- Email: [email protected]
Ab initio simulation of heat transfer through a mixture of rarefied gases
International Journal of Heat and Mass Transfer 71 (2014) 91–97
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Ab initio simulation of heat transfer through a mixture of rarefied gases José L. Strapasson ⇑, Felix Sharipov Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, Curitiba 81531-990, Brazil
a r t i c l e
i n f o
Article history: Received 21 September 2013 Received in revised form 28 November 2013 Accepted 2 December 2013
Keywords: Gaseous mixture Ab initio potential Heat transfer
a b s t r a c t The heat flux problem for a binary gaseous mixture confined between two parallel plates with different temperatures is studied on the basis of the direct simulation Monte Carlo method with an implementation of ab initio potential. The calculations were carried for a wide range of the gas rarefaction, for several values of the mole fraction and for two values of the temperature difference. The smaller value of the difference corresponds to the limit when the nonlinear terms are negligible, while the larger value describes a nonlinear heat transfer. The heat flux, temperature, and mole fraction distributions are presented. To study the influence of the intermolecular potential, the same simulations are carried out for the hard sphere molecular model. A relative deviation of the results based on this model from those based on the ab initio potential is analyzed. It is pointed out that the difference between the heat flux of the two potentials is about 8% and 5% for the small and large temperature differences, respectively. The temperature distribution between plates is weakly affected by the molecular potential, while the chemical composition variation is the most sensitive quantity for the considered problem. The reported results can be used as benchmark data to test model kinetic equations for gaseous mixtures. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In our previous paper [1], we showed that any intermolecular potential can be implemented into the direct simulation Monte Carlo (DSMC) method [2] with the same computational effort as that for the hard sphere (HS) molecular model. Since reliable information about intermolecular potentials of many gases can be found in literature, see e.g. Refs. [3,4], it is not necessary to use the potentials like variable hard spheres [2], variable soft spheres [5] and generalized hard spheres [6] elaborated specifically for the DSMC method. Recently, the ab initio (AI) potentials were calculated for practically all noble gases and their mixtures, see e.g. Refs. [7–14]. An implementation of this potential into the DSMC [15] made this method completely free from tuned parameters usually extracted from experiments. Such an approach allows us to obtain benchmark data which can be used to test kinetic models and approximate methods in order to solve many practical problems of heat and mass transfer with modest computational effort, but without losing reliability. Our recent paper [16] reports benchmark data for the Couette flow of helium–argon mixture over the whole range of the gas rarefaction based on the AI intermolecular potential. A comparison of the results obtained for the AI potential with those for the HS potential showed that the Couette flow is weakly sensitive to the ⇑ Corresponding author. Tel.: +55 4136722356. E-mail addresses: jstrapasson@fisica.ufpr.br (J.L. Strapasson), sharipov@fisica.ufpr.br (F. Sharipov). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.011
potential. In the present paper, we continue to study the influence of the intermolecular potential on various types of mixture flows. More specifically, a heat transfer through a mixture confined between two plates is calculated applying the DSMC technique. The calculations are carried out over a wide range of the gas rarefaction for both AI and HS potentials. The flow to be considered here is a classical problem of fluid mechanics. Many researchers studied this problem in case of a single rarefied gas, see e.g. Refs. [17–26]. On our knowledge, there are very few papers [27–31] on heat transfer through a mixture of rarefied gases. The works [27,30,31] provide results based on the Boltzmann equation with the HS potential. The paper [28] reports results on the heat flux based on the linearized McCormack (MC) model [32] of the Boltzmann equation. The paper [29] is also based on the MC model, but two potentials were used, viz., HS and the socalled realistic potential (RP). A comparison of results based on these two potentials showed that the heat transfer is strongly sensitive to the potential. As was shown in Refs. [33–38], many other phenomena in gaseous mixtures are very sensitive to the potential of the intermolecular interaction. Thus, it is important to obtain benchmark results based on the ab initio calculations for a large number of phenomena. The aim of the present paper is to calculate the heat flux through a binary mixture of rarefied gases confined between two parallel plates based on the AI potential implemented into the DSMC method. To study the influence of the potential on heat flux, temperature and chemical composition distributions, the same problem will be solved for the HS molecular model too.
92
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
2. Statement of the problem Consider a binary gaseous mixture confined between two parallel plates fixed at x ¼ H=2 and having different temperatures T 0 DT=2, respectively. Thus, H is the distance between the plates and DT is the temperature difference. We are going to calculate the heat flux q0x , temperature TðxÞ profile and mole fraction CðxÞ distribution between the plates. Besides the relative temperature difference DT=T 0 , the solution of the problem is determined by two more parameters. The first one is the mole fraction defined as
C ¼ n1 =ðn1 þ n2 Þ;
ð1Þ
where n1 and n2 are the number density of species. Because of thermodiffusion phenomenon, the mole fraction varies between the plates so that we will distinguish the equilibrium value C 0 , i.e. its value at DT ¼ 0, and the local mole fraction CðxÞ which is a function of the coordinate x when DT – 0. The other parameter determining the solution is the rarefaction parameter defined as
d ¼ H=‘;
‘ ¼ l0 v 0 =p0 ;
ð2Þ
where ‘ is the equivalent mean-free-path, l0 is the viscosity of mixture at the equilibrium temperature T 0 ; p0 is the equilibrium pressure, and v 0 is the characteristic molecular speed of the mixture given as
v0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2kB T 0 =m;
ð3Þ
kB is the Boltzmann constant, m is the mean molecular mass of mixture given as
m ¼ C 0 m1 þ ð1 C 0 Þm2 ;
ð4Þ
m1 and m2 being the molecular masses of species. The solution of the problem is also determined by the gas-surface interaction law, but in the present work we are not interested in the influence of this interaction on the heat transfer. Therefore, the diffuse scattering of gaseous particles of both species on the plates is assumed. The results will be given in terms of the dimensionless heat flux q defined by
q ¼ q0x T 0 =ðp0 v 0 DTÞ;
ð5Þ
which is always positive. Since the reduced heat flux q weakly depends on relative temperature drop DT=T 0 , the knowledge of this quantity allows us to use data obtained for a specific value of DT=T 0 over a wide range of this ratio. We are going to calculate q as a function of the parameters DT=T 0 ; C 0 , and d for both HS and AI potentials with the numerical error less than 0.5%. Moreover, the temperature and mole fraction profiles will be reported. A comparison of the results based on these two different potentials will allow us to study their influence on the heat transfer through a mixture. 3. Free-molecular regime
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 4p0 v 0 m m h 1 h2 ¼ pffiffiffiffi C 0 þ ð1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ; m1 m2 p 1þhþ 1h
ð6Þ
The dimensionless heat flux (5) takes the form
The temperature of the mixture is given as the geometric average of the surface temperatures, i.e.
T ¼ T0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h2 :
ð9Þ
The mole fraction in this regime is equal to its equilibrium values C0 . In order to estimate the deviation of the non-linear solution (8) from the linearized one, the expansion with respect to the temperature difference DT=T 0 is obtained as
rffiffiffiffiffiffiffi" 2 4 !# rffiffiffiffiffiffiffi 1 m m 3 DT DT ; q ¼ pffiffiffiffi C 0 þO þ ð1 C 0 Þ 1 m1 m2 32 T 0 T0 p 2 4 ! T 1 DT DT ¼1 þO : T0 8 T0 T0
ð10Þ ð11Þ
4. Hydrodynamic regime In the hydrodynamic regime (d 1), the solution is based on the Fourier equation
q0x ¼ jðdT=dxÞ;
ð12Þ
where j is the heat conductivity of mixture. Substituting the Fourier Eq. (12) into the energy conservation law, the equation for the temperature distribution is obtained as
d dx
j
dT dx
¼ 0:
ð13Þ
The main difficulty to solve this equation is that the coefficient j is a function of both temperature T and chemical composition C of mixture. The fact is that, in a non-isothermal mixture being at rest, a mole fraction gradient is established which is related to the temperature gradient as
dC=dx ¼ ðkT =TÞðdT=dxÞ;
ð14Þ
where kT is the thermal diffusion ratio which is also a function of local temperature and mole fraction. Thus, the mole fraction distribution is not known a priori that makes a rigorous analytical solution of Eq. (13) for mixture not possible. The contribution of non-linear terms into the heat flux and temperature distribution can be estimated for a single gas. Like in the previous work [16], here we will restrict ourselves by the approximation assuming the following dependence of the thermal conductivity j on temperature
jðTÞ ¼ j0 ðT=T 0 Þx ; where
ð15Þ
j0 ¼ jðT 0 Þ. Then, Eq. (13) is easily solved as
T=T 0 ¼ ðA þ BxÞ1=ðxþ1Þ :
ð16Þ
q0x ¼ j0 B:
ð17Þ
The unknown constant A and B are found under the temperature continuity condition, i.e.
