Volume-averaged equations

The volume-averaged continuity equation for dispersed laminar multiphase flows is given by applying the volume-averaging theorems to Eq. (5.13) as 

Tk represents the rate of mass generation of phase k per unit volume. It is noted that the unit volume in this context is that of the gas-solid system. If a phase is defined in a physical sense such as the solid phase or the gas phase, Tk may be caused by chemical reactions or phase changes. On the other hand, when a phase is defined in a dynamic sense [Soo, 1965], Tk may result from the size change due to attrition or agglomeration in addition to the chemical reaction or phase change. From the mass balance of the mixture, we have 

Applying the averaging theorems to Eq. (5.14), the volume-averaged momentum equation of phase k is obtained as 

The volume-averaged equation of energy conservation in terms of internal energy is given by averaging Eq. (5.18) as 

The term fa) is the volume-averaged dissipation function for the energy dissipated by the viscous force, which is irreversible dissipation of mechanical work into thermal energy or heat. For the solid-particle phase, the kinetic energy loss by attrition or inelastic collision may be included in this term. 

---

The volume-averaged products in the preceding volume-averaged equations can be further expressed by the products of the volume averages. From Eq. (5.100), the volume-averaged mass flux of phase k is given by... 

Three dominant processes in the reaction diffusion in biofilms and cellular systems are (1) diffusion in a continuous extracellular phase B, (2) transport of solutes across the membrane, and (3) diffusion and reaction in the intracellular phase A. Consider aerobic growth on a single carbon source. The volume-averaged equations of a substrate S and oxygen O (electron acceptor) transport are... 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

The calculated values required to evaluate the performance index are determined from the solution of differential equations describing flow. Pressure and velocity values are obtained by solving the locally volume-averaged equation of continuity and differential momentum balance (Eqs (1) and (2)). For steady-state conditions, and incompressible flow, the equation of continuity becomes... 

Averaging methods to derive time and/or volume averaged equations for all phases... 

This completes the list of the mathematical tools required to reformulate the volume averaged equation (3.121) into a more practically useful form. Applying the Leibnitz s rule (3.123) to the first term in (3.121) yields ... 

The governing instantaneous volume averaged equations are examined next with focus on the principal approximations normally applied to the interfacial integral terms. 

In order to carry out computations with the volume averaged equations on the form (3.138) or (3.139), we need to relate the average of products to products of averages and derive constitutive equations for the interfacial coupling terms. 

It is easily seen that the principal closure problem is maintained since three undetermined terms appear in the volume averaged equations (3.186). 

To illustrate the remaining covariance modeling task required closing the model, the fairly rigorous instantaneous volume averaged equations expressed in terms of mass-weighted quantities are listed below. 

As in the volume averaged equations discussed in sect 3.4.1, three undetermined terms can be identified in the time averaged equations (3.259). The first term, —V (/3fc(pfcv V )rfc) denotes the covariance or correlation terms. The second term, Tkikj, accounts for the effects of interfacial stress. 

In accordance with (3.185) the convective term in the generic volume averaged equation (3.138) is manipulated introducing mass-weighted variables (3.182) and the concept of spatial decomposition (3.183). Hence, the volume averaged convection term is given by ... 

Quintard M, Whitaker S (1993) Transport in Ordered and Disordered Porous Media Volume-Averaged Equations, Closure Problems, and Comparisons with Experiments. Chem Eng Sci 48(14) 2537-2564... 

This leaves the rigorous continuum treatment of the near-surface hydrodynamics heat transfer to be rather impossible. The alternative has been the use of various area averages introduced into the volume-averaged equations. These area-averaging-continuum descrip-... 

Unlike conventionally used volume-averaged equations, equation 20 does reduce to the Brinkman equation 5 when the inertial effects are negligible and to Darcy s law, equation 3, when both inertial and multidimensional effects are negligible. 

Single-phase fluid flow in porous media is a well-studied case in the literature. It is important not only for its application, but the characterization of the porous medium itself is also dependent on the study of a single-phase flow. The parameters normally needed are porosity, areal porosity, tortuosity, and permeability. For flow of a constant viscosity Newtonian fluid in a rigid isotropic porous medium, the volume averaged equations can be reduced to the following the continuity equation,... 

Quintard M. and Whitaker S. 1993a. Transport in ordered and disordered porous media Volume averaged equations, closure problems, and comparison with experiment, Chem. Eng. Sci., 48, 2537-2564. 

In the case of macromoleculcs with excluded volume, averaging (Equations 174 and 175) is performed as integration over all the functions t(/i) (Oelfand and Yaglom, 1956 Freed, 1972). I o clarify this operation, consider a simple integral... 

---