Smoluchowski Coagulation Equation Research Articles

Analytical study of the time-fractional Smoluchowski coagulation equation in light of different integrodifferential operators

AbstractThis article presents the derivation of the fractional Smoluchowski coagulation equation via the variational principles technique. We use the variational iteration method to solve the Caputo-type fractional coagulation equation. Furthermore, we analyze the time-fractional coagulation equation using the homotopy perturbation transform approach, considering three different fractional operators: Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Our findings demonstrate that the solutions for the total number of particles during coagulation align well with existing literature, particularly in the short time limit. Additionally, we examine the impact of the time-fractional order on the dynamics of particle coagulation for each fractional operator.

Solving crystallization/precipitation population balance models in CADET, Part II: Size-based Smoluchowski coagulation and fragmentation equations in batch and continuous modes

A particle size-based Smoluchowski coagulation and fragmentation equation was solved in the free and open source process modeling package CADET. The WFV and MCNP schemes were selected to discretize the internal particle size coordinate. Weights in these schemes were modified to preserve and conserve the zeroth and third moments for size-based equations. Modified propositions and proofs for the scheme are provided. Analytical Jacobians were derived and implemented to reduce the solver’s runtime. A two-dimensional Smoluchowski coagulation and fragmentation equation with axial position as external coordinate was formulated and discretized to support simulations of continuous particulate processes in dispersive plug flow reactors. Five 1D and four 2D test cases were used to validate the implementation and benchmark the solver’s performance. The runtime, L1 error norm, L1 error rate, particle size distribution moments up to sixth order and several scalar metrics were analyzed in detail.

Gelation and Localization in Multicomponent Coagulation with Multiplicative Kernel Through Branching Processes

The multicomponent coagulation equation is a generalization of the Smoluchowski coagulation equation, where the size of a particle is described by a vector. Similar to the original Smoluchowski equation, the multicomponent coagulation equation exhibits gelation behavior when supplied with a multiplicative kernel. Additionally, a new type of behaviour called localization is observed due to the multivariate nature of the particle size distribution. Here we extend the branching process representation technique, which we introduced to study differential equations in our previous work, and apply it to find a concise probabilistic solution of the multicomponent coagulation equation supplied with monodisperse initial conditions. We also provide short proofs for the gelation time and characterisation the localization phenomenon.

Polynomial approaches in improving accuracy of probability distribution estimation using the method of moments

AbstractBACKGROUNDDetermination of a probability density function (PDF) is an area of active research in engineering sciences as it can improve process systems. A previously developed polynomial method‐of‐moments‐based PDF estimation model has been applied in the research to produce accurate approximations to both standard and more complex PDF. A model with a different polynomial basis than a monomial is still to be developed and evaluated. This is the work that is undertaken in this study.RESULTSA set of standard PDF (Normal, Weibull, Log Normal and Bimodal) and more complex distributions (solutions to the Smoluchowski coagulation equation and Population Balance equation) were approximated by the method‐of‐moments using Chebyshev, Hermite and Lagrange polynomial‐based density functions. Results show that Lagrange polynomial‐based models improve the fit compared to monomial based‐modeling in terms of RMSE and Kolmogorov–Smirnov test statistic estimates. The Kolmogorov–Smirnov test‐statistics decreased by 19% and the RMSE values were improved by around 85% compared to the standard monomial basis when using Lagrange polynomial basis.CONCLUSIONThis study indicates that the procedure using Lagrange polynomials with method‐of‐moments is a more reliable reconstruction procedure that calculates the approximate distribution using lesser number of moments, which is desirable. © 2024 The Authors. Journal of Chemical Technology and Biotechnology published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry (SCI).

Coagulation, non-associative algebras and binary trees

We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by: (i) grafting together the roots of all pairs of order-compatible trees at preceding orders, or (ii) attaching binary branches to all free branches of trees at the previous order. We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.

Coagulation equations with source leading to anomalous self-similarity

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity , and behaves like when with . Our analysis shows that the long-time behaviour of the solutions depends on the parameters γ and λ. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of and on whether or . In all these cases, the scaling differs from the usual one that has been previously obtained when or . In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case .

