Multi-environment Probability Density Function Research Articles

Numerical Study of the Influence of the Geometric Parameters of a Radial Crystalizer Using a Multiscale CFD Model

This work investigates how the configuration of the geometric parameters of a radial crystallizer influences the results of the crystallization of lovastatin by antisolvent and using a multi-scale computational fluid dynamics (CFD) model. The OPENFOAM open-source software uses macro and micromixing expressions for flow, and complete energy and population equilibrium equations during nucleation and crystal growth. The model is based on the Reynolds-Averaged-Navier-Stokes (RANS) equation, along with a multi-environment probability density function (PDF) model and the spatially semi-discretized population equilibrium equation, operating a high-resolution finite volume method. The variation crystallizer construction parameters provided another crystallizer design, and analyses demonstrated improved performance and effects on crystal distribution.

Large eddy simulation based multi-environment PDF modelling for mixing processes of transcritical and supercritical cryogenic nitrogen jets

Turbulent cryogenic nitrogen jets at transcritical and supercritical conditions are numerically investigated by LES-based multi-environment probability density function method together with the tabulated thermodynamic properties. To account for the non-ideal thermodynamic effects, the mixture properties are calculated by using Soave-Redlich-Kwong equation of state. The effect of sub-grid scale fluctuations of thermodynamic properties is modelled by the multi-environment PDF. Numerical results indicate that the present multi-environment PDF approach has the prediction capability to capture the essential features of cryogenic nitrogen jets under the supercritical pressure. Comparing with the supercritical jet, the transcritical jet features the longer and denser core which is relatively resilient to core destruction caused by turbulence. It is also found that the stronger pseudo-boiling strength and the larger density difference are directly tied with the longer cold-core length. In terms of the sub-grid scale scalar fluctuation, the transcritical jet shows the larger magnitude than the supercritical jet. In the transcritical nitrogen jet, it is identified that the strong sub-grid fluctuations of density and isobaric specific heat are closely tied with the pseudo boiling process. These numerical results suggest that the high-fidelity numerical modelling for transcritical flows must have the capability to realistically represent the combined effects of turbulence and thermodynamic nonlinearities, sub-grid scale fluctuations, and scalar conditional statistics.

Real-Fluid Modeling for Turbulent Mixing Processes of N-Dodecane Spray Jet Under Superciritical Pressure

The multi-environment probability density function (MEPDF) approach using the real-fluid Equation of State (EoS) has been developed to numerically investigate the physical processes of the non-cryogenic n-dodecane fuel spray jet under the supercritical pressure. The MEPDF approach together with the conserved scalar formulation and the real-fluid library is utilized to account for the scalar fluctuation effects on the turbulent mixing processes of real fluids over the transcritical and supercritical spray conditions. To correctly predict the highly non-linear thermodynamic properties over a wide range of pressures and temperatures, this approach has adopted the RK-PR equations of state and dense-fluid correction models. In order to realistically and efficiently predict the supercritical spray jet with the strong anisotropic turbulence, the present study has adopted the RANS-based v2-f model and the LES-based turbulence model. Simulations are made for the “Spray-A” benchmark case in the Engine Combustion Network. Based on numerical results, the detailed discussions are also made for the prediction capability of RANS/LES based real-fluid approach as well as the essential features of the real-fluid n-dodecane jet under the supercritical pressure.

Multi-phase particle-in-cell coupled with population balance equation (MP-PIC-PBE) method for multiscale computational fluid dynamics simulation

The ‘multiphase particle-in-cell coupled with population balance equation’ (MP-PIC-PBE) method is introduced for simulating multi-scale multiphase particulate flows. This method couples the meso‑scale fluid dynamics simulated by the MP-PIC method with the simulation of the micro-scale particle size distribution. The homogeneous population balance equation is calculated for each discrete particle tracked in a Lagrangian frame, after the MP-PIC numerical procedure is followed at each time instance. This approach allows the particulate phase to accommodate the particulate stresses using spatial gradients and allows the Lagrangian description to predict particle properties by the PBE. For the antisolvent crystallization of Lovastatin in a biradial mixer, the proposed method is compared to an existing method that simulates the spatiotemporal evolution of the particle distribution by combining a multi-environment probability density function with the spatially varying PBE. The MP-PIC-PBE method has lower computational cost and provides more detailed information, such as particle age and location.