T ¼ T 0 DT=2;
at x ¼ H=2;
ð18Þ
so that
where
h ¼ DT=2T 0 :
ð8Þ
Combining Eqs. (12) and (16), the heat flux is obtained as
In the free-molecular regime (d ! 0) the problem is easily solved for the diffuse scattering of gaseous particles on the walls, see e.g. Refs. [2,29]. In this case, the heat flux reads
q0x
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 1 h2 2 m m q ¼ pffiffiffiffi C 0 þ ð1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi : m m p 1 2 1þhþ 1h
ð7Þ
h i A ¼ ð1 þ hÞxþ1 þ ð1 hÞxþ1 =2; h i B ¼ ð1 þ hÞxþ1 ð1 hÞxþ1 =H;
ð19Þ ð20Þ
93
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
where h is given by Eq. (7). In order to evaluate the contribution of non-linear terms into the temperature distribution and heat flux, the coefficients A and B are expanded with respect to DT=T 0 and substituted into Eqs. (16) and (17), i.e.
" 2 # 2 3 ! T x DT x 2x DT DT ; þ ¼1þ þO 1 T0 H T0 H T0 T0 8 " 4 !# 5 xð1 xÞ DT 2 DT q¼ ; 1 þO 4dPr 24 T0 T0
ð21Þ ð22Þ
where the dimensionless heat flux q defined by Eq. (5) has been used, Pr ¼ cp l0 =j0 is the Prandtl number, and cp ¼ 5kB =2m is the specific heat at a constant pressure. In case of small temperature difference (DT=T 0 1), the analytical solution of the problem in question was obtained previously [29] for the temperature jump boundary condition [37,39]. In our notation, the solution reads
T x DT ¼1þ ; T0 Hð1 þ 2fT =dÞ T 0 5 q¼ ; 4ðd þ 2fT ÞPr
ð23Þ ð24Þ
where fT is the temperature jump coefficient. The values of Pr based on both HS and AI potentials given in Table 1 for the mixture under consideration show that this parameter is sensitive to the chemical composition and intermolecular potential. The jump coefficient fT obtained previously [37,39] for both HS potential and RP presented in Table 1 is also strongly affected by the potential. It means that even for the small temperature difference DT=T 0 , the heat flux given by Eq. (24) significantly depends on the chemical composition and is sensitive to the potential. The temperature distribution given by (23) is affected by the potential only via the temperature jump coefficient, but since its contribution is small, the influence of the potential on the temperature distribution is weak. In case of large temperature drop DT=T 0 , the heat flux q given by (22) and the temperature distribution (21) are affected by the potential also via the viscosity index x. Its value calculated via the empirical data [40] are given in Table 1, while its value based on the HS potential is always equal to 0.5. Thus, the influence of the potential on the heat flux is significant for any temperature difference and the same influence is expected in a wide range of the rarefaction parameter. 5. Transitional regime Like our previous papers [15,16], the DSMC [2] method based on the AI potential is applied. The same mixture, i.e. He–Ar, is considered here so that all data on the AI potential can be found in Ref. [15]. The calculations were carried for the HS potential too. The molecular diameters d1 and d2 needed for this potential were extracted from the exact expression of viscosity for a single gas of hard spheres
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l ¼ 0:126668 2mkB T =d2
ð25Þ
obtained in [41,42]. Assuming that in the limits of single gases C 0 ¼ 0 and C 0 ¼ 1 the viscosity for HS should be equal to that for AI, the diameters are obtained as d1 ¼ 0:2173 nm and d2 ¼ 0:3618 nm for He and Ar, respectively. The calculations were carried out with the following input data: rarefaction parameter d, equilibrium mole fraction C 0 , relative temperature difference DT=T 0 . The viscosity l0 used to calculate the rarefaction parameter by (2) is given in Table 1 for both HS and AI potentials. The numerical scheme applied to the problem under question is exactly the same as that used in Ref. [16], i.e. the number of cells was 400, the number of model particles was 4 104 , the time increment Dt ¼ 0:002 in H=v 0 units. The deflection angles were precalculated for 900 values of the impact parameter distributed in the range from 0 to 3 units of the distance d corresponding to the zero potential and for 600 values of the relative kinetic energy varying from 0 to 600 , where is the well depth of potential. Such parameters provide the numerical error less than 0.5%. Note, the matrix of deflection angles obtained once for the calculations reported previously [15,16] was used in the present work. This is a great advantage of the method proposed in Ref. [1] allowing to apply the same look-up table to quite different gas flows. Two values of the temperature difference, DT=T 0 ¼ 0:2 and 1:5 corresponding to the temperature ratios 1.22 and 7, respectively, were chosen for the calculations. For the first value of the temperature difference (DT=T 0 ¼ 0:2), the contributions of the nonlinear terms into the heat flux in the hydrodynamic (22) and free-molecular (10) regimes are about 0.04% and 0.38%, respectively. The influence of the quadratic term ðDT=T 0 Þ2 into the temperature distribution, see Eqs. (11) and (21), does not exceed 0.5%. Thus, the results for the temperature difference DT=T 0 ¼ 0:2 correspond to the linearized solution within the accuracy 0.5% assumed in the present work. For the second value of the temperature difference (DT=T 0 ¼ 1:5), a deviation from the linearized solution is expected. Five values of the equilibrium mole fraction C 0 were considered, viz., 0, 0.25, 0.5, 0.75 and 1. The rarefaction parameter d varied from 0.01 to 40 covering the transitional and temperature jump regimes. In the problem in question, the heat flux q0x is calculated via the kinetic energy E of particles crossing an area A during a time interval t, i.e. q0x ¼ E=At. In our simulations, the heat flux was calculated through the middle plane (x ¼ 0), i.e., the kinetic energies of all particles crossing the middle plane were summarized with the sign depending on the motion direction and then divided by the total time t of the simulation. In order to reduce the relative statistical scattering of the heat flux up to the value significantly smaller than the adopted uncertainty 0.5%, the simulation time t was 5 104 and 104 of the units H=v 0 for DT=T 0 ¼ 0:2 and 1.5, respectively. The local temperature T of the mixture is calculated via its definition, i.e. the average kinetic energy of all particles in each cell must be equal to ð3=2ÞkB T. During the simulation, the kinetic energy was averaged over all particles and over the whole time of the simulation. The local mole fraction C was calculated following its definition (1) as C ¼ N 1 =ðN 1 þ N 2 Þ, where N a is the number of particles of species a in each cell averaged over the simulation time.
Table 1 Properties of mixture He–Ar at T = 300 K C0
l (lPa s), [16] HS
AI
HS
AI
HS
AI
HS
RP
0. 0.25 0.5 0.75 1.
22.68 22.81 22.77 22.22 19.91
22.68 23.22 23.53 23.18 19.91
0.6607 0.5232 0.4523 0.4408 0.6607
0.6652 0.4782 0.4096 0.4139 0.6648
– 0.1026 0.1693 0.1692 –
– 0.0604 0.0962 0.0916 –
1.954 1.769 1.777 1.910 1.954
1.954 1.969 2.017 2.070 1.954
Pr, [16]
x, [40]
fT , [37]
kT , [16]
0.85 0.81 0.77 0.71 0.68
94
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97 Table 4 Deviation of McCormac model (MC) for RP, see Ref. [29], from that for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 at DT=T 0 ¼ 0:2.
