Correlation Between Gas Molecular Weight, Heating Value and Sonic Speed Under Variable Compositions of Natural Gas

by

Dr. L. Burstein, Dr. D. Ingman and Dr. Y. Michlin

[Published in ISA Transactions 38 (1999), 347-359]

Abstract

A simulation study of likely uncertainties in molecular weight and heating value of the gas mixture as predicted from measured or calculated sonic speed. The sonic speed, molecular weight and heating value of natural gas were studied as a function of random fluctuation of the gas fractions. A method of sonic speed prediction was developed and used for over 50,000 computer-simulated variants of component concentrations in four- and five-component mixtures. Comparison of the obtained and the reference data on binary (methane-ethane) and multicomponent (Gulf Coast) gas mixtures under standard pressure and moderate temperatures indicates predictability of sonic speed on the basis of the binary virial coefficients, sonic speeds and heat capacities of the pure components. The results for two natural gas mixtures - with and without nonflammable components - are reported. Bivariate distribution - and covariance elliptic zone plots are presented for three pairs of dependences of practical interest: molecular weight - sonic speed, heating value - sonic speed and heating value - molecular weight. The correlation coefficients, covariance, and regression equations are given for each pair of variance and mixture.

Keywords Sonic Speed; Molecular Weight; Heating Value; Composition Fluctuations; Natural Gas; Uncertainties; Statistical Analysis

1. Introduction

Sonic speed is widely used for improving the accuracy of gas meters, measurement of molecular weight, in BTU-analyzers, etc. The correlation between the molecular weight of natural gas and speed of sonic waves is utilized in modern flowmeter apparatus (e.g. VGN868 by Panametrics or GFS700 by Krohne) and are the subject of various inventions - e.g. US Patent No 5645071. A correlation is also known between molecular weight and gas heating value, so that the sequence "sonic speed - molecular weight - heating value" (see e.g. US patent No 4246773), or the direct correlation between sonic speed and gas heating value, can be used for heating value estimation. In that case significant uncertainty can arise due to the presence of a nonflammable component. To a lesser extent in the case of molecular weight determinations - and to a large extent in the case of heating value determination - uncertainty in the gas composition leads to errors in the determined values. This is also true for engineering calculations of these values based on the thermodynamic properties of gas mixtures. The most accurate values can be obtained or measured in the case of strict determination of the gas component concentration or by comparison with reference sample (in the case of measuring). Here we intend to study the influence of possible fluctuations of the component concentrations in natural gas on the "molecular weight - sonic speed" and "heating value - sonic speed" correlations with a view to applying sonic speed testing directly to the molecular weight and heating value measurement omitting the additional definition of gas content.

In view of the infinite diversity of gas mixtures, various attempts are made to predict their thermal and caloric properties on the basis of pure component data. Equations of state are widely used for this purpose - E. A. Mason[1], J. Hirshfelder et al. [2]. In the region of moderate pressures and temperatures the virial equation is adopted:

p = rRT(1+ ), (1)

where p, r and T are gas pressure, density and temperature respectively, R is the universal gas constant, B - the virial coefficient and n its number so that ²B - is the second virial, ³B - the third virial and so on; in the case of a gas mixture

ⁿB = , _. (2)

where x - component concentration; i, j and l the component numbers.

The virial coefficients of pure components - , , ... - are readily available for most technical gases, in particularly for the hydrocarbon gases - CRC Handbook [3]. By contrast, the cross virial coefficients , , ... - have been less studied and are determined as combinations of the virial coefficients of the pure gases, or of their intermolecular constants, using the dependence of the virial coefficient on the intermolecular potential of interaction [2]. Further, using Eq.(1), the differential equations of thermodynamics and ideal gas functions, the density r, enthalpy I, heat capacities c_p and c_v , and entropy s of gas mixtures are calculated.

The same approach may be used for determining the thermodynamic sonic speed

c² = , (4)

or

c² = k, (5)

where k is the ratio of the heat capacities c_p and c_v .

Under the ideal gas approach = M R T, where M is the molecular weight; thus c² = M k R T. Thus it would appear to be possible to use the sonic speed measurements/calculations for determining the molecular weight of pure gases and their mixtures - L. C. Lynaworth [4]. In this case besides the uncertainty due to idealization of the gas, further uncertainty is introduced by fluctuation of the composition. In reality it must be borne in mind that a gas mixture may have the same molecular weight but different sonic speeds owing to different gas component concentrations and conversely the same sonic speed may occur at different molecular weights for the same reason. Table 1 shows that the mole fraction concentration of secondary fractions may change by a whole order of magnitude at constant molecular weight or at constant sonic speed. Uncertainties in M or c for such cases remain unevaluated to date.

