## 1. Introduction

It is well known that the thermally-induced flow of rarefied gas is generated by the temperature gradient along the walls of the Knudsen pump and that the gas is driven to flow from the low-temperature side to the high-temperature side. That is the basic mechanism of the Knudsen pump which was first put forward by Danish physicist Martin Knudsen in 1909 [

1]. The Knudsen pump can provide consistent gas flow and has the advantage of having of no moving parts, a simple structure, ease of operation, long life span, low energy consumption, and wide energy sources. It is widely applied in Micro Electro Mechanical Systems (MEMS) such as gas separators [

2,

3], gas analysis [

4,

5,

6], micro combustors [

7,

8], and micro-air vehicle systems [

9,

10].

The classic rectangular Knudsen pump is composed of a series of alternately connected wide and narrow micro-channels [

1]. A tangential temperature gradient appears by imposing high-temperature and low-temperature heat resources for the two ends of the wide channels respectively. This generates a thermal creep effect for the gas flow. In recent years, with the development of materials technology and micro-machining technology, the pump structure can now be produced by using poly-silicon material, and using the inter-molecular gaps in porous materials such as aerogel membranes [

10,

11,

12], mixed cellulose ester (MCE) [

13,

14], zeolite [

15,

16], porous ceramics [

17,

18] and Bi

_{2}Te

_{3} [

19,

20] to construct the flowing channel of the Knudsen pump. Since the rectangular Knudsen pump has been proposed, many structures for the channel were successively designed and studied (

Figure 1), including the sinusoidal micro-channel [

21], matrix micro-channel [

21], curved micro-channel with different curvature radii [

22], alternately connected curved and straight micro-channel [

23,

24], tapered micro-channel [

25], and ratchet micro-channel [

26,

27,

28,

29].

It is well known that the Boltzmann equation is the basic equation for solving the continuous, transition, and free-molecular regimes. It has been developed in mathematical methods such as the moment method and model equation in recent years. Chapman-Enskog solution is the most important representative of the moment method, whose first-order solution is the Navier–Stokes–Fourier (NSF) equation [

30,

31]. By adding velocity slip and temperature jump conditions, the Chapman-Enskog solution can be applied to the rarefied gas flow within a small Knudsen number. However, the regularized 13-moment equations [

32] can be used for the rarefied gas flows with a large Knudsen number. The model equation simplifies the collision integral in the Boltzmann equation. The most famous model equations are the BGK model introduced by Bhatnagar, Gross and Krook [

33] and McCormack model [

34]. The BGK model is well applied to the study of transport characteristics of gas mixtures in micro-channels [

35,

36]. The McCormack model also shows good consistency with experimental results in studying the flow state of gas mixtures [

37]. The Direct Simulation Monte Carlo (DSMC) method is a direct numerical solution of the Boltzmann equation, which eliminates the disadvantages in the mathematical solution of the Boltzmann equation. Although the DSMC method requires a large amount of internal storage space and long calculation time, its calculation results are highly consistent with experimental results [

38,

39,

40].

Compared with other study methods [

21,

22,

23,

24,

41,

42,

43,

44,

45,

46], the DSMC method is widely used for heat and mass transfer in micro-channels [

47,

48,

49,

50]. There are many studies that apply the DSMC method in the flow of gas mixtures [

51,

52,

53]. It is found that Knudsen pump shows good capability in gas separation by the simulation of DSMC [

54]. While in the studies of gas flow in Knudsen pumps, the method of DSMC is widely employed [

28,

29,

55]. For example, DSMC is used to study and simulate flow patterns of the gas in rectangular channels [

55] and ratchet channels [

28,

29]. The studies of Knudsen pumps have generally been focused on innovation of the structure, optimization of performance, and practical application. The gas used in simulations is mostly monatomic noble gas. However, gas mixtures have been more widely applied than single gases, and the proportions of the noble gas in the air are very small. Moreover, the size of the micro-channel has already reached the nanometer level, ensuring that the Knudsen pump operates normally under atmospheric pressure.

In present study, the flow characteristics of gas mixture of N

_{2} and O

_{2} in Knudsen pump are simulated with the DSMC method. A classic rectangular channel is applied, that is more common and convenient to machine. The problem statement and the numerical method are presented in

Section 2. The simulation results for the gas mixtures of N

_{2} and O

_{2} in three different ratios are discussed in

Section 3, and the conclusions are in

Section 4.

## 3. Results and Discussion

This section mainly discusses the flowing characteristics of the gas mixtures of N

_{2} and O

_{2} in the ratios of 4:1, 1:1 and 1:4 in the rectangular channel Knudsen pump. The flow field, distribution of temperature gradient, distribution of velocity, and mass flow rate are thoroughly studied. The physical properties of the gases for the simulation particles of N

_{2} and O

_{2} in the VHS model are listed in

Table 2.

