Multidimensional simulations, reduced

B. Using Reduced Chemistry Models in Multidimensional Simulations without Introducing Error... 

Plasma reactor simulations range from zero-dimensional (well-mixed) to three-dimensional. Well mixed [104-106] and one-dimensional models (including plug flow models [107, 108]) are best for sorting out the complicated gas and surface chemistry to arrive at a reduced reaction set for use in multidimensional simulations. Two-dimensional simulations can address the important aspect of reaction uniformity across the wafer radius. Three-dimensional simulations are useful for studying azimuthal asymmetries in the reactor due to non-axisymmetric power deposition, or non-axisymmetric gas inlets and pumping ports [109, 110]. 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

Pakalapati, S., Yavuz, I., Elizalde-Blancas, F. and Celik, I. (2006) Comparison of a multidimensional model with a reduced order pseudo three-dimensional model for simulation of solid oxide fuel ells, in Proceedings of the 4th International ASME Conference on Fuel Cell Science, Engineering and Technology, Irvine, CA, June 19-21. 

In [541] the mode of action of free jets in tubular reactors was investigated in connection with a simple fast chemical reaction (neutralization of NaOH with HCl) by a multidimensional numerical simulation of the turbulent flow field with superposition of the reaction kinetics. The study succeeded in describing the l/d = /(Cadd/Caicaii) relationship (where l/d is the length of the jet core, at which complete decoloration occurred, related to the propulsion jet diameter d Cacid/ aicaii is the concentration ratio of acid in the main flow to alkali in the jet core at the moment of complete decoloration and hence is an excess ratio). From this the smallest length of the jet core was determined, which for the reaction being considered may not be reduced. 

When dealing with multidimensional geometries and complex chemistries, fullblown self-consisted plasma simulations pose a very challenging task. While brute force simulations (for example ones that solve the complete set of conservation equations for all particles) are feasible, they are also labor and time consuming. Judicious approximations, based on the physics of the problem, can reduce the simulation times dramatically. Examples are the space-time averaging used in the non-local approach [58, 59], or the approximations used to construct rapid plasma simulation tools [59, 101, 148, 152, 153]. 

Salis and Kaznessis proposed a hybrid stochastic algorithm that is based on a dynamical partitioning of the set of reactions into fast and slow subsets. The fast subset is treated as a continuous Markov process governed by a multidimensional Fokker-Planck equation, while the slow subset is considered to be a jump or discrete Markov process governed by a CME. The approximation of fast/continuous reactions as a continuous Markov process significantly reduces the computational intensity and introduces a marginal error when compared to the exact jump Markov simulation. This idea becomes very useful in systems where reactions with multiple reaction scales are constantly present. 

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