Category: COMPUTATIONS

Chemical engineers perform a wide variety of computations ranging from simple arithmetic calculations to solving partial differential equations and simulating entire process plants. Technological advances have enabled engineers to access high-performance computing machines and use advanced software for obtaining solutions rapidly. As envisioned by Leibniz, this has allowed them the freedom from time-consuming, laborious, repetitive computational…

The two components enabling the computations are the hardware—machines that perform the calculations—and the software—the instructions to run solution algorithms by the machines. Both these components are described in the sections that follow. 4.3.1 Computational Machines A counting frame, or an abacus, is one of the earliest devices and has been used for more than three…

As mentioned previously, numerical computation of an integral is needed when it is not possible to integrate the expression analytically. In other cases, discrete values of the function may be available at various points. Numerical integration of such functions involves summing up the weighted values of the function evaluated or observed at specified points. The fundamental…

The common basis for linear as well as multiple regression is the minimization of the sum of squared errors (SSE) between the experimentally observed values and the values predicted by the model, as shown in equation 4.19: In this equation, yi is the observed value, and f(xi) is the predicted value based on the presumed function f. The function can…

Some computational problems may involve calculating or obtaining derivatives of functions. Depending on the complexity of the function, it may not be possible to obtain an explicit analytical expression for the derivative. Similarly, some of the problems may involve obtaining the derivative from observed data. For example, an experiment conducted for determination of the kinetics…

The complexity of solutions for polynomial and transcendental equations increases with increasing nonlinearity. Quadratic equations can be readily solved using the quadratic formula, provided such equations can be readily rearranged in the appropriate form. Formulas exist for obtaining roots of a cubic equation, but these are rarely used. No such easy formulas are available for…

It should be clear that systems of linear algebraic equations can range in size from very small (fewer than five equations) to very large (several hundreds), depending on the number of components and complexity of operations. For example, a system consisting of four components being separated in a distillation column containing five stages yields a…

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…

The differential equations representing the behavior of the system are obtained by the application of conservation principles to a differential element. Integration of these differential equations leads to expressions that describe the overall behavior of the entire system. Many of the differential equations can be integrated analytically, yielding algebraic or transcendental equations. However, such analytical…

Properties of systems are frequently dependent on, or are functions of, more than one independent variable. Modeling of such systems leads to a partial differential equation [4]. Temperature within a rod, for example, may vary radially as well as axially. Similarly, concentration of a species within a system may depend on the location as well…