Calculation of Viscosity and Diffusion Coefficients in ...Calculation of viscosity and diffusion coefficients in binary mixtures 285 pure gas viscosity at various temperatures can

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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