Regression Analysis and Interpolation

Chemical engineers routinely collect discrete data through various experiments, which they further use for design, control, and optimization. This often requires obtaining the value of the function (or dependent variable) at some value of the independent variable within the domain of experimental data where direct measurement is not available. Regression analysis involves fitting a smooth curve that approximates the data, yielding a continuous function [4]. It is then possible to interpolate—obtain the function value at any intermediate value of the independent variable. It is also possible to extrapolate—obtain the function value at a value of the independent variable that is outside the data range used for regression analysis. Linear regression involves approximating the data using a straight line, whereas nonlinear regression involves using polynomial or transcendental functions for the same purpose. Multiple regression involves performing regression analysis involving two or more independent variables that determine the value of the function. For example, equation 4.10 can be integrated to obtain the following mathematical relationship between concentration and time:

To determine the rate constant k, experiments are conducted obtaining the concentration-time data and a linear regression carried out between ln(C_A) and t, as shown in Figure 4.7.

Figure 4.7 Example of linear regression for determination of rate constant.

It can readily be seen that a chemical engineer must have skills to deal with and solve problems ranging from simple arithmetic calculations to those requiring highly sophisticated and involved algorithms. Further, the solution must be obtained fairly rapidly for the individuals and organizations to maintain their competitive edge and respond to changing conditions. Section 4.2 presents a brief overview of solution algorithms developed for numerical solutions of different types of problems. Section 4.3 describes the different tools including the machines and software available to chemical engineers to perform these computations.