Common potholes in modeling solid–liquid reactions—methods for avoiding them

The reaction rate of a solid in fluids depends on the reactive surface area rather than its concentration. Even though, analytical techniques have been developed enormously during the past decades, the reliable quantification of solid surface areas during a reaction still remains a challenge. Due to this, still today indirect methods such as test plots play a key role in determining reaction mechanisms and kinetics. The modeling of solid–liquid reactions is a challenge as several assumptions and simplifications need to be made and distinguishing between different hypothesis is not always straightforward. The influence of the particle size distribution (PSD) and the change in the morphology of the solid phase during the reaction are one of the most crucial factors in determining the kinetics, as they are directly related to the quantification of the reactive surface area. Neglecting to consider these factors in modeling can cause misleading conclusions and wrongful parameter estimation with traditional methodology.Techniques for evaluation when it is adequate to use the traditional methodologies and when these factors need to be accounted for are provided in the current work. If the particle size distribution is close to the Gaussian distribution and if the particles are not very rough or porous, the traditional modeling practices can soundly be used. These properties are measured and quantified with the help of a variation coefficient (PSD) and a shape factor (particle morphology). It is demonstrated that how these factors influence the results obtained with traditional approaches. A practical technique for implementing the PSD into kinetic models by using the Gamma distribution is provided. Moreover, a method for taking into account different particle morphologies with the help of a shape factor and solid phase exponents is presented. The dissolution of gibbsite in NaOH is used as an example case. The methodology can be extended to unconventional changes in particle morphology during the reaction as well as different reaction mechanisms, e.g. product layer formation.     

Mechanistic modelling of kinetics and mass transfer for a solid–liquid system: Leaching of zinc with ferric iron

A systematic approach was developed to consider liquid–solid reactions with rough solid particles and shrinking particle model. The model is able to predict the reactivities of both non-porous and porous solid particles; the reaction order with respect to the solid material varies from zero (non-porous slab) to one (porous particle).As a model system, leaching of zinc sulphide (sphalerite) with ferric iron in an acidic environment was considered. The modelling was based on experimental data obtained in a batch reactor system, for which both conventional mixing and ultrasound was applied. Rival models based on plausible reaction mechanisms were derived and discriminated qualitatively and with regression analysis. The best model described the leaching reaction as a stepwise process, where ferric ions react with solid zinc sulphide in consecutive surface reaction steps. Shrinking particle model along with the surface roughness approach was used. The model predicts first order behaviour with respect to zinc sulphide, while the reaction order with respect to ferric iron varies from one to two as the reaction progresses. This is in accordance with experimental observations. The intrinsic kinetics, liquid–solid mass transfer and the effect of ultrasound were well described by the best kinetic model.     

New modelling approach to liquid–solid reaction kinetics: From ideal particles to real particles

Typical features of liquid–solid reactions were reviewed: reaction kinetics, mass transfer effects and particle morphology. It was concluded that classical liquid–solid models based on ideal, non-porous geometries (sphere, infinite cylinder, slab) cannot satisfactorily describe real reactive solid particles with various surface defects, such as cracks, craters and limited porosity. Typically a too low reaction order for the reactive solid is predicted by the classical models. The surface morphology can be revealed by electron microscopy, which gives inspiration to develop new mathematical models for reactive solids.     

Reactions of solid particles with nonuinform distribution of solid reactant: The volume reaction model

The effect of non-uniform solid reactant distribution on conversion of solid particles in gas-solid reactions is analyzed based on the volume reaction model. Certain special features of such systems are pointed out. The possibility of ash layer formation in the kinetically controlled regime is discussed. Conditions leading to single or double ash layer formation, both at the center and surface of the particle, in the intermediate regime of diffusion with simultaneous reaction are described. Detailed mathematical equations which are useful for calculation of the conversion-time relationship for particles with non-uniform solid reactant distribution are presented. Comparison is made to reaction of uniform particles and differences in required reaction time for desired conversion are outlined.     