6. Results and discussions 6.1. Heat flux
ðqMC qAI Þ=qAI (%)
d
The numerical values of the dimensionless heat flux q for the AI potential are presented in Table 2. The values of q in the freemolecular regime (d=0) were calculated from Eq. (8) and those in the last row were calculated from Eq. (24) substituting d ¼ 40 and values of Pr and fT given in Table 1 for AI and RP. A comparison of the numerical data of q for d ¼ 0:01 with those for d ¼ 0 shows that the relative difference does not exceed 1%, i.e., it has the order of the rarefaction parameter d. The difference between the DSMC data for d ¼ 40 and DT=T 0 ¼ 0:2 from those calculated from the linearized Fourier Eq. (24) is less than 0.3%, i.e. the discrepancy between the numerical results and analytical solution does not exceed the numerical error. Table 3 presents the relative deviation of heat flux q for the HS potential from that for the AI potential. As expected, the difference between the data for mixture based on these two potentials is significant in the hydrodynamic regime (d 1) and reaches 8%. At the same time, the data for single gas (C 0 ¼ 0 and 1) do not differ significantly. It is explained by the fact that the Prandtl number for a mixture, see Table 1, is quite sensitive to the potential, while it is not affected by the potential in case of single gas. The relative difference between the values corresponding to the different potentials decreases by decreasing the rarefaction parameter when the intermolecular collisions vanish. In order to determine the uncertainty of the McCormack model equation [32] applied to heat transfer, a comparison of the heat flux calculated in Ref. [29] based on this model with that calculated here for the AI potential at DT=T 0 = 0.2 is performed in Table 4. Note, according to Ref. [29], the heat flux based on the McCormack model is affected by the intermolecular potential so that the data given in Table 4 were obtained from this model with the RP. As expected, the difference is practically within the numerical uncertainty in the case of single gas, i.e. C 0 ¼ 0 and 1. However for
0.01 0.1 1 10 20 40
C0 ¼ 0
0.25
0.5
0.75
1
0.55 1.1 0.76 0.08 0.03 0.08
0.47 0.60 0.63 1.8 1.8 1.9
0.44 0.42 1.0 2.4 2.2 2.4
0.44 0.43 0.91 1.9 1.7 1.8
0.49 0.57 0.28 0.62 0.42 0.35
mixture, the difference between the data obtained from the model equation and those based on the DSMC method exceeds the numerical error and reaches 2.4%. This discrepancy is explained by the fact that the McCormack equation reproduces transport coefficients only in their first term of the Sonine expansion [4], while the other terms of the expansion modify the heat conductivity within few percent. It is worth to note that computational efforts to solve the model kinetic equation is much smaller than those to apply the DSMC method so that the McCormack model could provide reliable results on heat transfer within the uncertainty of 2% with modest computational effort. 6.2. Temperature distribution For the small value of temperature difference (DT=T 0 ¼ 0:2), the temperature profile TðxÞ is quite close to a straight line and always satisfies the condition Tð0Þ ¼ T 0 . Its gradient smoothly varies from zero in the free-molecular regime, see Eq. (11), to the value dT=dx ¼ DT=H in the hydrodynamic regime according to Eq. (21). The temperature profile at the large value of temperature difference DT=T ¼ 1:5 is more interesting and it is worth to analyze it in details. Typical temperature profiles TðxÞ for DT=T 0 ¼ 1:5 and C 0 ¼ 0:5 are shown in Fig. 1. As can be seen, in this case the profiles differ
Table 2 Dimensionless heat flux q for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 and temperature difference DT=T 0 . q
d
C0 ¼ 0 DT T0
0a 0.01 0.1 1 10 20 40 40b a
¼ 0:2
0.5621 0.5577 0.5294 0.3971 0.1347 0.07843 0.04266 0.04280
C 0 ¼0.25
C 0 ¼0.5
C 0 ¼0.75
C 0 ¼1
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.4094 0.4076 0.3913 0.3046 0.1220 0.07355 0.04083 –
0.7619 0.7570 0.7230 0.5495 0.1870 0.1088 0.05930 0.05949
0.5550 0.5527 0.5308 0.4145 0.1699 0.1038 0.05799 –
0.8669 0.8617 0.8254 0.6318 0.2175 0.1267 0.06926 0.06930
0.6315 0.6291 0.6051 0.4764 0.1988 0.1222 0.06849 –
0.8395 0.8345 0.7996 0.6139 0.2135 0.1248 0.06829 0.06842
0.6115 0.6095 0.5878 0.4663 0.1962 0.1206 0.06774 –
0.5621 0.5580 0.5321 0.4013 0.1357 0.07874 0.04284 0.04282
0.4094 0.4080 0.3941 0.3094 0.1233 0.07441 0.04129 –
Eq. (8) Eq. (24)
b
Table 3 Relative deviation of heat flux q for HS from that for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 and temperature difference DT=T 0 . d
ðqHS qAI Þ=qAI (%) C0 ¼ 0 DT T0
0.01 0.1 1 10 20 40
¼ 0.2
0.21 0.92 2.1 1.7 1.2 1.2
C 0 ¼0.25
C 0 ¼0.5
C 0 ¼0.75
C 0 ¼1
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.28 1.4 3.3 2.0 1.7 1.6
0.10 0.35 0.50 5.3 6.7 7.4
0.20 1.0 1.7 2.6 4.2 5.3
0.11 0.20 0.79 6.1 7.4 8.1
0.16 0.90 1.5 2.7 4.4 5.3
0.10 0.30 0.05 3.5 4.4 5.0
0.45 0.83 1.8 0.31 1.2 1.6
0.16 0.41 1.0 0.95 0.82 0.80
0.20 0.68 1.7 0.90 0.48 0.41
95
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
1.75 1.5 1.25 T/T0
0.6 δ =0.01 1 10 40
0.55
1
δ =0.01 1 10 40
C 0.5
0.75
0.45
0.5 0.25 -0.5
-0.25
0 x/H
0.25
0.4 -0.5
0.5
-0.25
0
0.25
0.5
x/H Fig. 1. Temperature T=T 0 vs coordinate x for AI potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5. Fig. 2. Mole fraction C vs coordinate x for AI potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5.
from a straight line. In the hydrodynamic limit (d ¼ 40) the temperature T smoothly varies from the value slightly higher than that on the left plate, i.e. T=T 0 ¼ 0:25, to the value slightly lower than that of the right plate, i.e. T=T 0 ¼ 1:75. In the free-molecular regime (d ¼ 0:01) the temperature profile tends to its constant value given by Eq. (9), i.e. T=T 0 ¼ 0:661. Thus, in the transition from the hydrodynamic regime to the free-molecular one, the temperature gradient decreases and vanishes. The values of temperature near the surfaces, i.e. at x ¼ H=2, based on the AI potential are given in Table 5 for the temperature difference DT=T 0 ¼ 1:5. These data show that the temperature is weakly affected by the chemical composition of the mixture. An analysis of the numerical data for the HS potential shows that the temperature distribution is also weakly affected by the molecular potential even for the large value of the temperature drop.
Table 6 Mole fraction C at the walls (x ¼ H=2) for AI potential vs its equilibrium value C 0 and rarefaction parameter d at DT=T 0 ¼ 1:5. CðH=2Þ
d
C 0 ¼ 0:25
0. 0.01 0.1 1 10 20 40
C 0 ¼ 0:5
C 0 ¼ 0:75
x ¼ H2
H 2
H2
H 2
H2
H 2
0.25 0.249 0.238 0.220 0.204 0.199 0.197
0.25 0.251 0.260 0.271 0.283 0.287 0.290
0.5 0.495 0.482 0.457 0.424 0.415 0.410
0.5 0.505 0.515 0.533 0.553 0.559 0.563
0.75 0.749 0.734 0.712 0.674 0.664 0.659
0.75 0.751 0.763 0.779 0.800 0.805 0.808
6.3. Mole fraction
0.6
According to Eq. (14), the mole fraction is non-uniform in a nonisothermal mixture because of the thermodiffusion phenomenon. Typical mole fraction distributions for the large value of the temperature difference DT=T 0 ¼ 1:5 are shown in Fig. 2. Since the thermal diffusion ratio kT is negative, the mole fraction gradient has the same sign as that of the temperature. Physically, it means that the mole fraction of light gaseous species is higher near the hotter wall than that near the colder one. The mole fraction gradient vanishes by decreasing the rarefaction parameter d. Numerical values of the mole fraction near the surfaces (x ¼ H=2) are given in Table 6 as a function of the rarefaction parameter d and equilibrium mole fraction C 0 . For all values of C 0 , the deviation of CðH=2Þ from C 0 increases by increasing of the rarefaction parameter d and tends to a constant value in the limit d ! 1.
0.5 C AI HS
0.4
0.3 -0.5
-0.25
0 x/H
Fig. 3. Mole fraction C vs coordinate x DT=T 0 ¼ 1:5; C 0 ¼ 0:5 and d ¼ 40.
