Two-dimensional modelling

Onsager L 1944 Orystal statistics I. A two-dimensional model with an order-disorder transition Phys. Rev. 65 117... 

Brasseur G and De Baets P 1986 Ions in the mesosphere and lower thermosphere a two-dimensional model J. Geophys. Res. 91 4025-46... 

Multidimensionality may also manifest itself in the rate coefficient as a consequence of anisotropy of the friction coefficient [M]- Weak friction transverse to the minimum energy reaction path causes a significant reduction of the effective friction and leads to a much weaker dependence of the rate constant on solvent viscosity. These conclusions based on two-dimensional models also have been shown to hold for the general multidimensional case [M, 59, and 61]. 

Two-dimensional models can be used to provide effective approximations in the modelling of polymer processes if the flow field variations in the remaining (third) direction are small. In particular, in axisymraetric domains it may be possible to ignore the circumferential variations of the field unlaiowns and analytically integrate the flow equations in that direction to reduce the numerical model to a two-dimensional form. 

In two-dimensional solids theory, the size of the solid in a fixed direction is assumed to be small as compared to the other ones. Therefore, all characteristics of the thin solid are referred to a so-called mid-surface, and one obtains the two-dimensional model. Let us give the construction of plate and shell models (Donnell, 1976 Vol mir, 1972 Lukasiewicz, 1979 Mikhailov, 1980). 

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

Since we are going to rather extensively use the Hamiltonian (4.40) in the sequel, as a simplest two-dimensional model for an exchange chemical reaction, it is beneficial to establish some of... 

Figure 7.10. (a) Zachariasen s two-dimensional model of an AiOj glass, after Zachariasen (1932). (b) Two-dimensional representation of a sodium silicate glass,... 

Both extreme models of surface heterogeneity presented above can be readily used in computer simulation studies. Application of the patch wise model is amazingly simple, if one recalls that adsorption on each patch occurs independently of adsorption on any other patch and that boundary effects are neglected in this model. For simplicity let us assume here the so-called two-dimensional model of adsorption, which is based on the assumption that the adsorbed layer forms an individual thermodynamic phase, being in thermal equilibrium with the bulk uniform gas. In such a case, adsorption on a uniform surface (a single patch) can be represented as... 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

Peles et al. (1998) and Khrustalev and Faghri (1996) considered two-phase laminar flow in a heated micro-channel with distinct evaporating meniscus in the frame of quasi-one-dimensional and two-dimensional models. 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

The results of calculations of the Nusselt number are presented in Fig. 10.19. Here also the data of the calculated heat transfer by the quasi-one-dimensional model by Khrustalev and Faghri (1996) is shown. The comparison of the results related to one and two-dimensional model shows that for relatively small values of wall superheat the agreement between the one and two-dimensional model is good enough (difference about 3%), whereas at large At the difference achieves 30%. 

In order to estimate the extent of ozone depletion caused by a given release of CFCs, computer models of the atmosphere are employed. These models incorporate information on atmospheric motions and on the rates of over a hundred chemical and photochemical reactions. The results of measurements of the various trace species in the atmosphere are then used to test the models. Because of the complexity of atmospheric transport, the calculations were carried out initially with one-dimensional models, averaging the motions and the concentrations of chemical species over latitude and longitude, leaving only their dependency on altitude and time. More recently, two-dimensional models have been developed, in which the averaging is over longitude only. 

Clean Air Models. Models developed to simulate clean air chemistry generally have the least amount of chemical parameterization. Several recent zero-dimensional models (95,155,156) and one-dimensional models (157,158) have presented calculated HO concentrations for clean air. Two dimensional models have also provided predictions for global [HO ] (58,159,160,161). Three dimensional models that provide information... 

Roesler, J. F., An Experimental and Two-Dimensional Modeling Investigation of Combustion Chemistry in a Laminar Non-Plug-Flow Reactor, Proc. 27th Symp. (Int.) Combust., 1, 287-293 (1998). 

The rate equation for the two-dimensional model is then given by (12), where A = pT ptlT and /3 is a constant related to the individual reaction, and Topt represents the temperature where the reaction occurs most efficiently. 

Figure 2.51 Two-dimensional model geometry of a micro channel with a reaction occurring at the lower channel wall.

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