Speed of sound as a source of accurate thermodynamic properties of gases.

M. Bijedic^† and N. Neimarlija^‡

^† Faculty of Technology, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina. muhamed.bijedic@untz.ba
^‡ Faculty of Mechanical Engineering, University of Zenica, 72000 Zenica, Bosnia and Herzegovina. nagibn@mf.unze.ba

Abstract— A procedure for deriving thermodynamic properties of gases (the compression factor, the specific heat capacity at constant volume, and the specific heat capacity at constant pressure) from the speed of sound is presented. It is based on numerical integration of differential equations connecting the speed of sound with other thermodynamic properties. The set of differential equations is solved as the initial-boundary-value problem. The initial values of the compression factor and the specific heat capacity at constant volume are specified along the isochore in the perfect-gas limit, while the boundary values of the compression factor are specified along two isotherms at the lowest temperatures of the range. The procedure is tested on methane, ethane, and carbon dioxide. The average absolute deviations of the compression factor, the specific heat capacity at constant volume, and the specific heat capacity at constant pressure, from respective reference values, are 0.0001%, 0.004%, and 0.008%, respectively.

Keywords— Speed of Sound; Compression Factor; Heat Capacity; Gases.

I. INTRODUCTION

Knowledge of thermodynamic properties of fluids appearing in chemical industry is of crucial importance for analysis of processes and for design of process equipment. For some substances these properties are not available at all (e.g., newly synthesized compounds), for some they are available only in limited range of pressure and/or temperature (e.g., at atmospheric pressure and surrounding temperature), while for some reliability of these properties is questionable (e.g., high uncertainty of measurement). Traditionally, the engineers and scientists have been using tables and charts from various handbooks as the main source of such data. However, when one decides to make its own compilation, the discrepancies among different sources may become obvious. In such situation the decision may be difficult. Besides, tabulated data are rather inappropriate for big scale calculations involving digital computers (e.g., Computer Aided Design, Computational Fluid Dynamics). For this purpose the equations of state are more appropriate.

Van der Waals was the first who posed the equation of state which took into account influence of intermolecular forces in gases under elevated pressures (van der Waals, 1873). Since that, literally hundreds of various modifications of his equation have appeared. Although some of these equations of state have accuracy which satisfies majority of technical and scientific applications (Span, 2000), a vast majority of them fail to represent thermodynamic properties of substances for which they are designed (not only thermal but caloric as well) with uncertainties not exceeding those of direct measurements. Only equations in reference quality (e.g., reference or fundamental equations of state) may represent thermodynamic surface (including critical region) as reliably as direct measurement. Unfortunately, only a few of such equations of state have been developed to date (e.g., for argon (Tegeler et al., 1999), nitrogen (Span et al., 2000), carbon dioxide (Span and Wagner, 1996), water (Wagner and Pruss, 2002), methane (Setzmann and Wagner, 1991) and ethylene (Smukala et al., 2000)).

Generally, the caloric properties are measured with the uncertainties which are several orders of magnitude higher than those of the thermal properties. The situation is especially critical with the heat capacity (e.g., both isobaric and isochoric) (Younglove, 1974; Bier et al., 1976; and Ernst et al., 1989), since this property is very important when the full thermodynamic equation of state (e.g., both thermal and caloric) is designed. Fortunately, there is one exception to this rule - the speed of sound. Advances in measurement of this caloric property in the last several decades made this property very attractive source of other thermodynamic properties. These thermodynamic properties may be derived from the speed of sound with accuracy exceeding that of respective direct measurements. Nowadays, the speed of sound is measured with outstanding accuracy (Trusler and Zarari, 1992; Estrada-Alexanders and Trusler, 1997; and Estrada-Alexanders and Trusler, 1998), even exceeding that of the thermal properties (Pieperbeck et al., 1991; Funke et al., 2002; and Klimeck et al., 2001). Thanking to this fact, this caloric property is used in the design of the reference equation of state in its final stage, that is, for fine adjustment of the equation parameters (Schmidt and Wagner, 1985; Setzmann and Wagner, 1989; and Span and Wagner, 2003). However, the thermodynamic properties may also be derived from the speed of sound directly. Unfortunately, the speed of sound is connected to other thermodynamic properties through the set of partial differential equations whose general solution has not been found yet. The only exception is a part of the thermodynamic surface comprising the region of dilute gases at low pressures where analytical solution may be obtained from the model of intermolecular potential energy (Maitland and Smith, 1973; Trusler et al., 1997; and Estela-Uribe and Trusler, 2000). However, if higher pressures are considered the numerical integration (Estrada-Alexanders et al., 1995), the finite differences (Estrada-Alexanders and Justo, 2004), and the recursive equations (Lago and Giuliano Albo, 2008) are among the methods worth of attention.

