Two-dimensional treatment

Systems that require two-dimensional treatment are sensitive to the parameters in the model, and, as a result, the transport coefficients (ke and De) must be well known. Consequently, a one-dimensional model is usually used for preliminary process design,... 

The factor (3/5) differs from that in the original paper of Kuhn and Kuhn (159). In this paper unfortunately a factor (1/2) is given, due to an approximate two-dimensional treatment. 

Wherever a two-dimensional treatment is effective in imparting the knowledge, it is used instead of a three-dimensional treatment, to ensure that the material is more understandable. 

A two-dimensional treatment has been presented. The model is currently being extended to three dimensions to account for surface flaws in a three-dimensional material subjected to a drawing process and bulk flaws or dispersed phase particles in a three-dimensional matrix. 

A qualitative description of proton dynamics in H-bonded crystals requires a two-dimensional treatment assuming strong coupling between the proton-transfer coordinate and a low-frequency vibration. Due to environmental effects, this coupling is much stronger in crystals than in isolated H-bonded species. DFT calculations with periodic boundary conditions show the barrier along the proton transfer coordinate increases with increase in the 0---0 distance much faster in crystals than in the gas phase system. 

The two-dimensional treatment does not consider the change in the intramolecular nuclear distance in the two states which has been taken into account as a three-dimensional representation by de Boer and Custers [19]. 

K from 4 10 to 2.10 cm mole sec in the temperature range 300-lOOO K considered. These results are in agreement with the relations (218.Ill) which have been derived from one-dimensional tunneling calculations. It should be recalled, however, that in a two-dimensional treatment the effect of the nonseparability of the reaction coordinate is also incorporated in the factor 3 (or... 

Srinivasan, Hurwitz, and Bockris [57] derived expressions for the current-distribution and the /x=o x = o relation in one- and two-dimensional models for a single pore with a flat meniscus. The influence of electrolytic resistance, diffusion, and an inhibited discharge step with a simplified kinetic equation were considered. It was shown that the one-dimensional treatment may be used for current densities lower by a factor of 4 than the limiting current density. The limiting current density was obtained by the two-dimensional treatment. Special cases of predominance of one or two processes were discussed. As expected, the conclusions for a special case are the same as those for the corresponding case of the flooded pore (see sections 1 to 3) in the one-dimensional model. 

An interesting feature observed in this equal-foot treatment of substrate and protein motions is the similarity observed between the dividing surface (or TS) obtained when only the chemical coordinate is employed and that obtained in the two-dimensional treatment (when protein motions are explicitly considered). This similarity is a consequence of the fact that in reactive trajectories slower environmental motions precede the H-transfer and thus the TS obtained under the equilibrium treatment for the chemical coordinate is a reasonable approximation to that obtained with a more complete definition of the reaction coordinate. It should be taken into account that in TST the equilibrium assumption is needed to describe only the reactants and transition states (allowing the determination of their relative probabilities or free energy difference) and not along the whole reaction path. 

The novelty in the aforementioned studies is the use of a comprehensive numerical model for the investigation of catalytic microscale reactors which includes, for the first time in the literature, detailed heterogeneous and homogeneous chemical reaction mechanisms, two-dimensional treatment for both the gas and solid wall phases and surface radiation heat transfer, under both steady and transient (quasisteady) conditions. Moreover, a validated chemical kinetics model for the coupled catalytic and gas-phase combustion of propane (a fuel of particular interest for portable applications) is presented for the first time. 

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