A pseudo-similarity solution to the integro-differential equation of batch grinding

A solution to the integro-differential equation of batch grinding has been presented which maps the trajectory of particle size spectra of continuous size and time domains. The solution, which is the most general analytical closed form expression available at present, employs Kapur's similarity solution to the grinding equation along with standard transformation techniques. The resulting size spectra, a modified form of the generalized gamma distribution, is not self-preserving and the energy—size reduction relationship does not conform to Walker's power law. These aspects have been illustrated by simulation studies.     

水泥颗粒分布方程RRB式的研讨

从RRB式模型来源、分布特点及应用方式等方面入手，探讨了RRB式在描述多峰状水泥颗粒分布时存在的问题，并提出了解决这类问题的建议。     

(炯)传递和转化的普遍化动力学方程

能量在传递和转化过程中是守恒的,但不同形式的?在传递和转化过程中有不同程度的损耗,对该过程的深入理解有助于合理而又高效地利用能源。任何形式的能和?都可以被表示为一对基本强度量和基本广延量的乘积。利用基本强度量乘以与其共轭的基本广延量的平衡方程,导出了?传递和转化的普遍化动力学方程。该普遍化动力学方程表示出了任意形式的?在传递过程中与其它形式的?之间的转化关系,由此导出了在工程领域常见的动?、化学?、压?、电?和热?的传递和转化动力学关系式,并给出了系统总?的表述式。这些动力学关系式不仅清楚地反映了系统内部不同形式?之间存在的相互转化关系,而且还定量地反映了各种转化过程的不可逆性。     

Heat transfer and kinetics in the pyrolysis of shrinking biomass particle

The impact of shrinkage on pyrolysis of biomass particles is studied employing a kinetic model coupled with heat transfer model using a practically significant kinetic scheme consisting of physically measurable parameters. The numerical model is used to examine the impact of shrinkage on particle size, pyrolysis time, product yields, specific heat capacity and Biot number considering cylindrical geometry. Finite difference pure implicit scheme utilizing tri-diagonal matrix algorithm (TDMA) is employed for solving heat transfer model equation. Runge-Kutta fourth-order method is used for chemical kinetics model equations. Simulations are carried out for radius ranging from 0.0000125 to , temperature ranging from 303 to and shrinkage factors ranging from 0.0 to 1.0. The results obtained using the model used in the present study are in excellent agreement with many experimental studies, much better than the agreement with the earlier models reported in the literature. Shrinkage affects both the pyrolysis time and the product yield in thermally thick regime. However, it is found that shrinkage has negligible affect on pyrolysis in the thermally thin regime. The impact of shrinkage reflects on pyrolysis in several ways. It includes reduction of the residence time of gases within the particle, cooling of the char layer due to higher mass flux rates of pyrolysis products and thinning the pyrolysis reaction region.     

New approaches to calculate the transfer function of particle mass analyzers

Abstract

This article provides an overview of methods to evaluate transfer functions for the Couette centrifugal particle mass analyzer (CPMA) and aerosol particle mass analyzer (APM). The work first considers finite difference approaches to solving the partial differential equation governing particle motion, which represents an accurate but computationally-demanding approach to evaluating the transfer function. This is used as a baseline to compare to particle tracking methods, which have been shown to yield closed form expressions for the transfer function. In this work, we extend on previous treatments by presenting a generalized framework that allows us to consider a range of representations of the particle migration velocity. As a result, we derive new closed form expressions for the exact representation of the particle migration velocity under APM conditions and provide significant improvements in the accuracy of the transfer function for CPMA conditions. In the latter case, for a CPMA, particle migration effects dominate, which makes the transfer function easier to approximate. We also show that Taylor series approximations to the particle migration velocity should be taken about the centerline radius rather than the equilibrium radius as was done previously. We end by extending the particle tracking approach and derive new closed form expressions for the transfer function that include diffusion.

Copyright © 2019 American Association for Aerosol Research     

The experimental transfer function of the Couette centrifugal particle mass analyzer

The Couette centrifugal particle mass analyzer (CPMA) classifies particles by their mass-to-charge ratio. Unlike the aerosol particle mass (APM) analyzer, the Couette CPMA uses a stable system of forces to improve the transfer function of the classifier. A prototype Couette CPMA has been built and tested. The experimental results from the prototype agree well with theory and it is found that indeed the transfer function of the Couette CPMA is better than the APM's. Experimental work is shown using a differential mobility analyzer (DMA) and Couette CPMA to classify polystyrene latex (PSL) and di-2-ethylhexyl sebacate (DEHS) particles. By measuring the mass of PSL particles the absolute uncertainty of the Couette CPMA was found to be 6.7–38% higher in terms of mass (or 2.2–11% higher in terms of equivalent diameter) than the expected value. However, when the DMA–CPMA system was calibrated with PSL particles the density of DEHS particles was measured to within approximately 3% of the expected value.     

