# Thermodynamic Properties of Vapors from Speed of Sound

Authors: Muhamed Bijedić; Sabina Begić
Journal: Journal of Thermodynamics
 (2014)

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## Abstract

A numerical procedure for deriving the thermodynamic properties(Z, cv, and cp) 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 T-ϕ domain (ϕ=ρ/ρsat). The initial values of Z and ∂Z/∂Tρ are specified along the isotherm T0 with the highest temperature, at a several values of ϕ [0.1, 1.0]. The values of Z are generated by the reference equation of state, while the values of ∂Z/∂Tρ are derived from the speed of sound, by solving another set of differential equations in T-ρ domain in the transcritical temperature range. This set of equations is solved as the initial-boundary-value problem. The initial values of Z and cv 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 Z are specified along the same isotherm T0 and along another isotherm with a higher temperature, at several values of ρ. The procedure is tested on Ar, N2, CH4, and CO2, with the mean AADs for Z, cv, and cp at 0.0003%, 0.0046%, and 0.0061%, respectively (0.0007%, 0.0130%, and 0.0189% along the saturation line).


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## Body

## 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]):
(1)u2=(∂p∂ρ)s,
where u is the speed of sound, p is the pressure, ρ is the density, and s is the specific entropy. However, since s 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]):
(2)(∂p∂ρ)s=cpcv(∂p∂ρ)T,
where T is the thermodynamic temperature, cp is the specific heat capacity at constant pressure, and cv is the specific heat capacity at constant volume. Now, (1) takes the following form:
(3)u2=cpcv(∂p∂ρ)T.
The isobaric heat capacity may be eliminated from (3) on account of the following thermodynamic relation (Moran and Shapiro [13]):
(4)cp-cv=-T(∂v∂T)p2(∂p∂v)T,
where v is the specific volume. If the specific volume is replaced with the density, (4) becomes
(5)cp-cv=Tρ2(∂ρ∂T)p2(∂p∂ρ)T.
According to the chain rule from the calculus, one may write
(6)(∂p∂ρ)T(∂ρ∂T)p(∂T∂p)ρ=-1.
From (6) it follows that
(7)(∂ρ∂T)p=-(∂ρ∂p)T(∂p∂T)ρ.
If the right hand side of (7) is introduced into (5) one obtains
(8)cp=cv+Tρ2(∂ρ∂p)T(∂p∂T)ρ2.
Introducing the right hand side of (8) into (3), one finally obtains
(9)u2=(∂p∂ρ)T+Tρ2cv(∂p∂T)ρ2.
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:
(10)(∂cv∂ρ)T=-Tρ2(∂2p∂T2)ρ.
If the density is replaced with a less varying compressibility factor
(11)Z=MpρRT,
where Z is the compressibility factor, M is the molar mass, and R is the universal gas constant, (9) and (10), after rearrangement, take the following forms, respectively,
(12)(∂Z∂ρ)T=1ρ{Mu2RT-Z-RMcv[Z+T(∂Z∂T)ρ]2}(13)(∂cv∂ρ)T=-RMρ[2T(∂Z∂T)ρ+T2(∂2Z∂T2)ρ].
Equations (12) and (13) may be solved simultaneously for Z and cv in the range of p and T 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 T-ρ 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,(14)ϕ=ρρs,
where ϕ is the new independent variable and ρs is the density of the saturated vapor at the observed temperature. In such a way new rectangular T-ϕ 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(15)(∂Z∂T)ϕ=(∂Z∂T)ρ+(∂Z∂ρ)T(∂ρ∂T)ϕ.
If new variable a is defined as
(16)a=(∂Z∂T)ρ,
(13) now looks like
(17)(∂cv∂ρ)T=-RMρ[2Ta+T2(∂a∂T)ρ].
According to the calculus, one may also write
(18)(∂a∂T)ϕ=(∂a∂T)ρ+(∂a∂ρ)T(∂ρ∂T)ϕ.
If (17) and (18) are combined one obtains
(19)(∂a∂T)ϕ=-1T[MρRT(∂cv∂ρ)T+2a]+(∂a∂ρ)T(∂ρ∂T)ϕ.
Now, (12), (15), and (19) may be solved simultaneously for Z and cv in the range of p and T in which accurate values of the speed of sound are available.When the integration is complete,cp is calculated from the following relation:
(20)cp=cvMu2ZRT[1-pZ(∂Z∂p)T]