II. THEORY

Isaac Newton was the first who posed the problem of theoretical calculation of the speed of sound in fluids. However, his formula (Newton, 1687)

(1)

where u is the speed of sound, p is the pressure, and r is the density, implied isothermal rather than adiabatic propagation of sound. Pierre Simon Laplace explained adiabatic nature of a sound wave and corrected Newton's formula to obtain the final solution (Laplace, 1816)

(2)

where c_p is the specific heat capacity at constant pressure and c_v is the specific heat capacity at constant volume. Since acoustic wave propagates adiabatically, an adiabate of a perfect gas in pressure - density coordinates is given by relationship of the form

(3)

where k is the ratio of the heat capacities at constant pressure and volume (c_p/c_v). Now, general expression (3) may be written in a more specific form

(4)

where p and r are the instantaneous pressure and density at any point, while p₀ and r₀ are equilibrium pressure and density of the fluid (Benedetto et al., 1999). If one differentiates Eq. (4) with respect to r (bearing in mind that p₀ and r₀ are constants because they have uniform values throughout the fluid), fundamental relation between the speed of sound and macroscopic thermodynamic properties of a fluid is obtained (Trusler, 1991)

(5)

where S denotes entropy. This expression is exact in the limits of small amplitude and low frequency. The former limit can always be achieved while the latter is usually, but not always, approached in practice. Because of that, a practical measurement of the speed of sound is corrected to the zero-frequency limit.

While expression (5) is one of the fundamental thermodynamic relations, it is inconvenient for use because of the isentropic partial derivative. If temperature and pressure are taken as the independent variables, Eq. (5) may be written in a more convenient form

(6)

where T is the temperature. When temperature and density are taken as the independent variables the following expression is obtained

(7)

In either case, auxiliary equations will be necessary because not only density, but also the heat capacities are unknown. For that reason, Eqs. (6) and (7) are usually coupled with the thermodynamic relations (Sychev, 1983)

	(8)
	(9)

respectively. If density is eliminated from Eq. (7) and specific volume from Eq. (9), in favor of a more slowly varying compression factor, the following set of partial differential equations is obtained:

	(10)
	(11)
	(12)

where Z = Mp/(ρRT) is the compression factor, R is the universal gas constant, and M is the molar mass. The set of Eqs. (10) to (12) may be solved numerically in the range of ρ and T in which accurate speed of sound values are available. This set of equations may be solved as the initial-value problem (IVP) for the set of ordinary differential equations if density derivatives (∂Z/∂ρ)_T and (∂c_v/∂ρ)_T are known. Initial values of Z and a are specified along the isotherm at the lowest temperature of the range (Estrada-Alexanders et al., 1995).

The main disadvantage of the IVP method is its requirement not only for the Dirichlet, but also for the Neumann boundary conditions. For these later, to be determined with sufficient accuracy for the results to equal or exceed the accuracy of direct measurements, values of Z specified along two closely spaced isotherms should be sufficient but, for monatomic gases, several isotherms may be needed (Estrada-Alexanders et al., 1995). One way to avoid the Neumann boundary conditions is to specify several values of Z along two isotherms and two isochores surrounding the area of interest (Estrada-Alexanders and Justo, 2004).

