- Email: [email protected]
Rarefied gas flow through a long rectangular channel of variable cross section
Vacuum 101 (2014) 328e332
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Rarefied gas flow through a long rectangular channel of variable cross section I. Graur, M.T. Ho* IUSTI UMR CNRS 7343 Laboratory, Polytech Marseille, Aix Marseille Université, 5 rue Enrico Fermi, 13453 Marseille, Provence, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 June 2013 Received in revised form 20 July 2013 Accepted 22 July 2013
The method, proposed previously by other authors, is applied here to calculate a mass flow rate of rarefied gas through a long rectangular channel of variable cross-section aspect ratio. The gas flow through this channel is generated by the pressure and/or temperature gradient. The method is based on the results obtained previously on the basis of the kinetic equation and it requires very modest computational efforts. As a demonstration of its application some examples of the mass flow rate calculations for the isothermal and non-isothermal flows through the channels with variable rectangular cross sections are given. The analytical expressions in the case of the hydrodynamic and free molecular flow regimes are proposed. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Rarefied gas Long rectangular channel Variable cross section Mass flow rate
1. Introduction The gas flow through a long channel of a rectangular cross section is a practical problem in the MEMS and vacuum technology applications. This kind of flow was largely studied on the basis of the kinetic theory and the detailed review may be found in Ref. [1]. However in several applications a cross section varies along the channel. As examples of such kind of flow the leakage through compressor valves [2] and the flow in the micro bearing [3,4], may be given. A few results of the numerical simulations of the flow through variable conical cross section [5] and rectangular and conical sections [6] were found in the literature. In this work we apply the approach, proposed in Ref. [5] and implemented to the flow through the tube of a variable radius, to the channel of a variable rectangular cross section. The proposed technique allows to calculate the mass flow rate through a long channel with variable rectangular cross section for arbitrary pressure and temperature ratios in large range of gas rarefaction. 2. Problem statement Consider two reservoirs containing the same gas and connected by a long rectangular channel of variable cross-section aspect ratio. The channel width w is supposed to be the same, but the channel
height h varies continuously along the channel. Its value is equal to h1 in the first reservoir and h2 in the second reservoir. We assume that the relation max(h) w keeps along the channel. The first reservoir is maintained at the pressure p1 and temperature T1, while the pressure and temperature in the second reservoir are p2 and T2, respectively. We will calculate the mass flow rate through this channel in the whole range of the gas rarefaction. The results will be given in terms of reduced mass flow rate defined as follows
L G ¼ 2 h1 wp1
0042-207X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.vacuum.2013.07.047
(1)
here L is the channel length, m is the molecular mass of the gas, k is the _ is the mass flow rate through the channel. Boltzmann constant, M We assume the channel to be long enough (max(h)L) that the end effects can be neglected. According to Refs. [7,8], the end influence in the hydrodynamic regime has the order of max(h)/L, while in the free molecular regime [9] its order is (max(h)/L)ln(L/ max(h)). These estimations are confirmed by the numerical results for a tube flow [10]. The second assumption is that the pressure and temperature gradients can be considered small in any cross section of the channel
xP ¼ * Corresponding author. Tel.: (þ33) 04 91 10 68 77. E-mail addresses: [email protected] (I. Graur), [email protected] (M.T. Ho).
rffiffiffiffiffiffiffiffiffiffi 2kT1 _ M; m
h dp ; p dx
jxP j 1;
xT ¼
h dT ; T dx
jxT j 1;
(2)
where x is the longitudinal coordinate in the flow direction with the origin in the first reservoir, h ¼ h(x) is a local channel’s height,
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
p ¼ p(x) and T ¼ T(x) are a local pressure and temperature, respectively. Under such a condition the mass flow rate in a cross section is calculated as [11]
GT ¼ GP =2:
rffiffiffiffiffiffiffiffiffi _ ¼ hwp m GP x þ GT x ; M p T 2kT
1 2w 1 þ ; GP ¼ pffiffiffiffi ln h 2 p
(3)
329
(9)
In the case h/w/0 the two coefficients become [11]
GT ¼ GP =2:
(10)
where the coefficients GP and GT depend on the local gas rarefaction parameter d, defined as
3.2. Hydrodynamic regime
rffiffiffiffiffiffiffiffiffi hp m d¼ ; mðTÞ 2kT
In the hydrodynamic flow regime (d/N) the reduced mass flow rate is equal to [18]
(4)
m is the gas viscosity, which depends on the local temperature T(x). The values of the coefficients GP ¼ GP(d) and GT ¼ GT(d) in the
case of the gas flow through a rectangular cross section channel for different flow regimes were obtained by different authors from the solution of the linearized BGK and S-model kinetic equations, or from the linearized Boltzmann equation for the diffuse or diffusespecular boundary conditions [12,13,14,15]. The detailed review on these numerical results may be found in Ref. [1]. The results used in this work will be given in the next section. Following Ref. [5], from Eqs. (1)e(4) the differential equation is obtained
rffiffiffiffiffi L h 2 T1 dp p dT þ GT ðd Þ : GP ðdÞ G ¼ p1 h1 dx T dx T
(5)
If we assume that the temperature distribution along the channel is known we can obtain the corresponding pressure distribution and the reduced total mass flow rate G through the channel. Eq. (5) is solved numerically using the following finite difference scheme [5]:
