Mass-based finite volume scheme for aggregation, growth and nucleation population balance equation

Abstract

In this paper, a new mass-based numerical method is developed using the notion of Forestier-Coste & Mancini (Forestier-Coste & Mancini 2012, SIAM J. Sci. Comput. 34, B840–B860. (doi:10.1137/110847998)) for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method.

1. Introduction

Particulate processes affect changes in the physical properties of a population of particles due to aggregation, growth, nucleation and other similar mechanisms. Many natural systems and industrial applications involve these mechanisms, for example, nucleation and growth of rain drops, polymerization [1,2], crystallization [3], pharmaceutical granulation [4], liquid–liquid dispersions [5] and flame synthesis of materials [6]. In aggregation, the smaller size particles merge with each other to build bigger size particles. Hence, the number of particles in the system decreases gradually with time while total mass is conserved. Growth takes place when molecular clusters are added to the surface of existing particles. In this case, the mass of the particulate system increases while the number of particles remains constant. Nucleation refers to the formation of new particles via condensation or precipitation from a fluid phase leading to an increase in the number of particles as well as total mass of the system. A pictorial representation of all mechanisms is demonstrated in figure 1. Mathematical models, generally known as population balances, are essential to track the change in particle properties in these systems.

Figure 1.

Schematic diagram of (a) aggregation, (b) growth and (c) nucleation processes. (Online version in colour.)

The population balance equation (PBE) under the simultaneous action of nucleation, growth and aggregation is a nonlinear integro-partial differential equation. The one-dimensional continuous PBE involving the above-mentioned processes for a well-mixed system can be written as follows (dimension here refers to the number of components tracked):

$$\begin{array}{rl}& \frac{\mathrm{\partial }n(x,t)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }[G(x,t)n(x,t)]}{\mathrm{\partial }x}=\frac{1}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }\\ & -{\int }_0^{\mathrm{\infty }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }+B_{\mathrm{src}}(x,t),\end{array}$$ 1.1

subject to the initial condition

$$n(x,0)=n_0(x),x\in ]0,\mathrm{\infty }[.$$

Here, n(x, t) denotes the number density distribution of particles having volume x > 0 at time t≥0, that is, the number distribution function in the small range [x, x + dx] is computed by n(x, t) dx. We take x to be the particle volume, an additive property, noting that additivity is not necessarily fulfilled by any generic measure of ‘size’. G(x, t) denotes the rate of growth of particles and B_src(x, t) is the rate of nucleation. The first integral of equation (1.1) denotes the gain of particles having property (volume) x due to the coalescence of particle volumes x − x′ and x′. Similarly, the second integral corresponds to the removal of particles with volume x due to the collision with particles with volume x′. The aggregation kernel β(x, x′, t) interprets the rate at which a successful merging of two particles having volumes x and x′ take place. It is a positive function (β(x, x′, t)≥0) and symmetric with respect to its volume arguments, that is, β(x, x′, t) = β(x′, x, t). In this study, we consider kernels that are independent of time and are functions of particle mass only, i.e. β = β(x, x′), though our methods apply to time-dependent kernels as well.

Many practical applications [7] require the prediction of the number density function n along with some integral properties, namely moments. The jth moment of the number density function n(x, t) is defined

$${\mu }_j(t)={\int }_0^{\mathrm{\infty }}{x}^{j}n(x,t)\hspace{0.17em}\mathrm{d}x,$$ 1.2

for integers j = 0, 1, 2, … . Of particular interest is the zeroth order moment μ₀, which represents the total concentration of the particulate population, and the first moment μ₁, which is its total mass.

Due to the presence of nonlinear integrals, it is extremely difficult to obtain analytical solutions except in certain special cases. Exact solutions have been obtained for pure aggregation for simple kernels, such as the constant, additive and multiplicative kernels [8–10]. The analytical solution for a simultaneous aggregation and growth PBE was derived by Hu et al. [11]. No exact solution is available for simultaneous aggregation and nucleation PBE in the literature. It is often easier to obtain solutions for the moments than for the distribution itself, but even those solutions are limited to certain forms of the rate constants that appear in the PBE [12,13]. Various numerical methods have been developed to solve equation (1.1), including stochastic methods [14–19], finite-element methods [20], quadrature method of moments [21,22], finite volume schemes [13,23–32], the Galerkin method [33], method of moments [34–36], as well as sectional methods like fixed pivot technique [10,37] and cell average technique (CAT) [38,39]. Among all the available techniques, sectional methods, such as fixed pivot and CATs, are very good at conserving the various integral properties of the system [10,38]. They are, however, computationally expensive due to complex formulations. Moreover, the extension of these numerical methods to approximate multidimensional problems having two or more property coordinates is complicated by the requirement to distribute particle properties to neighbouring nodes, which in turn necessitates the recalculation of the birth term in order to conserve the integral properties [40–42]. Both sectional and finite volume methods rely on the assumption that particles need to be accumulated at the mean (pivot) of the cell. But the formation of new particles can take place at any position of the cell. In order to accommodate the formation of new particles, sectional methods distribute the particle properties to the neighbouring pivots in order to conserve the important moments, whereas finite volume schemes introduce appropriate weights to conserve the integral moments [28,29,43].

To solve the aggregation equation with simultaneous nucleation and growth, the typical approach is to combine one of the above-mentioned methods for aggregation with the method of characteristics (MOC) for growth and nucleation [10,13,26,38]. The fixed pivot technique leads to the numerical diffusion [10]. The cell average method is more accurate but computationally intensive [38].

Qamar et al. [26] have provided a simpler method to solve the simultaneous PBEs by combining the finite volume scheme and the MOC, but finding the moments accurately is the major issue with this method. The methods developed by Qamar et al. [26] and Kumar & Warnecke [27] handle the integral quantities (moments in our case) explicitly by taking account of the function xn(x, t) in place of n(x, t) by rewriting the PBE in the form:

$$\begin{array}{rl}\frac{\mathrm{\partial }(xn(x,t))}{\mathrm{\partial }t}+\frac{\mathrm{\partial }[G(x,t)xn(x,t)]}{\mathrm{\partial }x}& =-\frac{\mathrm{\partial }}{\mathrm{\partial }x}[{\int }_0^x{\int }_{x-y}^{\mathrm{\infty }}y\beta (y,\nu ,t)n(y,t)n(\nu ,t)\hspace{0.17em}\mathrm{d}\nu \hspace{0.17em}\mathrm{d}y]\\ & +x{B}_{\mathrm{src}}(x,t),x,t\in [0,\mathrm{\infty }[.\end{array}$$ 1.3

Kumar et al. [13] developed a finite volume scheme by reformulating the CAT into conservative form (1.3) of the PBE but its complex formulation and the need for a large number of grid points are practical drawbacks. The issue of complex formulations is overcome by the class of finite volume schemes, whose extension to higher dimensional population balances is very straight-forward [44] when assuming particles to accumulate at the pivot of the cell.

