A differentially weighted Monte Carlo method for two-component coagulation

The direct simulation Monte Carlo (DSMC) method for population balance modeling is capable of retaining the history of each simulation particle and is thus able to deal with multivariate properties in a simple and straightforward manner. As opposed to conventional DSMC approaches that track equally weighted simulation particles, a differentially weighted Monte Carlo method is extended to simulate two-component coagulation processes and is thereby able to simulate the micromixing of the components. A new feature of the method for this bivariate population balance modeling is that it is possible to specify how the simulation particles are distributed over the compositional axis. This allows us to obtain information about particles in those regions of the size and composition distribution functions where the non-weighted MC methods place insufficient simulation particles to obtain an inaccurate solution. The new feature results in lower statistical noise for simulating two-component coagulation, which is validated by using two-component coagulation cases for which analytical solutions exist (a discrete process with sum kernel for initial monodisperse populations and a process with constant kernel for initial polydisperse populations).     

Exact solution of a coagulation equation with a product kernel in the multicomponent case

In this paper, we obtain the general solution for the continuous Smoluchowski equation in the multicomponent case with a product kernel as a series expansion. The solution of the problem involves the Laplace transform in several dimensions. We obtain a nonlinear partial differential equation (PDE) of the advective kind generalizing the one previously given by other authors for the mono-component case.As in its relative mono-component case, gelation is produced at some point, the conditions for its occurrence being the same as those for the mono-component case, though substituting a sum of derivatives by a derivative in the Laplace transform field. We demonstrate that for a multicomponent particle size distribution (PSD) of multiplicative form, it is sufficient for one of the marginal PSDs to generate instantaneous gelation for the occurrence of instantaneous gelation in the multicomponent PSD.The general solution is applied to several specific cases, a discrete case that recovers a previously known solution, and another two continuous cases which can be used to check numerical methods designed to directly solve the Smoluchowski equation in more general cases.We have compared the solutions for the multicomponent PSD for constant, additive and product kernels and we conjecture about the relation existing between the functional forms for the solutions both in the mono-component and the multicomponent case.Finally, we have analysed the shape of the solutions for multicomponent PSD for constant, additive and product kernels for very small masses of components, obtaining a qualitatively different behaviour for the product kernel. This has effects in the mixing state of the sol phase as time passes.     

Coalescence of Particles by Differential Sedimentation

We consider a three dimensional system consisting of a large number of small spherical particles, distributed in a range of sizes and heights (with uniform distribution in the horizontal direction). Particles move vertically at a size-dependent terminal velocity. They are either allowed to merge whenever they cross or there is a size ratio criterion enforced to account for collision efficiency. Such a system may be described, in mean field approximation, by the Smoluchowski kinetic equation with a differential sedimentation kernel. We obtain self-similar steady-state and time-dependent solutions to the kinetic equation, using methods borrowed from weak turbulence theory. Analytical results are compared with direct numerical simulations (DNS) of moving and merging particles, and a good agreement is found.     

Research on bimodal particle extinction coefficient during Brownian coagulation and condensation for the entire particle size regime

The extinction coefficient of atmospheric aerosol particles influences the earth’s radiation balance directly or indirectly, and it can be determined by the scattering and absorption characteristics of aerosol particles. The problem of estimating the change of extinction coefficient due to time evolution of bimodal particle size distribution is studied, and two improved methods for calculating the Brownian coagulation coefficient and the condensation growth rate are proposed, respectively. Through the improved method based on Otto kernel, the Brownian coagulation coefficient can be expressed simply in powers of particle volume for the entire particle size regime based on the fitted polynomials of the mean enhancement function. Meanwhile, the improved method based on Fuchs–Sutugin kernel is developed to obtain the condensation growth rate for the entire particle size regime. And then, the change of the overall extinction coefficient of bimodal distributions undergoing Brownian coagulation and condensation can be estimated comprehensively for the entire particle size regime. Simulation experiments indicate that the extinction coefficients obtained with the improved methods coincide fairly well with the true values, which provide a simple, reliable, and general method to estimate the change of extinction coefficient for the entire particle size regime during the bimodal particle dynamic processes.     

Strongly differentiable solutions of the discrete coagulation-fragmentation equation

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.     

Universality Classes in Burgers Turbulence

We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with Lévy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski’s coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of self-similar solutions for this equation.     