An exact solution to nonstationary Smoluchowski's coagulation equation complicated by Ostwald ripening

The concluding stage of a phase transformation, governed by particle coagulation and Ostwald ripening processes, is considered. An exact solution to the non-stationary integrodifferential kinetic equation is constructed in a parametric form. Namely, the particle-volume distribution function, time, number of particles, volume of the condensed phase, and the average particle radius are derived as functions of liquid supersaturation. The analytical solution is found (i) for a constant collision-frequency function, and (ii) using the averaging of this function over all possible combinations of particle volumes for various coagulation mechanisms. It is shown that the particle-volume distribution function and average radius of particles are substantially different in cases of constant and averaged collision-frequency functions for shear coagulation and coagulation under gravity.

Correction: Islam et al. Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property. Mathematics 2023, 11, 2770

There were some errors in the original publication [...]

Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property

In this paper, we develop a generalized moment method with a continuous weight function for the Smoluchowski coagulation equation in its continuous form to study the mass conservation property of this equation. We first establish some basic inequalities for the generalized moment and prove the mass conservation property under a sufficient condition on the kernel and an initial condition, utilizing these inequalities. Additionally, we provide some concrete examples of coagulation kernels that exhibit mass conservation properties and show that these kernels exhibit either polynomial or exponential growth along specific particular curves.

The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.

On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel

Abstract We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$ , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left ( 0,\frac{1}{2}\right ) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.

Rate of convergence of two moments consistent finite volume scheme for non-classical divergence coagulation equation

This study focuses on performing a convergence analysis of the recently developed mass based finite volume scheme [20] to simulate a Smoluchowski coagulation equation. The goal of this method is to retain the total number of particles in the system while maintaining a constant mass of the particles utilizing some weights in the formulation. The Lipschitz continuity and consistency of the MBFVS are examined and investigated thoroughly by providing theorems. The rate of convergence of the MBFVS is investigated on (a) uniform grid, (b) non-uniform smooth (geometric) grid, and (c) locally uniform grid. The numerical results in term of number density function and its moments are computed and compared with exact results to ensure the authenticity of the scheme for gelling kernels. In comparison to the sectional approaches such as fixed pivot technique (Giri and Hausenblas, 2013 [13]) and cell average technique (Giri and Nagar, 2015 [14]), this scheme is the best candidate for coupling with powerful tools such as CFD, Ansys, and gPROMS due to its simplified formulation, robust implementation on any grid and coagulation kernel, second order convergence rate and consistency with two moments.

A deep learning approximation of non-stationary solutions to wave kinetic equations

We present a deep learning approximation, stochastic optimization based, method for wave kinetic equations. To build confidence in our approach, we apply the method to a Smoluchowski coagulation equation with multiplicative kernel for which an analytic solution exists. Our deep learning approach is then used to approximate the non-stationary solution to a 3-wave kinetic equation corresponding to acoustic wave systems. To validate the neural network approximation, we compare the decay rate of the total energy with previously obtained theoretical results. A finite volume solution is presented and compared with the present method.

Mass-conserving weak solutions to Oort-Hulst-Safronov coagulation equation with singular rates

Existence of global weak solutions to the continuous Oort-Hulst-Safronov (OHS) coagulation equation is investigated for coagulation kernels capturing a singularity near zero and growing linearly at infinity. The proof mainly relies on a relation, between classical Smoluchowski coagulation equation (SCE) and OHS coagulation equation, which is introduced in [16] as generalized coagulation equation. Moreover, all weak solutions formulated in a suitable sense are demonstrated to be mass-conserving. We obtain here a similar result for OHS coagulation equation as the one in [6] for SCE.

Modelling the size distribution of aggregated volcanic ash and implications for operational atmospheric dispersion modelling

Abstract. We have developed an aggregation scheme for use with the Lagrangian atmospheric transport and dispersion model NAME (Numerical Atmospheric Dispersion modelling Environment), which is used by the London Volcanic Ash Advisory Centre (VAAC) to provide advice and guidance on the location of volcanic ash clouds to the aviation industry. The aggregation scheme uses the fixed pivot technique to solve the Smoluchowski coagulation equations to simulate aggregation processes in an eruption column. This represents the first attempt at modelling explicitly the change in the grain size distribution (GSD) of the ash due to aggregation in a model which is used for operational response. To understand the sensitivity of the output aggregated GSD to the model parameters, we conducted a simple parametric study and scaling analysis. We find that the modelled aggregated GSD is sensitive to the density distribution and grain size distribution assigned to the non-aggregated particles at the source. Our ability to accurately forecast the long-range transport of volcanic ash clouds is, therefore, still limited by real-time information on the physical characteristics of the ash. We assess the impact of using the aggregated GSD on model simulations of the 2010 Eyjafjallajökull ash cloud and consider the implications for operational forecasting. Using the time-evolving aggregated GSD at the top of the eruption column to initialize dispersion model simulations had little impact on the modelled extent and mass loadings in the distal ash cloud. Our aggregation scheme does not account for the density of the aggregates; however, if we assume that the aggregates have the same density of single grains of equivalent size, the modelled area of the Eyjafjallajökull ash cloud with high concentrations of ash, significant for aviation, is reduced by ∼ 2 %, 24 h after the start of the release. If we assume that the aggregates have a lower density (500 kg m−3) than the single grains of which they are composed and make up 75 % of the mass in the ash cloud, the extent is 1.1 times larger.

