**3. Discussion**

The models originated from RET and NET-IV are briefly introduced and tested on a particular experiment performed by Rhodes. It is apparent from the experimental data that the rarefaction in a Hydrogen gas shows substantial deviation from its dense state. Here, the speed of sound data can be evaluated without any obstacles in NET-IV as well, but this evaluation highlights some crucial aspects.

First, we note the difference between the theoretical and measured properties which has great importance in kinetic theory. In contrast, a continuum model does not propose this distinction, it is not possible to calculate the "physical" (theoretical) coefficient without a detailed micromechanism. Moreover, especially from practical point of view, the NET-IV model contains the "effective" (measurable) quantities which have density dependence as well, beside the temperature, according to the referred literature. Since the continuum approach does consist of the measurable transport coefficients, it keeps the continuum model on a more general level that is not restricted specifically, in contrast to the kinetic theory. In the present experimental evaluation, constant viscosities and thermal conductivity are used in order to keep the model compatible with the kinetic theory.

Second, it is advantageous if all the data are given in the experiment; the frequency-pressure ratio could be insufficient, especially when the effective transport coefficients are not constant respect to the pressure. In continuum theory, it does matter what is changed: the frequency, the pressure, or both. In that sense, it is confusing to use dimensionless frequency, especially when all the necessary data are

given for the appropriate scale. Besides that fact, the application of dimensionless quantities is still valid, but their use makes it a bit more difficult to interpret the experimental data. Moreover, in the end, even the kinetic approach requires the knowledge about the pressure in order to evaluate the relaxation times and the coupling coefficients.

Moreover, it is also shown that besides the frequency/pressure scaling of experimental data, it is not necessarily obtained in the dispersion relations of a continuum model, merely by restrictive assumptions of constant coefficients. However, it is worth noting that if every experimental data are given appropriately, that is, the frequency and pressure are given separately, then the experimental data become evaluable even if the *ω*/*p* scaling is not visible in the dispersion relations. Furthermore, recalling the experimental data of Itterbeek and Gracki et al. in which the viscosity depends on the mass density, this is clear that its inclusion into the models would strongly influence the scaling properties as well. Thus, one question remains: which transport property is important from the aspect of the scaling, the theoretical one or the measurable one?

The difference between the presented approaches originates in the preassumptions. Using kinetic theory, a detailed interaction is assumed, and it restricts some degrees of freedom. On the other hand, in NET-IV, it is not necessary to make any preassumption regarding the type of the fluid, while in kinetic theory it is fixed as a first step, and the derived model such as Equation (5) will be valid only for this restricted case, in the price of the number of free coefficients. As mentioned in the introduction, one of the main difficulties is the fitting of free parameters in the model of NET-IV (4), that is, 7 coefficients are to be fitted instead of 3 that presented by Equation (5). It can be reduced if one adopts some knowledge from the kinetic approach and applies in the continuum model, such as the ratio of relaxation times. This disadvantage could be an advantage if the number of equations is still enough to describe a three-dimensional process on the contrary to the kinetic theory where the momentum expansion generates a large system of equations for a more general situation, for instance in the low temperature heat conduction problems.

As a final remark, it is essential to realize that these approaches extend each other from different directions. While NET-IV keeps the models as general as possible, RET builds a detailed one, with much more knowledge about the coefficients. Although the underlying physical picture is different, the resulting system of equations can be compatible with each other.

**Funding:** This research was funded by National Research, Development and Innovation Office—NKFIH K116197, K124366 and KH130378. The research reported in this paper was also supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Nanotechnology research area of Budapest University of Technology and Economics (BME FIKP-NANO).

**Acknowledgments:** The author is thankful to Henning Struchtrup and Péter Ván for valuable discussions.

**Conflicts of Interest:** The author declares no conflict of interest.