If Eqs. (11) and (12) are solved for (∂Z/∂ρ)_T and (∂c_v/∂ρ)_T, respectively, bearing in mind Eq. (10), the following set of partial differential equations is obtained:

	(13)
	(14)

The set of Eqs. (13) and (14) may be solved numerically as the initial-boundary-value problem (IBVP) for the set of ordinary differential equations if temperature derivatives (∂Z/∂T)_ρ and (∂²Z/∂T²)_ρ are known. Initial values of Z and c_v are specified along the isochore at the lowest density of the range.

If the lowest density of the range is in the perfect-gas limit, then Z^pg = 1, and (Goodwin and Trusler, 2003)

(15)

Therefore, in the IBVP method, initial values of Z and c_v are obtained from the perfect-gas law and speed of sound measurements (extrapolated to the zero-pressure). Theoretically, these initial values should be sufficient to obtain particular solution of Eqs. (13) and (14) if accurate temperature derivatives (∂Z/∂T)_ρ and (∂²Z/∂T²)_ρ are available. However, this requirement is difficult to meet in practice, especially at the terminal temperature(s) of the range. For that reason boundary values of Z are specified along two isotherms at the lowest temperatures of the range.

Having calculated c_v in either way, in the density and temperature range of interest, the specific heat capacity at constant pressure may be obtained from (Sychev, 1983)

(16)

When a function is given as a set of discrete data-points, whose analytical expression is not known, its derivative upon an independent variable(s) may not be calculated exactly but rather approximately. This approximate derivative (estimate) is usually obtained from an interpolation polynomial. However, since interpolation polynomial passes exactly through each data-point this approach may give satisfactory results only if data-points are smooth enough (e.g., do not oscillate much). Accuracy of derivatives estimated from an interpolation polynomial also depends on a number of data-points and on their distribution. Increasing a number of data-points (and order of a polynomial as well) accuracy of derivatives increases, but to some extent, and after that it decreases. If data-points are equally spaced in a whole range of an independent variable, derivative estimates in terminal points will be the worst (e.g., the Runge phenomenon). This can be reduced by clustering data-points around the edges of a range (e.g., the Chebyshev nodes).

One of commonly used forms of an interpolation polynomial is that of Lagrange (Cheney and Kincaid, 1985). Its application in numerical interpolation and differentiation is particularly efficient if coupled with the barycentric interpolation formula. The first form of the barycentric interpolation formula is (Berrut and Trefethen, 2004)

(17)

where

(18)

and the barycentric weights

(19)

The second (e.g., true) form of the barycentric interpolation formula is (Berrut and Trefethen, 2004)

(20)

If compression factor Z is represented as a function of temperature T along any isochore, by a polynomial in Lagrange form, , then the first and second derivatives of Z, with respect to T, are (Berrut and Trefethen, 2004):

	(21)
	(22)
	(23)
	(24)

The Chebyshev points of the second kind are given by (Berrut and Trefethen, 2004)

(25)

if the range of x is -1 to 1, or by

(26)

for an arbitrary range a to b.

III. RESULTS AND DISCUSSION

The IBVP method is tested on methane, ethane, and carbon dioxide. Density, temperature and pressure ranges considered are given in Table 1. The density ranges are divided into 7 equally spaced isochores, while the temperature ranges are divided into 15 isotherms distributed according to the Chebyshev points of the second kind

Table 1. ρ-T-p ranges considered

(27)

where n = 14. However, sound-speed is measured along isotherms at different pressures rather than densities. For that reason, 90 pressure data-points (6 isochores x 15 isotherms) are estimated from the Peng-Robinson equation of state (Peng and Robinson, 1976), and sound-speed values for methane (Setzmann and Wagner, 1991; and Trusler and Zarari, 1992), ethane (Estrada-Alexanders and Trusler, 1997; and Friend et al., 1991), and carbon dioxide (Span and Wagner, 1996; and Estrada-Alexanders and Trusler, 1998) are specified at these points. Since pressures are obtained from the compression factors at each integration step, sound-speed values between these base points are interpolated with respect to pressure according to

(28)

where

(29)

The remaining 15 pressure data-points at the lowest density have zero value. Besides, 12 compression factor data-points (plus two data-points in the perfect-gas limit) are required to impose the boundary conditions along two isotherms at the lowest temperatures of the range. According to Table 1, temperature interval considered is 140 K in all cases. When Eq. (27) is applied, two isotherms at the lowest (and the highest) temperatures are spaced at about 2 K.

