The age of transported scalars in fluid mechanics
Abstract: Computational fluid dynamics (CFD) is applied to the reacting flow in the freeboard of a 30MW biofuel fired grate furnace and results are validated by comparison with data from an extensive measurement campaign. The influence of the fuel bed model on the results of the freeboard model is investigated. The importance of including chemical equilibrium effects in the modelling is assessed by comparison between an approach where simplified one-way finite-rate reactions are employed and one where equilibrium is assumed. Possible areas of improvement of the model are identified, in particular the expression of the local mixing rate, which is a crucial parameter in combustion rate modelling. In the vicinity of the fuel bed the mixing rate is mainly determined by the character of the flow through the bed itself. In the remaining part of the furnace, the mixing rate depends on turbulence. A model of turbulence that gives more accurate estimations of the characteristic turbulence time scale (i.e. the mixing rate) is sought, which has similar computational requirements as two-equation models. A three-equation model of turbulence that accounts for the evolution of the energy spectrum is proposed. The model solves the transport equations of the turbulent kinetic energy k, of its dissipation rate , and of the age of the turbulence . While the equations for k and are taken from an earlier model, the derivation of the -equation is provided in the present work. Subsequently, two hypotheses are made; first, that turbulence develops with age towards an ideal state of homogeneity and isotropicity, which resembles the inertial range. Second, that the effect of velocity correlation on turbulent viscosity is a function of , analogous to the function of time in the expression of the dispersion of a particle in a turbulent flow. Expressions for the characteristic time scale and the turbulent viscosity are formulated consequently. The results of the present model and those of the earlier model are validated with experimental data from various benchmark flows, and the present model is shown to perform generally better. The transport equation for the age of transported scalars is derived for a number of specific cases, in particular that of a chemical species transported in a turbulent flow. Substituting the mass fraction with the turbulent kinetic energy in the latter derivation, the derivation of the transport equation of the age of the turbulence is obtained. A comparison between the Lagrangian and Eulerian approaches to residence time modelling is presented and validated against measurements in a laminar flow. The application of the age transport equation for analysis of fluid residence time in a reactor is also exemplified. Keywords: grate furnace, fluid age, residence time, scalar age, turbulence age, velocity correlation, energy spectrum, turbulence model.
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