Eulerian Monte Carlo method for the joint velocity and mass-fraction probability density function in turbulent reactive gas flows

A new Eulerian (field) Monte Carlo method for solving an equation that describes the one-time one-point probability density of species mass fractions in turbulent reactive gas flows has been proposed in a previous article. In the present paper, this method is extended to an equation for the joint velocity and mass-fraction probability density function. The method is based on passing from Lagrangian variables used in Lagrangian Monte Carlo methods to Eulerian variables. In this manner, stochastic ordinary differential equations for the Lagrangian trajectories of fluid particles are transformed to partial stochastic equations. As compared to the classical hydrodynamics, the stochastic velocity field satisfies only the mean continuity constraint and not the instantaneous one. As a consequence, one has to introduce a stochastic density, which differs from the physical density but has the same mean value. The case of the mass-fraction probability density is revised. The equations differ from those derived previously: they can be written in divergent form. Both formulations, however, are statistically equivalent. __________ Translated from Fizika Goreniya i Vzryva, Vol. 42, No. 6, pp. 144–155, November–December, 2006.     

Auxiliary error and probability density function based neuro-fuzzy model and its application in batch processes

This paper focuses on resolving the identification problemof a neuro-fuzzymodel (NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary error model and the identification principle based on the probability density function (PDF). The main contribution is that the NFM parameter updating approach is transformed into the shape control for the PDF ofmodeling error.More specifically, a virtual adaptive control systemis constructed with the aid of the auxiliary errormodel and then the PDF shape control idea is used to tune NFM parameters so that the PDF of modeling error is controlled to follow a targeted PDF, which is in Gaussian or uniform distribution. Examples are used to validate the applicability of the proposed method and comparisons are made with the minimum mean square error based approaches.     

湍流非预混燃烧数值模拟的代数二阶矩模型

湍流燃烧数值模拟是研究燃烧的一种重要手段,采用的湍流燃烧模型是否恰当直接影响最终结果的准确性。在湍流燃烧中,化学反应速率不仅取决于当地的组分浓度和温度,而且与组分的湍流脉动也有密切关系。通过对湍流燃烧模型进行探讨,发现代数二阶矩模型（ASOM）能综合考虑湍流和反应动力学因素的影响,而且比其他复杂的模型简单。研究将组分混合速率对化学反应速率的影响在一个修正的代数二阶矩模型（RASOM）中进行考虑,更准确地计算出化学反应速率。为了验证模型的准确性,RASOM模型被应用到Sandia实验室测量的甲烷-空气非预混燃烧（Flame-D）的数值模拟中。模拟得到的结果与实验结果以及修正的涡破碎模型（EBU-A）和原ASOM模型的结果进行了对比,发现RASOM模型的效果较好。     

Geometric figures with the same chord length probability density function: Six examples

The chord length probability density functions for isotropic uniform random chords f(l) have been studied for 12 different geometric figures Ki. The detailed analysis shows: six pairs of different Ki possess the same f_i(l). In greater detail, for the following six figure-pairs (specific length parameter b),1. 60°-angle with side b ↔ equilateral triangle with side length b,2. 90°-angle with side b ↔ square with side length b,3. 60°-wedge of side length b ↔ triangular rod of side length b,4. 90°-wedge of side length b ↔ square rod of side length b,5. 90°-angle with one side b, one infinitely long side e → ∞, ↔ plane stripe of breadth b,6. 90°-wedge with one breadth b, two sides of length e → ∞, ↔ infinite Layer of constant thickness b, the respective functions f_i(l,b) are identical. Thus, without additional shape information, an identification of such a figure via its chord length probability density function (PDF) is not possible. However, in all the cases considered, the length parameter b, involved in the function f(l), can be recognized from the intrinsic behavior of f(l,b).Furthermore, the agreement of the first moments of the respective functions f_i can be verified by use of the extended Cauchy theorem for non-convex figures.     

