Time-property least-squares spectral method for population balance equations

In multiphase chemical reactor analysis the dispersed phase distribution plays a major role in obtaining reliable predictions. The population balance equation is a well established equation for describing the evolution of the dispersed phase. However, the numerical solution of this type of equations is computationally intensive. In this work, a time-property least squares spectral method is presented for solving the population balance equation including breakage and coalescence processes. In this problem, both property and time are coupled in the least squares minimization procedure. Spectral convergence of the L ² least squares functional and L ² error norms in time-property is verified using a smooth solution to the population balance equation.     

Particle size correlations in brownian agglomeration. Closure hypotheses for product density equations

In a previous paper the authors have demonstrated inaccuracies in the population balance equation, as applied to agglomerating particulate systems, which arise due to an inherent assumption regarding the statistical independence of particle pairs. When this assumption is removed, an intractable hierarchy of product density equations results. In this paper a closure hypothesis for the product density equations is proposed, evaluated against alternatives, and used in an approximate solution to the product density equations. The closure hypothesis is shown to improve significantly upon the population balance equation which is itself the result of the simplest closure.     

A study of the collisional fragmentation problem using the Gamma distribution approximation

The nonlinear fragmentation population balance formulation has been elevated in recent years from a prototype for studying nonlinear integro-differential equations to a vehicle for analyzing and understanding several physicochemical processes of technological interest. The so-called pure collisional fragmentation, which is the particular mode of nonlinear fragmentation induced by collisions between particles, is studied here. It is shown that the corresponding population balance equation admits large time asymptotic (self-similarity) solutions for homogeneous fragmentation and collision functions (kernels). The self-similar solutions are given in closed form for some simple kernels. Based on the shape of the self-similar solutions the method of moments with Gamma distribution approximation is employed for transient solution (from initial state to establishment of the asymptotic shape) of the collisional fragmentation equation. These solutions are presented for several sets of parameters and their behavior is discussed rather extensively. The present study is similar to the one has already been performed for the case of the much simpler linear fragmentation equation [G. Madras, B.J. McCoy, AIChE J. 44 (1998) 647].     

Destructive aggregation: aggregation with collision-induced breakage

Mean-field population balance equations are used to describe the evolution of particle size distributions in a wide variety of systems undergoing simultaneous aggregation and breakage. In this paper we develop a population balance that includes aggregation combined with collision-induced particle breakage for arbitrary fragment distribution functions, provided that this distribution function depends only on the total mass of the particles undergoing a collision. We then develop a specific distribution function for arbitrary two-body collisions by postulating that each collision produces a transition-state aggregate having the morphology of a linear polymer. The behavior of the resulting equation is then analyzed for the case in which the collision kernel is a constant, and partial analytical solutions are derived and compared to corresponding Monte-Carlo simulation results. The computer simulations are then used to validate a proposed scaling law for the steady-state particle size distribution. Lastly, the behavior of the aggregation with collision-induced-breakage population balance equation is compared and contrasted with the behavior of an analogous aggregation with linear-breakage population balance equation.     

Self-similar solution to the problem of vapor diffusion toward the droplet nucleated and growing in a vapor-gas medium

The problem of vapor diffusion toward a droplet nucleated and growing in the diffusion regime is exactly solved using the similarity theory. The surface motion of droplets is taken into account in the solution. The constructed nonstationary concentration field of vapor satisfies the diffusion equation, the boundary condition of equilibrium on the surface of growing droplet, and the initial homogeneous condition. According to the found solution, the radius of a droplet is proportional to the square root of the time of its growth. Far from the critical point, at a low ratio between the densities of excess vapor and a liquid droplet, the proportionality coefficient coincides with that resulting from an approximate solution. The balance between the numbers of molecules removed from vapor and those composing a growing droplet exactly corresponds to the obtained solution.     

Floc breakage in agitated suspensions: Theory and data processing strategy

Flow visualization of chemical flocs in a simple extensional flow field reveals two distinct mechanisms for their breakage: splitting into a relatively small number of daughter fragments whose sizes are comparable to the parent flocs, along with continual disintegration by erosion to produce extremely fine particles from the extremities of the parent floc along the axis of extension. In turbulent flow, these two mechanisms still occur, although the kinematics of flow are more complex. This work presents a formulation of the population balance equation that governs the floc size distribution in turbulent flow, incorporating both the splitting and erosion mechanisms discussed above. Experiments were conducted in which floc size distributions of dilute suspensions are measured by a combination of techniques, including computerized optical scanning of photographs and pulse height analysis of signals from a light blockage transducer. The experimentally determined size distributions are then fit to those computed from the population balance equation, using constrained nonlinear least squares. This yields best values of certain coefficients that appear in the governing equation, providing a strategy to obtain a data base to promote deeper theoretical analysis. The method is demonstrated by analyzing data for kaolin-Fe(OH)₃ flocs in aqueous suspensions.     

