多相分散体系中气泡/液滴聚并和破碎的群平衡模拟

多相分散体系中气泡/液滴聚并和破碎过程的模拟对于过程工业中离散相粒径分布的调控具有极为重要的意义。群平衡模型（population balance model, PBM）是模拟离散相聚并和破碎的常用方法。然而由于多相分散体系的复杂性,PBM中现有的聚并和破碎模型通常基于现象模型、统计分析、经验关联式或半理论半经验方法,尚未有模型能够全面地考虑流场和物性对聚并或破碎过程的物理约束,从而使准确预测各类工况的聚并与破碎现象成为挑战性的课题。本文总结了目前针对气-液/液-液体系破碎和聚并过程的机理分析、主流模型、群平衡方程的求解方法以及PBM在气-液和液-液体系中的应用,并评述了各种模型的研究现状和未来发展方向。     

Bubble Coalescence Modeling in the Population Balance Framework

In the churn-turbulent bubbly flow regime with highly nonuniform bubble size distributions, bubble breakage and coalescence are important processes because they govern the bubble size distribution and consequently directly affect the interfacial mass, momentum, and heat transfer fluxes through the renewal bubble surfaces. At present, accurate prediction of bubble size distributions of dispersed gas–liquid flows by use of the population balance (PB) equation is a difficult task. The modeling of bubble breakup and coalescence rates is very complex and is based on the knowledge of collision and breakup frequencies, breakage daughter size distributions, and probability of coalescence. In this work, we focus on the coalescence phenomenon. The coalescence models are still on an empirical level and the mechanisms are not fully understood. This motivates the analysis of the suitability of the coalescence closures for the prediction of experimental data obtained from coalescence dominated gas–liquid flows. For this task, a cross-sectional averaged combined multifluid-PB model is adopted. Based on different theories for the coalescence efficiency, the simulation results show a similar trend in the prediction of the experimental data. Good prediction of the Sauter mean diameter is achieved although the shape of the bubble size distribution is not completely reproduced. The second aim of this work is to review the PB framework. Here, focus is placed on the coalescence term and the combined multifluid-PB model based on kinetic theory approach.     

Population Balance Model for Miniemulsion Polymerization, 2

Summary: The predictions of the model developed in Part 1 of this series are compared with experimental values taken from literature. Initially, the method of solution of the population balance equation and the simulation algorithm are given. Various radical entry mechanisms are discussed in adequate detail. Plausible arguments are given to identify the correct radical entry mechanism. An expression to evaluate the radical exit coefficient is given. Model predictions of a number of variables are discussed. These include average number of radicals per particle, particle phase monomer volume fraction, average number of radicals averaged over all particles, monomer volume fraction averaged over all particles, variation of nucleation rate, variation of fraction of droplets nucleated, variation of average diameter, variation of standard deviation, variation of polydispersity index, and development of particle size distribution with time. Finally, model predictions for the variation of conversion with time for five different initiator concentrations, number average diameter, standard deviation and full distribution are compared with experimental values.

Variation in the average number of radicals per particle with time, at different collocation points.     

Coupled Single‐Particle and Population Balance Modeling for Particle Size Distribution of Poly(propylene) Produced in Loop Reactors

A comprehensive model was developed for the PSD of PP produced in loop reactors. The polymeric multilayer model (PMLM) was first applied to calculate the single particle growth rate under intraparticle transfer limitations. In order to obtain the comprehensive model, the PMLM was solved together with a steady‐state particle population equation to predict the PSD in the loop reactors. The simulated PSD data obtained under steady‐state polymerization conditions agreed with the actual data collected from industrial scale plant. The comprehensive model was also used to predict the effects of some critical factors, including the intraparticle mass and heat transfer limitations, the feed catalyst particle size and the catalyst deactivation, etc., on the PSD.

