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Influence of quantum intermolecular interaction on internal flows of rarefied gases
Vacuum 156 (2018) 146–153
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Influence of quantum intermolecular interaction on internal flows of rarefied gases
T
Felix Sharipov Departamento de Física, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Quantum scattering Orifice flow Direct simulation Monte Carlo ab initio potential
In order to model gaseous flows over the whole temperature range beginning from 1 K, the intermolecular interaction should be considered on the basis of quantum approach. Such a consideration becomes important in case of light gases like helium and hydrogen. Recently, the direct simulation Monte Carlo (DSMC) method widely used to calculate flows of gases has been generalized to implement the quantum approach to intermolecular collisions. To evaluate the influence of the quantum scattering on typical flows of light gases, a benchmark problem has been solved for two helium isotopes 3He and 4He using an ab initio potential. More specifically, the flow-rate and flow-field of helium flowing through an orifice have been calculated over the temperature range from 1 K to 300 K for various values of the pressure ratio with the numerical error of 0.5%. As expected, no influence of the quantum effects on the flow-rate has been detected for the temperature 300 K. Though, the quantum approach requires less computational effort than the classical one at this temperature. For temperatures lower than 300 K, the influence of the quantum effects exceed the numerical error and reaches 41% at the temperature of 3 K. In this case, the quantum interaction is the only approach to model gas flows.
1. Introduction The direct simulation Monte Carlo (DSMC) method [1] is widely used to calculate rarefied gas flows in vacuum systems, microsystems, around space vehicles, etc. The open codes SPARTA [2] and FOAM [3] based on this method became a widespread tool used in many technological fields. An essential part of this method is a simulation of intermolecular collisions that requires a physical potential in order to obtain reliable results. Recently, the DSMC method has been generalized to an arbitrary potential [4] using the phenomenological LennardJones potential as an example. Along with phenomenological potentials containing some adjustable parameters, the generalization of the DSMC method allowed us to apply ab initio (AI) potentials [5], which are free from such parameters. Nowadays, the ab initio potentials are available in the open literature, see e.g. Refs. [6–13], practically for all noble gases and their mixtures Thus, the DSMC method based on AI potential [5] became free from adjustable parameters and was used to study the influence of the interatomic potential on various phenomena in rarefied gases [14–18]. In all these works, the intermolecular interaction was considered basing on the classical mechanics which is well justified at high temperatures for heavy gases. As is known [19–25], the interaction of light gases, e.g. helium, hydrogen, at low relative velocity is not classical any more so that the quantum effects must be considered. In
the previous paper [26], it was shown that the classical interaction applied to transport phenomena through helium at low temperatures leads to a significant error of heat flux and shear stress. The aim of the present paper is to evaluate the influence of quantum effects in a benchmark problem of vacuum gas dynamics [27]. More specifically, we are interested in quantum effects only in interatomic iterations. Other effects, like high densities at low temperatures when the interatomic distance is comparable to the de Broglie wavelength, are disregarded here. The benchmark problem considered here is a rarefied gas flow through a thin orifice [28], which is solved for both quantum and classical approaches to intermolecular collisions. It is pointed out the temperature range where the classical approach fails and the quantum theory becomes an unique alternative to calculate gas flows. This is important in many technologies such as cryogenic pumps [29,30], cryogenic systems used in the huge fusion reactor ITER [31,32], monochromatic beams of helium [33,34], acoustic thermometry at a low temperature [35,36], experimental set-up to measure the neutrino mass [37,38], etc. A comparison of computational effort of both approaches shows that the quantum scattering requires less computational time in comparison with the classical approach at high temperatures. Thus, it is suggested to apply the quantum scattering for the whole range of the temperature.
E-mail address: sharipov@fisica.ufpr.br. URL: http://fisica.ufpr.br/sharipov. https://doi.org/10.1016/j.vacuum.2018.07.022 Received 28 May 2018; Received in revised form 27 June 2018; Accepted 15 July 2018 Available online 18 July 2018 0042-207X/ © 2018 Elsevier Ltd. All rights reserved.
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2. Statement of the problem
Table 1 Viscosity μ reported in Ref. [42], pressure pmax corresponding to Λ= 0.005, and equivalent-free-path ℓ at this pressure vs. temperature T.
As was pointed out in Ref. [27], the orifice flow is a good example for benchmark problem in rarefied gas dynamics. Previously, this problem has been solved by the DSMC method based on the hard sphere model [28], variable hard spheres [39], and ab initio potential [18,40] using the classical theory of intermolecular collisions. Consider an orifice of radius a in an infinitesimally thin partition, which separates two semi-infinite chambers. The left chamber contains a gas at a pressure , while the right chamber contains the same gas at a smaller pressure p1 < p0 . The temperatures of the gas in both chambers are equal to T0 . The flow-rate is given in the dimensionless form as
W = M˙ / M˙ 0,
M˙ 0 =
π a2p0 / v0,
T (K)
(1)
where M˙ 0 is the mass flow-rate in the free-molecular regime in the limit p1 / p0 → 0 , v0 = 2kB T0/ m is the most probable molecular speed, m is the atomic mass of the gas, and kB is the Boltzmann constant. The flowrate W is determined by the pressure ratio p1 / p0 and rarefaction parameter
δ = a/ℓ,
ℓ = μ0 v0/ p0 ,
4
He
He
μ
pmax
ℓ
μ
pmax
ℓ
(μPa⋅s)
(Pa)
(μm)
(μPa⋅s)
(Pa)
(μm)
1
0.55920
0.68 × 102
0.61
0.32885
0.10 × 103
0.20
2
0.98706
0.38 × 103
0.27
0.45832
0.59 × 103
0.71 × 10−1
3
1.16502
0.11 × 10 4
0.14
0.71367
0.16 × 10 4
0.49 × 10−1
10
1.93016
0.21 × 105
0.21 × 10−1
2.10186
0.33 × 105
0.13 × 10−1
20
2.97202
0.12 × 106
0.81 × 10−2
3.35477
0.19 × 106
0.52 × 10−2
50
5.31136
0.12 × 107
0.23 × 10−2
6.08412
0.18 × 107
0.15 × 10−2
3. Numerical method The quantum scattering is applied to the intermolecular interactions using the concepts of the differential cross-section (DCS) and total cross-section (TCS) of particles. In contrast to the classical interaction used in the previous works [4,5,14–18], the quantum TCS is not infinite, but it is determined by the relative speed g of two colliding particles. If we denote the DCS as σ (g , cosχ ) , then the TCS is calculated by the integral
(2)
where ℓ is the equivalent-free-path, μ0 is the gas viscosity at the temperature T0 . The calculations have been carried for three values of the pressure ratio p1 / p0 = 0.1, 0.5 and 0.7 in the rarefaction parameter range δ from 0.1 to 50. In contrast to the previous works [18,28,40], here the solution have been obtained for a wide temperature range varying from 1 to 300 K. The viscosities μ0 of 3He and 4He calculated ab initio for a wide range of the temperature are reported in the papers [41,42] and used here to calculate the rarefaction parameter. The DSMC method consists of a decoupling the free-motion of molecules and intermolecular collisions between them during each time steps Δt . As mentioned by Uehling and Uhlenbeck [19,20], the quantum theory introduces two modifications into the classical description of gases: (i) The equilibrium distribution function is changed from the Maxwell-Boltzmann to the Einstein-Bose or Fermi-Dirac distributions. This modification affects the free-motion of particles. (ii) The second modification changes only the elementary process of collision between two particles, which must be treated according to the quantum theory of collisions. Some aspects of solving the Uehling-Uhlenbeck equation [19] including both modifications can be found in Ref. [43]. The first quantum effect affecting the free-motion is significant when the typical de Broglie wavelength λ = h/ 2πmkB T is comparable with the average interatomic distance. Here, h is the Planck constant. The contribution of such quantum effect into the state equation of gas has the order [20,44,45].
