CFD simulation of aggregation and breakage processes in laminar Taylor-Couette flow

An experimental and computational investigation of the effects of local fluid shear rate on the aggregation and breakage of approximately 10 microm latex spheres suspended in an aqueous solution undergoing laminar Taylor-Couette flow was carried out according to the following program. First, computational fluid dynamics (CFD) simulations were performed and the flow field predictions were validated with data from particle image velocimetry experiments. Subsequently, the quadrature method of moments (QMOM) was implemented into the CFD code to obtain predictions for mean particle size that account for the effects of local shear rate on the aggregation and breakage. These predictions were then compared with experimental data for latex sphere aggregates (using an in situ optical imaging method) and with predictions using spatial average shear rates. The mean particle size evolution predicted by CFD and QMOM using appropriate kinetic expressions that incorporate information concerning the particle morphology (fractal dimension) and the local fluid viscous effects on aggregation collision efficiency match well with the experimental data.     

Analysis of the aggregation-fragmentation population balance equation with application to coagulation

Coagulation of small particles in agitated suspensions is governed by aggregation and breakage. These two processes control the time evolution of the cluster mass distribution (CMD) which is described through a population balance equation (PBE). In this work, a PBE model that includes an aggregation rate function, which is a superposition of Brownian and flow induced aggregation, and a power law breakage rate function is investigated. Both rate functions are formulated assuming the clusters are fractals. Further, two modes of breakage are considered: in the fragmentation mode a particles splits into w2 fragments of equal size, and in the erosion mode a particle splits into two fragments of different size. The scaling theory of the aggregation-breakage PBE is revised which leads to the result that under the negligence of Brownian aggregation the steady state CMD is self-similar with respect to a non-dimensional breakage coefficient theta. The self-similarity is confirmed by solving the PBE numerically. The self-similar CMD is found to deviate significantly from a log-normal distribution, and in the case of erosion it exhibits traces of multimodality. The model is compared to experimental data for the coagulation of a polystyrene latex. It is revealed that the model is not flexible enough to describe coagulation over an extended range of operation conditions with a unique set of parameters. In particular, it cannot predict the correct behavior for both a variation in the solid volume fraction of the suspension and in the agitation rate (shear rate).     

CFD simulation of shear-induced aggregation and breakage in turbulent Taylor-Couette flow

An experimental and computational investigation of the effects of local fluid shear rate on the aggregation and breakage of approximately 10 microm latex spheres suspended in an aqueous solution undergoing turbulent Taylor-Couette flow was carried out. First, computational fluid dynamics (CFD) simulations were performed and the flow field predictions were validated with data from particle image velocimetry experiments. Subsequently, the quadrature method of moments (QMOM) was implemented into the CFD code to obtain predictions for mean particle size that account for the effects of local shear rate on the aggregation and breakage. These predictions were then compared with experimental data for latex sphere aggregates (using an in situ optical imaging method). Excellent agreement between the CFD-QMOM and experimental results was observed for two Reynolds numbers in the turbulent-flow regime.     

Population balance modeling of aggregation and breakage in turbulent Taylor-Couette flow

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor-Couette apparatus is presented. To monitor the aggregation and breakage processes, an in situ optical imaging technique was used. Consequently, a computational study using a population balance model was carried out to test the various parameters in the aggregation and breakage models. Very good agreement was found between the time evolution of the cluster size distribution (CSD) calculated with the model and that obtained from experiment. In order to correctly model the left-hand side of the CSD (small clusters), it was necessary to use a highly unsymmetric fragment-distribution function for breakage. As another test of the model, measurements with different solid volume fractions were performed. Within the range of the solid volume fractions considered here, the steady-state CSD was not significantly affected. In order to correctly capture the right-hand side of the CSD (large aggregates) at the higher solid volume fraction, a modified aggregation rate prefactor was used in the population balance model.     

离散相系统群体平衡模型的求解算法

准确预测离散相系统中微观粒子的尺度分布演变对系统动态流动行为的准确确定起关键性作用. 粒子的尺度分布演变以及引起尺度分布变化的离散相微观行为(聚并、破碎、长大等)由群体平衡模型来描述. 该模型是关于数值密度函数的非线性双曲型方程, 数值求解为主要手段. 本文对群体平衡方程的直接离散方法、Monte Carlo、矩方法从实现难易程度、计算机资源消耗、计算精度三方面进行了详细阐述, 并着重介绍了几种性能优越的矩方法—— 矩积分方法(QMOM)、矩直接积分方法(DQMOM)、可调节矩积分方法(M-QMOM)、自适应矩直接积分方法(ADQMOM)、定点矩积分方法(FPQMOM)、粒子游动算法(MPEM)和局部定点矩积分方法(LFPQMOM). 最后根据算法的优缺点及其当前发展状况对不同算法的未来发展做了预测.     

