Density Function Method Research Articles

Estimating the state of health of lithium-ion batteries based on a probability density function

Accurate estimation of the state of health (SOH) of lithium-ion batteries is important to ensure safe operation. Although SOH models that extract health features (HFs) from incremental capacity (IC) curves have proven to be effective methods of evaluating battery SOH, the use of smooth IC curves that rely heavily on complicated algorithms reduces the reliability of SOH assessment to a certain extent. In this paper, a probability density function (PDF) method is integrated with Gaussian process regression (GPR) and used to build highly optimised SOH evaluation models for three different types of batteries. The PDF peak position and regional charging time are extracted from charging voltage data in the form of HFs using the PDF method. The proposed SOH estimation model shows good performance when these two HFs are both adopted. Our SOH estimation models for both cells and modules show good robustness for LiCoO2 (LCO), LiNi0.8Co0.15Al0.05O2 (NCA) and lithium iron phosphate (LFP) batteries.

The impact of hydrogen substitution by ammonia on low- and high-temperature combustion

The combustion behaviour of ammonia has attracted intermittent interest with the original patent by Lyon (US3900554A) relating to its use for nitric oxide reduction through selective non-catalytic reduction (SNCR) a focal point. The recent interest in ammonia stems from its use as a hydrogen rich energy carrier with practical use requiring a much wider parameter space. The corresponding challenges (e.g. Kobayashi et al., Proc. Combust. Inst. 37 (2019) 109–133) include different flame dynamics and high emissions of oxides of nitrogen. The current paper explores the complex nature of ammonia oxidation and provides a reduced size reaction mechanism that enables application, without approximation, to the computation of turbulent flames through a joint-scalar transported probability density function (JPDF) method. Comprehensive validation suggests similar accuracy to a reference mechanism (Glarborg et al., Prog. Energy Combust. Sci. 67 (2018) 31–68) and highlights some uncertainties. The selected turbulent flame configuration features auto-ignition stabilised flames supported by a coflow of hot combustion products. The base case features a H2/N2 fuel jet that permits flame stabilisation at 1045 K corresponding to the onset of the SNCR temperature window. The impact of a gradual substitution of hydrogen by ammonia on flame stabilisation, emissions of oxides of nitrogen and the flame structure is quantified. It is shown that ammonia substitution leads to more prevalent local extinction, a more distributed flame structure and requires substantially increased coflow temperatures to achieve a similar flame stabilisation point. A lowering of the coflow temperature to operate within the SNCR regime substantially reduces NOx and leads towards a homogeneous/distributed reaction mode. The reduced fuel reactivity highlights the importance of turbulence-chemistry interactions leading to complexities in the design of practical devices.

The impact of ammonia addition on soot formation in ethylene flames

The present work investigates computationally the impact of blending ammonia with ethylene on soot formation in four laminar diffusion flames and the corresponding turbulent partially premixed flames. The conditions conform to experimental investigations (Bennett et al., Combust. Flame, 2020, and Boyette et al., Combust. Flame, 2021) with both sets of flames featuring identical fuel blends. A mass and number density preserving sectional method was applied to all cases with the soot surface growth and oxidation models updated to include reactions with oxides of nitrogen. The gas phase chemistry was extended to include a comprehensively validated ammonia sub-mechanism. For the laminar flames, the inception of soot particles is based on detailed chemistry with pyrene treated as representative of the PAH pool. The accuracy of fitting a global acetylene-based inception step is evaluated for the different fuel blends and subsequently applied to the turbulent flames using a fully coupled transported joint probability density function method featuring an 84-dimensional joint-scalar space. Results obtained for laminar flames show that detailed chemistry coupled with the sectional soot model provides excellent agreement with the measured suppression of the soot volume fraction with increased use of ammonia. Computed particle size distributions (PSDs) show an increase in smaller soot particles under such conditions, consistent with experimental observations. The experimentally observed reduced impact in turbulent flames is also reproduced computationally. The suppression of soot is principally caused by changes in the radical pool leading to reduced soot surface growth and, to a lesser extent, soot inception. The contribution of oxides of nitrogen to soot oxidation is modest. Computed PSDs in laminar and turbulent flames highlight the importance of differences in flame structures and flowfield timescales.

