Joint Composition Probability Density Function Research Articles

Numerical analysis of nitrogen oxides in turbulent lifted H2/N2 cabra jet flame issuing into a vitiated coflow

Abstract This paper gives an in-depth insight into NOX (NO, NO2, and N2O) formation of H2/N2 turbulent Cabra jet flame issuing into a hot vitiated coflow. The joint composition probability density function (PDF) was employed to model the combustion and to specify the characteristics of the flame (i.e., scalar variables, concentration of species etc.). The turbulent transport term was modelled by Reynold-Average-Naiver-Stokes (RANS) SSG and molecular mixing was modelled by modified curl model. A combustion mechanism including 13 species and 34 reactions was employed to define the thermochemical state of the flame. The chemical reaction terms were resolved and accelerated by In Situ Adaptive Tabulation (ISAT). The simulation was performed at different equivalence ratios (ER), fuel jet nitrogen content ( Y N 2 , C ), coflow ( T C ) and jet temperatures ( T J ), coflow oxygen ( Y O 2 , C ) and water contents ( Y H 2 O , C ). Results reveal NOX is composed of 30% NO2 and 70% NO in the burner. Reaction rate analysis at different operating points in the ignition kernel demonstrates that N + O H ⇌ N O + H and N O 2 + H ⇌ N O + O H are dominant reactions in NO formation, while N O + H O 2 ⇌ N O 2 + O H is the main reaction in NO2 formation.

Multi-environment probability density function approach for turbulent CH4/H2 flames under the MILD combustion condition

The multi-environment probability density function approach has been applied to simulate turbulent CH4/H2 flames under moderate or intense low-oxygen dilution (MILD) conditions. In order to circumvent excessive computational burden, the direct quadrature method of moments (DQMOM) has been adopted as an alternative approach to solve the transported probability density function (PDF) equation. The multi-environment PDF approach has the form of a conventional Eulerian scheme and retains the desirable property of a particle-based method. In this study, the joint-composition PDF is approximated using the two-environment PDF, expressed via the combination of weights and abscissas on the composition and physical space. Micromixing is represented by the IEM model, and the chemical mechanism is based on detailed chemistry. In terms of the unconditional means and conditional statistics for temperature and mass fractions of species and pollutants, the predicted profiles are in reasonable agreement with experimental data. Numerical results clearly indicate that the two-environment PDF transport model has the capability to realistically predict the effects of oxygen mass fraction on the flame lift-off, auto-ignition, flame structure, and NOx formation characteristics in turbulent CH4/H2 jet flames issuing from a jet in hot coflow (JHC). It was also found that the two-environment PDF approach reproduces the sensitivity of the CO and CO2 peak levels versus the O2 concentration variation, as well as the peak levels of NO conditional fluctuations under the MILD combustion condition. Even if considerable discrepancies in the unconditional means exist mainly due to inconsistent treatment of weight combination and shortcomings of singularity treatment procedure, the present three-environment PDF model well captures the measured high-level conditional temperature and CO fluctuations on the fuel-lean side where the local extinction occurs via mixing of the shrouded cold air and the vitiated flame field.

A Detailed Validation Study of Multi-Environment Eulerian Probability Density Function Transport Method for Modeling Turbulent Nonpremixed Combustion

The current work involves the validation of presumed shape multi-environment Eulerian probability density function (PDF) transport method (MEPDF) using direct quadrature method of moments (DQMOM)-interaction by exchange with mean (IEM) approach for modeling turbulence chemistry interactions in nonpremixed combustion problems. The joint composition PDF is represented as a collection of finite number of Delta functions. The PDF shape is resolved by solving the governing transport equations for probability of occurrence of each environment and probability-weighted mass fraction of species and enthalpy in Eulerian frame for each environment. A generic implementation of the MEPDF approach is carried out for an arbitrary number of environments. In the current work, the MEPDF approach is used for a series of problems to validate each component of MEPDF approach in an isolated manner as well as their combined effect. First of all, a nonreactive turbulent mixing problem with two different Reynolds numbers (Re = 7000 and 11,900) is used for validation of the mixing and correction terms appear in the MEPDF approach. The second problem studied is a diffusion flame with infinitely fast chemistry having an analytical solution. The reaction component is validated by considering a 1D premixed laminar flame. In order to validate the combined effect of mixing and turbulence chemistry interactions, two different turbulent nonpremixed problems using global one-step chemistry are used. The first reactive problem used is H2 combustion (DLR Flame H3), while the second reactive validation case is a pilot-stabilized CH4 flame. The current predictions for all validation problems are compared with experimental data or published results. The study is further extended by modeling a turbulent nonpremixed H2 combustion using finite-rate chemistry effects and radiative heat transfer. The current model predictions for different flame lengths as well as minor species are compared with experimental data. The current model gave excellent predictions of minor species like OH. The differences in the current predictions with experimental data are discussed.

LES-based Eulerian PDF approach for the simulation of scramjet combustors

A probability density function (PDF) approach to account for turbulence–chemistry interaction in the context of large eddy simulation (LES) based simulation of scramjets is developed. To solve the high-dimensional joint-composition PDF transport equation robustly, the semi-discrete quadrature method of moments (SeQMOM) is employed. The SeQMOM approach addresses key numerical issues in LES related to the inaccuracies in computing filter-scale gradients, enabling an efficient and numerically consistent solution of the PDF transport equation. The computational tool is used to simulate a cavity-stabilized Mach 2.2 supersonic combustor. The LES–SeQMOM approach captures the pressure profiles qualitatively, in spite of the large uncertainties in the boundary conditions. Further, this flow configuration is found to exhibit low-frequency dynamics that lead to oscillatory motions of the shock-system inside the combustor. Detailed analyses of the LES data are used to describe the physical mechanism leading to these oscillations.

