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Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 6, 283 - 296 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7414 Calculation of Viscosity and Diffusion Coefficients in Binary Mixtures of Dilute Gases A. F. Bogatyrev, O. A. Makeenkova, V. R. Belalov and M. A. Kucherenko Smolensk Branch of National Research University “Moscow Power Engineering Institute” Energeticheskij proezd 1, 214013 Smolensk, Russian Federation Copyright © 2017 A. F. Bogatyrev, O. A. Makeenkova, V. R. Belalov and M. A. Kucherenko. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A method for calculating of viscosity and diffusion coefficients in binary mixtures of dilute gases was observed basing on molecular kinetic theory and intermolecular pair potential energy. The calculations were performed for the binary mixtures of gases containing methane, nitrogen and carbon dioxide, basing on the data on the pure gases viscosity. A comparison of our calculation results with experimental and theoretical data obtained by other investigators was made. Keywords: molecular kinetic theory, gas mixtures, intermolecular potential energy, collision integral, calculation method, viscosity, binary diffusion coefficient 1 Introduction Mixtures of dilute gases are widely spread in nature and are the basis of many industrial processes. Currently, some experimental data on the study of transport properties of pure dilute gases has been accumulated. Significantly fewer experimental materials on the transport properties have been obtained for gas mixtures, particularly their temperature dependence is studied insufficiently. For the calculation and generalization of transport properties of pure chemicals several methods of molecular kinetic theory are used. In general, their results give good agreement with the experimental data [6, 16, 20, 21, 27].
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Vol. 11, 2017, no. 6, 283 - 296
HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7414
Coefficients in Binary Mixtures of Dilute Gases
A. F. Bogatyrev, O. A. Makeenkova, V. R. Belalov and M. A. Kucherenko
Smolensk Branch of National Research University
“Moscow Power Engineering Institute”
Energeticheskij proezd 1, 214013 Smolensk, Russian Federation
Copyright © 2017 A. F. Bogatyrev, O. A. Makeenkova, V. R. Belalov and M. A. Kucherenko.
This article is distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Abstract
A method for calculating of viscosity and diffusion coefficients in binary mixtures
of dilute gases was observed basing on molecular kinetic theory and
intermolecular pair potential energy. The calculations were performed for the
binary mixtures of gases containing methane, nitrogen and carbon dioxide, basing
on the data on the pure gases viscosity. A comparison of our calculation results
with experimental and theoretical data obtained by other investigators was made.
Keywords: molecular kinetic theory, gas mixtures, intermolecular potential
energy, collision integral, calculation method, viscosity, binary diffusion
coefficient
1 Introduction
Mixtures of dilute gases are widely spread in nature and are the basis of many
industrial processes. Currently, some experimental data on the study of transport
properties of pure dilute gases has been accumulated. Significantly fewer
experimental materials on the transport properties have been obtained for gas
mixtures, particularly their temperature dependence is studied insufficiently.
For the calculation and generalization of transport properties of pure
chemicals several methods of molecular kinetic theory are used. In general, their
results give good agreement with the experimental data [6, 16, 20, 21, 27].
284 A. F. Bogatyrev et al.
The situation is quite different when we speak about gas mixtures. There are
several methods of generalization and calculation of transport properties of gas
mixtures within the molecular kinetic theory [7, 8, 11, 12, 16, 18, 21]. Currently,
however, it is still too early to talk about the advantages or disadvantages of the
particular method. Furthermore, methods that give good agreement with the
experimental data, generally require a large amount of computational effort,
especially for polyatomic gases.
Nowadays, various semi-empirical calculation methods based on the
molecular kinetic theory are widely used for the calculation of transport properties
of gas mixtures [1, 3 - 5, 9, 19, 21, 23, 28]. Polyatomic and polar gas molecules
are of a particular interest. In the calculation of transport properties of such gases
we must take into account the non-sphericity of intermolecular pair potential
energy and the effect of inelastic collisions and polarizability of individual
molecules. It is often considered that the intermolecular potential energies of these
molecules are spherically symmetric for like and unlike interactions. Thus, the
calculation of transport properties are performed by means of common method of
the molecular kinetic theory [14], taking into account some specific adjustments
[6, 7, 18, 20] or using a three-parameter intermolecular potential [8].