Solution of Smoluchowski coagulation equation for Brownian motion with TEMOM

The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole, and it can be decomposed into the following two functions by similarity transformation: one is a function of time (the particle k-th moments), and the other is a function of dimensionless volume (self-preserving size distribution). In this paper, a simple iterative direct numerical simulation (iDNS) is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment, which has been solved with the Taylor-series expansion method of moment (TEMOM) in our previous work. The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature, and then the method is extended to the field of Brownian agglomeration over the entire size range. The results show that the difference between the lognormal function and the self-preserving size distribution is significant. Moreover, the thermodynamic constraint of the algebraic mean volume is also investigated. In short, the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship; thus, a complete method to solve the Smoluchowski coagulation equation asymptotically is established.

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

Markov Models of Coarsening in Two-Dimensional Foams with Edge Rupture

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space $\ell_1(\mathbb N)$ which counts total numbers of cells with $n\ge 3$ sides ($n$-gons). Under a large cell limit, we show that number densities of $n$-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.

Inverse Gaussian distributed method of moments for agglomerate coagulation due to Brownian motion in the entire size regime

The objective of this paper is to derive a model for studying the collision–coalescence dynamics of agglomerates. Accordingly, we first extend the inverse Gaussian distributed method of moments (IGDMOM; J. Atmospheric Sci., 2020, 77(9), 3011–3031) to solve the Smoluchowski coagulation equation for fractal-like agglomerates undergoing Brownian coagulation in the entire size regime. The extended IGDMOM, denoted as E-IGDMOM, is integrated with two solutions, namely Dahneke's solution and the harmonic mean solution, for deriving moment ordinary differential equations (MODEs) for Brownian coagulation in the free molecular regime and in the continuum-slip regime. Comparative analyses demonstrate that the E-IGDMOM achieves nearly the same efficiency as that of the Taylor-series expansion MOM (TEMOM) and the lognormal MOM (log MOM); nevertheless, it has higher precision than that of the log MOM in the free molecular regime and nearly the same precision as that of the TEMOM and log MOM in the transient regime. Regarding the skewness and kurtosis of size distribution functions, compared with the log MOM and IGDMOM, we observe that the E-IGDMOM yields values that are much closer to those of the quadrature MOM.

PLUME-MoM-TSM 1.0.0: a volcanic column and umbrella cloud spreading model

Abstract. In this paper, we present a new version of PLUME-MoM, a 1-D integral volcanic plume model based on the method of moments for the description of the polydispersity in solid particles. The model describes the steady-state dynamics of a plume in a 3-D coordinate system, and a modification of the two-size moment (TSM) method is adopted to describe changes in grain size distribution along the plume, associated with particle loss from plume margins and with particle aggregation. For this reason, the new version is named PLUME-MoM-TSM. For the first time in a plume model, the full Smoluchowski coagulation equation is solved, allowing us to quantify the formation of aggregates during the rise of the plume. In addition, PLUME-MOM-TSM allows us to model the phase change of water, which can be either magmatic, added at the vent as liquid from external sources, or incorporated through ingestion of moist atmospheric air. Finally, the code includes the possibility to simulate the initial spreading of the umbrella cloud intruding from the volcanic column into the atmosphere. A transient shallow-water system of equations models the intrusive gravity current, allowing computation of the upwind spreading. The new model is applied first to the eruption of the Calbuco volcano in southern Chile in April 2015 and then to a sensitivity analysis of the upwind spreading of the umbrella cloud to mass flow rate and meteorological conditions (wind speed and humidity). This analysis provides an analytical relationship between the upwind spreading and some observable characteristic of the volcanic column (height of the neutral buoyancy level and plume bending), which can be used to better link plume models and volcanic-ash transport and dispersion models.