Maximum‐likelihood maximum‐entropy constrained probability density function estimation for prediction of rare events

This work addresses the problem of estimating complete probability density functions (PDFs) from historical process data that are incomplete (lack information on rare events), in the framework of Bayesian networks. In particular, this article presents a method of estimating the probabilities of events for which historical process data have no record. The rare‐event prediction problem becomes more difficult and interesting, when an accurate first‐principles model of the process is not available. To address this problem, a novel method of estimating complete multivariate PDFs is proposed. This method uses the maximum entropy and maximum likelihood principles. It is tested on mathematical and process examples, and the application and satisfactory performance of the method in risk assessment and fault detection are shown. Also, the proposed method is compared with a few copula methods and a nonparametric kernel method, in terms of performance, flexibility, interpretability, and rate of convergence. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1013–1026, 2014     

A USM turbulence-chemistry model for simulating NOx formation in turbulent combustion

A unified second-order moment (USM) turbulence-chemistry model for simulating NO_x formation in turbulent combustion is proposed. All the correlations, including the correlation of the reaction-rate coefficient fluctuation with the concentration fluctuation, are closed by the transport equations in the same form. This model abandons the series expansion approximation of the exponential term or the approximation of using a product of two single-variable PDFs instead of a joint PDF. The proposed model is used to simulate methane-air jet diffusion combustion and NO_x formation. The combustion prediction results are compared with those using the EBU-Arrhenius model and other two versions of the second-order moment model. The NO_x prediction results are compared with those using the pure presumed PDF model. Validation of predictions using the experimental data given by the Sandia National Laboratory, USA indicates that the proposed model gives better results than other models, and it is much economical than other refined models.     

Boundary layer model for char combustion and gasification

A non-steady boundary layer model is developed for numerical simulation of combustion and gasification of a single shrinking char particle. The model considers mass and energy conservation coupled with heterogeneous char reactions producing CO and homogeneous oxidation of CO to CO₂ in the boundary layer surrounding the char particle. Mass conservation includes accumulation, molecular diffusion, Stefan flow and generation by chemical reaction. Energy conservation includes radiation transfer at the particle surface and heat accumulation within the particle. Simulation results predict experimentally measured conversion and temperature profiles of a burning Spherocarb particle in a laminar flow reactor. Effects of bulk oxygen concentration and particle size on the combustion process are addressed. Predicted particle temperature is significantly affected by boundary layer combustion of CO to CO₂. With increasing particle size, char gasification to char combustion ratio increases, resulting in decreasing particle temperature and increasing peak boundary layer temperature.     

后台阶反应流概率密度函数-雷诺应力法模拟

后台阶流动是计算流体力学领域十分重要的课题,但是后台阶流动与化学反应耦合的数值研究则比较少见.文中使用雷诺应力模型模拟后台阶流动的速度场,使用标量联合概率密度函数(PDF)方程模拟化学反应项,并将化学反应项的结果作为源项引入雷诺应力模型,模拟了后台阶流动中的甲烷与氧气的缓慢氧化反应.结果发现:雷诺应力模型能够很好地模拟...     

用于混煤燃烧控制砷排放模型研究

混煤掺烧是控制燃煤砷排放的有效方式,但由于缺乏相关的配煤模型,限制了该技术的应用。根据燃煤过程中砷的挥发释放机制,提出一种采用砷的释放指数P表征煤燃烧过程砷的释放特性的配煤模型。该模型综合考虑煤的灰分、灰中主要矿物元素含量、各矿物元素对砷的固定系数以及煤中砷含量等因素。研究结果表明,随着煤灰固定系数由23.12增至50.90,煤灰的气相砷吸附量由3.39 mg/g增至6.14 mg/g;随着释放指数P增大,砷的固定率减小,且随着温度升高,两者相关性由900℃的0.67增至1 300℃的0.86。根据P值筛选煤种进行掺烧,当掺混煤种P值差异较大时,掺烧低P值煤种不仅可降低混煤中的砷含量,还能促进高P值煤的砷在灰中富集,促进率达77.14%;掺混煤种P值差异较小时,掺烧低P值煤种会促进砷的释放。本模型可较好地筛选煤种,为混煤掺烧控制砷等痕量元素的排放提供了新的思路。     

The modeling of the combustion of high-ash coal-char particles suitable for pressurised fluidized bed combustion: shrinking reacted core model

