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].

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

93.266ii

i

i

TM

, (1)

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

iki f (2)

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:

,1

1

1

1

1

Z

XYX

Z

YX

см

,

2

12

2

2

112

21

11

2

1

xxxxX (3)

,

4

2

5

3

1

2

12

2

2

1211

2

112

21

2

21

112

21

2

1

11

2

1*

12

M

Mx

MM

MMxx

M

MxAY

,14

25

3

1

22

2

12

112

11

112

21

2

2121

2

12

1

*

12

M

Mx

MM

MMxx

M

MxAZ

where x1 and x2 – the mole fractions of components 1 and 2;

11 and

12 – first-order approximation of viscosity coefficients for

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

*

12 A (4)

Value 112 is determined by the following expression:

.

293.26610

*

12

*22

12

2

12

21217

112T

MMTMM

(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 can be calculated. Using the values of *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 is evaluated using

the following equation:

.

2*22

2

2

2

*22

1

2

1

*22

2

2

2

*22

1

2

1*22

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

diffusion coefficient (BDC) can be written in the following form:

,001858.0*22

12

2

12

*

122

1

21212

3

12 Dfp

AMMMMTD

(7)

where p – pressure; *

12A and *22

12

2

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

17.87 16.64 15.34 14.67 13.98 12.59 11.14 [8]

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

35.80 33.64 31.37 30.19 28.99 26.49 23.87 [8]

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

44.11 41.53 38.74 37.30 35.83 32.78 29.59 [8]

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