Temperature derivatives of compression factor are estimated from Eqs. (21) to (24). For the first 8 isotherms polynomial of the 7th degree is used, and for the last 7 isotherms polynomial of the 14th degree (e.g., the first isotherm is the one at the lowest temperature). In this way the best results are obtained.

Numerical integration was conducted using the Runge-Kutta-Verner fifth-order and sixth-order method with adaptive step-size (Hull et al., 1976).

Deviations of the results obtained, from respective reference data for methane (Younglove, 1974; Pieperbeck et al., 1991; Setzmann and Wagner, 1991; and Klimeck et al., 2001), ethane (Michels et al., 1954; Bier et al., 1976; Douslin and Harrison, 1973; and Friend et al., 1991), and carbon dioxide (Ernst et al., 1989; Klimeck et al., 2001; and Span and Wagner, 1996) are given in Tables 2 to 5. Uncertainties of reference data used are given in Table 6.

Table 2. Maximum relative deviation (IBVP method)

Table 3. Absolute average deviation (IBVP method)

Table 4. Maximum relative deviation (IVP method).

Table 5. Absolute average deviation (IVP method).

Table 6. Uncertainties of reference data used.

The relative deviation of respective quantities is calculated according to

(30)

where X_cal is calculated value and X_ref is reference value of Z, c_v, and c_p. The absolute average deviation of the compression factor is calculated according to

(31)

and that of the heat capacities according to

(32)

where M is total number of isochores (e.g., 7), and N is total number of isotherms (e.g., 15) in the ranges considered. Since the initial values of the compression factor and the heat capacity are specified along the lowest isochore it is omitted in Eqs. (31) and (32). Similarly, the boundary values of the compression factor are specified along two isotherms at the lowest temperatures and because of that these isotherms are omitted in Eq. (31). Therefore, total number of the compression factor data-points is (7-1) ' (15-2) = 78, and that of the heat capacities (7-1) ' 15 = 90.

Generally, it is clear that the IBVP method gives much better results than the IVP method (two orders of magnitude in average), for the same set of initial/boundary conditions. There are only two ways for improving the results of the IVP method - by narrowing spacing between the first two isotherms or by specifying initial values of Z at additional temperature(s). It is established that the IVP method may give similar results as the IBVP method only when spacing between the first two isotherms is two times of magnitude smaller (e.g., 0.02 K). Comparing figures from Table (3) to those from Table (6) it arises that values of AAD are two to three times of magnitude smaller than corresponding values of uncertainty of reference data. Therefore, uncertainty of the results obtained is virtually the same as that of reference data used.

IV. CONCLUSIONS

The method of the initial-boundary-value problem is recommended for deriving the thermodynamic properties of gases from the speed of sound. The initial values (several data-points of Z and c_v) are specified along the isochore in the perfect-gas limit. These data may be obtained directly from the perfect-gas law and speed of sound measurements (extrapolated to the zero-pressure). The boundary conditions (several data-points of Z) are specified along two isotherms at the lowest temperatures of the range. It is the least demanding method to date because it requires just a few thermal data-points (e.g., Z, r, v) from other source. Besides, no Neumann boundary conditions are required. The results obtained show excellent agreement with respective reference data.