0.25
for
AI and
0.5
HS potentials
Table 5 Temperature T=T 0 at the walls (x ¼ H=2) for AI potential vs equilibrium mole fraction C 0 and rarefaction parameter d at DT=T 0 ¼ 1:5. T=T 0
d
C0 ¼ 0
a
0 0.01 0.1 1 10 20 40 40b a b
Eq. (9) Eq. (18)
0:25
0:5
0:75
1
x ¼ H2
H 2
H2
H 2
H2
H 2
H2
H 2
H2
H 2
0.661 0.656 0.571 0.442 0.320 0.297 0.280 0.250
0.661 0.669 0.788 1.09 1.55 1.64 1.69 1.75
0.661 0.657 0.575 0.446 0.322 0.298 0.282 0.250
0.661 0.669 0.781 1.07 1.55 1.64 1.69 1.75
0.661 0.639 0.579 0.451 0.324 0.300 0.283 0.250
0.661 0.689 0.774 1.05 1.55 1.63 1.69 1.75
0.661 0.657 0.583 0.456 0.326 0.301 0.283 0.250
0.661 0.668 0.768 1.04 1.54 1.63 1.68 1.75
0.661 0.657 0.584 0.458 0.324 0.298 0.280 0.250
0.661 0.668 0.774 1.07 1.54 1.63 1.68 1.75
at
96
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
According to our previous works [15,16], the thermal diffusion ratio kT determining the mole fraction distribution is very sensitive to the intermolecular potential, see Table 1, so that the chemical composition variation between the plates is strongly affected by the potential. Fig. 3 shows a comparison of the mole fraction distribution CðxÞ for the AI potential with that for the HS potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5, and d ¼ 40. According to Table 1, kT for HS is larger than that for the AI potential then the corresponding gradient of CðxÞ for HS is also larger. Thus, if one is interested in reliable results on the distribution of species in a non-isothermal mixture, a reliable potential, e.g. AI or Lennard–Jones, must be used instead of HS. 7. Conclusions A planar heat transfer through a mixture He–Ar has been calculated by the DSMC method based on the ab initio potential and on the hard sphere molecular model. The calculations have been carried out for small and large values of the temperature difference over a wide range of the gas rarefaction with the numerical error of the heat flux less than 0.5%. Analytical solutions have been obtained in the hydrodynamic and free-molecular limits. A comparison of the heat flux calculated for the ab initio potential and for hard spheres showed that the influence of the intermolecular potential is significant, i.e. about 8% for both small and large temperature difference. To check the reliability of the McCormack model equation, a comparison between the results obtained in the previous paper [29] from this model with the present results has been performed for the small value of the temperature difference. It has been found that the difference is about 2% in the hydrodynamic regime. Thus, if the temperature difference is small the McCormack model equation provides reliable results within the uncertainty of 2%. Such a conclusion is important because a numerical solution of the model equation requires quite modest computational effort in comparison to a numerical solution of the Boltzmann equation or application of the DSMC method. The temperature profile is presented graphically and its values at the surfaces are tabulated for the large pressure difference. It is found that the temperature profile is weakly affected by the chemical composition and intermolecular potential. The mole fraction profile is also presented graphically and in Tables. It is shown that the mole fraction varies significantly from one plate to the other in the transitional and hydrodynamic regimes. Moreover, it is strongly sensitive to the molecular potential so that a physical potential, e.g. ab initio, should be used if one is interested in rigorous results on the mole fraction distribution in a non-isothermal mixture. The reported data can be used as benchmark for new methods of rarefied gas dynamics and for new kinetic model equations. Acknowledgments The calculations were carried out in LCPAD (UFPR). The authors thank the Brazilian agencies CNPq and CAPES for support of their research. References [1] F. Sharipov, J.L. Strapasson, Direct simulation Monte Carlo method for an arbitrary intermolecular potential, Phys. Fluids 24 (1) (2012) 011703. [2] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, 1994. [3] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, The Molecular Theory of Gases and Liquids, Wiley, New York, 1954. [4] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland Publishing Company, Amsterdam, 1972.
[5] K. Koura, H. Matsumoto, Variable soft sphere molecular-model for inversepower-law of Lennard-Jones potential, Phys. Fluids 3 (10) (1991) 2459–2465. [6] H.A. Hassan, D. Hash, A generalized hard-sphere model for Monte Carlo simulation, Phys. Fluids A 5 (1993) 738–744. [7] R.A. Aziz, A highly accurate interatomic potential for argon, J. Chem. Phys. 99 (6) (1993) 4518–4525. [8] A. Janzen, R. Aziz, An accurate potential energy curve for helium based on ab initio calculations, J. Chem. Phys. 107 (3) (1997) 914–919. [9] 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. [10] P. Slavícˇek, R. Kalus, P. Paška, I. Odvárková, P. Hobza, A. Malijevsky´, State-ofthe-art correlated ab initio potential energy curves for heavy rare gas dimers: Ar2, Kr2, and Xe2, J. Chem. Phys. 119 (4) (2003) 2102–2119. [11] A. Nasrabad, U. Deiters, Prediction of thermodynamic properties of krypton by monte carlo simulation using ab initio interaction potentials, J. Chem. Phys. 119 (2) (2003) 947–952. [12] T.P. Haley, S.M. Cybulski, Ground state potential energy curves for He–Kr, Ne– Kr, Ar–Kr, and Kr2: coupled-cluster calculations and comparison with experiment, J. Chem. Phys. 119 (2003) 5487–5496. [13] 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. [14] 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. [15] F. Sharipov, J.L. Strapasson, Ab initio simulation of transport phenomena in rarefied gases, Phys. Rev. E 86 (2012) 031130. [16] F. Sharipov, J.L. Strapasson, Benchmark problems for mixtures of rarefied gases. I. Couette flow, Phys. Fluids 25 (2013) 027101. [17] E.P. Gross, S. Ziering, Heat flow between parallel plates, Phys. Fluids 2 (6) (1959) 701–712. [18] M.L. Lavin, J.K. Haviland, Application of a moment method to heat transfer in rarefied gases, Phys. Fluids 5 (3) (1962) 274–279. [19] K. Frankowski, Z. Alterman, C.L. Pekeris, Heat transport between parallel plates in a rarefied gas of rigid sphere molecules, Phys. Fluids 8 (2) (1965) 245–258. [20] P. Bassanini, C. Cercignani, C. Pagani, Comparison of kinetic theory analysis of linearized heat transfer between parallel plates, Int. J. Heat Mass Transfer 10 (1967) 447–460. [21] S.K. Hsu, T.F. Morse, Kinetic theory of parallel plate heat transfer in a polyatomic gas, Phys. Fluids 15 (4) (1972) 584–591. [22] J.R. Thomas, T.S. Chang, C.E. Siewert, Heat transfer between parallel plates with arbitrary surface accommodation, Phys. Fluids 16 (12) (1973) 2116–2120. [23] Y.I. Makeev, P.E. Suetin, V.G. Chernyak, Heat transfer in a rarefied gas between two parallel plates for four collision models, Izv. AN SSSR, Mekh. Zid. i Gaza 1 (1978) 176–180. [24] D. Valougeorgis, J.R. Thomas, The Fn-method in kinetic theory: II. Heat transfer between parallel plates, Transp. Theory Stat. Phys. 14 (1985) 497–512. [25] T. Ohwada, K. Aoki, Y. Sone, Heat transfer and temperature distribution in a rarefied gas between two parallel plates with different temperatures: numerical analysis of the Boltzmann equation for a hard sphere molecule, in: E.P. Muntz, D.P. Weaver, D.H. Campbell (Eds.), Rarefied Gas Dynamics: Theoretical and Computational Techniques, 20st Int. Symp., China, 1996, AIAA, Washington, 1989, pp. 70–81. [26] C.E. Siewert, A discrete-ordinates solution for heat transfer in a plane channel, J. Comput. Phys. 152 (1999) 251–263. [27] S. Kosuge, K. Aoki, S. Takata, Heat transfer in a gas mixture between two parallel plates: finite-difference analysis of the Boltzmann equation, in: T.J. Bartel, M.A. Gallis (Eds.), Rarefied Gas Dynamics, 22nd Int. Symp., AIP Conference Proc., vol. 585, Melvile, 2001, pp. 289–296. [28] R.D.M. Garcia, C.E. Siewert, The McCormack model for gas mixtures: heat transfer in a plane channel, Phys. Fluids 16 (9) (2004) 3393–3402. [29] F. Sharipov, L.M.G. Cumin, D. Kalempa, Heat flux through a binary gaseous mixture over the whole range of the Knudsen number, Physica A 378 (2007) 183–193. [30] R.D.M. Garcia, C.E. Siewert, Heat transfer between parallel plates: an approach based on the linearized Boltzmann equation for a binary mixture of rigidsphere gases, Phys. Fluids 19 (2) (2007) 027102. [31] A. Raines, Heat transfer in a gas mixture between parallel plates, Comput. Math. Math. Phys. 48 (2) (2008) 306–313. [32] F.J. McCormack, Construction of linearized kinetic models for gaseous mixture and molecular gases, Phys. Fluids 16 (1973) 2095–2105. [33] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient, Phys. Fluids 16 (3) (2004) 759–764. [34] S. Naris, D. Valougeorgis, D. Kalempa, F. Sharipov, Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction, Physica A 336 (3–4) (2004) 294–318. [35] S. Naris, D. Valougeorgis, F. Sharipov, D. Kalempa, Discrete velocity modelling of gaseous mixture flows in MEMS, Superlattices Microstruct. 35 (2004) 629– 643. [36] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. III. Diffusion slip coefficient, Phys. Fluids 16 (10) (2004) 3779–3785. [37] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. IV. Temperature jump coefficient, Int. J. Heat Mass Transfer 48 (6) (2005) 1076–1083.
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97 [38] F. Sharipov, Gaseous mixtures in vacuum systems and microfluidics, J. Vac. Sci. Technol. A 31 (2013) 050806. [39] F. Sharipov, Data on the velocity slip and temperature jump on a gas-solid interface, J. Phys. Chem. Ref. Data 40 (2) (2011) 023101. [40] J. Kestin, K. Knierim, E.A. Mason, B. Najafi, S.T. Ro, M. Waldman, Equilibrium and transport properties of the noble gases and their mixture at low densities, J. Phys. Chem. Ref. Data 13 (1) (1984) 229–303.
