Rate equation
Rate equations describing the temporal concentration profiles of different species, for example in coupled chemical reactions, or in the presence and Abregungsprozessen various levels in atoms or molecules:
With
The rate equations are, in general, to a system of coupled, stiff, non-linear first order differential equations for which the block conservation must apply. In the stationary case, there is the law of mass action.
- 4.1 derivation
- 4.2 Example of the matrix
Ratenkoeffizenten
The reaction rate coefficients appearing in the rate equations can (see also Plasma Physics: Thermal equilibrium ) commonly known as arbitrary functions of the corresponding, possibly time- dependent temperature are considered. In general, rates of coefficients have to be taken for chemical processes of heavy particles from the literature ( " rate constant ' a chemical reaction ), the rate coefficient for the electron impact induced processes can be obtained with the aid of the Electron Kinetics.
The kinetic basis for treatment of the electrons, both used to calculate rates of such coefficients, as well as electronic transport processes (electric conductivity ) is the Boltzmann equation for the electron energy distribution.
Example
Hydrogen oxidation
For clarity, the hydrogen oxidation is used:
Dissociated part
The rate equations ( equation 1 ) for the five species are:
The concentrations of the species:
Numerical solution methods
Since this is a system of stiff differential equations in the rate equations, is a procedure with the greatest possible stability region is forced to choose, so that the integration steps are not too small. The cheapest are A- stable methods.
For the rate equations means ' stiff ' that differ greatly, the time constants of the various species: In relation to other some concentrations change only very slowly. Two examples of absolutely rigid - stable integration methods are the implicit trapezoidal method and the implicit Euler method, as well as some BDF methods (backward differentiation formula ) are suitable.
Block conservation
The principle of conservation module provides a way to verify the quality of the numerical solutions because it is at all times:
In which
Derivation
A species i, written here as is in this case from the building blocks as follows:
Used in the rate equation ( equation 1 ) and summing over all species, provides for the above module preservation.