Mathematical Problems in Engineering

Volume 2016, Article ID 1735897, 8 pages

http://dx.doi.org/10.1155/2016/1735897

## Analytical Solutions for Composition-Dependent Coagulation

^{1}Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China^{2}Jiyang College, Zhejiang Agriculture and Forestry University, Zhuji 311800, China

Received 6 January 2016; Accepted 4 April 2016

Academic Editor: Babak Shotorban

Copyright © 2016 Manli Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Exact solutions of the bicomponent Smoluchowski’s equation with a composition-dependent additive kernel are derived by using the Laplace transform for any initial particle size distribution. The exact solution for an exponential initial distribution is then used to analyse the effects of parameter on mixing degree of such bicomponent mixtures and the conditional distribution of the first component for particles with given mass. The main finding is that the conditional distribution of large particles at larger time is a Gaussian function which is independent of the parameter .

#### 1. Introduction

Modelling of a number of industrially important processes such as coagulation and growth of aerosols [1], granulation of powders [2], crystallization [3], crystal shape engineering [4], and synthesis of nanoparticles [5] requires particles to be identified with two or more of their attributes, such as mass for two or more different compositions, mass and surface area, mass of primary particles and binder volume, particle volume, and uncapped surface area. In general cases, the coagulation kernel is a function of both size and composition of the particles. In granulation, for instance, the surface properties (surface energy, roughness) of granules determine the efficiency by which granules are coated by the binder [6]. Therefore, components with different wetting properties may exhibit markedly different behavior during coagulation. Compositional effects introduce yet another dimension in the interaction between particles in coagulation.

This multicomponent coagulation problem was brought to focus by Lushnikov [7] and later by Krapivsky and Ben-Naim [8] for systems in which the coagulation kernel is independent of composition. Vigil and Ziff [9] summarized the solutions of Lushnikov and showed that in these cases the compositional distribution is a Gaussian function. More recently, Matsoukas et al. formulated the bicomponent problem in terms of one population balance equation for the size distribution and another for the distribution of components and provided solutions for kernels that are independent of composition [10–12]. These solutions have shown that for such kernels the distribution of components follows Gaussian scaling that is independent of the details of the kernel.

Without loss of generality, only two-component coagulation problem is considered and two components are given by their mass (or volume) .

The governing equation for this coagulation problem is the following population balance equation (PBE):which is an extension of Smoluchowski’s equation for one-component coagulation, where is the number of density functions at time such that represents the number concentration of particles in the size range of -component, to , and the size range of -component, to ; is the coagulation rate coefficient. Recently, Fernández-Díaz and Gómez-García, by using Laplace transform, obtained an exact analytical solution for (1) with the additive kernel (which is independent of composition) for any initial particle size distribution (PSD) [13]. They further analysed the behavior of the solution for larger sizes and time and found that the scaling solution cannot be used to describe the behavior of the number of the particle size distributions. In this study, we extend Fernández-Díaz and Gómez-García’s procedure to solve (1) with a composition-dependent kernel (see (2) in next section) and analyse the effects of parameter on the properties of bicomponent coagulation.

#### 2. Exact Solution for a Composition-Dependent Kernel

The kernel considered in this study is given aswhere parameter determines the relative contribution of composition to the coagulation. Evidently the additive kernel studied in [13] is recovered if we are letting of the kernel given in (2).

To seek the exact solution of (1) with kernel given by (2), we can, from (1), obtain the equation for the total number of particles:where is the total number of particles, is the mass of -component, and is the mass of -component. The solution of (3) is easily obtained: , with , , and the characteristic coagulation time .

Following [9], the number concentration distribution can be given as below:

Substituting (4) into (1), we have

For (5) we can use two-dimensional Laplace transform

Taking the derivative in (6) follows

This is Burgers’ equation in multidimension without a diffusive term. It can be solved in the transformed space by the Lagrange-Charpit method [14]:

With multidimensional Lagrange inversion [15], we obtainwith

That is,when naming .

Applying Laplace inverse transform, we obtain

By rearranging the multiple series in one, we arrive at

With (4), we finally obtain the general solutions of (1)This solution can be applied for any initial distributions by determining the multidimensional Laplace transform . If exponential initial PSD is assumed

The Laplace transform for this function isand we can obtain the explicit solution as below:It is easily verified that the solution in [13] is recovered if we are letting . We will further analyse some interesting properties of solution (17) in the next section and attention is paid particularly to the effects of parameter on the coagulation properties of such systems.

#### 3. The Effects of on the Coagulation

##### 3.1. The Total Number and Mass of Particles with the Concentration

Mixing degree of the mixtures is one of the key issues in bicomponent coagulation problems. To this end, we define two important magnitudes introduced in [7, 13]. One is the total number of particles having the concentration of the first component:and the mass of these particlesWith the compositional-dependent additive kernel (1) and the initial condition (15), we obtainwithAs suggested in [13], we observe the process at different average particles size with .

Figures 1–3 show the evolution of total number and total mass with concentration for , 1, and 20, respectively. It is shown that the overall behavior of the evolution of total number and mass is similar for different , while the evolution for total number ((a) in Figures 1–3) and total mass ((b) in Figures 1–3) is obviously different. The curves of total number do not tend to a Dirac- function, whereas the curves of total mass do, and this result is consistent with the result in [13]. The effect of can be clearly seen from comparison between Figures 2(b) and 3(b), and it is shown that the maxima for the total mass approach the overall fraction from the left end for , but from the right end for . This is due to the fact that the larger the parameter , the bigger the contribution of the -component to the coagulation, which leads to the quicker growth of particles with higher concentration.