Recent Nonlinear Fitting Methods

The next evolution in the development of equations of state was the jump from the quasi-nonlinear algorithm to fully-nonlinear methods. Nonlinear fitting has 

A reasonable preliminary equation is required as a starting point for nonlinear fitting. The exponents for density and temperature, along with the coefficients and exponents in critical region terms, are determined simultaneously with the coefficients of the equation. In addition, the terms in the ideal gas heat capacity equation and the reducing parameters (critical temperature and density) of the equation of state can also be fitted. Thus, with an 18-term equation, there are at times up to 90 values being fitted simultaneously to derive the equation. 

The nonlinear algorithm adjusts the parameters of the equation of state to minimize the overall sum of squares of the deviations of calculated properties from the input data, where the residual sum of squares is represented as 

Other fitting techniques and criteria that can be used include proper handling of the second and third virial coefficients, elimination of the curvature of low temperature isotherms in the vapour phase, control of the two-phase loops and the number of false two-phase solutions, convergence of the extremely high temperature isotherms to a single line, and proper control of the ideal curves with, for example, the Joule inversion curve that will be discussed later. The work of Lemmon and Jacobsen for pentafluoroethane and Lemmon et al. for propane describe the properties that can be added to the sum of squares so that the equation of state meets the criteria. 

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Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

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