Joint Velocity-composition Probability Density Function Research Articles

PDF modeling and simulations of pulverized coal combustion – Part 1: Theory and modeling

A transported probability density function (PDF) method is developed for pulverized coal combustion. Two separate PDF transport equations, one for the gas phase (joint velocity-composition PDF) and one for the coal particle phase, are solved by means of stochastic Lagrangian methods. The gas composition is described by two mixture fractions (one that tracks the devolatilization products and one that tracks the char combustion products) and the total specific enthalpy. Two different models are proposed for the unclosed terms in the gas-phase PDF transport equation due to the mass transfer from the solid to the gas phase through devolatilization and char combustion. To close the transport equation for the coal particle PDF, the gas phase velocity and composition sampled along the coal particle trajectories are modeled. Radiative heat transfer is modeled by a discrete transfer method (DTM) to solve for the Reynolds averaged radiative heat transfer equation for a gray absorbing emitting and scattering gas–particle mixture. The turbulence–radiation interaction of the emission is given in closed form in the proposed two-phase PDF approach.

Stochastic modeling of evaporating sprays within a consistent hybrid joint PDF framework

In the present study, a framework for modeling two-phase evaporating flow is presented, which employs an Eulerian–Lagrangian–Lagrangian approach. For the continuous phase, a joint velocity-composition probability density function (PDF) method is used. Opposed to other approaches, such PDF methods require no modeling for turbulent convection and chemical source terms. For the dispersed phase, the PDF of velocity, diameter, temperature, seen gas velocity and seen gas composition is calculated. This provides a unified formulation, which allows to consistently address the different modeling issues associated with such a system. Because of the high dimensionality, particle methods are employed to solve the PDF transport equations. To further enhance computational efficiency, a local particle time-stepping algorithm is implemented and a particle time-averaging technique is employed to reduce statistical and bias errors. In comparison to previous studies, a significantly smaller number of droplet particles per grid cell can be employed for the computations, which rely on two-way coupling between the droplet and gas phases. The framework was validated using established experimental data and a good overall agreement can be observed.

Embedding quasi laminar 1D flame profiles to model turbulent premixed combustion with a joint PDF method

AbstractA new model for turbulent premixed combustion is presented which is based on a joint velocity composition probability density function (JPDF) method. The key idea is a scale separation approach. The method combines the model by Bray, Moss and Libby [1] (BML) for premixed combustion with the flamelet approach for nonpremixed combustion. Here, a Lagrangian formulation of the BML model is considered. The progress variable used by the BML model becomes a computational particle property and its value is triggered by the arrival of the flame front at the particle's position. Similar as in the flamelet approach we assume that the smallest eddies are not small enough to disturb the reactive diffusive flame structure. To resolve the (embedded) quasi laminar flame structure, a flame residence time is introduced. With that residence time, the evolution of the particle composition, including enthalpy, can be determined from precomputed laminar 1D flames. The main challenge with this approach is to model the probability that an embedded flamefront arrives at the particle location, which is necessary to close the chemical source term. Numerical experiments of a turbulent premixed flame show good agreement with experimental data. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling of a turbulent reacting gas/particle flow

A mathematical model for two-phase turbulent reactive flows is presented which is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of “particles” representing the system. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is not smeared over computational cells but restricted to the vicinity of solid particles by the definition of an “action-sphere” which is attached to every solid particle. Applications of the method to isotropic, homogeneous turbulence indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase, and it provides extensive statistics of various correlations of properties.

Modelling of Turbulent Gas/Particle Combustion by a Lagrangian PDF Method

A mathematical model for two-phase turbulent reactive flows is presented which is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of ”particles” representing the system. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is not smeared over computational ceils but restricted to the vicinity of solid particles by the definition of an “action-sphere“ which is attached to every solid particle. Applications of the method to isotropic, homogeneous turbulence indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase and it provides extensive statistics of various correlations of properties.

Computations of steady-state and transient premixed turbulent flames using pdf methods

Premixed propagating turbulent flames are modeled using a one-point, single time, joint velocity-composition probability density function (pdf) closure. The pdf evolution equation is solved using a Monte Carlo method. The unclosed terms in the pdf equation are modeled using a modified version of the binomial Langevin model for scalar mixing of Valiño and Dopazo, and the Haworth and Pope (HP) and Lagrangian Speziale-Sarkar-Gatski (LSSG) models for the viscous dissipation of velocity and the fluctuating pressure gradient. The source terms for the presumed one-step chemical reaction are extracted from the rate of fuel consumption in laminar premixed hydrocarbon flames, computed using a detailed chemical kinetic mechanism. Steady-state and transient solutions are obtained for planar turbulent methane-air and propane-air flames. The transient solution method features a coupling with a Finite Volume (FV) code to obtain the mean pressure field. The results are compared with the burning velocity measurements of Abdel-Gayed et al. and with velocity measurements obtained in freely propagating propane-air flames by Videto and Santavicca. The effects of different upstream turbulence fields, chemical source terms (different fuels and strained/unstrained laminar flames) and the influence of the velocity statistics models (HP and LSSG) are assessed.