piþ1
sffiffiffiffiffi! Dx pi Tiþ1 Ti p1 G h1 2 Ti : GT ðdi Þ ¼ pi þ Dx L Ti hi T1 GP ðdi Þ
(6)
In the previous equation Dx ¼ L/N is the grid step in the x direction, 0 i N, pi, Ti, hi are the pressure, temperature and channel height in i grid point, respectively. The rarefaction parameter di is calculated as
sffiffiffiffiffi pi hi mðT1 Þ T1 di ¼ d1 ; p1 h1 mðTi Þ Ti
rffiffiffiffiffiffiffiffiffiffi p h m d1 ¼ 1 1 : mðT1 Þ 2kT1
(7)
Eq. (6) is solved numerically by the shooting method (Ref. [16], Section 7.3) with the boundary condition p(x ¼ 0) ¼ p1. G is a parameter of eq. (6) which is found satisfying the second boundary condition p(x ¼ L) ¼ p2. 3. Determination of coefficients GP and GT 3.1. Free molecular regime In the free molecular regime (d ¼ 0) the coefficient GP was found analytically in Ref. [17]
0 0 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 4w @ h h2 A w w2 @ ln þ 1 þ 2 þ ln þ 1þ 2A GP ¼ pffiffiffiffi w h w h p h 3 1 w w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: 3 w þ w2 þ h2 h þ w2 þ h2 The coefficient GT for the case of d ¼ 0 is equal to
(8)
GPh ¼
d
1
6
! N 192 h X tan hð0:5pð2n þ 1Þw=hÞ : p5 w n ¼ 0 ð2n þ 1Þ5
(11)
In the slip flow regime the slip correction must be added, which was found in Refs. [17,19,20]. N 4 256 h X tan hð0:5pð2n þ 1Þw=hÞ 5 3 p w n¼0 ð1 þ 2nÞ5 ! X N 32 h tan hð0:5pð2n þ 1Þw=hÞ 4 1 : w n¼0 p ð1 þ 2nÞ4
GPslip ¼ sp
(12)
Therefore the coefficient GP for the hydrodynamic and slip regimes (d > 40) becomes
GP ¼ GPh þ GPslip :
(13)
The thermal creep coefficient in these flow regimes is equal to [11]
GT ¼
sT : d
(14)
The coefficients sp and sT in Eqs. (12) and (14) are the velocity slip and the thermal slip coefficients, equal to 1.016 [21] and 1.175 [22], respectively. It is to be noticed that expressions (12) and (14) are valid for any slip coefficients, e.g. for non-diffuse scattering or gaseous mixtures [23]. 3.3. Transitional flow regime In order to solve numerically Eq. (6) we need to known the values of the rarefaction parameter at arbitrary couple of two values: the rarefaction parameter and the channel cross-section aspect ratio (d,h/w). The two-dimensional grid of values GP(d,h/w) and GT(d,h/w) is formed from the numerical data of Ref. [11], where the quantities GP(d) and GT(d) are calculated for large range of the gas rarefaction parameter and for four different h/w aspect ratios h/ w ¼ 0,0.05,0.1,1, where h/w ¼ 0 corresponds to the case of the flow between two parallel plates. To complete the data of Ref. [11] the same approach is applied to obtain a solution of the linearized S-model kinetic equation [24]. The additional values of the channel cross-section aspect ratios h/ w ¼ 0.0367,0.25,0.5,0.75 and for the rarefaction parameter varies from 0.001 to 40 are calculated. The 1000 1000 grid in the physical space is implemented, the polar coordinates are used in the molecular velocity space with 200 points for the velocity directions and 25 points in the velocity magnitude distributed according to the Gaussian rule. The values of the GP and GT coefficients obtained in the present work are presented in Tables 1 and 2. The method of two-dimensional interpolation is employed to calculate the values GP and GT in an arbitrary point (d*,(h/w)*) from two-dimensional grid of GP(d,h/w) and GT(d,h/w) values. Below we explain the interpolation steps realized to calculate the value of
330
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
GP(d*,(h/w)*) in a point (d*, (h/w)*), which does not coincide with any values of d and h/w reported in two-dimensional Table 1. The interpolation is accomplished by the following steps.
Table 2 Coefficient GT obtained in Ref. [11] and in present work* from the numerical solution of the linearized S-model kinetic equation.
d
A block of the reduced mass flow rate GP(di,(h/ w)j)i ¼ 1.m þ 1,j ¼ 1.n þ 1 values containing a point (d*,(h/w)*) is searched and extracted using the bisection method (Chapter 3 of Ref. [25]) from the ordered tabular data, see Table 1, which contains the numerical values of GP obtained on the basis of kinetic equation [11]. Here, i and j are the indexes for the row d and the column h/w, respectively. The values m and n are the orders of interpolation in d-direction and in h/w-direction, respectively. Using n þ 1 values of GP in h/w-direction, the nth order interpolation with Neville’s algorithm (given in Chapter 3 of Ref. [25]) is carried out for each of m þ 1 extracted values in d-direction and a set of values GP(di,(h/w)*)i ¼ 1.m þ 1 is obtained. Then mth order interpolation in the d-direction is performed in order to obtain the value of GP(d*,(h/w)*). The value of GT at an arbitrary point (d*,(h/w)*) is calculated by the interpolation in the same fashion as for the value of GP. The accuracy of the interpolation in the h/w-direction is evaluated by comparing the results of the numerical solution of the linearized S-model kinetic equation for the h/w ¼ 0.0367 and the interpolation results for the same values of h/w ratio, see Table 3. Analyzing the results of Table 3 one can see that the accuracy of the interpolation is of the order of 2%. The similar verification was carried out for the interpolation in the d direction. The same type of the comparison was carried out by comparing the solution of the linearized kinetic equation and the interpolation results. When analyzing the results given in Table 4 one can see that the difference between the corresponding values does not also exceed 2%. However, when d/0 (for the small values of h/w) or d/N the coefficients GP and GT change steeply and the use of the sparse grid rises up the interpolation error. In the following Sections the results of the isothermal and nonisothermal flows simulations are given.