On the impact of differential diffusion between soot and gas phase species in turbulent flames

The molecular diffusivities of larger PAHs and soot particles approach zero leading to differential diffusion with gas-phase species. The present work systematically quantifies the impact on soot moments, soot related statistical correlations and Particle Size Distributions (PSDs) using a fully coupled transported joint probability density function (JPDF) method featuring a 78-dimensional joint-scalar space, including enthalpy, gas phase species with the PSD discretised using 62 size classes via a mass and number density preserving sectional method. Differential diffusion of soot (DDS) is treated via a gradual decline of diffusivity among soot sections maintaining realisability and the expected exponential decay of variance. The solution of the flow field features a time-dependent second moment closure and an elliptic solver. The turbulent non-premixed Sandia C2H4 flame from the International Sooting Flame (ISF) data base was selected as a target along with the KAUST (C2H4/N2) variant of the same flame. Results show that reduced soot diffusion leads to a significant increase in the soot volume fraction RMS and that the correlation coefficient between soot volume fraction and temperature is further reduced in a particle-size-dependent manner. Similar observations are made for correlations between the soot volume fraction and the mass fractions of gas-phase species such as CO, OH, H and C2H2. The results suggest that computational methods that presume explicit (e.g. flame structure related) correlations between such scalars and with soot face leading order modeling challenges. It is also shown that the correlation between CO and soot increases due to oxidation of soot and that DDS leads to a modest downstream shift of PSDs towards larger particles.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

A consistent scheme of the high-speed source term in probability density function methods for supersonic flows

To improve the particle energy accuracy of the probability density function (PDF) method in supersonic flows, a consistent numerical scheme for the high-speed source term in the particle energy equation is put forward. The proposed scheme is designed and computed based on the characteristic format, which shares the same right and left eigenvectors for the Jacobian of the convection term of the system. Therefore, this scheme is fully consistent with the spatial discretization of the inviscid term in the finite-difference solver of compressible flows. To show the advantages of the proposed scheme, a redundant energy equation is solved along with the Euler equation. The consistent scheme and the quasi scheme are numerically tested and compared in several canonical flows. The results show that the high-speed source term calculated by the consistent scheme generates much less numerical oscillation than the quasi scheme around discontinuities. Due to the accumulation of the numerical errors of the high-speed source term in every single iteration, the energy redundantly computed by the consistent scheme agrees better with the Euler results than the other scheme. Following this new high-speed source computation scheme, the large eddy simulation-PDF method is further developed and tested in a shock tube problem interacting with an isotropic turbulent flow and a supersonic temporally developing mixing layer. The results show that PDF with this consistent high-speed source scheme can improve the energy accuracy as well as turbulent combustion in supersonic flows.

Performance Assessment of Chemical Kinetics Neural Ordinary Differential Equations in Pairwise Mixing Stirred Reactor

Abstract The present study aims to assess the potential of the neural ordinary differential equations (NODE) network for reliable and computationally efficient implementation of chemistry in combustion simulations. Investigations are performed using a hydrogen-air pairwise mixing stirred reactor (PMSR). The PMSR is a zero-dimensional case affordable to study combustion chemistry entailing a similar numerical solution procedure as probability density function methods for turbulent combustion simulations. A systematic approach is presented to apply the NODE, solely trained on canonical constant pressure homogeneous reactor data, to predict complex chemistry and mixing interactions in PMSR. The reactor involves combustion of hydrogen in air described by a finite-rate mechanism with 9 chemical species and 21 reaction steps. The NODE network is shown to accurately capture the evolution of thermochemical variables for different mixing and chemical timescales. It also exhibits a significant reduction in numerical stiffness resulting in improving the computational efficiency and enabling the use of explicit solvers for the integration of chemical kinetics. The assessment results based on PMSR show that compared to direct integration of detailed kinetics, the NODE can achieve significant computational time speedup for a comparable accuracy.

Investigation of reaction-induced subgrid scalar mixing in LES/FDF simulations of turbulent premixed flames

Modeling molecular mixing remains a key issue for the filtered density function (FDF) method. We propose a new closure for the mixing timescale of each individual chemical species. A posteriori tests demonstrate the advantages of the proposed model, highlighting its potential to be employed in FDF simulations of turbulent reacting flows.

