Coagulation Equation Research Articles

Contractivity for Smoluchowski's coagulation equation with solvable kernels

We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski's coagulation equation.

Existence of self-similar solutions to Smoluchowski’s coagulation equation with product kernel

Existence of self-similar solutions to Smoluchowski’s coagulation equation with product kernel

The scaling hypothesis for Smoluchowski's coagulation equation with bounded perturbations of the constant kernel

We consider Smoluchowski's coagulation equation with a kernel of the form K=2+ϵW, where W is a bounded kernel of homogeneity zero. For small ϵ, we prove that solutions approach a universal, unique self-similar profile for large times, at almost the same speed as the constant kernel case (the speed is exponential when self-similar variables are considered). All the constants we use can be explicitly estimated. Our method is a constructive perturbation analysis of the equation, based on spectral results on the linearisation of the constant kernel case. To our knowledge, this is the first time the scaling hypothesis can be fully proved for a family of kernels which are not explicitly solvable.

Efficient Method of Moments for Simulating Atmospheric Aerosol Growth: Model Description, Verification, and Application

AbstractThe atmospheric aerosol dynamics model (AADM) has been widely used in both comprehensive air quality model systems and chemical transport modeling globally. The AADM consists of Smoluchowski's coagulation equation (SCE), whose solution undergoing Brownian coagulation in the free molecular regime is a challenge because it is inconsistent with aerosols whose size distribution cannot exactly follow the lognormal size distribution. Thus, a new method for solving the SCE without assuming lognormal size distribution is proposed and developed. The underlying principle of this method is that the hybridization of the well‐established method of moments with the assumed lognormal size distribution (log MOM) and Taylor‐series expansion method of moments (TEMOM) is implemented. This method shows excellent agreement with the sectional method (SM) which is used as reference. The accuracy of these two specific models closely approaches that of the TEMOM, but overcomes the limitation of the classical log MOM. The computational time of this scheme is considerably lower than that of the SM. The new method was successfully implemented to reveal the formation and growth of secondary particles emitted from a vehicle exhaust tailpipe. It was found that the formation of new particles only occurs in the interface region of the turbulent exhaust jet (which is very close to the tailpipe exit), whereas no new particles are formed in the mixture of the exhaust jet plume and the surrounding cold air downstream. The new method is verified as an efficient and reliable numerical scheme for studying atmospheric aerosol dynamics.

Parametric Method of Moments for Solving the Smoluchowski Coagulation Equation in the Theory of Accumulation of Dust Bodies in a Protoplanetary Disk

In relation to the problem of accumulation of dust particles, which are the main structure-forming element of planetesimals in a protoplanetary cloud, we propose a parametric method of moments for solving the Smoluchowski integro-differential equation that describes dispersed coagulation of disk matter. We consider a parametric approach to finding the size distribution function of protoplanetary bodies based on the Pearson diagram, with the aid of which the corresponding distributions are quite satisfactorily found by their first four moments. This approach is especially effective when it is necessary to know only the general properties of the volume distribution functions of coagulating bodies and their temporal evolution. Since the kinetics of the processes of aggregation of protoplanetary bodies substantially depends on the specific type of coagulation kernels, a fairly general method for their approximation is proposed in the study, which allows one to obtain simplified expressions. As a practical application, the parametric method of moments is demonstrated by a number of examples of the growth of protoplanetary bodies. The results provide a new productive approach to solving the key problem of stellar-planetary cosmogony associated with an explanation of the process of growth of interstellar dust particles to large planetesimals.

