Volume-averaged Transport Equations

As far as modeling of transport phenomena in porous media is concerned, the task is to provide a generic description which is applicable to as broad a class of materials as possible. The models should to some extent be idealized, allowing them to capture a broad class of phenomena without the need to model all geometric details of the pore space and allowing for a fundamental understanding of transport processes in porous media. 

Following the line of arguments developed by CarboneU and Whitaker [188] and Nozad et al. [189], a volume-averaged temperature is introduced as 

Assuming local thermal equilibrium, i.e. the equality of the averaged fluid and solid temperature, a transport equation for the average temperature results which still contains and integral over the fluctuating component. In order to close the equation, a relationship between the fluctuating component and the spatial derivatives of the average temperature of the form 

The effective thermal conductivity tensor depends on the transformation vec- 

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The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

In order to carry out computations with the time after volume averaged transport equations on the form (3.329) and (3.330), we need to relate the average of products to products of averages and derive constitutive equations for the interfacial coupling terms. 

The dynamics of a heterogeneously catalyzed gas-phase reaction occurring in a nanoporous medium in combination with heat and mass transfer was simulated using a finite-volume approach. In contrast to other studies of similar type, heat and mass transfer in the nanoporous medium was explicitly accounted for by solving volume-averaged transport equations in the porous medium. Such an approach made it possible to compare the transport resistances in the gas phase and in the porous medium and to study the tradeoff between maximization of... 

A similar procedure can be applied to the energy transport terms. When the volume-averaged energy equation is expressed in terms of the internal energy and time averaging... 

Here we have emphasized the intrinsic nature of our area-averaged transport equation, and this is especially clear with respect to the last term which represents the rate of reaction per unit volume of the fluid phase. In the study of diffusion and reaction in real porous media (Whitaker, 1986a, 1987), it is traditional to work with the rate of reaction per unit volume of the porous medium. Since the ratio of the fluid volume to the volume of the porous medium is the porosity, i.e. 

The EMA method is similar to the volume-averaging technique in the sense that an effective transport coefficient is determined. However, it is less empirical and more general, an assessment that will become clear in a moment. Taking mass diffusion as an example, the fundamental equation to solve is... 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

It is extremely difficult to model macroscopic transport of mass, energy, and momentum in porous media commonly encountered in various fields of science and engineering based on microscopic transport models that account for variation of velocity and temperature as well as other quantities of interest past individual solid particles. The basic idea of porous media theory, therefore, is to volume average the quantities of interest and develop field equations based on these average quantities. 

In this review, a set of balance equations for transport of heat, mass, and momentum in stationary and moving porous media has been derived based on a local volume averaging approach. The advantage of this method is that it allows precise definition of average temperature, velocity, and pressure. Moreover, equations are derived rigorously from first principles. 

Calibration of FAGE1 from a static reactor (a Teflon film bag that collapses as sample is withdrawn) has been reported (78). In static decay, HO reacts with a tracer T that has a loss that can be measured by an independent technique T necessarily has no sinks other than HO reaction (see Table I) and no sources within the reactor. From equation 17, the instantaneous HO concentration is calculated from the instantaneous slope of a plot of ln[T] versus time. The presence of other reagents may be necessary to ensure sufficient HO however, the mechanisms by which HO is generated and lost are of no concern, because the loss of the tracer by reaction with whatever HO is present is what is observed. Turbulent transport must keep the reactor s contents well mixed so that the analytically measured HO concentration is representative of the volume-averaged HO concentration reflected by the tracer consumption. If the HO concentration is constant, the random error in [HO] calculated from the tracer decay slope can be obtained from the slope uncertainty of a least squares fit. Systematic error would arise from uncertainties in the rate constant for the T + HO reaction, but several tracers may be employed concurrently. In general, HO may be nonconstant in the reactor, so its concentration variation must be separated from noise associated with the [T] measurement, which must therefore be determined separately. 

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... 

The zeroth order moments of the volume averaged bubble population equations, i.e., the balances on the total bubble density in flowing and stationary foam, have the form of the usual transport equations and can be readily incorporated into a suitable reservoir simulator. 

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