Journal of Thermodynamics

Volume 2016 (2016), Article ID 2035704, 8 pages

http://dx.doi.org/10.1155/2016/2035704

## Density and Heat Capacity of Liquids from Speed of Sound

Faculty of Technology, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

Received 5 October 2015; Revised 25 December 2015; Accepted 30 December 2015

Academic Editor: Felix Sharipov

Copyright © 2016 Muhamed Bijedić and Sabina Begić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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%.

#### 1. Introduction

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 second-order (Trusler [1]). A general solution to this set of equations is available only for dilute imperfect gases (i.e., gases at low pressures) where the pressure effect on density may be satisfactorily described by the virial expansion truncated after the second virial coefficient, which may be obtained from the speed of sound through a model of the intermolecular potential energy. While the virial expansion does not work when applied to liquids (Allen and Tildesley [2]), it is still possible to obtain a particular solution if appropriate initial/boundary values are available from corresponding direct measurements (e.g., volumetric and caloric). In an open literature one can find several different approaches for deriving the thermodynamic properties of liquids (e.g., the density, heat capacity, isothermal compressibility, isentropic compressibility, and isobaric thermal expansivity) from the speed of sound. The majority of them are based on numerical integration of corresponding differential equations connecting these properties (Muringer et al. [3] and Sun et al. [4–6]). Other approaches include an iterative method of calculation (Petitet et al. [7]), a grid algorithm (Khasanshin et al. [8]), and a heuristic approach (Scalabrin et al. [9]).

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. [10]). This approach is applicable only to rectangular domain, that is, the one where the temperature range is constant across the entire pressure range (e.g., the supercritical pressure range). Since the IVP is highly sensitive to the initial values and since the experimental uncertainty of direct measurement of these properties is decreasing with increasing the pressure, it is preferable to specify them at pressure as close as possible to the atmospheric one. However, by decreasing the lowest pressure of the range the temperature range between a minimum temperature and that at the saturation also decreases. This may be partly overcome by imposing the initial values at the highest pressure of the range and performing the integration down to the lowest pressure along lines having similar shape to that of the saturation line. While this approach may give satisfactory results (Bijedić and Neimarlija [11]), it may not be suitable for substances with relatively high critical pressures.

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.

#### 2. Theory

When the density and heat capacity are derived from the speed of sound in rectangular domain, the following set of equations may be used (Benedetto et al. [10]):where is the pressure, is the temperature, is the speed of sound, is the density, is the specific heat capacity at constant pressure, and is the thermal expansion coefficient:However, if the domain is not of rectangular shape (e.g., in the subcritical pressure range), the following set of equations is suitable (Bijedić and Neimarlija [11]):where represents a path connecting the points with temperaturewhere is the lowest temperature of the range and is the highest temperature at observed pressure (e.g., the saturation temperature). Using instead of as a path of integration enables one to shape the domain arbitrarily.

The set of (4) and (5) may be solved simultaneously for and if their initial values are specified at the lowest pressure of the range. However, for the solution to be stable across the whole pressure range, the boundary values of and need to be specified along connecting the points with the highest temperatures at each pressure (i.e., along the saturation line). If all the temperature derivatives are estimated, say, from an interpolation polynomial, the set of equations may be solved as the initial boundary value problem (IBVP) for the set of ordinary differential equations. In order to ensure that the temperature derivatives are estimated as accurate as possible, the lines of constant should be distributed so as to avoid Runge’s phenomenon (e.g., at the Chebyshev knots).

While the set of (1) and (2) is suitable for the supercritical pressure range, it may be used for the subcritical one as well, if the temperature range is being extended to the saturation line after each integration step. This set of equations may also be solved as the IBVP with the same set of initial/boundary values. However, the procedure of solution is more robust because the results from the current pressure have to be interpolated to a new set of temperatures in a wider temperature range at the next pressure.

#### 3. Results and Discussion

The methods described are tested with several different substances. Their list, along with the ranges covered, are given in Tables 1 and 2 for the subcritical and transcritical domain, respectively. These domains are also presented graphically in coordinates at Figure 1. While this graph is given for CO_{2}, respective graphs for other substances are qualitatively the same.