Field quantity

A standard approach to modeling transport phenomena in the field of chemical engineering is based on convection-diffusion equations. Equations of that type describe the transport of a certain field quantity, for example momentum or enthalpy, as the sum of a convective and a diffusive term. A well-known example is the Navier-Stokes equation, which in the case of compressible media is given as... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

In general terms, the phenomena described above belong to the class of phase transitions and critical phenomena in confined spaces. From the field of statistical physics, some far-reaching results applying to such problems are knovm. One fruitful concept used in statistical physics is the correlation length (see, e.g., [64]). The correlation length describes how a local field quantity evaluated at one point in space is correlated with the same quantity at another point. As an example, the correlation length crfor density fluctuations in a fluid is defined via... 

Figure 2.67 Microstructure of a porous medium together with sample volume over which field quantities are averaged.

Pressure drop Grid spacing Field quantity Diffusivity... 

The coordinates are not varied 6L is due entirely to the variation of the field quantities. The variation can be rewritten in the form... 

Note that the estimation of particle-field quantities such as

mixing models based on the estimates is nearly identical for both types of codes. 

From the general solutions for particle-shaped states, integrated field quantities... 

Case 2 of a dissipative medium is now considered where x = 0 defines the vacuum interface in a frame (x,y, z). The orientation of the xy plane is chosen such as to coincide with the plane of wave propagation, and all field quantities are then independent on z as shown in Fig. 3. In the denser medium (region I) with the refractive index = n > 1 and defined by x < 0, an incident (7) EM wave is assumed to give rise to a reflected (r) EM wave. Here is the angle between the normal direction of the vacuum boundary and the wave normals of the incident and reflected waves. Vacuum region (II) is defined by x > 0 and has a refractive index of = 1. The wavenumber [35] and the phase (47) of the weakly damped EM waves then yield... 

It has thus been shown that the present theory of charged particle equilibria necessarily leads to point-like configurations with an excessively small characteristic radius r0, permitted, in principle, even to approach the limit r0 = 0. In this way the integrated field quantities can be rendered finite and nonzero. As pointed out in Section V.A.l.b, a strictly vanishing radius would not become physically acceptable, whereas a nonzero but very small radius is reconcilable both with experiments and with the present analysis. It would leave space for some form of internal particle structure. A small but lower limit of the radius would also be supported by considerations based on general relativity [15,20]. 

S, I, P, L, one obtains a set of coupled field equations for the determination of the following set of unknown field quantities ... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

The relationship between e and h and between r and q is then assumed. The assumed relation between r and q will usually involve the values of these quantities at the wall. The integrals in Eqs. (5.77) and (5.78) can then be related and eliminated between the two equations leaving a relationship between the wall heat transfer rate and the wall shear stress and certain mean flow field quantities. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

In order to determine the field quantities such as displacements, strains and stresses inside the laminate, away from the discretized boundary, the first part of equation (21) is considered ... 

The equation (33) and the subsequent equations show the main benefit of the Boundary Finite Element Method The field quantities inside the continuum can be calculated easily in closed-form analytical manner. 

It is significant that the energy density and intensity depend of the square of field quantities. We will exploit an analogous relationship in the interpretation of the wavefunction in quantum mechanics. 

Another vitally important question is how to model the turbulence within the canopy. It is clear by intuition that a number of vortices exists there, each shed from an individual obstruction. Figure 1.18 provides an experimental evidence to this. The known possible approach would be to introduce the effective turbulent viscosity v, and to model it as a function of coordinates and flow field quantities. The simplest case of a constant effective viscosity vf = const is known as the quasilaminar flow model. It will precede, in the Chapter 3, to more sophisticated models of turbulence considered in the Chapters 2, 4-8. 

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