Transport Equations For Mixture Fraction Research Articles

Numerical simulation of micro-mixing in gas–liquid and solid–liquid stirred tanks with the coupled CFD-E-model

The coupled CFD-E-model for multiphase micro-mixing was developed, and used to predict the micro-mixing effects on the parallel competing chemical reactions in semi-batch gas–liquid and solid–liquid stirred tanks. Based on the multiphase macro-flow field, the key parameters of the micro-mixing E-model were obtained with solving the Reynolds-averaged transport equations of mixture fraction and its variance at low computational costs. Compared with experimental data, the multiphase numerical method shows the satisfactory predicting ability. For the gas–liquid system, the segregated reaction zone is mainly near the feed point, and shrinks to the exit of feed-pipe when the feed position is closer to the impeller. Besides, surface feed requires more time to completely exhaust the added H + solution than that of impeller region feed at the same operating condition. For the solid–liquid system, when the solid suspension cloud is formed at high solid holdups, the flow velocity in the clear liquid layer above the cloud is notably reduced and the reactions proceed slowly in this almost stagnant zone. Therefore, the segregation index in this case is larger than that in the dilute solid–liquid system. The coupled CFD-E-model for multiphase micro-mixing is developed based on the Eulerian–Eulerian multiphase flow. The micro-mixing effects on the parallel competing chemical reactions occurring in solid–liquid and gas–liquid stirred tanks are investigated using the coupled CFD-E-model with better model parameters in terms of the concept of the mixture fraction and its variance. The predicted segregation index under different operation conditions is in reasonable agreement with the experimental data.

Modelling the generation of temperature inhomogeneities by a premixed flame

The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is investigated and modelled, with emphasis on the generation of ‘entropy waves’, i.e. entropic inhomogeneities of downstream temperature. Transient computational fluid dynamics simulations of two adiabatic lean methane-air flames of different Péclet numbers provide guidance and validation data for subsequent modelling. The respective entropy transfer functions, which describe the production of temperature inhomogeneities, as well as transfer functions for the variation of the heat release, are determined from the computational fluid dynamics time series data by means of system identification. The processes governing the dynamics of the entropy transfer functions are segregated into two sub-problems: (1) heat release due to chemical reaction at the flame front and (2) advective and diffusive transport. By adopting a formulation in terms of a mixture fraction variable, these two sub-problems can be treated independently from each other. Models for both phenomena are derived and analysed using simple 0- and 1-dimensional configurations. The heat release process (1) is represented by a fast-reaction-zone model, which takes into account variations of the specific heat capacity with equivalence ratio in order to evaluate the magnitude of downstream temperature fluctuations with quantitative accuracy. For the transport processes (2), two types of models based on mean field data from the computational fluid dynamics simulation are proposed: A semi-analytical, low-order formulation based on stream lines, and a state-space formulation, which is constructed by Finite Elements discretisation of the transport equation for mixture fraction. Model predictions for the entropy transfer functions are found to agree well with the computational fluid dynamics reference data at very low computational costs.

A flamelet model for transcritical LOx/GCH4 flames

This work presents a numerical framework to efficiently simulate methane combustion at supercritical pressures. A LES flamelet approach is adapted to account for real-gas thermodynamics effects which are a prominent feature of flames at near-critical injection conditions. The thermodynamics model is based on the Peng-Robinson equation of state (PR-EoS) in conjunction with a novel volume-translation method to correct deficiencies in the transcritical regime. The resulting formulation is more accurate than standard cubic EoSs without deteriorating their good computational performance. To consistently account for pressure and strain fluctuations in the flamelet model, an additional enthalpy equation is solved along with the transport equations for mixture fraction and mixture fraction variance. The method is validated against available experimental data for a laboratory scale LOx/GCH4 flame at conditions that resemble those in liquid-propellant rocket engines. The LES result is in good agreement with the measured OH* radiation.