Forestier-Coste & Mancini [45] developed a finite volume scheme for an aggregation PBE on non-uniform grids, which is more suitable for the practical applications as it does not require particles to be assigned to the pivot of the cell, which is an essential assumption for both sectional and existing finite volume methods. In general, the probability of formation of new particles of different sizes due to various processes at the pivot of the cell is very low. Forestier-Coste & Mancini [45] overcome the issue of conservation of total mass in the system by introducing a proportionality constant by modifying the aggregation kernel. Their method was shown to predict the zeroth- and first-order moments as well as the distribution itself with very high accuracy. This implementation requires a fine grid, and even then accuracy deteriorates once 2% of the volume leaves the discretized domain, a condition that can be reached early in the simulation (in their case, at t = 1.5). Forestier-Coste & Mancini [45] have demonstrated that their finite volume scheme can be easily adapted to solve a PBE in higher dimensions due to its simpler mathematical formulation.

In this paper, a finite volume scheme is developed for solving the mass form of the aggregation PBE different from equation (1.3) using the notion of Forestier-Coste & Mancini [45]. It will be shown that the new scheme predicts the number density function and its moments more accurately and efficiently than the existing scheme. Moreover, the existing and new schemes will be extended to approximate the simultaneous aggregation-growth and simultaneous aggregation-nucleation problems using a much coarser grid but retaining the accuracy of the numerical results.

The rest of the paper is structured as follows: §2 begins with a brief description of the existing finite volume method of Forestier-Coste & Mancini [45] before introducing the mass-based finite volume scheme (MFVS) for solving the pure aggregation, simultaneous aggregation-growth and simultaneous aggregation-nucleation PBEs. Some theoretical results and a condition for the positivity of the solution are also derived in §3. In §§3–5, the numerical results obtained by the mass-based method for some standard (benchmark) kernels are analysed with the existing finite volume scheme in order to verify the efficiency as well as the accuracy. Finally, §6 highlights important issues that arose during this work.

2. Finite volume schemes for pure aggregation PBE

In this section, the mathematical formulations of the existing finite volume scheme by Forestier-Coste & Mancini [45] (labelled henceforward as FCM2012) is briefly reviewed and an MFVS on non-uniform meshes is introduced and analysed. A thorough study of the usage of a non-uniform mesh has been done by Forestier-Coste & Mancini [45] who found that a few non-uniform grids may be used instead of a large number of uniform grids but still retain accuracy. This entails much fewer numerical operations and hence, lower computational cost.

The property coordinate x in the second integral of the PBE (1.1) ranges from 0 to ∞ but, since we are working with the numerical methods, we fix the volume domain by replacing ∞ with a large positive number, say x_max in our case. Thus, the truncated PBE can be written as follows:

$$\frac{\mathrm{\partial }n(x,t)}{\mathrm{\partial }t}=\frac{1}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }-{\int }_0^{x_{\max }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime },$$ 2.1

subject to the initial condition

$$n(x,0)=n_0(x),x\in ]0,x_{\max }[.$$

The choice of x_max depends on the type of problem considered and should be selected in such a fashion that the total volume loss from the system is very small for each simulation. The finite 1D computational domain with the upper limit, x_I+1/2 < ∞, is shown in figure 2. The domain is partitioned into I number of small cells having x_j representing the jth cell, Λ_j: = [x_j−1/2,  x_j+1/2), j∈1, …, I. Denoting x_j−1/2 and x_j+1/2 as the lower and upper ends of the jth cell, respectively, the midpoints and the step size of the cells are defined by

$$x_{1/2}=0,x_j=\frac{x_{j-1/2}+x_{j+1/2}}{2}\mathrm{and}\mathrm{\Delta }{x}_j=x_{j+1/2}-x_{j-1/2}.$$ 2.2

Figure 2.

One-dimensional domain discretization.

(a). Existing finite volume scheme (FCM2012)

For the derivation of the existing method, coalescence of cells i and k leads to a new cell (i + k) having lower and upper boundaries as x_i−1/2 + x_k−1/2 and x_i+1/2 + x_k+1/2, respectively. For non-uniform meshes, the possibility that the merging cells (i + k) fall exactly within any cell j is extremely low. This signifies that the newly formed cell after coalescing can intersect with multiple cells. The schematic of three basic configurations for overlapping is demonstrated in figure 3. In order to derive the formulation, the above notion of overlapping cells is used to define the set of indices as follows:

$$S^j=\{(i,k)\in \mathbb{N}\times \mathbb{N}:(i+k)\cap j\ne \mathrm{\varnothing }\}$$ 2.3

and

$$R_{i,k}=\{j\in \mathbb{N}:j\cap (i+k)\ne \mathrm{\varnothing }\}.$$ 2.4

The set Sⁱ denotes the intersection of the sum of mesh couples (i, k) with the mesh j, whereas R_i,k expresses the non-empty intersection of the set of mesh j and the sum of the meshes i and k. Further discretize the time into t^p+1 = t^p + Δt for p∈{0, …, N − 1}. Now suppose n^p_j denotes the average value of n at time t^p in the cell j, which is used to approximate n(x, t^p), given by

$${n_j}^p=\frac{1}{\mathrm{\Delta }{x}_j}{\int }_{x_{j-1/2}}^{x_{j+1/2}}n(x,t^p)\hspace{0.17em}\mathrm{d}x,j\in \{1,\dots ,I\}.$$ 2.5

Figure 3.

Different possibilities for overlaps.

