Coagulation Kernel Research Articles

Stationary solutions to coagulation-fragmentation equations

Existence of stationary solutions to the coagulation-fragmentation equation is shown when the coagulation kernel K and the overall fragmentation rate a are given by K(x,y)=xαyβ+xβyα and a(x)=xγ, respectively, with 0≤α≤β≤1, α+β∈[0,1), and γ>0. The proof requires two steps: a dynamical approach is first used to construct stationary solutions under the additional assumption that the coagulation kernel and the overall fragmentation rate are bounded from below by a positive constant. The general case is then handled by a compactness argument.

A conservative method for numerical solution of the population balance equation, and application to soot formation

The objective of this paper is to present a finite volume method for the discretisation of the population balance equation with coagulation, growth and nucleation that combines: (a) accurate prediction of the distribution with a small number of sections, (b) conservation of the first moment (or any other single moment) in a coagulation process, (c) applicability to an arbitrary non-uniform grid, and (d) speed and robustness that make it suitable for combining with a CFD code for solving problems such as soot formation in flames. The conservation of the first moment of a distribution with respect to particle volume is of particular importance for two reasons: it is an invariant during a coagulation process and it represents conservation of mass. The method is based on a geometric evaluation of the double integrals arising from the finite volume discretisation of the coagulation terms and an exact balance of coagulation source and sink terms to ensure moment conservation. Extensive testing is performed by comparison with analytical solutions and direct numerical solutions of the discrete PBE for both theoretical and physically important coagulation kernels. Finally, the method is applied to the simulation of a laminar co-flow diffusion sooting flame, in order to assess its potential for coupling with CFD, chemical kinetics, transport and radiation models. The results show that accurate solutions can be obtained with a small number of sections, and that the PBE solution requires less than one fourth of the time of the complete simulation, only half of which is spent on the discretisation (the remaining being for the evaluation of the temperature dependence of the coagulation kernel).

Smoluchowski’s discrete coagulation equation with forcing

In this article we study an extension of Smoluchowski’s discrete coagulation equation, where particle in- and output takes place. This model is frequently used to describe aggregation processes in combination with sedimentation of clusters. More precisely, we show that the evolution equation is well-posed for a large class of coagulation kernels and output rates. Additionally, in the long-time limit we prove that solutions converge to a unique equilibrium with exponential rate under a suitable smallness condition on the coefficients.

A novel moment method using the log skew normal distribution for particle coagulation

A novel moment method is proposed to predict the detailed size distribution for particle coagulation. The method uses the log skew normal distribution (LSND) to approximate the actual size distribution. The LSND is a four-parameter distribution that generalizes the conventional log-normal distribution to allow for asymmetrical characteristics. The moment closure is achieved by applying the Gauss-Hermite quadrature for the integral terms in the moment equations. With this approach, no further treatments are needed for specific coagulation kernels. Then the method is validated by comparing it with other recognized numerical methods for Brownian coagulation in the continuum regime and the free-molecular regime. The results show that the present method well predicts the detailed size distribution over the entire coagulation time for both regimes. Moreover, the tails of the self-preserving size distribution are well reproduced by the present method. The asymptotic characteristics of Brownian coagulation are also analyzed based on the LSND. The results validate the LSND in analytically representing the self-preserving size distribution and predicting the asymptotic geometric standard deviation.

Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

AbstractGlobal weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Modeling the colloidal dispersions: flocculation kinetics though particle dynamics

The problem of stability of colloids recently begins to attract extra attention in construction materials science. This is due to numerous attempts to employ different kind of nanoscale modifiers for production of building materials with enhanced operational properties. Problems of stability and coagulation in colloidal dispersions are studied for several decades, and numerous results were already obtained within framework of Smoluchowski coagulation theory. In the present work we have performed numerical study of the flocculation process and compared the results with well-known ones. It was shown that even for complex pairwise potential the kinetics of number density for isolated particles is not very different from the kinetics which corresponds to constant coagulation kernel. However, for number density of many-particle aggregates we have observed number of peculiarities, including semi-periodic behavior.

Existence of mass conserving solution for the coagulation–fragmentation equation with singular kernel

In this article, we investigate the existence of a solution to the initial value problem associated to a singular coagulation–fragmentation equation. We analyze the problem with singular coagulation kernel and multiple fragmentation kernel. It is also shown that the solution satisfies the mass conservation law.