Table 1. Example of Molar Concentration Fluctuations:

Five Component Gas Including a Nonflammable

Fraction

Methane	Ethane	Propane	Butane	Nitrogen
	Molecular weight 17.26±0.02 kmol/kg
0.912	0.014	0.001	0.004	0.069
0.912	0.031	0.001	0.003	0.052
0.914	0.029	0.001	0.004	0.053
0.925	0.044	0.010	0.002	0.018
0.925	0.031	0.013	0.001	0.029
0.926	0.036	0.010	0.003	0.025
0.930	0.039	0.010	0.005	0.016
0.933	0.016	0.018	0.003	0.030
0.934	0.048	0.012	0.004	0.001
0.934	0.033	0.018	0.002	0.012
0.935	0.036	0.014	0.004	0.011
0.935	0.038	0.014	0.004	0.009
	Sonic speed 438±2 m/s
0.907	0.027	0.002	0.001	0.063
0.920	0.043	0.007	0.002	0.028
0.921	0.038	0.007	0.003	0.031
0.922	0.038	0.008	0.003	0.029
0.922	0.033	0.008	0.003	0.034
0.923	0.034	0.011	0.001	0.030
0.923	0.031	0.006	0.005	0.035
0.924	0.026	0.010	0.003	0.037
0.926	0.039	0.010	0.003	0.022
0.927	0.034	0.009	0.004	0.026
0.931	0.031	0.016	0.002	0.021
0.939	0.036	0.016	0.005	0.004

It was established that the heating value H of natural gases for light hydrocarbons is an almost linear function of the molecular weight, though this fact is of no practical use, as the presence of nonflammable components in the gas mixture reduces the real heating value and exaggerates the measured/calculated value at the same time.

So the object of this research is prediction of the sonic speed, molecular weight and heating value of natural gas mixtures on the basis of those and other thermodynamic data of the pure components, for a large number of random molar concentrations; determination of the correlation between the obtained values; statistical analysis of the results.

2. Multicomponent Gas Mixture Model in Approach to Sonic speed.

Here we use the classical approach, in which the thermodynamic properties of a real mixture are constituted from those of the pure components with corrections for the ideal gas mixture approach. Such a concept was used by L. M. Burshtein [5] for the equation of state of a gas mixture, and can be extended to the sonic speed in it.

From Eq.(4)

= . (6)

For m-component mixtures of constant composition, the derivative in (6) can be determined from the equation of state (1) and the virial coefficient equation (2).

(7)

Under normal conditions we can only use the second virial coefficient, which for m=2 (binary mixture, x₂ = 1 - x₁) equals

²B = ²B₁ x₁² + ²B₂ x₂² + 2 ²B₁₂ x₁ x₂ = ²B₁ x₁ + ²B₂ x₂ + (2 ²B₁₂ - ²B₁ - ²B₂) x₁ x_{2 .} (8)

Here and further on we use the notations .

Taking into account that -= rRT(²B₁ - ²B₂) we obtain

= x₁ + x₂ + x₁ x₂ , (9)

where = (2 ²B₁₂ - ²B₁ - ²B₂)/( ²B₁ - ²B₂).

Thus Eq.(9) comprises the pure component values , the correction - and the coefficient A representing a combination of the second virial coefficients.

Extending the described operations to the m-component case we have

= x_i + x_i x_j , (10)

where = (2 ²B_ij - ²B_i - ²B_j)/( ²B_i - ²B_j) .

Repeating this type of transformation with the third, fourth and higher virial coefficients we obtain the heat capacities ratio

. (11)

and the sonic speed in an m - component gas mixture is

c = . (12)

where , , are given in the Appendix.

In the case of an ideal gas mixture =0 , and for the ideal gas capacity ratio k = k_i = Const , eq.(12) reduces to .