Magnetic and vibrational amplitude dependences of MRE grid composite sandwich plates

In this work, both magnetic and vibration amplitude dependent properties of magnetorheological elastomer (MRE) grid composite sandwich plates (MREGCSPs) are investigated. Initially, to prove such a nonlinearly dependent phenomenon, a series of characterization tests are performed on the MREGCSP specimens with different magnetic field intensities and base vibrational amplitudes. Then, using the Jones-Nelson nonlinear material theory, the strain energy density function method, the complex modulus approach, and the Biot-Savart law, the nonlinear material assumption of MRE is defined. A theoretical model consisting of an MRE grid core and two fiber-reinforced polymer (FRP) skins is also proposed to obtain the solutions for the nonlinear frequency, damping, and vibration response parameters, which is based on the modified first-order shear deformation theory, the energy principle, the eigenvalue increment method, the Newmark- beta approach, etc. Finally, a detailed comparison of magnetic and vibrational amplitude dependent natural frequencies, loss factors, and vibration responses is performed to confirm the effectiveness and superiority of the current model over a linear model. This provides a solid basis to reveal the nonlinear dynamic mechanism of the studied smart structure subjected to complex excitation loads. Also, some practical suggestions are summarized for improving its active vibration control capability.

Zone-adaptive modeling of turbulent flames with multiple chemical mechanisms

A novel zone-adaptive modeling method (AdaCM) with multiple chemical mechanisms is proposed for turbulent flames to achieve high-fidelity, yet computationally efficient, predictions. Specifically, a computational economical species transport model, e.g., the well-mixed model, via finite volume algorithm is employed with a simple mechanism as the base model for the whole computational domain, while the advanced transported probability density function (TPDF) method via Lagrangian particle tracking is employed with a detailed mechanism only for spatial regions with intense turbulence-chemistry interaction (TCI), denoted as the “PDF regions”. The PDF regions are dynamically identified based on local flow and flame characteristics and may evolve with time. A two-way particle/finite volume submodel coupling is formulated to ensure the composition consistency between submodels in the PDF regions and impose the correct interface conditions for composition and mass flow rate on the boundary of the PDF regions. With regard to transformation between different species representations in the mechanisms, a species reconstruction/reduction approach based on constrained chemical equilibrium is proposed to ensure element conservation and an adequate specification of unrepresented species at the model interface. The proposed adaptive modeling method has been applied to the well-known Sandia Flame D, in which the well-mixed combustion model with a 6-species, two-step global mechanism is employed as a base model and the high-fidelity TPDF with a 25-species skeletal mechanism is employed for regions with intense TCI. Results demonstrate the consistency in PDF regions between submodels with two distinct mechanisms. The predictions from the adaptive modeling are almost identical to those of TPDF and agree well with experimental measurements, illustrating the preservation of prediction accuracy in the adaptive method. In addition, the total number of computational particles in AdaCM for Flame D is only 18% of that for the stand-alone TPDF, and the recorded computational speedup is about 2.8.

Transported JPDF modelling and measurements of soot at elevated pressures

Accurate measurements and modelling of soot formation in turbulent flames at elevated pressures form a crucial step towards design methods that can support the development of practical combustion devices. A mass and number density preserving sectional model is here combined with a transported joint-scalar probability density function (JDPF) method that enables a fully coupled scalar space of soot, gas-phase species and enthalpy. The approach is extended to the KAUST turbulent non-premixed ethylene-nitrogen flames at pressures from 1 to 5 bar via an updated global bimolecular (second order) nucleation step from acetylene to pyrene. The latter accounts for pressure-induced density effects with the rate fitted using comparisons with full detailed chemistry up to 20 bar pressure and with experimental data from a WSR/PFR configuration and laminar premixed flames. Soot surface growth is treated via a PAH analogy and soot oxidation is considered via O, OH and O2 using a Hertz-Knudsen approach. The impact of differential diffusion between soot and gas-phase particles is included by a gradual decline of diffusivity among soot sections. Comparisons with normalised experimental OH-PLIF and PAH-PLIF signals suggest good predictions of the evolution of the flame structure. Good agreement was also found for predicted soot volume statistics at all pressures. The importance of differential diffusion between soot and gas-phase species intensifies with pressure with the impact on PSDs more evident for larger particles which tend to be transported towards the fuel rich centreline leading to reduced soot oxidation.