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 formu=u(T,p) and not in the form u=u(T,ρ). However, the numerical integration is performed with T and ρ/ρs as the independent variables, rather than T and p/ps, because more accurate results are obtained (Bijedić and Neimarlija [10]). For that reason, the lines of constant ϕ are generated in advance by the Peng-Robinson equation of state (Peng and Robinson [14]), except the line ϕ=1.0 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 (p=ZρRT/M). Then, M×N partial derivatives (∂ρ/∂T)ϕ are estimated from the interpolating polynomial in the Lagrange form. The total number of the isotherms, M, is 7 to 10, and the total number of the lines with constant ϕ, N, is 10. The values of (∂ρ/∂T)ϕ off the isotherms are interpolated along the lines of constant ϕ with respect to the temperature.The calculation procedure is initialized by specifying values ofZ and a along the isotherm T0 with the highest temperature (see Table 1), at 10 equidistant values of ϕ [0.1, 1.0]. The values of Z are generated by the corresponding reference equation of state, while values of a are derived from the speed of sound, by solving the set of (12) and (13) in T-ρ domain in the transcritical temperature range (see Tables 2 and 3). Then, the partial derivatives (∂Z/∂ρ)T and (∂a/∂ρ)T are estimated along T0 from the interpolating polynomial in the Lagrange form. Next, cv is calculated from (12) and the partial derivatives (∂cv/∂ρ)T are estimated along T0 also from the interpolating polynomial in the Lagrange form. Now, values of Z and a are calculated from (15) and (19), along the isotherm T1=T0-ΔT, by the Runge-Kutta-Verner fifth-order and six-order method with the adaptive step-size ΔT (Hull et al. [15]). The whole procedure is repeated until the lowest temperature of the range is reached (see Table 1).Table 1
ρ-T-p ranges covered for the subcritical domain.
Substance
ρ [kg × m−3]
T [K]
p [MPa]
min
max
min
max
min
max
Ar
0
178.85835
90
140
0
3.16822
N2
0
87.29420
70
115
0
1.93704
CH4
0
61.37508
100
180
0
3.28518
CO2
0
171.96269
220
290
0
5.31772Table 2
ρ-T-p ranges covered for the transcritical domain.
Substance
ρ [kg × m−3]
T [K]
p [MPa]
min
max
min
max
min
max
Ar
0
178.85835
140
280
0
9.75263
N2
0
87.29420
115
255
0
6.39904
CH4
0
61.37508
180
320
0
9.14395
CO2
0
171.96269
290
430
0
11.84715Table 3
The calculated and the reference(∂Z/∂T)ρ at T0.
Substance
p [MPa]
(
∂
Z
/
∂
T
)
ρ  ×  103 [K−1]
Calculated
Reference
ArT0 = 140 K
0.81075631.49936902.07292602.53872002.90423203.1682270
0.98731.96812.94533.92344.91175.8890
0.98731.96812.94553.92404.91345.8933
N2 T0 = 115 K
0.47881110.89321181.24620201.54083001.78012301.9370410
1.10372.20793.31524.43495.58296.5622
1.10372.20813.31584.43645.58656.5692
CH4 T0 = 180 K
0.85981061.57963302.16919302.63854802.99776803.2851810
0.84671.68412.50783.32684.15755.1380
0.84671.68412.50793.32734.15895.1412
CO2 T0 = 290 K
1.50051402.72993003.71141404.46854005.02476305.3177280
0.69731.37832.04452.70043.35533.8461
0.69731.37832.04452.70053.35553.8472The results are assessed by means of the relative deviation (RD) and the absolute average deviation (AAD), which are computed from the following relations:(21)RD=Xcal-XrefXref×100,[%]AAD=100(M-1)N∑i=2M∑j=1N|Xcal-Xref|Xref,[%],
where Xcal is the calculated value and Xref is the reference value of Z, cv, and cp (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 (ϕ=1.0), because it exhibits the highest nonlinearity of Z with respect to T, 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).Table 4
The maximum RD in the subcritical domain.
Substance
RD [%]
Z
c
v
c
p
Ar
+0.0014−0.0001
+0.0012−0.0532
+0.0016−0.0801
N2
+0.0025−0.0000
+0.0135−0.0875
+0.0142−0.1085
CH4
+0.0021−0.0004
+0.0352−0.0482
+0.0361−0.0787
CO2
+0.0018−0.0003
+0.0073−0.0211
+0.0074−0.0562Table 5
AAD in the subcritical domain.
Substance
AAD [%]
Z
c
v
c
p
Ar
0.0001
0.0025
0.0037
N2
0.0003
0.0048
0.0061
CH4
0.0003
0.0062
0.0078
CO2
0.0005
0.0048
0.0069
Mean
0.0003
0.0046
0.0061Table 6
AAD along the saturation line.
Substance
AAD [%]
Z
c
v
c
p
Ar
0.0004
0.0093
0.0142
N2
0.0009
0.0180
0.0235
CH4
0.0009
0.0155
0.0220
CO2
0.0004
0.0093
0.0160
Mean
0.0007
0.0130
0.0189Table 7
AAD in the transcritical domain.
Substance
AAD [%]
Z
c
v
c
p
Ar
0.0031
0.0573
0.0680
N2
0.0039
0.0671
0.0672
CH4
0.0016
0.0435
0.0957
CO2
0.0009
0.0222
0.0943
Mean
0.0024
0.0475
0.0813Table 8
The uncertainty of the reference data.
Substance
Uncertainty [%]
Z
c
v
c
p
Ar
±0.02
±0.30
±0.30
N2
±0.02
±0.30
±0.30
CH4
±0.03
±1.00
±1.00
CO2
±0.03
±1.00
±1.00


## 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.



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