Macroscale equation

To simplify the presentation, we limit this discussion to two phases, liquid and solid, with N constituents per phase, and restrict our discussion to results pertaining to the continuity equations and momenta balance. Interfacial effects are assumed negligible, although these effects have been incorporated into HMT [1, 12], In [4] the following macroscale equations are derived. 

We make the average for the microscale domain Qz and for the mesoscale domain 2, then we have the following macroscale equation [MaSE] ... 

The macroscale equation [MaSE] can be solved together with the boundary conditions (13) and (14) and the initial condition (15). 

A mathematical scheme that can treat a micro-inhomogeneous material uniformly at the microscale and the macroscale is referred to as Homogenization Analysis (HA) (see Sanchez-Palencia 1980 Bakhvalov Panasenko 1984). In the HA method, we introduce a perturbation scheme by using both a macroscale coordinate system and a microscale one, and derive a microscale equation, which represents the geometry and material properties in the micro-domain. Then, using the solution of the microscale equation, we determine the macroscale equation (Fig. 1.2). However, since the HA method is implemented within a framework of continuum mechanics, it also experiences difficulties when the material properties of micro-inhomogeneous materials need to be obtained. 

In conclusion, the homogenization analysis procedure can be stated as follows (1) The microscale equation (7.12) is first solved under the periodic boundary condition, which gives the characteristic function Ai(xi). (2) Using the characteristic function Ai(xi) we then calculate the averaged elastic modulus E. (3) The macroscale equation (7.14) can then be solved, giving the first perturbed term o(- °). Sinee Mi(x ,x1) is calculated by (7.11), the first order approximation of M (x) can be represented as... 

In order to solve the multiscale HA seepage problem, we proceed as follows First we solve the microscale equations (8.49) and (8.53), and obtain the characteristic functions vf and p. We next determine the permeabilities Kfj and Kij in the meso-domain and macro-domain using (8.54) and (8.60). Then we can solve the macroscale equation (8.59), and obtain the pressure p. By substituting these into... 

Thus the macroscale equation in the macro-domain is reduced to the problem of obtaining c by solving (9.33) under the homogenized Djj, vj and /, which are given by (9.31) and (9.32). The local distribution of concentration is calculated as... 

Furthermore, we introduce an integration average (- j in the meso-domain J2i, and obtain the following macroscale equation ... 

Clearly, there are two quite different types of models for a gas flow the continuum models and the molecular models. Although the molecular models can, in principle, be used to any length scale, it has been almost exclusively applied to the microscale because of the limitation of computing capacity at present. The continuum models present the main stream of engineering applications and are more flexible when applying to different macroscale gas flows however, they are not suited for microscale flows. The gap between the continuum and molecular models can be bridged by the kinetic theory that is based on the Boltzmann equation. 

For the purpose of illustration, in this paper we use a viscosity-capillarity model (Truskinovsky, 1982 Slemrod, 1983) as an artificial "micromodel",and investigate how the information about the behavior of solutions at the microscale can be used to narrow the nonuniqueness at the macroscale. The viscosity-capillarity model contains a parameter -Je with a scale of length, and the nonlinear wave equation is viewed as a limit of this "micromodel" obtained when this parameter tends to zero. As we show, the localized perturbations of the form x /-4I) can influence the choice of attractor for this type of perturbation, support (but not amplitude) vanishes as the small parameter goes to zero. Another manifestation of this effect is the essential dependence of the limiting solution on the... 

Hybrid mixture theory is a hybridization of classical volume averaging of field equations (conservation of mass, momenta, energy) and classical theory of mixtures [8] whose theory of constitution results from the exploitation of the entropy inequality in the sense of Coleman and Noll [9], In [4] the microscale field equations for each species of each phase, modified appropriately to include charges, polarization, and an electric field, are averaged to the macroscale, defined to be the scale where the phases are indistinguishable. Thus at the macroscale the porous media is viewed a mixture, with each thermodynamic property for each constituent of each phase defined at each point in space. 

Bennethum, L.S. and Cushman, J.H. (2002) Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics I. Macroscale field equations, Transport in Porous Media, 47(3), 309-336... 

In order to quantify the structure-property relations at each scale, a multiscale hierarchy of numerical simulations was performed, coupled with experiments, to determine the internal state variable equations of macroscale plasticity and damage... 

In a macroscale channel, gravitational force has an effect on the flow pattern of a biphasic system consequently, the flow pattern varies between vertical and horizontal channels. However, in a microchannel, the gravity effect is dominated by the viscous forces that are expressed by the ratio of gravity force and the surface tension using the Bond number (Bo) as expressed in Equation 4.5 where Ap is the density difference between two immiscible liquids, g is acceleration due to gravity, dh is the channel dimension and ct the surface tension. 

In all of the above cases, a strong non-linear coupling exists between reaction and transport at micro- and mesoscales, and the reactor performance at the macroscale. As a result, the physics at small scales influences the reactor and hence the process performance significantly. As stated in the introduction, such small-scale effects could be quantified by numerically solving the full CDR equation from the macro down to the microscale. However, the solution of the CDR equation from the reactor (macro) scale down to the local diffusional (micro) scale using CFD is prohibitive in terms of numerical effort, and impractical for the purpose of reactor control and optimization. Our focus here is how to obtain accurate low-dimensional models of these multi-scale systems in terms of average (and measurable) variables. 

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

Continuum models encompass both micro and macro scales and in li-ion models the microscale is governed by the solid phase diffusion equation. The coupling of the microscale and the macroscale variables pose computational limitations. 

The most powerful property of the detailed microbalance models, especially in combination with visualization techniques, is the a priori prediction of (observable) macroscale phenomena. This can be particularly helpful in reducing the required experimental effort. Important problems are the amount of detailed information required for the microscale transport equations and the large progranuning and computational efforts required to solve specific problems. Nevertheless, these types of models, by generating insight in the micro- and... 

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