An extension of the spline fit technique and applications to thermodynamic data

Abstract

T h e spline fit technique i s extended to permit the smoothing of experimental doto. It i s applied to thermodynamic data which are otherwise difficult to treat.There is always a need for methods to smooth, interpolate, differentiate, or otherwise treat experimental data. Sometimes equations are available from theoretical considerations which contain a certain number of undetermined constants whose best values may be calculated from the experimental data. In other c a s e s empirical equations have come into u s e where experience has shown them to correlate adequately a certain kind of experimental data. A familiar example i s the use of the Antoine equation to correlate vapor pressure data a s a function of temperature (1). In the absence of either a theoretical or an empirical correlating equation, polynomials have frequently been used. Experience has shown that they often correlate experimental data very well. Moreover, if the polynomial coefficients are viewed a s the unknown quantities to be determined from experimental data, the fitting procedure is linear, a distinct computational advantage. There are a number of numerical procedures for this type of calculation.One of the most powerful is that due to Forsythe (2). This has been generalized and applied to various types of thermodynamic data (3).T o be acceptable, a representation should be reasonably low-ordered, correlate the data to within the experimental error, and introduce no unwarranted inflection points. The latter is particularly important if the data are to be differentiated. Unfortunately there are a number of instances where the search for an adequate polynomial representation of the complete s e t of data has been unsuccessful. Certain alcohol-hydrocarbon heat of mixing and vapor pressure data as a function of liquid mole fraction are of this type. The difficulty in representing these data seems to be related to the steepness of the required curves in one or more parts of the composition interval a s compared with other parts. THE SPLINE FIT TECHNIQUEOne way to circumvent these problems is to use the spline fit technique which i s discussed by Landis and Nilsen ( 4 ) . This method puts a different cubic between every two successive data points such that the curves p a s s exactly through each data point and that the first two derivatives of the curve on the right-hand side of a data point are equal, respectively, to the first two derivatives of the curve on the left-hand side of the data point, all derivatives evaluated a t the data point.Since some of the same equations apply in the extended method to be presented later, the derivations of the equations of the original method are summarized below.Since the curves in every interval* are cubics, the second derivative in a given interval is. linear; that is *The k'h interval in these derivations is taken t o b e Xk sx < X k + l , where xk is the independent variable for the kch data point,

Vapor‐Liquid equilibrium: Part I. An appraisal of data reduction methods

“…The feasibility of vapor-liquid equilibrium (VLE) determination from measurements at constant temperature of just the solution vapor pressure as a function of liquid composition has already been shown (Ljunglin and Van Ness, 1962;Van Ness et al, 1967), and an apparatus has been developed (Gibbs and Van Ness, 1972) for the rapid accumulation of such data. The question arises as to whether this method, which does not require measurement of vapor compositions, is in some way deficient in comparison with methods based on measurement of all the variables P, x, and y.…”

Least Square Polynomial Spline Approximation

“…Plots of the analogous approximations for another data set of similar interest (cf. [9]) are given in Fig. 2.…”

Two methods of selecting smoothing splines applied to fermentation process data

“…The use of smoothing splines, especially cubic splines, is widespread in the analysis of process and laboratory data (Klaus and van Ness, 1967;Dunfield and Read, 1972;Tao and Watson, 1988;Steemson and White, 1988). Oner et al (1986) reported the use of cubic spline functions in the analysis of data from anaerobic fermentation processes.…”