Table 1 Coefficient GP obtained in Ref. [11] and in present work* from the numerical solution of the linearized S-model kinetic equation.
d
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
h/w
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
0 [11]
0.0367*
0.05 [11]
0.1 [11]
0.25*
0.5*
0.75*
1 [11]
4.273 3.050 2.713 2.400 2.306 2.120 2.038 1.823 1.651 1.613 1.561 1.549 1.565 1.606 1.855 2.000 2.456 2.772 3.577 4.393 6.040 7.695
2.510 2.345 2.238 2.101 2.051 1.940 1.886 1.724 1.591 1.560 1.517 1.511 1.530 1.573 1.820 1.961 2.411 2.722 3.513 4.316 5.916 7.533
2.344 2.218 2.132 2.018 1.975 1.879 1.832 1.688 1.567 1.537 1.498 1.493 1.513 1.556 1.801 1.941 2.384 2.690 3.470 4.262 5.847 7.451
1.978 1.910 1.860 1.790 1.763 1.702 1.671 1.573 1.486 1.465 1.438 1.437 1.462 1.505 1.746 1.882 2.313 2.609 3.364 4.129 5.639 7.214
1.496 1.465 1.442 1.412 1.400 1.373 1.359 1.315 1.279 1.271 1.270 1.277 1.312 1.359 1.584 1.709 2.105 2.378 3.071 3.771 5.171 6.501
1.150 1.134 1.122 1.105 1.099 1.084 1.076 1.053 1.037 1.036 1.043 1.054 1.088 1.131 1.324 1.429 1.758 1.987 2.563 3.146 4.301 5.336
0.962 0.951 0.942 0.930 0.925 0.915 0.909 0.892 0.880 0.879 0.885 0.892 0.920 0.953 1.114 1.210 1.491 1.692 2.174 2.660 3.602 4.269
0.837 0.831 0.826 0.816 0.812 0.801 0.796 0.779 0.768 0.766 0.769 0.774 0.793 0.818 0.933 0.995 1.189 1.323 1.662 2.006 2.698 3.395
0.0367* 0.05 [11] 0.1 [11] 0.25*
0.5*
0.75*
1 [11]
1.8550 1.2460 1.0780 0.9200 0.8719 0.7754 0.7320 0.6105 0.4955 0.4620 0.3953 0.3633 0.3092 0.2719 0.1870 0.1621 0.1154 0.0966 0.0682 0.0526 0.0358 0.0270
1.2390 1.0860 0.9975 0.8854 0.8462 0.7615 0.7212 0.6006 0.4911 0.4583 0.3923 0.3621 0.3086 0.2718 0.1876 0.1628 0.1161 0.0972 0.0688 0.0530 0.0367 0.0279
0.5736 0.5558 0.5416 0.5210 0.5128 0.4925 0.4815 0.4408 0.3910 0.3731 0.3327 0.3122 0.2732 0.2443 0.1738 0.1521 0.1097 0.0921 0.0654 0.0505 0.0347 0.0274
0.4800 0.4678 0.4576 0.4425 0.4364 0.4214 0.4132 0.3825 0.3441 0.3301 0.2982 0.2818 0.2499 0.2258 0.1644 0.1443 0.1049 0.0881 0.0628 0.0486 0.0336 0.0272
0.4181 0.4110 0.4037 0.3912 0.3857 0.3716 0.3637 0.3390 0.3071 0.2953 0.2684 0.2545 0.2275 0.2070 0.1539 0.1366 0.1017 0.0868 0.0632 0.0495 0.0344 0.0263
1.1620 1.0440 0.9662 0.8656 0.8291 0.7483 0.7089 0.5968 0.4881 0.4453 0.3894 0.3593 0.3060 0.2693 0.1856 0.1609 0.1148 0.0961 0.0680 0.0524 0.0357 0.0270
0.9839 0.9165 0.8658 0.7960 0.7695 0.7078 0.6763 0.5814 0.4806 0.4490 0.3848 0.3553 0.3029 0.2667 0.1842 0.1598 0.1141 0.0956 0.0677 0.0522 0.0354 0.0269
0.7451 0.7120 0.6874 0.6524 0.6385 0.6051 0.5871 0.5231 0.4492 0.4239 0.3692 0.3428 0.2946 0.2606 0.1818 0.1582 0.1131 0.0947 0.0671 0.0517 0.0354 0.0276
4. Isothermal flow The most simple case is the isothermal flow, when the gas temperature anywhere in a system is equal to the temperature in the first reservoir: T(x) ¼ T1. In the case, when the both inlet and outlet channel’s cross sections are in the hydrodynamic flow regime (d1/N and d2/N) and the outlet section is large enough (h2/w/0, two parallel plates), the analytical solution may be found and the reduced mass flow rate becomes
G ¼
d1 6
p2 1 p1
2
0 11 ZL h1 3 A @1 dx : L hðxÞ
(15)
0
Table 3 Coefficients GP,GT obtained from the numerical solution of the S-model kinetic equation for case h/w ¼ 0.0367 and from 2nd order interpolation in h/w -direction.
d
h/w
0 [11]
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
GP
GT
S-model
Interpolation
S-model
Interpolation
2.510 2.345 2.238 2.101 2.051 1.940 1.886 1.724 1.591 1.560 1.517 1.511 1.530 1.573 1.820 1.961 2.411 2.722 3.513 4.316 5.916 7.533
2.706 2.389 2.257 2.105 2.052 1.937 1.883 1.722 1.589 1.557 1.515 1.508 1.527 1.569 1.815 1.957 2.403 2.712 3.498 4.297 5.900 7.515
1.2390 1.0860 0.9975 0.8854 0.8462 0.7615 0.7212 0.6006 0.4911 0.4583 0.3923 0.3621 0.3086 0.2718 0.1876 0.1628 0.1161 0.0972 0.0688 0.0530 0.0367 0.0279
1.2965 1.0906 0.9949 0.8816 0.8422 0.7568 0.7160 0.6006 0.4901 0.4478 0.3908 0.3604 0.3068 0.2700 0.1860 0.1612 0.1150 0.0962 0.0681 0.0525 0.0357 0.0270
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332 Table 4 Coefficients GP,GT obtained from the numerical solution of the S-model kinetic equation for case h/w ¼ 0.0367 and from 2nd order interpolation in d-direction.
d
GP
0.03 3.0 25.0
Interpolation
S-model
Interpolation
2.1614 1.6876 5.1115
2.1567 1.6815 5.0543
0.9340 0.2215 0.0435
0.9306 0.2171 0.0416
To obtain the previous expression we used Eq. (11) which gives GP ¼ d/6 when h/w/0. If we assume the linear variation of the channel height
x hðxÞ ¼ h1 þ ðh2 h1 Þ; L
(16)
from the integration of Eq. (15) one obtains
G ¼
d1 6
1
p2 p1
2
ðh2 =h1 Þ2 : ðh2 =h1 Þ þ 1
(17)
The same results (17) may be obtained from the solution of the Stokes equation subjected to the non-slip boundary conditions, see Ref. [3]. In the case of the free molecular flow regime (d/0) and always for the case of the flow between two parallel plates (h2/w/0) the integration of Eq. (5) can be done analytically. In the case of h/w/0 the coefficient GP may be found using Eq. (10) and the reduced mass flow rate becomes:
0 11 ZL h21 dx 1 p2 @1 A : G ¼ pffiffiffiffi 1 L p1 p hðxÞ2 ðlnð2w=hðxÞÞ þ 0:5Þ
(18)
0
The last integral in previous expression may be evaluated numerically. In order to obtain the values of the reduced mass flow rate in the transitional flow regime Eq. (5) must be solved numerically. In Table 5 the results on the reduced mass flow rate G are given for the case of the flow in very large channel (h2/w/0, two parallel plates) and when the channel height increases in the flow direction by 10 times (h2/h1 ¼ 10). In Table 6 the results on the reduced mass flow rate G are given for the case of the flow through the channel of the square outlet cross section (h2/w ¼ 1) and when the channel height increases in the flow direction by 10 times (h2/h1 ¼ 10).