流体-固体两相流的数值模拟

鉴于流体 -固体两相流及其数值模拟在化学工程中愈来愈广泛的应用 ,本文综述了Euler坐标系下流体相湍流模型、颗粒轨道模型以及流体 -颗粒双流体模型的基本原理和数值模拟方法 ;对相间耦合和颗粒间的相互作用也进行了介绍 ,特别是对能详细描述多颗粒间相互作用的颗粒离散单元法在流体 -固体两相流中的应用进行了详细描述 ;在评述各模型的优缺点和分析目前存在的问题的基础上 ,提出了今后的发展方向     

高炉铝酸钙炉渣浸出过程动力学

研究了高炉铝酸钙炉渣的浸出动力学,考察了搅拌强度、浸出反应温度、浸出剂初始浓度及炉渣粒度对浸出速率的影响. 结果表明,浸出过程符合一级反应的收缩未反应核模型,宏观动力学方程为1+2(1-xB)-3(1-xB)2/3= 1.108exp(-1906/T)t,表观活化能为15.84 kJ/mol,过程速率为固膜内扩散速率控制. 通过实验数据验证,表明所得模型能较好地描述炉渣的浸出过程.     

A unified Eulerian theory of turbulent deposition to smooth and rough surfaces

This paper presents a simple, unified theory of deposition that is applicable for particles of any size, and reproduces very closely experimentally measured variation in deposition velocity with particle relaxation time. Apart from providing physical insight, the theory offers a simple, fast and reliable computational tool of practical use to aerosol engineers. The predictions are at least as accurate as the state-of-the-art particle-tracking calculations but require much less computational time. The theory includes the effects of thermophoresis, turbophoresis, electrostatic forces, gravity, lift force and surface roughness. The theory consists of writing the particle continuity and momentum conservation equations in their proper form and then performing Reynolds averaging. This procedure results in an expression for the particle flux which consists of three distinct terms for each of which a clear physical interpretation is available. The first term is a diffusive flux due to Brownian motion and turbulent fluctuation, the second is a diffusive flux due to temperature gradient (thermophoresis), and the third is a convective flux that arises primarily as an interaction between particle inertia and the inhomogeneity of the fluid turbulence field (turbophoresis). The lift force and electrostatic forces also contribute to this convective flux. It is shown that it is crucial to include the particle momentum equation in the analysis as this gives an estimate of the mentioned convective slip velocity of the particles. Absence of this equation in many previous studies which included only the particle continuity equation necessitated postulations such as stopping distance models. Only the dominant terms in the continuity and momentum equations are retained in the present analysis which give almost the same answer as with a calculation retaining all terms, but the former is more amenable to direct physical interpretation. The method of Reynolds averaging is general, and other effects not included in this study, e.g. pressure diffusion can easily be incorporated by including the appropriate forces in the particle momentum equation. The present study includes the effects of surface roughness, and the calculations show that the presence of small surface roughness even in the hydraulically smooth regime significantly enhances deposition especially of small particles. Thermophoresis can have equally strong effects, even with a modest temperature difference between the wall and the bulk fluid. For particles of the intermediate size range, turbophoresis, thermophoresis and roughness are all important contributors to the overall deposition rate.     

Computationally efficient incorporation of microkinetics into resolved-particle CFD simulations of fixed-bed reactors

A method is developed to couple microkinetics with fluid flow in a fixed bed and transport inside the catalyst particles using computational fluid dynamics. Initially, the microkinetics model is solved for a wide range of different temperatures, and partial pressures. Next, reaction rates are mapped into quadratic multivariate splines. Splines coefficients are then imported into our user-defined function, and consequently the reaction rates are evaluated at each iteration simultaneously with the CFD simulations. This method has been applied to our solid particle model to investigate the effects of fluid flow, transport and elementary reaction steps on each other for ethylene and methanol partial oxidations. Reaction rates of all elementary steps as well as species surfaces sites and compositions are evaluated inside the particle. The suggested method can couple complex reaction mechanisms with detailed CFD simulations without increasing the computational time compared with global kinetics methods.     

