Modeling Continuous Powder Mixing by Means of the Theory of Markov Chains

This article demonstrates the efficiency of the application of the theory of Markov chains as a tool to model and simulate continuous powder mixing to aid in better design of such equipment. Markov chain models allow calculating practically all parameters of the process necessary for its characterization, and in particular those related to particle residence time distribution (RTD). Some numerical examples from the model, which are important for better understanding the process, are also included. It is shown that the main factor defining the efficiency of continuous mixing, through the variance reduction ratio (VRR), is the ratio of the mean residence time and the period of inflows fluctuation, rather than the variance of the RTD. Also, the influence of the dimensions of the mixer outlet on the mean residence time, and in turn on the VRR, is examined as another way of improving the design.     

Modeling Continuous Powder Mixing by Means of the Theory of Markov Chains

This article demonstrates the efficiency of the application of the theory of Markov chains as a tool to model and simulate continuous powder mixing to aid in better design of such equipment. Markov chain models allow calculating practically all parameters of the process necessary for its characterization, and in particular those related to particle residence time distribution (RTD). Some numerical examples from the model, which are important for better understanding the process, are also included. It is shown that the main factor defining the efficiency of continuous mixing, through the variance reduction ratio (VRR), is the ratio of the mean residence time and the period of inflows fluctuation, rather than the variance of the RTD. Also, the influence of the dimensions of the mixer outlet on the mean residence time, and in turn on the VRR, is examined as another way of improving the design.     

大型循环流化床底部区域颗粒浓度分布研究

为获得内径400mm、高9100mm的大型循环流化床底部区域颗粒浓度分布规律,利用光导纤维探头研究了该区域颗粒浓度的径向分布。实验结果表明,颗粒浓度径向分布不均匀,呈现出典型的环-核结构。r/R=0.8可以作为环-核分界点。增大颗粒循环速率,颗粒浓度增大,边壁区域增大幅度比中心区域大。增大表观气速,颗粒浓度减小,边壁区域减小幅度比中心区域大。径向不同位置颗粒浓度随轴向高度增加均减小,边壁区域减小幅度比中心区域大。     

Modeling the Multi-Stage Classification in a Fluidized Bed Jet Mill

The research and modeling the classification effects in a fluidized bed opposed jet mill are presented. Particle distribution is the result of the interaction between two stages of classification in this mill: gravity classification above the bed, and centrifugal classification in a mill classifier. The objective of the study is to develop a model of multi-stage classification and apply in to the process in question. The universal algorithm of constructing the matrix model of arbitrary complex classification circuit is proposed and the example of its application to modeling the classification in the mill is presented. The experimental identification of the model parameters was carried out.     

粒径分布对循环床内颗粒速度分布的影响

利用激光相位多普勒粒子分析仪（PDPA），在截面尺寸为100mm×15mm的二维循环流化床实验装置上测量比较了平均粒径基本相同的3种不同粒径分布的玻璃珠颗粒在相同操作气速下的颗粒时均速度、脉动速度与数量密度的截面分布特性；初步考察了粒径分布对气固并流上行两相湍流流动行为的影响问题。     

流化床法制备太阳能级多晶硅用沉积硅粒的性能研究

在流化过程中,为保证硅粒在流化床中分布状态良好,提高硅的沉积产率,同时要达到产生太阳能级纯度的多晶硅,对流化床中流化的硅粒的粒径和纯度要求是非常严格的,本文采用球磨技术,通过正交实验,对球磨时间、球磨介质、球的均匀度工艺参数对沉积硅粒的粒径影响及球磨污染程度和形貌特征进行试验研究。试验表明：选择合适的球磨罐和球磨介质,以高纯工业硅进行实验,制得的硅粒平均粒径处于145—166um和166—180um之间时,在进行流态化试验时,完全满足太阳能级多晶硅用沉积硅粒在流化床中流态化分布的要求。     

Factorial Modeling for Transient Solid Particle Velocity in a Fluidized Bed Combustor Cold Model

ABSTRACT

Transient hydrodynamics phenomena in the fluidized bed combustor (FBC) freeboard have been critical in the past two decades. Within a 152 mm ID FBC cold model, solid particle transient velocities were measured and analyzed with the assistance of advanced laser-based particle image velocimetry (PIV) instrumentation. Two layers of swirling secondary air were injected into the cold model. The PIV system was applied to the FBC cold model to visualize transient solid particle velocity. A series of transient solid particle velocity profiles were generated for the factorial analysis. In each profile, the solid particle velocity vectors (Vx and Vy) for 10 × 10 grids were generated. Analysis of variance (ANOVA) was used to determine the significant factors that affect transient solid particle velocities, time, and position coordinates. Then, the 10¹⁰ factorial design method was used to develop a specific empirical model of transient solid particle velocity in the FBC freeboard, which was in the shape of V_x = f₁(t, x, y) and V_y = f₂(t, x, y).

This unique factorial analysis method proved to be a very effective and practical method to evaluate experimental conditions and analyze experimental results in the FBC systems.     