2 Method of calculation
Viscosity
As part of a kinetic theory [14] the first approximation for viscosity of a pure
gas can be simply written as follows:
*2221
where Mi – molecular weight;
T – temperature, K; *22 – reduced collision integral at reduced temperature T* = kT/εi;
σi and εi/k – energy and length scaling parameters characterizing the
interaction of gas molecules.
According to this theory, k-approximation for viscosity can be written as:
, 1
k
where 11 f . It should be noted that the function
kf weakly depends on the
reduced temperature, T*, and differs slightly from unity, not more than 0.8%.
Currently, temperature dependence of viscosity has been measured for many
of the dilute gases. Certain information on the intermolecular forces can be
obtained from these measurements. Thus, the generalizing formula to calculate the
Calculation of viscosity and diffusion coefficients in binary mixtures 285
pure gas viscosity at various temperatures can be evaluated [6, 8, 14, 18, 20, 22].
According to [14], viscosity of dilute binary gas mixture, as a first
approximation, can be calculated by the following relationships:

M xAZ
where x1 and x2 – the mole fractions of components 1 and 2;
11 and
components 1 and 2;
M1 and M2 – molecular weights of the components 1 and 2; *
12A – ratio of reduced collision integrals, weakly depends on T*: .*11
12
*22
12

.
(5)
This value can be regarded as viscosity of hypothetical pure gas which
molecules have a molecular weight equal to 2M1M2/(M1+M2) and interact
according to the potential curve defined by parameters σ12 and ε/k12. If in Eq. (5)
index 2 is replaced by 1, then we obtain an equation similar to the Eq. (1) for pure
gas in a first approximation.
With the known potential energy of interaction between the molecules the
value *22
i as a function of T*
and viscosity values at different temperatures, processed by non-linear least
squares method, it is possible to obtain the values σi and εi/k, assuming the types
of approximating functions as shown in [18].
Values σ12 and ε12/k are commonly found basing on various combination
rules [14, 22, 23], using the values of σ1, σ2 and ε1/k, ε2/k obtained from pure gas
viscosity or other way. In the paper we propose a different method for finding the
parameters σ12 and ε12/k. The values *222
ii for both components of the gas mix-
286 A. F. Bogatyrev et al.
ture are calculated using Eq. (1), and then the value *22
12
2
12
2
12
(6)
It is worth noting that the Eq. (6) is similar to the equation for calculating the
mass of molecules of a hypothetical gas.
Substituting the value *22
12
2
12 to the Eq. (5), the value of 112 can be
found. With the values of *22
12
2
12 at various temperatures, and using tabulated
values of collision integrals for this type of intermolecular interaction, we can
obtain the values of ε12/k and *
12A with the help of non-linear least squares
method. Then, using Eq. (3) we can calculate binary gas mixture viscosity for any
mixture composition and temperature.
Binary diffusion coefficient
According to kinetic theory [14], the equation for calculating the binary

12 are found according to Eqs. (4) and (6);
fD – correction factor [14] having a value of the order of unity, and for
most of the gases is in the range 1.00 – 1.03; in special cases, when the
molecular weights of the gases differ greatly, fD is even greater but does
not exceed 1.10 [17].
Thus, the Eq. (7) allows calculating the BDC values basing on pure gases
viscosity for this particular type of intermolecular potential energy.
3 Calculation and comparison results
Gas mixtures containing polyatomic molecules of methane, nitrogen and
carbon dioxide were taken to approve the calculation results for viscosity and
diffusion coefficients for binary mixtures. The choice of the gases was due to the
fact that viscosity of these pure gases and their mixtures, as well as the BDCs had
been calculated and generalized in a wide range of compositions and temperatures
on the basis of experimental data by different authors. Calculations were
performed using different methods [6 - 8, 10 - 13, 18, 20, 25] within the kinetic
theory [14]. The analysis of [6 - 8, 10 - 13, 18, 20, 25] has shown that the
generalization and the calculation of pure gas viscosity give the results that agree
fairly well with each other. In general, we can say that there is an agreement
within 0.3 – 1.5% in the temperature range of 200 – 1300 K.
