Global Model
If the Diffusion Model property is set to Global, the model equations solved are greatly simplified because the spatial information of the different quantities in the plasma reactor are treated as volume-averaged. Without the spatial derivatives the numerical solution of the equation set becomes considerably simpler and the computational time is greatly reduced. This type of model is useful when investigating a broad region of parameters with complex plasma chemistries.
To avoid unnecessary complexity in the notation the same symbols which were used to represent the space-dependent quantities are used to represent the volume-averaged quantities. In the following the different densities, mass fractions and other quantities are to be thought as volume-averaged.
For a mixture consisting of k = 1, …, Q species and j = 1, …, N reactions the mass-fraction balance equations for the first Q − 1 species is given by
where:
•
mf is the total feed mass-flow rate (SI unit: kg/s)
•
mo is the outlet mass-flow rate (SI unit: kg/s)
•
wf,k is the mass fraction of species kth in the feed (1)
•
hl is a correction factor of surface l (1)
•
Al is the area of surface l (SI unit: m2)
•
V is the reactor volume (SI unit: m3),
•
Rsurf,k,l is the surface rate expression of surface l (SI unit: mol/(m2·s)).
•
Mf,l is the inward mass flux of surface l (SI unit: kg/(m2·s)).
The sum in the last two terms is over surfaces where species are lost or created. The fourth term on the right hand side accounts for surface reactions of species kth. The last term on the right hand side is introduced because the species mass-balance equations are written in the nonconservative form and it is used the mass-continuity equation to replace for the mass density time derivative.
To take into account possible variations of the system total mass or pressure the mass-continuity equation can also be solved
(5-23).
Closed reactor
In the Reactor property it is possible to choose three different types of reactor models. If the Reactor Type is set to Closed Reactor the mass-flow feeds are set to zero
(5-24)
and Equation 5-23 is solved to take into account possible mass changes in the system caused by surface reactions.
Constant mass
If the Reactor Type is set to Constant Mass the outlet mass-flow is set such that the feed and surface reactions cannot change the mass of the system
(5-25)
and the pressure in the reactor is found in order to keep the mass-density constant
(5-26).
Constant pressure
If in the Reactor property the Reactor Type is set to Constant Pressure Equation 5-23 is not solved and the outlet mass-flow feed is set such that the mass feed and surface reactions cannot change the mass of the system
(5-27).
This condition ensures that the feed and surface reactions cannot change the mass density. However, the mass density can still change as a result of a change in the mean molar mass of the gas mixture in order to maintain a constant pressure.
Global Model Surface reactions
A large part of a successful implementation of a plasma global modal depends on how the surface losses are estimates. For positive ions the forward rate constant can be estimate to be equal to the Bohm velocity
(5-28)
where Te is the electron temperature (V).
For neutral species the forward rate constant can be estimated by
(5-29)
(5-30).
When the Motz-Wise correction option is set to On
(5-31)
and when the Motz-Wise correction option is set to Off, the forward rate constant is given via:
(5-32).
Equation 5-30 is an estimation of the diffusive losses to the wall, where Λeff is the effective diffusion length, and Dk.m is the mixture-average diffusion coefficient of species k.
Surface reactions can be adjusted using the Correction factor hl. It is common practice to correct the surface ion losses by a factor that takes into account the spatial profile of the ions. Models for the ion correction factor can be found in literature Ref. 11 and Ref. 12.