Factorial Modeling for Transient Solid Particle Velocity in a Fluidized Bed Combustor Cold Model

Transient hydrodynamics phenomena in the fluidized bed combustor (FBC) freeboard have been critical in the past two decades. Within a 152 mm ID FBC cold model, solid particle transient velocities were measured and analyzed with the assistance of advanced laser-based particle image velocimetry (PIV) instrumentation. Two layers of swirling secondary air were injected into the cold model. The PIV system was applied to the FBC cold model to visualize transient solid particle velocity. A series of transient solid particle velocity profiles were generated for the factorial analysis. In each profile, the solid particle velocity vectors (Vx and Vy) for 10 × 10 grids were generated. Analysis of variance (ANOVA) was used to determine the significant factors that affect transient solid particle velocities, time, and position coordinates. Then, the 10¹⁰ factorial design method was used to develop a specific empirical model of transient solid particle velocity in the FBC freeboard, which was in the shape of V_x = f₁(t, x, y) and V_y = f₂(t, x, y).

This unique factorial analysis method proved to be a very effective and practical method to evaluate experimental conditions and analyze experimental results in the FBC systems.     

流化床中颗粒尺寸的模糊测量模型

本建立了一种对流化床中颗粒尺寸进行的模糊模型，在实验室条件下对气－固流化床的颗粒进行了测量，通过功率谱密度对所获得的压力波动进行了分析，并订对颗粒尺寸与压力波动动及其它影响参数的关系进行了研究。     

Modeling of the Transient Particle Velocity Distribution in the Fluidized Bed Combustor (FBC) Riser

The geometrical distributions of transient particle velocity in a fluidized bed combustor (FBC) riser are critical to FBC design and manufacturing. Particle image velocimetry (PIV) instrumentation was applied to visualize the particle transient movement in the area of interest (AOI) of an experimental cold model (152 mm ID × 610 mm height) of an FBC. Sixteen (16) PIV particle velocity profiles were generated for the AOI in a 1.6-second time period. A ten (columns) × five (rows) grid was set for each profile. Three levels of fluidizing air velocity were set for the experiments at 2.477 m/s, 2.677 m/s, and 2.823 m/s respectively. A knowledge-based regression method was applied to generate the empirical model of the two-dimensional particle transient velocity with the consideration of four independent variables, x (x coordinate), y (y coordinate), V_g (fluidizing air velocity), and t (time). This model had a data fitting accuracy of 88%. In addition, this model had a very good validation performance with the data obtained from other tests at different experimental setups.

The modeling method is applied for the first time to the particle movement in FBC risers. The research work proved that the design of the experiment and regression analysis are very effective and practical for evaluating experimental conditions and analyzing experimental results in FBC systems.     

Modeling of the Transient Particle Velocity Distribution in the Fluidized Bed Combustor (FBC) Riser

ABSTRACT

The geometrical distributions of transient particle velocity in a fluidized bed combustor (FBC) riser are critical to FBC design and manufacturing. Particle image velocimetry (PIV) instrumentation was applied to visualize the particle transient movement in the area of interest (AOI) of an experimental cold model (152 mm ID × 610 mm height) of an FBC. Sixteen (16) PIV particle velocity profiles were generated for the AOI in a 1.6-second time period. A ten (columns) × five (rows) grid was set for each profile. Three levels of fluidizing air velocity were set for the experiments at 2.477 m/s, 2.677 m/s, and 2.823 m/s respectively. A knowledge-based regression method was applied to generate the empirical model of the two-dimensional particle transient velocity with the consideration of four independent variables, x (x coordinate), y (y coordinate), V_g (fluidizing air velocity), and t (time). This model had a data fitting accuracy of 88%. In addition, this model had a very good validation performance with the data obtained from other tests at different experimental setups.

The modeling method is applied for the first time to the particle movement in FBC risers. The research work proved that the design of the experiment and regression analysis are very effective and practical for evaluating experimental conditions and analyzing experimental results in FBC systems.     

On Perturbation Bounds for the Joint Stationary Distribution of Multivariate Markov Chain Models

Multivariate Markov chain models have previously been proposed in for studying dependent multiple categorical data sequences. For a given multivariate Markov chain model, an important problem is to study its joint stationary distribution. In this paper, we use two techniques to present some perturbation bounds for the joint stationary distribution vector of a multivariate Markov chain with s categorical sequences. Numerical examples demonstrate the stability of the model and the effectiveness of our perturbation bounds.     