Calculation of viscosity and diffusion coefficients in binary mixtures 287
In this case, our calculation of *
12A was performed using the tabulated
collision integrals for the Lennard-Jones (6, 12) model potential shown in [14].
This technique allows using other potential models for unlike interactions, too.
Tables 1 and 2 show the results of viscosity calculation for the mixture of
methane and nitrogen (CH4-N2) at various compositions and temperatures.
Table 1: Viscosity of mixture of methane and nitrogen (CH4-N2) at various
compositions and temperatures. Smoothed experiment [15] and our calculation
results (OC). Average deviation – 0.46%, max. – 0.60%
T, K Mole fraction x(CH4)
Ref. 0.0 0.2 0.4 0.6 0.8 1.0
298.16 17.79 16.55 15.26 13.92 12.54 11.13 [15]
16.50 15.18 13.85 12.50 OC
323.16 18.93 17.63 16.28 14.88 13.43 11.93 [15]
17.57 16.19 14.79 13.37 OC
348.16 20.03 18.68 17.27 15.80 14.28 12.71 [15]
18.62 17.17 15.71 14.22 OC
373.16 21.10 19.69 18.22 16.69 15.10 13.46 [15]
19.63 18.12 16.59 15.04 OC
473.16 25.07 23.46 21.78 20.02 18.18 16.28 [15]
23.39 21.67 19.91 18.11 OC
573.16 28.68 26.89 25.01 23.04 21.00 18.83 [15]
26.80 24.88 22.91 20.89 OC
673.16 32.02 30.06 28.00 25.83 23.56 21.19 [15]
29.97 27.86 25.69 23.46 OC
773.16 35.16 33.04 30.80 28.44 25.97 23.39 [15]
32.93 30.64 28.29 25.87 OC
Table 2: Viscosity of mixture of methane and nitrogen (CH4-N2) at various
compositions and temperatures. Calculation results
T, K Mole fraction x(CH4)
Ref. 0.0 0.2 0.4 0.5 0.6 0.8 1.0
300
16.56 15.23 14.56 13.88 12.54 OC
17.80 16.58 15.30 14.65 13.98 12.61 11.19 [13]
16.52 15.22 14.56 13.89 12.55 OC
800
33.55 31.22 30.04 28.84 26.38 OC
35.66 33.48 31.22 30.05 28.87 26.44 23.94 [13]
33.45 31.18 30.01 28.83 26.42 OC
288 A. F. Bogatyrev et al.
Table 2: (Continued): Viscosity of mixture of methane and nitrogen (CH4-N2) at
various compositions and temperatures. Calculation results
1100
41.38 38.56 37.11 35.65 32.68 OC
44.08 41.42 38.66 37.25 35.81 32.87 29.82 [13]
41.41 38.64 37.23 35.79 32.85 OC
Table 1 also shows the smoothed experimental data obtained in [15], and the
data, we calculated by the proposed method. As pure gas viscosity, we used data
on the viscosity of the pure gases obtained in this work. Also the maximum and
average deviations of experimental values compared with our calculation results
are given. As seen from the table, the average deviation is 0.46%, the maximum
deviation is 0.60%.
In Table 2, the results of viscosity calculation for the mixture of methane and
nitrogen (CH4-N2) taken from [8, 13] and our calculations results basing on the
pure gases viscosity given in [8, 13] are shown. As seen from the table, viscosity
values from [8, 13] and our calculation results agree within 0.6%.
Tables 3 and 4 contain the smoothed experimental data [15] and viscosity
calculation results obtained using the proposed method for mixture of methane
and carbon dioxide (CH4-CO2) and mixture of nitrogen and carbon dioxide
(N2-CO2) at various compositions and temperatures. As seen from the tables 3 and
4, the average deviation for CH4-CO2 gas system is 0.8% maximum – 1.0%, and
for N2-CO2 gas system – 0.5% and 1.0%, respectively, that is actually within the
experimental and calculation error.