Self-similar theory of nonisothermal vapor condensation on a droplet growing in a vapor-gas medium

Rigorous self-similar solutions to the joint problems of vapor diffusion toward a droplet growing in a vapor-gas medium and the removal of heat released during vapor condensation are found. An equation for the temperature of a droplet ensuring the existence of a self-similar solution is derived. This equation sets the constancy of the temperature of a droplet throughout the time of its growth and unambiguously determines this temperature. In the case of the strong heat effects, when the rate of droplet growth decreases substantially, the analytical solution to this equation is obtained. This temperature coincides precisely with the temperature, which is established in the droplet at the diffusion regime of its growth. At the found droplet temperature, interconnected fields of vapor concentration and temperature of vapor-gas medium around the droplet are expressed through the initial (prior to the droplet nucleation) parameters of a vapor-gas medium. These parameters are used to express the dependence of the radius of a droplet on the time at the diffusion regime of its growth and the time required to establish the diffusion regime of droplet growth. The case of weak heat effects is also studied.     

Quadrature method of moments for aggregation-breakage processes

Investigation of particulate systems often requires the solution of a population balance, which is a continuity statement written in terms of the number density function. In turn, the number density function is defined in terms of an internal coordinate (e.g., particle length, particle volume) and it generates integral and derivative terms. Different methods exist for numerically solving the population balance equation. For many processes of industrial significance, due to the strong coupling between particle interactions and fluid dynamics, the population balance must be solved as part of a computational fluid dynamics (CFD) simulation. Such an approach requires the addition of a large number of scalars and the associated transport equations. This increases the CPU time required for the simulation, and thus it is clear that it is very important to use as few scalars as possible. In this work the quadrature method of moments (QMOM) is used. The QMOM has already been validated for crystal growth and aggregation; here the method is extended to include breakage. QMOM performance is tested for 10 different cases in which the competition between aggregation and breakage leads to asymptotic solutions.     

A combined CFD modeling with population balance equation to predict pressure drop in venturi scrubbers

A venturi scrubber is one of the most important devices for air pollution control. Although there are different models for predicting the pressure drop in venturi scrubbers, most of them have some defects and cannot predict the pressure drop correctly. In this study, for the first time, an Eulerian–Eulerian computational fluid dynamics (CFD) model is combined with a population balance equation to predict the pressure drop in venturi scrubbers. This simulation takes into account a multiple size group model for droplet dispersion and droplet size distribution, which is based on a population balance equation. Flow field has been calculated by solving the time averaged continuity and Navier–Stokes equations along with the standard k–ε turbulence model. The equations included drag, turbulent dispersion, and buoyancy forces. The calculated pressure drop with and without considering the population balance equation was compared with the experimental data to evaluate the accuracy of the CFD modeling. The size distribution of droplets in the venturi scrubber was studied at different points for different liquid to gas ratios and throat gas velocities. The results show that the maximum break-up of droplets happens at the liquid injection point. Finally, the effects of nozzle diameter and nozzle arrangement on pressure drop in venturi scrubbers were investigated.     

Extended cell average technique for the solution of coagulation equation

Sectional (zero order) methods constitute a very important class of methods for the solution of the population balance equation offering distinct advantages compared to their competitors, namely, higher order and moment methods. For the last ten years a particular sectional method, the so-called fixed pivot technique has been the most extensively used in the scientific community for the solution of the coagulation equation because it offers arbitrary grid choice and conservation of two moments of the particle size distribution. Very recently, a new method (called cell average technique) has been developed which gives more accurate results than the fixed point technique. In the present work, the extension of this new method in order to conserve three moments is attempted. A stable algorithm for the solution of the coagulation equation is developed. Although the new method allows improved computation of moments of practical interest, this is not always the case with respect to complete particle size distribution.     

A Theory of Homogeneous Boiling-Up of Liquid Solutions: 1. Kinetic Equation of Boiling-Up

Homogeneous formation of bubbles of a stable gas phase in a liquid solution is studied. The intermediate (between diffusion and free molecular) regime of molecule exchange between a bubble and the solution is taken into account. A two-dimensional kinetic equation describing the homogeneous boiling-up of the solution is derived. The variables selected for describing an emerging bubble are shown to be very convenient for studying the resultant equation. One of the variables is thermodynamically stable and rapidly changes with time; on the contrary, the other variable is thermodynamically unstable and slowly changes with time. The mobility of the boundary of an emerging bubble has a merely slight effect on the establishment of the stationary rate of the change in the number of molecules in the bubble; the smallness of this effect is substantiated.     

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.     

Kinetics of binary condensation on the droplet growing in diffusion regime

A relaxation equation determining the regular tendency of the concentration of binary solution in the growing droplet to the stationary value, at which there is a self-similar solution to the problem of the condensation in a binary mixture, is derived. An analytical solution of the relaxation equation is obtained and it is demonstrated that the stationary value of concentration is achieved via the power law. The time interval that elapses from the emergence of the droplet until the diffusion regime of droplet growth and derived relaxation equation become effective is revealed. The stationary value of concentration is found for the model of ideal solution.     