Application of the theory of Markov chains to theoretical study of processes in a circulating fluidized bed

Abstract

The objective of the study is to develop a simple yet informative model to investigate the features of the process of particulate flows formation in a circulating gas-solid fluidized bed. The proposed model is based on the theory of Markov chains. The riser is presented as a chain of perfectly mixed cells. The conversion of particles of a source material into particles of an end product occurs in these cells due to this or that chemical or thermo-physical process. The transportation of material over the chain is controlled by a matrix of transition probabilities. The return flow that comes from a separator is routed to the downer, which is described as a plug flow apparatus that has a time delay between the inflow and outflow issuing from it. The outflow is routed back into the riser. The numerical experiments were carried out for the batch fluidization. It is found that the batch circulating fluidization only has advantages over the dense bed under a small time delay in the downer and a relatively high gas flow velocity in the riser. The results of laboratory tests performed at the circulating fluidized bed demonstrated a good qualitative correlation with the theoretical data.     

Manufacture of Slow-Release Matrix Granules by Wet Granulation with an Aqueous Dispersion of Quaternary Poly(meth)acrylates in the Fluidized Bed

ABSTRACT

Slow-release matrix granules were manufactured in the fluidized bed using an aqueous dispersion of quaternary poly(meth)acrylates (Eudragit® RS 30 D) as binder for granulation. A factorial design was carried out to investigate the influence of the following parameters, spraying rate, applied polymer amount, and inlet air temperature, on various granule properties. Prerequisites for a slow release of the model drug theophylline are high spraying rate, high amount of polymer, and low inlet air temperature. No considerable decrease of the drug release rate can be achieved without a subsequent curing of the dry granules. A clear correlation exists between the moisture content of the fluidized bed, indicated by the terminal moisture content (TMC), and the mean dissolution time for 80% of the drug (MDT₈₀).     

Application of the theory of Markov chains to model heat and mass transfer between stochastically moving particulate and gas flows

A cell model to describe heat and mass transfer between two media, one or both of which are involved in stochastic motion, is proposed. The model consists of two chains of cells to describe 1D stochastic motion of the flows. Each couple of corresponding cells experiences heat and mass exchange at every transition. The model allows describing the transient process as well as the steady-state distribution of parameters of the flows. It allows taking into account such non-linear phenomena as phase transformation (boiling, for example) and the internal heat sources caused, for example, by chemical reactions. The examples of modelling given in the paper demonstrate the ability of the model to describe all necessary features of the process.     

A Model for the Prediction of Concentration and Particle Size Distribution for the Flow of Multisized Particulate Suspensions through Closed Ducts and Open Channels

The model proposed by Karabelas has been modified for the prediction of solid concentration and particle size distributions for the flow of multi size particulate slurries through pipe, 2-D duct, and open channel. The experimental data reported in the literature have been used to establish the limitations of the Karabelas model. Based on extensive analysis of the experimental data, modifications have been proposed in the model to overcome the deficiencies. In particular, the effect of solid concentration on particle settling and turbulent diffusivity has been incorporated into the modified model. The modified model predicts the solid concentration and particle size distribution with reasonable accuracy for all the geometries considered.     

A Model for the Prediction of Concentration and Particle Size Distribution for the Flow of Multisized Particulate Suspensions through Closed Ducts and Open Channels

The model proposed by Karabelas has been modified for the prediction of solid concentration and particle size distributions for the flow of multi size particulate slurries through pipe, 2-D duct, and open channel. The experimental data reported in the literature have been used to establish the limitations of the Karabelas model. Based on extensive analysis of the experimental data, modifications have been proposed in the model to overcome the deficiencies. In particular, the effect of solid concentration on particle settling and turbulent diffusivity has been incorporated into the modified model. The modified model predicts the solid concentration and particle size distribution with reasonable accuracy for all the geometries considered.     

Bayesian Inference and Life Testing Plan for the Weibull Distribution in Presence of Progressive Censoring

This article deals with the Bayesian inference of unknown parameters of the progressively censored Weibull distribution. It is well known that for a Weibull distribution, while computing the Bayes estimates, the continuous conjugate joint prior distribution of the shape and scale parameters does not exist. In this article it is assumed that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameter has a conjugate prior distribution. As expected, when the shape parameter is unknown, the closed-form expressions of the Bayes estimators cannot be obtained. We use Lindley's approximation to compute the Bayes estimates and the Gibbs sampling procedure to calculate the credible intervals. For given priors, we also provide a methodology to compare two different censoring schemes and thus find the optimal Bayesian censoring scheme. Monte Carlo simulations are performed to observe the behavior of the proposed methods, and a data analysis is onducted for illustrative purposes.     

Modeling of the Mechanical Behavior of a Nickel Alloy by Using a Time-Dependent Thermodynamic Approach to Relaxations of Continuous Media

In this paper, we show the capabilities of a thermodynamicapproach to the relaxation of continuous media to describe the behaviorof the Nickel-based alloy IN738LC submitted to various mechanicalloadings at high temperature (850°C). A preliminarysimple modeling (with one spectrum of relaxation times) does not providesatisfactory results. Microscopic observations of the alloy (Sievert etal., 1997) show the existence of several viscosity domains and we takeone of them into account in our modeling of two spectra of relaxationtimes. The results are much better. Finally, we compare our simulationsto those obtained with the classical elastoviscoplastic modeling ofChaboche and we link these two approaches at the conceptual level.