Using the above-defined sets, the finite volume approximation [45] can be written as

$$n_j^{p+1}=n_j^p+\mathrm{\Delta }t(\underset{B_{\mathrm{agg},j}^{\mathrm{FCM}2012}}{\underbrace{\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}{n}_i^p{n}_k^p{\lambda }_{i,k}^j\frac{\mathrm{\Delta }{x}_i\mathrm{\Delta }{x}_k}{\mathrm{\Delta }{x}_j}}}-\underset{D_{\mathrm{agg},j}^{\mathrm{FCM}2012}}{\underbrace{\sum _{j=1}^I{\beta }_{j,i}\hspace{0.17em}{n}_j^p\hspace{0.17em}{n}_i^p\mathrm{\Delta }{x}_i}}),$$ 2.6

where β_i,k = β(x_i, x_k) and λ^j_i,k are correction factors used to describe the overlapping of cells, which is defined by

$${\lambda }_{i,k}^j=\left(\frac{\overline{m_{i,k}^j}-\underset{\_}{m_{i,k}^j}}{\mathrm{\Delta }{x}_i+\mathrm{\Delta }{x}_k}\right).$$ 2.7

Here, $\overline{m_{i,k}^j}$ and $\underset{\_}{m_{i,k}^j}$ indicate the maximum and minimum bounds of the intersection of the cell i + k with a given cell j:

$$\overline{m_{i,k}^j}=min(x_{j+1/2},x_{i+1/2}+x_{k+1/2})$$

and

$$\underset{\_}{m_{i,k}^j}=max(x_{j-1/2},x_{i-1/2}+x_{k-1/2}).$$

The method formulated in equation (2.6) is not volume conserving. Forestier-Coste & Mancini [45] accomplished the volume conservation property by modifying the aggregation kernel in the first term of the right-hand side of equation (2.6) as follows:

$${\hat{\beta }}_{i,k}={\beta }_{i,k}\frac{2(x_i+x_k)}{\sum _{j\in R_{i,k}}{x}_j({\lambda }_{i,k}^j+{\lambda }_{k,i}^j)}.$$ 2.8

The theoretical proof of the conservation of the total volume in the system and its detailed formulation are provided in Forestier-Coste & Mancini [45].

(b). Mass-based finite volume scheme

In this subsection, the finite volume scheme of Forestier-Coste & Mancini [45] is extended to enhance the accuracy of integral moments as well as the number density function. It will be shown that the conversion of the original number-based PBE to mass-based PBE improves the accuracy of the various order moments and number density function by 50%. The mass form PBE can be obtained from the original equation (2.1) by multiplying it with the property coordinate x, which gives the following expression:

$$\frac{\mathrm{\partial }xn(x,t)}{\mathrm{\partial }t}=\frac{x}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }-x{\int }_0^{x_{\max }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }.$$ 2.9

In order to derive the formulation for solving the above equation (2.9), the same set of indices defined in equations (2.3) and (2.4) are used. Using the same time discretization defined earlier, and supposing that M^p_j is the average value of M(x, t^p) at time t^p in the cell j, then

$${M_j}^p=\frac{1}{\mathrm{\Delta }{x}_j}{\int }_{x_{j-1/2}}^{x_{j+1/2}}M(x,t^p)dx,j\in \{1,\dots ,I\}.$$ 2.10

Proceeding the same way as in the previous section, we propose the modified finite volume scheme as

$$M_j^{p+1}=M_j^p+\mathrm{\Delta }t(\underset{B_{\mathrm{agg},j}^{\mathrm{MFVS}}}{\underbrace{\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}(x_i+x_k){\lambda }_{i,k}^j}}-\underset{D_{\mathrm{agg},j}^{\mathrm{MFVS}}}{\underbrace{\sum _{i=1}^I{\beta }_{j,i}\frac{M_i^p}{x_i}{M}_j^p}}).$$ 2.11

The notion of overlapping of the cells is incorporated using the parameter λ^j_i,k in the above formulation. The idea of a finite volume scheme is based on conservation of the total volume in the system; however, the formulation (2.11) takes no measure for the volume conservation property. Similar to the existing method, the modification of the kernel is required, which helps in achieving the conservation of total volume. This is given by the following expression:

$${\hat{\beta }}_{i,k}={\beta }_{i,k}\frac{2}{\sum _{j\in R_{i,k}}({\lambda }_{i,k}^j+{\lambda }_{k,i}^j)}.$$ 2.12

Now our main motive is to prove that the formulation satisfies the volume conservation using the modified aggregation kernel (2.12). Thus, we have to prove that

$$\sum _{j=1}^I{M}_j^{p+1}=\sum _{j=1}^I{M}_j^p,$$

where M_j denotes the total mass in jth cell at time t^p.

Proposition 2.1. —

The numerical formulation (2.11) holds the volume conservation property, that is, no loss of volume takes place from the computational domain.

Proof. —

Taking summation over j on both sides in (2.11), the following is obtained:

$$\sum _{j=1}^I{M}_j^{p+1}=\sum _{j=1}^I{M}_j^p+\mathrm{\Delta }{t}^{p}T,$$ 2.13

where

$$T=\sum _{j=1}^I(\underset{T_1}{\underbrace{\frac{1}{2}\sum _{(i,k)\in S^j}{\beta }_{i,k}\frac{M_i^p{M}_k^p}{x_i{x}_k}(x_i+x_k)\frac{2{\lambda }_{i,k}^j}{\sum _{j\in R_{i,k}}({\lambda }_{i,k}^j+{\lambda }_{k,i}^j)}}}-\underset{T_2}{\underbrace{\sum _{i=1}^I{\beta }_{j,i}\frac{M_j^p}{x_j}{M}_i^p}}).$$ 2.14

In order to prove the volume conservation property, it is required to show that T₁ = T₂.