REFERENCES
1. Benedetto, G., R.M. Gavioso and R. Spagnolo, "Precision measurements of the speed of sound and thermodynamic properties of gases," Rivista Del Nuovo Cimento, 22, 1-42 (1999).         [ Links ]
2. Berrut, J.-P. and L.N. Trefethen, "Barycentric Lagrange interpolation," SIAM Rev., 46, 501-517 (2004).         [ Links ]
3. Bier, K., J. Kunze and G. Maurer, "Thermodynamic properties of ethane from calorimetric measurements," J. Chem. Thermodynamics, 8, 857-868 (1976).         [ Links ]
4. Cheney, W. and D. Kincaid, Numerical Mathematics and Computing, Brooks/Cole Publishing Company, Pacific Grove (1985).         [ Links ]
5. Douslin, D.R. and R.H. Harrison, "Pressure, volume, temperature relations of ethane," J. Chem. Thermodynamics, 5, 491-512 (1973).         [ Links ]
6. Ernst, G., G. Maurer and E. Wiederuh, "Flow calorimeter for the accurate determination of the isobaric heat capacity at high pressures; Results for carbon dioxide," J. Chem. Thermodynamics, 21, 53-65 (1989).         [ Links ]
7. Estela-Uribe, J.F. and J.P.M. Trusler, "Acoustic and volumetric virial coefficients of nitrogen," Int. J. Thermophys., 21, 1033-1044 (2000).         [ Links ]
8. Estrada-Alexanders, A.F., J.P.M. Trusler and M.P. Zarari, "Determination of thermodynamic properties from the speed of sound," Int. J. Thermophys., 16, 663-673 (1995).         [ Links ]
9. Estrada-Alexanders, A.F. and J.P.M. Trusler, "The speed of sound and derived thermodynamic properties of ethane at temperatures between 220 K and 450 K and pressures up to 10.5 MPa," J. Chem. Thermodynamics, 29, 991-1015 (1997).         [ Links ]
10. Estrada-Alexanders, A.F. and J.P.M. Trusler, "Speed of sound in carbon dioxide at temperatures between (220 and 450) K and pressures up to 14 MPa," J. Chem. Thermodynamics, 30, 1589-1601 (1998).         [ Links ]
11. Estrada-Alexanders, A.F. and D. Justo, "New method for deriving accurate thermodynamic properties from speed-of-sound," J. Chem. Thermodynamics, 36, 419-429 (2004).         [ Links ]
12. Friend, D.G., H. Ingham and J.F. Ely, "Thermophysical properties of ethane," J. Phys. Chem. Ref. Data, 20, 275-347 (1991).         [ Links ]
13. Funke, M., R. Kleinrahm and W. Wagner, "Measurements and correlation of the (p,r,T) relation of ethane. I. The homogeneous gas and liquid regions in the temperature range from 95 K to 340 K at ressures up to 12 MPa," J. Chem. Thermodynamics, 34, 2001-2015 (2002).         [ Links ]
14. Goodwin, A.R.H. and J.P.M. Trusler, Speed of Sound, in: Measurement of the Thermodynamic Properties of Single Phases (Eds. Goodwin, A.R.H., K.N. Marsh and W.A. Wakeham), Elsevier, Amsterdam (2003).         [ Links ]
15. Hull, T.E., W.H. Enright and K.R. Jackson, User's Guide for DVERK - A Subroutine for Solving Non-Stiff ODEs, Technical Report 100, University of Toronto, Toronto (1976).         [ Links ]
16. Klimeck, J., R. Kleinrahm and W. Wagner, "Measurements of the (p,r,T) relation of methane and carbon dioxide in the temperature range 240 K to 520 K at pressures up to 30 MPa using a new accurate ingle-sinker densitometer," J. Chem. Thermodynamics, 33, 251-267 (2001).         [ Links ]
17. Lago, S. and P.A. Giuliano Albo, "A new method to calculate the thermodynamic properties of liquids from accurate speed-of-sound measurements," J. Chem. Thermodynamics, 40, 1558-1564 (2008).         [ Links ]
18. Laplace, P.-S., "On the speed of sound in the air and water," Ann. de Chim. et de Phys., 3, 238-241 (1816).         [ Links ]
19. Maitland, G.C. and E.B. Smith, "A simplified representation of intermolecular potential energy," Chem. Phys. Letters, 22, 443-446 (1973).         [ Links ]
20. Michels, A., W. Van Straaten and J. Dawson, "Isotherms and thermodynamical functions of ethane at temperatures between 0°C and 150°C and pressures up to 200 atm," Physica, 20, 17-23 (1954).         [ Links ]
