One-dimensional formulation

The TDE solute module is formulated with one equation describing pollutant mass balance of the species in a representative soil volume dV = dxdydz. The solute module is frequently known as the dispersive, convective differential mass transport equation, in porous media, because of the wide employment of this equation, that may also contain an adsorptive, a decay and a source or sink term. The one dimensional formulation of the module is ... 

The pseudo-homogeneous fixed bed dispersion models are divided into three categories The axial dispersion model, the conventional two-dimensional dispersion model, and the full two-dimensional axi-symmetrical model formulation. The heterogeneous fixed bed dispersion models can be grouped in a similar way, but one dimensional formulations are employed in most cases. 

In a one-dimensional formulation with moisture moving in the direction normal to a specimen of a slice of wood of thickness 2a, the diffusion equation can be written as ... 

Let J (M/T) denote the tracer discharge from a rock volume. The conventional modelling approach is based on a one-dimensional formulation and has the solution in the Laplace domain (Shapiro, 2001)... 

The temperature distribution is assumed to be steady and the transferred heat is used mainly to evaporate water (Allaf 2009). By neglecting possible shrinkage phenomena, one can assume that ps = constant and Vs = 0 then Equations 22.6 and 22.7 may be transformed into one-dimensional formulation (r) as follows ... 

An appropriate set of coupled differential equations analogous to the relations (7.46) may be developed for the one-dimensional formulation of the transport equation by performing a series of operations on the general result (7.39). First we integrate this equation over all so as to collect the contributions to the neutron population of all processes which yield neutrons whose direction cosines are irrespective of their azimuthal orientation. This step is analogous to the integration performed in (7.76). If we use the fact that for a one-dimensional system... 

The model equations for mass, momentum, and energy balance and the EoS that will be useful in the following discussions are summarized in the following paragraphs. Any number of textbooks have detailed derivations and discussions (eg, Todreas and Kazimi, 1990 Collier and Thome, 1996). An area-averaged, transient, one-dimensional formulation for compressible fluids will be sufficient for the applications considered in this chapter. Some aspects of accounting for separate speeds for the vapor and liquid phases in a two-phase mixture are also included. 

The independent variables in the one-dimensional formulation are vertical height z and time t. The fluid occupies fraction e of the control volume fluid and particle velocities are Uf and Up respectively the fluid-particle interaction force per unit volume of suspension F/ is regarded as... 

The LST, on the other hand, explicitly takes into account all correlations (up to an arbitrary order) that arise between different cells on a given lattice, by considering the probabilities of local blocks of N sites. For one dimensional lattices, for example, it is simply formulated as a set of recursive equations expressing the time evolution of the probabilities of blocks of length N (to be defined below). As the order of the LST increases, so does the accuracy with which the LST is able to predict the statistical behavior of a given rule. 

A one-dimensional isothermal plug-flow model is used because the inner diameter of the reactor is 4 mm. Although the apparent gas flow rate is small, axial dispersion can be neglected because the catalj st is closely compacted and the concentration profile is placid. With the assumption of Langmuir adsorption, the reactor model can be formulated as. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

In the formulation of the boundary conditions, it is presumed that there is no dispersion in the feed line and that the entering fluid is uniform in temperature and composition. In addition to the above boundary conditions, it is also necessary to formulate appropriate equations to express the energy transfer constraints imposed on the system (e.g., adiabatic, isothermal, or nonisothermal-nonadiabatic operation). For the one-dimensional models, boundary conditions 12.7.34 and 12.7.35 hold for all R, and not just at R = 0. 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

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