Multidimensional models

In general, comprehensive, multidimensional modeling of turbulent combustion is recognized as being difficult because of the problems associated with solving the differential equations and the complexities involved in describing the interactions between chemical reactions and turbulence. A number of computational models are available commercially that can do such work. These include FLUENT, FLOW-3D, and PCGC-2. 

The PC version runs comparatively slow on large problems. FIRAC can perform lumped parameter/control volume-type analysis but is limited in detailed multidimensional modeling of a room or gaa dome space. Diffusion and turbulence within a control volume is not modeled. Multi-gas species are not included in the equations of state. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

Multidimensional Data Intercomparisons. Estimation of reliable uncertainty intervals becomes quite complex for non-linear operations and for some of the more sophisticated multidimensional models. For this reason, "chemometric" validation, using common, carefully-constructed test data sets, is of increasing importance. Data evaluation intercomparison exercises are thus analogous to Standard Reference Material (SRM) laboratory intercomparisons, except that the final, data evaluation step of the chemical measurement process is being tested. 

The first major CFD models were those by Liu and co-workers " at the University of Miami. They are nonisothermal and the first multidimensional models. They allowed for a more in-depth study of the effects along the channels than the models described above. While the original model by Gurau et al. did not include liquid-water transport, it did have a variable water content in the membrane. To study... 

Meng, H. Wang, C. Y. Multidimensional modeling of polymer electrolyte fuel cells under current density boundary condition. Submitted for publication. 

It has been demonstrated that kg can be estimated by analogy with the Graetz-Nusselt problem governing heat transfer to a fiuid in a duct with constant wall temperature (SH= Nut) [30] and that the axial concentration profiles of NO and of N H 3 provided by the 1D model are equivalent and almost superimposed with those of a rigorous multidimensional model of the SCR monolith reactor in the case of square channels and of ER kinetics, which must be introduced to comply with industrial conditions for steady-state applications characterized by substoichiometric NH3 NO feed ratio, that is, a[Pg.401]

It must be mentioned, however, that the one-dimensional model gives only a qualitative explanation of thermal quenching. A quantitatively valid explanation can by obtained only by a multidimensional model. 

Hales, J. M A Generalized Multidimensional Model for Precipitation Scavenging and Atmospheric Chemistry, Atmos. Environ., 23, 2017-2031 (1989). 

A brief review of the development history of monolith reactor models for TWC applications can be found in Koltsakis and Stamatelos (1997). Various workers have looked at 1-, 2- and 3-dimensional models considering both the whole monolith and just a single channel. A multidimensional model for the whole monolith is required for investigating the effects of a flow maldistribution across the front face of the monolith, but is probably unnecessary when the flow is uniform. Other workers have studied multidimensional single channel models, where the gas flow within the channel is modelled in detail. In general, for a model to be useful in practice, some compromise has to be made between having a reasonable runtime versus detail/complexity, both in terms of the chemical kinetics and the description of the flow field within the channels of and across the monolith. 

Mvmmi topic 21,4 Dynamic Properties of Linear Multidimensional Models... 

For this model the master equation corresponding to Eq. (6) for the multidimensional model discussed earlier is... 

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. 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

H. Schapertons and W. Lee, Multidimensional Modelling of Knocking Combustion in SI Engines, SAE Technical Paper 850502 (1985). 

B. Natarajan and F.V. Bracco, On Multidimensional Modeling of Auto-Ignition in Spark-Ignition Engines, Comb, and Flame 57 (1984) 179. 

In this section we present an approximate solution for the rate associated with the multidimensional model, Eqs. (4.1) and (4.2). Here we focus on the equivalent one-dimensional model which disregards all modes but the reaction coordinate... 

The outlines of methods found in this book describe the use of multidimensional models in synthetic chemistry, and they can therefore be seen as methods for quantification of chemical intuition. Models derived from experiments where several factors have been jointly considered, furnish quantitative relations and discernable patterns. They must, however, be analyzed by statistical methods to ensure that they represent real phenomena and not merely spurious correlations due to random events and experimental noise. 

Gosman AD, Lekakou C, Polits S, Issa RI, Looney MK (1992) Multidimensional Modeling of Turbulent Two-Phase Flows in Stirred Vessels. AIChE J 38(12) 1946-1956... 

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