Numerical simulation of micro-mixing in stirred reactors using the engulfment model coupled with CFD

Owing to computational efficiency and model simplicity, the engulfment model (E-model) is more suitable for studying micro-mixing in industrial scale reactors. However, the E-model does not contain the information of macro-mixing on the reactor scale. A CFD method coupled with the E-model is proposed and implemented to predict the micro-mixing in stirred tanks. The new modeling approach describes the small-scale segregation of fluid by solving the transport equations of mixture fraction and its variance based on the simulated macro-flow, and calculates the representative turbulence kinetic energy and dissipation in the reaction zone via the weighted average of mixture-fraction variance for the E-model. The predictions of segregation index compare with experimental results satisfactorily.

A comparative study of instabilities in forced reacting plumes of nonpremixed flames

A comparative study has been performed to investigate the flow instabilities and their interaction for nonpremixed methane-air flames using experimental results, numerical simulations and theoretical analyses. The effects of buoyancy and the strong external perturbations on the vortex dynamics and instabilities of forced reacting plumes are studied. Results suggest that the flame surface breaks in forced reacting plumes and two flame fronts are formed eventually due to the convective instability coupled with buoyancy-driven instability. Flame pinch-off could occur in a short time for the cases with the strong perturbations of low frequencies. This indicates that the nonpremixed flame system exhibits a low-pass characteristic, which is sensitive to the low frequency perturbations. In addition, the buoyancy instability can be observed from the comparisons, which is of an absolute unstable nature. Despite the fact that theoretical analyses are derived from the single transport equation for mixture fraction with a number of assumptions, the results are still in good agreement with those obtained from the experiments and numerical simulations.

Evaluation of a steady flamelet approach for use in oxy-fuel combustion

The present work investigates CFD analysis of an 11.5kW lab-scale furnace with oxy-natural-gas combustion for high temperature processes. In recent years many researches were done for oxy-fuel combustion with flue gas recycle in power plants. However, this paper considers combustion models which are applicable for oxy-fuel environment in melting and annealing furnaces. The main emphasis was to use the steady laminar flamelet model (SFM) associated with three detailed chemical mechanisms to predict the combustion process in the furnace. This approach was chosen to avoid time expensive chemical calculations in CFD while calculating the flow field, temperature and species concentrations. Just two additional transport equations for mixture fraction and mixture fraction variance have to be solved to determine the reactions in the flow field. It was found, that the skeletal mechanism with 17 species and 25 reversible reactions generated the best results with SFM. CFD calculations with SFM were compared with results from a 2-step eddy dissipation model (EDM) and a numerically expensive eddy dissipation concept (EDC) model with 17 species and 46 reversible reactions. The EDM was exposed to be unsuitable for modelling oxy-fuel combustion due to dissociation effects and forming of radicals in high temperature applications. In contrast to the EDM results, temperatures calculated by EDC and SFM showed close agreement with measured data in the furnace. Also for the species concentrations, obtained with both models, no significant differences were observed. A significant improvement was the reduction of computational time from 3weeks to 4days on 8 CPU-cores by using the SFM. The P1 and discrete ordinates (DO) model were also applied to solve the radiative transfer equation. A weighted sum of grey-gases model (WSGGM) was applied to model the radiative properties of the flue gas. Due to the special composition of the flue gas in oxy-fuel combustion, which consists mainly of H2O and CO2, the radiative properties should be treated with a non-grey gas approach. Nevertheless, the WSGGM was used in this study because of the small beam length of the furnace. Simulation with the P1 model overestimates the radiation, which leads to lower temperatures in the furnace. The DO model calculated temperatures in good agreement with the measurement with the SFM and EDC.

Numerical Study on Liquid Fuel Spray Behavior in Turbulent Flames Using LES

In the present study, Large-Eddy Simulation (LES) modeling for turbulent spray combustion flows has been developed in conjunction with the Eulerian/Lagrangian method for the gas phase and dispersed phase representation and flamelet approach for combustion modeling. Moreover, a new thermal energy coupling model is proposed to estimate the thermal interaction between the gas phase and the dispersed phase. The governing equations for the gas phase are the mass and momentum conservation equations and the mixture fraction transport equation. The governing equations for spray droplets are the mass, momentum, and thermal energy equations for each droplet. We applied the numerical procedure to a laboratory-sized spray combustor and found that procedure can predict key features of turbulent spray combustion phenomena, such as the temperature distribution of the combustion gas, individual droplet behavior, and droplet vaporization phenomena. Moreover, the flame temperature drop caused by the thermal impact from the spray droplets is expressed properly by employing the coupling model.