Table 5 Reduced mass flow rate G for the isothermal case and for the h2/w ¼ 0 channel, the channel’s height aspect ratio is h2/h1 ¼ 10.
d1
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
Table 6 Reduced mass flow rate G for the isothermal case and for the h2/w ¼ 1 channel, the channel’s height aspect ratio is h2/h1 ¼ 10.
d1
GT
S-model
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
0
0.01
0.1
0.5
0.9
93.34 33.47 24.51 21.63 16.80 15.83 19.06 25.82 85.35 161.1
92.41 32.18 23.87 21.10 16.44 15.54 18.93 25.74 85.30 161.0
38.46 27.45 20.75 18.34 14.52 13.94 18.00 24.95 84.15 159.2
20.72 13.95 10.62 9.448 7.897 7.951 11.88 17.40 62.58 119.4
4.024 2.674 2.035 1.824 1.591 1.656 2.725 4.140 15.61 30.02
p2/p1 0
0.01
0.1
0.5
0.9
15.36 15.02 14.14 13.67 12.68 12.60 16.35 22.53 74.86 141.3
15.21 14.86 13.98 13.52 12.54 12.47 16.27 22.47 74.82 141.3
13.83 13.47 12.62 12.20 11.36 11.41 15.54 21.82 73.76 139.6
7.670 7.418 6.897 6.681 6.393 6.657 10.35 15.26 54.88 104.8
1.532 1.474 1.368 1.327 1.306 1.401 2.381 3.631 13.70 26.34
5. Non-isothermal case The non-isothermal flow is considered: the temperature in the second reservoir is supposed to be higher and equal to T2 ¼ 1.5T1. The temperature gradient along the channel is supposed to be linear. In Table 7 the results on the reduced mass flow rate are given for the case of the flow through the square (h2/w ¼ 1) outlet crosssection channel and when the channel height increases in the flow direction in 10 times (h2/h1 ¼ 10), as for the second isothermal case, see previous section. If comparing the results of Tables 6 and 7, obtained under isothermal and non-isothermal conditions, we can observed very small influence of the temperature gradient on the reduced mass flow rate for small values of the gas rarefaction and the small values of the pressure ratio between the reservoirs. However, for the larger values of the rarefaction parameter the difference in the reduced mass flow rate becomes significant even for the gas expansion into vacuum. The influence of the temperature gradient along the channel is also large for the pressure ratio close to 1 and small values of the rarefaction parameter. In this case the mass flow rate due to the thermal creep phenomenon (due to the temperature gradient) is directed from the cold reservoir to the hot one, that means in the same direction as the pressure-driven Poiseuille flow. 6. Conclusion The simple method, proposed previously by other authors, is applied here to calculate the gas mass flow rate through the channel of the variable rectangular cross section. The calculations are based on the results of the numerical solution of the linearized S-model kinetic equation obtained by other authors and completed using the same approach in the present paper. The explicit analytical expressions are proposed in the case of the Table 7 Reduced mass flow rate G for the non-isothermal case and for the h2/w ¼ 1 channel, the channel’s height aspect ratio is h2/h1 ¼ 10, T2/T1 ¼ 1.5.
d1
p2/p1
331
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
p2/p1 0
0.01
0.1
0.5
0.9
15.35 15.01 14.12 13.61 12.51 12.35 15.56 21.17 69.44 130.8
15.23 14.88 13.98 13.49 12.39 12.23 15.48 21.10 69.39 130.7
14.11 13.74 12.84 12.38 11.36 11.27 14.73 20.43 68.29 129.1
9.073 8.739 8.020 7.668 6.907 6.894 9.818 14.27 50.75 96.86
4.046 3.797 3.314 3.038 2.295 2.047 2.418 3.472 12.68 24.36
332
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
hydrodynamic and free molecular flow regimes. The simple interpolation method is realized to calculate the mass flow rate in the transitional flow regime. Acknowledgments The authors thank the Mesocentre of Aix-Marseille University for the supercomputing resources under Project No. 13b015. References [1] Sharipov F, Seleznev V. Data on internal rarefied gas flows. J Phys Chem Ref Data 1998;27(3):657e706. [2] de Silva LR, Deschamps CJ. Modeling of gas leakage through compressor valves. In: International compressor engineering conference 2012. [3] Stevanovic ND. A new analytical solution of microchannel gas flow. J Micromech MicroEng 2007;17:1695e702. [4] Stevanovi c ND, Djordjevi c VD. The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime. Publ de l’Institut Math 2012;91(105):83e93. [5] Sharipov F, Bertoldo G. Rarefied gas flow through a long tube of variable radius. J Vac Sci Technol A 2006;24(3):2087e93. [6] Titarev VA, Utyuzhnikov SV, Shakhov EM. Rarefied gas flow into vacuum through square pipe with variable cross section. Comput Math Math Phys 53(8). [7] Weissberg HL. End correction for slow viscous flow through long tubes. Phys Fluids 1962;5(5):1033e6. [8] Pantazis S, Valougeorgis D, Sharipov F. Rarefied gas flow through a long tube of variable radius. J Vac Sci Technol A 2005;23(3):531e3. [9] Berman AS. Free molecular transmission probabilities. J Appl Phys 1965;36(10):3356. [10] Titarev VA. Rarefied gas flow in a circular pipe of finite length. Vacuum 2013;94:92e103.
[11] Sharipov F. Non-isothermal gas flow through rectangular microchannels. J Micromech Microeng 1999;9(4):394e401. [12] Cercignani C, Daneri A. Flow of a rarefied gas between two parallel plates. Phys Fluids 1963;6:993e6. [13] Huang AB, Giddens DP, Bagual CW. Rarefied gas flow between parallel plates based on the discrete ordinate method. Phys Fluids 1967;10(3):498e 502. [14] Loyalka SK, Petrellis N, Storvik TS. Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates. Z Angew Math Phys (ZAMP) 1979;30:514e21. [15] Ohwada T, Sone Y, Aoki K. Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard sphere molecules. Phys Fluids A 1989;1(12):2042e9. [16] Stoer J, Bulirsch R. Introduction to numerical analysis. New York: Springer Verlag; 1980. [17] Titarev VA, Shakhov EM. Kinetic analysis of the isothermal flow in a long rectangular microchannel. Comput Math Math Phys 2010;50(7):1221e37. [18] Sharipov F. Rarefied gas flow through a long rectangular channel. J Vac Sci Technol A 1999;17(5):3062e6. [19] Méolans JG, Nacer MH, Rojas M, Perrier P, Graur I. Effects of two transversal finite dimensions in long microchannel: analytical approach in slip regime. Phys Fluids 2012;24(11). [20] Zhvick VV, Friedlander OG. Poiseuille flow in a capillary at all regimes of rarefied gas flows. In: AIP conference proceedings, vol. 1501. American Institute of Physics; 2012. p. 830e6. [21] Albertoni S, Cercignani C, Gotusso L. Numerical evaluation of the slip coefficient. Phys Fluids 6:993e996. [22] Siewert C, Sharipov F. Model equations in rarefied gas dynamics: viscous-slip and thermal-slip coefficients. Phys Fluids 2002;14(12):4123e9. [23] Sharipov F. Data on the velocity slip and temperature jump on a gas-solid interface. J Phys Chem Ref Data 2011;40(2):023101-1e023101-28. [24] Shakhov EM. Generalization of the Krook kinetic relaxation equation. Fluid Dyn 1968;3(5):95e6. [25] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes: the art of scientific computing. 3rd ed. New York, NY, USA: Cambridge University Press; 2007.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Rarefied gas flow through a long rectangular channel of variable cross section I. Graur, M.T. Ho* IUSTI UMR CNRS 7343 Laboratory, Polytech Marseille, Aix Marseille Université, 5 rue Enrico Fermi, 13453 Marseille, Provence, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 June 2013 Received in revised form 20 July 2013 Accepted 22 July 2013
The method, proposed previously by other authors, is applied here to calculate a mass flow rate of rarefied gas through a long rectangular channel of variable cross-section aspect ratio. The gas flow through this channel is generated by the pressure and/or temperature gradient. The method is based on the results obtained previously on the basis of the kinetic equation and it requires very modest computational efforts. As a demonstration of its application some examples of the mass flow rate calculations for the isothermal and non-isothermal flows through the channels with variable rectangular cross sections are given. The analytical expressions in the case of the hydrodynamic and free molecular flow regimes are proposed. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Rarefied gas Long rectangular channel Variable cross section Mass flow rate
1. Introduction The gas flow through a long channel of a rectangular cross section is a practical problem in the MEMS and vacuum technology applications. This kind of flow was largely studied on the basis of the kinetic theory and the detailed review may be found in Ref. [1]. However in several applications a cross section varies along the channel. As examples of such kind of flow the leakage through compressor valves [2] and the flow in the micro bearing [3,4], may be given. A few results of the numerical simulations of the flow through variable conical cross section [5] and rectangular and conical sections [6] were found in the literature. In this work we apply the approach, proposed in Ref. [5] and implemented to the flow through the tube of a variable radius, to the channel of a variable rectangular cross section. The proposed technique allows to calculate the mass flow rate through a long channel with variable rectangular cross section for arbitrary pressure and temperature ratios in large range of gas rarefaction. 2. Problem statement Consider two reservoirs containing the same gas and connected by a long rectangular channel of variable cross-section aspect ratio. The channel width w is supposed to be the same, but the channel
height h varies continuously along the channel. Its value is equal to h1 in the first reservoir and h2 in the second reservoir. We assume that the relation max(h) w keeps along the channel. The first reservoir is maintained at the pressure p1 and temperature T1, while the pressure and temperature in the second reservoir are p2 and T2, respectively. We will calculate the mass flow rate through this channel in the whole range of the gas rarefaction. The results will be given in terms of reduced mass flow rate defined as follows
L G ¼ 2 h1 wp1
0042-207X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.vacuum.2013.07.047
(1)
here L is the channel length, m is the molecular mass of the gas, k is the _ is the mass flow rate through the channel. Boltzmann constant, M We assume the channel to be long enough (max(h)L) that the end effects can be neglected. According to Refs. [7,8], the end influence in the hydrodynamic regime has the order of max(h)/L, while in the free molecular regime [9] its order is (max(h)/L)ln(L/ max(h)). These estimations are confirmed by the numerical results for a tube flow [10]. The second assumption is that the pressure and temperature gradients can be considered small in any cross section of the channel
xP ¼ * Corresponding author. Tel.: (þ33) 04 91 10 68 77. E-mail addresses: [email protected] (I. Graur), [email protected] (M.T. Ho).
rffiffiffiffiffiffiffiffiffiffi 2kT1 _ M; m
h dp ; p dx
jxP j 1;
xT ¼
h dT ; T dx
jxT j 1;
(2)
where x is the longitudinal coordinate in the flow direction with the origin in the first reservoir, h ¼ h(x) is a local channel’s height,
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
p ¼ p(x) and T ¼ T(x) are a local pressure and temperature, respectively. Under such a condition the mass flow rate in a cross section is calculated as [11]
GT ¼ GP =2:
rffiffiffiffiffiffiffiffiffi _ ¼ hwp m GP x þ GT x ; M p T 2kT
1 2w 1 þ ; GP ¼ pffiffiffiffi ln h 2 p
(3)
329
(9)
In the case h/w/0 the two coefficients become [11]
GT ¼ GP =2:
(10)
where the coefficients GP and GT depend on the local gas rarefaction parameter d, defined as
3.2. Hydrodynamic regime
rffiffiffiffiffiffiffiffiffi hp m d¼ ; mðTÞ 2kT
In the hydrodynamic flow regime (d/N) the reduced mass flow rate is equal to [18]
(4)
m is the gas viscosity, which depends on the local temperature T(x). The values of the coefficients GP ¼ GP(d) and GT ¼ GT(d) in the
case of the gas flow through a rectangular cross section channel for different flow regimes were obtained by different authors from the solution of the linearized BGK and S-model kinetic equations, or from the linearized Boltzmann equation for the diffuse or diffusespecular boundary conditions [12,13,14,15]. The detailed review on these numerical results may be found in Ref. [1]. The results used in this work will be given in the next section. Following Ref. [5], from Eqs. (1)e(4) the differential equation is obtained