97
[41] C.L. Pekeris, Z. Alterman, Solution of the Boltzmann–Hilbert integral equation. II. The coefficients of viscosity and heat conduction, Proc. Nat. Acad. Sci. 43 (1957) 998–1007. [42] F. Sharipov, G. Bertoldo, Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential, J. Comp. Phys. 228 (9) (2009) 3345–3357.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Ab initio simulation of heat transfer through a mixture of rarefied gases José L. Strapasson ⇑, Felix Sharipov Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, Curitiba 81531-990, Brazil
a r t i c l e
i n f o
Article history: Received 21 September 2013 Received in revised form 28 November 2013 Accepted 2 December 2013
Keywords: Gaseous mixture Ab initio potential Heat transfer
a b s t r a c t The heat flux problem for a binary gaseous mixture confined between two parallel plates with different temperatures is studied on the basis of the direct simulation Monte Carlo method with an implementation of ab initio potential. The calculations were carried for a wide range of the gas rarefaction, for several values of the mole fraction and for two values of the temperature difference. The smaller value of the difference corresponds to the limit when the nonlinear terms are negligible, while the larger value describes a nonlinear heat transfer. The heat flux, temperature, and mole fraction distributions are presented. To study the influence of the intermolecular potential, the same simulations are carried out for the hard sphere molecular model. A relative deviation of the results based on this model from those based on the ab initio potential is analyzed. It is pointed out that the difference between the heat flux of the two potentials is about 8% and 5% for the small and large temperature differences, respectively. The temperature distribution between plates is weakly affected by the molecular potential, while the chemical composition variation is the most sensitive quantity for the considered problem. The reported results can be used as benchmark data to test model kinetic equations for gaseous mixtures. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In our previous paper [1], we showed that any intermolecular potential can be implemented into the direct simulation Monte Carlo (DSMC) method [2] with the same computational effort as that for the hard sphere (HS) molecular model. Since reliable information about intermolecular potentials of many gases can be found in literature, see e.g. Refs. [3,4], it is not necessary to use the potentials like variable hard spheres [2], variable soft spheres [5] and generalized hard spheres [6] elaborated specifically for the DSMC method. Recently, the ab initio (AI) potentials were calculated for practically all noble gases and their mixtures, see e.g. Refs. [7–14]. An implementation of this potential into the DSMC [15] made this method completely free from tuned parameters usually extracted from experiments. Such an approach allows us to obtain benchmark data which can be used to test kinetic models and approximate methods in order to solve many practical problems of heat and mass transfer with modest computational effort, but without losing reliability. Our recent paper [16] reports benchmark data for the Couette flow of helium–argon mixture over the whole range of the gas rarefaction based on the AI intermolecular potential. A comparison of the results obtained for the AI potential with those for the HS potential showed that the Couette flow is weakly sensitive to the ⇑ Corresponding author. Tel.: +55 4136722356. E-mail addresses: jstrapasson@fisica.ufpr.br (J.L. Strapasson), sharipov@fisica.ufpr.br (F. Sharipov). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.011
potential. In the present paper, we continue to study the influence of the intermolecular potential on various types of mixture flows. More specifically, a heat transfer through a mixture confined between two plates is calculated applying the DSMC technique. The calculations are carried out over a wide range of the gas rarefaction for both AI and HS potentials. The flow to be considered here is a classical problem of fluid mechanics. Many researchers studied this problem in case of a single rarefied gas, see e.g. Refs. [17–26]. On our knowledge, there are very few papers [27–31] on heat transfer through a mixture of rarefied gases. The works [27,30,31] provide results based on the Boltzmann equation with the HS potential. The paper [28] reports results on the heat flux based on the linearized McCormack (MC) model [32] of the Boltzmann equation. The paper [29] is also based on the MC model, but two potentials were used, viz., HS and the socalled realistic potential (RP). A comparison of results based on these two potentials showed that the heat transfer is strongly sensitive to the potential. As was shown in Refs. [33–38], many other phenomena in gaseous mixtures are very sensitive to the potential of the intermolecular interaction. Thus, it is important to obtain benchmark results based on the ab initio calculations for a large number of phenomena. The aim of the present paper is to calculate the heat flux through a binary mixture of rarefied gases confined between two parallel plates based on the AI potential implemented into the DSMC method. To study the influence of the potential on heat flux, temperature and chemical composition distributions, the same problem will be solved for the HS molecular model too.
92
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
2. Statement of the problem Consider a binary gaseous mixture confined between two parallel plates fixed at x ¼ H=2 and having different temperatures T 0 DT=2, respectively. Thus, H is the distance between the plates and DT is the temperature difference. We are going to calculate the heat flux q0x , temperature TðxÞ profile and mole fraction CðxÞ distribution between the plates. Besides the relative temperature difference DT=T 0 , the solution of the problem is determined by two more parameters. The first one is the mole fraction defined as
C ¼ n1 =ðn1 þ n2 Þ;
ð1Þ
where n1 and n2 are the number density of species. Because of thermodiffusion phenomenon, the mole fraction varies between the plates so that we will distinguish the equilibrium value C 0 , i.e. its value at DT ¼ 0, and the local mole fraction CðxÞ which is a function of the coordinate x when DT – 0. The other parameter determining the solution is the rarefaction parameter defined as
d ¼ H=‘;
‘ ¼ l0 v 0 =p0 ;
ð2Þ
where ‘ is the equivalent mean-free-path, l0 is the viscosity of mixture at the equilibrium temperature T 0 ; p0 is the equilibrium pressure, and v 0 is the characteristic molecular speed of the mixture given as
v0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2kB T 0 =m;
ð3Þ
kB is the Boltzmann constant, m is the mean molecular mass of mixture given as
m ¼ C 0 m1 þ ð1 C 0 Þm2 ;
ð4Þ
m1 and m2 being the molecular masses of species. The solution of the problem is also determined by the gas-surface interaction law, but in the present work we are not interested in the influence of this interaction on the heat transfer. Therefore, the diffuse scattering of gaseous particles of both species on the plates is assumed. The results will be given in terms of the dimensionless heat flux q defined by
q ¼ q0x T 0 =ðp0 v 0 DTÞ;
ð5Þ
which is always positive. Since the reduced heat flux q weakly depends on relative temperature drop DT=T 0 , the knowledge of this quantity allows us to use data obtained for a specific value of DT=T 0 over a wide range of this ratio. We are going to calculate q as a function of the parameters DT=T 0 ; C 0 , and d for both HS and AI potentials with the numerical error less than 0.5%. Moreover, the temperature and mole fraction profiles will be reported. A comparison of the results based on these two different potentials will allow us to study their influence on the heat transfer through a mixture. 3. Free-molecular regime
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 4p0 v 0 m m h 1 h2 ¼ pffiffiffiffi C 0 þ ð1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ; m1 m2 p 1þhþ 1h
ð6Þ
The dimensionless heat flux (5) takes the form
The temperature of the mixture is given as the geometric average of the surface temperatures, i.e.
T ¼ T0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h2 :
ð9Þ
The mole fraction in this regime is equal to its equilibrium values C0 . In order to estimate the deviation of the non-linear solution (8) from the linearized one, the expansion with respect to the temperature difference DT=T 0 is obtained as
rffiffiffiffiffiffiffi" 2 4 !# rffiffiffiffiffiffiffi 1 m m 3 DT DT ; q ¼ pffiffiffiffi C 0 þO þ ð1 C 0 Þ 1 m1 m2 32 T 0 T0 p 2 4 ! T 1 DT DT ¼1 þO : T0 8 T0 T0
ð10Þ ð11Þ
4. Hydrodynamic regime In the hydrodynamic regime (d 1), the solution is based on the Fourier equation
q0x ¼ jðdT=dxÞ;
ð12Þ
where j is the heat conductivity of mixture. Substituting the Fourier Eq. (12) into the energy conservation law, the equation for the temperature distribution is obtained as
d dx
j
dT dx
¼ 0:
ð13Þ
The main difficulty to solve this equation is that the coefficient j is a function of both temperature T and chemical composition C of mixture. The fact is that, in a non-isothermal mixture being at rest, a mole fraction gradient is established which is related to the temperature gradient as
dC=dx ¼ ðkT =TÞðdT=dxÞ;
ð14Þ
where kT is the thermal diffusion ratio which is also a function of local temperature and mole fraction. Thus, the mole fraction distribution is not known a priori that makes a rigorous analytical solution of Eq. (13) for mixture not possible. The contribution of non-linear terms into the heat flux and temperature distribution can be estimated for a single gas. Like in the previous work [16], here we will restrict ourselves by the approximation assuming the following dependence of the thermal conductivity j on temperature
jðTÞ ¼ j0 ðT=T 0 Þx ; where
ð15Þ
j0 ¼ jðT 0 Þ. Then, Eq. (13) is easily solved as
T=T 0 ¼ ðA þ BxÞ1=ðxþ1Þ :
ð16Þ
q0x ¼ j0 B:
ð17Þ
The unknown constant A and B are found under the temperature continuity condition, i.e.