An analytical solution of the population balance equation for simultaneous Brownian and shear coagulation in the continuum regime

Brownian coagulation of aerosol particles can take place in both laminar and turbulent flows. Thus, the simultaneous Brownian and shear coagulation will always occur in practical applications. This study presents an analytical solution to describe the size evolution of polydisperse particles undergoing simultaneous Brownian and shear coagulation. The analytical solution is derived using the log-normal method of moments (LNMOM) with some approximations. Then, the analytical solution is validated by comparing with previous analytical solutions derived for limiting cases. The results show that the present analytical solution is consistent with previous analytical solutions for these limiting cases. Further, the time trajectories of the total particle number concentration, the geometric standard deviation and the geometric number mean particle volume predicted by the present analytical solution are compared with those of a numerical LNMOM model. The results show that the present analytical solution gives good predictions of the total particle number concentration but less accurate predictions of the geometric standard deviation and the geometric number mean particle volume. A dimensionless analysis shows that the coagulation rate ratio and the initial polydispersity are two important factors for characterizing the size evolution of simultaneous Brownian and shear coagulation.

Global classical solutions to the continuous coagulation equation with collisional breakage

Existence and uniqueness of global classical solutions to the continuous coagulation equation with collisional breakage are investigated for collision kernels, growing indefinitely for large volumes, and a particular distribution function which gives the case of binary collisional breakage. The existence proof relies on the compactness method. Moreover, it is shown that the unique global classical solution is mass conserving.

A new method for solving population balance equations using a radial basis function network

This study presents a new method for solving the population balance equation (PBE) for particle coagulation. The method introduces a radial basis function network to approximate the number density ...

Weak solutions to the continuous coagulation model with collisional breakage

A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernel and distribution functions. The model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each of these processes may take place with a suitably assigned probability depending on the volume of particles participating in the collision.

Including Dust Coagulation in Hydrodynamic Models of Protoplanetary Disks: Dust Evolution in the Vicinity of a Jupiter-mass Planet

Abstract Dust growth is often neglected when building models of protoplanetary disks due to its complexity and computational expense. However, it does play a major role in shaping the evolution of protoplanetary dust and planet formation. In this paper, we present a numerical model coupling 2D hydrodynamic evolution of a protoplanetary disk, including a Jupiter-mass planet, and dust coagulation. This is obtained by including multiple dust fluids in a single grid-based hydrodynamic simulation and solving the Smoluchowski equation for dust coagulation on top of solving for the hydrodynamic evolution. We find that fragmentation of dust aggregates trapped in a pressure bump outside of the planetary gap leads to an enhancement in the density of small grains. We compare the results obtained from the full-coagulation treatment to the commonly used, fixed-dust-size approach and to previously applied, less computationally intensive methods for including dust coagulation. We find that the full-coagulation results cannot be reproduced using the fixed-size treatment, but some can be mimicked using a relatively simple method for estimating the characteristic dust size in every grid cell.

The Effect of a Monomer Source on the Oscillation Period in an Irreversible Coagulation Model

The article examines the kinetic equations of irreversible coagulation with a source of monomers and a sink of particles that exceed the maximum allowed size. Time-periodic solutions are known for the class of Brownian kernels. In this study, we analyze the effect of the monomer source intensity on the period and the amplitude of the particle concentration oscillations over time. The numerical results suggest that as the source intensity is increased, the oscillation amplitude increases while the oscillation period decreases, so that no qualitative changes are observed in the solution structure. A change in source intensity does not produce scaling of the model time and model concentrations of the particles per unit volume of the medium.

Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation

In this paper we study the continuous coagulation and multiple fragmentation equation for the mean-field description of a system of particles taking into account the combined effect of the coagulation and the fragmentation processes in which a system of particles growing by successive mergers to form a bigger one and a larger particle splits into a finite number of smaller pieces. We demonstrate the global existence of mass-conserving weak solutions for a wide class of coagulation rate, selection rate and breakage function. Here, both the breakage function and the coagulation rate may have algebraic singularity on both the coordinate axes. The proof of the existence result is based on a weak $ L^1 $ compactness method for two different suitable approximations to the original problem, namely, the conservative and non-conservative approximations. Moreover, the mass-conservation property of solutions is established for both approximations.

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.