rffiffiffiffiffi L h 2 T1 dp p dT þ GT ðd Þ : GP ðdÞ G ¼ p1 h1 dx T dx T
(5)
If we assume that the temperature distribution along the channel is known we can obtain the corresponding pressure distribution and the reduced total mass flow rate G through the channel. Eq. (5) is solved numerically using the following finite difference scheme [5]:
piþ1
sffiffiffiffiffi! Dx pi Tiþ1 Ti p1 G h1 2 Ti : GT ðdi Þ ¼ pi þ Dx L Ti hi T1 GP ðdi Þ
(6)
In the previous equation Dx ¼ L/N is the grid step in the x direction, 0 i N, pi, Ti, hi are the pressure, temperature and channel height in i grid point, respectively. The rarefaction parameter di is calculated as
sffiffiffiffiffi pi hi mðT1 Þ T1 di ¼ d1 ; p1 h1 mðTi Þ Ti
rffiffiffiffiffiffiffiffiffiffi p h m d1 ¼ 1 1 : mðT1 Þ 2kT1
(7)
Eq. (6) is solved numerically by the shooting method (Ref. [16], Section 7.3) with the boundary condition p(x ¼ 0) ¼ p1. G is a parameter of eq. (6) which is found satisfying the second boundary condition p(x ¼ L) ¼ p2. 3. Determination of coefficients GP and GT 3.1. Free molecular regime In the free molecular regime (d ¼ 0) the coefficient GP was found analytically in Ref. [17]
0 0 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 4w @ h h2 A w w2 @ ln þ 1 þ 2 þ ln þ 1þ 2A GP ¼ pffiffiffiffi w h w h p h 3 1 w w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: 3 w þ w2 þ h2 h þ w2 þ h2 The coefficient GT for the case of d ¼ 0 is equal to
(8)
GPh ¼
d
1
6
! N 192 h X tan hð0:5pð2n þ 1Þw=hÞ : p5 w n ¼ 0 ð2n þ 1Þ5
(11)
In the slip flow regime the slip correction must be added, which was found in Refs. [17,19,20]. N 4 256 h X tan hð0:5pð2n þ 1Þw=hÞ 5 3 p w n¼0 ð1 þ 2nÞ5 ! X N 32 h tan hð0:5pð2n þ 1Þw=hÞ 4 1 : w n¼0 p ð1 þ 2nÞ4
GPslip ¼ sp
(12)
Therefore the coefficient GP for the hydrodynamic and slip regimes (d > 40) becomes
GP ¼ GPh þ GPslip :
(13)
The thermal creep coefficient in these flow regimes is equal to [11]
GT ¼
sT : d
(14)
The coefficients sp and sT in Eqs. (12) and (14) are the velocity slip and the thermal slip coefficients, equal to 1.016 [21] and 1.175 [22], respectively. It is to be noticed that expressions (12) and (14) are valid for any slip coefficients, e.g. for non-diffuse scattering or gaseous mixtures [23]. 3.3. Transitional flow regime In order to solve numerically Eq. (6) we need to known the values of the rarefaction parameter at arbitrary couple of two values: the rarefaction parameter and the channel cross-section aspect ratio (d,h/w). The two-dimensional grid of values GP(d,h/w) and GT(d,h/w) is formed from the numerical data of Ref. [11], where the quantities GP(d) and GT(d) are calculated for large range of the gas rarefaction parameter and for four different h/w aspect ratios h/ w ¼ 0,0.05,0.1,1, where h/w ¼ 0 corresponds to the case of the flow between two parallel plates. To complete the data of Ref. [11] the same approach is applied to obtain a solution of the linearized S-model kinetic equation [24]. The additional values of the channel cross-section aspect ratios h/ w ¼ 0.0367,0.25,0.5,0.75 and for the rarefaction parameter varies from 0.001 to 40 are calculated. The 1000 1000 grid in the physical space is implemented, the polar coordinates are used in the molecular velocity space with 200 points for the velocity directions and 25 points in the velocity magnitude distributed according to the Gaussian rule. The values of the GP and GT coefficients obtained in the present work are presented in Tables 1 and 2. The method of two-dimensional interpolation is employed to calculate the values GP and GT in an arbitrary point (d*,(h/w)*) from two-dimensional grid of GP(d,h/w) and GT(d,h/w) values. Below we explain the interpolation steps realized to calculate the value of
330
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
GP(d*,(h/w)*) in a point (d*, (h/w)*), which does not coincide with any values of d and h/w reported in two-dimensional Table 1. The interpolation is accomplished by the following steps.
Table 2 Coefficient GT obtained in Ref. [11] and in present work* from the numerical solution of the linearized S-model kinetic equation.
d
A block of the reduced mass flow rate GP(di,(h/ w)j)i ¼ 1.m þ 1,j ¼ 1.n þ 1 values containing a point (d*,(h/w)*) is searched and extracted using the bisection method (Chapter 3 of Ref. [25]) from the ordered tabular data, see Table 1, which contains the numerical values of GP obtained on the basis of kinetic equation [11]. Here, i and j are the indexes for the row d and the column h/w, respectively. The values m and n are the orders of interpolation in d-direction and in h/w-direction, respectively. Using n þ 1 values of GP in h/w-direction, the nth order interpolation with Neville’s algorithm (given in Chapter 3 of Ref. [25]) is carried out for each of m þ 1 extracted values in d-direction and a set of values GP(di,(h/w)*)i ¼ 1.m þ 1 is obtained. Then mth order interpolation in the d-direction is performed in order to obtain the value of GP(d*,(h/w)*). The value of GT at an arbitrary point (d*,(h/w)*) is calculated by the interpolation in the same fashion as for the value of GP. The accuracy of the interpolation in the h/w-direction is evaluated by comparing the results of the numerical solution of the linearized S-model kinetic equation for the h/w ¼ 0.0367 and the interpolation results for the same values of h/w ratio, see Table 3. Analyzing the results of Table 3 one can see that the accuracy of the interpolation is of the order of 2%. The similar verification was carried out for the interpolation in the d direction. The same type of the comparison was carried out by comparing the solution of the linearized kinetic equation and the interpolation results. When analyzing the results given in Table 4 one can see that the difference between the corresponding values does not also exceed 2%. However, when d/0 (for the small values of h/w) or d/N the coefficients GP and GT change steeply and the use of the sparse grid rises up the interpolation error. In the following Sections the results of the isothermal and nonisothermal flows simulations are given.
