Abstract
In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients, σ1,T and σ2,T, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (αt, αn) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (αt, αn) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent γ has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since γ mainly depends on αt, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (αt, αn), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of αt. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature.
Highlights
From the pioneering works of Reynolds [1], Maxwell [2] and Knudsen [3,4] it is known that a gas flow is generated through a capillary if the temperature is different at its ends, and this flow is directed toward the hot extremity of the capillary
The present investigation is mostly guided by the desire to analyze recent experimental studies, described in detail in Section 4, in order to propose a reliable procedure of extraction for the tangential momentum and normal energy accommodation coefficients
A variational technique applied to the Boltzmann equation for hard-sphere molecules has been used to derive analytical expressions for the first- and second-order thermal slip coefficients in terms of the tangential momentum and the normal energy accommodation coefficients, defined in the frame of the Cercignani-Lampis model of boundary conditions
Summary
From the pioneering works of Reynolds [1], Maxwell [2] and Knudsen [3,4] it is known that a gas flow is generated through a capillary if the temperature is different at its ends, and this flow is directed toward the hot extremity of the capillary This phenomenon is called the thermal transpiration or thermal creep and the mass flow rate generated by applying a temperature gradient along the capillary walls is called the temperature driven mass flow rate, denoted in the following by MT. The pressure difference between the reservoirs, which corresponds to this final stage, is called the Thermal molecular Pressure Difference (TPD) These phenomena become important when a gas flowing through a capillary is rarefied, i.e., the ratio between the molecular mean free path, λ, and a characteristic flow dimension, a, (in this case the capillary radius) called the Knudsen number, Kn = λ/a, is larger than 0.01. The thermal creep phenomenon is important either at low pressure or for the gas flows at small scales
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