Average transport equation

The reduction of the turbulent-reacting-flow problem to a turbulent-scalar-mixing problem represents a significant computational simplification. However, at high Reynolds numbers, the direct numerical simulation (DNS) of (5.100) is still intractable.86 Instead, for most practical applications, the Reynolds-averaged transport equation developed in... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

Given (5.292) and the mixture-fraction PDF, the chemical source term in the Reynolds-averaged transport equation for (7) is closed ... 

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. 

It is, for example, very common in reactors to have flow predominantly in one direction, say, z in a tubular reactor. The major gradients then occur in that direction. For many cases the cross-sectional average transport equations may thus be very useful. 

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 two-phase k — e model analyzed was based on the Favre averaged transport equation for turbulent kinetic energy developed by [73, 74]. The resulting transport equation for kinetic energy is similar to the one obtained from the single phase model (5.2), supporting the semi-empirical modification introduced in that model. 

When dealing with porous catalysts, one generally does not work with the intrinsic average transport equation given by... 

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 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... 

The quantitative description of such phenomena is commonly done by solving the averaged transport-equations for momentum, enthalpy and species-concentration. The calculation of the averaged source-term due to chemical reaction is usually based on various assumptions. E. g., the assumption of infinite fast chemistry in premixed configurations leads to the so called Eddy-Break-Up-model. An other example is the assumption of flame-sheet combus-... 

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