Consistent submodel coupling in hybrid particle/finite volume algorithms for zone-adaptive modelling of turbulent reactive flows

A hybrid particle/finite volume algorithm has been formulated for zone-adaptive modelling of turbulent reactive flows to achieve both high fidelity in predictions and high computational efficiency. Specifically, a computationally economical species transport model via finite volume algorithm is employed as the base model for the whole computational domain, while the advanced transported probability density function (TPDF) method via Lagrangian particle tracking is employed only for regions with intense turbulence-chemistry interaction. The ‘PDF regions’ can be updated dynamically based on local flow and flame characteristics, and are compatible with complex geometric structures such as separated multi blocks, non-convex, and multi-connected regions. A two-way particle/finite volume submodel coupling is formulated to ensure the composition consistency in submodels in the PDF regions and to impose the correct interface conditions for composition and mass flow rate on the boundary of the PDF regions. The spatial partition and particle algorithms for time-varying PDF regions are demonstrated and the convergence characteristics of the adaptive modelling are investigated specifically for the variation of statistical error and bias with the number of particles per cell. The proposed zone-adaptive hybrid particle/finite volume algorithm has been numerically validated in a turbulent hydrogen/air non-premixed jet flame. It is shown that the predictions from zone-adaptive modelling are almost identical to those of stand-alone TPDF, illustrating the preservation of prediction accuracy but with significantly less computational cost.

A computational study of the inclusion of β-cyclodextrin and nicotinic acid: DFT, DFT-D, NPA, NBO, QTAIM, and NCI-RDG studies.

Forming complexes with cyclodextrins can protect nicotinic acid (vitamin B3) from premature metabolism and enhance the solubility and stability of this drug. In this work, the formation of the inclusion complex of the neutral form of nicotinic acid and β-cyclodextrin was achieved. The complex is modeled using PM3, PM6-D4H3, and PM7, by considering two orientations of the guest: A and B, one is from wide to narrow rim, and the second is from narrow to the wide rim, respectively. The global minima positions were re-optimized using three density function methods: MN-15, B3LYP, and PW6B95-D3 with polarized Pople basis set 6-31G(d) in gas and aqueous phase. Orientation A showed the minimum complexation energy where the carboxylic functional group of nicotinic acid is located on the primary hydroxyl rim of β-cyclodextrin and the pyridine ring is totally embedded in the cavity. To further our study on the nature of complexation and the interactions of this host-guest system, different calculations were done. The reactivity indices showed that orientation A is harder than B and more electrophilic; the charge transfer occurred from the host to the guest and was confirmed by the natural population analysis (NPA). The natural bond orbitals (NBO) reveal the delocalization of orbitals between the host and the guest, quantum theory of atoms in molecules (QTAIM) analysis, and non-covalent interaction (NCI) analysis based on a reduced density gradient (RDG) give a detailed description of the nature of interactions between the host and the guest such as the hydrogen bonding and van der Waals interaction, and confirmed the stability of the complex given by the orientation A.

Machine learning tabulation of thermochemistry of fuel blends

The objective of the present work is to develop a machine learning tabulation methodology for thermochemistry that accounts for fuel blends. The approach is based on the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) methodology (Ding et al., 2021), the essence of which is to train a set of artificial neural networks (ANNs) using random data so as to anticipate the composition space encountered in turbulent flame simulations. As such, it is applicable to any combustion modelling approach that involves direct coupling of chemistry and flow, such as transported probability density function (PDF) methods, direct numerical simulation (DNS), conditional moment closure (CMC), unsteady flamelet, multiple mapping closure (MMC), thickened flame model, linear eddy model (LEM), partially stirred reactor (PaSR) as in OpenFOAM and laminar flame computation. In this paper, the HFRD approach is further developed to generate data of varying fuel ratios. Furthermore, radiative heat losses are included and it is shown that the ANN-based simulations are able to account for it. The ANNs generated are first tested on 1-D laminar flame simulations and then applied to two turbulent flames with different fuel compositions: a pure methane flame, Sandia flame D, and Sydney flame HM1, which is a methane/hydrogen flame. The results of species mass fraction and temperature are compared between ANN and direct integration, and excellent agreement are achieved. These results indicate that the methodology has great capacity for generalisation and is applicable to a range of blended fuels. Furthermore, a speed-up ratio of 14 to 17 is attained for the reaction step compared with direct integration, which greatly reduces the computational cost of turbulent combustion simulations.