T ¼ T 0 DT=2;
at x ¼ H=2;
ð18Þ
so that
where
h ¼ DT=2T 0 :
ð8Þ
Combining Eqs. (12) and (16), the heat flux is obtained as
In the free-molecular regime (d ! 0) the problem is easily solved for the diffuse scattering of gaseous particles on the walls, see e.g. Refs. [2,29]. In this case, the heat flux reads
q0x
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi 1 h2 2 m m q ¼ pffiffiffiffi C 0 þ ð1 C 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi : m m p 1 2 1þhþ 1h
ð7Þ
h i A ¼ ð1 þ hÞxþ1 þ ð1 hÞxþ1 =2; h i B ¼ ð1 þ hÞxþ1 ð1 hÞxþ1 =H;
ð19Þ ð20Þ
93
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
where h is given by Eq. (7). In order to evaluate the contribution of non-linear terms into the temperature distribution and heat flux, the coefficients A and B are expanded with respect to DT=T 0 and substituted into Eqs. (16) and (17), i.e.
" 2 # 2 3 ! T x DT x 2x DT DT ; þ ¼1þ þO 1 T0 H T0 H T0 T0 8 " 4 !# 5 xð1 xÞ DT 2 DT q¼ ; 1 þO 4dPr 24 T0 T0
ð21Þ ð22Þ
where the dimensionless heat flux q defined by Eq. (5) has been used, Pr ¼ cp l0 =j0 is the Prandtl number, and cp ¼ 5kB =2m is the specific heat at a constant pressure. In case of small temperature difference (DT=T 0 1), the analytical solution of the problem in question was obtained previously [29] for the temperature jump boundary condition [37,39]. In our notation, the solution reads
T x DT ¼1þ ; T0 Hð1 þ 2fT =dÞ T 0 5 q¼ ; 4ðd þ 2fT ÞPr
ð23Þ ð24Þ
where fT is the temperature jump coefficient. The values of Pr based on both HS and AI potentials given in Table 1 for the mixture under consideration show that this parameter is sensitive to the chemical composition and intermolecular potential. The jump coefficient fT obtained previously [37,39] for both HS potential and RP presented in Table 1 is also strongly affected by the potential. It means that even for the small temperature difference DT=T 0 , the heat flux given by Eq. (24) significantly depends on the chemical composition and is sensitive to the potential. The temperature distribution given by (23) is affected by the potential only via the temperature jump coefficient, but since its contribution is small, the influence of the potential on the temperature distribution is weak. In case of large temperature drop DT=T 0 , the heat flux q given by (22) and the temperature distribution (21) are affected by the potential also via the viscosity index x. Its value calculated via the empirical data [40] are given in Table 1, while its value based on the HS potential is always equal to 0.5. Thus, the influence of the potential on the heat flux is significant for any temperature difference and the same influence is expected in a wide range of the rarefaction parameter. 5. Transitional regime Like our previous papers [15,16], the DSMC [2] method based on the AI potential is applied. The same mixture, i.e. He–Ar, is considered here so that all data on the AI potential can be found in Ref. [15]. The calculations were carried for the HS potential too. The molecular diameters d1 and d2 needed for this potential were extracted from the exact expression of viscosity for a single gas of hard spheres
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l ¼ 0:126668 2mkB T =d2
ð25Þ
obtained in [41,42]. Assuming that in the limits of single gases C 0 ¼ 0 and C 0 ¼ 1 the viscosity for HS should be equal to that for AI, the diameters are obtained as d1 ¼ 0:2173 nm and d2 ¼ 0:3618 nm for He and Ar, respectively. The calculations were carried out with the following input data: rarefaction parameter d, equilibrium mole fraction C 0 , relative temperature difference DT=T 0 . The viscosity l0 used to calculate the rarefaction parameter by (2) is given in Table 1 for both HS and AI potentials. The numerical scheme applied to the problem under question is exactly the same as that used in Ref. [16], i.e. the number of cells was 400, the number of model particles was 4 104 , the time increment Dt ¼ 0:002 in H=v 0 units. The deflection angles were precalculated for 900 values of the impact parameter distributed in the range from 0 to 3 units of the distance d corresponding to the zero potential and for 600 values of the relative kinetic energy varying from 0 to 600 , where is the well depth of potential. Such parameters provide the numerical error less than 0.5%. Note, the matrix of deflection angles obtained once for the calculations reported previously [15,16] was used in the present work. This is a great advantage of the method proposed in Ref. [1] allowing to apply the same look-up table to quite different gas flows. Two values of the temperature difference, DT=T 0 ¼ 0:2 and 1:5 corresponding to the temperature ratios 1.22 and 7, respectively, were chosen for the calculations. For the first value of the temperature difference (DT=T 0 ¼ 0:2), the contributions of the nonlinear terms into the heat flux in the hydrodynamic (22) and free-molecular (10) regimes are about 0.04% and 0.38%, respectively. The influence of the quadratic term ðDT=T 0 Þ2 into the temperature distribution, see Eqs. (11) and (21), does not exceed 0.5%. Thus, the results for the temperature difference DT=T 0 ¼ 0:2 correspond to the linearized solution within the accuracy 0.5% assumed in the present work. For the second value of the temperature difference (DT=T 0 ¼ 1:5), a deviation from the linearized solution is expected. Five values of the equilibrium mole fraction C 0 were considered, viz., 0, 0.25, 0.5, 0.75 and 1. The rarefaction parameter d varied from 0.01 to 40 covering the transitional and temperature jump regimes. In the problem in question, the heat flux q0x is calculated via the kinetic energy E of particles crossing an area A during a time interval t, i.e. q0x ¼ E=At. In our simulations, the heat flux was calculated through the middle plane (x ¼ 0), i.e., the kinetic energies of all particles crossing the middle plane were summarized with the sign depending on the motion direction and then divided by the total time t of the simulation. In order to reduce the relative statistical scattering of the heat flux up to the value significantly smaller than the adopted uncertainty 0.5%, the simulation time t was 5 104 and 104 of the units H=v 0 for DT=T 0 ¼ 0:2 and 1.5, respectively. The local temperature T of the mixture is calculated via its definition, i.e. the average kinetic energy of all particles in each cell must be equal to ð3=2ÞkB T. During the simulation, the kinetic energy was averaged over all particles and over the whole time of the simulation. The local mole fraction C was calculated following its definition (1) as C ¼ N 1 =ðN 1 þ N 2 Þ, where N a is the number of particles of species a in each cell averaged over the simulation time.
Table 1 Properties of mixture He–Ar at T = 300 K C0
l (lPa s), [16] HS
AI
HS
AI
HS
AI
HS
RP
0. 0.25 0.5 0.75 1.
22.68 22.81 22.77 22.22 19.91
22.68 23.22 23.53 23.18 19.91
0.6607 0.5232 0.4523 0.4408 0.6607
0.6652 0.4782 0.4096 0.4139 0.6648
– 0.1026 0.1693 0.1692 –
– 0.0604 0.0962 0.0916 –
1.954 1.769 1.777 1.910 1.954
1.954 1.969 2.017 2.070 1.954
Pr, [16]
x, [40]
fT , [37]
kT , [16]
0.85 0.81 0.77 0.71 0.68
94
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97 Table 4 Deviation of McCormac model (MC) for RP, see Ref. [29], from that for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 at DT=T 0 ¼ 0:2.