A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows

This paper investigates the first part of a two-stage methodology for the detailed fully coupled modelling of nanoparticle formation in turbulent reacting flows. We use a projected fields (PF) method to approximate the joint composition probability density function (PDF) transport equation that describes the evolution of the nanoparticles. The method combines detailed chemistry and the method of moments with interpolative closure (MoMIC) population balance model in a commercial computational fluid dynamics (CFD) code. We show details of the implementation and present an extensive set of numerical experiments and validation. We consider the example of the chloride process for the industrial synthesis of titania. We show good agreement with experimental data and present fully coupled detailed chemistry CFD simulations of nanoparticle formation in a representative ‘slot’ reactor geometry. The simulations show that inception occurs in a mixing zone near the reactor inlets. Most of the nanoparticle mass is due to surface growth downstream of the mixing zone with a narrower size distribution occurring in the regions of higher surface growth. The predicted temperature and particle properties are compared to a perfect mixing case. The implications for the second part of the methodology, where it is proposed to post-process the data using a more detailed particle model, are discussed critically.

Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Partially Stirred Reactor Model: Analytical Solutions and Numerical Convergence Study of a PDF/Monte Carlo Method

We investigate the partially stirred reactor (PaSR), which is based on a simplified joint composition probability density function (PDF) transport equation. Analytical solutions for the first four moments of the mass density function (MDF) obtained from the PaSR model are presented. The Monte Carlo particle method with first order time splitting algorithm is implemented to obtain the first four moments of the MDF numerically. The dynamics of the stochastic particle system is determined by inflow-outflow, chemical reaction, and mixing events. Three different inflow-outflow algorithms are investigated: an algorithm based on the inflow-outflow event modeled as a Poisson process, an inflow-outflow algorithm mentioned in the literature, and a novel algorithm derived on the basis of analytical solutions. It is demonstrated that the inflow-outflow algorithm used in the literature can be explained by considering a deterministic waiting time parameter of a corresponding stochastic process, and also forms a specific case of the new algorithm. The number of particles in the ensemble, N, the nondimensional time step, $\Delta t^{*}$ (ratio of the global time step to the characteristic time of an event), and the number of independent simulation trials, L, are the three sources of the numerical error. The split analytical solutions and the numerical experiments suggest that the systematic error converges as $\Delta t^{*}$ and N-1 . The statistical error scales as L-1/2 and N-1/2 . The significance of the numerical parameters and the inflow-outflow algorithms is also studied by applying the PaSR model to a practical case of premixed kerosene and air combustion.

Detailed modeling of soot formation in a partially stirred plug flow reactor

The purpose of this work is to propose a detailed model for the formation of soot in turbulent reacting flow and to use this model to study a carbon black furnace. The model is based on a combination of a detailed reaction mechanism to calculate the gas phase chemistry, a detailed kinetic soot model based on the method of moments, and the joint composition probability density function (PDF) of these scalar quantities.Two problems, which arise when modeling the formation of soot in turbulent flows using a PDF approach, are studied. A consistency study of the combined scalar-soot moment approach reveals that the molecular diffusion term in the PDF-equation can be closed by the IEM and Curl-type mixing models. An investigation of different kernels for the collision frequency of soot particles shows that the influence of turbulence on particle coagulation is negligible for typical flame conditions and the particle size range considered.The model is used as a simple tool to simulate a furnace black process, which is the most important industrial process for the production of carbon blacks. Despite the simplifications in the modeling of the turbulent flow reasonable agreement between the calculated soot yield and data measured in an industrial furnace black reactor is achieved although no adjustments were made to the kinetic parameters of the soot model. The effect of the mixing intensity on soot yield and different soot formation rates is investigated. In addition the influence of different operating conditions such as temperature and equivalence ratio in the primary zone of the reactor is studied.

The BMC/GIEM Model for Micromixing in Non-Premixed Turbulent Reacting Flows

A novel beta mapping closure (BMC)/generalized IEM (GIEM) model is proposed to close the micromixing term in the joint composition probability density function (PDF) transport equation for nonpremixed, multiscalar turbulent reacting flows. Because the beta PDF gives an excellent description of binary inert scalar mixing, it is used to approximate the mixture fraction PDF. The proposed model thus effectively extends presumed PDF methods to handle finite-rate chemistry without introducing ad hoc assumptions about the form of the joint composition PDF. The BMC/GIEM model can be solved efficiently using a particle-based Monte-Carlo code. The model is first compared with the amplitude mapping closure (AMC)/GIEM model and homogeneous reaction-diffusion data for scalar mixing with a two-step reaction. Its application to inhomogeneous flows is demonstrated by simulating a reacting scalar mixing layer with a one-step reaction using a Lagrangian composition PDF (LCPDF) code. The linear-mean-square-estimation (LMSE) model is also evaluated for comparison. The results from the BMC/GIEM model are in excellent agreement with experimental data, whereas the LMSE model gives errors in higher-order statistics.