Cluster aggregation and fragmentation kinetics model for gelation

Gelation can occur in polymer, hydrogel, and colloid systems that undergo reversible aggregation-fragmentation (crosslinking accompanied by breakage). Gelation, characterized by rapid divergence of weight-average molecular weight and viscosity due to initial network formation, can be reversed if conditions change. In this paper, reversible aggregation and fragmentation in the pre-gelation time period are modeled with distribution kinetics. Moment equations are obtained from the population balance equation, and solved for eight different rate kernels. We identify the cases for which gelation is possible and obtain the critical values for the rate constants that allow gelation. The model provides a good simulation of published experimental data for aggregation and degradation of plasticized wheat gluten during thermo-mechanical treatments. We also evaluate two closure approximations based on Gamma and log-normal distributions, and conclude that log-normal closure predicts all five possible steady states, in agreement with the Vigil-Ziff criterion, and Gamma closure predicts only three. However, Gamma closure approximates the steady state either closely or exactly, whereas log-normal closure only poorly approximates the steady-state distribution.     

Experimental investigation and population balance equation modeling of solid lipid nanoparticle aggregation dynamics

Solid lipid nanoparticles (SLNs) have applications in drug delivery and the encapsulation of bioactive, lipophilic compounds. However, SLNs tend to aggregate when stored due to the lipid crystals undergoing a polymorphic transformation from the unstable α form to the stable β form. We developed a population balance equation (PBE) model for prediction of average polymorph content and aggregate size distribution to better understand this undesirable behavior. Experiments with SLNs stored at room temperature showed that polymorphic transformation was the rate determining step for our system, SLNs with smaller initial size distributions aggregated more rapidly, and aggregates contained particles with both α and β crystals. Using parameter values estimated from our data, the PBE model was able to capture the bimodal nature of aggregate size distributions, the α-to-β polymorph ratio, and the faster aggregation dynamics of SLNs with smaller initial size distributions. However, the model was unable to adequately capture the fast disappearance rate of primary particles, the broad size distributions of formed aggregates, and the significant α content of aggregating particles. These discrepancies suggest that a PBE model which accounts for polymorph content as an internal variable along with aggregate size may be required to better reproduce experimental observations.     

An improved collision efficiency model for particle aggregation

A generalized geometric model is presented which describes the collision efficiency factor of aggregation (the probability of a binary particle or aggregate collision resulting in adhesion) for systems comprised of two oppositely charged species. Application of the general model to specific systems requires calculation of the area of each species available for collision with a second species. This is in contrast to previous models developed for polymer-particle flocculation that are based on the fractional surface coverage of adsorbed polymer. The difference between these approaches is suggested as an explanation for previously observed discrepancies between theory and observation. In the current work the specific case of oppositely charged nondeformable spherical particles (heteroaggregation) is quantitatively addressed. The optimum concentration of oppositely charged particles for rapid aggregation (maximum collision efficiency) as a function of relative particle size is calculated and an excellent correlation is found with data taken from literature.     

Dynamic simulation of floc breakage with a random breakage distribution

The transient behavior of floc breakage in a lean, batch-stirred tank is investigated. The stochastic nature of breakage distribution (BD), which includes breakage mode (BM) and daughter-floc size distribution (DFSD), makes a deterministic approach unrealistic. The difficulty of representing BM and DFSD through deterministic probability distributions is circumvented by adopting- the Monte Carlo simulation approach to determine their functional forms numerically. On the basis of the conservation of the number of primary particles, the population balance equation describing the transient behavior of the system is solved analytically. The present model involves fewer adjustable parameters than those reported in the literature, yet it is able to take account or the random feature of the breakage phenomenon and the fractal geometry of floc structure.     

The equal-size binary breakage problem: evolution toward a steady shape or periodic behavior?

During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.     

Advances in numerical methods for the solution of population balance equations for disperse phase systems

Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system. A population balance equation (PBE), a non-linear hyperbolic equation of the number density function, is usually employed to describe the micro-behavior (aggregation, breakage, growth, etc.) of a disperse phase and its effect on particle size distribution. Numerical solution is the only choice in most cases. In this paper, three different numerical methods (direct discretization methods, Monte Carlo methods, and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation, computational load and numerical accuracy. Special attention is paid to the relatively new and superior moment methods including quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), modified quadrature method of moments (M-QMOM), adaptive direct quadrature method of moments (ADQMOM), fixed pivot quadrature method of moments (FPQMOM), moving particle ensemble method (MPEM) and local fixed pivot quadrature method of moments (LFPQMOM). The prospects of these methods are discussed in the final section, based on their individual merits and current state of development of the field. Supported by the National Basic Research Program of China (Grant No. 2004CB720208), the National Natural Science Foundation of China (Grant Nos. 40675011 & 10872159), and the Key Laboratory of Mechanics on Disaster and Environment in Western China     