Table 3: Viscosity of mixture of methane and carbon dioxide (CH4-CO2) at
various compositions and temperatures. Smoothed experiment [15] and
calculation results. Average deviation – 0.8%, max – 1.0%
T, K Mole fraction x(CH4)
Ref. 0.0 0.2 0.4 0.6 0.8 1.0
298.16 14.98 14.61 14.09 13.37 12.41 11.13 [15]
14.50 13.96 13.24 12.30 OC
323.16 16.22 15.79 15.20 14.39 13.33 11.93 [15]
15.68 15.01 14.25 13.21 OC
348.16 17.42 16.94 16.27 15.39 14.22 12.71 [15]
16.82 16.12 15.24 14.09 OC
373.16 18.60 18.05 17.32 16.35 15.09 13.46 [15]
17.93 17.16 16.13 14.93 OC
473.16 23.00 22.23 21.23 19.95 18.33 16.28 [15]
22.10 21.04 19.76 18.18 OC
Calculation of viscosity and diffusion coefficients in binary mixtures 289
Table 3: (Continued): Viscosity of mixture of methane and carbon dioxide
(CH4-CO2) at various compositions and temperatures. Smoothed experiment [15]
and calculation results. Average deviation – 0.8%, max – 1.0%
573.16 27.02 26.04 24.80 23.23 21.27 18.83 [15]
25.90 24.59 23.01 21.09 OC
673.16 30.74 29.57 28.10 26.26 23.99 21.19 [15]
29.42 27.85 26.00 23.80 OC
773.16 34.22 32.87 31.18 29.10 26.53 23.39 [15]
32.70 30.92 28.81 26.33 OC
Table 4: As for table 3 but for mixture of nitrogen and carbon dioxide
(N2-CO2). Average deviation – 0.5%, max – 1.0%
T, K Mole fraction x(CO2)
Ref. 0.0 0.2 0.4 0.6 0.8 1.0
298.16 17.79 17.39 16.87 16.28 15.64 14.98 [15]
17.26 16.72 16.12 15.55 OC
323.16 18.93 18.56 18.07 17.50 16.87 16.22 [15]
18.43 17.91 17.34 16.78 OC
348.16 20.03 19.70 19.23 18.68 18.07 17.42 [15]
19.56 19.05 18.51 17.96 OC
373.16 21.10 20.80 20.36 19.83 19.23 18.60 [15]
20.66 20.17 19.66 19.13 OC
473.16 25.07 24.89 24.55 24.11 23.58 23.00 [15]
24.75 24.36 23.93 23.47 OC
673.16 32.02 32.04 31.88 31.60 31.21 30.74 [15]
31.90 31.70 31.43 31.10 OC
873.16 38.14 38.31 38.29 38.14 37.86 37.50 [15]
38.19 38.13 37.99 37.77 OC
1073.16 43.71 44.00 44.10 44.04 43.86 43.59 [15]
43.90 43.96 43.92 43.79 OC
1273.16 48.91 49.29 49.48 49.50 49.39 49.18 [15]
49.21 49.36 49.39 49.32 OC
Deviation values of viscosity calculated in [7, 8, 11, 13, 18] from the ones
obtained using the proposed method on the base of pure gas viscosity from [6] for
equimolar mixtures of these gas systems are given on Figs. 1 – 3. The pure gas
viscosities shown in [6] are greater than cited elsewhere, and greater than
experimental data from [15, 24, 25] for almost all temperatures.
290 A. F. Bogatyrev et al.
Figures 1 – 3 also show the values of the deviation of experimental data of
[15] and our viscosity calculations basing on pure gases viscosities taken from [7,
8, 11, 13, 18] compared to the ones obtained on the data from [6].