For natural gases under normal conditions it suffices to use eqs.(11,12) with the coefficients , which include only the second virials. The cross coefficients and can be found from the rule (see ASTM Standard [6]):

(13)

Thus for the second residual virial coefficients we have orin the case of a pressure-series equation of state ; now, as an example, the equation of is obtained as

, (14)

where B must have the same sign for different fractions; in the case of negative B we can use its absolute value with a negative sign of the square root. The relationship for calculating the virial coefficient for a pure gas can be found from the equation

, (15)

where T_o = 298.15 and the coefficients a_i are given in the CRC Handbook[3].

Applicability of the model was checked on the B. A. Younglove et al. [7] sonic speed data for binary methane-ethane gas mixtures and multicomponent Gulf Coast gas, which consists of flammable and nonflammable components. We use the data for 0.1013 MPa in the temperature range 250...400 K. As can be seen from Tables 2 and 3 the model gives more accurate values of sonic speed throughout the data range; the one-sided uncertainty in c is not worse than ±1.4 percent, which falls within the inaccuracy limits of pure gas speed (dw up to 2%) , and heat capacities (dc_p above 3%) data. For a multicomponent gas the inaccuracies are significantly smaller than in the case of a binary mixture.

Table 2. Calculated and Measured Values of Sonic Speed , m/s. Methane - Ethane

Gas Mixture at Standard Pressure

T, K	Data [7]	Ideal Mixture	Deviation %	Eqs.(28,29)	Deviation %
250	362	373.8	3.3	369.9	2.2
275	375	390.9	4.2	386.0	2.9
300	392	406.9	3.8	402.1	2.6
325	406	421.8	3.9	416.7	2.6
350	420	435.9	3.8	430.6	2.5

Table 3. Calculated and Measured Values of Sonic Speed, m/s. Multicomponent

Mixture (Gulf Coast) at Standard Pressure

T, K	Data [7]	Ideal Mixture	Deviation %	Eqs.(28,29)	Deviation %
275	422	428	1.4	414	1.9
300	437	445	1.8	445	1.8
325	455	461	1.3	451	0.9
350	470	477	1.5	468	0.4

3. Modeling of Component Fluctuations in Natural Gas

The composition of natural gases fluctuates according to both the number of components and their concentrations. In Table 4 the mean values of component concentrations in natural gas for the average composition of 120 natural gases based on GRI (report 82/0037) and MTI1 data. As can be seen, methane, ethane and a certain mount of nitrogen are always present in natural gases. The actual concentrations of these and other components vary over a wide range, in particular for small components.

Table 4. Average Composition of Natural Gases*, mole %
Component	Mean Value	Deviation	Min. Value
Methane	93.0	5.5	80.0
Ethane	3.0	2.6	0.4
Propane	1.0	1.4	0.0
Butanes	0.5	1.0	0.0
Pentanes	0.1	0.3	0.0
Hexane	0.1	0.1	0.0
Nitrogen	1.5	2.9	0.1
Carbon Dioxide	0.5	0.5	0.0

*based on GRI and MTI-data.

It is natural to suggest that the molar concentration is a random value, which fluctuates within certain limits. As a result, the molecular weight of a specific natural gas may be the same while the molar concentrations of the fractions differ. In that case the measured sonic speed also differs and we observe a number of molecular weights instead of the single true value. Obviously, a series of molecular weight values can correspond to a single value of sonic speed as a result of accidental variation of the gas component concentrations. Sonic speed calculations for the large possible number of component concentrations have to be carried out to estimate the inaccuracy in molecular weight or sonic speed determination due to the random fluctuations of the molar concentration.

Such a procedure was created under the following premises:

· the gas mixture has not less than four flammable components, with methane as basic component;

· the concentrations of m-1 components vary as a random value normally distributed with given dispersion s; the concentration of the last component varies between 1 and the sum of all the others.

· zero concentrations and those less than 0.05% are disregarded.

· if the sum of the concentrations exceeded unity, these variants were not included in the calculations.

The MathCAD rnorm-function was used to generate random normally distributed values of concentrations of the pure gas components.

The molecular weight of the gas mixture

, (16)

the sonic speed - as per Eq.(12), and the gas heating value

, (17)

were calculated from the generated values of x_i.

As the gas heating value of the nonflammable component is zero, the value for the gas mixture determined by sonic speed measurements is erroneous both in terms of the concentrations and of the presence of such contamination. The suggested method permits determination of the relationship between these errors and indicates when the accuracy in determining the heating value is sufficient.