Now consider the term

$$T_1=\sum _{j=1}^I\frac{1}{2}\sum _{(i,k)\in S^j}\sum _{j\in R_{i,k}}{\beta }_{i,k}\frac{M_i^p{M}_k^p}{x_i{x}_k}(x_i+x_k)\frac{2{\lambda }_{i,k}^j}{({\lambda }_{i,k}^j+{\lambda }_{k,i}^j)}.$$ 2.15

The expression $\sum _{j=1}^I\sum _{(i,k)\in S^j}$ shows the combination of all cells having indices i and k that intersect with cells of index j. Therefore, it can also be written as follows:

$$\sum _{j=1}^I\sum _{(i,k)\in S^j}=\sum _{i=1}^I\sum _{k=1}^I.$$

After simplification, equation (2.16) takes the following form:

$$T_1=\sum _{i=1}^I\sum _{k=1}\sum _{j\in R_{i,k}}{\beta }_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}{x}_i\frac{2{\lambda }_{i,k}^j}{({\lambda }_{i,k}^j+{\lambda }_{k,i}^j)}.$$ 2.16

The parameters λ^j_i,k and β are symmetric, that is, λ^j_i,k = λ^j_k,i and β_i,k = β_k,i. This further simplifies to the following expression:

$$T_1=\sum _{i=1}^I\sum _{k=1}^I{\beta }_{k,i}\frac{M_k^p}{x_k}{M}_i^p.$$ 2.17

Change the indices from $k\to j$ in the above equation and it reduces to the following form:

$$T_1=\sum _{i=1}^I\sum _{j=1}^I{\beta }_{j,i}\frac{M_j^p}{x_j}{M}_i^p.$$ 2.18

This implies that T₁ = T₂, that is, the mass-based formulation satisfies the volume conservation formulation. ▪

Proposition 2.2. —

The modified aggregation kernel ${\hat{\beta }}_{i,j}$ is equivalent to β_i,j for the case of uniform grids.

Proof. —

Consider uniform meshes such that x_j = jh, where h > 0 is a constant. Moreover, the integers i and k are considered in such a fashion that (x_i−1/2 + x_k−1/2, x_i+1/2 + x_k+1/2) falls completely within the jth cell having boundaries (x_j−1/2, x_j+1/2). This implies

$$\overline{m_{i,k}^j}=\frac{x_{i+1/2}+x_{k+1/2}}{\mathrm{\Delta }{x}_i+\mathrm{\Delta }{x}_k}$$

and

$$\underset{\_}{m_{i,k}^j}=\frac{x_{i-1/2}+x_{k-1/2}}{\mathrm{\Delta }{x}_i+\mathrm{\Delta }{x}_k}.$$

Using the above two relations in equation (2.7), we obtain the following

$$\begin{array}{rl}{\lambda }_{i,k}^j& =\frac{x_{i+1/2}+x_{k+1/2}-x_{i-1/2}-x_{k-1/2}}{\mathrm{\Delta }{x}_i+\mathrm{\Delta }{x}_k}\\ & =1.\end{array}$$

Further equation (2.7) implies

$${\hat{\beta }}_{i,k}={\beta }_{i,k}.$$

Hence, the result is proved. ▪

Since the mass-based method is an explicit time-stepping numerical method, certain constraints have to be considered for the positivity of the numerical solutions. Consider the positive initial condition n⁰_j > 0 for all j. To assure the positive numerical solution, a stability Courant–Friedrichs–Lewy (CFL) condition for the time step similar to Forestier-Coste & Mancini [45] is required. The CFL condition for the mass-based method is given by

$$\mathrm{\Delta }{t}^p<\underset{i}{min}\left(\right|\frac{n_j^p}{B_j^p-D_j^p}\left|\right),$$ 2.19

where the discrete birth (B^p_j) and death (D^p_j) terms are given by

$$B_j^p=\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}(x_i+x_k){\lambda }_{i,k}^j$$

and

$$D_j^p=\sum _{i=1}^I{\beta }_{j,i}\frac{M_i^p}{x_i}{M}_j^p.$$

Hence, the mass-based formulation with a positive initial condition ensured the positivity of the solution under the constraint (2.20) on the time step Δt^p.

(c). Numerical approximation for simultaneous aggregation and growth

In this section, the first ever numerical approximation is developed for solving simultaneous growth-aggregation processes based on the notion of overlapping of the cells [45]. The mathematical model that describes these mechanisms simultaneously is given by

$$\begin{array}{rl}\frac{\mathrm{\partial }n(x,t)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }[G(x,t)n(x,t)]}{\mathrm{\partial }x}& =\frac{1}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }\\ & -{\int }_0^{\mathrm{\infty }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }.\end{array}$$ 2.20

Here, G(x, t) denotes the rate of growth of particles and can be defined as G(x, t) = dx/dt.

Using the calculations from Kumar & Ramkrishna [10], the above equation can be further simplified to the following expression:

$$\begin{array}{rl}\frac{\mathrm{\partial }n(x,t)}{\mathrm{\partial }t}+n(x,t)\frac{\mathrm{\partial }[G(x,t)]}{\mathrm{\partial }x}& =\frac{1}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }\\ & -{\int }_0^{\mathrm{\infty }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }.\end{array}$$ 2.21

The mass form PBE for simultaneous growth and aggregation processes will take the following form:

$$\begin{array}{rl}\frac{\mathrm{\partial }xn(x,t)}{\mathrm{\partial }t}+xn(x,t)\frac{\mathrm{\partial }[G(x,t)]}{\mathrm{\partial }x}& =\frac{x}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }\\ & -x{\int }_0^{\mathrm{\infty }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }.\end{array}$$ 2.22

Earlier, Kumar et al. [38] implemented the CAT to solve the simultaneous growth-aggregation problem. The major issue with that method was the inability to predict the various order moments accurately. Later, Qamar & Warnecke [24] developed a finite volume scheme combined with the MOC to solve this problem but the number density obtained by this method deviated far from the exact result. Moreover, for employing this method, the original PBE was changed to the mass conservative form [23] to stabilize the solution. It is shown in the literature that the growth process can be handled in a similar way like aggregation (discrete events), but the numerical diffusion becomes a big issue [38]. This inability of the existing sectional methods and finite volume scheme to couple with the MOC leads us to propose a solution to this simultaneous aggregation-growth PBE which is more accurate and less expensive. Therefore, a moving pivot method is considered as the best approach to solve the growth problem. The typical representation of the moving grid is shown in figure 4.

Figure 4.

Schematic representation of the moving grid.