Use of numerical modeling in design for co-firing biomass in wall-fired burners

Co-firing biomass with coal or gas in the existing units has gained increasing interest in the recent past to increase the production of environmentally friendly, renewable green power. This paper presents design considerations for co-firing biomass with natural gas in wall-fired burners by use of numerical modeling. The models currently used to predict solid fuel combustion rely on a spherical particle shape assumption, which may deviate a lot from reality for big biomass particles. A sphere gives a minimum in terms of the surface-area-to-volume ratio, which impacts significantly both motion and reaction of a particle. To better understand the biomass combustion and thus improve the design for co-firing biomass in wall-fired burners, non-sphericity of biomass particles is considered. To ease comparison, two cases are numerically studied in a long gas/biomass co-fired burner model. (1) The biomass particles are assumed as solid or hollow cylinders in shape, depending on the particle group. To model accurately the motion of biomass particles, the forces that could be important are all considered in the particle force balance, which includes a drag for non-spherical particles, an additional lift due to particle non-sphericity, and a “virtual-mass” force due to relatively light biomass particles, as well as gravity and a pressure-gradient force. Since the drag and lift forces are both shape factor- and orientation-dependent, coupled particle rotation equations are resolved to update particle orientation. To better model the reaction of biomass particles, the actual particle surface area available and the average oxygen mass flux at particle surface are considered, both of which are shape factor-dependent. (2) The non-spherical biomass particles are simplified as equal-volume spheres, without any modification to the motion and reaction due to their non-sphericity. The simulation results show a big difference between the two cases and indicate it is very significant to take into account the non-sphericity of biomass particles in order to model biomass combustion more accurately. Methods to improve the design for co-firing biomass in wall-fired burners are finally suggested.     

A UNIFIED COLLOCATION ALGORITHM FOR PACKED-BED CHEMICAL REACTOR SIMULATION

Mathematical models of packed-bed catalytic reactors are aimed to predict the conversions and temperature profiles in both fluid and solid phases within the reactor. Although very general models can be mathematically formulated, usually several simplifying hypothesis are introduced for the fluid phase and/or the solid phase, in order to overcome computational difficulties

We describe in this paper a computational algorithm based on Orthogonal Collocation Method on finite elements, with elimination of the knot unknown functions, coupled with an integration method for stiff ordinary differential equations. This has been used in the development of a computer code, which allows us to find the transient behavior of the reactor by solving the equation relative to the external field, coupled with those describing the transient behavior in the catalyst particles, for a wide class of reactor models. The most general examined model includes axial dispersion in the external fluid phase, interphase mass and heat transfer resistances, intraphase mass resistance and any given kinetic scheme with complex reaction rate expressions.     

The role of colloid chemistry in modeling deep bed liquid filtration

From mass balance on the suspension and filter and a charge balance on the filter, a set of hyperbolic partial differential equations is derived which describes how the suspension concentration, surface charges of the filter and particles, porosity of the filter, and the pressure drop vary with filter depth and time. These equations include a local deposition term which is evaluated by considering transport of the suspension particles to the filter particles by Brownian diffusion, interception, and sedimentation. The effect of the surface forces due to electrical double layer and van der Waals interactions was taken into account by treating the surface of the filter particles as possessing first order intermediate reaction kinetics, for which the rate constant is a function of the stability ratio of colloid chemistry.The governing equations were solved numerically, and the results compared with experimental data for unflocculated particles.The proposed filtration model is an advance over present models in that it contains no empirical factors which must be evaluated from filter runs and the effects of surfactants, pH, and ionic strength are accounted for.     

非稳态非均相扩散反应模型及解法

非稳态非均相流体固体扩散反应模型具有较大的适应性.通常使用的收缩核心模型、两阶段模型和均相模型都是它的特殊情况.对于自钢渣中用碳酸钠和(或)碳酸氢钠溶液浸取钒的过程,考虑了波相双组份(CO_3~(2-) ,HCO_3~-及固相双组份(Ca(VO_3)_2,Ca(OH)_2)以及固体产物覆盖层对钒浸出率的影响,建立了模型方程.模型方程采用正交配置法和半隐态龙格库塔(Runge-Kutta)法求解,参数估算采用复合形法,计算结果和实验值符合良好.多相非催化流体固体反应的数学模型在广泛的领域内有重要的实用意义.对于这一类数学模型已有较好的综述文章.在冶金及其他工业上待提取的固相物料往往嵌布在惰性介质(脉石)上,体现在微观结构上较大的分散性,宜用体积反应的数学模型.     