4. Results and Discussion

The heat capacity ratio, sonic speed, molecular weight, and gas heating value were obtained by Eqs.(11,12,16,17) using CRC data [3] for the virial coefficients, S. S. Chen et al. [8], J. Chao et al.[9], and R. T. Jakobson et al.[10] data for the ideal gas function and tables of J. Chao et al. [10], R. T. Jacobson et al.[11] for methane, ethane, propane, butane, and nitrogen. The mean values of the components were taken from Table 1, the dispersions of these values were 0.017, 0.008, 0.004 and 0.001 for methane, ethane, propane, and butane respectively.

The results of the M, c, H - calculations and obtained statistical characteristics for four- and five-component gases are presented in Figs.1 through 8 and in Table 5.

The dependences between the molecular weight and sonic speed (Fig.1), between the gas heating value and sonic speed (Fig.2), and between the gas heating value and molecular weight (Fig.3) were found from 46110 computer-simulated random concentrations for the four-component and 48970 computer-simulated random concentrations for the five-component gas.

Figure 1. Molecular Weight Versus Sonic Speed for Two Natural Gases Mixtures:

1- Methane + Ethane + Propane + Butanes;

2- Methane + Ethane + Propane + Butanes + Nitrogen.

Figure 2. Heating Value Versus Sonic speed for Two Natural Gas Mixtures:

1- Methane + Ethane + Propane + Butanes;

2 - Methane + Ethane + Propane + Butanes + Nitrogen

Figure 3. Heating Value Versus Molecular Weight for Two Natural Gas Mixtures:

1 - Methane + Ethane + Propane + Butanes;

2 - Methane + Ethane + Propane + Butanes + Nitrogen

As we can see from Fig.1, the changes in the molecular weight of the gases through random fluctuation of the fraction concentrations amounted to 20% at sonic speed fluctuation above only 3.5%. It is interesting that with the added nonflammable nitrogen fraction (five-component gas), the maximal molecular weight fluctuation range is half that for the same rate of variation of the sonic speed. The accuracy of M evaluation by sonic speed measurement or calculation is more than double in this case. The maximum inaccuracy in molecular weight by sonic speed data is above 1.3% at sonic speed uncertainty not more than 0.1%.

Figs.2, 3 illustrate the dependence between heating value, sonic speed and molecular weight for four- and five-component gases. Though the nitrogen fraction strongly influences the results, even in this case, as it can be seen we have an interval of speeds or weights where owing to the concentration fluctuations, the inaccuracies exceed those due to addition of the nonflammable component.

From Fig.2,3 it follows that with an added nonflammable component the natural gas heating value is reduced up to 30% through variation of the gas contents; the uncertainty of the gas heating value is about ± 2.5% for gas with inflammable components only. Moreover, there is the probability of correct determination of the heating value from the c data, as the range of cross-data involves almost 20% of all possible component concentrations of the four- and five-component mixtures .As for the H data, although the inaccuracy increases in the presence of the nonflammable component, cross data exist here as well, with equal heating values for both mixtures, with and without the nonflammable component. In other words, the error in heating value or molecular weight determined from sonic speed data, due to ignorance of the number of gas components, may be negligible over a sufficiently wide range, unlike the error due to concentration disturbances. The concrete values of their errors, the bivariate distribution of variables, the covariance areas and the regression equations are discussed further on.

Figs.4,5 show bivariate bar and surface histograms for four (Figs.4A, 5A) and five (Figs.4B, 5B) - component gas mixtures, with the molecular weight as X, and the sonic speed and heating value as Y.

(A)

(B)

Figure 4. Bivariate Sonic Speed-Molecular Weights Distributions

A) Methane-Ethane-Propane and Butanes Mixture;

B) Methane-Ethane-Propane- Butanes and Nitrogen Mixture.

(A)

(B)

Figure 5. Bivariate Gas Heating-Molecular Weight Distributions.

A) Methane-Ethane-Propane and Butanes Mixture;

B) Methane-Ethane-Propane- Butanes and Nitrogen Mixture

Figs.6 through 8 present the covariance 68% -, 95% - areas and 100% - point population for the M,c - , H,c -, and H,M - pairs of variables. Points lying in the 68% and 95% zones were screened from the values calculated in accordance with the covariance (error) ellipse equation [12] of the bivariate normal distribution:

, (18)

where X is c or M and Y is M or H in our cases, s is the standard deviation, m_X and m_Y are the means of the frequency distributions, is the correlation coefficient.