6. Results and discussions 6.1. Heat flux
ðqMC qAI Þ=qAI (%)
d
The numerical values of the dimensionless heat flux q for the AI potential are presented in Table 2. The values of q in the freemolecular regime (d=0) were calculated from Eq. (8) and those in the last row were calculated from Eq. (24) substituting d ¼ 40 and values of Pr and fT given in Table 1 for AI and RP. A comparison of the numerical data of q for d ¼ 0:01 with those for d ¼ 0 shows that the relative difference does not exceed 1%, i.e., it has the order of the rarefaction parameter d. The difference between the DSMC data for d ¼ 40 and DT=T 0 ¼ 0:2 from those calculated from the linearized Fourier Eq. (24) is less than 0.3%, i.e. the discrepancy between the numerical results and analytical solution does not exceed the numerical error. Table 3 presents the relative deviation of heat flux q for the HS potential from that for the AI potential. As expected, the difference between the data for mixture based on these two potentials is significant in the hydrodynamic regime (d 1) and reaches 8%. At the same time, the data for single gas (C 0 ¼ 0 and 1) do not differ significantly. It is explained by the fact that the Prandtl number for a mixture, see Table 1, is quite sensitive to the potential, while it is not affected by the potential in case of single gas. The relative difference between the values corresponding to the different potentials decreases by decreasing the rarefaction parameter when the intermolecular collisions vanish. In order to determine the uncertainty of the McCormack model equation [32] applied to heat transfer, a comparison of the heat flux calculated in Ref. [29] based on this model with that calculated here for the AI potential at DT=T 0 = 0.2 is performed in Table 4. Note, according to Ref. [29], the heat flux based on the McCormack model is affected by the intermolecular potential so that the data given in Table 4 were obtained from this model with the RP. As expected, the difference is practically within the numerical uncertainty in the case of single gas, i.e. C 0 ¼ 0 and 1. However for
0.01 0.1 1 10 20 40
C0 ¼ 0
0.25
0.5
0.75
1
0.55 1.1 0.76 0.08 0.03 0.08
0.47 0.60 0.63 1.8 1.8 1.9
0.44 0.42 1.0 2.4 2.2 2.4
0.44 0.43 0.91 1.9 1.7 1.8
0.49 0.57 0.28 0.62 0.42 0.35
mixture, the difference between the data obtained from the model equation and those based on the DSMC method exceeds the numerical error and reaches 2.4%. This discrepancy is explained by the fact that the McCormack equation reproduces transport coefficients only in their first term of the Sonine expansion [4], while the other terms of the expansion modify the heat conductivity within few percent. It is worth to note that computational efforts to solve the model kinetic equation is much smaller than those to apply the DSMC method so that the McCormack model could provide reliable results on heat transfer within the uncertainty of 2% with modest computational effort. 6.2. Temperature distribution For the small value of temperature difference (DT=T 0 ¼ 0:2), the temperature profile TðxÞ is quite close to a straight line and always satisfies the condition Tð0Þ ¼ T 0 . Its gradient smoothly varies from zero in the free-molecular regime, see Eq. (11), to the value dT=dx ¼ DT=H in the hydrodynamic regime according to Eq. (21). The temperature profile at the large value of temperature difference DT=T ¼ 1:5 is more interesting and it is worth to analyze it in details. Typical temperature profiles TðxÞ for DT=T 0 ¼ 1:5 and C 0 ¼ 0:5 are shown in Fig. 1. As can be seen, in this case the profiles differ
Table 2 Dimensionless heat flux q for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 and temperature difference DT=T 0 . q
d
C0 ¼ 0 DT T0
0a 0.01 0.1 1 10 20 40 40b a
¼ 0:2
0.5621 0.5577 0.5294 0.3971 0.1347 0.07843 0.04266 0.04280
C 0 ¼0.25
C 0 ¼0.5
C 0 ¼0.75
C 0 ¼1
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.4094 0.4076 0.3913 0.3046 0.1220 0.07355 0.04083 –
0.7619 0.7570 0.7230 0.5495 0.1870 0.1088 0.05930 0.05949
0.5550 0.5527 0.5308 0.4145 0.1699 0.1038 0.05799 –
0.8669 0.8617 0.8254 0.6318 0.2175 0.1267 0.06926 0.06930
0.6315 0.6291 0.6051 0.4764 0.1988 0.1222 0.06849 –
0.8395 0.8345 0.7996 0.6139 0.2135 0.1248 0.06829 0.06842
0.6115 0.6095 0.5878 0.4663 0.1962 0.1206 0.06774 –
0.5621 0.5580 0.5321 0.4013 0.1357 0.07874 0.04284 0.04282
0.4094 0.4080 0.3941 0.3094 0.1233 0.07441 0.04129 –
Eq. (8) Eq. (24)
b
Table 3 Relative deviation of heat flux q for HS from that for AI potential vs rarefaction parameter d, equilibrium mole fraction C 0 and temperature difference DT=T 0 . d
ðqHS qAI Þ=qAI (%) C0 ¼ 0 DT T0
0.01 0.1 1 10 20 40
¼ 0.2
0.21 0.92 2.1 1.7 1.2 1.2
C 0 ¼0.25
C 0 ¼0.5
C 0 ¼0.75
C 0 ¼1
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.2
1.5
0.28 1.4 3.3 2.0 1.7 1.6
0.10 0.35 0.50 5.3 6.7 7.4
0.20 1.0 1.7 2.6 4.2 5.3
0.11 0.20 0.79 6.1 7.4 8.1
0.16 0.90 1.5 2.7 4.4 5.3
0.10 0.30 0.05 3.5 4.4 5.0
0.45 0.83 1.8 0.31 1.2 1.6
0.16 0.41 1.0 0.95 0.82 0.80
0.20 0.68 1.7 0.90 0.48 0.41
95
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
1.75 1.5 1.25 T/T0
0.6 δ =0.01 1 10 40
0.55
1
δ =0.01 1 10 40
C 0.5
0.75
0.45
0.5 0.25 -0.5
-0.25
0 x/H
0.25
0.4 -0.5
0.5
-0.25
0
0.25
0.5
x/H Fig. 1. Temperature T=T 0 vs coordinate x for AI potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5. Fig. 2. Mole fraction C vs coordinate x for AI potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5.
from a straight line. In the hydrodynamic limit (d ¼ 40) the temperature T smoothly varies from the value slightly higher than that on the left plate, i.e. T=T 0 ¼ 0:25, to the value slightly lower than that of the right plate, i.e. T=T 0 ¼ 1:75. In the free-molecular regime (d ¼ 0:01) the temperature profile tends to its constant value given by Eq. (9), i.e. T=T 0 ¼ 0:661. Thus, in the transition from the hydrodynamic regime to the free-molecular one, the temperature gradient decreases and vanishes. The values of temperature near the surfaces, i.e. at x ¼ H=2, based on the AI potential are given in Table 5 for the temperature difference DT=T 0 ¼ 1:5. These data show that the temperature is weakly affected by the chemical composition of the mixture. An analysis of the numerical data for the HS potential shows that the temperature distribution is also weakly affected by the molecular potential even for the large value of the temperature drop.
Table 6 Mole fraction C at the walls (x ¼ H=2) for AI potential vs its equilibrium value C 0 and rarefaction parameter d at DT=T 0 ¼ 1:5. CðH=2Þ
d
C 0 ¼ 0:25
0. 0.01 0.1 1 10 20 40
C 0 ¼ 0:5
C 0 ¼ 0:75
x ¼ H2
H 2
H2
H 2
H2
H 2
0.25 0.249 0.238 0.220 0.204 0.199 0.197
0.25 0.251 0.260 0.271 0.283 0.287 0.290
0.5 0.495 0.482 0.457 0.424 0.415 0.410
0.5 0.505 0.515 0.533 0.553 0.559 0.563
0.75 0.749 0.734 0.712 0.674 0.664 0.659
0.75 0.751 0.763 0.779 0.800 0.805 0.808
6.3. Mole fraction
0.6
According to Eq. (14), the mole fraction is non-uniform in a nonisothermal mixture because of the thermodiffusion phenomenon. Typical mole fraction distributions for the large value of the temperature difference DT=T 0 ¼ 1:5 are shown in Fig. 2. Since the thermal diffusion ratio kT is negative, the mole fraction gradient has the same sign as that of the temperature. Physically, it means that the mole fraction of light gaseous species is higher near the hotter wall than that near the colder one. The mole fraction gradient vanishes by decreasing the rarefaction parameter d. Numerical values of the mole fraction near the surfaces (x ¼ H=2) are given in Table 6 as a function of the rarefaction parameter d and equilibrium mole fraction C 0 . For all values of C 0 , the deviation of CðH=2Þ from C 0 increases by increasing of the rarefaction parameter d and tends to a constant value in the limit d ! 1.
0.5 C AI HS
0.4
0.3 -0.5
-0.25
0 x/H
Fig. 3. Mole fraction C vs coordinate x DT=T 0 ¼ 1:5; C 0 ¼ 0:5 and d ¼ 40.
0.25
for
AI and
0.5
HS potentials
Table 5 Temperature T=T 0 at the walls (x ¼ H=2) for AI potential vs equilibrium mole fraction C 0 and rarefaction parameter d at DT=T 0 ¼ 1:5. T=T 0
d
C0 ¼ 0
a
0 0.01 0.1 1 10 20 40 40b a b
Eq. (9) Eq. (18)
0:25
0:5
0:75
1
x ¼ H2
H 2
H2
H 2
H2
H 2
H2
H 2
H2
H 2
0.661 0.656 0.571 0.442 0.320 0.297 0.280 0.250
0.661 0.669 0.788 1.09 1.55 1.64 1.69 1.75
0.661 0.657 0.575 0.446 0.322 0.298 0.282 0.250
0.661 0.669 0.781 1.07 1.55 1.64 1.69 1.75
0.661 0.639 0.579 0.451 0.324 0.300 0.283 0.250
0.661 0.689 0.774 1.05 1.55 1.63 1.69 1.75
0.661 0.657 0.583 0.456 0.326 0.301 0.283 0.250
0.661 0.668 0.768 1.04 1.54 1.63 1.68 1.75
0.661 0.657 0.584 0.458 0.324 0.298 0.280 0.250
0.661 0.668 0.774 1.07 1.54 1.63 1.68 1.75
at
96
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97
According to our previous works [15,16], the thermal diffusion ratio kT determining the mole fraction distribution is very sensitive to the intermolecular potential, see Table 1, so that the chemical composition variation between the plates is strongly affected by the potential. Fig. 3 shows a comparison of the mole fraction distribution CðxÞ for the AI potential with that for the HS potential at DT=T 0 ¼ 1:5; C 0 ¼ 0:5, and d ¼ 40. According to Table 1, kT for HS is larger than that for the AI potential then the corresponding gradient of CðxÞ for HS is also larger. Thus, if one is interested in reliable results on the distribution of species in a non-isothermal mixture, a reliable potential, e.g. AI or Lennard–Jones, must be used instead of HS. 7. Conclusions A planar heat transfer through a mixture He–Ar has been calculated by the DSMC method based on the ab initio potential and on the hard sphere molecular model. The calculations have been carried out for small and large values of the temperature difference over a wide range of the gas rarefaction with the numerical error of the heat flux less than 0.5%. Analytical solutions have been obtained in the hydrodynamic and free-molecular limits. A comparison of the heat flux calculated for the ab initio potential and for hard spheres showed that the influence of the intermolecular potential is significant, i.e. about 8% for both small and large temperature difference. To check the reliability of the McCormack model equation, a comparison between the results obtained in the previous paper [29] from this model with the present results has been performed for the small value of the temperature difference. It has been found that the difference is about 2% in the hydrodynamic regime. Thus, if the temperature difference is small the McCormack model equation provides reliable results within the uncertainty of 2%. Such a conclusion is important because a numerical solution of the model equation requires quite modest computational effort in comparison to a numerical solution of the Boltzmann equation or application of the DSMC method. The temperature profile is presented graphically and its values at the surfaces are tabulated for the large pressure difference. It is found that the temperature profile is weakly affected by the chemical composition and intermolecular potential. The mole fraction profile is also presented graphically and in Tables. It is shown that the mole fraction varies significantly from one plate to the other in the transitional and hydrodynamic regimes. Moreover, it is strongly sensitive to the molecular potential so that a physical potential, e.g. ab initio, should be used if one is interested in rigorous results on the mole fraction distribution in a non-isothermal mixture. The reported data can be used as benchmark for new methods of rarefied gas dynamics and for new kinetic model equations. Acknowledgments The calculations were carried out in LCPAD (UFPR). The authors thank the Brazilian agencies CNPq and CAPES for support of their research. References [1] F. Sharipov, J.L. Strapasson, Direct simulation Monte Carlo method for an arbitrary intermolecular potential, Phys. Fluids 24 (1) (2012) 011703. [2] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, 1994. [3] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, The Molecular Theory of Gases and Liquids, Wiley, New York, 1954. [4] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland Publishing Company, Amsterdam, 1972.