CFD Evaluation of Droplet Drying Models in a Spray Dryer Fitted with a Rotary Atomizer

Computational fluid dynamics (CFD) modeling of spray dryers requires a simple but sufficiently realistic drying model. This work evaluates two such models that are currently in discussion; reaction engineering approach (REA) and characteristic drying curve (CDC). Two versions of the CDC, linear and convex, drop in drying rate were included. Simulation results were compared to the overall outlet conditions obtained from our pilot-scale experiments. The REA and CDC with a linear drop in drying rate predicted the outlet conditions reasonably well. This is contrary to the kinetics determined previously. Analysis shows that the models exhibit different responses to changes in the initial feed moisture content. Utilizing different models did not result in significantly different particle trajectories. This is due to the low relaxation time of the particles. Despite the slight differences in the drying curves, both models predicted similar particle rigidity depositing the wall. For the first time in a CFD simulation, the REA model was extended to calculate the particle surface moisture, which showed promising results for wet particles. Room for improvement was identified when applying this concept for relatively dry particles.     

连续多级串联完全混合槽中固体颗粒反应产率计算方法

本文针对级内颗粒完全混合一级间无颗粒返混的多级反应器,探讨了按停留时间分布计算固体颗粒反应率的方法。首先分析了用δ（0）函数示踪法与多元独立随机变量和的分布密度函数法求颗粒停留时间分布的特点;应用后一种方法推得了计算颗粒反应产率的一般公式。从固体颗粒固相反应物浓度分布密度函数也可以推得此公式。在此基础上对一类特殊的反应速度方程,解析求出了直接计算任何一级反应产率的通式;对求解计算通式困难的反应速度方程,论证了两种逐级计算方法:以反应时间作计算参量及以颗粒固相反应物浓度作计算参量。     

Design and simulation of a solar chemical reactor for the thermal reduction of metal oxides: Case study of zinc oxide dissociation

A laboratory-scale solar reactor was designed and simulated for the thermal reduction of metal oxides involved in water-splitting thermochemical cycles for hydrogen production. This reactor features a cavity-receiver directly heated by concentrated solar energy, in which solid particles are continuously injected. A computational model was developed by coupling the fluid flow, heat and mass transfer, and the chemical reaction. The reactive particle-laden flow was simulated, accounting for a multiphase model (solid-gas flow). A discrete phase model based on a Lagrangian approach was developed. The kinetics of the chemical reaction was considered in the specific case of zinc oxide dissociation for which reliable data are available. The complete model predicts temperature and gas velocity distributions, species concentration profiles inside the reactor, particle trajectories and fates, and conversion rate assessing the reaction degree of completion. The reaction extent is highly dependent on temperature of the radiation-absorbing particles. Initial diameter of injected particles is also a key parameter because it determines the available surface area for a given particle mass feed rate. The higher the particle surface area, the higher the conversion rate. As a result, reaction completion can be achieved when particle temperature exceeds 2200 K for a initial particle diameter.     

PET固相缩聚反应器动态模型

PET固相缩聚过程涉及反应固体颗粒和反应器床层两个空间尺度的传质,是一个复杂的多维多相对象。以反应过程控制为目标,在已有的反应动力学模型基础上,同时考虑可逆化学反应和小分子产物内外扩散两个控制因素,并借用固定床拟均相模型的建模思想,将小分子产物在颗粒内外扩散形成的浓度梯度用一个有效系数来表示,从而建立了简化的PET固相缩聚反应器一维动态模型。在相关文献的实验条件下求解分布参数模型的数值解,并进行启动过程仿真和过程动态分析,计算反应器出口质量指标,与文献曲线和数据对照,验证了模型的正确性。