21. Newton, I., The Mathematical Principles of Natural Philosophy, (1687).         [ Links ]
22. Peng, D.Y. and D.B. Robinson, "A new two-constant equation of state," Ind. Eng. Chem. Fund., 15, 59-64 (1976).         [ Links ]
23. Pieperbeck, N., R. Kleinrahm, W. Wagner and M. Jaeschke, "Results of (pressure, density, temperature) measurements on methane and on nitrogen in the temperature range from 273.15 K to 323.15 K at pressures up to 12 MPa sing a new apparatus for accurate gas-density measurements," J. Chem. Thermodynamics, 23, 175-194 (1991).         [ Links ]
24. Schmidt, R. and W. Wagner, "A new form of the equation of state for pure substances and its application to oxygen," Fluid Phase Equil., 19, 175-200 (1985).         [ Links ]
25. Setzmann, U. and W. Wagner, "A new method for optimizing the structure of thermodynamic correlation equations," Int. J. Thermophys., 10, 1103-1126 (1989).         [ Links ]
26. Setzmann, U. and W. Wagner, "A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 K and pressures up to 1000 MPa," J. Phys. Chem. Ref. Data, 0, 1061-1155 (1991).         [ Links ]
27. Smukala, J., R. Span and W. Wagner, "A new equation of state for ethylene covering the fluid region for temperatures from the melting line to 450 K at pressures up to 300 MPa," J. Phys. Chem. Ref. Data, 29, 1053-1121 (2000).         [ Links ]
28. Span, R., Multiparameter Equations of State - An Accurate Source of Thermodynamic Property Data, Springer-Verlag, Berlin (2000).         [ Links ]
29. Span, R. and W. Wagner, "A new equation of state for carbon dioxide covering the fluid region for temperature from the triple-point temperature to 1100 K at pressures up to 800 MPa," J. Phys. Chem. Ref. Data, 25 , 1509-1596 (1996).         [ Links ]
30. Span, R., E.W. Lemmon, R.T. Jacobsen, W. Wagner and A. Yokozeki, "A reference equation of state for the thermodynamic properties of nitrogen for temperatures from 63.151 to 1000 K and pressures to 2200 MPa," J. Phys. Chem. Ref. Data, 29, 1361-1433 (2000).         [ Links ]
31. Span, R. and W. Wagner, "Equations of state for technical applications. I. Simultaneously optimized functional forms for nonpolar and polar fluids," Int. J. Thermophys., 24, 1-39 (2003).         [ Links ]
32. Sychev, V.V., The Differential Equations of Thermodynamics, Mir Publishers, Moscow (1983).         [ Links ]
33. Tegeler, C., R. Span and W. Wagner, "A new equation of state for argon covering the fluid region for temperature from the melting line to 700 K at pressures up to 1000 MPa," J. Phys. Chem. Ref. Data, 28, 779-850 (1999).         [ Links ]
34. Trusler, J.P.M., Physical Acoustics and Metrology of Fluids, Adam Hilger, Bristol (1991).         [ Links ]
35. Trusler, J.P.M. and M. Zarari, "The speed of sound and derived thermodynamic properties of methane at temperatures between 275 K and 375 K and pressures up to 10 MPa," J. Chem. Thermodynamics, 24, 973-991 (1992).         [ Links ]
36. Trusler, J.P.M., W.A. Wakeham and M.P. Zarari, "Model intermolecular potentials and virial coefficients determined from the speed of sound," Molecular Physics, 90, 695-703 (1997).         [ Links ]
37. Van der Waals, J.D., On the Continuity of the Gas and Liquid State, Ph. D. thesis, Leiden (1873).         [ Links ]
38. Wagner, W. and A. Pruss, "The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use," J. Phys. Chem. Ref. Data, 31, 387-535 (2002).         [ Links ]
39. Younglove, B.A., "The specific heats c_p and c_v of compressed and liquefied methane," J. Res. Natl. Bur. Stand., 78, 401-410 (1974).         [ Links ]

Received: August 28, 2012
Accepted: March 20, 2013.
Recommended by Subject Editor: Pedro de Alcântara Pessôa.