Study on the potential of BML-approach and G-equation concept-based models for predicting swirling partially premixed combustion systems: URANS computations

In this work the potential of two combustion modeling approaches (BML and G-equation based models) for partially premixed flames in combustion systems of various complexities is investigated using URANS computations. The first configuration consists of a nonconfined swirled premixed methane/air flame (swirl number 0.75) exhibiting partially premixed effects due to coflowing. The system is studied either in the isothermal case or in the reacting mode and for different thermal powers. The second configuration represents a model GT combustion chamber and features the main properties of real GT combustors: a confined swirled flow with multiple recirculation zones and reattachment points, resulting in a partially premixed methane/air aerodynamically stabilized flame and an additional diffusion flame formed by the fuel and oxidizer not consumed in the premixed flame. This makes it possible to subject the modeling to variation of different parameters, such as confinement, Re-number or flame power, or adiabatic or nonadiabatic conditions. For this purpose an extended Bray–Moss–Libby model and a G-equation-based approach, both coupled to the mixture fraction transport equation to account for partially premixed effects, are used following the so-called conditional progress variable approach (CPVA). The radiation effects are also taken into account. To account for the turbulence–chemistry interaction, a (multivariate) presumed PDF approach is applied. The results are compared with LDV, Raman, and PLIF measurements. Beyond a pure validation, the URANS is used to capture the presence of the precessing vortex core and to analyze the performance of different modeling strategies of partially premixed combustion in capturing the expansion ratio, species formation conditioned on the flame front, and flame front stabilization. It appears that the combustion models used are able to achieve plausible results in the complex combustion systems under study, while the BML-based model affords less computational time.

A hybrid scalar model for sooting turbulent flames

A Lagrangian Monte Carlo solution of the joint scalar pdf transport equation for mixture fraction and representative soot properties, coupled with an Eulerian solution for the turbulent flow field and here described as a “hybrid model,” has been developed. The modeling of soot formation and destruction employs an existing description of the key processes based on two soot variables—the soot volume fraction (or mass concentration) and the particle number density. The gas-phase chemistry is introduced through flamelet-state relationships. The simulation strategy is based on tracing the evolution of reactive stochastic particles within the computational domain. The ensemble of these particles at a fixed location and time then describes the joint scalar pdf. Soot rate equations, represented as functions of mixture fraction, soot mass concentration, and number density, are solved exactly in terms of the scalar values of each individual stochastic particle and the associated gas-phase properties derived from laminar flamelet-state relationships. The solution for the turbulent flow field provides the mean velocity and mixing frequency required for the transport of the stochastic particles in both physical and compositional space, while the Monte Carlo simulation returns the computed mean density field to the CFD code. Density-weighted mean values are approximated by ensemble averages over the scalar values of the stochastic particles in individual computational cells. The principal objective of the hybrid model is the improved treatment of nonlinear soot formation and oxidation, in particular, the capture of the intermittency in the oxidation process associated with the noncoexistence of soot and the principal oxidizing species. Significant computational economies accompany the adoption of the laminar flamelet approach for the source terms in the soot rate equations and the reduced number of scalars computed stochastically.

Quantification of differential diffusion in nonpremixed systems

Most attempts to quantify differential diffusion (DD) are based on the difference between different definitions of the mixture fraction. This paper presents a general method for evaluating differential diffusion in premixed or nonpremixed systems based on conservation equations for the elemental mass fractions. These measures form a basis for analysing differential diffusion. Casting these in terms of a mixture fraction gives particular insight into differential diffusion for nonpremixed systems, and provides a single DD measure. Furthermore, it allows direct evaluation of the validity of the traditional assumptions involved in writing a mixture fraction transport equation. Results are presented for one-dimensional opposed flow simulations of hydrogen and methane flames as well as direct numerical simulations (DNS) of CH4/H2–air and CO/H2–air flames. For a common definition of the mixture fraction, the DD measure can be approximated well by considering only the contribution of H2 and CH4 in methane–air flames. Differential diffusion is largely driven by production of H2 in the flame zone for hydrocarbon flames. Effects of strain rate and filter width on the relative importance of differential diffusion are examined.

