Two different methods for deriving the density and isobaric heat capacity of liquids in the subcritical pressure range, from the speed of sound, are recommended. In each method, corresponding set of differential equations relating these properties is solved as the initial boundary value problem (IBVP). The initial values are specified at the lowest pressure of the range and the boundary values along the saturation line. In the first method, numerical integration is performed along the paths connecting the Chebyshev points of the second kind between the minimum and maximum temperature at each pressure. In the second method, numerical integration is performed along the isotherms distributed in the same way, with the temperature range being extended to the saturation line after each integration step. The methods are tested with the following substances: Ar, N_{2}, CO_{2}, and CH_{4}. The results obtained for the density and isobaric heat capacity have the average absolute deviation from the reference data of 0.0005% and 0.0219%, respectively. These results served as the initial values for deriving the same properties in the transcritical pressure range up to the pressure approximately twice as large as the critical pressure. The results obtained in this pressure range have respective deviations of 0.0019% and 0.1303%.
The relations between the thermodynamic speed of sound (i.e., the mechanical disturbance of a small amplitude and a low frequency) and other thermodynamic properties (e.g., the density and heat capacity) comprise the set of nonlinear partial differential equations of the secondorder (Trusler [
Deriving the density and heat capacity of liquids from the speed of sound is usually performed by solving the set of differential equations, which relates these properties, as the initial value problem (IVP) for the set of ordinary differential equations. The initial values are usually specified at the lowest pressure of the range considered (Benedetto et al. [
In this paper an attempt is made to reconcile these two opposites, that is, to specify the initial values at the lowest pressure and, at the same time, to cover the maximum temperature range possible (i.e., to the saturation line). This lower pressure limit is chosen so as to enable sufficiently wide temperature range to accommodate a reasonable number of integration paths. Unfortunately, as it turned out, there is a compromise which has to be done. Namely, in order for the solution to be stable across the whole pressure range, the boundary values must be specified along the saturation line. Still, the majority of these values are specified at pressures which are considerably below the upper limit of the pressure range. Two different methods based on the initial boundary value problem (IBVP) are recommended. The main difference between them is in the paths of integration. In one method, these paths change their shape progressively from that of an isotherm to that of the saturation line, while in the other one they follow isotherms which are modified in each integration step to suit consecutively broader temperature ranges. To ensure that the results obtained are reliable enough they are not only compared to respective reference data but also used as the initial values for deriving the same properties in the transcritical domain up to the pressure twice of that in the critical point.
When the density and heat capacity are derived from the speed of sound in rectangular
The set of (
While the set of (
The methods described are tested with several different substances. Their list, along with the
Substance 




min  max  min  max  
Ar  0.70  3.40  100.00  110.78/141.69 
N_{2}  0.20  2.20  70.000  83.626/117.41 
CO_{2}  1.00  6.00  220.00  233.03/295.13 
CH_{4}  0.10  3.40  100.00  111.51/181.05 
Substance 




min  max  min  max  
Ar  3.40  10.0  100.00  141.69 
N_{2}  2.20  7.00  70.000  117.41 
CO_{2}  6.00  15.0  220.00  295.13 
CH_{4}  3.40  10.0  100.00  181.05 
The domains of integration for CO_{2}: (red line): the boiling line, (blue line): the melting line, C.P.: the critical point, and T.P.: the triple point.
The lowest pressure of the subcritical domain is chosen so as to enable a reasonable temperature range, and the highest one corresponds to the saturation pressure for the temperature approximately 10 K below the critical point. The lowest temperature is constant across both the domains, while the highest one in the subcritical domain corresponds to the saturation temperature at each pressure.
The lowest pressure of the transcritical domain coincides with the highest pressure of the subcritical domain, and the highest one is approximately twice as large as the critical pressure. The highest temperature is constant across the domain and corresponds to the saturation temperature at the lowest pressure of the domain or to the highest temperature of the subcritical domain.
Four sets of calculations are performed for each substance. Two of them cover the subcritical domain and the other two the transcritical domain. The calculations are performed first in the subcritical domain, and the results obtained along the highest pressure of the domain served as the initial values for the transcritical domain.
For both methods to be implemented in both domains, the reference values of the density and heat capacity, as well as the speed of sound, are specified along several mainly equidistant isobars, at temperatures distributed along each isobar according to the Chebyshev points of the second kind (Berrut and Trefethen [
In the first method (IBVP1) the initial values of
Number of initial/boundary datapoints of
Substance  Initial/boundary values  Speed of sound  

At lowest 
At saturation  Subcritical  Transcritical  
Ar  30  18  150  105 
N_{2}  30  20  165  120 
CO_{2}  30  20  165  135 
CH_{4}  30  28  225  105 
The Lagrange polynomial orders used in the subcritical domain.
Substance  Interpolation with respect to  Derivation with respect to  






Ar  9  14, 15 
9  2 
N_{2}  10  14, 15  10  2, 11 
CO_{2}  10  14, 15  10  2, 11 
CH_{4}  14  14, 15  14  2, 11 
In the second method (IBVP2) the initial values of
In the transcritical domain, the set of (
The Lagrange polynomial orders used in the transcritical domain.
Substance  Interpolation with respect to  Derivation with respect to  




