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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…

Modeling—developing a set of governing equations—of systems of interest to chemical engineers often starts with defining a differential element of the system. This differential element is a subset of the larger system, but with infinitesimally small dimensions. All the processes and phenomena occurring in the larger system are represented in the differential element. The modeling approach involves…

Many of the equations in chemical engineering involve functions of variables more complex than simple powers. An equation containing exponential, logarithmic, trigonometric, and other similar functions is not amenable to solution by algebraic means—that is, by simple addition, multiplication, or root extraction operations. Such equations “transcend” algebra and are called transcendental equations [4]. Equation 4.7, the Nikuradse equation, often…

Algebraic equations comprise the most common group of problems in chemical engineering. Linear algebraic equations are algebraic equations in which all the terms are either a constant or a first-order variable [1]. The straight line is represented by a linear algebraic equation. Linear algebraic equations are often encountered in phase equilibrium problems associated with separation processes. Figure 4.1 is…

Chemical engineers deal with a multitude of equations ranging in complexity from simple linear equations to highly involved partial differential equations. The solution techniques accordingly range from simple calculations to very large computer programs. The classification of the problems based on the mathematical nature is presented in the following sections.