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Solution of the Adiabatic Sound Wave Equation as a Nonlinear Least Squares Problem

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Abstract

A new method for deriving accurate thermodynamic properties of gases and vapors (the compression factor and heat capacities) from the speed of sound is recommended. A set of PDEs connecting speed of sound with other thermodynamic properties is solved as a nonlinear least squares problem, using a modified Levenberg–Marquardt algorithm. In supercritical domain, boundary values of compression factor are imposed along two isotherms (one slightly above Tc and another several times Tc) and two isochores (one at zero density and another slightly above ρc). In subcritical domain, the upper isochore is replaced by the saturation line, the lower isotherm is slightly above the triple point, and the upper isotherm is slightly below the critical point. Initial values of compression factor inside the domains are obtained from the boundary values by a cubic spline interpolation with respect to density. All the partial derivation with respect to density and temperature, as well as speed of sound interpolation with respect to pressure, is also conducted by a cubic spline. The method is tested with Ar, CH4 and CO2. The average absolute deviation of compression factor and heat capacities is better than 0.002 % and 0.1 %, respectively.

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Authors and Affiliations

  1. Department of Chemical Engineering, Faculty of Technology, University of Tuzla, Univerzitetska 8, 75000, Tuzla, Bosnia and Herzegovina

    Muhamed Bijedić & Sabina Begić

Authors
  1. Muhamed Bijedić
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  2. Sabina Begić
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Corresponding author

Correspondence to Muhamed Bijedić.

Electronic supplementary material

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Supplementary material 1 (TXT 68 kb)

Appendix

Appendix

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Table 1 ρ–T–p ranges covered in supercritical domain
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Table 2 ρ–T–p ranges covered in subcritical domain
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Table 3 Other data for supercritical domain
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Table 4 Other data for subcritical domain
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Table 5 Average absolute deviation (AAD) in supercritical domain
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Table 6 Average absolute deviation (AAD) in subcritical domain
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Table 7 AAD in supercritical domain when boundary conditions are perturbed + 0.1 %/− 0.1 %
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Table 8 AAD in subcritical domain when boundary conditions are perturbed + 0.1 %/− 0.1 %
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Table 9 AAD in supercritical domain when speed of sound data are perturbed + 0.1 %/− 0.1 %
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Table 10 AAD in subcritical domain when speed of sound data are perturbed + 0.1 %/− 0.1 %
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1.1 Argon in Supercritical Domain

See Figs. 1, 2, 3, 4, 5 and 6.

Fig. 1
figure 1

Relative deviation of Z versus ρ with respect to [18]

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Fig. 2
figure 2

Relative deviation of Z versus T with respect to [18]

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Fig. 3
figure 3

Relative deviation of cv versus ρ with respect to [18]

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Fig. 4
figure 4

Relative deviation of cv versus T with respect to [18]

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Fig. 5
figure 5

Relative deviation of cp versus ρ with respect to [18]

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Fig. 6
figure 6

Relative deviation of cp versus T with respect to [18]

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1.2 Argon in Subcritical Domain

See Figs. 7, 8, 9, 10, 11 and 12.

Fig. 7
figure 7

Relative deviation of Z versus ϕ with respect to [18]

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Fig. 8
figure 8

Relative deviation of Z versus T with respect to [18]

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Fig. 9
figure 9

Relative deviation of cv versus ϕ with respect to [18]

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Fig. 10
figure 10

Relative deviation of cv versus T with respect to [18]

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Fig. 11
figure 11

Relative deviation of cp versus ϕ with respect to [18]

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Fig. 12
figure 12

Relative deviation of cp versus T with respect to [18]

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1.3 Methane in Supercritical Domain

See Figs. 13, 14, 15, 16, 17 and 18.

Fig. 13
figure 13

Relative deviation of Z versus ρ with respect to [19]

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Fig. 14
figure 14

Relative deviation of Z versus T with respect to [19]

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Fig. 15
figure 15

Relative deviation of cv versus ρ with respect to [19]

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Fig. 16
figure 16

Relative deviation of cv versus T with respect to [19]

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Fig. 17
figure 17

Relative deviation of cp versus ρ with respect to [19]

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Fig. 18
figure 18

Relative deviation of cp versus T with respect to [19]

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1.4 Methane in Subcritical Domain

See Figs. 19, 20, 21, 22, 23 and 24.

Fig. 19
figure 19

Relative deviation of Z versus ϕ with respect to [19]

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Fig. 20
figure 20

Relative deviation of Z versus T with respect to [19]

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Fig. 21
figure 21

Relative deviation of cv versus ϕ with respect to [19]

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Fig. 22
figure 22

Relative deviation of cv versus T with respect to [19]

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Fig. 23
figure 23

Relative deviation of cp versus ϕ with respect to [19]

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Fig. 24
figure 24

Relative deviation of cp versus T with respect to [19]

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1.5 Carbon Dioxide in Supercritical Domain

See Figs. 25, 26, 27, 28, 29 and 30.

Fig. 25
figure 25

Relative deviation of Z versus ρ with respect to [20]

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Fig. 26
figure 26

Relative deviation of Z versus T with respect to [20]

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Fig. 27
figure 27

Relative deviation of cv versus ρ with respect to [20]

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Fig. 28
figure 28

Relative deviation of cv versus T with respect to [20]

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Fig. 29
figure 29

Relative deviation of cp versus ρ with respect to [20]

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Fig. 30
figure 30

Relative deviation of cp versus T with respect to [20]

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1.6 Carbon Dioxide in Subcritical Domain

See Figs. 31, 32, 33, 34, 35 and 36.

Fig. 31
figure 31

Relative deviation of Z versus ϕ with respect to [20]

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Fig. 32
figure 32

Relative deviation of Z versus T with respect to [20]

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Fig. 33
figure 33

Relative deviation of cv versus ϕ with respect to [20]

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Fig. 34
figure 34

Relative deviation of cv versus T with respect to [20]

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Fig. 35
figure 35

Relative deviation of cp versus ϕ with respect to [20]

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Fig. 36
figure 36

Relative deviation of cp versus T with respect to [20]

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Bijedić, M., Begić, S. Solution of the Adiabatic Sound Wave Equation as a Nonlinear Least Squares Problem. Int J Thermophys 40, 15 (2019). https://doi.org/10.1007/s10765-018-2479-8

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