Conditions for the size spectrum of aerosols to be self-preserving

A one-dimensional population balance governing the particle size distribution in a system where particles change their sizes due to both coagulation and individual growth is studied for the possibility of a similarity solution. It is found that when a dimensionless parameter γ is a constant, then the population balance can be transformed into an ordinary integrodifferential equation. Conditions under which the latter equation may have a self-preserving solution are established. Examples related to aerosol dynamics in the continuum regime, the free-molecule regime and the intermediate regime are discussed in detail.     

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.     

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.     

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.     

Asymptotic Particle Size Distributions Attained during Coagulation Processes

The effects of the collision kernel on the self-preserving size distribution and on the gelation phenomenon of aerosol coagulation were investigated. An analytical solution for the asymptotic width of log-normally preserving size distribution during coagulation was obtained as a function of the degree of homogeneity using arbitrary shape of homogeneous collision kernels. From the solution obtained, it was shown that when the degree of homogeneity is larger than 1, self-preserving size distribution does not exist, and gelation occurs. A very accurate numerical coagulation simulation method, the sectional method, was also used for calculating the self-preserving particle size distribution for some specific classes of coagulation kernels and the results were compared with the analytical solution obtained by the log-normal method. Copyright 2001 Academic Press.     

Analytical Solutions for Bikinetic Coagulation: Incorporation of a Maximum Size Class

Analytical solutions are derived for both orthokinetic and perikinetic coagulation rate mechanisms by considering bikinetic collisions of the agglomerating particles. The bikinetic behavior assumes that only equal-sized particles collide to form larger aggregates. In addition to the bikinetic assumption used to simplify the rate equations, the coagulating particles are assumed to approach a maximum size class with increasing time. Mechanisms of perikinetic, orthokinetic, differential sedimentation, and flow-induced breakup were examined in the analyses. A solution incorporating flow-induced breakup forced the equilibrium population to a maximum size class that was less than an initially assumed maximum size class, consistent with coagulation modeling. This breakup equation is shown to be beneficial to researchers seeking to develop predictive models for orthokinetic coagulation by establishing estimates for breakup constants. The new solutions are contrasted with the existing simplified models and found to be more characteristic of actual coagulation behavior. The new solutions approach a maximum particle size class and conserve mass at increasing time, rather than an asymptotic approach to zero particles or zero mass as predicted by the existing models. Applications of the new coagulation rate solutions are useful in estimating particle collision efficiency and breakup rate constants and in determining the accuracy of numerical simulations of coagulation processes. Copyright 2000 Academic Press.     

The size of particle aggregates produced by flocculation with PNIPAM, as a function of temperature

This work investigates the effect of temperature on the size of alumina aggregates formed by flocculation with temperature responsive Poly(N-Isopropylacrylamide)(PNIPAM). The results are discussed in terms of the effects of temperature on particle collision, particle adhesion and aggregate breakage. It was found that the size of alumina aggregates increases with increasing solution temperature. Particle/particle collision and aggregate breakage are largely unaffected by increasing solution temperature and therefore could not account for the change in aggregate size. The dominant factor in aggregate growth with increasing temperature was found to be the increase in the force of adhesion between alumina particles. The appearance of the adhesive force is triggered by the increase in temperature above the lower critical solution temperature of PNIPAM.     

Brownian coagulation at high concentration

Particle growth by Brownian coagulation at high concentration in the continuum regime is investigated by solving the Langevin dynamics (LD) equations for each particle trajectory of polydisperse suspensions. By monitoring the LD attainment of the self-preserving size distribution (SPSD), it is shown that the classic Smoluchowski collision frequency function is accurate for dilute particle volume fractions, phis, below 0.1%. At higher phis, coagulation is about 4 and 10 times faster than for the classic theory at phis = 10 and 20%, respectively. For complete particle coalescence upon collision, SPSDs develop even in highly concentrated suspensions (up to phis = 35%), as with dilute ones, but are broadened with increasing phis. At high particle concentration, an overall coagulation rate is proposed that reduces to the classic one at low concentration. Detailed collision frequency functions are also obtained at various phis values. Fractal-like agglomerates undergoing coagulation at constant fractal dimension attain an SPSD only temporarily because their effective volume fraction continuously increases, approaching gelation in the absence of restructuring or fragmentation.