Mass-conserving solutions to coagulation-fragmentation equations with nonintegrable fragment distribution function

Existence of mass-conserving weak solutions to the coagulation-fragmentation equation is established when the fragmentation mechanism produces an infinite number of fragments after splitting. The coagulation kernel is assumed to increase at most linearly for large sizes and no assumption is made on the growth of the overall fragmentation rate for large sizes. However, they are both required to vanish for small sizes at a rate which is prescribed by the (nonintegrable) singularity of the fragment distribution.

Parallel Numerical Algorithm for Solving Advection Equation for Coagulating Particles

In this work we present a parallel implementation of numerical algorithm solving the Cauchy problem for equation of advection of coagulating particles. This equation describes time-evolution of the concentration f(x, v, t) of particles of size v at the point x at the time-moment t. Our numerical algorithm is based on use of total variation diminishing (TVD) scheme and perfectly matching layers (PML) for approximation of advection operator along spatial coordinate x and utilization of the fast numerical method for evaluation of coagulation integrals exploiting low-rank decomposition of coagulation kernel coefficients and fast FFT-based implementation of convolution operation along particle size coordinate v. In our work we exploit one-dimensional domain decomposition approach along spatial coordinate x because it allows to avoid use of parallel FFT implementations which are very expensive in terms of data exchanges and have poor parallel scalability. Moreover, locality of finite-difference operator from TVD-scheme along x coordinate allows to obtain good scalability even for computing clusters with slow network interconnect due to modest volumes of data necessary for synchronization exchanges between times integration steps.

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 $\alpha>0$.

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

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.

A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.

Exact solutions of the population balance equation including particle transport, using group analysis

The population balance equation (PBE) has received an unprecedented amount of attention in recent years from both academics and industrial practitioners because of its long history, widespread use in engineering, and applicability to a wide variety of particulate and discrete-phase processes. However it is typically impossible to obtain analytical solutions, although in almost every case a numerical solution of the PBEs can be obtained. In this article, the symmetries of PBEs with homogeneous coagulation kernels involving aggregation, breakage and growth processes and particle transport in one dimension are found by direct solving the determining equations. Using the optimal system of one and two-dimensional subalgebras, all invariant solutions and reduced equations are obtained. In particular, an explicit analytical physical solution is also presented.

An efficient numerical method for a mathematical model of a transport of coagulating particles

A new computational algorithm based on a fast way of computing integral operators describing the coagulation process is proposed for a mathematical model of coagulating particles. Using this algorithm, the computational complexity of each timestep of an explicit difference scheme can be substantially reduced. For each step, the complexity of execution is reduced from O(NM2) arithmetic operations to O(NMRlnM), where N is the number of mesh points along the physical coordinates of particles, M is the number of mesh points in a grid corresponding to sizes of coagulating particles, and R is the rank of a matrix corresponding to the values of the function of a coagulation kernel at mesh points. Using this approach, computations can be greatly accelerated, provided that kernel rank R is small.

Study and modification of a size-discrete semi-implicit simulation model for polydisperse aerosol coagulation

Aerosols in many natural and industrial processes are affected by coagulation, the merging of aerosol particles. The present work focuses on a semi-implicit size-discrete method to simulate the evolution of the particle size distribution by coagulation. The derivation of the method is retraced and generalized in terms of the size definition and discretization schemes. Based on this, new forms for the partition coefficients, that implement the redistribution of coagulated particles into the discrete size intervals, are proposed. For this, either constant or linear particle size distribution functions in terms of number and volume concentrations are assumed in each size interval. These assumptions also impact the calculation of the mean volume of the particles in each discretization interval. The inclusion of the coagulation kernel, which describes the physical conditions and the collision mechanism, in the partition coefficients is also tested. The modifications are compared against existing methods in the reproduction of analytical solutions for certain coagulation problems. It was found that certain combinations of modifications were able to reduce the numerical error caused by particle size discretization by more than 80% in the tested cases.