An investigation was undertaken involving the combustion of high-ash coal/char particles under conditions suitable for pressurised fluidised bed combustion, in order to evaluate an overall combustion model. The use of very poor quality feedstocks (greater than 40% ash, low calorific value and high sulphur content) in conventional pulverised fuel combustors (PFC) could be technically difficult and un-economical, and has the associated disadvantage of generating gaseous pollutants. Pressurised fluidised bed combustion (PFBC) which is an attractive alternative process and which uses millimetre-sized coal particles is increasing in use on a commercial scale and is the basis for several clean coal technology processes. A Thermogravimetic Analyser (TGA) was used for the experimentation, which was capable of handling relatively large coal/char particles at high pressures and temperatures. Experimentation with prepared coal/chars particles with a diameter of 3 mm at a pressure of 487 kPa and temperatures between 750 and 950 °C was carried out. For the determination of the overall kinetics of combustion it was found necessary to deviate from the established methods (surface-based reaction) and that it was essential to incorporate diffusion in the overall reaction model. Also, the concept of carbon concentration variation in the particle is introduced to account for the effect of high ash content (a mixture of carbon and minerals), instead of assuming pure carbon. This model, which consists essentially of a shrinking reactive core, was found to agree very well with experimental results and all relevant parameters required for an overall rate equation were evaluated. It is also shown that at high temperatures the shrinking reacted core model results approached the results obtained from the conventional shrinking unreacted core model.     

A kinetic-empirical model for particle size distribution evolution during pulverised fuel combustion

Particle size is an essential parameter in pulverised fuel (PF) combustion as many of the problems or further areas of development in these systems are strongly influenced by the fuel and ash size distribution. This is particularly true for dynamic processes like pollutant formation, corrosion, erosion, slagging and fouling and the related decrease of the combustion and boiler efficiency. The evolution of particle size distribution (PSD) is a complex interaction of various competing chemical and physical transformations. Char oxidation, devolatilization and fragmentation, etc. represent first line physical and chemical transformations which can amend the particle size in the radiation zone. The evolution of the PSD represents the convolution of all of these physical and chemical transformations, operating over the entire size distribution. As a consequence, it is difficult to extract the relative importance of all competing size altering processes from the experiments. Various models such as break-up, thermal stress, shrinking core, percolation and particle-population model have been developed by incorporating numerous ash transformation mechanisms to predict the particle size evolution during the pulverised fuel combustion. The present work describes an adaptation of the numerical kinetic-based particle-population balance for predicting particle size evolution during PF combustion developed by Dunn-Rankin and Mitchell. The model is further simplified analytically and validated against experimental results. Several empirical parameters derived from the experiments are incorporated into the model. The resulting simplified PSD evolution model shows good agreement with literature and experimental results, with maximum 10% absolute standard deviation.     

Calculation of the flux density function for protein crystals from small scale settling and filtration experiments

Development and engineering of protein crystals regarding mechanical stability and crystallizability occurs on a small scale. Later in the process chain of industrial production however, filtration properties are important to separate the crystals from mother liquor. Many protein crystals are sensitive to mechanical stress which is why it is important to know the filtration behavior early on. In this study, we analyze settling and filtration behavior of isometric, rod-like and needle shaped lysozyme and rod-like alcohol dehydrogenase (ADH) crystals on a small scale using an optical analytical centrifuge. Needle shaped lysozyme and rod-like ADH crystals show compressible material behavior. With the results from settling and filtration experiments, the flux density function is calculated and modeled which can be used to describe the whole settling and permeation process in dependency of the solids volume fraction. This is also an issue for simulations of industrial processes.     

A double-k-ε and multi-δ pdf model for turbulent gas-liquid reacting flows

The gas-liquid reacting two-phase flow of a gaseous SF₆ jet submerged in liquid metal Li is numerically simulated by a two-fluid model. To describe the changes of the flow pattern in the two-phase flow, a double-k-ε two-phase turbulence model is developed and adopted. A multi-δ-pdf model is proposed and applied to simulate the turbulent combustion of gaseous SF₆ with liquid metal Li. The gas and liquid velocities, void fraction, shape of the penetrating region (the region containing the gas phase) and the mixture fraction related to gas temperature and species concentration are predicted. Comparison with the results using a single-fluid model shows that current predictions are reasonable in reflecting the influence of velocity slip between the two phases and capable of predicting Li/SF₆ submerged combustion.     

Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains

The 2D probability-generating function technique is a powerful method for modeling bivariate distributions of polymer properties. It is based on the transformation of bivariate population balance equations using 2D probability generating functions (pgf) followed by a recovery of the distributions from the transform domain by numerical inversion. A key step of this method is the inversion of the pgf transforms. Available numerical inversion methods yield excellent results for pgf transforms of distributions with independent dimensions with similar orders of magnitude, for example bivariate molecular weight distributions in copolymerization systems. However, numerical problems are found for 2D distributions in which the independent dimensions have very different ranges of values, such as the molecular weight distribution-branching distribution in branched polymers. In this work, two new 2D pgf inversion methods are developed, which regard the pgf as a complex variable. The superior accuracy of these innovative methods makes them suitable for recovering any type of bivariate distribution. This enhances the capabilities of the 2D pgf modeling technique for simulation and optimization of polymer processes. An application example of the technique in a polymeric system of industrial interest is presented.     

Theoretical model for multiple breakup of fluid particles in turbulent flow field

Generalized phenomenological model, based on the theories of probability and isotropic turbulence, is developed for multiple breakup of fluid particles in turbulent flow field. The approach uses a series of successive binary breakup events occur at a time scale comparable to the colliding eddy turnover time. It was found that the use of energy density, instead of energy, will increase the predicted binary breakup rate which is usually underestimated by the existing models in the literature. Generalization of the binary breakup model for multiple fragmentations is performed by defining a “remaining energy function” for the colliding eddy which means the contribution of original eddy to the later breakup events. For ternary breakage, the model shows a reasonably good agreement with the experimental data. The quaternary fragmentation frequency, however, is of negligible importance at lower energy dissipation rates but its contribution to breakage fraction at higher energy dissipation rates becomes considerable. The results also show that ternary and quaternary breakups have a considerable 90% contribution to the overall fragmentation, while pentenary and further fragmentations are of lower importance at low energy dissipation rates. At higher levels of energy dissipation rate, fragmentations up to six daughter particles contribute to more than 95% of the overall fragmentations. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4508–4525, 2016     

富氧燃烧含灰烟气辐射特性的部分光谱k模型

针对煤粉在富氧燃烧方式下烟气为炭黑、飞灰、高浓度CO₂和H₂O蒸气混合物的辐射换热问题,建立了描述混合物非灰辐射特性的部分光谱k分布模型。为了验证模型的有效性,以文献中煤粉在富氧燃烧方式下烟气生成状况为例,采用该模型与离散坐标法相结合的方法,计算了一维无限大平行平板、单光谱行程和二维矩形封闭系统内混合物的辐射源项和壁面热流量,并与逐线计算、全光谱k分布模型和宽带k分布模型的计算结果进行了比较。结果表明：所提出的部分光谱k分布模型与逐线计算的结果吻合较好;与文献报道的两个模型相比,计算精度有所提高,因此适用于富氧燃烧方式下含灰烟气辐射特性的计算。     

A reduced NOx reaction model for pulverized coal combustion under fuel-rich conditions

A reduced NO_x reaction model was developed for analysis of industrial pulverized coal firing boilers. The model was developed from experiments of laminar premixed combustion under a variety of stoichiometric ratios, burning temperatures, coal ranks (from sub-bituminous coal to anthracite) and particle diameters. Calculations agreed with experimental results for NO_x and nitrogen species (NH₃ and HCN), if the model assumed that the hydrocarbon radicals were formed not only from pyrolysis of volatile matter, but also from char oxidation and gasification. The presence of hydrogen in char at the final burnout stage supported this assumption. NO_x reduction by hydrocarbon radicals was the most important reaction in high temperature (>1500 K), fuel-rich, char combustion regions. NO_x reduction from nitrogen species was sensitive to peak NO_x concentration in volatile combustion regions, but NO_x emission downstream had little influence from the peak NO_x concentration. The heterogeneous reaction between char and NO_x was important for fuel-lean or low-temperature conditions.     

基于改进PDF技术的间歇过程NFM模型

间歇过程是一类具有典型复杂非线性特性的生产过程,可以利用模糊神经网络(NFM)建立其输入输出的非线性映射关系。在前期的研究中曾提出过基于概率密度函数(PDF)技术的模型训练方法,成功解决了传统的基于MSE准则训练方法模型泛化能力弱的问题,但又产生了概率密度难以估计及目标PDF未知时模型性能不稳定的问题。针对这两个问题,引入了新的概率密度窗宽估计方法,并提出了在目标PDF未知时采用PDF预估器及其收缩策略的算法。仿真实验表明:该方法能够保证足够的概率密度估计精度和模型预测性能。     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.