PDF simulation of a turbulent series—parallel reaction in an axisymmetric reactor

Because both the turbulent mixing and chemical reaction terms are modeled in the traditional moment approach, it is very difficult to isolate the effect of turbulent mixing on chemical reactions. In contrast, these two terms appear in closed form in the joint velocity—composition probability density function (PDF) formulation. A joint velocity—composition PDF code was to study the effect of the scalar mixing rate on a series—parallel reaction (A + B → R, R + B → S) in a single-jet tubular reactor and the results are compared with experimental data. The turbulent flow field was obtained from the k-ɛ model and the chemical reaction terms were efficiently dealt with using a look-up table. Using PDF simulations, the effects of different scalar mixing rate models on the chemical reaction were investigated under different flow conditions. Since the first reaction step is highly sensitive to the scalar mixing rate, by comparing the yield of the desired product ( R) from simulations with the experimental data, the best scalar mixing rate model is determined.

Raman measurements and joint PDF modeling of a nonpremixed bluff-body-stabilized methane flame

Mixture fraction, temperature, and major species in a recirculation-stabilized nonpremixed methane-air flame are (1) measured using laser Raman scattering and (2) calculated using the joint velocity-composition probability density function (PDF) model, appropriately combined with an elliptic mean flow solver. The Raman system was modified to account for the significant levels of laser-induced fluorescence (LIF) and incandescence encountered in the rich sooty zone of the flame. The joint PDF contains three velocity components, which are modeled by Langevin equations, and five thermochemical scalar variables with mixing given by linear deterministic relaxation to the mean and chemistry by a four-step steady-state reduced scheme. The flame is characterized by significant finite-rate chemistry, including, unlike a CO/H 2 /N 2 fuel in the same apparatus, strong bimodality in the temperature-mixture fraction scatter plots. Calculations are compared with Raman data on temperature and major species. The agreement is in general reasonable, with the largest discrepancies being caused by the breakdown of the assumption of a chemical steady state for the cool fuel-rich gas at the core of the flame. Large discrepancies are found on the peak CO, as in other similar studies, confirming a need for better measurements of CO than Raman spectroscopy can provide. The model overpredicts the degree of extinction. The radial flux of the mixture fraction calculated directly from the joint PDF is compared with the flux given by an a posteriori gradient diffusion calculation, indicating no (radial) counter-gradient diffusion and a “turbulent Schmidt number” of about 0.4.

PDF calculations of turbulent nonpremixed flames of [formula omitted] using reduced chemical mechanisms

A three-step chemical kinetic scheme, reduced systematically from a more detailed mechanism is used to represent chemical reaction in the Monte Carlo calculations of the joint velocity-composition probability density function (pdf) for pilot-stabilized turbulent jet nonpremixed flames of H 2 CO 2 fuel mixtures. Flames with a range of fuel jet velocities and hence varying degrees of finite rate chemical kinetic effects have been calculated. The independent variables used are mixture fraction, ξ and the specific molar abundance of three reactive scalars, Γ H 2O , Γ CO 2 , and Γ H. Three multidimensional look-up tables are generated for density, other dependent properties and the composition increments due to chemical reaction. The use of these tables in all subsequent calculations is computationally efficient. Terms representing the effects of pressure fluctuations, mechanical dissipation, and scalar dissipation are modeled using an improved particle interaction model. Solutions are obtained on a Sun-Sparc1 station using 30 000 Monte Carlo particles for the low jet velocity flames and this number is increased to 50 000 at high jet velocities. The calculated velocity and turbulence fields as well as centerline velocity decay rates have not been compared with measurements but are consistent with recent experimental data collected in similar piloted jet flames of methane fuel. The calculated mixing field compares well with experimental data and the calculated flame length is slightly shorter than the observed visible flame length. For the low- and intermediate-velocity flames, the calculated temperature and mass fractions of the stable species agree reasonably well with instantaneous experimental data collected in similar flames. At high jet velocities, when the flames are close to blowoff and the finite rate chemical kinetic effects are very significant, discrepancies occur between measurements and computations and the blowoff jet velocity is underpredicted by about 20%.