Our idea is that one may use the finite volume scheme for approximating the aggregation problem (see §2b), whereas a Lagrangian approach can be implemented to tackle the growth problem in order to obtain a more accurate numerical result. Thus, the final set of ordinary differential equations required to solve the simultaneous aggregation-growth problem using a number-based finite volume scheme is given below:

$$\begin{array}{rl}\frac{\mathrm{d}{n}_j}{\mathrm{d}t}& =\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}{n}_i^p{n}_k^p{\lambda }_{i,k}^j\frac{\mathrm{\Delta }{x}_i\mathrm{\Delta }{x}_k}{\mathrm{\Delta }{x}_j}-\sum _{j=0}^I{\beta }_{j,i}{n}_j^p{n}_i^p\mathrm{\Delta }{x}_i,\end{array}$$ 2.23

$$\begin{array}{rl}\frac{\mathrm{d}{x}_j}{\mathrm{d}t}& =G(x_j),\end{array}$$ 2.24

$$\begin{array}{rl}\text{and}\frac{\mathrm{d}{x}_{j+1/2}}{\mathrm{d}t}& =G(x_{j+1/2}).\end{array}$$ 2.25

Similarly, the set of ordinary differential equations obtained for the MFVS to approximate simultaneous aggregation-growth PBE is given as:

$$\frac{\mathrm{d}{M}_j}{\mathrm{d}t}=\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}(x_i+x_k){\lambda }_{i,k}^j-\sum _{i=1}^I{\beta }_{j,i}\frac{M_i^p}{x_i}{M}_j^p,$$ 2.26

together with

$$\frac{\mathrm{d}{x}_j}{\mathrm{d}t}=G(x_j),$$ 2.27

and

$$\frac{\mathrm{d}{x}_{j+1/2}}{\mathrm{d}t}=G(x_{j+1/2}).$$ 2.28

Here, equations (2.24) and (2.27) describe the rate of change of particles due to the aggregation process obtained using number-based finite volume schemes and MFVSs, respectively, while equations (2.25)/(2.28) and (2.26)/(2.29) describe the change of representative (or pivot) and boundaries of the jth cell, respectively.

(d). Numerical approximation for simultaneous aggregation and nucleation

In this section, the numerical treatment of the population balances involving simultaneous aggregation–nucleation processes is developed. The mathematical model required to describe these mechanisms simultaneously is described as follows:

$$\begin{array}{rl}\frac{x\mathrm{\partial }n(x,t)}{\mathrm{\partial }t}& =\frac{x}{2}{\int }_0^x\beta (x^{\prime },x-x^{\prime },t)n(x-x^{\prime },t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }\\ & -x{\int }_0^{\mathrm{\infty }}\beta (x,x^{\prime },t)n(x,t)n(x^{\prime },t)\hspace{0.17em}\mathrm{d}{x}^{\prime }+x{B}_{\mathrm{src}(x,t)}.\end{array}$$ 2.29

There are two possible ways to handle nucleation: (i) if the nucleation is considered to be monodisperse then by setting a boundary condition near to the initial boundary of the given domain (at particle size zero) and (ii) an alternative approach will be to consider a source located near particle size zero. For our study, we chose the second approach.

The conventional way of coupling the aggregation and nucleation processes and their solution (using the fixed pivot technique) was proposed by Kumar & Ramkrishna [10] and can be written as

$$\frac{\mathrm{d}{n}_j}{\mathrm{d}t}\hspace{0.17em}\text{o}r\hspace{0.17em}\frac{\mathrm{d}{M}_j}{\mathrm{d}t}=B_{\mathrm{agg},j}^{\mathrm{Num}}-D_{\mathrm{agg},j}^{\mathrm{Num}}+{\int }_{x_{j-1/2}}^{x_{j+1/2}}{B}_{\mathrm{src}}(t,x)\hspace{0.17em}\mathrm{d}x,$$ 2.30

where the superscript ‘Num’ denotes the numerical methods used to solve the aggregation PBE. The major drawback of this formulation is that it may lose or gain mass as it is assumed that the nucleation takes place exactly at the representative of the cell. This problem was overcome by Kumar et al. [38] using the CAT and provides a more efficient way of coupling the aggregation and nucleation processes as follows:

$$\frac{\mathrm{d}{n}_i}{\mathrm{d}t}=B_{\mathrm{agg}+\mathrm{src},j}^{\mathrm{Num}}-D_{\mathrm{agg},j}^{\mathrm{Num}},$$ 2.31

where

$$B_{\mathrm{agg}+\mathrm{nuc},j}^{\mathrm{Num}}=B_{\mathrm{agg},j}^{\mathrm{Num}}+{\int }_{x_{j-1/2}}^{x_{j+1/2}}{B}_{\mathrm{src}}(x,t)\hspace{0.17em}\mathrm{d}x.$$ 2.32

The above formulation (2.32) can be used to solve a general polydisperse nucleation but for a monodisperse nucleation, the formation of the smallest particle and the smallest pivot (or representative) is considered in such a manner that it overlaps with the appearing monodisperse particles. Therefore, the above formulation takes the following form for the number-based finite volume scheme:

$$\frac{\mathrm{d}{n}_1}{\mathrm{d}t}=\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}{n}_i^p{n}_k^p{\lambda }_{i,k}^j\frac{\mathrm{\Delta }{x}_i\mathrm{\Delta }{x}_k}{\mathrm{\Delta }{x}_j}-\sum _{j=0}^I{\beta }_{j,i}{n}_j^p{n}_i^p\mathrm{\Delta }{x}_i+B_{\mathrm{src}}(x_j,t),\mathrm{for}\hspace{0.17em}j=1$$ 2.33

and

$$\frac{\mathrm{d}{n}_j}{\mathrm{d}t}=\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}{n}_i^p{n}_k^p{\lambda }_{i,k}^j\frac{\mathrm{\Delta }{x}_i\mathrm{\Delta }{x}_k}{\mathrm{\Delta }{x}_j}-\sum _{j=0}^I{\beta }_{j,i}{n}_j^p{n}_i^p\mathrm{\Delta }{x}_i,\mathrm{for}\hspace{0.17em}j=2,3,\dots ,I.$$ 2.34

Similarly, the MFVSs for approximating the simultaneous aggregation and nucleation PBE takes the following form:

$$\frac{\mathrm{d}{M}_1}{\mathrm{d}t}=\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}(x_i+x_k){\lambda }_{i,k}^j-\sum _{i=1}^I{\beta }_{j,i}\frac{M_i^p}{x_i}{M}_j^p+B_{\mathrm{src}}(x_j,t),\mathrm{for}\hspace{0.17em}j=1$$ 2.35

and

$$\frac{\mathrm{d}{M}_j}{\mathrm{d}t}=\frac{1}{2}\sum _{(i,k)\in S^j}{\hat{\beta }}_{i,k}\frac{M_i^p}{x_i}\frac{M_k^p}{x_k}(x_i+x_k){\lambda }_{i,k}^j-\sum _{i=1}^I{\beta }_{j,i}\frac{M_i^p}{x_i}{M}_j^p,\mathrm{for}\hspace{0.17em}j=2,3,\dots ,I.$$ 2.36

3. Numerical results for pure aggregation PBE

The accuracy and efficiency of the mass-based method is examined by comparison with the numerical results obtained by the existing method in terms of various order moments. Additionally, the number density functions obtained numerically are also compared with exact results available in the literature. The comparison is conducted for the various analytically solvable kernels such as constant, additive and product (multiplicative) kernels with the known exact solutions, and certain physically relevant kernels for which the analytical (exact) solutions of number density functions are not available in the literature.

