Evolution of Particle Size Distributions due to Turbulent and Brownian Coagulation

The time evolution of particle size distribution due to Brownian and turbulent coagulation (using the kernel of Kruis and Kusters (1997)) was systematically investigated. Using a new definition of dimensionless size distribution parameters based on the geometric mean values, self-preserving particle size distributions for turbulent coagulation were found to exist. The width of such distributions depends on the initial size distribution as well as the turbulence intensity. When starting with submicron aerosols, however, only the turbulence intensity plays a role in determining the final self-preserving form, whereas the initial conditions have no influence. Typically, broad particle size distributions with σ g in the 1.5-1.9 range are obtained. Because of the importance of scavenging by the largest particles in the size distribution, the possibility of developing a "runaway mass" exists, for which some experimental indications in turbulent systems exist.     

Breakage probability of irregularly shaped particles

The investigation of breakage probability by compression of single particles was carried out. The spherical glass particles and irregularly shaped particles of NaCl, sugar, basalt and marble were subjected to a breakage test. The breakage test includes the compression up to breakage of 100 particles to obtain the distribution of the breakage probability depending on the breakage force or compression work. The breakage test was conducted for five particle size fractions from each individual material, at two stressing rates. Thus obtained 50 breakage force distributions and corresponding 50 breakage work distributions were fitted with log-normal distribution function.Usually, the breakage probability distribution can be found by means of stress or energy approach. The first one uses the stress to calculate the breakage probability distribution. The second approach uses the mass-related work done to break the particle. We prefer to use the breakage force and energy as essential variables. The correlation between the force and energy at their breakage points is obtained by integrating the characteristic force–displacement curve, i.e. the constitutive function of elastic–plastic mechanical behavior of the particle. The irregularly shaped particle is approximated by comparatively “large” hemispherical asperities. In terms of elastic–plastic deformation of the contacting asperities with the plate, a transition from elastic to inelastic deformation behavior was considered. Thus, one may apply the model of soft contact behavior of comparatively stiff hemispheres. Based on this model a relationship between the breakage force distributions and corresponding energy distributions was analyzed. Every tested material exhibits a linear relationship between average breakage energy and average breakage force calculated for every size fraction.For future consideration both force and energy distributions were normalized by division by average force or energy, consequently. The relationship between the fit parameters of normalized energy distribution and corresponding fit parameters of normalized force distribution was established. The mean value and standard deviation of normalized force distribution can be found from mean value and standard deviation of normalized energy distribution by means of system of two linear equations. The coefficients of those linear equations remain the same for all of the above tested materials; particle size fractions and stressing rates. As a result the simple transformation algorithm of distributions is developed. According to this algorithm the force distribution can be transformed into energy distribution and vice versa.     

On the general theory of solid granulation

The general population balance equation for the kinetics of granulation by simultaneous coalescence and crushing and layering mechanisms is posed and self-preserving solutions are obtained. A general similarity transformation, which contains in it all the physical information concerning the granulation process, is presented. It is clearly shown that self-preserving granule size distributions are possible for granulation by simultaneous coalescence and crushing and layering.     

Some studies on glass fiber-reinforced polypropylene. Part I: Reduction in fiber length during processing

The reduction in fiber length during extrusion and injection molding of two commercial glass fiber-reinforced polypropylene products containing 30 percent by weight of glass fibers was studied. The first product had very small fibers of average length around 0.5 mm and also contained a coupling agent. The second product contained relatively longer glass fibers of 9 mm length and no coupling agent. In both cases, fiber attrition occurs predominantly at the solid-melt interface in the meiting zone of the extruder. However, in the short fiber granules, the maximum of the length distribution, which for the initial sample is around 0.5 mm, moved to shorter fiber lengths along the screw channels further from the hopper. In the long fiber granules, a bimodal length distribution was obtained in the intermediate channels; the first maximum was around the original length of 9 mm and the second centered around 0.5 mm. Thus, the forces at the solid-melt interface result in fiber breakage to lengths which are predominantly around 0.5 mm. The fiber attrition was observed to be more severe in injection molding apparently because of higher shear rates and also because the fibers had to pass through narrow channels. The measured distributions of fiber length along the screw channels for the two products are presented, and the possible mechanisms of fiber breakage are discussed. The mechanical properties of samples containing different fiber length distributions and the effects of fiber length and interfacial adhesion on properties are presented and discussed in Part II.     

Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation

The dynamic behavior of concentrated suspensions (up to a solids volume fraction of 20%) of non-spherical particles is investigated theoretically by coupling a rheological law to a population balance model accounting for coagulation and fragmentation of the detailed particle size distribution. In these suspensions, the immobilization of matrix liquid renders the viscosity dependent on the particle aggregation state. The effect of initial solids volume concentration and shear rate on the transient behavior of particle size distribution and suspension viscosity is examined. Power law correlations for the equilibrium flow curves of aggregating suspensions are deduced and compared to experimental data. Steady-state or equilibrium particle size distributions are found to be self-preserving with respect to solids volume fraction and shear rate.     

Enhancement of Silica Aerosol Coagulation by Van Der Waals Forces

The interaction energy between small silica particles has been calculated for a range of diameters between 2 and 30 nm. The enhancement of the coagulation rate has been calculated for these particles. The enhancement factor ranges from a maximum of about 2.62 for a 2-nm diameter pair down to 1.96 for a 30-nm diameter pair. Calculations of the aerosol size distribution indicate that a nearly self-preserving form is attained after the cessation of nucleation.     

氧化硼研磨的实验与理论研究

用行星式球磨机对13个不同尺寸区间氧化硼的研磨过程进行了实验与理论研究. 实验测量了13个不同尺寸区间氧化硼的粉碎速率常数及其分布系数,通过对测量结果的分析得到了不同尺寸氧化硼的粉碎速率函数及其分布系数函数,进而建立了粉碎过程质量分数的积分微分方程. 用四阶龙格-库塔法对氧化硼研磨过程的质量分数积分微分方程进行了数值计算,并与实验结果进行了比较. 计算与比较结果表明,氧化硼的研磨过程具有时变特征.     

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.     

Determination of Breakage Parameters in Turbulent Fluid‐Fluid Breakage

Numerous sets of single‐particle breakage experiments are required in order to provide a sufficient database for improving the modeling of fluid particle breakage mechanisms. This work focuses on the interpretation of the physical breakage events captured on video. In order to extract the necessary information required for modeling the mechanisms of the fluid particle breakage events in turbulent flows, a well‐defined image analysis procedure is necessary. Two breakage event definitions are considered, namely, initial breakup and cascade breakup. The reported breakage time, the number of daughter particles created, and the daughter size distribution are significantly affected by the definition used. For each breakage event definition, an image analysis procedure is presented.     

Nano-milling of pigment agglomerates using a wet stirred media mill: Elucidation of the kinetics and breakage mechanisms

The present study concerns the production of pigment nanoparticles in a wet-batch stirred media mill with polymeric media. The breakage kinetics and mechanisms were investigated using size-discrete population balance models (PBMs). The temporal variation of the particle size distribution was measured via dynamic light scattering. Considering the G-H model, a time-invariant PBM, and a time-variant PBM, the specific breakage rate parameters and breakage distribution parameters were identified. It is found that the breakage rate is not first-order and that a delay time exists for the breakage of nanoparticles. The time-variant PBM captures all these features and suggests a transition from deagglomeration of agglomerates to the breakage of primary particles. The analysis of the breakage distribution parameters suggests splitting as the dominant mechanism as opposed to attrition or massive fracture.     

Parametric dependence of particle breakage mechanisms

It has been observed that the pattern of particle impact breakage in two-dimensional systems is a result of two mechanisms. “Mechanism I” accounts for the breakage induced by the stresses that appear in unbroken particles. “Mechanism II” breakage is due to the buckling of the Mechanism I fragments. The purpose of this paper is to try and understand the parameters that govern the magnitude of the breakage induced by the various mechanisms—in particular, to understand the specific effects of impact velocity on the size distribution. To that end, a dimensionless parameter that governs the magnitude of the breakage induced by Mechanism I in both two and three dimensions is developed. The two-dimensional analysis demonstrates a velocity dependence for the Mechanism I breakage that accounts for the observed velocity effect on the generated size distributions. However, the three-dimensional analysis demonstrates no such velocity effect. Both findings are supported by simulation results.     