Table 1 Coefficient GP obtained in Ref. [11] and in present work* from the numerical solution of the linearized S-model kinetic equation.
d
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
h/w
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
0 [11]
0.0367*
0.05 [11]
0.1 [11]
0.25*
0.5*
0.75*
1 [11]
4.273 3.050 2.713 2.400 2.306 2.120 2.038 1.823 1.651 1.613 1.561 1.549 1.565 1.606 1.855 2.000 2.456 2.772 3.577 4.393 6.040 7.695
2.510 2.345 2.238 2.101 2.051 1.940 1.886 1.724 1.591 1.560 1.517 1.511 1.530 1.573 1.820 1.961 2.411 2.722 3.513 4.316 5.916 7.533
2.344 2.218 2.132 2.018 1.975 1.879 1.832 1.688 1.567 1.537 1.498 1.493 1.513 1.556 1.801 1.941 2.384 2.690 3.470 4.262 5.847 7.451
1.978 1.910 1.860 1.790 1.763 1.702 1.671 1.573 1.486 1.465 1.438 1.437 1.462 1.505 1.746 1.882 2.313 2.609 3.364 4.129 5.639 7.214
1.496 1.465 1.442 1.412 1.400 1.373 1.359 1.315 1.279 1.271 1.270 1.277 1.312 1.359 1.584 1.709 2.105 2.378 3.071 3.771 5.171 6.501
1.150 1.134 1.122 1.105 1.099 1.084 1.076 1.053 1.037 1.036 1.043 1.054 1.088 1.131 1.324 1.429 1.758 1.987 2.563 3.146 4.301 5.336
0.962 0.951 0.942 0.930 0.925 0.915 0.909 0.892 0.880 0.879 0.885 0.892 0.920 0.953 1.114 1.210 1.491 1.692 2.174 2.660 3.602 4.269
0.837 0.831 0.826 0.816 0.812 0.801 0.796 0.779 0.768 0.766 0.769 0.774 0.793 0.818 0.933 0.995 1.189 1.323 1.662 2.006 2.698 3.395
0.0367* 0.05 [11] 0.1 [11] 0.25*
0.5*
0.75*
1 [11]
1.8550 1.2460 1.0780 0.9200 0.8719 0.7754 0.7320 0.6105 0.4955 0.4620 0.3953 0.3633 0.3092 0.2719 0.1870 0.1621 0.1154 0.0966 0.0682 0.0526 0.0358 0.0270
1.2390 1.0860 0.9975 0.8854 0.8462 0.7615 0.7212 0.6006 0.4911 0.4583 0.3923 0.3621 0.3086 0.2718 0.1876 0.1628 0.1161 0.0972 0.0688 0.0530 0.0367 0.0279
0.5736 0.5558 0.5416 0.5210 0.5128 0.4925 0.4815 0.4408 0.3910 0.3731 0.3327 0.3122 0.2732 0.2443 0.1738 0.1521 0.1097 0.0921 0.0654 0.0505 0.0347 0.0274
0.4800 0.4678 0.4576 0.4425 0.4364 0.4214 0.4132 0.3825 0.3441 0.3301 0.2982 0.2818 0.2499 0.2258 0.1644 0.1443 0.1049 0.0881 0.0628 0.0486 0.0336 0.0272
0.4181 0.4110 0.4037 0.3912 0.3857 0.3716 0.3637 0.3390 0.3071 0.2953 0.2684 0.2545 0.2275 0.2070 0.1539 0.1366 0.1017 0.0868 0.0632 0.0495 0.0344 0.0263
1.1620 1.0440 0.9662 0.8656 0.8291 0.7483 0.7089 0.5968 0.4881 0.4453 0.3894 0.3593 0.3060 0.2693 0.1856 0.1609 0.1148 0.0961 0.0680 0.0524 0.0357 0.0270
0.9839 0.9165 0.8658 0.7960 0.7695 0.7078 0.6763 0.5814 0.4806 0.4490 0.3848 0.3553 0.3029 0.2667 0.1842 0.1598 0.1141 0.0956 0.0677 0.0522 0.0354 0.0269
0.7451 0.7120 0.6874 0.6524 0.6385 0.6051 0.5871 0.5231 0.4492 0.4239 0.3692 0.3428 0.2946 0.2606 0.1818 0.1582 0.1131 0.0947 0.0671 0.0517 0.0354 0.0276
4. Isothermal flow The most simple case is the isothermal flow, when the gas temperature anywhere in a system is equal to the temperature in the first reservoir: T(x) ¼ T1. In the case, when the both inlet and outlet channel’s cross sections are in the hydrodynamic flow regime (d1/N and d2/N) and the outlet section is large enough (h2/w/0, two parallel plates), the analytical solution may be found and the reduced mass flow rate becomes
G ¼
d1 6
p2 1 p1
2
0 11 ZL h1 3 A @1 dx : L hðxÞ
(15)
0
Table 3 Coefficients GP,GT obtained from the numerical solution of the S-model kinetic equation for case h/w ¼ 0.0367 and from 2nd order interpolation in h/w -direction.
d
h/w
0 [11]
0.001 0.01 0.02 0.04 0.05 0.08 0.1 0.2 0.4 0.5 0.8 1 1.5 2 4 5 8 10 15 20 30 40
GP
GT
S-model
Interpolation
S-model
Interpolation
2.510 2.345 2.238 2.101 2.051 1.940 1.886 1.724 1.591 1.560 1.517 1.511 1.530 1.573 1.820 1.961 2.411 2.722 3.513 4.316 5.916 7.533
2.706 2.389 2.257 2.105 2.052 1.937 1.883 1.722 1.589 1.557 1.515 1.508 1.527 1.569 1.815 1.957 2.403 2.712 3.498 4.297 5.900 7.515
1.2390 1.0860 0.9975 0.8854 0.8462 0.7615 0.7212 0.6006 0.4911 0.4583 0.3923 0.3621 0.3086 0.2718 0.1876 0.1628 0.1161 0.0972 0.0688 0.0530 0.0367 0.0279
1.2965 1.0906 0.9949 0.8816 0.8422 0.7568 0.7160 0.6006 0.4901 0.4478 0.3908 0.3604 0.3068 0.2700 0.1860 0.1612 0.1150 0.0962 0.0681 0.0525 0.0357 0.0270
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332 Table 4 Coefficients GP,GT obtained from the numerical solution of the S-model kinetic equation for case h/w ¼ 0.0367 and from 2nd order interpolation in d-direction.
d
GP
0.03 3.0 25.0
Interpolation
S-model
Interpolation
2.1614 1.6876 5.1115
2.1567 1.6815 5.0543
0.9340 0.2215 0.0435
0.9306 0.2171 0.0416
To obtain the previous expression we used Eq. (11) which gives GP ¼ d/6 when h/w/0. If we assume the linear variation of the channel height
x hðxÞ ¼ h1 þ ðh2 h1 Þ; L
(16)
from the integration of Eq. (15) one obtains
G ¼
d1 6
1
p2 p1
2
ðh2 =h1 Þ2 : ðh2 =h1 Þ þ 1
(17)
The same results (17) may be obtained from the solution of the Stokes equation subjected to the non-slip boundary conditions, see Ref. [3]. In the case of the free molecular flow regime (d/0) and always for the case of the flow between two parallel plates (h2/w/0) the integration of Eq. (5) can be done analytically. In the case of h/w/0 the coefficient GP may be found using Eq. (10) and the reduced mass flow rate becomes:
0 11 ZL h21 dx 1 p2 @1 A : G ¼ pffiffiffiffi 1 L p1 p hðxÞ2 ðlnð2w=hðxÞÞ þ 0:5Þ
(18)
0
The last integral in previous expression may be evaluated numerically. In order to obtain the values of the reduced mass flow rate in the transitional flow regime Eq. (5) must be solved numerically. In Table 5 the results on the reduced mass flow rate G are given for the case of the flow in very large channel (h2/w/0, two parallel plates) and when the channel height increases in the flow direction by 10 times (h2/h1 ¼ 10). In Table 6 the results on the reduced mass flow rate G are given for the case of the flow through the channel of the square outlet cross section (h2/w ¼ 1) and when the channel height increases in the flow direction by 10 times (h2/h1 ¼ 10).
