Assay Curve Fitting Procedures

Assay characterization divides an assay into cuts that boil at different temperatures. The average properties of the cuts (such as boiling temperature) are modelled using pseudo-components (TBPCUTS). Before a curve is cut into pseudo-components, the distillation data must be fitted to a smoothed, continuous curve that can be integrated to determine the average boiling points for the pseudo-components. This is necessary because, in general, the supplied data points do not correspond to the desired cut points.

Curve fitting also must extrapolate and interpolate the distillation data to include all the material in the range. It also is important that the lower end of the curve meshes well with the shape of any light ends that may be present. Different methods make different assumptions about the curves, and particularly affect the shape of the upper and lower end regions of the fitted curve. These options are supported in keywords by the FIT entry on the ASSAY statement. PRO/II offers four methods for fitting distillation curves.

Cubic Spline (SPLINE method)

All supplied internal points (percentages) percents are fit to a curve using a cubic spline function. A normal probability function is used to extrapolate beyond the first and last data point if necessary. When not provided, the initial point and end point are defined as the 0.01 % and 99.99 % points, respectively. If only two points are provided, the entire curve is defined with a probability fit.

This method usually results in the best possible fit for nearly all laboratory curves. Curves with step functions may cause fitting problems and should this occur the validity of the data should be checked. (see Alternative Cubic Spline below.)

Alternate Cubic Spline (default or ALTERNATE)

This is an alternative implementation of the default SPLINE method (described above). In certain situations, when partitioning an assay, the SPLINE option selects an integration block that is inappropriately large, resulting in inaccurate cut ranges.

The ALTERNATE option almost always avoids this situation, resulting in a better model of the assay. For this reason, starting in version 9.1, the ALTERNATE cubic spline is the default method in PRO/II .

Quadratic Polynomials (QUADRATIC)

Successive quadratic approximations are applied to the shape of the input curve. This method is not as elegant as the Cubic Spline method, but does exhibit more stability for curves with step functions.

Note: This method was exclusively used in the SimSci Process Simulation Program for Versions earlier than 3.01.

Probability Density Function (PDF)

PDF is a least squares probability density function which is fit to all points on the curve that are supplied. This method is recommended when errors are suspected in the distillation data. Unlike the other fitting methods, the generated curve does not actually pass through the supplied data points except as noted below. The square of the errors in fitting the supplied data to a probability shaped curve is minimized, however.

The PDF curve may be forced through the initial point and/or the end point by using the Include in PDF feature. Unless this feature is used the initial and end points are handled in similar fashion to the other points on the curve. When the Initial Point option is checked and the initial point is supplied, the PDF curve is forced through the supplied initial point. In similar fashion, the PDF curve may be forced through a supplied end point by checking the End Point option. When the Initial Point option is used and the initial point is not supplied, the curve is first extrapolated linearly to determine an initial point and this point is then held constant for the PDF fitting. Undefined end points are handled in similar fashion when the End Point option is used.

Additional Information

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