In discussing the numerical results, we use the degree of aggregation I_agg:

$$I_{\mathrm{agg}}(t)=1-\frac{{\mu }_0(t)}{{\mu }_0(t=0)},t\ge 0,$$ 3.1

which describes the decrease in total number of initial primary particles due to aggregation events. In the beginning of simulations, I_agg = 0 and in the limit $t\to \mathrm{\infty }$, I_agg will approach unity with all primary particles forming one aggregate.

Errors are quantified using the notion of maximum relative errors and can be calculated by the following expression:

$${\eta }_j(t)=\left|\frac{{\mu }_j^{\mathrm{exc}}-{\mu }_j^{\mathrm{num}}}{{\mu }_j^{\mathrm{exc}}}\right|.$$ 3.2

The superscripts 'exc' and 'num' refer to the exact and numerical values of moment, respectively. Here, η₀ and η₁ express the relative errors in zeroth-order moment and first-order moment, respectively. It is significant to note that the relative errors in the various order moments are calculated at the end of the simulations (end time). Errors in the number density function are calculated using weighted sectional errors. Calculating these errors is only possible for those kernels whose exact solutions of number density functions are available in the literature. Weighted sectional errors in number density functions can be computed using the expression given as follows:

$${\sigma }_i(t)=\frac{\sum _{j=1}^I|n_j^{\mathrm{exc}}-n_j^{\mathrm{num}}|x_j^i\mathrm{\Delta }{x}_j^i}{\sum _{j=1}^I{n}_j^{\mathrm{exc}}{x}_j^i\mathrm{\Delta }{x}_j^i}.$$ 3.3

For example, σ₀ describes the relative weighted error in predicting the number density function over the whole volume domain. Similarly, other quantities can be measured.

(a). Analytically solvable kernels

The aggregation kernels whose exact solution of number density functions and different order moments are available in the literature play a significant role in verifying the numerical methods. Among many kernels, the PBE (1.1) for the kernels β(x, x′) = 1, x + x′ and xx′ can be solved exactly [9,46]. For comparison of the various results, the exponential initial condition, that is, n₀(x) = e^−x, is considered in all these test cases. In the case of the product kernel, β(x, x′) = xx′, which is known to lead to gelation [47], the solutions presented are in the pre-gel region.

(i). Constant kernel

The comparison begins with the constant kernel, β(x, x′) = 1. The computational domain spans the range x_min = 10⁻⁶ ≤ x ≤ x_max = 300 and is divided into 30 non-uniform cells. This computational domain is used to run the simulations for both numerical methods. The numerical simulations are run till the degree of aggregation I_agg = 0.90.

Figure 5a shows that the total number of particles (zeroth-order moment) computed by MFVS is closer to the exact result than FCM2012. The first-order moments obtained by MFVS and FCM2012 show equal accuracy and agree well with the exact moment. The second-order moment (μ₂) is more accurately calculated by the MFVS even though no measure is taken for this order moment, whereas the FCM2012 shows a large deviation from the exact result (see figure 5c). The maximum relative errors in various moments calculated using the expression (3.2) are listed in table 1. It reveals that the maximum relative error in different order moments obtained by the MFVS is approximately 50% lesser than the FCM2012.

Figure 5.

Comparisons of normalized integral properties and number density function for the constant kernel: (a) zeroth-order moment, (b) first-order moment, (c) second-order moment and (d) number density function. (Online version in colour.)

Table 1.

Maximum error in different moments for the constant kernel.

η	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
η0	0.06324	0.02871	0.02430	0.01515
η1	1.7 × 10−10	4.7 × 10−12	9.8 × 10−16	2.7 × 10−16
η2	0.37822	0.17112	0.08142	0.03975

The comparison of number density functions is illustrated in figure 5d. It shows that the number density function was obtained with more precision by the MFVS than the FCM2012, especially at large sizes. The sectional weighted errors in the number density functions determined using equation (3.3) are listed in table 2, and it can be observed that the MFVS calculations for sectional errors are 50% more accurate than the FCM2012. The errors can be further reduced to any desired level by refining the grid, but this decision must be weighed against the computational cost. For the cases shown here the computational time taken by the FCM2012 and MFVS are 0.1931 s and 0.1468 s, respectively.

Table 2.

Weighted error of number distribution for the constant kernel.

σ	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
σ0	0.14614	0.07049	0.05748	0.03465
σ1	0.22948	0.11457	0.06512	0.02982
σ2	0.43653	0.20165	0.10336	0.04445

(b). Additive kernel

In the next comparison, we consider the additive kernel, β(x, x′) = x + x′. The computational domain for the comparison is from x_min = 10⁻⁴ and x_max = 4000 and is divided into 30 non-uniform cells. The numerical results are obtained by running the simulations until the degree of aggregation, I_agg = 0.80.

Figure 6a shows the graphical comparison of the zeroth-order moments obtained by MFVS with the exact moment and it shows better agreement with the exact result than the FCM2012. The first-order moment approximated by MFVS and FCM2012 matches well with the exact moment. The behaviour of the second-order moment shows that the MFVS approximates this moment with higher precision, whereas FCM2012 significantly deviates from the exact result (see figure 6c). The comparison of the number density computed by both numerical methods with the exact result is shown qualitatively in figure 6d. The results confirm that the MFVS approximates the number density function very well, whereas the number density function determined by the FCM2012 deviates considerably from the exact result, also justifying the reason of deviation of integral properties for the case of FCM2012. The maximum relative and weighted section errors in moments and number density functions are quantified in tables 3 and 4, respectively. The behaviour of the results is similar to the constant kernel, as the MFVS calculated these errors more accurately than the FCM2012. In terms of CPU time, the time consumed by the MFVS and FCM2012 are 0.3167 s and 0.3976 s, respectively, for computing all numerical results.