Λ= nλ3 = nh3/(2πmkB T )3/2 ,
3
σT (g ) = 2π
∫0
π
σ (g , cosχ )sinχ dχ ,
(4)
where χ is the deflection angle of collisions. Calculations of both σ (g , cosχ ) and σ T based on quantum theory are described in the previous papers [26,46]. According to the no-time-counter version of the DSMC method [1], the number of pairs to be tested for collisions during a time step Δt in a cell of volume Vc having Np model particles reads
Ncoll =
1 Δt Np (Np−1) FN (σT g )max , 2 Vc
(5)
where FN is the number of real particles represented by one model particle and the quantity (σ T g ) max represents a maximum value of the product σ T g in each specific cell. Then, Ncoll pairs within the cell are chosen randomly. If a selected pair of particles satisfies the condition
σT g /(σT g )max > Rf ,
(6)
the post-collision velocities are calculated; otherwise, the pre-collision velocities are left unchanged. Here, Rf is a random fraction varying uniformly from 0 to 1. The relation of the post-collision velocities to pre-collision ones contains the deflection angle χ and impact angle ε, see Eqs.(8.32)–(8.35) from Ref. [47]. The angle ε is chosen randomly from the interval [0,2π], while the deflection angle χ should be calculated using the DCS. To avoid calculations of the DCS for each collisions, look-up tables of the deflection angle χ generated once for some specific gas can be used for any flows of this gas. In the previous work [26], the matrix of the deflection angles has been generated using the ab initio potential reported in Ref. [6] by such a way that all its elements for some specific speed g are equally probable and can be chosen just randomly. The matrices for 3He and 4He provided in Supplementary material to Ref. [26] are based on Ng = 800 nodes of the speed g distributed as
(3)
hence this effect can be neglected under the condition Λ≪ 1. An estimation for the normal conditions (T = 273.15 K and p = 1 atm) leads to the value of this criterion Λ= 6.0 × 10−6 and, surely, the quantum effect is negligible. For low temperatures, say T = 1 K, the condition Λ≪ 1 is met at pressures lower then the atmospheric one. If we assume an uncertainty of 0.5%, then the maximum pressure when the quantum effect can be still neglected is given by pmax = 0.005k B T / λ3. Its values for some temperatures and the equivalent-free-path calculated by (2) with the pressure pmax are given in Table 1. These data show that even for extremely low temperature (T = 1 K), the quantum effect of free-motion is negligible for typical pressures of vacuum systems, i.e. pmax ≈ 1 mbar. At the same time, the equivalent-free-path is still smaller than 1 micron, therefore the intermolecular conditions are not negligible. The condensation of helium happens at pressures quite higher than pmax given in Table 1 so that this phenomenon is also disregarded here. Thereby, under conditions Λ≪ 1, the free-motion of particles is described by the classical mechanics, but the intermolecular interaction is considered on the basis on the quantum theory described below.
gj (m/s) = 400⋅(1.005 j − 1),
1 ≤ j ≤ 800,
(7)
and on Nχ = 100 values of the deflection angles for each speed. The 101 t h column of each matrix provides the values of σ T in Å2 and the last columns contain the corresponding speed values gj in m/s. In classical theory of collisions, the intermolecular potential must be cut-off, i.e. we assume that two particles do not interact with each other when the distance between them exceeds some limit quantity d max . In 147
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Table 2 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.1. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.9756 0.9717 0.9695 0.9720 0.9720 0.9713 0.9726 0.9730
δ=1 4
3
0.9675 0.9732 0.9767 0.9725 0.9731 0.9723 0.9721 0.9723
1.0406 1.0355 1.0325 1.0358 1.0363 1.0365 1.0366 1.0369
He
He
δ=2 4
He
1.0296 1.0394 1.0446 1.0368 1.0363 1.0366 1.0360 1.0371
δ=5
3
4
3
1.1469 1.1408 1.1386 1.1423 1.1424 1.1422 1.1418 1.1424
1.1347 1.1464 1.1502 1.1422 1.1423 1.1419 1.1428 1.1424
1.3286 1.3286 1.3273 1.3283 1.3281 1.3272 1.3281 1.3273
He
He
δ = 10 4
He
He
1.3247 1.3293 1.3291 1.3264 1.3272 1.3277 1.3274 1.3277
3
4
1.4361 1.4363 1.4397 1.4388 1.4389 1.4390 1.4380 1.4381
1.4390 1.4396 1.4346 1.4374 1.4387 1.4381 1.4387 1.4380
He
He
Table 3 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.5. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.5605 0.5575 0.5559 0.5571 0.5578 0.5575 0.5579 0.5581
δ=1 4
3
0.5549 0.5570 0.5623 0.5582 0.5576 0.5577 0.5579 0.5577
0.6195 0.6140 0.6117 0.6146 0.6154 0.6155 0.6148 0.6155
He
He
δ=2 4
He
0.6110 0.6174 0.6227 0.6158 0.6154 0.6150 0.6150 0.6158
δ=5
3
4
3
0.7322 0.7259 0.7214 0.7253 0.7258 0.7259 0.7260 0.7265
0.7195 0.7299 0.7371 0.7271 0.7263 0.7259 0.7260 0.7268
0.9987 0.9888 0.9903 0.9937 0.9941 0.9945 0.9941 0.9943
3
4
3
0.4659 0.4609 0.4581 0.4609 0.4616 0.4616 0.4618 0.4612
0.4566 0.4646 0.4696 0.4618 0.4613 0.4613 0.4618 0.4617
0.6932 0.6879 0.6853 0.6877 0.6881 0.6887 0.6882 0.6885
He
He
δ = 10 4
He
3
4
0.9886 0.9948 0.9883 0.9944 0.9937 0.9934 0.9934 0.9941
1.1938 1.1862 1.1911 1.1928 1.1927 1.1926 1.1926 1.1925
1.1893 1.1910 1.1943 1.1928 1.1919 1.1918 1.1920 1.1924
4
3
4
0.9191 0.9151 0.9150 0.9160 0.9164 0.9172 0.9166 0.9168
0.9129 0.9171 0.9207 0.9163 0.9163 0.9159 0.9165 0.9169
He
He
He
Table 4 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.7. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.3417 0.3405 0.3388 0.3391 0.3394 0.3395 0.3397 0.3399
δ=1 4
3
0.3376 0.3404 0.3438 0.3397 0.3392 0.3398 0.3399 0.3400
0.3834 0.3791 0.3775 0.3797 0.3795 0.3805 0.3803 0.3806
He
He
δ=2 4
He
0.3769 0.3813 0.3851 0.3811 0.3806 0.3800 0.3801 0.3806
He
this case, the TCS is constant and equal to
σT =
2 πd max
Ncoll
He
δ = 10
He
He
0.6828 0.6923 0.6973 0.6885 0.6881 0.6882 0.6886 0.6885
He
He
Table 5 Ratio of computational time applying the classical approach tc to that using the quantum calculation tq at p1 / p0 = 0.1, and δ = 50 .