Modelling the morphological evolution of nanosuspension droplet in constant-rate drying stage

A novel mathematical model of constant-rate stage of nanosuspension droplet drying is proposed. In contrast to previously published literature studies, the developed model considers two morphologically different periods of the constant-rate drying: before the shell formation and after the shell formation; the latter was named “transition period”. The point of initial “locking” between nanoparticles on the droplet surface and beginning of the shell formation is associated with theoretical maximum of solid volume fraction. It is postulated that shrinking and thickening shell of nanoparticles occurs fast, and thus the shell virtually remains submerged in the liquid during the overall transition period. Because of the submerged shrinking shell, in the transition period, the evaporation process still takes place from the droplet surface and the drying rate remains unchanged as it was before the shell formation. Correspondingly, the droplet temperature retains at the level of equilibrium evaporation temperature. The developed theory was successfully validated by the published experiment of silica nanosuspension droplet drying. Finally, the developed model proposes a simple morphology criterion based on comparison between the calculated droplet volume at the end of transition period and the corresponding volume of solid final particle with the given porosity.     

外场作用下多组分质量传递Maxwell-Stefan方程及其应用：电场作用下球形颗粒中的扩散

Fick扩散定律只能用于没有外场作用下的二组分扩散过程,Maxwell-Stefan（MS）方程适用于外场作用下多组分系统中的质量传递过程。阐述了MS方程的物理意义,并由此将外场作用下普遍化MS方程的表达式改写为清晰且易于使用的形式。给出了在MS方程中常用的外场作用力的数学形式。Fick扩散定律是MS方程在没有外场作用下对二元理想流体扩散的应用;MS方程对离心分离和电解质扩散等过程的描述比传统的描述方法更全面。利用MS方程导出了欧姆定律,利用该结论可以通过测量电导率计算带电粒子的扩散系数。利用MS方程研究了电场作用下球形颗粒中的扩散过程,结果表明,该方程可以对电场强化质量传递过程给出有效描述。MS方程为外场作用下的质量传递过程提供了一种有效的理论研究方法。     

Heat transfer between particle beds and submerged surfaces

Theoretical modelling of heat transfer to particle beds comprises two sequential steps: transfer from the heating surface to contacting particles followed by transfer to the interior of the bed. Two different limiting case can be formulated for the second step: unmixed and homogeneously mixed bed. In the case, heat is transferred gradually via a repeated sequence of heat transfer in the gap between adjacent particles and conduction in the particulate material. In the second case, heat is transferred to the interior of the bed by mixing of particles which have previously attained the temperature of the heating surface. On the other hand, the mixing motion maintains a homogeneous lower temperature throughout the bed. Theory predicts a significant and easily measurable difference in the behaviour of heat transfer coefficients for the two regimes at long contact times t: unmixed beds ∝ $$ \sqrt t $$ and homogeneously mixed beds ∝ 1/t. For short times t, both regimes show the same behaviour, namely of t. From a theoretical standpoint, it makes sense to differentiate further between the behaviour patterns of unmixed beds: at long times t, instantaneous heat transfer coefficients are independent of heat transfer form the heating surface to adjacent particles. Comparison with experimental result from literature shows that the derived models, which are consistent, are suitable for describing the heat transfer from submerged surfaces to unmixed and mixed beds of particles.     

A general particle stress equation for fixed and fluidised beds

A general constitutive equation is proposed for the solid particle stress in a fluidised bed. This equation reduces in a straightforward manner to describe the particle stress in a fixed bed, and it implies a Mohr-Coulomb yield criterion for the transition from the fixed-bed to the fluidised-bed state. Experimental evidence is given to show that at incipient fluidisation, the particle stress assumes a form agreeing with that predicted by theory.     

Effect of chemical reaction on particle to gas heat transfer

Particle to gas heat transfer studies were carried out by employing the endothermic cyclohexane dehydrogenation reaction. The calculated heat transfer coefficients were on the lower side of those previously reported in the literature for unreactive systems. Previous studies of other investigators with reactive systems suggest that particle to gas heat transfer coefficients may increase with exothermic reaction due to the product molecules leaving the catalyst surface with excess vibrational energy. For an endothermic reaction this effect would result in the calculation of a lower heat transfer coefficient.     