A mixed Weibull model for size reduction of particulate and fibrous materials

This paper describes a simple alternative to the classical population balance breakage model, which characterizes and controls the size distribution of particles submitted to a reduction process. The new approach is based on cumulative distribution functions of mixed random variables. Results indicate that a Weibull mixture distribution function adequately models the size of particles submitted to various breakage processes. The model was further applied to experimental reduction processes with apparently random breakage probability and yielded good estimates of the unbroken particle and fragment distributions. Use of these results for direct and indirect prediction of the size alteration under dimensional reduction processes is discussed.     

Mathematical modeling of fluid energy milling based on a stochastic approach

In this study, the stochastic method is used to simulate the grinding process in a fluid energy mill: the product particle size distribution is regarded as the result of repeating elementary breakage events, i.e. M_p=M₀[T_m]^m, where M₀ is the row vector of the size distribution of feed particles, M_p is the row vector of the size distribution of product particles, m is the number of elementary steps, and T_m is the matrix of transition probabilities representing the elementary breakage event. The matrix of transition probabilities can be related to the breakage rate function and the breakage distribution function of the elementary breakage event. A specially designed apparatus, named single-event fluid mill, was employed to experimentally estimate those two breakage functions of the elementary breakage event with a breakage rate correction factor θ. The classification effect is taken into consideration by defining a cutting size under which the particle will not break any more. Using this strategy, the product particle size distribution is calculated. The good consistency between the simulation and the experimental results indicates that this model is valid to quantitatively estimate the grinding performance of the fluid energy mill.     

Strength of short fiber reinforced polymers: Effect of fiber length distribution

In this study, the prediction of mechanical strength of short fiber reinforced plastics (SFRPs) is made possible by obtaining a Fiber Length Distribution (FLD) efficiency factor, η_FLD, from the formerly known twofold discrete strengthening equation of Kelly–Tyson. The unified parameter η_FLD is developed involving both the effects of fiber breakage and resulting distribution, fiber volume fraction and fiber and interface properties, so that they can be incorporated into modified rule of mixtures (MROM). This procedure helps to clarify the experimentally observed loss in strengthening rate with increasing fiber fraction. By adapting a few experimentally determined distributions to a Weibull type function, the analytical solutions described in this study establish the exploration of the strength of SFRPs in the entire fiber content range or can reveal the interfacial bond strength. After investigating the effects of fiber and interface parameters on strengthening efficiency, it is found that common fiber‐matrix combinations possessing intermediate critical fiber lengths show a significant decrease in strengthening efficiency with increasing fiber content at low fiber loadings. On the contrary, higher and lower critical fiber lengths yield less significant losses. POLYM. COMPOS., 2008. © 2008 Society of Plastics Engineers     

Continuous grinding in a small wet rod mill Part II. Breakage of some common ore minerals

A study has been made of the breakage of galena, marmatite, chalcopyrite and quartz, separately, in a calcite environment in a rod mill. The behaviour of all may be described by the same basic dynamic breakage model, which involves a breakage function, a rate function and a distribution of residence time function. The same breakage and distribution of residence time functions applied to all the minerals but the mean residence times decreased in the order galena > marmatite > chalcopyrite > quartz. Size for size, the values of the rate of breakage constants followed the same sequence.     

Short Fiber orientation and its Effects on the Properties of Thermoplastic Composite Materials

I. INTRODUCTION

While it is true that preform processes involving the use of long or continuous fibers are known and used in the manufacture of reinforced thermoplastic articles–Azdel [1] or STX sheet [2], for example–it is generally the case that such articles are formed by injection molding. Both the feedstock requirements for this process and the occurrence of high melt shear during it ensure that only short fibers will be present in the finished article. Although the use of slow screw speeds, slow injection rates, low back pressure, wide sprues, runners, and gates, and large radii of curvature avoids fiber breakage during molding, such conditions are not often found in practice. Furthermore, the necessity of incorporating reground material into the feedstock also ensures short fiber lengths in the final part, lengths not greatly in excess of the critical length required for effective stress transfer from polymer matrix to reinforcing fiber. In a practical part, design uncertainties caused by fiber length attrition are further compounded by the effects of fiber orientation. Although length distribution effects have been studied by a number of workers, both experimentally [3] and theoretically [4], relatively little has been reported on orientation effects in short fiber reinforced thermoplastics.