Roots of a Single Nonlinear Equation

Many engineering problems require the solution of a single nonlinear equation. Such an equation can always be cast into the form 

The objective of this chapter is to study methods and learn of Excel tools for finding the root(s) of a nonlinear equation, that is, for finding x such that/(x) = 0. 

Simple algebra provides the root for a linear equation. However, for more complex (nonlinear or transcendental) equations, it is often the case that no analytical solution is available, or is difficult to obtain, so that numerical methods must be used. 

An example of a nonlinear equation is the van der Waals equation of state, which 

A typical problem is to find the molar volume, V, given the temperature and pressure (and the type of gas). While it is possible to find analytic solutions to the van der Waals equation of state for V (this is simply a cubic equation), a numerical solution is often preferred. The equation of state in the form/(x) = 0 is obtained by simple rearrangement  

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Solving problems in chemical engineering and science often requires finding the real root of a single nonlinear equation. Examples of such computations are in fluid flow, where pressure loss of an incompressible turbulent fluid is evaluated. The Colebrook [8] implicit equation for the Darcy friction factor, f, for turbulent flow is expressed... 

There are many problems which require the finding of the root of a single nonlinear equation in one unknown. For example, the heat capacity of carbon dioxide is given as a function of temperature as... 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

A fairly thorough look has been taken at Newton s method for solving a single nonlinear (or linear) equation for a root (or roots) of a function. If a function has multiple roots, Newton s method must be started close to each root in order to find more than one solution. It is also seen that for some formulation of a physical problem, it is essentially impossible to get Newton s method to converge to a solution while a rearrangement of the basic equation can rapidly lead to valid solutions with Newton s method. The more one knows about a particular physical problem... 

In contrast, nonlinear equations that contain a single unknown variable may have any number of real roots (as well as imaginary and complex roots). For example,... 

Equation 1-81 is highly nonlinear and must be solved for 0, the Underwood parameter, or the root of the equation. It has a singularity at each value of 0 = aj. If we can determine with some precision the degree of vaporization of the feed and the distillation composition, then we can also obtain a single value of 0 by an iterative solution. The value of 0 is generally bounded between the relative volatilites of the light and heavy keys. 

The purpose of this paper is to describe steady flow of water and transport of solutes across single and series arrays of arbitrary numbers of membranes. Differential forms of the flow and transport equations are used as the point of departure and from these the incremental forms are derived. This theory allows one to state concisely some general properties of series arrays of membranes, with regard to nonlinearity, polarity, and changes of the ordering of individual membranes. This study is motivated by the problems of flow of water and transport of solutes in clay soils [1] and of simultaneous uptake of water and solutes by plant roots [2], Some of the conclusions are generalizations of results obtained earlier for special cases see [3] for a detailed evaluation of the literature. 

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