Integral formulation of the population balance equation: Application to particulate systems with particle growth

Numerical solution of the population balance equation (PBE) is widely used in many scientific and engineering applications. Available numerical methods, which are based on tracking population moments instead of the distribution, depend on quadrature methods that destroy the distribution itself. The reconstruction of the distribution from these moments is a well-known ill-posed problem and still unresolved question. The present integral formulation of the PBE comes to resolve this problem. As a closure rule, a Cumulative QMOM (CQMOM) is derived in terms of the monotone increasing cumulative moments of the number density function, which allows a complete distribution reconstruction. Numerical analysis of the method show two unique properties: first, the method can be considered as a mesh-free method. Second, the accuracy of the targeted low-order cumulative moments depends only on order of the CQMOM, but not on the discrete grid points used to sample the cumulative moments.     

Nanoparticles for convective heat transfer enhancement: heat transfer coefficient and the effects of particle size and zeta potential

This work presents a study of using the Wilson Plot method to determine the convective heat transfer coefficient (CHTC) of the following nanoparticles in water as the base fluid: SiO₂, TiO₂, and Al₂O₃. The experiments were carried out in a double layer concentric glass tube in which the hot fluid and nanofluids exchange heat in a counter current fashion without direct contact. Attention was also given to the volumetric concentration, flow rate, and the size of nanoparticles to investigate their effects on CHTC. From the experiments, it was found that by adding nanoparticles, the CHTC of water can generally be enhanced and a 45% increase has been achieved with a 0.5?vol% concentration of Al₂O₃ nanoparticles at an intermediate Reynolds number around 4100. Moreover, simply reducing nanoparticle size and increasing the nanofluid flow rate do not necessarily lead to the CHTC enhancement, rather, they have adverse effects. It is concluded that the enhancement depends on the stability of the dispersed nanoparticles that can be characterized by their overall mean size and zeta potential as useful measures.     

Modification of the dusty-gas equation to predict mass transfer in general poruos media

Volume-mean correction factors are used in the dusty-gas equation to account for heteroporosity effects on isobaric and nonisobaric mass transfer in general porous media. Using conventional information on pore-size distribution, the modified equation maintains reliance on the original dusty-gas constants for characterizing the porous structure. The modified equation was tested with experimental data for a binary helium-nitrogen system in macroporous α-alumina and microporous γ-alumina.     

Experimental equation on intensity of transmitted light through particle suspension of higher concentration

For a given medium and wave and path lengths of incident light, the relationship was experimentally investigated among the intensity of transmitted light, the suspension concentration of particles and the median size on particle volume basis. The numerical correlations were examined and an approximate equation was proposed over a wider range of concentrations up to about 3 vol.%. This expression was found to be valid not only for uniformly sized particles but also for mixtures of different particle sizes as compared with the previous equations for low concentration of particles of uniform size.     

Fluid—particle heat transfer in fixed and fluidized beds

Fluid-particle heat transfer in fixed and fluidized beds is re-examined. The heat transfer factor depends not only on the particle Reynolds number but also on the void fraction of the bed. The available experimental data has been correlated to yield a non-linear relationship between ?J_h and the particle Reynolds number. The resulting expression is:

.     

Study of mass transfer in a vertically moving particle bed electrode

A study was made of the influence of process parameters on the mass-transfer coefficient in a flow-through cell with a cascade of rotating drums partially filled with conductive particles (called the vertically moving particle bed). Copper deposition from an acidic sodium sulphate solution was used as the model reaction. To evaluate the experimental data a macrohomogeneous mathematical model of potential and current density distribution inside the cell was developed. The electrolyte flow distribution between the empty space above the particle bed and through the bed was evaluated. On the basis of these results the following correlation is proposed:

Heat and mass transfer of single droplet/wet particle drying

A theoretical model, which considers the fully unsteady character of both heat and mass transfer during the drying of single droplet/wet particle, is presented. The model enables prediction of pressure and fraction distributions of air-vapour mixture within the capillary pores of the wet particle crust. The simulations of the drying process of a single silica droplet under different conditions show a permanent rising of pressure within the capillary pores, but the corresponding vapour fraction remains less than unity. The comparison between the drying histories of the silica droplet, predicted by the present model with the data, calculated by the model which assumes a quasi-steady-state mass transfer and linear pressure profile within the capillary pores, shows inconsiderable differences between the droplet/wet particle temperature and mass time-changes. At the same time, the present model predicts pressure build-up and temperature rising within the particle wet core. However, in the studied cases the temperature of the wet core temperature does not exceed the liquid saturation temperature and therefore no boiling of liquid within the particle wet core is observed.