Differential coupling equations, hydrogen

Simultaneous solution of the so-called differential-algebraic equation (DAE) set requires coupling of the ODE and algebraic equation solvers, the latter which are not discussed here, but can be found in detail elsewhere [ 1 ]. Description of a DAE set and its solution in the context of a one-dimensional (ID) heterogeneous packed-bed reactor model for autothermal conversion of methane to hydrogen is available in the literature [7]. It is also worth noting that packages such as DASSL and DAEPACK are also available for the solution of coupled DAE sets. 

In this bimolecular reaction the stoichiometric coefficients are equal to one, meaning that one hydrogen atom collides with one oxygen molecule. Once the reaction mechanism and all the rate constants for the elementary reactions are known, the reaction rates for all species are given by a simple set of coupled first-order differential equations. These equations can be solved quite easily on a computer, and give the concentrations of all species as a function of time. These results may then be compared with experimental results. 

We may anticipate here that all what has been said with respect to finite-nucleus effects for hydrogen-like atoms also holds for many-electron atoms to be discussed in chapter 9. This is possible as we shall see because one-electron functions are determined by solution of coupled first-order differential equations of the type of the Dirac equation for a single electron. In the case of... 

The coupled differential equations are solved numerically by the BD method implemented in the software LSODE. The main program and the model subroutine written in Fortran 90 for the toluene hydrogenation process are listed below ... 

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