Unsteady-state particle-size distributions at the coagulation stage of phase transformations

In this paper, a new analytical solution of nonstationary integro-differential coagulation equation, which describes the concluding stage of a phase transformation process in metastable liquids, is deduced. This solution takes into account the initial distribution function, which is found on the basis of previously developed theory for the intermediate stage. It is demonstrated that the derived particle-size distribution function representing a bell-shaped curve decreases with time and describes the long-term experimental data. The developed theoretical approach describes the transition of a phase transformation process from its intermediate stage to the stage of particle coagulation.

Modeling simultaneous coagulation and charging of nanoparticles at high temperatures using the method of moments

A large number of chemically and thermally ionized species are produced in flames. During flame synthesis of nanoparticles, these ions collide with the particles as do the particles amongst themselves. Both charging and particle-particle collisions decide the particle size distribution, but existing models in flame synthesis often do not consider the coupling of these two effects. In this work, a model simulating simultaneous charging and coagulation is developed using the method of moments (MoM), with the help of asymptotic methods and perturbation theory. This model considers different charged states, as well as the particle size distribution in each charged state and their interactions. To achieve this, first, a simplified polynomial expression for the charging coefficient is derived from Fuchs’ theory. This expression is found to be in good agreement with the complete Fuchs theory expression at high temperatures in the free molecular regime. Next, this expression is used in the general dynamic equation for simultaneous charging and coagulation to derive population balance equations of volume moments. A simplified modeling method, named the monodisperse model (MdM), was used to compare the simulation results. Both the MoM and MdM showed good agreement in different simulated cases. It was found that for constant bipolar ion environment, the collisional growth increases as the ion concentration increases, and flattens out for high ion concentration (>108 #/cm3). Simulated results also showed that for a unipolar ion environment, MoM predicted that the particle growth by collisions would be more suppressed, resulting in particles with a lower polydispersity index. The simplified expressions for the ion-attachment coefficient used in the MoM works well at high temperatures in the free molecular regime, and that MdM is able to capture the physics of the system well.

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.

Extended log-normal method of moments for solving the population balance equation for Brownian coagulation

An extended log-normal method of moments (ELNMOM) is presented in this study for solving the population balance equation (PBE) for Brownian coagulation. The method is an extension of the log-normal...

Self-Similar Solutions to Coagulation Equations with Time-Dependent Tails: The Case of Homogeneity One

We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski’s coagulation equation, for a class of kernels $${K(x,y)}$$ which are homogeneous of degree one and satisfy $${K(x,1) \to k_0 > 0}$$ as $${x\to 0}$$ . In particular, we establish the existence of a critical $${\rho_* > 0}$$ with the property that for all $${\rho\in(0,\rho_*)}$$ there is a positive and differentiable self-similar solution with finite mass M and decay $${A(t)x^{-(2+\rho)}}$$ as $${x\to\infty}$$ , with $${A(t)=e^{M(1+\rho)t}}$$ . Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter $${\rho}$$ .

Параметрический метод моментов решения уравнения коагуляции Смолуховского в теории аккумуляции пылевых тел в допланетном диске

Параметрический метод моментов решения уравнения коагуляции Смолуховского в теории аккумуляции пылевых тел в допланетном диске

A COMPARATIVE STUDY OF NUMERICAL APPROXIMATIONS FOR SOLVING THE SMOLUCHOWSKI COAGULATION EQUATION

ABSTRACT In this work, numerical approximations for solving the one dimensional Smoluchowski coagulation equation on non-uniform meshes has been analyzed. Among the various available numerical methods, finite volume and sectional methods have explicit advantage such as mass conservation and an accurate prediction of different order moments. Here, a recently developed efficient finite volume scheme (Singh et al., 2015) and the cell average technique (Kumar et al., 2006) are compared. The numerical comparison is established for both analytically tractable as well as physically relevant kernels. It is concluded that the finite volume scheme predicts both number density as well as different order moments with higher accuracy than the cell average technique. Moreover, the finite volume scheme is computationally less expensive than the cell average technique.