     

A Novel Approach to Population Balance Modeling of Reactive Polymer Modification Leading to Branching

The mathematical treatment of polymer modification systems, described by population balances containing convolution is discussed. The two‐dimensional case (molecular weight vs. number of branch points) was considered by utilizing approximations of distributions, expanding them in terms of Gaussian basis functions. Three branching reactions were addressed: chain backbone to chain end point coupling; three‐functional coupling of chain ends; and crosslinking. The results were compared to those of Monte Carlo (MC) simulations. Good agreement was observed, although the quality of a distribution as generated by the numerical approach is much better in view of the strong scatter in the MC data.

     

Population Balance Model for Emulsion Polymerisation Under Pseudo-Bulk Conditions: Combined Particle Size Distribution and Molecular Weight Distribution

A comprehensive model for emulsion polymerisation is presented, accounting for particle size distribution (PSD) and molecular weight distribution (MWD). The PSD information is incorporated through a population balance framework. A mechanistic formulation is adopted in modelling the average number of radicals/particle under pseudo-bulk compartmentalisation conditions. The method of moments is adopted to simplify the MWD equations over each discrete size class. The impact of the pseudo-bulk assumption on the PSD and MWD results is assessed. An identification of potential manipulated variables for control of PSD and MWD is done through sensitivity analysis.     

Numerical Solution of the Drop Population Balance Equation Using Weighted Residual and Finite Volume Methods

This article presents a comparison of numerical results obtained by two different approximations of population balances—the spectral orthogonal collocation and finite volume methods. In particular, the population balance equation for a homogeneous dispersed liquid–liquid system in a batch reactor was considered in the present numerical study. The focus was placed on the accuracy of the numerical approximation of the particle property density distribution. An advantage of the finite volume method is the easy of distributing the points in a nonuniform discretization. It is supposed that the spectral-element orthogonal collocation method may benefit by dividing the computational domain into elements of various polynomial orders. For the present problems studied, the orthogonal collocation in the spectral framework does not perform as well as the finite volume method.     

Population Balance Model for Batch Gravity Separation of Crude Oil and Water Emulsions. Part I: Model Formulation

A population balance model for the separation of emulsions in a batch gravity settler is presented. Within the context of the model development, possible methodologies were elucidated for incorporating a) the physical properties of the bulk liquids, b) the physical properties of the phase interface, and c) the presence and functioning of interfacially active compounds. The model presented explicitly accounts for interfacial coalescence and the deformation of the emulsion zone due to the dynamic growth of the resolved dispersed phase; interfacial coalescence is modeled in terms of physically meaningful film drainage models and an approach for incorporating the accumulated buoyancy force in the dense packed layer is also discussed. Hydrodynamically hindered sedimentation is also considered in the model. The model is well suited to predict experimental batch settling data, especially data obtained from low-frequency NMR measurements of emulsion destabilization. The model predicts the evolution of the volume fraction of the dispersed phase at any axial position and time in the separator. Furthermore, the model predicts the location of the resolved dispersed phase interface as a function of time. Additionally, for any axial position and time in the settler, the model predicts the evolution of the average number density of droplets, the average volume/radius of droplets, the standard deviation of the droplet volume/radius, and the rate of growth of the droplets. The model is compared directly with experimental data for crude oil separations in Part II of this article.     

Chemo- and Regioselective Additions of Nucleophiles to Cyclic Carbonates for the Preparation of Self-Blowing Non-Isocyanate Polyurethane Foams

Polyurethane (PU) foams are indisputably daily essential materials found in many applications, notably for comfort (for example, matrasses) or energy saving (for example, thermal insulation). Today, greener routes for their production are intensively searched for to avoid the use of toxic isocyanates. An easily scalable process for the simple construction of self-blown isocyanate-free PU foams by exploiting the organocatalyzed chemo- and regioselective additions of amines and thiols to easily accessible cyclic carbonates is described. These reactions are first validated on model compounds and rationalized by DFT calculations. Various foams are then prepared and characterized in terms of morphology and mechanical properties, and the scope of the process is illustrated by modulating the composition of the reactive formulation. With impressive diversity and accessibility of the main components of the formulations, this new robust and solvent-free process could open avenues for construction of more sustainable PU foams, and offers the first realistic alternative to the traditional isocyanate route.     

