1. Introduction
Population balance equations (PBEs) describe the behavior of particle properties’ changes due to phenomena such as nucleation, growth, aggregation and breakage [
1]. Many applications of PBEs can be found in the area of chemical engineering [
2], depolymerization [
3,
4], waste water treatment [
5], bubble columns [
6], physics [
7] and pharmaceutical sciences [
8,
9,
10,
11,
12], where these mechanisms have a significant impact on the particle properties in the system. In the aggregation process, two or more smaller particles merge at a specific rate to build larger size particles. Aggregation-driven wet granulation processes such as twin-screw granulation [
11,
13], high shear granulation [
14] and fluidized-bed granulation [
15] are extensively used in pharmaceutical manufacturing when characterized using PBEs, mostly assuming that only one property of the particles is changing. However, the aggregation mechanism is caused by the mixing-driven presence of liquid binder in the system. Therefore, for such processes, the ability to mechanistically characterize aggregation requires consideration of the degree of mixing between particles and the binder phase. Iveson [
16] has shown that the univariate PBEs are inadequate to capture the actual particle behavior in granulation and therefore suggested the application of multivariate (higher dimensional) PBEs. Such PBEs can be used for the simultaneous prediction of several components, the compositional distribution and hence the accurate prediction of the mixing state in the system. Quantification of the degree of mixing in binary component aggregation has been discussed by Matsoukas et al. [
17]. The study concluded that the scaling of the variance indicates that the mixing of components is not characterized by a time scale but by a size scale. This emphasizes the need for accurate prediction of particle size distribution evolution, which depends on the selection of a suitable numerical scheme to solve the PBEs and aggregation.
In this study, we focus on the relevance of different numerical schemes for multivariate PBE solution in terms of their capability to capture the degree of mixing during aggregation. In the aggregation mechanism, the total number of particles reduces with time, but the total mass of the system remains constant. A pure bivariate aggregation population balance equation is an integro-partial differential equation which can be written as follows [
18].
subject to the initial condition
Here,
is the number distribution functions having size
at time
t (s) and
(
m) is a vector given in terms of the summation of properties like the mass or volume of the components. The bold notations are used to denote vector quantities, i.e.,
, where
expresses the property of the particle in
direction. Moreover, the birth term in Equation (
1) represents the formation of new particles of properties
due to the aggregation of smaller particles of properties
and
. Similarly, the death term describes the loss of particles with properties
due to the collision of particles with properties
. The aggregation kernel
describes the rate of merging of two particles of properties
and
. It can be noted that the aggregation kernel is non-negative and symmetric in nature with respect to size variable, i.e.,
. Moreover, for simplicity, we assume that the aggregation kernels chosen for this study are time-independent, i.e., only size dependency is taken into consideration.
Apart from finding the number distribution function
n, different properties of the system, namely integral moments corresponding to the number distribution function, are also important for understanding the complete dynamics of the system [
19]. For bivariate PBE, the
order moment corresponding to the number distribution function is defined as
Here, denotes the total number of particles in the system, which is also known as zeroth-order moment, whereas gives the total mass of the pth property. Similarly, other moments can also be obtained.
Finding analytical (exact) solutions of the population balance equation (PBE) (
1) is difficult due to the presence of a nonlinear integral in the equation. However, still, for some simple structured kernels, a few analytical solutions are listed in [
20,
21,
22,
23]. Therefore, in this exercise, we choose numerical approximations to solve bivariate pure aggregation PBE (
1). To date, many numerical methods have been proposed by various authors, including fixed pivot techniques [
24,
25,
26], method of projection [
27], Euler method [
28], finite volume schemes [
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39], quadrature method of moments [
40,
41,
42,
43], finite element method [
44,
45], sectional methods [
18,
46,
47,
48,
49,
50] and Monte Carlo method [
19,
51,
52,
53].
Among all numerical schemes, the cell average technique (CAT) is well known for the accurate prediction of the number distribution function and their moments. However, the mathematical formulation for CAT is very complex; hence, it is computationally expensive [
54,
55]. The finite volume schemes (FVS) are well recognized in the literature as being able to accurately and efficiently obtain these results. The studies conducted in the last decade restricted their comparison only to the number distribution functions and their integral moments [
18,
35,
46,
47,
49,
56]. To examine the accuracy of the prediction of the component mixing for a higher-dimensional PBE, the cell average technique [
46] and the finite volume scheme [
57] are implemented to approximate a multi-dimensional pure aggregation PBE and compared. The verification of the numerical results is also conducted by comparing the average size of particles predicted in the system using the exact solution.
The paper is organized as follows: to start, a brief introduction of the existing CAT as well as the FVS for solving bivariate pure aggregation PBE on non-uniform meshes is provided in
Section 2. In
Section 3, the qualitative and quantitative numerical results, particularly various order moments and number density functions computed by both numerical schemes, are analyzed. Finally,
Section 4 summarizes the conclusions and discussion of this study.