(8)
and the expression (5) becomes
1 Δt = Np (Np−1) FN σT gmax , 2 Vc
δ=5
T (K)
tc / tq 3
4
4.06 4.17 4.56
3.78 4.05 4.45
He
(9) 50 100 300
while the condition (6) is reduced to g / g max > Rf . The quantity d max should be sufficiently large so that its further increase could not change results of simulation within an adopted error. To estimate the discrepancy of the flow-rate and flow-field due to the classical intermolecular interaction, additional calculations have been carried out applying the classical approach following the technique described in Ref. [4] for the same nodes (7) of the speed g. The number of nodes with respect to the impact parameter b was 500 in the range from 0 to d max . The cut-off distance d max needed to calculate the classical total cross-section (8) was assumed to be 3d 0 , where d 0 is the zero point of the potential V (d 0) = 0 , i.e. where the potential changes it
He
own sign. The value d 0 = 2.64095 Å correspondes to the potential [6] used here so that the TCS is equal to σT = 197. 20 Å2 for all values of the speed gj . The direct simulation Monte Carlo (DSMC) method [1] is applied with the same computational domain described in Ref. [40]. The parameters of the numerical scheme are also the same, namely: the number of model particles is 30 millions, the time increment Δt is equal 148
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Fig. 2. Dimensionless flow-rate W vs. rarefaction parameter δ at T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 1. Dimensionless flow-rate W vs. temperature T0 at δ = 2 : solid line quantum approach, dashed line - classical approach.
computational time needed to solve the problem under consideration applying the classical interaction tc to that using the quantum calculation tq . It shows that the quantum approach reduces four times the computational effort to simulate the orifice flow at the room temperature providing the same results as the classical approach. Since the largest difference between the quantum and classical approach is expected in the transitional regime, the comparison of the flow-rate at δ = 2 obtained by both approaches is performed in Fig. 1. It is not surprise that all four results (3He, 4He, quantum and classical interactions) tend to the same value by increasing the temperature and they are in agreement with each other within the numerical error of 0.5%. However, the deviation of the classical results from the quantum ones increases by decreasing the temperature. The flow-rate obtained by quantum approach weakly varies with the temperature and it has an undulatory behavior at the low temperature, while the flow-rates obtained by the classical approach sharply increase by decreasing the temperature. The difference of the flow-rate obtained by both approaches exceeds the numerical error at the temperature T0 = 50 K and lower. It reaches 41% for 3He at δ = 2 , p1 / p0 = 0.7 , and T0 = 2 K. The difference of the flow-rate for 3He from that for 4He is relatively small and only slightly exceeds the numerical error of 0.5%. Its maximum value of 2.4% corresponds to δ = 2 , p1 / p0 = 0.7 and T0 = 3 K. To show the discrepancy between the quantum and classical
to 0.005a/ v0 , the number of time steps is 105, 4 × 105 and 106 for p1 / p0 = 0.1, 0.5 and 0.7, respectively. These parameters of the computational scheme provide the numerical error of the flow-rate smaller than 0.5%.
4. Results and discussions 4.1. Flow-rate Numerical values of the reduced flow-rate W based on the quantum scattering are given in Tables 2–4 for the pressure ratios p1 / p0 = 0.1, 0.5, and 0.7, respectively. These data show that the dependence of the flow-rate W on the temperature T0 based on the quantum scattering is weak. The relative variation of the flow-rate W due to the temperature variation reaches its maximum in the transitional regime δ = 2 . It is lager for 4He than for 3He and represents 1.4%, 2.5%, and 2.9% at p1 / p0 = 0.1, 0.5 and 0.7, respectively. The relative difference of the flow-rate W for 3He from that for 4He has the same order being 1.2%, 2.2%, and 2.5% at p1 / p0 = 0.1, 0.5 and 0.7, respectively. As mentioned above, the same gas flow has been calculated applying the classical approach too. Table 5 contains the ratio of the
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Fig. 3. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.1, δ = 2 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 4. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.5, δ = 2 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
approaches as a function of the rarefaction parameter, the reduced flow-rate W at T0 = 3 K is plotted against the rarefaction parameter δ in Fig. 2 which shows that the largest difference is observed in the transition regime 0.5 ≤ δ ≤ 10 for all pressure ratios considered here. As expected, the difference vanishes in the free-molecular regime (δ → 0 ), where the intermolecular collisions do not matter, and in the limit of continuous medium (δ → ∞), where the flow-rate does not depend on the intermolecular frequency. Comparing the flow-rate calculated here with the heat flux and shear stress reported in Ref. [26], we can notice a certain similarity in behaviours of these different quantities. First, all quantities based on the quantum approach have the undulatory dependence on the temperature in the range from 1 K to 10 K. However, the heat flux and shear stress Ref. [26] based on the classical interaction are smaller than the corresponding quantities based on the quantum theory. On the contrary, the flow-rate based on classical theory represents the overstated value in comparison with the quantum approach. The fact is that, the dimensionless heat flux and shear stress decrease by increasing the rarefaction parameter, while the reduced flow-rate defined by (1) increases in the same situation. Thus, the tendency of a quantity deviation based on the classical interaction can be predicted from the dependence of this quantity on the rarefaction parameter.