Figure 6.

Comparisons of normalized integral properties and number density function for the additive kernel: (a) zeroth-order moment, (b) first-order moment, (c) second-order moment and (d) number density function. (Online version in colour.)

Table 3.

Maximum error in different moments for the additive kernel.

η	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
η0	0.04323	0.01128	0.01464	0.00601
η1	5.5 × 10−5	2.7 × 10−7	2.04 × 10−6	2.6 × 10−10
η2	3.29372	0.74289	0.41126	0.12224

Table 4.

Weighted error of number distribution for the additive kernel.

σ	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
σ0	0.14714	0.06025	0.05791	0.03180
σ1	0.53947	0.20924	0.16139	0.04977
σ2	3.95423	0.90389	0.59017	0.15199

(c). Multiplicative kernel

In this test case, we consider the multiplicative kernel, β(x, x′) = xx′, an aggressive kernel that promotes the growth of large sizes. The computational domain consists of 30 non-uniform cells, ranging from x_min = 10⁻⁴ to x_max = 4000. The simulations are run until the multiplicative kernel exhibits gelling behaviour as indicated by Smit et al. [48].

Figure 7 demonstrates that the zeroth-order moment predicted by MFVS shows very good agreement with the exact result, whereas the FCM2012 underpredicts the moment, similar to the previous cases. The first-order moment predicted by MFVS is also showing a better agreement with the exact moment than the FCM2012, as the FCM2012 starts to deviate at a very early stage, as shown in figure 7b. The second-order moment (μ₂) is well captured by the MFVS, whereas the FCM2012 shows a large deviation from the exact result, as seen in figure 7c.

Figure 7.

Comparisons of normalized integral properties and number density function for the multiplicative kernel: (a) zeroth-order moment, (b) first-order moment, (c) second-order moment and (d) number density function. (Online version in colour.)

The number density function is shown in figure 7d. MFVS shows a small deviation from the exact number density function for larger size particles, whereas a large deviation is observed with the FCM2012. It is important to note that the existing scheme FCM2012 requires 70 non-uniform cells to predict both moments and number distribution functions accurately. On the other side, MFVS obtained moments as well as a distribution with high accuracy using 30 non-uniform cells. In order to estimate the errors in the moments as well as number density functions, the maximum relative errors in various order moments and weighted sectional errors are calculated using the expressions (3.2) and (3.3), respectively. The results reveal that the errors computed by the MFVS are lower compared with the FCM2012 by 50% (see tables 5 and 6). These errors can be reduced further to the desired level by adding more grid points in the given computational domain. Additionally, the computational time taken by the MFVS and FCM2012 to calculate the numerical results are 0.2672 s and 0.2745 s, respectively.

Table 5.

Maximum error in different moments for the multiplicative kernel.

η	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
η0	0.00663	2.7 × 10−8	0.00159	0.00601
η1	0.00187	2.7 × 10−7	9.6 × 10−9	1.5 × 10−11
η2	3.00013	0.58157	0.19112	0.05852

Table 6.

Weighted error of number distribution for the multiplicative kernel.

σ	FCM2012 30 cells	MFVS 30 cells	FCM2012 60 cells	MFVS 60 cells
σ0	0.05658	0.04492	0.01465	0.01119
σ1	0.18563	0.12123	0.05193	0.02795
σ2	6.53898	1.95209	0.61222	0.18475

(d). Application-oriented kernels

The constant, sum and multiplicative kernels examined above represent benchmark cases for which analytical solutions are available. For the last two comparisons, we consider two kernels associated with specific physical models, the coalescencence kernel of Ruckenstein & Pulvermacher [49] and the gravitational kernel used to model the gravitational coalescencence of large drops [50].

Exact results are not available for the number density functions, although Piskunov & Golubev [50] have derived the zeroth- and second-order moments with the generalized approximation (GA) for special initial conditions. Piskunov & Golubev [50] have also shown that the GA method is the most accurate approximation for the computation of different moments when compared with other numerical methods. The following initial condition was considered by Piskunov & Golubev [50] for the comparison:

$$f_0(x)=\frac{1}{\sqrt{2\pi }}\frac{N_0}{x\sigma }\hspace{0.17em}{\mathrm{e}}^{-{\ln }^2(x/x_0)/2{\sigma }^2},$$

with N₀ = 1, $x_0=\sqrt{3}/2$ and $\sigma =\sqrt{\ln (4/3)}$. The accuracy of the various moments relies on a parameter I that addresses the number of cells considered in a given computational domain. For the comparison of results, parameter I = 4 is used.

(i). Coalescence kernel β₊

The coalescence kernel, defined as

$${\beta }_{+}(x,x^{\prime })=x^{2/3}+x^{\prime 2/3},$$ 3.4

has been used to model migration and coalescence on a heated substrate [49]. The computational domain is from x_min = 10⁻⁵ to x_max = 8 × 10⁵ and is divided into 30 non-uniform cells. Numerical simulations are performed for t = [0, 50].

The quantitative comparison of the numerical zeroth- as well as second-order moments and exact values obtained by the GA method for these moments is shown in table 7. The zeroth moment by MFVS is in good agreement with the GA method, whereas overprediction is shown by FCM2012. Similarly, the second-order moments computed by the MFVS are closer to the values obtained by the GA method. The results clearly demonstrate that the higher order moments, μ₂ in particular, obtained by the FCM2012 deviate significantly from the exact moment.

Table 7.

Comparison of the zeroth and second moments for the coalescencence kernel β₊.

	F CM2012		M FVS		G A
t	μ0(t)	μ2(t)	μ0(t)	μ2(t)	μ0(t)	μ2(t)
0	1.00	1.00	1.00	1.00	1.00	1.00
1	0.418	8.077	0.443	5.022	0.443	3.88
5	6.93 × 10−2	86.85	7.06 × 10−2	2.19 × 102	7.05 × 10−2	54.83
10	2.26 × 10−2	1.33 × 103	2.09 × 10−2	4.68 × 102	2.07 × 10−2	2.75 × 102
50	1 × 10−3	8.25 × 104	6.65 × 10−4	3.71 × 104	6.84 × 10−4	2.27 × 104

(ii). Gravitational kernel β_g

Analogous to the coalescence kernel β₊, the comparison of zeroth- and second-order moments is also conducted for the gravitational coagulation kernel, defined as

$${\beta }_g(x,x^{\prime })={(x^{1/3}+x^{\prime 1/3})}^2|x^{1/6}-x^{\prime 1/6}|.$$ 3.5

This particular kernel is used to model large drops [50]. Similar to the previous case, numerical moments are compared at different dimensionless time points ranging from t = [0, 50]. The computational domain is from x_min = 10⁻⁵ to x_max = 2 × 10⁶ and is partitioned into 100 grid points.

