%), and there is a need for further refinement in the variational calculations. The effect of partial accommodation is found more significant in the free molecule region than in the near continuum region. Thus to minimize the surface effects, the experiments on the thermal conductivity measurements via thermal transpiration would best be conducted in the near continuum region.

Knudsen heat capacity

“…The heat capacity is 5/2 in the hydrodynamic limit, and with increasing Kn the heat capacity decreases. In free molecular flow conditions, c p/T γ /k B reaches the values at Kn = 10 of 1.99051 for Sharipov, 14 2.01725 for Loyalka, 18 and 2.04608 for Ohwada et al 19 These are reasonably close to the theoretical value of c p/T γ /k B = 2 that we calculated above.…”

“…(7) is substituted into Eq. (15), with coefficients from Table I, the best match with the calculated Knudsen heat capacities in Table II 18 data capture the heat capacities within about 5%. The differences between the results slightly increase for Ar, Kr, and Xe because of deviation from the ideal gas approximation for larger atoms.…”

“…In Refs. 14, 18, and 19, rarefied gas flows at different Knudsen numbers in different geometries were investigated using the S-model, 14 the BGK model, 18 and the hard-sphere gas model. 19 The γ relation can be obtained from logarithmic curve fitting to this published data in the range Kn = 0.1-10, i.e.,…”

“…For the flow characteristic dependency of the heat capacity, we need a more general relation between Kn and γ that can cover mid-range Knudsen numbers, however, the Kn-γ relations proposed in the literature are not always successful for these systems. [15][16][17] Because of the lack of a welldeveloped analytical expression for the Kn-γ relation over all flow regimes from the hydrodynamic to the free molecular, we have curve-fitted a relationship from data in the literature, 14,18,19 as follows. In Refs.…”

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

“…As a cornerstone rarefaction phenomenon, this mechanism has been widely discussed and applied. Thermal creep was first studied by Maxwell (1879), and is the dominant physical effect driving steady flow in the Knudsen compressor (Knudsen 1909b(Knudsen , 1910Loyalka 1971;Vargo et al 1999;Sone 2007). More recently, steady thermal creep has also been used to explain motion of (volatile) Leidenfrost drops on a ratchet surface; see Lagubeau et al (2011) andWürger (2011).…”