4.2. Flow-field First, let us compare the flow-field in the transitional regime (δ = 2 ) at T0 = 3 K, when the influence of quantum effects on the flow-rate is maximum. The density, temperature and velocity distributions for the pressure ratio p1 / p0 = 0.1 at δ = 2 and T0 = 3 are plotted in Fig. 3. As expected, the difference between the distributions of 3He and 4He based on the quantum collisions is small, while the distribution of 3He based on the classical collisions represents the large deviation from all other situations especially for the temperature and velocity distributions. The behaviors of the same distributions for the pressure ratio p1 / p0 = 0.5 shown in Fig. 4 are similar to those for p1 / p0 = 0.1 with the difference that the temperature T / T0 deviates less from unity and the velocity u x / v0 is smaller. Now, let us consider the flow-field in the hydrodynamic regime (δ = 50 ) where the flow-rates are not affected by the quantum effects according to Fig. 2. The corresponding distributions of density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 for p1 / p0 = 0.1 and p1 / p0 = 0.5 are depicted in Figs. 5 and 6, respectively. For both pressure ratios, the distributions obtained by the quantum and classical approaches for 3He and 4He are exactly the same in the region close to the orifice, i.e. x / a < 2 for p1 / p0 = 0.1 and x / a < 1 for p1 / p0 = 0.5. That is
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Fig. 5. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.1, δ = 50 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 6. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.5, δ = 50 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
why the flow-rates are the same for the large rarefaction parameter. However, in the region far from the orifice, the distributions diverge. The first case ( p1 / p0 = 0.1) represents a formation of the Mach disk past the orifice, i.e. the region where the all quantities sharply change. The bulk velocity in this region varies from a supersonic value to a subsonic one. As Fig. 5 shows, the classical approach leads to sharper variations of all quantities. The behaviours of 3He and 4He based on the quantum collisions are similar to each other, but the difference of the distributions is larger than in the transitional regime (δ = 2 ). The second case ( p1 / p0 = 0.5) presented in Fig. 6 corresponds to a jet formation past the orifice with a small variation of the density. The temperature monotone tends to its equilibrium values, while the velocity also monotone vanishes. The slopes of all curves obtained by the quantum approach are larger than those obtained by the classical theory. To show the whole pattern in the hydrodynamic regime, the flowfields of 3He at δ = 50 , T0 = 3 K for the pressure ratios p1 / p0 = 0.1 and p1 / p0 = 0.5 are shown in Figs. 7 and 8, respectively. The upper part of each graph corresponds to the quantum approach, while the lower part represents the classical interaction. The last pictures of each Figure shows the local Mach number defined as
Ma = u 3m /5k B T ,
(10)
where u is the local bulk speed and T is the local temperature. In case of low pressure in the downstream chamber p1 / p0 = 0.1, the difference of the flow-fields is significant. Far from the orifice x / a > 2 , the variations of all quantities n/ n 0 , T / T0 , and Ma are significantly larger for the classical approach than those for the quantum one. We may say that two approaches lead to a qualitative difference in the flow-fields. The flow-fields at the pressure ratio p1 / p0 = 0.5 qualitatively are similar to each other, but the classical approach leads to a stronger jet in the downstream chamber. 5. Conclusions To estimate the influence of the quantum approach to interatomic interaction, a benchmark problem of rarefied gas dynamics, namely, the orifice flow has been solved for two helium isotopes 3He and 4He over the temperature range from 1 K to 300 K. The calculations have been carried out by both quantum and classical approaches. As expected, no influence of the quantum effects on the flow-rate has been detected within the numerical error of 0.5% for the temperature 300 K.
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Fig. 7. Fields of density n/ n 0 , temperature T / T0 and local Mach number at p1 / p0 = 0.1, δ = 50 and T0 = 3 K.
Fig. 8. Fields of density n/ n 0 , temperature T / T0 and local Mach number at p1 / p0 = 0.5, δ = 50 and T0 = 3 K.
However, the quantum approach requires significantly less computational effort than the classical one at this temperature. Such a difference is due to the fact that the total cross-section calculated by the quantum approach is relatively small, while in the frame of classical theory, the total cross-section is not well determined so that a relatively large value is usually assumed for it. For temperatures lower than 300 K, the influence of the quantum effects exceeds the numerical error and increases by decreasing the temperature. The flow-rates of fermions 3He and bosons 4He are close to each other with the maximum relative difference equal 2.5% reached at p1 / p0 = 0.7 , δ = 2 and T0 = 3 K. The flow-rates for both isotopes have an undulatory dependence on the temperature at T < 10 K. The difference between the flow-rates obtained by the quantum and classical approaches can reach 41%. The flow-field obtained by these two approached is also different even in situations when the corresponding flow-rates are the same. It should be noted that the influence of quantum effect can be larger for flows with a larger temperature variation, e.g., supersonic flows. Usually, flows of gaseous mixtures [14,15] are more sensitive to intermolecular potential than single gas flows so that the influence of quantum effect in case of mixtures also can be larger. To obtain the results reported here, the matrices of the deflection angle for 3He and 4He provided in Supplemental Material to Ref. [26] have been used. The matrices are universal and can be used to model any flow of helium in the temperature range from 1 K to any temperature when no ionization happens. An implementation of the deflection angle matrices into a DSMC codes like SPARTA [2] or FOAM [3] requires a modification of few lines.