A hybrid stochastic Lagrangian – cellular automata framework for modelling fire propagation in inhomogeneous terrains

A stochastic model motivated by the Lagrangian transported probability density function method for turbulent reacting flows and the cellular automata approach for forest fires was put together to simulate propagation of fires in terrains with inhomogeneous composition. In contrast to the usual cellular automata models for fires where the probability of ignition is prescribed, here the ignition of cells is determined by a random walk that mimics turbulent convection and diffusion of the hot gases and firebrands from upwind and neighbouring fire fronts. Radiation is also included. The model is aimed at speed of computation while approximating the key physics through only a few terrain-related inputs and tunable parameters representing fire intensity, hot gas and ember decay timescales, cell ignition delay and local turbulence. These parameters were calibrated against controlled fire experiments and the model was then used to give reasonable predictions for fires of increasing complexity. The presented framework allows improvements for more accurate representation of the flammable material characteristics, fire-induced flow modifications, and most other phenomena present in fires, hence providing an extendable and simple yet physically-realistic novel modelling approach.

Study on spatial-temporal characteristics and influencing factors of urban environmental resource efficiency in the Yangtze River Basin of China

There is a close relationship between environmental resource efficiency and the high-quality development of the river basin economy. Improving urban environmental resource efficiency is of great significance to the high-quality development of the river basin. Based on the environmental panel data of cities in the Yangtze River Basin from 2004 to 2020, this paper applies the DEA-Malmquist index model to explore the static characteristics and dynamic changes in the overall environmental resource efficiency of cities in the Yangtze River Basin. Then, the spatial and temporal aspects of the environment are discussed by combining the kernel density function method. The Tobit regression model is used to analyze the factors affecting the environmental resource efficiency of cities in the Yangtze River Basin and its importance. Finally, the grey prediction model is utilized to predict the undesirable output data of cities in the Yangtze River Basin from 2023 to 2025. The results show that the overall urban environmental resource efficiency level in the Yangtze River Basin is high. However, the number of cities that achieve DEA efficiency is less than that of non-DEA efficient cities. The Total Factor Productivity (TFP) of cities shows a “high-low-high” trend, and the technical efficiency change (Effch) and pure technical efficiency change (Pech) have an improvement trend. Industrial structure, regional factors, and openness are positively correlated with environmental resource efficiency, during the economic scale and environmental governance level negatively moderate environmental resource efficiency. The forecast results show that the undesired output of cities in the Yangtze River Basin has been somewhat controlled.

An indicator to quantify the complexity of signals and surfaces based on scaling behaviors transcending fractal

In the well-established fractal scheme, the fractal dimension (D) is a central indicator of the complexity of fractal features. The D values of non-fractal signals and surfaces are 1 and 2, respectively, while there can be varieties in their complexities. In this study, the scaling characteristics of root-mean-squared roughness could exhibit a continuous variation transcending the boundary between fractal and non-fractal by using the roughness scaling extraction (RSE) method proposed in previous study, and an universal indicator (HRSE, Hurst exponent calculated by RSE method) to quantify the complexity of both fractal and non-fractal features is demonstrated. The actual signals (milling vibration) and surfaces (silver thin films) together with the artificial ones generated through Weierstrass–Mandelbrot (W–M) function were analyzed. Within the fractal scope, the calculated results with RSE method could be close to the ideal D values of W–M function with an accuracy higher than those of the traditional fractal methods (including Box-Counting, Higuchi, Katz, power spectral density, structure function, and autocorrelation function methods). For the non-fractal features, the complexity could also be quantified effectively by HRSE. Chatter could be recognize by HRSE of milling vibration signals, because it was larger than 1, from 0.5 to 1, and less than 0.5 in idling, stable milling and chatter milling states, respectively; For thin film surfaces, HRSE increased monotonically from 0.79 to 1.32 along with Sq increasing, indicating a strong positive correlation. The findings indicated that the scaling analysis could be utilized for both fractal and non-fractal features, which would be beneficial for various engineering applications.