#### 3.1. Velocity and Temperature Distribution

For N

_{2} and O

_{2} in the three different mixed ratios, the representative velocity and temperature distribution in the rectangular channel for

Kn = 0.055,

Kn = 0.387, and

Kn = 3.87 are illustrated respectively in

Figure 5.

By observing the figures, it can be seen that gases flow forward from left to right in the narrow channels without differences, though the Knudsen number increases. However, large differences are triggered in the wide channels, and the rarefied degree of the gas is enhanced. A larger circular-flow vortex appears respectively in the upper side and the lower side of the wide channel for Kn = 0.055. When the Knudsen number increases, the velocity stream of gases near the central axis expands towards the upper side and the lower side. Within the larger vortex, two anticlockwise secondary vortexes are generated, illustrated by white arrows for Kn = 0.387. With the higher rarefied degree of the gas, the secondary vortexes completely replace the main vortexes. The secondary vortexes individually exist near the corners of the wide channel, shown by white arrows for Kn = 3.87.

In terms of the temperature field, as the rarefied degree of the gas increases, the total number of gas molecules decreases. The thermal conductivity is weakened. Thus, the energy transmitting from the walls to the field decreases dramatically, as does the temperature difference. For different ratios of the gas mixtures with the same Knudsen number, the general distributions of the temperature field and the velocity field do not change dramatically with the variations of the ratio.

#### 3.2. Temperature Gradient

In the DSMC method, the overall temperature of the gas

$T$ is as follows [

38,

39,

40],

where

p is the species of the simulation particle,

q is the amount of the species of the simulation particle,

$\sum {N}^{\u2033}$ is the weighted number of simulated molecules, and the temperature of species

p is

where,

${T}_{\mathrm{tr},p}$,

${T}_{\mathrm{rot},p}$,

${T}_{\mathrm{vib},p}$, and

${T}_{\mathrm{el},p}$ are translational temperature, rotational temperature, vibrational temperature and electronic temperature, respectively.

${\zeta}_{\mathrm{rot},p}$,

${\zeta}_{\mathrm{vib},p}$, and

${\zeta}_{\mathrm{el},p}$ are the corresponding degrees of freedom. The effective number of degrees of freedom

${\zeta}_{p}$ of species

p is

Due to no consideration of the vibrational energy and the electronic energy (

${\zeta}_{\mathrm{vib},p}={\zeta}_{\mathrm{el},p}=0$), the temperature of species

p is

where,

where,

$\Sigma {\epsilon}_{\mathrm{rot},p}^{\u2033}$ is the weighted sum of the rotational energy of the simulated molecules of species

p.

By substituting Equations (9) and (10) into Equation (5), the overall temperature is obtained by the summation of all species of the simulation particles. Similarly, in the dsmcFoam, the overall temperature of the gas

$T$ is obtained by the mean value of the relevant macroscopic physical properties in the cell. The representation is as follows by dsmcFoam code [

46],

where

${\rho}_{\mathrm{lKE}}$ is mean value of the linear kinetic energy density,

${\rho}_{\mathrm{m}}$ is the mean value of the mass density,

$U$ is the mean value of gas velocity,

${\rho}_{\mathrm{ilE}}$ is the mean value of the internal energy density, and

${\rho}_{\mathrm{N}}$ is the mean value of the real gas-molecular number density in the cell.

${\rho}_{\mathrm{iDF}}$ is the mean value of the internal degree of the freedom density, but the influence of internal degree of freedom is not considered in the VHS model; here, it is assumed that

${\rho}_{\mathrm{iDF}}$ = 0. To simulate the molecular motion, the random numbers are generated in the DSMC method. The average values of macroscopic physical quantities are applied to maximize the simulation accuracy.

Figure 6 illustrates the distribution of the temperature gradient for three gas mixtures and three Knudsen numbers on the central axis of the channel (indicated by the red full line in

Figure 6) within one structure unit. It can be seen that the changes in the composition of the gas mixtures do not make a difference in the distribution of the temperature gradient on the central axis. The distribution patterns are similar to the asymmetric sinusoid shown in

Figure 6a. This is because the size of the narrow channel is much smaller, and more energy is transferred to the central axis.

Moreover, it was found that for different Knudsen numbers, these distribution patterns always remain as asymmetric sinusoidal patterns. The maximum temperature gradients respectively reach the medium points of the narrow channel and the wide channel (shown as

Figure 6b). For the reason that the rarefied degree of the gas increases and thermal conductivity weakens, the maximal values of temperature gradient decrease with increasing Knudsen numbers. However, the velocity for

Kn = 0.387 is larger than the velocity for

Kn = 0.155 (Discussion later). This is because the thermal creep effect hardly appears near the slip regime. It is not obvious, though the temperature gradient is larger. Therefore, we conclude that the value of the temperature gradient does not correspond to the performance of the thermal creep effect near the slip regime.