Acknowledgments The author thanks CNPq (Brazil) for the support of his research, grant 303697/2014-8. References [1] G.A. Bird, The DSMC Method, (2013). [2] A. Klothakis, I. Nikolos, T. Koehler, M. Gallis, S. Plimpton, Validation simulations of the DSMC code SPARTA, AIP Conference Proceedings 1786, 2016, p. 050016. [3] T. Scanlon, E. Roohi, C. White, M. Darbandi, J. Reese, An open source, parallel DSMC code for rarefied gas flows in arbitrary geometries, Comput. Fluids 39 (2010) 2078–2089. [4] F. Sharipov, J.L. Strapasson, Direct simulation Monte Carlo method for an arbitrary intermolecular potential, Phys. Fluids 24 (1) (2012) 011703. [5] F. Sharipov, J.L. Strapasson, Ab initio simulation of transport phenomena in rarefied gases, Phys. Rev. E 86 (3) (2012) 031130. [6] M. Przybytek, W. Cencek, J. Komasa, G. Łach, B. Jeziorski, K. Szalewicz, Relativistic and quantum electrodynamics effects in the helium pair potential, Phys. Rev. Lett. 104 (2010) 183003 Erratum in Phys. Rev. Lett. 108 , 129902 (2012). [7] S.M. Cybulski, R.R. Toczylowski, Ground state potential energy curves for He2, Ne2, Ar2, He-Ne, He-Ar, and Ne-Ar: a coupled-cluster study, J. Chem. Phys. 111 (23) (1999) 10520–10528. [8] A. Baranowska, S.B. Capelo, B. Fernandez, New basis sets for the evaluation of interaction energies: an ab initio study of the He-He, Ne-Ne, Ar-Ar, He-Ne, He-Ar and Ne-Ar van der Waals complex internuclear potentials and ro-vibrational spectra, Phys. Chem. Chem. Phys. 12 (41) (2010) 13586–13596. [9] R. Hellmann, E. Bich, E. Vogel, Ab initio potential energy curve for the helium atom pair and thermophysical properties of dilute helium gas. I. Helium-helium interatomic potential, Mol. Phys. 105 (23–24) (2007) 3013–3023. [10] R. Hellmann, E. Bich, E. Vogel, Ab initio potential energy curve for the neon atom pair and thermophysical properties of the dilute neon gas. I. Neon-neon interatomic potential and rovibrational spectra, Mol. Phys. 106 (1) (2008) 133–140. [11] B. Jäger, R. Hellmann, E. Bich, E. Vogel, Ab initio pair potential energy curve for the
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Influence of quantum intermolecular interaction on internal flows of rarefied gases
T
Felix Sharipov Departamento de Física, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Quantum scattering Orifice flow Direct simulation Monte Carlo ab initio potential
In order to model gaseous flows over the whole temperature range beginning from 1 K, the intermolecular interaction should be considered on the basis of quantum approach. Such a consideration becomes important in case of light gases like helium and hydrogen. Recently, the direct simulation Monte Carlo (DSMC) method widely used to calculate flows of gases has been generalized to implement the quantum approach to intermolecular collisions. To evaluate the influence of the quantum scattering on typical flows of light gases, a benchmark problem has been solved for two helium isotopes 3He and 4He using an ab initio potential. More specifically, the flow-rate and flow-field of helium flowing through an orifice have been calculated over the temperature range from 1 K to 300 K for various values of the pressure ratio with the numerical error of 0.5%. As expected, no influence of the quantum effects on the flow-rate has been detected for the temperature 300 K. Though, the quantum approach requires less computational effort than the classical one at this temperature. For temperatures lower than 300 K, the influence of the quantum effects exceed the numerical error and reaches 41% at the temperature of 3 K. In this case, the quantum interaction is the only approach to model gas flows.
1. Introduction The direct simulation Monte Carlo (DSMC) method [1] is widely used to calculate rarefied gas flows in vacuum systems, microsystems, around space vehicles, etc. The open codes SPARTA [2] and FOAM [3] based on this method became a widespread tool used in many technological fields. An essential part of this method is a simulation of intermolecular collisions that requires a physical potential in order to obtain reliable results. Recently, the DSMC method has been generalized to an arbitrary potential [4] using the phenomenological LennardJones potential as an example. Along with phenomenological potentials containing some adjustable parameters, the generalization of the DSMC method allowed us to apply ab initio (AI) potentials [5], which are free from such parameters. Nowadays, the ab initio potentials are available in the open literature, see e.g. Refs. [6–13], practically for all noble gases and their mixtures Thus, the DSMC method based on AI potential [5] became free from adjustable parameters and was used to study the influence of the interatomic potential on various phenomena in rarefied gases [14–18]. In all these works, the intermolecular interaction was considered basing on the classical mechanics which is well justified at high temperatures for heavy gases. As is known [19–25], the interaction of light gases, e.g. helium, hydrogen, at low relative velocity is not classical any more so that the quantum effects must be considered. In
the previous paper [26], it was shown that the classical interaction applied to transport phenomena through helium at low temperatures leads to a significant error of heat flux and shear stress. The aim of the present paper is to evaluate the influence of quantum effects in a benchmark problem of vacuum gas dynamics [27]. More specifically, we are interested in quantum effects only in interatomic iterations. Other effects, like high densities at low temperatures when the interatomic distance is comparable to the de Broglie wavelength, are disregarded here. The benchmark problem considered here is a rarefied gas flow through a thin orifice [28], which is solved for both quantum and classical approaches to intermolecular collisions. It is pointed out the temperature range where the classical approach fails and the quantum theory becomes an unique alternative to calculate gas flows. This is important in many technologies such as cryogenic pumps [29,30], cryogenic systems used in the huge fusion reactor ITER [31,32], monochromatic beams of helium [33,34], acoustic thermometry at a low temperature [35,36], experimental set-up to measure the neutrino mass [37,38], etc. A comparison of computational effort of both approaches shows that the quantum scattering requires less computational time in comparison with the classical approach at high temperatures. Thus, it is suggested to apply the quantum scattering for the whole range of the temperature.
E-mail address: sharipov@fisica.ufpr.br. URL: http://fisica.ufpr.br/sharipov. https://doi.org/10.1016/j.vacuum.2018.07.022 Received 28 May 2018; Received in revised form 27 June 2018; Accepted 15 July 2018 Available online 18 July 2018 0042-207X/ © 2018 Elsevier Ltd. All rights reserved.
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2. Statement of the problem
Table 1 Viscosity μ reported in Ref. [42], pressure pmax corresponding to Λ= 0.005, and equivalent-free-path ℓ at this pressure vs. temperature T.