Table 5 Reduced mass flow rate G for the isothermal case and for the h2/w ¼ 0 channel, the channel’s height aspect ratio is h2/h1 ¼ 10.
d1
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
Table 6 Reduced mass flow rate G for the isothermal case and for the h2/w ¼ 1 channel, the channel’s height aspect ratio is h2/h1 ¼ 10.
d1
GT
S-model
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
0
0.01
0.1
0.5
0.9
93.34 33.47 24.51 21.63 16.80 15.83 19.06 25.82 85.35 161.1
92.41 32.18 23.87 21.10 16.44 15.54 18.93 25.74 85.30 161.0
38.46 27.45 20.75 18.34 14.52 13.94 18.00 24.95 84.15 159.2
20.72 13.95 10.62 9.448 7.897 7.951 11.88 17.40 62.58 119.4
4.024 2.674 2.035 1.824 1.591 1.656 2.725 4.140 15.61 30.02
p2/p1 0
0.01
0.1
0.5
0.9
15.36 15.02 14.14 13.67 12.68 12.60 16.35 22.53 74.86 141.3
15.21 14.86 13.98 13.52 12.54 12.47 16.27 22.47 74.82 141.3
13.83 13.47 12.62 12.20 11.36 11.41 15.54 21.82 73.76 139.6
7.670 7.418 6.897 6.681 6.393 6.657 10.35 15.26 54.88 104.8
1.532 1.474 1.368 1.327 1.306 1.401 2.381 3.631 13.70 26.34
5. Non-isothermal case The non-isothermal flow is considered: the temperature in the second reservoir is supposed to be higher and equal to T2 ¼ 1.5T1. The temperature gradient along the channel is supposed to be linear. In Table 7 the results on the reduced mass flow rate are given for the case of the flow through the square (h2/w ¼ 1) outlet crosssection channel and when the channel height increases in the flow direction in 10 times (h2/h1 ¼ 10), as for the second isothermal case, see previous section. If comparing the results of Tables 6 and 7, obtained under isothermal and non-isothermal conditions, we can observed very small influence of the temperature gradient on the reduced mass flow rate for small values of the gas rarefaction and the small values of the pressure ratio between the reservoirs. However, for the larger values of the rarefaction parameter the difference in the reduced mass flow rate becomes significant even for the gas expansion into vacuum. The influence of the temperature gradient along the channel is also large for the pressure ratio close to 1 and small values of the rarefaction parameter. In this case the mass flow rate due to the thermal creep phenomenon (due to the temperature gradient) is directed from the cold reservoir to the hot one, that means in the same direction as the pressure-driven Poiseuille flow. 6. Conclusion The simple method, proposed previously by other authors, is applied here to calculate the gas mass flow rate through the channel of the variable rectangular cross section. The calculations are based on the results of the numerical solution of the linearized S-model kinetic equation obtained by other authors and completed using the same approach in the present paper. The explicit analytical expressions are proposed in the case of the Table 7 Reduced mass flow rate G for the non-isothermal case and for the h2/w ¼ 1 channel, the channel’s height aspect ratio is h2/h1 ¼ 10, T2/T1 ¼ 1.5.
d1
p2/p1
331
0.001 0.01 0.05 0.1 0.5 1 5 10 50 100
p2/p1 0
0.01
0.1
0.5
0.9
15.35 15.01 14.12 13.61 12.51 12.35 15.56 21.17 69.44 130.8
15.23 14.88 13.98 13.49 12.39 12.23 15.48 21.10 69.39 130.7
14.11 13.74 12.84 12.38 11.36 11.27 14.73 20.43 68.29 129.1
9.073 8.739 8.020 7.668 6.907 6.894 9.818 14.27 50.75 96.86
4.046 3.797 3.314 3.038 2.295 2.047 2.418 3.472 12.68 24.36
332
I. Graur, M.T. Ho / Vacuum 101 (2014) 328e332
hydrodynamic and free molecular flow regimes. The simple interpolation method is realized to calculate the mass flow rate in the transitional flow regime. Acknowledgments The authors thank the Mesocentre of Aix-Marseille University for the supercomputing resources under Project No. 13b015. References [1] Sharipov F, Seleznev V. Data on internal rarefied gas flows. J Phys Chem Ref Data 1998;27(3):657e706. [2] de Silva LR, Deschamps CJ. Modeling of gas leakage through compressor valves. In: International compressor engineering conference 2012. [3] Stevanovic ND. A new analytical solution of microchannel gas flow. J Micromech MicroEng 2007;17:1695e702. [4] Stevanovi c ND, Djordjevi c VD. The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime. Publ de l’Institut Math 2012;91(105):83e93. [5] Sharipov F, Bertoldo G. Rarefied gas flow through a long tube of variable radius. J Vac Sci Technol A 2006;24(3):2087e93. [6] Titarev VA, Utyuzhnikov SV, Shakhov EM. Rarefied gas flow into vacuum through square pipe with variable cross section. Comput Math Math Phys 53(8). [7] Weissberg HL. End correction for slow viscous flow through long tubes. Phys Fluids 1962;5(5):1033e6. [8] Pantazis S, Valougeorgis D, Sharipov F. Rarefied gas flow through a long tube of variable radius. J Vac Sci Technol A 2005;23(3):531e3. [9] Berman AS. Free molecular transmission probabilities. J Appl Phys 1965;36(10):3356. [10] Titarev VA. Rarefied gas flow in a circular pipe of finite length. Vacuum 2013;94:92e103.
[11] Sharipov F. Non-isothermal gas flow through rectangular microchannels. J Micromech Microeng 1999;9(4):394e401. [12] Cercignani C, Daneri A. Flow of a rarefied gas between two parallel plates. Phys Fluids 1963;6:993e6. [13] Huang AB, Giddens DP, Bagual CW. Rarefied gas flow between parallel plates based on the discrete ordinate method. Phys Fluids 1967;10(3):498e 502. [14] Loyalka SK, Petrellis N, Storvik TS. Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates. Z Angew Math Phys (ZAMP) 1979;30:514e21. [15] Ohwada T, Sone Y, Aoki K. Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard sphere molecules. Phys Fluids A 1989;1(12):2042e9. [16] Stoer J, Bulirsch R. Introduction to numerical analysis. New York: Springer Verlag; 1980. [17] Titarev VA, Shakhov EM. Kinetic analysis of the isothermal flow in a long rectangular microchannel. Comput Math Math Phys 2010;50(7):1221e37. [18] Sharipov F. Rarefied gas flow through a long rectangular channel. J Vac Sci Technol A 1999;17(5):3062e6. [19] Méolans JG, Nacer MH, Rojas M, Perrier P, Graur I. Effects of two transversal finite dimensions in long microchannel: analytical approach in slip regime. Phys Fluids 2012;24(11). [20] Zhvick VV, Friedlander OG. Poiseuille flow in a capillary at all regimes of rarefied gas flows. In: AIP conference proceedings, vol. 1501. American Institute of Physics; 2012. p. 830e6. [21] Albertoni S, Cercignani C, Gotusso L. Numerical evaluation of the slip coefficient. Phys Fluids 6:993e996. [22] Siewert C, Sharipov F. Model equations in rarefied gas dynamics: viscous-slip and thermal-slip coefficients. Phys Fluids 2002;14(12):4123e9. [23] Sharipov F. Data on the velocity slip and temperature jump on a gas-solid interface. J Phys Chem Ref Data 2011;40(2):023101-1e023101-28. [24] Shakhov EM. Generalization of the Krook kinetic relaxation equation. Fluid Dyn 1968;3(5):95e6. [25] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes: the art of scientific computing. 3rd ed. New York, NY, USA: Cambridge University Press; 2007.