#### 3.3. Mean Velocity

Figure 7 presents the mean velocity of each composition with the different gas mixture in a cross-section at

x = 1 μm (indicated by the red full line in

Figure 2) for different Knudsen numbers. Even in a mixture of N

_{2} and O

_{2} with a similar molecular mass, the velocity of N

_{2} is larger than that of O

_{2}. The close relation between the thermal creep effect and the gas-molecular mass is clearly proved [

60]. When the ratio of N

_{2} rises in the gas mixtures, the thermal creep effect is enhanced. This not only causes

n increase in the velocity of N

_{2} itself, but also contributes to an increase in the velocity of O

_{2}. That is, N

_{2} can promote the movement of O

_{2}.

It was also found that the mean velocity of each composition in different gas mixtures reaches the maximum for Kn = 0.387. When the gas mixtures are in the slip regime (Kn < 0.077) and free-molecular regime (Kn > 7.75), no matter how changeable the compositions of the gas mixtures are, the mean velocities of each composition are almost the same. That is because the thermal creep effect cannot be effectively induced by the temperature gradient in the slip regime, while in the free-molecular regime the thermal creep effect is weaker. Therefore, regarding systems that process gas separation using the thermal creep effect, it should be guaranteed that gas mixtures are in a transition regime in order to improve the efficiency and quality of gas separation.

#### 3.4. Mass Flow Rate

During one physical time

${t}_{\mathrm{avg}}$ which used for calculating the average of the macroscopic physical quantities, the mass flow rate is obtained by calculating the total mass of all particles that pass through the cross section in

x = 1 μm (indicated by a red full line in

Figure 2). This is demonstrated by the following equations:

where

$\Delta N$ is the total number of the simulating particles passing through the cross section within

${t}_{\mathrm{avg}}$,

${m}_{i}$ is the mass of the molecules;

${\overrightarrow{c}}_{i}$ is the velocity of particles;

${\overrightarrow{n}}_{f}$ is the unit normal vector of the cross-section, and the positive and negative directions are the same as the

x-axis. Likewise, in order to improve the accuracy of the simulation in the DSMC, the paper decreases the numerical error through the mean value of the mass flow rate for a large number of cycle numbers. The sampling interval is usually the same as the physical time,

${t}_{\mathrm{avg}}$.

Therefore, the total mass flow rates of these three different N

_{2}–O

_{2} gas mixes can be obtained. The mass flow rate of each composition for the different Knudsen numbers is illustrated in

Figure 8. Mass flow rate decreases with the increase of the Knudsen number, and the maximum and minimum values occur respectively at

Kn = 0.055 and

Kn = 7.75. This is mainly because the enhancement of the gas rarefied degree leads to a decrease in the number of molecules. The variation of the mass flow rate is the largest in the Knudsen number range from 0.155 to 0.387. It is determined by the stronger thermal creep effect within this range.

Furthermore, though O_{2} molecules are heavier and have higher densities in each of the gas mixtures, N_{2} has the larger flowing velocity, and the number of N_{2} gas molecules passing through the channel per unit time increases. Thus, the increase of the ratio of N_{2} weakens the influence of the differences in the value of the densities. This increases the mass flow rate to some degree. The thermal creep effect is stronger, especially when the value of the Knudsen number is in the range of 0.155–0.387. The total mass flow rates of these three N_{2}–O_{2} gas mixtures are much closer to each other.

## 4. Conclusions

The flow characteristics of N_{2}–O_{2} gas mixtures in the rectangular Knudsen pump are studied by using the DSMC method. By exerting temperature gradient boundary conditions on the walls, a thermally induced flow (thermal creep flow) is successfully generated. The influences of N_{2}–O_{2} gas mixtures in different ratios, and different gas rarefied degrees (different Knudsen numbers) on the flow characteristics of gases are well studied. The following conclusions can be drawn:

(a) Under the same Knudsen number, the flow fields of the three different gas mixtures in the Knudsen pump channel are highly similar. The distribution of the temperature gradient is all asymmetric sinusoid in nature on the central axis of the channel. That is, the variations of the gas compositions do not make a difference in the distribution of the flow field in the Knudsen pump channels.

(b) Even in N_{2} and O_{2} gas mixtures with similar molecular masses, N_{2} is found to have a stronger thermal creep effect. It was successfully verified that the thermal creep effect has a relationship with the weight of the gas molecules.

(c) In gas mixtures, N_{2} has a larger velocity than O_{2}. If the proportion of N_{2} increases, the overall velocity also increases. The lighter gas can promote the movement of the heavier gas.

(d) The lighter gas and heavier gas respectively correspond to a larger volume flow rate and a larger mass flow rate. With the ratio of the lighter gas increasing, the incremental volume flow rate somewhat weakens the difference of the mass flow rate resulting from the difference of densities. Even though the ratios of each composition of the gas mixtures differ greatly, the total mass flow rates are almost equal, especially when the thermal creep effect is the strongest.