
The Eulerian unsteady laminar flamelet model can be used to predict slowforming intermediate and product species which are not in chemical equilibrium. Typical examples of slowforming species are gasphase pollutants like NOx, and product compounds in liquid reactors. By reducing the chemistry computation to one dimension, detailed kinetics with multiple species and stiff reactions can be economically simulated in complex 3D geometries.
The model, following the work of Barths et al. [ 23], postprocesses an unsteady marker probability equation on a steadystate, converged flow field. In FLUENT, the steady flow solution must be computed with the steady laminar flamelet model (see Section 15.4). Since the unsteady flamelet equations are postprocessed on a steadystate, steady flamelet solution, the effect of the unsteady flamelet species on the flowfield are neglected.
The transport equation for the unsteady flamelet marker probability, , is
Equation 15.51 is always solved unsteady, and is initialized as
where is the mean mixture fraction and is a user supplied constant, which should be set greater than the stoichiometric mixture fraction. At inlet boundaries, FLUENT always sets toward zero, and hence the field decreases to zero with time as is convected and diffused out of the domain (for cases with outlet boundaries).
The unsteady flamelet species equations (Equation 15.36) are integrated simultaneously with the marker probability equation, . For liquidphase chemistry, the initial flamelet field is the mixedbutunburnt flamelet, as liquid reactions are assumed to proceed immediately upon mixing. However, gasphase chemistry invariably requires ignition, so the initial flamelet field is calculated from a steady flamelet solution. All slowforming species, such as NOx, are zeroed in this initial flamelet profile since, at ignition, little time has elapsed for any significant formation. The slowforming species are identified by the user before solution of the unsteady flamelet equations.
The scalar dissipation at stoichiometric mixture fraction ( ) is required by the flamelet species equation. This is calculated from the steadystate FLUENT field at each time step as a probabilityweighted volume integral:
where is defined in Equation 15.53, and denotes the fluid volume. FLUENT provides the option of limiting to a userspecified maximum value, which should be approximately equal to the flamelet extinction scalar dissipation (the steady flamelet solver can be used to calculate this extinction scalar dissipation in a separate simulation).
The unsteady flamelet energy equation is not solved in order to avoid flamelet extinction for high scalar dissipation, and to account for nonadiabatic heat loss or gain. For adiabatic cases, the flamelet temperature is calculated at each time step from the steady flamelet library at the probabilityweighted scalar dissipation from Equation 15.53. For nonadiabatic cases, the flamelet temperature at time is calculated from
where
In Equation 15.55, denotes the FLUENTsteadystate mean cell temperature conditioned on the local cell mixture fraction.
Unsteady flamelet mean species mass fractions in each cell are accumulated over time as
where is the 'th species unsteady flamelet mass fraction, and denotes the Beta PDF.
The probability marker equation (Equation 15.51) and the flamelet species equation (Equation 15.36) are advanced together in time until the probability marker has substantially convected and diffused out of the domain. The unsteady flamelet mean species, calculated from Equation 15.55, reaches steadystate as the probability marker vanishes.
Liquid Reactions
Liquid reactors are typically characterized by:
The Eulerian unsteady laminar flamelet model can be used to model liquid reactions. When the Liquid MicroMixing model is invoked, FLUENT uses the volumeweightedmixinglaw formula to calculate the density.
The effect of high Sc is to decrease mixing at the smallest (micro) scales and increase the mixture fraction variance, which is modeled with the Turbulent Mixer Model [ 17]. Three transport equations are solved for the inertialconvective ( ), viscousconvective ( ) and viscousdiffusive ( ) subranges of the turbulent scalar spectrum,
where the constants through have values of 2, 1.86, 0.058, 0.303 and 17050 respectively. The total mixture fraction variance is the sum of , and .
In Equation 15.59, the cell Schmidt number, , is calculated as where is the viscosity, the density and the mass diffusivity as defined for the pdfmixture material.
Since liquid species are invariably in chemical nonequilibrium, the unsteady laminar flamelet model should be used, postprocessing on a converged, steadystate, steady flamelet FLUENTcase and data.