 
Ar  7  14  —  2, 11 
N_{2}  8  14  —  2, 11 
CO_{2}  9  14  —  2, 11 
CH_{4}  7  14  —  2, 11 
The average stepsize and number of steps taken in the subcritical domain.
Substance  Average stepsize, MPa  Number of steps taken  

IBVP1 
IBVP2 
IBVP1  IBVP2  
Ar  0.073  0.01  37  270 
N_{2}  0.056  0.01  36  200 
CO_{2}  0.114  0.01  44  500 
CH_{4}  0.065  0.001  51  3300 
The average stepsize and number of steps taken in the transcritical domain.
Substance  Average stepsize, MPa  Number of steps taken  

IBVP1 
IBVP2 
IBVP1  IBVP2  
Ar  0.471  0.440  14  15 
N_{2}  0.369  0.343  13  14 
CO_{2}  0.529  0.500  17  18 
CH_{4}  0.471  0.413  14  16 
The results obtained are assessed by the average absolute deviation (AAD) from corresponding reference data (Tegeler et al. [
The average absolute deviation (AAD) in the subcritical domain.
Substance  AAD, %  


 
IBVP1  IBVP2  IBVP1  IBVP2  
Ar  0.0001  0.0001  0.0084  0.0042 
N_{2}  0.0003  0.0003  0.0066  0.0064 
CO_{2}  0.0004  0.0002  0.0903  0.0175 
CH_{4}  0.0010  0.0010  0.0260  0.0154 
Average  0.0005  0.0004  0.0328  0.0109 
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
Relative deviation of calculated
In order to test accuracy of the results additionally, their values along the highest pressure (the most unfavorable conditions) served as the initial values for deriving the same properties in the transcritical domain. Here, the results for the heat capacity obtained with the initial values from both methods have almost the same AADs, while the results for the density obtained with the initial values from the IBVP1 method have AAD two times lower than that of the IBVP2 method (see Table
The average absolute deviation (AAD) in the transcritical domain.
Substance  AAD, %  


 
IBVP1  IBVP2  IBVP1  IBVP2  
Ar  0.0003  0.0003  0.0418  0.0367 
N_{2}  0.0005  0.0016  0.0346  0.1078 
CO_{2}  0.0029  0.0032  0.2815  0.2264 
CH_{4}  0.0016  0.0046  0.0955  0.2178 
Average  0.0013  0.0024  0.1134  0.1472 
The uncertainty of the reference data.
Substance  Uncertainty, %  



 
Ar  0.02  2.0  1.0 
N_{2}  0.02  0.8  1.0 
CO_{2}  0.04  1.5  0.5 
CH_{4}  0.03  1.0  0.3 
Average  0.03  1.3  0.7 
Influence of uncertainties of the initial and boundary values and speed of sound values on the results is investigated as well. Corresponding AADs of the results obtained after changing these values in the limits of their uncertainties (see Table
Influence of the uncertainty of the initial and boundary values of
Substance  AAD, %  


 
IBVP1  IBVP2  IBVP1  IBVP2  
Ar  0.0199  0.0199  0.0085  0.0043 
N_{2}  0.0200  0.0200  0.0066  0.0064 
CO_{2}  0.0398  0.0160  0.0904  0.0036 
CH_{4}  0.0299  0.0299  0.0260  0.0154 
Average  0.0274  0.0215  0.0329  0.0074 
Influence of the uncertainty of the initial and boundary values of
Substance  AAD, %  


 
IBVP1  IBVP2  IBVP1  IBVP2  
Ar  0.0076  0.0076  2.0425  2.0427 
N_{2}  0.0018  0.0018  0.8087  0.8087 
CO_{2}  0.0061  0.0007  1.5357  0.6034 
CH_{4}  0.0028  0.0028  1.0133  1.0134 
Average  0.0046  0.0032  1.3501  1.1171 
Influence of the uncertainty of the speed of sound on AAD.
Substance  AAD, %  


 
IBVP1  IBVP2  IBVP1  IBVP2  
Ar  0.0042  0.0042  0.0090  0.0058 
N_{2}  0.0038  0.0038  0.0067  0.0069 
CO_{2}  0.0030  0.0030  0.0905  0.0183 
CH_{4}  0.0019  0.0019  0.0261  0.0154 
Average  0.0032  0.0032  0.0331  0.0116 
The density and heat capacity of a liquid may be derived from the speed of sound in the subcritical pressure range for the temperatures up to the saturation line. This may be accomplished by the use of a standard numerical procedure for solving an initial boundary value problem (IBVP) for the set of ordinary differential equations and an interpolation polynomial. The initial values are specified along the lowest pressure of the range and the boundary values along the saturation line. For the results to have uncertainty not higher than that of corresponding direct measurements, numerical derivatives must be estimated as accurately as possible prior to each integration step. This precondition may be difficult to fulfill for two reasons. The first one emerges from the fact that the set of equations is of the secondorder with respect to the temperature, and the second one is that the density and heat capacity (whose temperature derivatives are estimated) change with temperature abruptly in the vicinity of the saturation line. For this to be overcome the number of these temperatures should be optimal and they have to be distributed so as to avoid Runge’s phenomenon (e.g., according to the Chebyshev knots).
See Figures
The authors declare that there is no conflict of interests regarding the publication of this paper.