Numerical modeling for multiple combustion modes in turbulent partially premixed jet flames

In this study, the multi-environment probability density function (MEPDF) model has been used to numerically investigate the flame structure under the multiple combustion modes with nearly homogeneous and inhomogeneous inlets. For nearly homogeneous and inhomogeneous cases, the MEPDF approach shows a good conformity with experimental data both conditioned statistics, and unconditioned mean and variance scalars. However, there exist the distinct discrepancies in the edge of reaction zone mainly due to the shortcomings of the RANS-based turbulence model. Predicted results indicate that the used model shows the limitation to describe the combustion processes with the local extinction in the downstream regions. With regard to the predicted conditioned environment scatters, the multienvironment PDF model has shown the capability to precisely simulate the nearly perpendicular shift from the fuel-rich side to the stoichiometric side. Based on the numerical results, detailed discussion was made for the characteristics of turbulent partially premixed jet flames with multiple combustion modes and the shortcomings of present model.

Large eddy simulation of piloted turbulent jet flames using the multi-environment PDF model based on the flamelet generated manifold

Using the multi-environment probability density function (MEPDF) approach, large Eddy simulation (LES) has been carried out for the piloted turbulent jet flames Sandia flames D and F, which are well documented for the validation of turbulent combustion model. In the present approach, the detailed chemistry is represented by the flamelet generated manifold (FGM) and the turbulence-chemistry interaction is modeled by the MEPDF approach. In this LES-based MEPDF model, the joint scalar PDF is modeled as a sum of weighted Dirac delta functions. As a result, filtered reaction term is closed without additional modellings. Moreover, a finite number of high order moments can be accurately reconstructed with direct quadrature method of moments. In terms of the unconditional and conditional means for mixture fraction, temperature, major and key radical species, numerical results are compared with experimental data. For flame D, the present LES-based MEPDF approach yields a good agreement with experimental data. However, for flame F, the conformity with measurements is not quite satisfactory compared to flame D. It is also found that the much finer grid refinement yields the noticeably improved agreements with experimental data in terms of the unconditional and conditional means. These numerical results have demonstrated that the present LES-MEPDF approach using the FGM tabulated chemistry has the prediction capability of the physically complex turbulent flames as well as computational efficiency and robustness.

Multi-environment PDF modeling for n-dodecane spray combustion processes using tabulated chemistry

The multi-environment probability density function approach together with a tabulated chemistry model has been developed to numerically simulate spray combustion processes under Diesel-like combustion conditions. In this study, the joint composition PDF is approximated using the multi-environment PDF, consisting of a combination of weights and weighted scalars in both chemical composition space and physical space. To account for the auto-ignition process and subsequent flame propagation, a flamelet generated manifold is generated by utilizing steady and unsteady flamelet solutions. The spray dynamics are represented by the state-of-art physical models in context with the Eulerian–Lagrangian approach. The RANS-based computations are made for the n-dodecane non-reacting and reacting spray jets. In terms of spray and vapor penetration, the mean and rms mixture fraction, ignition delay, and lift-off height, the numerical results are in reasonably good agreement with the available experimental data of the Engine Combustion Network. Based on the numerical results, detailed discussions are made for the effects of the ambient conditions on the structures and characteristics of high-pressure Diesel-fueled spray jet flames.

Multi-environment PDF modeling for non-catalytic partial oxidation process under MILD oxy-combustion condition

The multi-environment probability density function approach has been applied to numerically investigate the Moderate or Intense Low-Oxygen (MILD) oxy-combustion processes encountered in the non-catalytic partial oxidation (POX) gasifier. The multi-environment PDF approach has the form of a conventional Eulerian scheme and retains the desirable property of a particle-based method. Micro-mixing is represented via the IEM model, and the detailed chemistry is based on GRI 3.0 mechanism without NOx chemistry. In terms of the mean temperature, the present multi-environment PDF approach yields the overall agreement with the measurements in the highly fuel-rich MILD oxy-combustion situation with the strong flue gas recirculation even if there exist the certain discrepancies in the upstream region. Special emphasis is given to the effects of the fuel/oxygen injection velocity and O2/CH4 ratio on the characteristics of the strongly recirculated MILD oxy-combustion processes. Depending on injection velocity or O2/CH4 ratio, the present MEPDF approach well reproduces the qualitative flame transition characteristics from MILD combustion to conventional combustion. The higher fuel/oxygen injection velocity leads to the much longer jet penetration and the much higher SDR level which makes the ignition to occur at further downstream region. The relatively lower O2/CH4 ratios maintain the basic characteristics of the MILD combustion while the highest O2/CH4 ratio locally creates the oxy-flame like structure rather than the non-visible flame field. Based on numerical results, the detailed discussions are made for flame stabilization, auto-ignition process and precise flame structure in terms of recirculation rate, distribution of turbulent Damköhler number, scalar dissipation rate, mean temperature and mole fraction of CH2O and OH.