As can be seen from the figures, results of viscosity calculation performed by
various authors and the ones obtained using our proposed method are in a good
agreement. Thus, for CH4-N2 gas system, for which the greatest number of
calculations have been performed in the temperature range 200 – 700 K, all the
data are consistent with each other within 0.7%, and in the temperature range
700 – 1300 K, within 1%, respectively. This corresponds to the experimental and
calculation errors of generalizations carried out by other authors. For CH4-CO2
and N2-CO2 gas systems experimental and calculation results agree within 1.5%.
In accordance with Eq. (7) using pure gas viscosity given in the papers by
various authors, we have calculated BDCs for all the listed gas systems. Figs.
4 – 6 give deviations of BDCs for equimolar mixtures of these gas systems from
our calculation results basing on the data on the pure gases viscosity given in [6].
The figures also show the deviation of experimental data of [2, 26] from our
calculation results.
Figure 1: Deviation plot for viscosity values obtained by other authors compared
to our calculation results using the proposed method on the base of pure gas
viscosity from [6] for equimolar mixture of methane and nitrogen (CH4-N2) gas
system
Calculation of viscosity and diffusion coefficients in binary mixtures 291
Figure 2: As for Fig.1 but for methane and carbon dioxide (CH4-CO2) gas system
Figure 3: As for Fig.1 but for nitrogen and carbon dioxide (N2-CO2) gas system
Figure 4: Deviation plot for binary diffusion coefficients obtained by other
authors compared to our calculation results using the proposed method on the base
of pure gas viscosity from [6] for equimolar mixture of methane and nitrogen
(CH4-N2) system
292 A. F. Bogatyrev et al.
Figure 5: As for Fig. 4 but for methane and carbon dioxide (CH4-CO2) gas
system
Figure 6: As for Fig. 4 but for nitrogen and carbon dioxide (N2-CO2) gas system
As seen in Fig. 4, for CH4-N2 gas system in the temperature range of
300 – 1300 K all the calculated data agree with each other within 6%. All the
calculated data agree with experimental data within 7%. It should be noted, that the BDCs calculated in [7, 8, 13, 15, 18] are in reasonable agreement with each other
Calculation of viscosity and diffusion coefficients in binary mixtures 293
within 7%. In addition, the BDCs obtained using Eq. (7) basing on pure gases
viscosity from various authors agree with each other within 1%.
For the CH4-CO2 gas system there are only two experimental values of the
BDC, the remaining data were calculated by us basing on pure gases viscosity
from [7, 13, 15, 18]. Calculated data agree with each other within 1.5%.
Fig. 6 shows the deviation of BDCs for equimolar mixture of nitrogen and
carbon dioxide (N2-CO2) gas system. Deviations of the experimental data for this
system are taken from [2], they are consistent with each other in the range of 14%.
Unfortunately, experimental data with deviations beyond the scale of the figure
are not presented on it to avoid unreasonable scale reduction. It should also be
noted that the data obtained in [7], have greater deviations at low temperatures
than at higher ones. Besides the data calculated by us basing on the pure gases
viscosity from [7, 13, 15, 18] are consistent with each other within 2%.
The calculation results of the BDCs in [7, 8, 11, 13, 18] in the temperature
range 200 – 300 K have an error of 2 – 5% according to the authors, and deviation
value increases with temperature increasing.
4 Conclusion
Analysis of the proposed method of calculation of the viscosity and diffusion
coefficients in binary mixtures of dilute gases had been performed. As a result, a
number of problems arising in the calculation process of transport properties of
gas mixtures had been revealed. One problem of the calculation and
generalization of transport properties of dilute gases comes down to the presence
or absence of experimental data at various temperatures, the experimental error
and reliability of that data.
Unfortunately, the amount of experimental data on the transport properties of
gas mixtures is relatively small [8, 13, 15], besides only little data, especially on
the diffusion coefficients, have an acceptable error within 2 – 3%, which is
necessary for further calculations. Hopefully, method of calculation and
generalization we proposed in the paper will prove itself well in the calculation of
transport properties of other gas mixtures.
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Received: May 4, 2017; Published: May 23, 2017