影响高浓度氧化石墨烯分散液流变行为的重要因素及群体平衡动力学分析

氧化石墨烯(GO)片的基面和边缘上存在大量的含氧官能团,能很好地分散在水中,因而具有很好的加工性和广阔的应用前景。在较高浓度范围下,GO水分散液中存在着强烈的竞争性相互作用,从而对流变行为产生较大影响。在本文中,通过稳态、动态等流变实验以及理论分析,研究了pH值、温度和不同的有机溶剂对GO分散液流变行为的影响。结果表明,降低pH值、适当增加温度以及加入吡啶均可促进GO水分散液从粘弹性液体到凝胶态的转变。利用DLVO (Deryagin-Landau-Verwey-Overbeek)理论,探讨了GO片之间的范德华作用力以及双电层排斥作用的相互关系,及其对流变性能的影响。通过群体平衡模型(PBE)分析了GO分散液的屈服应力与体积分数的正相关关系。同时,通过蠕变和松弛实验发现,高浓度的GO分散液中结构变化及流变行为在很多方面与高聚物相似,利用Poyting-Thomson模型能较好地拟合其粘弹性行为。上述研究结果为深入研究复杂的GO分散体系提供理论支撑和实验依据。     

Chemo‐ and Regioselective Additions of Nucleophiles to Cyclic Carbonates for the Preparation of Self‐Blowing Non‐Isocyanate Polyurethane Foams

Polyurethane (PU) foams are indisputably daily essential materials found in many applications, notably for comfort (for example, matrasses) or energy saving (for example, thermal insulation). Today, greener routes for their production are intensively searched for to avoid the use of toxic isocyanates. An easily scalable process for the simple construction of self‐blown isocyanate‐free PU foams by exploiting the organocatalyzed chemo‐ and regioselective additions of amines and thiols to easily accessible cyclic carbonates is described. These reactions are first validated on model compounds and rationalized by DFT calculations. Various foams are then prepared and characterized in terms of morphology and mechanical properties, and the scope of the process is illustrated by modulating the composition of the reactive formulation. With impressive diversity and accessibility of the main components of the formulations, this new robust and solvent‐free process could open avenues for construction of more sustainable PU foams, and offers the first realistic alternative to the traditional isocyanate route.     

Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of “how well do we know our parameters, along with their uncertainties?.” The results reveal that Bayesian inversion statistical analysis on an example, prototype nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a “Gold Standard” Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.     

A kinetic modeling of particle formation by gas antisolvent process: Precipitation of aspirin

Population balance equations (PBEs) are often integro-partial differential in nature due to complexities involved in nonconventional crystallization processes, especially gas antisolvent (GAS). The reason is that they include phenomena such as primary nucleation, secondary nucleation, crystal growth, agglomeration, and/or breakage of crystals. Therefore, the solution to such models has become rather difficult. Considering these difficulties, a powerful numerical algorithm was adopted in this paper to treat the population balance model for the precipitation of aspirin by the GAS process. This method was the combination of Lax–Wendroff and Crank–Nicholson numerical methods. It was used to investigate the effect of significant operating parameters, that is, antisolvent addition rate, process temperature, and solute concentration, on the final product properties for two solvents. The antisolvent addition rate was varied between 8 and 40 bar/min, the process temperature was kept constant at levels 37°C and 42°C, the solute concentration was manipulated at two levels, namely, 0.2 and 0.27 g solute/g solution, and methanol and acetone were used as the organic solvents. The results indicated the successful performance of the applied method in treating PBE, since smooth particle size distributions were produced, which were in an acceptable agreement with the experimental data of the investigated system.     