As was pointed out in Ref. [27], the orifice flow is a good example for benchmark problem in rarefied gas dynamics. Previously, this problem has been solved by the DSMC method based on the hard sphere model [28], variable hard spheres [39], and ab initio potential [18,40] using the classical theory of intermolecular collisions. Consider an orifice of radius a in an infinitesimally thin partition, which separates two semi-infinite chambers. The left chamber contains a gas at a pressure , while the right chamber contains the same gas at a smaller pressure p1 < p0 . The temperatures of the gas in both chambers are equal to T0 . The flow-rate is given in the dimensionless form as
W = M˙ / M˙ 0,
M˙ 0 =
π a2p0 / v0,
T (K)
(1)
where M˙ 0 is the mass flow-rate in the free-molecular regime in the limit p1 / p0 → 0 , v0 = 2kB T0/ m is the most probable molecular speed, m is the atomic mass of the gas, and kB is the Boltzmann constant. The flowrate W is determined by the pressure ratio p1 / p0 and rarefaction parameter
δ = a/ℓ,
ℓ = μ0 v0/ p0 ,
4
He
He
μ
pmax
ℓ
μ
pmax
ℓ
(μPa⋅s)
(Pa)
(μm)
(μPa⋅s)
(Pa)
(μm)
1
0.55920
0.68 × 102
0.61
0.32885
0.10 × 103
0.20
2
0.98706
0.38 × 103
0.27
0.45832
0.59 × 103
0.71 × 10−1
3
1.16502
0.11 × 10 4
0.14
0.71367
0.16 × 10 4
0.49 × 10−1
10
1.93016
0.21 × 105
0.21 × 10−1
2.10186
0.33 × 105
0.13 × 10−1
20
2.97202
0.12 × 106
0.81 × 10−2
3.35477
0.19 × 106
0.52 × 10−2
50
5.31136
0.12 × 107
0.23 × 10−2
6.08412
0.18 × 107
0.15 × 10−2
3. Numerical method The quantum scattering is applied to the intermolecular interactions using the concepts of the differential cross-section (DCS) and total cross-section (TCS) of particles. In contrast to the classical interaction used in the previous works [4,5,14–18], the quantum TCS is not infinite, but it is determined by the relative speed g of two colliding particles. If we denote the DCS as σ (g , cosχ ) , then the TCS is calculated by the integral
(2)
where ℓ is the equivalent-free-path, μ0 is the gas viscosity at the temperature T0 . The calculations have been carried for three values of the pressure ratio p1 / p0 = 0.1, 0.5 and 0.7 in the rarefaction parameter range δ from 0.1 to 50. In contrast to the previous works [18,28,40], here the solution have been obtained for a wide temperature range varying from 1 to 300 K. The viscosities μ0 of 3He and 4He calculated ab initio for a wide range of the temperature are reported in the papers [41,42] and used here to calculate the rarefaction parameter. The DSMC method consists of a decoupling the free-motion of molecules and intermolecular collisions between them during each time steps Δt . As mentioned by Uehling and Uhlenbeck [19,20], the quantum theory introduces two modifications into the classical description of gases: (i) The equilibrium distribution function is changed from the Maxwell-Boltzmann to the Einstein-Bose or Fermi-Dirac distributions. This modification affects the free-motion of particles. (ii) The second modification changes only the elementary process of collision between two particles, which must be treated according to the quantum theory of collisions. Some aspects of solving the Uehling-Uhlenbeck equation [19] including both modifications can be found in Ref. [43]. The first quantum effect affecting the free-motion is significant when the typical de Broglie wavelength λ = h/ 2πmkB T is comparable with the average interatomic distance. Here, h is the Planck constant. The contribution of such quantum effect into the state equation of gas has the order [20,44,45].
Λ= nλ3 = nh3/(2πmkB T )3/2 ,
3
σT (g ) = 2π
∫0
π
σ (g , cosχ )sinχ dχ ,
(4)
where χ is the deflection angle of collisions. Calculations of both σ (g , cosχ ) and σ T based on quantum theory are described in the previous papers [26,46]. According to the no-time-counter version of the DSMC method [1], the number of pairs to be tested for collisions during a time step Δt in a cell of volume Vc having Np model particles reads
Ncoll =
1 Δt Np (Np−1) FN (σT g )max , 2 Vc
(5)
where FN is the number of real particles represented by one model particle and the quantity (σ T g ) max represents a maximum value of the product σ T g in each specific cell. Then, Ncoll pairs within the cell are chosen randomly. If a selected pair of particles satisfies the condition
σT g /(σT g )max > Rf ,
(6)
the post-collision velocities are calculated; otherwise, the pre-collision velocities are left unchanged. Here, Rf is a random fraction varying uniformly from 0 to 1. The relation of the post-collision velocities to pre-collision ones contains the deflection angle χ and impact angle ε, see Eqs.(8.32)–(8.35) from Ref. [47]. The angle ε is chosen randomly from the interval [0,2π], while the deflection angle χ should be calculated using the DCS. To avoid calculations of the DCS for each collisions, look-up tables of the deflection angle χ generated once for some specific gas can be used for any flows of this gas. In the previous work [26], the matrix of the deflection angles has been generated using the ab initio potential reported in Ref. [6] by such a way that all its elements for some specific speed g are equally probable and can be chosen just randomly. The matrices for 3He and 4He provided in Supplementary material to Ref. [26] are based on Ng = 800 nodes of the speed g distributed as
(3)
hence this effect can be neglected under the condition Λ≪ 1. An estimation for the normal conditions (T = 273.15 K and p = 1 atm) leads to the value of this criterion Λ= 6.0 × 10−6 and, surely, the quantum effect is negligible. For low temperatures, say T = 1 K, the condition Λ≪ 1 is met at pressures lower then the atmospheric one. If we assume an uncertainty of 0.5%, then the maximum pressure when the quantum effect can be still neglected is given by pmax = 0.005k B T / λ3. Its values for some temperatures and the equivalent-free-path calculated by (2) with the pressure pmax are given in Table 1. These data show that even for extremely low temperature (T = 1 K), the quantum effect of free-motion is negligible for typical pressures of vacuum systems, i.e. pmax ≈ 1 mbar. At the same time, the equivalent-free-path is still smaller than 1 micron, therefore the intermolecular conditions are not negligible. The condensation of helium happens at pressures quite higher than pmax given in Table 1 so that this phenomenon is also disregarded here. Thereby, under conditions Λ≪ 1, the free-motion of particles is described by the classical mechanics, but the intermolecular interaction is considered on the basis on the quantum theory described below.