FGM-based non-adiabatic multi-environment PDF modeling for MILD combustion processes with the strong flow reversal

A Multi-environment probability density function (MEPDF) approach coupled with a tabulated chemistry has been developed to numerically investigate the flame structure under Moderate and intense low oxygen dilution (MILD) combustion with strong flow reversal. The tabulated chemistry is represented by Flamelet generated manifolds (FGM). To realistically represent the highly recirculated flameless combustion, we employed the chemistry database using laminar premixed flamelets rather than laminar diffusion flamelets. Moreover, the present FGM-based MEPDF realistically accounts for the heat loss effects caused by convective heat transfer at the wall. Numerical results in this study were validated against experimental data. Based on numerical results, detailed discussions were made for the essential features of the highly recirculated flameless combustion.

Multi-environment Probability Density Function Modeling for Turbulent CH4 Flames under Moderate or Intense Low-Oxygen Dilution Combustion Conditions with Recirculated Flue Gases

The present study has adopted the multi-environment probability density function (MEPDF) approach to simulate the turbulent CH4 flames under flameless combustion conditions with recirculated flue gases. The MEPDF approach is based on an Eulerian PDF formulation with computational efficiency. Micromixing can be represented via the IEM (interaction by exchange with the mean) model, and the chemistry is based on the GRI 2.11 mechanism including the NOx chemistry. Special emphasis is given to the effects of the micromixing model constants on the structure and characteristics of recirculated moderate or intense low-oxygen dilution (MILD) combustion processes. In terms of the temperature and species mole fraction, the multi-environment PDF model with a micromixing constant of 0.5 yields reasonably good agreement with experimental data. In terms of the integrated NO production rate for the MILD combustion condition, the N2O path yields the highest level, followed by the prompt, reburn, NO2, thermal, and NNH mech...

Multi-environment probability density function approach for turbulent partially-premixed methane/air flame with inhomogeneous inlets

The present study aims to systematically evaluate the capability of a multi-environment PDF approach using non-premixed and premixed tabulated chemistry to predict the fundamental characteristics of turbulent partially-premixed piloted flames with near-homogeneous and inhomogeneous inlets. For the near-homogeneous case, the non-premixed manifold yields better agreement with measurements in terms of conditional mean, unconditional mean, and rms scalars; however, the premixed manifold generates a narrower hot flame zone, which is mainly attributed to its inability to account for diffusion in the mixture fraction space. For the inhomogeneous flame, the premixed and non-premixed manifolds are limited in their ability to reproduce the measured flame at upstream locations with multiple combustion modes, while the non-premixed manifold is reasonably good at predicting the unconditional and conditional profiles of all scalars in the downstream regions with dominant non-premixed combustion. In terms of the predicted environment-conditioned scalar profiles, the present multi-environment PDF approach demonstrated the ability to realistically predict the near-vertical transition from the fuel-rich flammable mixture condition to the stoichiometric pilot condition. Unlike the near-homogeneous case, the inhomogeneous case generates a distinctly different distribution of CO conditional means, where the peak conditional CO level is leaning to the richer side along the downstream region. This tendency is a clear indication of the flame transition from a premixed-dominated flame to a diffusion-dominated flame. Based on our numerical results, detailed discussions about the essential features of turbulent partially-premixed flames with multiple combustion modes are presented, as well as the limitations of the proposed approach.

Large-eddy simulation of piloted diffusion flames using multi-environment probability density function models

Two piloted partially premixed jet flames (Sandia Flames D and F) are calculated using large-eddy simulation. A multi-environment turbulent combustion model, which depicts the sub-grid probability density function (pdf) as a weighted summation of a small number of Dirac delta functions in composition space, is used for modeling of turbulence chemistry interaction. It was coupled with a 19-species reduced mechanism and an In Situ adaptive tabulation algorithm for chemical source term integration. The impact of number of environments is investigated in this study. While predictions of Flame D by both two- and three-environment models are in good agreement with experimental data, only the three-environment model correctly captures the strong extinction and re-ignition in Flame F. The current study highlights the capability of the multi-environment pdf model for modeling of turbulent flames that feature strong local extinction and re-ignition.