2. Numerical Methods and System Analysis
In this section, the mathematical formulations of the existing cell average technique [
46] and the finite volume scheme [
57] for solving a bivariate aggregation PBE (
1) on non-uniform grids are outlined (
Figure 1). For developing the expressions of both numerical methods, it is assumed that particles within a grid cell are concentrated on its representative (or mean of the cell). Before giving the description of the numerical methods, it is necessary to fix the computational domain. For the numerical approximations, we define the size variables
in PBE (
1) ranges from
to
∞; thus, a large sufficient vector such as
is replaced with
∞ in the second integral of PBE (
1). Thus, the original PBE (
1) takes the following form:
with the modified initial condition
The above expression well suited to numerical simulations; however, it does not capture the property of mass conservation. During numerical computations, a sufficiently large
is chosen to minimize loss of mass from the system. Further, we assume that the whole domain is divided into
, where
is the number of grids in
r direction for
. Now, for any
r, the mesh points and the step size can be defined by
2.1. Cell Average Technique (CAT)
First, the mathematical explanation of the CAT developed by Kumar et al. [
46] on the non-uniform grids is presented. Let us suppose that
defines the number of particles in the
ith cell, which can be expressed as
where
and
for
Now, let us express the number distribution function in terms of dirac delta functions, i.e.,
Substituting the above expression in the original PBE (
1), we obtain the following set of ordinary differential equations:
Here, the discrete forms of birth and death terms are expressed as
and
Here,
and
are
p-dimensional vectors. The corresponding mass or flux of the particles, which takes birth in the
ith cell, can be computed as follows:
Further, once the aggregation events of the particles in the
ith cell are completed, it is necessary to compute the average properties of all birth events in the cell using the following expression:
It is assumed that the particle properties are concentrated on the representative of the cell; however, the possibility of the aggregating particles falling on the representative is low. Therefore, if the birth
takes place in the
ith cell and is not represented by the nodes, then the averaging properties, such as
, are distributed to the neighboring nodes in such a fashion that the integral properties, particularly the zeroth and first order moments for our case, remain conserved. The assigned values
can be calculated from the following relations:
where
are the coordinate vectors of the vertices of the
ith cell. Hence, the final set of discrete equations can be written as
It can be noted that the cell average technique is computationally very expensive as it is necessary to determine the average properties of the aggregating particles in each cell and then redistribution of particle properties is carried out to the neighboring representative of the cell. The distribution is carried out in such a way that the zeroth and first order moments of the system are conserved. A detailed description of the cell average technique can be found in Kumar et al. [
46] and its schematic flowchart is provided in
Figure 2a.
2.2. Finite Volume Scheme (FVS)
Now, we provide the mathematical formulation of the finite volume scheme [
57] for solving a generalized aggregation PBE. The finite volume scheme is based on the idea of preserving the total number of particles and conserving the total mass in the system by just adding two correction factors in the formulation. For developing the mathematical expression of the FVS in a similar domain, it is necessary to define the following set of indices:
where
and
denote the boundaries of the
ith cell, respectively, and
represents the mean value of the
ith cell. The representation of the
defines all those pairs of cell indices
j and
k with particle properties
and
, such that the addition of their particle properties
will fall in any cell having representative
after the aggregation of particles.
Similar to the CAT, for the FVS, it is presumed that the point masses are concentrated on the representatives. Therefore, by proceeding in a similar way as in the CAT, the expression for the FVS is obtained, which is given by
We know that the finite volume schemes are well known for the conservation of the various properties. The above formulation (
14) takes into account the preservation of the zeroth order moment but not the conservation of the total mass (first order moment) of the system. Nevertheless, this can be resolved by introducing only two weights in the above equation, leading to the following expression:
where the correction factors
and
are defined as
and
where
denotes the sum of the components of the vector
, i.e.,
and
is the index of the cell where
falls. The theoretical proof of the preservation of the total number of particles as well as total mass of the particles in the system along with the CFL conditon can be found in Kaur et al. [
57] and its flowchart is depicted in
Figure 2b.
2.3. Kernel Selection
The efficiency and accuracy of the numerical methods is tested against the analytically tractable kernels corresponding to 2D PBE. In particular, two kernels, namely constant (size-independent) and sum (size-dependent) kernels, are chosen for the comparison.
2.3.1. Size-Independent Kernel
For a size-independent kernel
, the analytical solution corresponding to the size-independent kernel was formulated by Gelbard and Seinfeld [
22] and is listed in
Table 1.
2.3.2. Size-Dependent Kernel
Mathematically, the sum kernel can be expressed as
and is heavily dependent on the size of the particles. The analytical solution in this case is formulated by Fernández-Díaz and Gómez-García [
21] and is listed in
Table 2.
2.4. Model Initialization and Post-Processing
For the initial condition
, the exact results of number density as well as various order moments for constant and sum kernels are provided in the literature by Gelbard and Seinfeld [
22] and Fernández-Díaz and Gómez-García [
21], respectively. Before comparing the numerical results, it is important to define the degree of aggregation
:
which describes the decrease in the number of initial primary particles due to the aggregation process. At time
,
, and as it approaches large values,
, with all primary particles forming one large particle.
Furthermore, the weighted relative error is also calculated to test the accuracy of the number distribution quantitatively [
49]:
The superscripts and represent the exact and numerical solutions, respectively. Here, represents the relative error in the distribution of the number of particles over the whole size domain. The two solutions corresponding to the two numerical schemes may identify the same prediction for the total number of particles, even though the distribution of particle populations may disagree considerably. This is well captured by . Similarly, the expresses the relative error in the distribution of the total mass of the system. All the simulations and computations for both schemes were carried out using “MATLAB” on a CPU with GHz and 16 GB RAM.
2.5. Average Size Particles
The average size of particles formed in the system along
u and
v components is determined using the following expression:
Moreover, the total average size of particles formed in the system is given by
2.6. Quantification of Mixing
For identifying the mixing of components in a bivariate PBE, Matsoukas et al. [
17] provided a theory to show that the mixing of components is calculated using the
parameter corresponding to the variance of excess binder. The underlying principle relies on the assumption that if the components are perfectly mixed among all aggregates, the amount of binder in an aggregate of size
v would be
. If the actual amount of binder in the aggregate is
, the difference
defines the excess binder
.
For kernels independent of both composition and initial conditions, the
parameter should be constant at all times for constant and sum kernels. The expression for a
parameter is given below:
where
. A detailed description of the theory of mixing of components can be found in Matsoukas et al. [
17] and Matsoukas and Marshall Jr [
19].