Comment on “Data on Internal Rarefied Gas Flows” [J. Phys. Chem. Ref. Data 27, 657 (1998)]

Reference 1 is a comprehensive review of an important rarefied gas dynamics problem. Researchers actively use the review, which is confirmed by its high citation rate. The review presents data on various internal gas flows from hydrodynamic to free-molecular flow regime. In particular, Sec. 5.2 of the review discusses gas flow through capillaries of finite length under free-molecular regime. Under this regime, gas flow through a capillary is commonly described with a so-called transmission probability W. The physical significance of this value is that when a molecule arrives at the entrance cross section of a capillary connecting containers I and II, this molecule will pass through the capillary from container I to container II with the probability of W_I→II. Similarly, W_II→I is the probability that the molecule will pass the same capillary in the other direction. The review postulates condition (5.5), according to which in the case of isotropic capillary W_I→II = W_II→I.

Section 5.2.4 of the review examines the influence of the capillary’s wall roughness on the transmission probability. It is claimed that if the wall roughness has the sawlike shape, then an asymmetric “saw” can make the capillary anisotropic, and, consequently, condition (5.5) can be violated. In our opinion, this statement is debatable.

Let us consider an isolated system of two equal containers, both initially filled with an equal amount of gas molecules. The containers are connected to each other with a narrow rough-surfaced capillary, for which condition (5.5) is false. It is not difficult to demonstrate that when a sufficiently long period of time passes after opening up the capillary, the ratio of the quantity of molecules in the containers will be described by the equation N_II/N_I = W_I→II/W_II→I, where N_I and N_II are the quantity of molecules in containers I and II, respectively. If the values of transmission probability W_I→II and W_II→I differ significantly, the quantities of molecules in the containers will be noticeably different as well. Since the system is isolated, this conjecture comes into contradiction with the second law of thermodynamics and, therefore, with the representation of the equilibrium state of the gas.

This paper aims to conduct a numerical experiment in order to evaluate the correctness of the claim made in section 5.2.4 concerning violation of condition (5.5) in the case of anisotropic capillary. To this end, the test particle Monte Carlo method² (TPMC) will be used. The TPMC method relies on the physical nature of molecular transport and can be regarded as a numerical experiment. The total number of generated particles in each individual computation equals 10⁹, which allows for the computation error of less than 0.1%.

The geometry of the modeled rough-surfaced capillary is demonstrated in Fig. 1. For our computations, we have chosen the rectangular shape of the capillary (channel) and limited the model to two dimensions. As shown in the figure, the channel of length l and height h has the wall roughness in the shape of an asymmetric saw with uniform teeth; Λ is the height of a tooth, whereas α and β are the tooth’s inclination angles. For the gas-surface scattering law, the diffuse and specular as well as Maxwell³ and Cercignani–Lampis⁴ models have been used.

In our computations, the values of the diffuse fraction ε in the Maxwell model as well as the accommodation coefficient for the part of the kinetic energy corresponding to the motion normal to the wall α_n and the accommodation coefficient of the tangential momentum α_t in the Cercignani–Lampis model are equivalent to the values commonly encountered in reality.⁵ The roughness height used in computations is also close to real values.

Table 1 presents the results of computations for rough-surfaced channels with the length to height ratio l/h = 1 and 10 in the case of the relative height of the roughness Λ/h = 1 × 10⁻⁴ and inclination angles α = 90° and β = 10° (asymmetric “saw”). Table 1 consists of two parts, upper and lower. The upper part shows the parameters of the free-molecular gas flow through the channel from container I to container II, and the lower part shows the same for the opposite direction of the flow. The following parameters have been computed: transmission probability W, mean path 〈S〉 in the channel and time 〈t〉 of traveling through the channel for the particles that passed through the entire channel and made at least one contact with the wall; also, the mean quantity of particle’s collisions 〈n〉 with the rough surface during contact.

The physical significance of n may be gathered from the equation offered by Roberts,⁶ who first pointed out the effect of the surface’s roughness upon the value of the energy accommodation coefficient. Representing the relationship between the energy accommodation coefficient for a smooth surface $α E 1$ with the effective energy accommodation coefficient α_E for a surface of such roughness that each molecule on the average strikes it in n different places before returning to the body of the gas, Roberts’ equation is as follows: $α E =1− ( 1 − α E 1 ) n$⁠. Also, v_m in Table 1 is the most probable molecular velocity.

Let us recall that, in the case of a smooth-surfaced channel and diffuse scattering, the value of transmission probability W can be arrived at using the reliable Berman’s formula,⁷ according to which the value of W equals 0.684 38 and 0.240 80 for l/h = 1 and 10, respectively.

As it is obvious from Table 1, in each computation the parameters of gas flow from container I to container II coincide with the parameters of the opposite direction flow within computation error. In particular, this is true for the values of transmission probabilities W_I→II and W_II→I. Thus, in the case of anisotropic capillary, condition (5.5) has not been violated.

It is understood that a numerical experiment cannot be accepted as a rigorous proof of our statement. However, upon conducting numerous computations with other geometrical parameters of the asymmetric saw, at no time was a violation of condition (5.5) observed, even in the case of the saw height Λ approaching the channel height h.

It should be noted that a violation of condition (5.5) occurred upon examination of the thermal transpiration phenomenon.⁸ However, in this case the containers I and II have different temperatures, meaning that the entire system cannot be considered isolated.

The support by the Ministry of Education and Science of the Russian Federation through the base part of the State task for high educational institutions (the Research Project No. 2189) is gratefully acknowledged.