Abstract
A numerical procedure for deriving the thermodynamic properties , , and of the vapor phase in the subcritical temperature range from the speed of sound is presented. The set of differential equations connecting these properties with the speed of sound is solved as the initial-value problem in domain . The initial values of and are specified along the isotherm with the highest temperature, at a several values of [0.1, 1.0]. The values of are generated by the reference equation of state, while the values of are derived from the speed of sound, by solving another set of differential equations in domain in the transcritical temperature range. This set of equations is solved as the initial-boundary-value problem. The initial values of and are specified along the isochore in the limit of the ideal gas, at several isotherms distributed according to the Chebyshev points of the second kind. The boundary values of are specified along the same isotherm and along another isotherm with a higher temperature, at several values of . The procedure is tested on Ar, N2, CH4, and CO2, with the mean AADs for , , and at 0.0003%, 0.0046%, and 0.0061%, respectively (0.0007%, 0.0130%, and 0.0189% along the saturation line).
1. Introduction
The speed of sound is the property of a fluid which is measured with an exceptional accuracy (Ewing and Goodwin [1], Estrada-Alexanders and Trusler [2], Costa Gomes and Trusler [3], Trusler and Zarari [4], and Estrada-Alexanders and Trusler [5]). While the speed of sound itself is a piece of useful information about a fluid, other thermodynamic properties like the density and the heat capacity, which may be derived from it, make the speed of sound even more useful. However, the speed of sound in a fluid is connected to these properties through the set of nonlinear partial differential equations of the second order. The general solution to this set of equations has not been found yet, except for the gaseous phase under low pressures (Trusler et al. [6] and Estela-Uribe and Trusler [7]). At moderate and higher pressures a procedure based on the numerical integration may be used for obtaining the particular solution (Estrada-Alexanders et al. [8] and Bijedić and Neimarlija [9–11]). The main disadvantage of this approach is the need for the initial and/or the boundary values of the quantities being calculated (the Dirichlet conditions) and their derivatives with respect to an independent variable (the Neumann conditions).
In the previous paper (Bijedić and Neimarlija [11]) it was shown that the compressibility factor and the heat capacity of a gas may be derived from the speed of sound in the supercritical temperature range only with the initial/boundary conditions of the Dirichlet type. In this paper, that procedure is applied to the transcritical temperature range in order to generate the initial conditions of the Neumann type, which are then used for deriving the compressibility factor and the heat capacity of a vapor in the subcritical temperature range. In this way, the main disadvantage of the approach based on the numerical integration is minimized, because only a few data points of the compressibility factor from other sources are needed to cover the whole gaseous phase except the region around the critical point. The boundary values of the compressibility factor are specified along two isotherms, several degrees below the critical point. The maximum density is chosen so as to correspond to that of the saturated vapor at the initial temperature. In this way, the saturation line becomes a boundary of the domain of integration, and the thermodynamic properties of the saturated vapor are derived as well. While the derived properties of a gas in the transcritical temperature range have a very good agreement with the corresponding reference data they are not discussed in this paper (see Table 7).
2. Theory
The thermodynamic speed of sound (the mechanical disturbance of small amplitude and low frequency) is connected to other macroscopic thermodynamic properties through the following fundamental relation (Trusler [12]): where is the speed of sound, is the pressure, is the density, and is the specific entropy. However, since is not a measurable property it is convenient to avoid it. This can be accomplished with the help of the following thermodynamic relation (Moran and Shapiro [13]): where is the thermodynamic temperature, is the specific heat capacity at constant pressure, and is the specific heat capacity at constant volume. Now, (1) takes the following form: The isobaric heat capacity may be eliminated from (3) on account of the following thermodynamic relation (Moran and Shapiro [13]): where is the specific volume. If the specific volume is replaced with the density, (4) becomes According to the chain rule from the calculus, one may write From (6) it follows that If the right hand side of (7) is introduced into (5) one obtains Introducing the right hand side of (8) into (3), one finally obtains In (9), not only the density but also the isochoric heat capacity is unknown. Therefore, another equation, which connects the heat capacity with the thermal properties, is necessary. For this purpose, the following thermodynamic relation is suitable: If the density is replaced with a less varying compressibility factor where is the compressibility factor, is the molar mass, and is the universal gas constant, (9) and (10), after rearrangement, take the following forms, respectively, Equations (12) and (13) may be solved simultaneously for and in the range of and in which accurate values of the speed of sound are available (Bijedić and Neimarlija [11]). This set of equations is designed to be solved in rectangular domain.