A) B)

Figure 6. Molecular Weight - Sonic Speed Covariance Ellipses for

Four - Component (A) and Five - Component (B) Mixtures.

1- All point population

2- 95% points area

3- 68% points area

A) B)

Figure 7. Heating Value - Sonic Speed Covariance Ellipses for

Four - Component (A) and Five - Component (B) Mixtures.

1- All point population

2- 95% points area

3- 68% points area

A) B)

Figure 8. Heating Value - Molecular Weight Covariance Ellipses for:

Four - Component (A) and Five - Component (B) Mixtures.

1- All point population

2- 95% points area

3- 68% points area

Figs.6,7 can be used for determining the uncertainty in the molecular weight or heating value from measured data of sonic speed. For this we must draw a perpendicular from the point of the measured value of c up to say the circumference of the 95% ellipse, which gives us the limits of error caused by gas composition fluctuations. For example, for a measured value of c=435±0.4m/s, from Fig.6A (flammable components only) we obtain the molecular weight limits 17.7...18.2 or 17.95±0.25 kg/kmol and from Fig.6B (nonflammable component present) 17.6...17.8 or 17.7±0.1 kg/kmol. This example shows also that a nonflammable component reduces the uncertainty in the molecular weight by more than half.

The same tendency appears in the case of the heating value as well. For example, for aforesaid value of c, from Fig.7A (flammable components only) we obtain the heating value limits 1040...1120 or 1080±40 BTU/ft³ and from Fig.7B (nonflammable component present) 960...990 or 975±15 BTU/ft³. As a nonflammable component is present in more than 120 natural gases (Table 2), even more accurate determination of the molecular weight or heating value from the sonic speed measurements or calculations can be expected than for gases with flammable components only.

All the above justifies also determination of the heating value and its error from molecular weight data - in this case Fig.8 must be used. With the regression equation as a sort of calibration curve, and using the covariance ellipse with the appropriate statistical parameters, we can obtain the limits of error for every concrete case. Thus, in Table 5 the regression equations and some statistical characteristics of four (upper line), and five (lower line) component gas mixtures are presented for three pairs of dependences - molecular weight/sonic speed, heating value/sonic speed, and heating value/molecular weight.

Table 5. Four - and Five - Component (lower line) Gas Mixture Statistical Characteristics
Pair of	M, kg/kmol - c, m/s	H, Btu/ft^3 - c, m/s	H, Btu/ft^3 - M, kg/kmol
Variables
Correlation	-0.99	-0.95	0.97
Coefficient	-0.95	0.74	-0.57
Covariance	-1.63	-86.97	23.42
	-0.37	-2.19	28.01
Regression	M=131.246-0.261 c	H=7130-13.906 c	H=119.887+53.919 M
Equation	M=59.542-0.096 c	H=-2188+7.216 c	H=23.115-0.005824 M
Mean	18.10	1096	1096
Value	17.26	1004	1004
Standard	0.096	12.09	9.63
Error	0.062	13.16	0.16

The correlation coefficients are 0.99 and 0.95 for the gas mixture without- and with the nonflammable component respectively, showing good reliability for the weight - sound measurements. Thus the given regression M(c) equations can be used in practice. The standard error in the molecular weight (found from sonic speed data) is here only ±0.3% for the five - component as against ±0.5% for the four - component gas mixture.

The heating value can be determined more precisely from molecular weight data than from sonic speed (see Table 5). In this case the standard error in H is only about 0.02% for the five - component mixture.

5. Conclusion

Our calculations estimate that the proposed method for obtaining the sonic speed of a gas mixture on the basis of pure component data with correction for the interaction of heterogeneous atoms of the gas components gives an uncertainty about ±1.4% in the case of selected variations of natural gas.

The used simulation method of random concentration fluctuation enables one to obtain the standard error in the molecular weight, sonic speed and heating values.

The uncertainty range in determining the molecular weight and heating value appears reduced on addition of a nonflammable component.

6. Acknowledgments

The authors are indebted to Select Corporation (9300 Wilshire Blvd., Suite 600 Beverly Hills, CA 90212, U.S.A) for encouraging this research.