BZ反应系非理想性对化学振荡动力学区域的影响

基于Debye-Huckel强电解质溶液活度系数理论、反应速度的活化络合物理论以及三变量微分方程奇点理论,探讨了非理想性对BZ反应体系振荡动力学区域的影响。以Oregonator为模型建立了均匀体系非理想BZ反应的系统动力学方程,计算并绘制了化学振荡的临界"溴酸钠浓度-离子强度"阈值图谱,划分了不同的动力学区域,并进行了一系列实验研究,得到了与理论分析一致的结论.     

面向过程的分析化学酸碱平衡体系定量解析策略

在微机高度普及和数值计算软件迅速发展的背景下,本文提出了面向过程的解析策略,用于酸碱平衡体系的定量计算。证明了酸碱平衡体系中"质子条件"不独立于"物料平衡"和"电荷平衡",而且不如后二者直观、容错性高。因此,面向过程的解析策略基于"物料平衡"和"电荷平衡"两种基本定量关系式。相对于传统的公式计算,面向过程的解析策略的重点是对平衡体系的分析以及定量关系式的建立,数值计算则由软件完成。结果表明,面向过程的解析策略方便直观、容易实施,不需要任何近似条件,不需要记忆公式,原则上能解决所有酸碱平衡体系的计算问题。在教学中应用这种策略,不仅能够显著减轻学生的记忆负担,提升学习兴趣,教师也会有更多的课时用于复杂酸碱平衡体系的深入讲解。本研究作为当前形势下分析化学教学改革的一个尝试,对传统课程内容进行优化重组,去冗存精,使其简明凝练,重点突出,定量分析的主旨更加明晰。     

Electroosmosis in porous solids for high zeta potentials

When surface potentials (or the surface charges) are high, the exponential term on the right-hand side of the Poisson-Boltzmann equation cannot be linearized. This nonlinear regime is systematically studied for various porous media and various physicochemical conditions. As in the linear regime, the numerical data for the electroosmotic coefficient when made dimensionless are shown to follow the semianalytical solution derived for a plane Poiseuille flow. Therefore, this coefficient can be deduced either from the specific surface or from the permeability and the formation factor.     

The electric double layer structure around charged spherical interfaces

We derive a formally simple approximate analytical solution to the Poisson-Boltzmann equation for the spherical system via a geometric mapping. Its regime of applicability in the parameter space of the spherical radius and the surface potential is determined, and its superiority over the linearized solution is demonstrated.     

Stochastic diffusion interactions and coarsening in a system of droplets growing from a supersaturated gas mixture

In this work we study diffusion interactions among liquid droplets growing in stochastic population by condensation from supersaturated binary gas mixture. During the postnucleation transient regime collective growth of liquid droplets competing for the available water vapor decreases local supersaturation leading to the increase of critical radius and the onset of coarsening process. In coarsening regime the growth of larger droplets is prevailing noticeably broadening the droplet size-distribution function when the condensation process becomes more intensive than the supersaturation yield. Modifications in the kinetic equation are discussed and formulated for a stochastic population of liquid droplets when diffusional interactions among droplets become noteworthy. The kinetic equation for the droplet size-distribution function is solved together with field equations for the mass fraction of disperse liquid phase, mass fraction of water vapor component of moist air, and temperature during diffusion-dominated regime of droplet coarsening. The droplet size and mass distributions are found as functions of the liquid volume fraction, showing considerable broadening of droplet spectra. It is demonstrated that the effect of latent heat of condensation considerably changes coarsening process. The coarsening rate constant, the droplet density (number of droplets per unit volume), the screening length, the mean droplet size, and mass are determined as functions of the temperature, pressure, and liquid volume fraction.     

Thermophysical Characterization of Paraffin Wax Based on Mass-Accommodation Methods Applied to a Cylindrical Thermal Energy-Storage Unit

Two mass-accommodation methods are proposed to describe the melting of paraffin wax used as a phase-change material in a centrally heated annular region. The two methods are presented as models where volume changes produced during the phase transition are incorporated through total mass conservation. The mass of the phase-change material is imposed as a constant, which brings an additional equation of motion. Volume changes in a cylindrical unit are pictured in two different ways. On the one hand, volume changes in the radial direction are proposed through an equation of motion where the outer radius of the cylindrical unit is promoted as a dynamical variable of motion. On the other hand, volume changes along the axial symmetry axis of the cylindrical unit are proposed through an equation of motion, where the excess volume of liquid constitutes the dynamical variable. The energy–mass balance at the liquid–solid interface is obtained according to each method of conceiving volume changes. The resulting energy–mass balance at the interface constitutes an equation of motion for the radius of the region delimited by the liquid–solid interface. Subtle differences are found between the equations of motion for the interface. The differences are consistent with mass conservation and local mass balance at the interface. Stationary states for volume changes and the radius of the region delimited by the liquid–solid interface are obtained for each mass-accommodation method. We show that the relationship between these steady states is proportional to the relationship between liquid and solid densities when the system is close to the high melting regime. Experimental tests are performed in a vertical annular region occupied by a paraffin wax. The boundary conditions used in the experimental tests produce a thin liquid layer during a melting process. The experimental results are used to characterize the phase-change material through the proposed models in this work. Finally, the thermodynamic properties of the paraffin wax are estimated by minimizing the quadratic error between the temperature readings within the phase-change material and the temperature field predicted by the proposed model.