Diffusion mediated coagulation and fragmentation based study of domain formation in lipid bilayer membrane

We estimate the equilibrium size distribution of cholesterol rich micro-domains on a lipid bilayer by solving Smoluchowski equation for coagulation and fragmentation. Towards this aim, we first derive the coagulation kernels based on the diffusion behaviour of domains moving in a two dimensional membrane sheet, as this represents the reality better. We incorporate three different diffusion scenarios of domain diffusion into our coagulation kernel. Subsequently, we investigate the influence of the parameters in our model on the coagulation and fragmentation behaviour. The observed behaviours of the coagulation and fragmentation kernels are also manifested in the equilibrium domain size distribution and its first moment. Finally, considering the liquid domains diffusing in a supported lipid bilayer, we fit the equilibrium domain size distribution to a benchmark solution.

Reynolds-number dependence of turbulence enhancement on collision growth

Abstract. This study investigates the Reynolds-number dependence of turbulence enhancement on the collision growth of cloud droplets. The Onishi turbulent coagulation kernel proposed in Onishi et al. (2015) is updated by using the direct numerical simulation (DNS) results for the Taylor-microscale-based Reynolds number (Reλ) up to 1140. The DNS results for particles with a small Stokes number (St) show a consistent Reynolds-number dependence of the so-called clustering effect with the locality theory proposed by Onishi et al. (2015). It is confirmed that the present Onishi kernel is more robust for a wider St range and has better agreement with the Reynolds-number dependence shown by the DNS results. The present Onishi kernel is then compared with the Ayala–Wang kernel (Ayala et al., 2008a; Wang et al., 2008). At low and moderate Reynolds numbers, both kernels show similar values except for r2 ∼ r1, for which the Ayala–Wang kernel shows much larger values due to its large turbulence enhancement on collision efficiency. A large difference is observed for the Reynolds-number dependences between the two kernels. The Ayala–Wang kernel increases for the autoconversion region (r1, r2 < 40 µm) and for the accretion region (r1 < 40 and r2 > 40 µm; r1 > 40 and r2 < 40 µm) as Reλ increases. In contrast, the Onishi kernel decreases for the autoconversion region and increases for the rain–rain self-collection region (r1, r2 > 40 µm). Stochastic collision–coalescence equation (SCE) simulations are also conducted to investigate the turbulence enhancement on particle size evolutions. The SCE with the Ayala–Wang kernel (SCE-Ayala) and that with the present Onishi kernel (SCE-Onishi) are compared with results from the Lagrangian Cloud Simulator (LCS; Onishi et al., 2015), which tracks individual particle motions and size evolutions in homogeneous isotropic turbulence. The SCE-Ayala and SCE-Onishi kernels show consistent results with the LCS results for small Reλ. The two SCE simulations, however, show different Reynolds-number dependences, indicating possible large differences in atmospheric turbulent clouds with large Reλ.

Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation

In this paper we present a novel numerical algorithm for the space-homogeneous multicomponent (multidimensional) Smoluchowski coagulation equation, the number of components is considered as dimensionality. The new methodology is based on the classical finite-difference predictor-corrector scheme. In a straightforward implementation of this scheme, however, one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid. We propose to use special low-parametric representations for the grid functions and as well for the coagulation kernel. The corresponding multidimensional arrays are approximated by low-rank tensor-train decompositions reducing them to combinations of small low-dimensional arrays, eventually to matrices for which we can use fast algorithms of linear algebra. Instead of O ( N 2 d ) operations in the classical scheme, we propose a new method that requires only O ( d 2 N log ? N ) operations, where N is the number of nodes per axis in the space grid and d is the number of components. In this work we accelerate the predictor-corrector time-scheme and use the trapezoidal rule for the computation of multidimensional integral operators. Thus, the accuracy of the new method is O ( h 2 + ? 2 ) , where h is the space grid step and ? is the time step.

Advancement of the kinetic approach to Brownian coagulation on the base of the Langevin theory

The kinetic approach to calculation of the coagulation kernel, derived in the two subsequent approximations of the random walk theory in the authors’ previous papers avoiding semi-empirical assumptions of the traditional models, is further advanced by application of the Langevin equation for Brownian particle migration. The interpolation expressions for the collision rate derived in the simple random walk approximations were additionally justified and further improved.

A Metropolis scheme for Monte Carlo methods for the solution of particle population balance

A Metropolis scheme that can be used by various Monte Carlo methods for solving particle population balance equations has been proposed and verified. A detailed derivation of the scheme is presented. The computational accuracy of the proposed scheme is tested under various coagulation kernels. The computational performance of the scheme is also measured and compared to that of Inverse and acceptance–rejection scheme.