7. References

1. E. A. Mason. The Virial Equation of State. Oxford, Pergamon Press, 1969, 297p.

2. J. Hirshfelder, C. Curtiss, R. Bird. Molecular Theory of Gases and Liquids. University of Wisconsin. N.Y., L. 1954.

3. CRC Handbook of Chemistry and Physics. A Ready Reference Book of Chemical and Physical Data. 78^th Edition, 1997, pp.6-23 - 6-42.

4. L. C. Lynaworth. Ultrasonic Gas Flowmeters. Measurements & Control. Vol.29, No.5, Oct. 1995, pp.92-101.

5. L. M. Burshtein. The Equation of State of a Gas Mixture for Gas-Analysis calculations. Measurement Techniques. Plenum Publishing Corporation, Vol.18, No.12, Dec.1975, pp.1818-1812.

6. ASTM Standard. D3588 -- 91 Standard Practices for Calculating Heat Value, Compressibility Factor, and Relative Density (Specific Gravity) of Gaseous Fuels.

7. B. A. Younglove, N.V.Frederick. Sonic speed Measurements on Gas Mixtures of Natural Gas Components Using a Cylindrical Resonator. Int. Journal of Thermophysics. Vol. 11, No 5, Dec. 1990, pp.897-910.

8. S. S. Chen, R.C.Wilhoit, B. J. Zwolinski. Ideal Gas Thermodynamic Properties and Isomerization of n-Butane and Isobutane. J. Phys. Chem. Ref. Data, Vol. 4, No 4, 1975, pp.859-867.

9. J. Chao, R.C.Wilhoit, B. J. Zwolinski. Ideal Gas Thermodynamic Properties of Ethane and Propane. J. Phys. Chem. Ref. Data, Vol.2, No 2,1973, pp. 427-436.

10.R. T. Jacobson, R. B. Stewart. Thermodynamic Properties of Nitrogen from the Freezing Line to 2000 K at pressures to 1000 Mpa. J.Phys. Chem. Ref. Data, Vol.15, No 2, 1986, pp. 735-867.

11.B. A. Younglove, J. F. Ely. Thermophysical Properties of Fluids. II. Methane, Ethane, Propane, Isobutene, and Normal Butane. J. Phys. Chem. Ref. Data, Vol. 16, No 4, 1987, pp. 577-813.

12.D. Freedman, R. Pisani, R. Purves. Statistics. 3rd Edition, 1997, New York.

Appendix

Sonic Speed for Multicomponent Gas Mixture through the Complete Virial Equation

At moderate and higher pressures the third and higher virial coefficients have to be taken into account. In that case we must execute with every virial coefficient of mixture the same conversion as it was done for the second virial. So for the third virial for instance we have

³B_mix = ³B₁ x₁ + ³B₂ x₂ + (3 ³B₁₁₂ - 2 ³B₁ - ³B₂) x₁ x₂ + [(³B₁ + ³B₂) - 3(³B₁₁₂ - ³B₁₂₂)] x₁ x₂² . (A1)

Taking into account the fourth and higher virials we obtain

= x_i + . (A2)

Here x_i > x_j and

,

(A3)

,

...,

,

and we shall call the residual virial coefficients.

Dropping the terms with higher degrees of x_i and x_j (as the residual virials and their products with the higher degrees of molar concentration are very small) we obtain

= x_i + . (A4)

In this equation the mixture heat capacity ratio is still undetermined. Thus the capacities c_p and c_v have to be rewritten in terms of those of the pure components. From the differential relation

(A5)

we have

= - T dp , (A6)

where - the heat capacity of the ideal gas and v = 1/r.

Using the virial equation of state expanded in a pressure series:

v = ( 1 + ) (A7)

in which , , ... , we have

(A8)

and

- RT. (A9)

Using now the relation ⁿB for the mixture and the equation

= + RTp. (A10)

with = _i -_j , we obtain for the m component mixture

, (A11)

where

And as so by analogy with the derivation of c we obtain

. (A13)

For the heat capacity at constant volume

= + T dv (A14)

using equation of state (1) we can obtain

(A15)

or

. (A16)

with

. (A17)

Thus the heat capacity ratio for the gas mixture has the form

. (A18)

From eq.(A2), the sonic speed in an m-component gas mixture is

c = . (A19)

So far eqs. (A1, A7, A11, A15) were constructed on the basis of the virial equations of state without any simplifications, as they are rigorous throughout the range of applicability of the virial equation.

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