Efficient treatment of secondary kinetic processes for pre-partitioned adaptive chemistry approaches

Probability Density Function (PDF) methods, which allow for the direct integration of chemical kinetics, are well established to accurately simulate turbulent flames with strong turbulence-chemistry interactions. While adaptive chemistry techniques have been proven effective in reducing the high CPU cost and memory requirements associated with the handling of chemistry in such simulations, performance metrics have mostly been focussed on the primary oxidation pathways converting fuel to major products. In contrast, this work investigates the ability of adaptive techniques, in this case, the pre-partitioned adaptive chemistry (PPAC) approach, to handle secondary kinetics pathways that are parallel, but tightly coupled to the main oxidation process, taking formation as a case study. PPAC relies on a partitioning of the composition space into a user-specified number of regions, on which specialised reduced models are generated using the Directed Relation Graph with Error Propagation (DRGEP) reduction technique. The direct application of that methodology to a mix of hydrocarbon oxidation and nitrogen-related targets is shown to yield excessively detailed region-specific reduced mechanisms in order to properly capture both the main oxidation and the secondary formation processes, thereby decreasing the benefits of the adaptive approach. To address this issue, a sequential approach is proposed for the generation of the region-specific reduced mechanisms, in which the primary combustion pathways relevant for each region are identified first, followed by the selective addition, directly at the reduced level, of any secondary pathways relevant for that region using a recently developed build-up technique. This new strategy is assessed in the context of propane combustion in a partially stirred reactor (PaSR) and methane combustion in the Sandia Flame D configuration, demonstrating in both cases the benefits of the sequential approach for reduced model generation.

Using a flamelet generated manifold method in transported probability density function modeling of soot formation and thermal radiation

The simple semi-empirical precursor soot model of Brookes and Moss based on the soot number density and soot mass concentration is adopted in a transported probability density function (PDF) method for turbulent diffusion flames. The gas phase chemistry is described by a flamelet generated manifold (FGM) based on the mixture fraction, progress variable and enthalpy loss. The accuracy of the FGM method is validated by using flamelet solutions that are not included in the generating set of the FGM. To account for the radiative heat transfer in the flames, we use a non-gray weighted sum of gray gases model for the gas radiation and a gray soot radiation model. Turbulence–radiation interaction is closed at the level of the optically thin fluctuation approximation and the Reynolds averaged radiative transfer equation is solved by means of a discrete transfer method. The proposed modeling approach is applied in simulations of two turbulent non-premixed methane–air flames at one bar and three bar pressure, respectively. Predictions of the mean temperature and mean soot volume fraction are in good agreement with the measurements in the one bar flame. In the higher pressure flame the mean soot volume fraction is over predicted. For this flame, simulation results using the semi-empirical model of Lindstedt provided better agreement with the measurements. The main difference between the Brookes and Moss model and the Lindstedt model is the nine-times increased soot particle agglomeration rate of the latter. When using the same increased agglomeration rate parameter in the Brookes and Moss model the results become virtually identical. The negligible molecular diffusion of the soot was accounted for by neglecting mean molecular diffusion of the soot variables and by greatly reducing their micro-mixing. The effect of this differential soot diffusion on the mean soot volume fraction is found to be small, but it is significant for the variance of the soot volume fraction.

A mixing timescale model for differential mixing in transported probability density function simulations of turbulent non-premixed flames

The modeling of scalar mixing timescale remains a primary challenge in the transported probability density function (TPDF) method. The variation of scalar mixing timescale among species, i.e., differential mixing, results from the difference in molecular diffusivity and reaction-induced scalar gradient. Nevertheless, the vast majority of TPDF studies on turbulent non-premixed flames simply apply a single mixing timescale determined by the mixture fraction. In this work, a reaction-induced differential mixing timescale (RIDM) model for the mixing timescale of individual species in turbulent non-premixed flames is proposed. The key idea of the RIDM model is to approximate the relative magnitude of the species dissipation rates by using their values in laminar flamelets. A direct numerical simulation dataset of a temporally evolving non-premixed ethylene flame is employed to thoroughly evaluate the model performance via a priori and a posteriori tests. Results show that specifying a single mixing timescale for all species results in a poor prediction of the species dissipation rate and thus the failure to predict the overall combustion process. By accounting for the difference in molecular diffusivity, a slightly better prediction can be obtained, but the improvement is very limited, illustrating that simply modeling the difference due to molecular diffusivities for differential mixing is not sufficient. In comparison, the RIDM model exhibits superior performance in both a priori and a posteriori tests. Moreover, all the components of the RIDM model are readily available in the TPDF method, making the RIDM model a promising candidate employed in practice.