Large Eddy simulation of a turbulent non-premixed flame

Large Eddy Simulation (LES) has been applied to the calculation of a turbulent hydrogen diffusion flame using a conserved scalar formalism. The Favre filtered Navier-Stokes equations have been closed using the Smagorinsky model and its dynamically calibrated variant For the Favre-filtered mixture fraction transport equation, closure has been obtained through combining the subgrid scale eddy viscosity with a constant-valued subgrid scale Schmidt number. A dynamically calibrated gradient transport model has also been used, and in a third simulation, the equation for ξ is left unclosed in order to compare the effects of the dissipation supplied by the model and that produced by the TVD scheme, needed to ensure the boundedness of the mixture fraction, ξ. An assumption of equilibrium chemistry has been employed to express the thermochemical variables (temperature, density, and species concentrations) as functions of ξ. The effects of fluctuations in the subgrid scales on the resolved scale values of these scalars is accounted for through the use of a presumed shape subgrid probability density function (sgpdf) for ξ; a β-function has been used in the present case. Knowledge of the subgrid scale mixture fraction variance is needed to fix the shape of the β-sgpdf; here a simple equilibrium model has been employed. The results of the simulations show that LES is capable of producing good agreement with measurements of mean velocity, Reynolds stresses and fluxes. The spreading of the mixture fraction field appears to have been slightly overpredicted in all cases. Though this discrepancy is not large, it has a noticeable effect on the predicted mean thermochemical fields in physical space. When plotted against mixture fraction, however, the predicted mean temperature and species concentrations give much better agreement with measurements, and show the combustion model to be working well in the present case.

Large-Eddy Simulation of a Turbulent Hydrogen Diffusion Flame

In this work a large-eddy simulation (LES) of a turbulent hydrogen jet diffusion flame is presented. The numerical method handles fluctuations of density in space and in time, but assumes density to be independent of pressure (incompressibility). The chemical composition of the fluid is described by solving the filtered transport equation for mixture fraction f. Density, viscosity and temperature are evaluated assuming chemical equilibrium. To account for sub-grid fluctuations of f, its sub-grid distribution is presumed to have the shape of a β-function. The results of the simulation are discussed extensively. The influence of inlet boundary conditions is addressed and radial profiles at different axial positions are shown for a complete set of one-point statistical data. Agreement of numerical results and experimental data is very good. Furthermore, a comparison of Reynolds- and Favre-averages is done and energy spectra at different locations in the flame are discussed.

A Consistent Flamelet Formulation for Non-Premixed Combustion Considering Differential Diffusion Effects

A flamelet formulation for non-premixed combustion that allows an exact description of differential diffusion has been developed. The main difference to previous formulations is the definition of a mixture fraction variable, which is not related directly to any combination of the reactive scalars, but defined from the solution of a conservation equation with an arbitrary diffusion coefficient and appropriate boundary conditions. Using this definition flamelet equations with the mixture fraction as the independent coordinate are derived without any assumptions about the Lewis numbers for chemical species. The formulation is shown to be exact if the scalar dissipation rate is prescribed as a function of the mixture fraction. Different approximations of the scalar dissipation rate that had been derived from analytic solutions for special cases are investigated by varying the diffusion coefficient of the mixture fraction transport equation. As examples, counterflow flames of hydrogen and n-heptane, which have much larger and much smaller diffusivities than oxygen and nitrogen, are considered. It is shown that the use of equal thermal and mixture fraction diffusivities yields a sufficiently well-described flame structure and is therefore recommended for the definition of the mixture fraction diffusion coefficient. Finally, the possibility of using constant species Lewis numbers has been examined. It has been found that once an appropriate set of Lewis numbers is determined, good results are achieved over wide ranges of the parameters, such as scalar dissipation rate, pressure, and oxidizer temperature.