Simulation of the Knudsen Pump by means of Quasi Gasdynamic Equation System

The use of miniature micropumps based on the effect of thermal gas sliding along the walls with an applied temperature gradient (Knudsen pump) is very relevant when creating modern miniature devices. Currently, algorithmically complex methods such as DSMC are usually used to simulate gas flows in such pumps. Progress in the development of algorithms based on a quasi-gasdynamic system of equations, pushed the authors to apply such algorithm to modeling the flow of a binary gas mixture gases in one variant of a micropump. The equations of the QGD system are conservation laws with additional dissipative terms. These terms contain a coefficient determined by the characteristic time between particles collisions. In this paper, the problem of flow in a rectangular Knudsen pump is solved based on this algorithm for a gas mixture in a single-fluid approximation. Different variants of the composition of the gas mixture are considered. The Knudsen numbers for which it is possible to describe gas flows by the QGD system correspond to flows of dense or moderately rarefied gases. Calculations for not too large Knudsen numbers demonstrated a good agreement with the results obtained by the DSMC method which is much more complex and expensive.


Introduction
The development of micro-and nanotechnologies makes it possible to create miniature thermomolecular micro pumps based on the effect of thermal gas sliding along the walls with an applied temperature gradient.Such devices can be widely used in the creation of modern mass spectrometers, optical spectrometers, gas analyzers and other miniature devices.The operation of such devices is impossible without high-vacuum pumps.In micro and nanoscales, the use of traditional pumps becomes inefficient, since their operation is associated with the mechanical movement of gas.Such pumps have a complex design, large size and many moving parts.Their technical implementation is practically impossible at the micro and nanoscale.The advantages of micro-devices created on the basis of the effect of thermal sliding of gas are that they are durable, have no moving mechanical parts, are environmentally friendly, and have the possibility of simpler technological implementation at the micro level compared to traditional high-vacuum pumps.
Knudsen pumps have a periodic structure consisting of alternating, sequentially connected tubes of two types.The temperature distribution in the Knudsen pump is periodic with the same period that the structure has.The temperature along the tubes of the first type increases linearly from a certain value T1 to T2, and along the tubes of the second type decreases linearly from T2 to T1.
In this paper, the simulation of the operation of one of the variants of the Knudsen pump is carried out based on the quasi-gas dynamic (QGD) equation system for a gas mixture [1].Initially, this system was obtained basing on the use of the collisionless spread -instant Maxwellization model and used for dense gases simulation.The resulting equations represent conservation laws with additional dissipative terms.These terms contain a coefficient determined by the characteristic time between particles collisions.
In the early 2000s, the first attempts were made to create a system of equations for the gas dynamics of a mixture focused on describing the flows of rarefied gases [2].It was a two-fluid model in which each component of the mixture had its own density, velocity and internal energy, and the relationship between the equations was carried out using exchange terms.Later, the system of equations was improved and written in the form of conservation laws, and the exchange terms were generalized to the case of polyatomic gases [3].As a result, the differences between the QGD system of equations and the Navier-Stokes system associated with these additional terms, are most clearly manifested when modelling flows of moderately rarefied gases.The advantages of the QGD system for modelling rarefied gases are demonstrated, in particular, by the example of flows in micro channels [4 -5].However, in numerical modelling, the two-fluid model turns out to be much more cumbersome from an algorithmic point of view.At the same time, the applicability of such models is significantly limited by the scope of applicability of the gas-dynamic models themselves for nonequilibrium flows.
In 2019, two new systems of equations were constructed in [6] for a homogeneous mixture of viscous compressible gases without chemical reactions in a single-fluid approximation.Homogeneous mixtures are understood to mean a mixture in which there are no dedicated interface surfaces between the components.In the one-liquid approximation, it is assumed that the velocities and temperatures of the components of the mixture are in thermodynamic equilibrium.Regularized equations for binary mixtures are considered and a system of equations for moderately rarefied gases is obtained by summing previously derived general equations.From the constructed equations, the balance equations for total mass, kinetic and internal energy were derived, as well as new balance equations for total entropy.It was proved that entropy production is non-negative However, as the practice of numerical calculations has shown, within the framework of this approach, when modelling the flows of mixtures with very different adiabatic indices of the components, non-physical oscillations of the densities of the components of the mixture occur, while the overall density of the mixture remains fairly smooth.A similar problem arises when using other numerical algorithms.The elimination of this disadvantage leads to a significant complication of computational schemes.For example, in [7][8] the double flux method is used to eliminate this disadvantage, in which special modifications of numerical methods are used to approximate convective fluxes at cell boundaries.In [9], a method was given for constructing an elegant and very easy-to-implement QGD algorithm for modeling the flows of a homogeneous binary mixture of gases in a single-fluid approximation.This algorithm allows to obtain monotone profiles of component densities even with a strong difference in their adiabatic indices.A variant of this algorithm has recently been implemented within the framework of the OpenFOAM open software package [10].In this paper, the algorithm [9] is applied to the simulation of a gas flow in a classical rectangular Knudsen pump.

