Numerical Solution of Nonlinear Equations in One Variable

Here is the mathematical problem you must solve Given a set of chemicals, temperature and pressure, find the specific volume of the mixture. To do this, you must find the critical temperature and pressure of each chemical. Once you have the parameters, you must solve the cubic equation, Eq. (2.8), which is a nonlinear equation in one variable. Because it is a cubic equation, it is possible to find the solution in a series of analytical steps (Perry and Green, 1997, p. 3-114), but this is not usually done because it is quicker to find the solution numerically, albeit iteratively. 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

Equation (8.25) represents a set of partial differential equations (PDEs) that must be solved in one space dimension and propagated in time. Typically, one discretizes the space variable 0 < x < 1 into n intervals. If the equations are linear, one can write difference equations at each point and solve the resulting matrix equation. The solution is progressed in time by also discretizing the time derivative with a Crank-Nicholson-like scheme. If the equations are nonlinear, the situation is more complicated. The appendix of Electrochemical Systems (Newman and Thomas-Alyea [25]) gives a discussion of numerical solution of partial differential equations. 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

The well mixed hypothesis for the chemostat does not allow a nutrient gradient to be generated. A basic tenet is that the nutrient concentration is the same everywhere hence any advantage in nutrient consumption is present everywhere. The model that incorporates a true gradient would be one involving partial differential equations a new variable, space, must be accommodated. Systems of nonlinear partial differential equations are difficult mathematical objects to understand and analyze. Even numerical solutions pose added and significant difficulties. Moreover, even if an experimental gradient is devised, measurements that do not disturb the local environment take on new difficulties. 

The numerical method of lines was used to solve linear and nonlinear elliptic partial differential equations in section 6.1.7. This method involves using finite differences in one direction and solving the resulting system of boundary value problems in y using Maple s dsolve numeric command. This method provides a numerical solution for both the dependent variables and its derivative in the y-direction. 

With the limitations and the problems associated with both the perturbation analysis and the one-dimensional models, the full nonlinear equations of motion for the jet are solved numerically. One such solution is by Ashgriz and Mashayek [75]. They studied the temporal instability of an axisymmetric incompressible Newtonian liquid jet in vacuum and zero gravity. The variables are nondimensio-nalized by the radius of undisturbed jet, a, and a characteristic time (pa" jof. ... 

One approach to solving this equation is to specify a reactor temperature T and then solve for the corresponding exit concentration c that satisfies f(c, Tjjj) = 0. Solution of f, given a value for T, involves the solution of a nonlinear function in a single variable (c in this instance). This may be achieved with any suitable numerical nonlinear solver. Figure 7.27(a) shows a representative plot of f versus c, when T is specified at 450 K. 

A brief discussion is perhaps in order on the selected form for representing the equation set as in Eq. (10.25). Some numerical packages used to solve differential equations require that a set of functions be defined that return the derivative values of the differential equations when evaluated. This has several disadvantages. First such a formulation does not allow one to have a set of equations expressed in terms of combinations of the derivative terms. Second, the general case of nonlinear derivative terms or transcendental functions precludes such a simple formulation. Lastly, some important sets of equations, as discussed later, are formulated in terms of a combined set of differential and algebraic equations where some equations do not have a derivative term. These important cases can not be handled if the defining equation formulism simply requires the return of the derivative value. The form selected for representation here has no such limitations and is in keeping with other problem formulations in this work where equations were coded such that the function returns zero when satisfied by a set of solution variables. 

Incorporation of the superposition approximation leads inevitably to a closed set of several non-linear integro-differential equations. Their nonlinearity excludes the use of analytical methods, except for several cases of asymptotical automodel-like solutions at long reaction time. The kinetic equations derived are solved mainly by means of computers and this imposes limits on the approximations used. For instance, we could derive the kinetic equations for the A + B — C reaction employing the higher-order superposition approximation with mo = 3,4,... rather than mo = 2 for the Kirkwood one. (How to realize this for the simple reaction A + B —> B will be shown in Chapter 6.) However, even computer calculations involve great practical difficulties due to numerous coordinate variables entering these non-linear partial equations. 

Nonlinear superposition. Very often in pressure transient testing, the pressure (or flow rate) is changed in time for liquids, flow rate (or pressure) response is obtained by linear superposition of elementary solutions. For gases, superposition is not possible because nonlinear solutions are not linearly additive. How does one calculate the response when pressure or flow rate at the well vary, say stepwise, in time Fortunately, the governing equations can be numerically integrated with respect to t. It remains for us to represent stepwise changes in any particular variable using convenient mathematical devices. 

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