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

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.