Polynomial curves

The results obtained by Katz et al. [15] are shown as experimental points on the curves relating the distribution coefficient of the solute against volume fraction of methanol added to the original mixture in Figure 31. Due to the difficulty of measuring the distribution coefficient of each solute between pure water and hexadecane (because of their extremely high retention), the values were obtained from a polynomial curve fit to the data which gave a value for (K) at a = 0. 

It is now necessary to identify the correct functions of the capacity factor (k") to be used in equation (12) to evaluate the diffusivity. Knox [7] suggested the following approach which required a polynomial curve fitting procedure to identify the necessary constants. Rearranging equation (12),... 

Kragten, J., Least-Squares Polynomial Curve-Fitting for Calibration Purposes (STATCAL-CALIBRA), Ana/yrica Chimica Acta 241, 1990, 1-13. 

In engineering practice we are often faced with the task of fitting a low-order polynomial curve to a set of data. Namely, given a set of N pair data, (y , x ), i=l,...,N, we are interested in the following cases,... 

The best and easiest way to smooth the data and avoid misuse of the polynomial curve fitting is by employing smooth cubic splines. IMSL provides two routines for this purpose CSSCV and CSSMH. The latter is more versatile as it gives the option to the user to apply different levels of smoothing by controlling a single parameter. Furthermore, IMSL routines CSVAL and CSDER can be used once the coefficients of the cubic spines have been computed by CSSMH to calculate the smoothed values of the state variables and their derivatives respectively. 

This expression can be modified to apply directly to any of various techniques used to measure the interaction parameter, including membrane and vapor osmometry, freezing point depression, light scattering, viscometry, and inverse gas chromatography [89], A polynomial curve fit is typically used for the concentration dependence of %, while the temperature dependence can usually be fit over a limited temperature range to the form [47]... 

Fit a polynomial curve of the desired type (degree) to the data, using least-square curve fitting. 

Details can be found in the references at the end of this chapter. Where relationships are not straight lines, polynomial curve fitting... 

Muj taba, I. M. and S. Macchietto. Efficient Optimization of Batch Distillation with Chemical Reaction Using Polynomial Curve Fitting Techniques. Ind Eng Chem Res 36 2287-2295 (1997). 

Specialty calculations for gel permeation chromatography (GPC) where the integrator would calculate the molecular weight distribution, molecular weight average, and polynomial curves for calibration. 

Equilibrium combustion product compositions and properties may be readily calculated using thermochemical computer codes which minimize the Gibbs free energy and use thermodynamic databases containing polynomial curve-fits of physical properties. Two widely used versions are those developed at NASA Lewis (Gordon and McBride, NASA SP-273, 1971) and at Stanford University (Reynolds, STANJAN Chemical Equilibrium Solver, Stanford University, 1987). 

According to the DFA method, the time series y t) is first integrated and then divided into boxes of equal length, At. In each box, a least squares line (or polynomial curve of order /, DFA-/) is then fitted, in order to detrend the integrated time series by subtracting the locally fitted trend in each box. The root-mean-square (rms) fluctuation Fd(At) of this integrated and detrended time series is calculated over all timescales (box sizes). 

Instead of using a triangular weight function, the Savitsky-Golay (2) method uses a selectable nonlinear function on each side of the central point. The use of an odd number of points remains the same, but each side of the center weight is fitted with a polynomial curve, which can be a quad-... 

It is well known that using an exponential or power function can also describe the portion of a polynomial curve. Indeed, these types of functions, which can represent the relationships between the process variables, accept to be developed into a Taylor expansion. This procedure can also be applied to the example of the statistical process modelling given by the general relation (5.3) [5.20]. 

In turn, the values of d can be converted to the radial location in the pipe so that the velocity profile in a pipe can be obtained. The velocity profile is used to calculate velocity gradients (shear rates), (dv/dr), at specific locations using an even-order polynomial curve fit to the velocity data. 

Resolution of the velocity data and removal of data points near the center of the tube which are distorted by noise aid robustness of the curve fit the polynomial curve fit introduced a systematic error when plug-like flow existed at radial positions smaller than 4 mm in a tube of 22 mm diameter. The curve fit method correctly fit the velocity data of Newtonian and shear-thinning behaviors but was unable to produce accurate results for shear-thickening fluids (Arola et ah, 1999). 

Grenier and Westrura (6) measured low temperature heat capacities from 5 to 346 K. The adopted Cp are derived from their experimental heat capacities by a polynomial curve fitting technique. The derived entropy, S°(298.15 K), is 17.576 l 0.02 cal K ... 

We have adopted the Cp data of Turdakin (60-295 K) and Shmidt (307-600 K) which are smoothly Joined at 298.15 K by a polynomial curve fitting technique. The drop calorimetric data of Southard in this region 381 to 600 K are less reliable since... 

A polynomial curve fitting technique is used for the Pankratz et al. (21) and Levinson (17) data, with the above mentioned constraints, to obtain a smooth joining of the results near 298 K and to provide an equation suitable for extrapolation to... 

We have fitted a smooth polynomial curve through their data and obtained S (298.15 K) 0.021 cal... 

The adopted Cp values for T > 298.15 K are obtained by a combination of graphical and polynomial curve fitting techniques. A... 

Fig. 7.20. Plot of absolute electrophoretic mobility versus gel concentration for a typical data set, derived here from 23 separate analyses of Mycophage PSi-III, a double-stranded linear RNA. The curve represents the second-order polynomial curve derived from this data set by regression analysis.

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