Numerical Solution of the Direct Problem in Chemical Kinetics
Previously discussed analytical methods for solving the direct problem in chemical kinetics are not sufficient for analysis of different reaction kinetic schemes. First, even given the mathematical model represented by an ODE system, it is not always poss
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Numerical Solution of the Direct Problem in Chemical Kinetics
3.1
Given/Odesolve Solver in Mathcad System
Among the built-in tools of the Mathcad suite an important role belongs to those designed for the numerical solving ordinary differential equations (ODEs) and their systems. Let’s look through these tools starting with the solving unit Given/ Odesolve. The Odesolve function first appeared in Mathcad 2000 Pro. On default this function used Runge–Kutta method of the fourth-level of precision.1 The organization of the Given/Odesolve solver resembles greatly that of the Given/Find solving block: it starts with the Given keyword. An ODE or a system as well as the initial conditions should be placed in the solver body. The solving is performed with a call up of built-in function Odesolve using the following format:
Odesolve(x, b, [steps]), where x is the unknown, and b is the upper integration limit. The last parameter, steps, determines the number of integration steps and is optional. If this parameter is absent, the number of steps is set up automatically. As an example, let’s calculate a kinetic curve for an intermediate in a consecutive reaction where the second step is of the second order: k1
A ! B; k2
2B ! P: Analytical solution of this problem was discussed in the Sect. 1.3.2, so we can compare numerically- and analytically-obtained results. The way of using Given/Odesolve solver and the results of its work are shown in Fig. 3.1. The following values were used: initial concentration A ¼ 0.5 M,
1
In this book we will not discuss the essence of such and such numerical integration methods. This information is available in virtually all handbooks on numerical methods.
V.I. Korobov and V.F. Ochkov, Chemical Kinetics with Mathcad and Maple, DOI 10.1007/978-3-7091-0531-3_3, # Springer-Verlag/Wien 2011
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3 Numerical Solution of the Direct Problem in Chemical Kinetics
Fig. 3.1 Kinetic curve calculation for an intermediate in a consecutive second-order reaction using Odesolve function (on-line calculation http://twt.mpei.ac.ru/MCS/Worksheets/Chem/ChemKin-3-01.xmcd)
rate constants k1 . Referring to Fig. 3.1, the kinetic curve calculated numerically (dots) matches the curve calculated using analytical formula (line). The authors want to point out that during the assigning of the results of numerical computation one should use only the name of the desired function (without argument). For the construction of a graph both function name and its argument are specified (Fig. 3.1). The results of this example allow one to think that the Odesolve function is a sufficient tool for solving the direct kinetic problem. Now we proceed to a numerical solution of an ordinary differential equation (ODE) set using a solver. Solving process for an ODE set does not differ much from that for a single ODE: all equations in the set as well as the starting condition equalities are given in the solver. The only difference is in the number of arguments that Odesolve function should have. An array of desired function names is
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