[5] K. Koura, H. Matsumoto, Variable soft sphere molecular-model for inversepower-law of Lennard-Jones potential, Phys. Fluids 3 (10) (1991) 2459–2465. [6] H.A. Hassan, D. Hash, A generalized hard-sphere model for Monte Carlo simulation, Phys. Fluids A 5 (1993) 738–744. [7] R.A. Aziz, A highly accurate interatomic potential for argon, J. Chem. Phys. 99 (6) (1993) 4518–4525. [8] A. Janzen, R. Aziz, An accurate potential energy curve for helium based on ab initio calculations, J. Chem. Phys. 107 (3) (1997) 914–919. [9] 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. [10] P. Slavícˇek, R. Kalus, P. Paška, I. Odvárková, P. Hobza, A. Malijevsky´, State-ofthe-art correlated ab initio potential energy curves for heavy rare gas dimers: Ar2, Kr2, and Xe2, J. Chem. Phys. 119 (4) (2003) 2102–2119. [11] A. Nasrabad, U. Deiters, Prediction of thermodynamic properties of krypton by monte carlo simulation using ab initio interaction potentials, J. Chem. Phys. 119 (2) (2003) 947–952. [12] T.P. Haley, S.M. Cybulski, Ground state potential energy curves for He–Kr, Ne– Kr, Ar–Kr, and Kr2: coupled-cluster calculations and comparison with experiment, J. Chem. Phys. 119 (2003) 5487–5496. [13] 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. [14] 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. [15] F. Sharipov, J.L. Strapasson, Ab initio simulation of transport phenomena in rarefied gases, Phys. Rev. E 86 (2012) 031130. [16] F. Sharipov, J.L. Strapasson, Benchmark problems for mixtures of rarefied gases. I. Couette flow, Phys. Fluids 25 (2013) 027101. [17] E.P. Gross, S. Ziering, Heat flow between parallel plates, Phys. Fluids 2 (6) (1959) 701–712. [18] M.L. Lavin, J.K. Haviland, Application of a moment method to heat transfer in rarefied gases, Phys. Fluids 5 (3) (1962) 274–279. [19] K. Frankowski, Z. Alterman, C.L. Pekeris, Heat transport between parallel plates in a rarefied gas of rigid sphere molecules, Phys. Fluids 8 (2) (1965) 245–258. [20] P. Bassanini, C. Cercignani, C. Pagani, Comparison of kinetic theory analysis of linearized heat transfer between parallel plates, Int. J. Heat Mass Transfer 10 (1967) 447–460. [21] S.K. Hsu, T.F. Morse, Kinetic theory of parallel plate heat transfer in a polyatomic gas, Phys. Fluids 15 (4) (1972) 584–591. [22] J.R. Thomas, T.S. Chang, C.E. Siewert, Heat transfer between parallel plates with arbitrary surface accommodation, Phys. Fluids 16 (12) (1973) 2116–2120. [23] Y.I. Makeev, P.E. Suetin, V.G. Chernyak, Heat transfer in a rarefied gas between two parallel plates for four collision models, Izv. AN SSSR, Mekh. Zid. i Gaza 1 (1978) 176–180. [24] D. Valougeorgis, J.R. Thomas, The Fn-method in kinetic theory: II. Heat transfer between parallel plates, Transp. Theory Stat. Phys. 14 (1985) 497–512. [25] T. Ohwada, K. Aoki, Y. Sone, Heat transfer and temperature distribution in a rarefied gas between two parallel plates with different temperatures: numerical analysis of the Boltzmann equation for a hard sphere molecule, in: E.P. Muntz, D.P. Weaver, D.H. Campbell (Eds.), Rarefied Gas Dynamics: Theoretical and Computational Techniques, 20st Int. Symp., China, 1996, AIAA, Washington, 1989, pp. 70–81. [26] C.E. Siewert, A discrete-ordinates solution for heat transfer in a plane channel, J. Comput. Phys. 152 (1999) 251–263. [27] S. Kosuge, K. Aoki, S. Takata, Heat transfer in a gas mixture between two parallel plates: finite-difference analysis of the Boltzmann equation, in: T.J. Bartel, M.A. Gallis (Eds.), Rarefied Gas Dynamics, 22nd Int. Symp., AIP Conference Proc., vol. 585, Melvile, 2001, pp. 289–296. [28] R.D.M. Garcia, C.E. Siewert, The McCormack model for gas mixtures: heat transfer in a plane channel, Phys. Fluids 16 (9) (2004) 3393–3402. [29] F. Sharipov, L.M.G. Cumin, D. Kalempa, Heat flux through a binary gaseous mixture over the whole range of the Knudsen number, Physica A 378 (2007) 183–193. [30] R.D.M. Garcia, C.E. Siewert, Heat transfer between parallel plates: an approach based on the linearized Boltzmann equation for a binary mixture of rigidsphere gases, Phys. Fluids 19 (2) (2007) 027102. [31] A. Raines, Heat transfer in a gas mixture between parallel plates, Comput. Math. Math. Phys. 48 (2) (2008) 306–313. [32] F.J. McCormack, Construction of linearized kinetic models for gaseous mixture and molecular gases, Phys. Fluids 16 (1973) 2095–2105. [33] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient, Phys. Fluids 16 (3) (2004) 759–764. [34] S. Naris, D. Valougeorgis, D. Kalempa, F. Sharipov, Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction, Physica A 336 (3–4) (2004) 294–318. [35] S. Naris, D. Valougeorgis, F. Sharipov, D. Kalempa, Discrete velocity modelling of gaseous mixture flows in MEMS, Superlattices Microstruct. 35 (2004) 629– 643. [36] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. III. Diffusion slip coefficient, Phys. Fluids 16 (10) (2004) 3779–3785. [37] F. Sharipov, D. Kalempa, Velocity slip and temperature jump coefficients for gaseous mixtures. IV. Temperature jump coefficient, Int. J. Heat Mass Transfer 48 (6) (2005) 1076–1083.
J.L. Strapasson, F. Sharipov / International Journal of Heat and Mass Transfer 71 (2014) 91–97 [38] F. Sharipov, Gaseous mixtures in vacuum systems and microfluidics, J. Vac. Sci. Technol. A 31 (2013) 050806. [39] F. Sharipov, Data on the velocity slip and temperature jump on a gas-solid interface, J. Phys. Chem. Ref. Data 40 (2) (2011) 023101. [40] J. Kestin, K. Knierim, E.A. Mason, B. Najafi, S.T. Ro, M. Waldman, Equilibrium and transport properties of the noble gases and their mixture at low densities, J. Phys. Chem. Ref. Data 13 (1) (1984) 229–303.
97
[41] C.L. Pekeris, Z. Alterman, Solution of the Boltzmann–Hilbert integral equation. II. The coefficients of viscosity and heat conduction, Proc. Nat. Acad. Sci. 43 (1957) 998–1007. [42] F. Sharipov, G. Bertoldo, Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential, J. Comp. Phys. 228 (9) (2009) 3345–3357.