gj (m/s) = 400⋅(1.005 j − 1),
1 ≤ j ≤ 800,
(7)
and on Nχ = 100 values of the deflection angles for each speed. The 101 t h column of each matrix provides the values of σ T in Å2 and the last columns contain the corresponding speed values gj in m/s. In classical theory of collisions, the intermolecular potential must be cut-off, i.e. we assume that two particles do not interact with each other when the distance between them exceeds some limit quantity d max . In 147
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Table 2 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.1. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.9756 0.9717 0.9695 0.9720 0.9720 0.9713 0.9726 0.9730
δ=1 4
3
0.9675 0.9732 0.9767 0.9725 0.9731 0.9723 0.9721 0.9723
1.0406 1.0355 1.0325 1.0358 1.0363 1.0365 1.0366 1.0369
He
He
δ=2 4
He
1.0296 1.0394 1.0446 1.0368 1.0363 1.0366 1.0360 1.0371
δ=5
3
4
3
1.1469 1.1408 1.1386 1.1423 1.1424 1.1422 1.1418 1.1424
1.1347 1.1464 1.1502 1.1422 1.1423 1.1419 1.1428 1.1424
1.3286 1.3286 1.3273 1.3283 1.3281 1.3272 1.3281 1.3273
He
He
δ = 10 4
He
He
1.3247 1.3293 1.3291 1.3264 1.3272 1.3277 1.3274 1.3277
3
4
1.4361 1.4363 1.4397 1.4388 1.4389 1.4390 1.4380 1.4381
1.4390 1.4396 1.4346 1.4374 1.4387 1.4381 1.4387 1.4380
He
He
Table 3 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.5. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.5605 0.5575 0.5559 0.5571 0.5578 0.5575 0.5579 0.5581
δ=1 4
3
0.5549 0.5570 0.5623 0.5582 0.5576 0.5577 0.5579 0.5577
0.6195 0.6140 0.6117 0.6146 0.6154 0.6155 0.6148 0.6155
He
He
δ=2 4
He
0.6110 0.6174 0.6227 0.6158 0.6154 0.6150 0.6150 0.6158
δ=5
3
4
3
0.7322 0.7259 0.7214 0.7253 0.7258 0.7259 0.7260 0.7265
0.7195 0.7299 0.7371 0.7271 0.7263 0.7259 0.7260 0.7268
0.9987 0.9888 0.9903 0.9937 0.9941 0.9945 0.9941 0.9943
3
4
3
0.4659 0.4609 0.4581 0.4609 0.4616 0.4616 0.4618 0.4612
0.4566 0.4646 0.4696 0.4618 0.4613 0.4613 0.4618 0.4617
0.6932 0.6879 0.6853 0.6877 0.6881 0.6887 0.6882 0.6885
He
He
δ = 10 4
He
3
4
0.9886 0.9948 0.9883 0.9944 0.9937 0.9934 0.9934 0.9941
1.1938 1.1862 1.1911 1.1928 1.1927 1.1926 1.1926 1.1925
1.1893 1.1910 1.1943 1.1928 1.1919 1.1918 1.1920 1.1924
4
3
4
0.9191 0.9151 0.9150 0.9160 0.9164 0.9172 0.9166 0.9168
0.9129 0.9171 0.9207 0.9163 0.9163 0.9159 0.9165 0.9169
He
He
He
Table 4 Reduced flow-rate W vs. rarefaction parameter δ and temperature T0 at p1 / p0 = 0.7. T0 (K)
W δ = 0.5 3
1 2 3 10 20 50 100 300
He
0.3417 0.3405 0.3388 0.3391 0.3394 0.3395 0.3397 0.3399
δ=1 4
3
0.3376 0.3404 0.3438 0.3397 0.3392 0.3398 0.3399 0.3400
0.3834 0.3791 0.3775 0.3797 0.3795 0.3805 0.3803 0.3806
He
He
δ=2 4
He
0.3769 0.3813 0.3851 0.3811 0.3806 0.3800 0.3801 0.3806
He
this case, the TCS is constant and equal to
σT =
2 πd max
Ncoll
He
δ = 10
He
He
0.6828 0.6923 0.6973 0.6885 0.6881 0.6882 0.6886 0.6885
He
He
Table 5 Ratio of computational time applying the classical approach tc to that using the quantum calculation tq at p1 / p0 = 0.1, and δ = 50 .
(8)
and the expression (5) becomes
1 Δt = Np (Np−1) FN σT gmax , 2 Vc
δ=5
T (K)
tc / tq 3
4
4.06 4.17 4.56
3.78 4.05 4.45
He
(9) 50 100 300
while the condition (6) is reduced to g / g max > Rf . The quantity d max should be sufficiently large so that its further increase could not change results of simulation within an adopted error. To estimate the discrepancy of the flow-rate and flow-field due to the classical intermolecular interaction, additional calculations have been carried out applying the classical approach following the technique described in Ref. [4] for the same nodes (7) of the speed g. The number of nodes with respect to the impact parameter b was 500 in the range from 0 to d max . The cut-off distance d max needed to calculate the classical total cross-section (8) was assumed to be 3d 0 , where d 0 is the zero point of the potential V (d 0) = 0 , i.e. where the potential changes it
He
own sign. The value d 0 = 2.64095 Å correspondes to the potential [6] used here so that the TCS is equal to σT = 197. 20 Å2 for all values of the speed gj . The direct simulation Monte Carlo (DSMC) method [1] is applied with the same computational domain described in Ref. [40]. The parameters of the numerical scheme are also the same, namely: the number of model particles is 30 millions, the time increment Δt is equal 148
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Fig. 2. Dimensionless flow-rate W vs. rarefaction parameter δ at T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 1. Dimensionless flow-rate W vs. temperature T0 at δ = 2 : solid line quantum approach, dashed line - classical approach.
computational time needed to solve the problem under consideration applying the classical interaction tc to that using the quantum calculation tq . It shows that the quantum approach reduces four times the computational effort to simulate the orifice flow at the room temperature providing the same results as the classical approach. Since the largest difference between the quantum and classical approach is expected in the transitional regime, the comparison of the flow-rate at δ = 2 obtained by both approaches is performed in Fig. 1. It is not surprise that all four results (3He, 4He, quantum and classical interactions) tend to the same value by increasing the temperature and they are in agreement with each other within the numerical error of 0.5%. However, the deviation of the classical results from the quantum ones increases by decreasing the temperature. The flow-rate obtained by quantum approach weakly varies with the temperature and it has an undulatory behavior at the low temperature, while the flow-rates obtained by the classical approach sharply increase by decreasing the temperature. The difference of the flow-rate obtained by both approaches exceeds the numerical error at the temperature T0 = 50 K and lower. It reaches 41% for 3He at δ = 2 , p1 / p0 = 0.7 , and T0 = 2 K. The difference of the flow-rate for 3He from that for 4He is relatively small and only slightly exceeds the numerical error of 0.5%. Its maximum value of 2.4% corresponds to δ = 2 , p1 / p0 = 0.7 and T0 = 3 K. To show the discrepancy between the quantum and classical
to 0.005a/ v0 , the number of time steps is 105, 4 × 105 and 106 for p1 / p0 = 0.1, 0.5 and 0.7, respectively. These parameters of the computational scheme provide the numerical error of the flow-rate smaller than 0.5%.
