Macroscopic transport equations

Introducing the reformulated forms of the terms examined above, the exact macroscopic transport equation (3.276) (i.e., without including any constitutive equations for the unknown terms) can be expressed as ... 

Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Berlin Springer. 

To derive the macroscopic transport equations, the conservation Relation [10] and [11] must be converted to differential equations. The main assumption needed is that the mean density and the mean velocity vary slowly in space and in time. Starting from Eq. [10] and [11], the macro dynamic equations describing the large-scale behavior of the lattice gas are obtained by multiple-scale perturbation expansion technique (Frisch et al., 1987). We shall not derive this formalism here. In the continuous limit, Eq. [10] leads to the macro dynamical conservation of mass or Euler equation... 

The use of macroscopic transport equations for the determination of time-correlation functions of fluctuating quantities is equivalent to the Onsager regression hypothesis outlined in Section 10.2 and discussed in Chapter 11. 

Macroscopic transport equations are commonly used to describe the semiconductor and the electrolyte in the liquid-junction cell. A microscopic model of the semiconductor-electrolyte interface couples the equations governing the macroscopic systems. 

Quantitative optimization or prediction of the performance of photoelectrochemical cell configurations requires solution of the macroscopic transport equations for the bulk phases coupled with the equations associated with the microscopic models of the interfacial regions. Coupled phenomena govern the system, and the equations describing their interaction cannot, in general, be solved analytically. Two approaches have been taken in developing a mathematical model of the liquid-junction photovoltaic cell approximate analytic solution of the governing equations and numerical solution. 

For zeolite membranes, the separation of water and alcohol molecules can be explained by strong interactions between the water molecules and ionic sites in the zeolite crystal lattice and the partial sieving achieved by the zeolite channels (Shah, Kissick, Ghorpade, Hannah, Bhattacharyya, 2000). Macroscopic transport equations describing the mass transfer through such composite membranes are often Maxwell— Stefan based (Krishna van Den Broeke, 1995). Wee, Tye, and Bhatia (2008) listed several zeolite materials for the dehydration of alcohols, such as silicalite or mordenite. Most of these materials were supported by an a-Al203 porous support membrane. 

D macroscopic transport equations are then solved in this domain with the transport coefficients, which take into account the nature of every computational cell (direct numerical simulation (DNS) model). Physically, this approach enables to calculate transport parameters of a porous media, for example, porosity or the Bruggemann exponent (Mukherjee and Wang, 2006 Wang et al., 2006a). 

The macroscopic transport equation for this thermal conduction process is... 

It was noted above that the mathematical formulation of the transport equation in the presence of IPV/surface exclusion for polymer was represented by the extended Equation 7.3 which included the factor /. However, this model is not rigorous, and its assumptions have been investigated within different models by various authors (Alishaeva and Entov, 1984 Gilman and MacMillan, 1985). Here the problem of deriving a consistent macroscopic transport equation in the presence of IPV/surface exclusion effects and... 

Comparing these expressions for the current densities with the macroscopic transport equations, given in eqs. (4) and (5), we obtain the following expressions for the electrical resistivity, p, the thermal conductivity, A, and the thermopower, S, in the case of a... 

Other macroscopic transport equations can be used to derive rela--tions for other transport coefficients as functionals of time-correlation functions. For example, the Navier-Stokes equation can be used to derive a time-correlation function expression for the coefficient of shear viscosity. 

Having at one s disposal kinetic equations for adsorbate and gas phase one has an opportimity to consider different regimes and derive for these regimes appropriate macroscopic transport equations by the kinetic theory methods. The investigations of transport processes at interphase boundary in the framework of UGAL - type models have been presented elsewhere (Borisov et al. 1988 Borman et al. 1988). 

In the following discussion we will throughout assume the existence of macroscopic transport equations like (5) without trying to justify them from statistical mechanical models. 

Macroscopic-mixture. These variables describe the state of the REV. They are the main variables in the mixture macroscopic transport equations. Appropriate definitions are. Mixture enthalpy ... 

The success in deriving the Maxwellian type macroscopic transport equations and in obtaining appropriate source and flux closures based on the kinetic theory of granular flow relies heavily on the knowledge about the distribution function involved. In principle, for reactive systems the single particle distribution function should satisfy... 

Following the standard procedure, a Maxwellian type macroscopic transport equation for the granular fluid properties can be obtained by multiplying the Boltzmann type equation (4.187) by m-ij) and integrate over the total phase space performing... 

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