If the subcritical vapor phase is considered, it is convenient to introduce new independent variable instead of the density. For this purpose, the ratio of the density and its maximum value at the observed temperature (at the saturation) is suitable, that is, where is the new independent variable and is the density of the saturated vapor at the observed temperature. In such a way new rectangular domain is obtained, which allows one to cover the temperature range from the critical point down to the triple point and the density range from the ideal gas limit up to the saturation line.
According to the calculus, one may write If new variable is defined as (13) now looks like According to the calculus, one may also write If (17) and (18) are combined one obtains Now, (12), (15), and (19) may be solved simultaneously for and in the range of and in which accurate values of the speed of sound are available.
When the integration is complete, is calculated from the following relation: which is obtained from (3) when the density is replaced with the compressibility factor.
3. Results and Discussion
The speed of sound is usually available in the form and not in the form . However, the numerical integration is performed with and as the independent variables, rather than and , because more accurate results are obtained (Bijedić and Neimarlija [10]). For that reason, the lines of constant are generated in advance by the Peng-Robinson equation of state (Peng and Robinson [14]), except the line which is generated by the corresponding reference equation of state. Since the compressibility factor is calculated before the speed of sound is required, in each integration step, the latter is interpolated along the isotherms with respect to the pressure (). Then, partial derivatives are estimated from the interpolating polynomial in the Lagrange form. The total number of the isotherms, , is 7 to 10, and the total number of the lines with constant , , is 10. The values of off the isotherms are interpolated along the lines of constant with respect to the temperature.
The calculation procedure is initialized by specifying values of and along the isotherm with the highest temperature (see Table 1), at 10 equidistant values of [0.1, 1.0]. The values of are generated by the corresponding reference equation of state, while values of are derived from the speed of sound, by solving the set of (12) and (13) in domain in the transcritical temperature range (see Tables 2 and 3). Then, the partial derivatives and are estimated along from the interpolating polynomial in the Lagrange form. Next, is calculated from (12) and the partial derivatives are estimated along also from the interpolating polynomial in the Lagrange form. Now, values of and are calculated from (15) and (19), along the isotherm , by the Runge-Kutta-Verner fifth-order and six-order method with the adaptive step-size (Hull et al. [15]). The whole procedure is repeated until the lowest temperature of the range is reached (see Table 1).
The results are assessed by means of the relative deviation (RD) and the absolute average deviation (AAD), which are computed from the following relations: where is the calculated value and is the reference value of , , and (Tegeler et al. [16], Span et al. [17], Setzmann and Wagner [18], and Span and Wagner [19]).
As one could anticipate, the worst results are obtained along the saturation line (), because it exhibits the highest nonlinearity of with respect to , and because all the partial derivatives with respect to the density are the worst exactly along this boundary line (Runge’s phenomenon). For that reason, the maximum relative deviations given in Table 4 actually refer to the saturation line. While these results are worse than the rest, they are still in the limits of the experimental uncertainties of the direct measurements (see Table 8). However, the mean AADs of the results are 2 orders of magnitude (1-2 along the saturation line) better than the experimental uncertainties of the corresponding direct measurements (see Tables 5 and 6).
4. Conclusions
The compressibility factor and the heat capacity (the isochoric and the isobaric) of a vapor in the subcritical temperature range (including the saturation line) may be derived from the speed of sound with the AADs two orders of magnitude better than the experimental uncertainties of the corresponding direct measurements. This can be accomplished on account of just a few data points of the compressibility factor obtained from the direct measurement of the density. Moreover, the temperature derivatives of the compressibility factor, needed for initialization of the numerical integration, may also be derived from the speed of sound. Practically, the whole gaseous phase from the triple point to far above the critical point and from the ideal gas limit to the saturation line may be covered with just a few thermal data points from other source.
Appendix
See Tables 1, 2, 3, 4, 5, 6, 7, and 8.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.