4. Results and discussions 4.1. Flow-rate Numerical values of the reduced flow-rate W based on the quantum scattering are given in Tables 2–4 for the pressure ratios p1 / p0 = 0.1, 0.5, and 0.7, respectively. These data show that the dependence of the flow-rate W on the temperature T0 based on the quantum scattering is weak. The relative variation of the flow-rate W due to the temperature variation reaches its maximum in the transitional regime δ = 2 . It is lager for 4He than for 3He and represents 1.4%, 2.5%, and 2.9% at p1 / p0 = 0.1, 0.5 and 0.7, respectively. The relative difference of the flow-rate W for 3He from that for 4He has the same order being 1.2%, 2.2%, and 2.5% at p1 / p0 = 0.1, 0.5 and 0.7, respectively. As mentioned above, the same gas flow has been calculated applying the classical approach too. Table 5 contains the ratio of the
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Fig. 3. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.1, δ = 2 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 4. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.5, δ = 2 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
approaches as a function of the rarefaction parameter, the reduced flow-rate W at T0 = 3 K is plotted against the rarefaction parameter δ in Fig. 2 which shows that the largest difference is observed in the transition regime 0.5 ≤ δ ≤ 10 for all pressure ratios considered here. As expected, the difference vanishes in the free-molecular regime (δ → 0 ), where the intermolecular collisions do not matter, and in the limit of continuous medium (δ → ∞), where the flow-rate does not depend on the intermolecular frequency. Comparing the flow-rate calculated here with the heat flux and shear stress reported in Ref. [26], we can notice a certain similarity in behaviours of these different quantities. First, all quantities based on the quantum approach have the undulatory dependence on the temperature in the range from 1 K to 10 K. However, the heat flux and shear stress Ref. [26] based on the classical interaction are smaller than the corresponding quantities based on the quantum theory. On the contrary, the flow-rate based on classical theory represents the overstated value in comparison with the quantum approach. The fact is that, the dimensionless heat flux and shear stress decrease by increasing the rarefaction parameter, while the reduced flow-rate defined by (1) increases in the same situation. Thus, the tendency of a quantity deviation based on the classical interaction can be predicted from the dependence of this quantity on the rarefaction parameter.
4.2. Flow-field First, let us compare the flow-field in the transitional regime (δ = 2 ) at T0 = 3 K, when the influence of quantum effects on the flow-rate is maximum. The density, temperature and velocity distributions for the pressure ratio p1 / p0 = 0.1 at δ = 2 and T0 = 3 are plotted in Fig. 3. As expected, the difference between the distributions of 3He and 4He based on the quantum collisions is small, while the distribution of 3He based on the classical collisions represents the large deviation from all other situations especially for the temperature and velocity distributions. The behaviors of the same distributions for the pressure ratio p1 / p0 = 0.5 shown in Fig. 4 are similar to those for p1 / p0 = 0.1 with the difference that the temperature T / T0 deviates less from unity and the velocity u x / v0 is smaller. Now, let us consider the flow-field in the hydrodynamic regime (δ = 50 ) where the flow-rates are not affected by the quantum effects according to Fig. 2. The corresponding distributions of density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 for p1 / p0 = 0.1 and p1 / p0 = 0.5 are depicted in Figs. 5 and 6, respectively. For both pressure ratios, the distributions obtained by the quantum and classical approaches for 3He and 4He are exactly the same in the region close to the orifice, i.e. x / a < 2 for p1 / p0 = 0.1 and x / a < 1 for p1 / p0 = 0.5. That is
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Fig. 5. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.1, δ = 50 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
Fig. 6. Density n/ n 0 , temperature T / T0 and bulk velocity u x / v0 distributions along the axis at p1 / p0 = 0.5, δ = 50 and T0 = 3 K: solid line - quantum approach, dashed line - classical approach.
why the flow-rates are the same for the large rarefaction parameter. However, in the region far from the orifice, the distributions diverge. The first case ( p1 / p0 = 0.1) represents a formation of the Mach disk past the orifice, i.e. the region where the all quantities sharply change. The bulk velocity in this region varies from a supersonic value to a subsonic one. As Fig. 5 shows, the classical approach leads to sharper variations of all quantities. The behaviours of 3He and 4He based on the quantum collisions are similar to each other, but the difference of the distributions is larger than in the transitional regime (δ = 2 ). The second case ( p1 / p0 = 0.5) presented in Fig. 6 corresponds to a jet formation past the orifice with a small variation of the density. The temperature monotone tends to its equilibrium values, while the velocity also monotone vanishes. The slopes of all curves obtained by the quantum approach are larger than those obtained by the classical theory. To show the whole pattern in the hydrodynamic regime, the flowfields of 3He at δ = 50 , T0 = 3 K for the pressure ratios p1 / p0 = 0.1 and p1 / p0 = 0.5 are shown in Figs. 7 and 8, respectively. The upper part of each graph corresponds to the quantum approach, while the lower part represents the classical interaction. The last pictures of each Figure shows the local Mach number defined as
Ma = u 3m /5k B T ,
(10)
where u is the local bulk speed and T is the local temperature. In case of low pressure in the downstream chamber p1 / p0 = 0.1, the difference of the flow-fields is significant. Far from the orifice x / a > 2 , the variations of all quantities n/ n 0 , T / T0 , and Ma are significantly larger for the classical approach than those for the quantum one. We may say that two approaches lead to a qualitative difference in the flow-fields. The flow-fields at the pressure ratio p1 / p0 = 0.5 qualitatively are similar to each other, but the classical approach leads to a stronger jet in the downstream chamber. 5. Conclusions To estimate the influence of the quantum approach to interatomic interaction, a benchmark problem of rarefied gas dynamics, namely, the orifice flow has been solved for two helium isotopes 3He and 4He over the temperature range from 1 K to 300 K. The calculations have been carried out by both quantum and classical approaches. As expected, no influence of the quantum effects on the flow-rate has been detected within the numerical error of 0.5% for the temperature 300 K.
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Fig. 7. Fields of density n/ n 0 , temperature T / T0 and local Mach number at p1 / p0 = 0.1, δ = 50 and T0 = 3 K.
Fig. 8. Fields of density n/ n 0 , temperature T / T0 and local Mach number at p1 / p0 = 0.5, δ = 50 and T0 = 3 K.
However, the quantum approach requires significantly less computational effort than the classical one at this temperature. Such a difference is due to the fact that the total cross-section calculated by the quantum approach is relatively small, while in the frame of classical theory, the total cross-section is not well determined so that a relatively large value is usually assumed for it. For temperatures lower than 300 K, the influence of the quantum effects exceeds the numerical error and increases by decreasing the temperature. The flow-rates of fermions 3He and bosons 4He are close to each other with the maximum relative difference equal 2.5% reached at p1 / p0 = 0.7 , δ = 2 and T0 = 3 K. The flow-rates for both isotopes have an undulatory dependence on the temperature at T < 10 K. The difference between the flow-rates obtained by the quantum and classical approaches can reach 41%. The flow-field obtained by these two approached is also different even in situations when the corresponding flow-rates are the same. It should be noted that the influence of quantum effect can be larger for flows with a larger temperature variation, e.g., supersonic flows. Usually, flows of gaseous mixtures [14,15] are more sensitive to intermolecular potential than single gas flows so that the influence of quantum effect in case of mixtures also can be larger. To obtain the results reported here, the matrices of the deflection angle for 3He and 4He provided in Supplemental Material to Ref. [26] have been used. The matrices are universal and can be used to model any flow of helium in the temperature range from 1 K to any temperature when no ionization happens. An implementation of the deflection angle matrices into a DSMC codes like SPARTA [2] or FOAM [3] requires a modification of few lines.
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