DQMOM approach for poly-dispersed soot formation processes in a turbulent non-premixed ethylene/air flame

To realistically treat the poly-dispersed soot formation system, the Direct Quadrature Method of Moment (DQMOM) in conjunction with the transient flamelet approach was adopted to simulate a turbulent non-premixed C2H4/air flame. The population balance equation of the soot particle distribution was approximated using the multi-environment probability density function which consists of multiple weights and abscissas in the physical space. The transient flamelet model was employed to treat the turbulence-chemistry interactions and the two-equation soot model was used to account for the soot/gas chemistry coupling allowing mass and energy exchange. In the framework of the transient flamelet model, the physical soot model terms such as nucleation, surface growth, and oxidation for the population balance equation of soot particle distribution were closed. In terms of the mean temperature, soot volume fraction, primary soot particle number density, primary soot aggregate number density, mean number of primary particle per aggregate, mean radius of soot aggregate, and mean primary soot particle diameter, the predicted profiles agreed reasonably well with the experimental data.

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.

Multi-environment probability density function method for modelling turbulent combustion using realistic chemical kinetics

A computational fluid dynamics (CFD) tool for performing turbulent combustion simulations that require finite-rate chemistry is developed and tested by modelling a series of bluff-body stabilized flames that exhibit different levels of finite-rate chemistry effects ranging from near equilibrium to near global extinction. The new modelling tool is based on the multi-environment probability density function (MEPDF) methodology and combines the following: the direct quadrature method of moments (DQMOM); the interaction-by-exchange-with-the-mean (IEM) mixing model; and realistic combustion chemistry. Using DQMOM, the MEPDF model can be derived from the transport PDF equation by depicting the joint composition PDF as a weighted summation of a finite number of multi-dimensional Dirac delta functions in the composition space. The MEPDF method with multiple reactive scalars retains the unique property of the joint PDF method of treating chemical reactions exactly. However, unlike the joint PDF methods that typically must resort to particle-based Monte-Carlo solution schemes, the MEPDF equations (i.e. the transport equations of the weighted delta-peaks) can be solved by traditional Eulerian grid-based techniques. In the current study, a pseudo time-splitting scheme is adopted to solve the MEPDF equations; the reaction source terms are computed with a highly efficient and accurate in-situ adaptive tabulation (ISAT) algorithm. A 19-species reduced mechanism based on quasi-steady state assumptions is used in the simulations of the bluff-body flames. The modelling results are compared with the experimental data, including mixing, temperature, major species and important minor species such as CO and NO. Compared with simulations using a Monte-Carlo joint PDF method, the new approach shows comparable accuracy.

Simulation of Mixing Effects in Antisolvent Crystallization Using a Coupled CFD-PDF-PBE Approach

Antisolvent crystallization is widely used in the production of pharmaceuticals. Although it has been observed experimentally that the crystal size distribution is strongly influenced by the imperfect mixing of the antisolvent with the solution, these effects have not been adequately quantified. In this work, a turbulent computational fluid dynamics (CFD) code was coupled with a multienvironment probability density function (PDF) model, which captures the micromixing in the subgrid scale, and the population balance equation, which models the evolution of the crystal size distribution. The population balance equation (PBE) was discretized along the internal coordinate using a high-resolution central scheme. The presence of solids was addressed by treating the suspension as a pseudo-homogeneous phase with a spatial variation in the effective viscosity. This coupled CFD-PDF-PBE algorithm was applied to an antisolvent crystallization process in an agitated semibatch vessel, where the rising liquid level was modeled by a dynamic mesh. The effects of agitation speed, addition mode, and scale-up on the local primary nucleation and size-dependent growth and dissolution rates, as well as the crystal size distribution, were numerically investigated.

A multienvironment conditional probability density function model for turbulent reacting flows

The multienvironment conditional probability density function (MECPDF) model was first proposed by Fox [Computational Models for Turbulent Reacting Flows (Cambridge University Press, Cambridge, 2003)] as a simple extension of multienvironment probability density function models for turbulent reacting flows. Like the conditional moment closure (CMC) and the laminar flamelet model (LFM), the MECPDF model describes the reacting scalars conditioned on the value of the mixture fraction. However, unlike CMC and LFM, the new model provides a consistent description of conditional fluctuations in both the scalar dissipation rate and the reacting scalars, and hence can be used to model partial extinction and reignition in homogeneous turbulent reacting flows. In this work, a general derivation of the MECPDF model is presented for a single reaction-progress variable using the direct quadrature method of moments. Extensions of the model to multiple reaction-progress variables and conditioning on the mixture-fraction vector are also discussed. After deriving the model, the closure assumptions are validated using direct simulations for pure diffusion of two randomly distributed, initially correlated scalar fields. Two homogeneous applications are then considered: nonreactive mixing starting from nontrivial initial conditions, and reactive mixing with partial extinction and reignition.