Chemistry Reference
In-Depth Information
discretized by dividing the solution region into a finite number of elements that are
equipped with basis functions of the finite element space. When expressing the test
functions and the unknown solution function as a linear combination of the basis
functions and when employing the time step discretization as described earlier, a
system of equations to determine the values of the solution function at the nodes for
each time step is obtained. The time step constraints of the FEM are similar to the
FDM. An advantage of the FEM is its high flexibility in node placement anywhere
in the solution region to make optimal use of geometric and physical properties of
the system. The method is used for example, in the software package COMSOL
Multiphysics [25].
In the finite volume method, the function values are calculated at discrete
locations on a meshed geometry similar to FDM [26]. Therefore, the solution domain
is divided into nonoverlapping small control volumes surrounding each node point
on the mesh and the partial differential equation is integrated over each control
volume. Volume integrals in the partial differential equation that contain a diver-
gence term are converted to surface integrals using the divergence theorem of Gauß
and derivatives of higher order are replaced by difference quotients analogous to
FDM. This procedure results in a system of algebraic equations for the determi-
nation of the cell averages of the unknown function values within each control
volume. From a physical point of view, FVM provides the most natural scheme of
discretization for conservation laws because it makes use of the integral form of
the conservation law and ensures conservation of the quantities considered. Another
advantage of FVM is that unstructured meshes and complex domains can easily be
handled since each control volume is treated independently from its neighboring
volume as in FEM. The method is used in many computational fluid dynamics pack-
ages like ANSYS CFX [27], CFD-ACE+ [28], Fluent [29], PHOENICS [30], and
STAR-CD [31].
9.1.5.2 Solution of the Boltzmann Equation
With knowledge of the external force and collision cross sections, the Boltzmann
equation (9.14) can be used to determine the velocity distribution function
f
α
)
of species α. Depending on the complexity of the plasma physics, an equation with up
to seven dimensions has to be solved for each species. For weakly ionized plasmas, the
densities of the charge carriers are low in comparison with the densities of the neutral
particles. Thus, collisions of charge carriers with neutral particles have marginal
influence on the velocity distribution function of neutral particles, which possess a
Maxwellian velocity distribution
(
x
,
v
,
t
m
α
2π
k
B
T
α
3
/
2
exp
m
α
2
k
B
T
α
2
f
α
(
x
,
v
,
t
)
=
n
α
(
x
,
t
)
−
(
v
−
u
α
)
(9.47)
because of their intensive energetic contact in elastic collisions.
A standard technique for solving the Boltzmann equation of the electrons
(α
=
e
) consists in an orthogonal expansion of the velocity distribution function