The zeroth- and second-order moments calculated by both numerical methods along with the GA method are listed in table 8. The zeroth-order moment shows better accuracy when approximated by the MFVS, in contrast to the FCM2012, which exhibits significant deviation from the exact values. The second-order moments (μ₂) show a similar trend as the zeroth-order moments, that is, the second-order moment computed by the MFVS shows less deviation from the GA methods than the FCM2012.

Table 8.

Comparison of the zeroth and second moments for the gravitational kernel β_g.

	F CM2012		M FVS		G A
t	μ0(t)	μ2(t)	μ0(t)	μ2(t)	μ0(t)	μ2(t)
0	1.00	1.00	1.00	1.00	1.00	1.00
1	0.716	1.68	0.720	1.63	0.832	1.47
5	0.195	16.53	0.201	13.03	0.314	9.02
10	4.25 × 10−2	163.44	4.51 × 10−2	105.66	7.90 × 10−2	67.07
50	3.87 × 10−5	2.18 × 105	4.71 × 10−5	1.27 × 105	1.34 × 10−4	1.83 × 104

4. Numerical results for simultaneous aggregation-growth PBE

In this part of the paper, we consider the simultaneous aggregation-growth PBE. The computational domain is taken from x_min = 10⁻⁶ to x_max = 1500 and is partitioned into 30 non-uniform cells. The simulation is run from time 0 to 20 for the constant aggregation kernel and linear growth function, whereas, for the sum aggregation kernel and linear growth function, simulation is run from time 0 to 5. For the purpose of checking, the accuracy of methods for the simultaneous aggregation-growth problem, constant and sum aggregation kernels with linear growth function (G(t, x) = G₀x) are considered and their exact solutions for number density functions and moments are obtained from Hu et al. [11].

Figures 8 and 9 demonstrate the comparisons of the normalized moments and number density function obtained by both numerical methods with the exact solutions for cases of (i) the constant aggregation kernel and linear growth function and (ii) the sum aggregation kernel and linear growth function, respectively. It can be seen that both zeroth- and first-order moments are predicted more accurately by the MFVS than the FCM2012 and match well with the exact results (see figures 8a,b and 9a,b). The number density function obtained by the MFVS overlaps with the exact result, whereas the FCM2012 shows over-prediction for these results. In terms of computational CPU time, the mass-based method took less time than the existing method, as shown in table 9.

Figure 8.

Comparisons of numerical results for the constant kernel with linear growth: (a) normalized zeroth moment, (b) normalized first moment and (c) number density function. (Online version in colour.)

Figure 9.

Comparisons of numerical results for the sum aggregation kernel with linear growth: (a) normalized zeroth moment, (b) normalized first moment and (c) number density function. (Online version in colour.)

Table 9.

Comparison of the computational time taken by both methods.

constant kernel with linear growth			sum kernel with linear growth
cells	methods	time taken (s)	cells	methods	time taken (s)
30	MFVS	0.7279	30	MFVS	0.5149
30	FCM2012	0.8224	30	FCM2012	0.5987

It can also be observed from the literature that the numerical methods developed by Kumar et al. [38], Qamar et al. [26], Kumar & Warnecke [27] and Kumar et al. [13] require 65, 100, 60 and 60 cells, respectively, to obtain the results accurately.

5. Numerical results for simultaneous aggregation-nucleation PBE

In this final test, we consider the case of simultaneous aggregation and nucleation. For this particular problem, a constant aggregation kernel with negative exponential function (B_src(x, t) = e^−x) is considered. The exact solution for the number density function is not available in the literature, but the solution moments are known and were given by Kumar & Ramkrishna [10]. The computational domain x_min = 10⁻⁶ to x_max = 3 × 10⁴ is divided into 30 non-uniform cells, and the simulation is run from time 0 to 100.

Figure 10 shows the comparison of the zeroth- and first-order moments obtained using the numerical methods by solving a simultaneous aggregation–nucleation PBE with exact moments. The zeroth-order moment computed by the MFVS shows excellent agreement with the exact results, whereas the FCM2012 shows deviation after time t = 1. The first-order moment predicted by the MFVS matches very well with the analytical solution, whereas the FCM2012 starts losing mass from the system after time t = 1. In terms of CPU computation time, the MFVS is less expensive as it took 2.7187 s to calculate all of the numerical results while the FCM2012 took 3.8076 s. The existing method [51] requires 60 cells to compute the moments and number density function accurately.

Figure 10.

Comparisons of numerical results for the constant kernel with negative exponential nucleation function: (a) normalized zeroth moment and (b) normalized first moment. (Online version in colour.)

6. Conclusion

In this work, new MFVSs for the solution of a univariate PBE have been developed. For the pure aggregation PBE, the accuracy and efficiency of the mass-based method has been verified for five aggregation kernels. In contrast to the existing finite volume method (FCM2012), the MFVS shows exceptional performance at a lower computational cost for different order moments and number density functions. It has been illustrated that the mass-based method exhibits 50% more accuracy in terms of zeroth- and second-order moments than the existing method. Additionally, the modified method presents an edge over the old method in predicting the number density functions. In addition, it was shown that the new approximations developed for approximating the simultaneous aggregation-growth and simultaneous aggregation-nucleation PBEs are highly accurate in terms of predicting the moments and number density function on a grid which is much coarser than in existing methods.

Data accessibility

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Authors' contributions

M.S. conceived the presented idea, and developed the theory and code. H.Y.I. helped in building the code and analysing the results. T.M. encouraged investigation, discussed and validated the findings of this work and supervised the project. A.B.A. and G.W. provided technical details and physical interpretation of the proposed problem and finalized the manuscript. All the authors of this article have contributed to the preparation and revision of the paper.

Competing interests

We declare we have no competing interests.

Funding

The authors gratefully acknowledge the financial support provided by H2020 Marie Skłodowska-Curie Fellowship no. 841906 to M.S.