A Graphite‐Polyurethane Composite Electrode for the Analysis of Furosemide

A graphite‐polyurethane composite electrode has been used for the determination of furosemide, a antihypertensive drug, in pharmaceutical samples by anodic oxidation. Cyclic voltammetry and electrochemical impedance spectroscopy were used to characterize the electrooxidation process at +1.0 V vs. SCE over a wide pH range, with the result that no adsorption of analyte or products occurs, unlike at other carbon‐based electrode materials. Quantification was carried out using cyclic voltammetry, differential pulse voltammetry, and square‐wave voltammetry. Linear ranges were determined (up to 21 μmol L^?1 with cyclic voltammetry) as well as limits of detection (0.15 μmol L^?1 by differential pulse voltammetry). Four different types of commercial samples were successfully analyzed. Recovery tests were performed which agreed with those obtained by spectrophotometric evaluation. The advantages of this electrode material for repetitive analyzes, due to the fact that no electrode surface renewal is needed owing to the lack of adsorption, are highlighted.     

表面作用力孔流通模型计算高分子滤过膜孔径大小及分布的方法

综述了一种建立在表面作用力孔流通模型基础上计算高分子多孔滤过膜孔径大小及分布的方法。     

糖酵解模型双曲型反应-扩散方程的非线性行为

The Stability and chemical oscillation of the hyperbolic reaction-diffusion equations for glycolysis model are studied and compared with that of the corresponding parabolic equations. The results show that the parabolic equation is the limiting case of the hyperbolic system when the reaction-diffusion number Nrd →∞, and that the divergence of the wave speed, which exists in the parabolic system, does not appear in the hyperbolic one. The stabilities of these two systems are significantly different. The hyperbolic system may exist in chaos state under certain conditions. It is shown that the hyperbolic system is more suitatle to be used as the model for studying chemical oscillations.     

Initial stages of soot formation in thermal pyrolysis of acetylene. II. A model for the incipience and growth of soot particles

A model is developed which describes simultaneously occurring processes of the initial hydrocarbon pyrolysis, nucleation, surface growth, and coagulation of soot particles. The model permits one to find the size distribution of the primary soot particles up to size 30–40 nm using a relatively small set of equations. The computed time dependence of soot particle concentration agrees satisfactorily with available experimental data. The existence of two limiting stages of the soot formation is revealed.     

基于新球型微观结构模型模拟PEMFC催化层

建立了一个新球型催化层微观结构模型, 并基于此模型对质子交换膜燃料电池(PEMFC)性能进行了模拟. 该模型中假设催化层由Pt/C 颗粒和离子聚合物-孔混合相组成. 假设Pt/C 颗粒为球形结构, 其直径符合正态分布, 用不同直径的球来表示随机分散在电极中的Pt/C 颗粒. 计算了催化层内的传递和电化学反应, 研究了质子和氧气及电化学反应速率在电极厚度方向上的分布, 并且通过对比氧气浓度、过电位和电化学反应速率的分布、极化曲线及催化剂利用率等证明了适当的电极厚度与Pt/C颗粒粒径有利于提高电池性能.     

Thermal Dispersion Model Compared with Euler-Lagrange Approach in Simulation of Convective Heat Transfer for Nanoparticle Suspensions

Convective heat transfer characteristics of water/Al₂O₃ nanofluid flow inside a tube were evaluated in this study. A non-uniform concentration distribution was used in thermal dispersion model. Meanwhile, an experimental study was done to find the dispersion coefficient in addition to assess the accuracy of simulation results. The accuracy of the results of thermal dispersion model was compared with the numerical solution using discrete phase modeling and homogenous method, while the effective parameters on particle migration were considered to find the particle distribution for being used in the dispersion model. Non-uniformity of the particle distribution is increased by raising volume fraction and Reynolds number. Concentration distribution was obtained using discrete phase method and was compared with the distribution employed for the dispersion model. When a uniform concentration is used in the dispersion model, error of prediction is expected to be increased. The thermal dispersion model, in which the particles have followed a non-uniform distribution, provides acceptable results in spite of its lower calculational time as compared to the two-phase approach.