Nondiffusive coagulation of microparticles in solid-state dispersions

An expression is derived for the size distribution of dispersed particles and other relations are derived as well, on the basis of a more general equation for the rate of change of the upper and lower limits of the size spectrum. An analysis of these relations reveals the characteristics of transition between two processes, namely from formation of a dispersed phase (crystallization) to coagulation of particles through a disperse system where no coagulation of microparticles occurs. It is shown that in many real disperse systems (dispersion-hardened alloys) microparticles grow by the mechanism of nondiffusive coagulation, the latter being limited by the rate at which atoms build up at the boundaries of particles.     

超声促进胶体聚沉作用的研究

本文研究了超声促进胶体聚沉作用，选用了具有代表性的原糖溶液和老抽酱油为研究对象，研究结果表明，适宜的超声参数能十分显著加速胶体的聚沉，为物理场手段降去胶体提供一个新的途径。     

Diffusional coagulation of superparamagnetic particles in the presence of an external magnetic field

In this paper, we investigate the diffusional coagulation of colloidal superparamagnetic (SP) latex particles that are under the influence of an external magnetic field. The cluster size distributions (CSDs) that evolve with time were determined using an optical set-up that permits the direct visualization of particle clusters. Following the dynamic scaling analysis of van Dongen and Ernst (Phys. Rev. Lett. 54 (1985) 1396), we find that the CSDs all collapse onto a master curve when properly scaled. The bell-shape of this master curve indicates that large clusters preferentially scavenge small clusters in our system. From the time evolution of the average cluster size we infer that the reactivity between large clusters diminishes with increasing cluster size. These results are consistent with a simple mathematical formulation of the coagulation rate constant, or kernel, for the Brownian coagulation of magnetic particles. Moreover, our results support a growing body of evidence that the dynamic scaling theory developed by van Dongen and Ernst is a useful framework with which to study the microscale processes governing particle coagulation.     

Exact solutions for random coagulation processes

Smoluchowski's coagulation equation can only derive physical validity as the limit of a random coagulation process. For coagulation rateK _ij =a+b(i+j) and no fragmentation, random coagulation is exactly solvable and has the coagulation equation as its limit as the system sizeM, with deviations for finite systems of relative orderO(M ^–1/2). The same is true for constant coagulation and non-zero fragmentation rate, independent ofi andj (the sizes of the coagulation clusters) in which case the stochastic equations (master equation) are also exactly solvable.     

Coarse-grained computation for particle coagulation and sintering processes by linking Quadrature Method of Moments with Monte-Carlo

The study of particle coagulation and sintering processes is important in a variety of research studies ranging from cell fusion and dust motion to aerosol formation applications. These processes are traditionally simulated using either Monte-Carlo methods or integro-differential equations for particle number density functions. In this paper, we present a computational technique for cases where we believe that accurate closed evolution equations for a finite number of moments of the density function exist in principle, but are not explicitly available. The so-called equation-free computational framework is then employed to numerically obtain the solution of these unavailable closed moment equations by exploiting (through intelligent design of computational experiments) the corresponding fine-scale (here, Monte-Carlo) simulation. We illustrate the use of this method by accelerating the computation of evolving moments of uni- and bivariate particle coagulation and sintering through short simulation bursts of a constant-number Monte-Carlo scheme.     

Uniqueness of Mass-Conserving Self-similar Solutions to Smoluchowski’s Coagulation Equation with Inverse Power Law Kernels

Uniqueness of mass-conserving self-similar solutions to Smoluchowski’s coagulation equation is shown when the coagulation kernel K is given by \(K(x,x_*)=2(x x_*)^{-\alpha }\), \((x,x_*)\in (0,\infty )^2\), for some \(\alpha >0\).     

Assessing the application and downstream effects of pulsed mode ultrasound as a pre-treatment for alum coagulation

The application of pulsed mode ultrasound (PMU) as a pre-treatment for alum coagulation was investigated at various alum dosages and pH levels. The effects of the treatments on turbidity and dissolved organic carbon (DOC) removal and residual Al were evaluated. Response surface methodology (RSM) was utilized to optimize the operating conditions of the applied treatments. The results showed that PMU pre-treatment increased turbidity and DOC removal percentages from maximum of 96.6% and 43% to 98.8% and 52%, respectively. It also helped decrease the minimum residual Al from 0.100 to 0.094 ppm. The multiple response optimization was carried out using the desirability function. A desirability value of >0.97 estimated respective turbidity removal, DOC removal and Al residual of 89.24%, 45.66% and ∼0.1 ppm for coagulation (control) and 90.61%, >55% and ∼0 for coagulation preceded by PMU. These figures were validated via confirmatory experiments. PMU pre-treatment increased total coliform removal from 80% to >98% and decreased trihalomethane formation potential (THMFP) from 250 to 200 ppb CH₃Cl. Additionally, PMU application prior to coagulation improved the settleability of sludge due to the degassing effects. The results of this study confirms that PMU pre-treatment can significantly improve coagulation performance.     

