Model two dimensional

Suppose that a body is strongly elongated in some direction and with sufficient accuracy it can be treated as the two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the field at the observation points. We will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, and z, as is shown in Fig. 4.5a, so that the body is elongated along the y-axis. It is clear that if at any plane y — constant the behavior of the field is the same. To carry out calculations we will preliminarily perform two procedures, namely, 

Mentally divide the cross section of a body into many elementary areas. Correspondingly we can treat the model as a system of many elementary prisms. The dimensions of every elementary cross section are much smaller than the distance between an observation point and any point in this area. 

Each elementary prism is replaced by an infinitely thin line directed along the y-axis with the same mass per unit length as that of the prism. These two steps allow us to replace the two-dimensional body by a system of infinitely thin lines which are parallel to each other, and the distribution of mass on them is defined from the equality 

Derivation of the formula for the attraction field caused by an infinitely thin line with the density X is very simple, and is illustrated in Fig. 4.5b. We will consider the field at the plane y = 0. Due to the symmetry of the mass distribution, we can always find a pair of elementary masses Xdy and —Xdy, which when summed do not create the field component gy directed along the y-axis, and respectively the total field generated by all elements of the line has only the component located in the plane y — 0. Here r is the coordinate of the cylindrical system with its origin at the point 0, and the line with masses is directed along its axis. As is seen from Fig. 4.5b the component dg at the point located at the distance r from the origin 0 is 

At the beginning, we assume that the line length is 21. Then, summation of fields caused by all elementary masses of the line gives 

The stages of migration of adsorbed A and B particles are written as (5) jZf+YZg -+YZf+jZg, where j — A, B / and g are adjacent sites, V is a vacant site (a vacancy). The index a corresponds to the indicated stage numbers. It is enough to consider the interactions of the first and second neighbors in the quasi-chemical approximation. There are two possibilities of the equation constructions for the distributed two-dimensional model, and for point models. In the last subsection the next question will be discussed - How the form of the systems of equations alters for a great difference in the mobilities of the reactants  

The kinetic equations for the local concentrations of the adsorbed particles that are having comparable migration rates or for initial time interval, have the form 

All two-site elementary processes are realized on the two nearest neighboring sites / and g. So the index r — 1 is omitted in U]g(r a) and below in multipliers KJ (r a). The addends on the right-hand sides of Eq. (41) are the rates of the elementary processes (whose stage numbers a are indicated in parentheses), the particles participating in an elementary process are defined by superscripts, and the site numbers, by subscripts. 

For the reverse direction of a jump of a particle i from site g to the vacant site /, the places of the indices i and V should be exchanged in formula (45). 

The system of equations for the local concentrations is supplemented with equations for the probabilities of the two-body configurations djg(r) at a distance of r, where r — 1 and 2, that obey normalizing relations (27). With a view to these relations for 5 = 3, it is sufficient to write the equations relative to the functions 0 jg(r), where ij — AA, AB, BA, and BB. For particles at the distance of first neighbors, the equations have the following form  

Now we consider an Nx AT-site square lattice with cyclic boundary conditions. We replace each site of the lattice by a square (Fig.7) with spins s = 1/2 at its corners, making the total number of spins equal to 41V2. To avoid misunderstanding, however, from now on we continue to refer to these squares as sites. The wave 

The singlet wave function (60) is conveniently identified graphically with a square lattice, each site corresponding to a fourth-rank spinor (whose form is identical for all sites), and each segment linking sites corresponds to a metric spinor gXfi (Fig.8). 

To completely define the wave function (60), it is necessary to know the form of the site spinor xpup. The specific form of the fourth-rank spinor tyxPlJp [and, hence, the wave function (60)] describing the system of four spins s = 1/2 is governed by 14 quantities [31], which are parameters of the model. 

We now choose a Hamiltonian H for which the wave function (60) is an exact ground state wave function. As in the one-dimensional case, we seek the required Hamiltonian in the form of a sum of cell Hamiltonians acting in the space of two nearest neighbor spin quartets  

The first term in Eq. (61) is the sum of the cell Hamiltonians in the horizontal direction, and the second term is the same for the vertical. The cell Hamiltonians 

Stoichiometric coefficients ( ) and the logarithm of the corresponding values of the concentrations of the master species plus the logarithm of the corresponding equilibrium constant  

The mass balance for each master species corresponds to the sum over one column of the product of the stoichiometric coefficients E) and the concentration values of the species (C). In this step, the concentrations of the precipitated species are also considered. The mass balance of protons (normally located at the first column) is exchanged with the electroneutrality condition, and then the column corresponding to the electrical charge is exchanged with the column corresponding to the protons. The mass balance residues are calculated for the Newton-Raphson method  

Finally, the saturation indexes of each precipitated species are calculated. When a precipitated species is present, the corresponding value of saturation index is appended to the matrix of the residues for the Newton-Raphson method. The saturation index values are calculated by the sum of the product of the stoichiometric coefficients corresponding to the precipitates (Ep) and the logarithm of the corresponding values of the concentrations of the master species minus the logarithm of the corresponding solubility product constant  

Up to this time, numerous one-dimensional models for electrokinetic soil remediation have been presented (Alshawabkeh and Acar, 1992 Choi and Lui, 1995  

Of course, if an analytical solution is not feasible for a one-dimensional model, there will be little chance of finding an analytical solution for a two-dimensional arrangement. Therefore, a numerical solution is developed following the same procedures as outlined in the latter subsection. In this case, a horizontal surface of saturated soil with a depth of Az is considered (Fig. 25.4). The area is divided into a two-dimensional grid of Ni rows and Nj columns resulting in (Ni x Nj) volume elements with horizontal dimensions of Ax by Ay. The concentration of the kth ion in the cell (/,/) is denoted by Ctjk, where i and j are row and column indices into the grid. The electrodes are placed perpendicular to the horizontal surface of the soil, the anode compartment is located in the ER, AC) cell, and the cathode compartment in the (ER, CC) cell. 

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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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