QGD equations system
In the single-fluid model, it is assumed that the gas mixture has a uniform velocity u and temperature T, and the density of the mixture, its pressure, specific total energy and other parameters are determined through the parameters of its components as follows: , , Here F and Q denote the specific power of external forces and heat sources, respectively.Auxiliary QGD-values are usually small additions to the velocity, viscous stress tensor and heat flow and have the following form: When summing equations ( 1) and ( 2), a regularized continuity equation of the QGD system for a single-component gas is obtained The Navier-Stokes viscous stress tensor and the heat flux vector have the traditional form The regularization parameter τ in the case of dense gases is usually associated with the step of the spatial grid h and the speed of sound in the mixture с s .However, for moderately rarefied gases, it seems more natural to take the free path length λ instead of the grid step. .
Here α is a free parameter used to adjust the dissipative properties of the algorithm and its stability.The difference scheme for the QGD system of equations on an orthogonal grid is based on the finite volume method and has a second order of approximation.

Test problem formulation
The classic rectangular Knudsen pump is considered, which is composed of a series of alternately connected wide and narrow micro-channels.The geometrical dimensions and initial parameters of the mixture of two gases (oxygen and nitrogen) are taken from [11].These gases have the same adiabatic index 1.4   , therefore, modeling the flows of such a mixture using the QGD system does not face difficulties.The configuration is shown in figure 1.

Figure 1. Configuration of the computational region.
A periodic structure is well-established in the x-direction.For decreasing the calculation time a basic unit of the pump (wide and narrow channels) between two dash-dotted lines in figure 1 is considered.The length of the section is L = 4 μm, the width of the narrow and wide parts of the channel are H =1 μm and D = 3 μm respectively.The lengths of the narrow and wide channels in the x-direction are both equal to L m = L/2 = 2 μm.The temperature at the edges of the section is 225 K, in the middle of it is 375 K and change between them linearly.The problem is solved in a twodimensional formulation assuming that the third direction is infinite.External forces F and heat sources Q are equal to zero.
The Knudsen number Kn / H   is defined by the width of the narrow channel H and the molecules free path length λ.QGD system is designed to simulate flows of dense or moderately rarefied gases, so Knudsen number cannot be too large.In this regard, one of the goals of the work was to clarify the limits of applicability of the approach used.
The boundary conditions on the left and right ends of the computational region (dash-dotted lines) are periodic.The slip conditions described in [1] is established on all solid walls.It is connected with the tangential derivative ( / s   ) of the wall temperature and normal derivative ( / n   ) of the tangential velocity [1]: Calculations were carried out for three values of the Knudsen number: 0.05, 0.087 and 3.87.The corresponding values of the initial total pressure a b p p  are given in table 1.

Results and discussion
Calculations results for Knudsen numbers equal to 0.055 and 0.0387 are presented in figures 2 -5.The results for the case of Kn = 0.387 are not so good.In [11] each vortex in the wide channel splits into two ones.In our results such splitting does not take place.One can see in figure 3 that the centres of the vortices shifted to the middle of the channel, but their decay does not occur.
Figure 4 and figure 5 show velocity profiles in the vertical section in the middle of the narrow channel.Molecular mass of N 2 is greater than molecular mass of O 2 , so the velocity of a mixture with a predominance of nitrogen is greater than the velocity of a mixture with a predominance of oxygen.
Average velocities in this section for different calculation variants are given in table 2. Note that these values coincide well with the results of [11] for all variants.As a result we may conclude that our approach gives good results even for Kn = 0.387 with respect to speeds in the narrow part of the channel, which determines mass flow rate in the pump.However, the flow picture in the wide channel is correct only for small enough Knudsen numbers.A calculation was carried out for the case of a rarefied gas with Kn = 3.87.One can see in figure 6 that the decay of the central vortices still does not occur although the velocity stream near the central axis expands towards the upper side and the lower side of the channel.In addition the average velocity in the narrow channel became several times more than indicated in [11].So, one can concluded that for such large Knudsen numbers our method is not applicable, at least in its current form.Perhaps these disadvantages are associated with a simplified boundary condition (11) and with an insufficiently good choice of α coefficient in (10).In any case, this is a subject for further research.

Conclusion
The problem of gas mixture flow in a rectangular Knudsen pump is solved based on QGD algorithm for a gas mixture in a single-fluid approximation.It is shown that this algorithm is quite applicable to flows with not very large Knudsen numbers and demonstrates a good agreement with the results [11] obtained by the DSMC method which is much more complex and expensive.But it faces serious difficulties in the case of highly rarefied gases.This is a subject for further investigations.

Figure 2
and figure3demonstrate velocity streamlines on the background of the temperature distribution in the computational region.Here are the results for only one composition of the mixture ( the second composition gives almost the same picture.The flow picture for the case of Kn = 0.055 is very similar to one, presented in[11].

Figure 4 .
Figure 4. Velocity profiles in the middle of the narrow channel.Kn = 0.055.

Figure 5 .
Figure 5. Velocity profiles in the middle of the narrow channel.Kn = 0.387.

Table 1 .
Initial pressure for different Knudsen numbers.
Two compositions of a mixture of nitrogen (gas a) and oxygen (gas b) are considered: :4:1

Table 2 .
Average velocities in the middle of the narrow channel.