Structural stability of disperse systems and finite nature of a coagulation front

A new discrete model of coagulation, which is a discrete analog of the Oort-van de Hulst-Safronov equation, is derived. It is shown that the familiar version, in contrast with Smoluchowski’s equation, can be used to calculate the propagation of a coagulation front. The relationship between compliance to the mass conservation law and the finite nature of the coagulation front is established, and then estimates of the time of violation of the mass conservation law are made for several classes of coagulation kernels. One of the conclusions is that the mass conservation law can be violated in cases where particles of roughly equal mass cannot coagulate, as occurs, for example, in gravitational coagulation. Estimates of the time for the appearance of structural instability of the system are made for multiplicative coagulation kernels. Zh. éksp. Teor. Fiz. 116, 717–730 (August 1999)     

Tail Behaviour of Self-Similar Profiles with Infinite Mass for Smoluchowski’s Coagulation Equation

In this article, we consider self-similar profiles to Smoluchowski’s coagulation equation for which we derive the precise asymptotic behaviour at infinity. More precisely, we look at so-called fat-tailed profiles which decay algebraically and as a consequence have infinite total mass. The results only require mild assumptions on the coagulation kernel and thus cover a large class of rate kernels.     

Filamentary coagulation of particles from water-heterogeneous systems in the field of an acoustic resonator

The coagulation of particles from water-heterogeneous systems in the field of a confocal ultrasonic resonator is studied. It is found that, at frequencies of several megahertz, when acoustic power of about 1 W is applied to the resonator, long stable filaments consisting of the material of the heterogeneous system are formed in the vicinity of the resonator axis. The filaments consist of thin disks formed by coalescent particles spaced at intervals strictly equal to half of the sound wavelength. The features of this coagulation are determined for suspensions of various nature (metal and dielectric particles, colloidal solutions, and oil emulsions). It is established that the coagulation in a standing acoustic wave occurs faster than under natural conditions (under the influence of gravity). The possibility of using this effect for cleaning liquids from impurities and separating hyperfine particles without employing filter materials is discussed.     

An investigation of mechanisms for the enhanced coagulation removal of Microcystis aeruginosa by low-frequency ultrasound under different ultrasound energy densities

There is a lack of studies elaborating the differences in mechanisms of low-frequency ultrasound-enhanced coagulation for algae removal among different ultrasound energy densities, which are essential to optimizing the economy of the ultrasound technology for practical application. The performance and mechanisms of low-frequency ultrasound (29.4 kHz, horn type, maximum output amplitude = 10 μm) -coagulation process in removing a typical species of cyanobacteria, Microcystis aeruginosa, at different ultrasound energy densities were studied based on a set of comprehensive characterization approaches. The turbidity removal ratio of coagulation (with polymeric aluminum salt coagulant at a dosage of 4 mg Al/L) was considerably increased from 44.1% to 59.7%, 67.0%, and 74.9% with 30 s of ultrasonic pretreatment at energy densities of 0.6, 1.11, and 2.22 J/mL, respectively, indicating that low-frequency ultrasound-coagulation is a potential alternative to effectively control unexpected blooms of M. aeruginosa. However, the energy density of ultrasound should be deliberately considered because a high energy density (≥18 J/mL) results in a significant release of algal organic matter, which may threaten water quality security. The specific mechanisms for the enhanced coagulation removal by low-frequency ultrasonic pretreatment under different energy densities can be summarized as the reduction of cell activity (energy density ≥ 0.6 J/mL), the slight release of negatively charged algal organic matter from cells (energy density ≥ 1.11 J/mL), and the aggregation of M. aeruginosa cells (energy density ≥ 1.11 J/mL). This study provides new insights for the ongoing study of ultrasonic pretreatment for the removal of algae via coagulation.     

On the theory of Brownian coagulation

The traditional diffusion approach for calculation of the collision frequency function for coagulation of Brownian particles is critically analyzed and shown to be valid only in the particular case of coalescence of small particles with large ones and inapplicable to calculation of the coalescence rate for particles of comparable sizes. It is shown that coalescence of Brownian particles generally occurs in the kinetic regime (realized under condition of homogeneous spatial distribution of particles), however, the expression for the collision frequency function in the continuum mode of the kinetic regime formally coincides with the standard expression derived in the diffusion regime for the particular case of large and small particles. This explains the validity of the traditional form of the coagulation rate equation in a wide range of parameters, corresponding to the continuum mode. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical approach of existing models.