1. Introduction
Fragmentation, breakage or attrition, describe processes in which a single object is separated into at least two new objects. The reasons for breakage can be manifold but are often linked to some kind of stress exerted on the object, for instance thermal stress from heating and rapid cooling (or vice versa and cyclically)—a natural process specifically observed in deserts, leading to disintegration of rocks; or mechanical stress, applied for millenia in the process of grain milling. Nowadays, fragmentation plays a key role in several industrial sectors like mineral processing (e.g., comminution of ores [
1,
2,
3,
4]), reaction engineering (e.g., break-up of bubbles in reacting bubble columns for separation processes [
5,
6,
7,
8] or steel-casting [
9]) or pharmaceutical industries (e.g., milling of active pharmaceutical ingredients to increase their solubility and uptake capacity in the human or animal body [
10,
11,
12]).
Many objects, e.g., particles, bubbles or even rain drops, consist of different components resulting in their anisotropic structure. The probability of fragmentation upon stress therefore depends on the distribution of the components within the objects, i.e., each component adds an independent dimension to the fragmentation problem. In an attempt to describe these complex processes and make them accessible for model- and knowledge-based process design, optimization and control, multidimensional fragmentation equations have been proposed and used in different fields of application, see, for instance, the works [
13,
14,
15,
16,
17].
Theoretical aspects on the existence of scaling solutions and their behavior at the onset of “shattering” transition have been discussed for instance in the works of [
18,
19,
20,
21]. Fragmentation models are particularly challenging as they consist of partial-integro differential equations as will be shown in the following. Analytical results are scarce and often of very limited practical relevance, strongly motivating the development of numerical methods for approximation of the solution to (multidimensional) fragmentation problems.
As a prototype, consider the conservative formulation of the multiple fragmentation equation given by [
22,
23]: The initial value problem for
is formulated as
with the initial condition
The
flux function is defined by
In Equation (
1), the internal variables
x and
t denote the particle property and the time component, respectively. On the left hand side, the function
is defined by
, where
denotes the distribution of particle volume
x in a system at time
t. The rate of selection of an
x-volume cluster to undergo breakage is denoted by
, and the distribution of daughter particles
y due to the breakage of large particle
x is denoted by
. The breakage function
satisfies the following relations:
The first relation defines that
number of fragments are produced during the breakup of a large
x-cluster, and the second relation defines that the total volume of the daughter
y-clusters is exactly the same as the volume of the mother
x-cluster. Note that the formulation (
1) is well-known in the literature as the volume conservative form. Integration of Equation (
1) over the volume variable
x from 0 to
∞, with the help of relation (
4), yields
It should be noted that Equation (
1) is a first-order hyperbolic, initial value partial differential equation. In this regard, the representation (
1) gathered importance because the divergent nature allows the model to obey the volume conservation laws. The coefficient
S belongs to
and
. Here and below, the notation
stands for the space of the Lebesgue measurable real-valued functions on
which are integrable with respect to the measure
.
In most of the previous studies it is assumed that a single parameter, which is usually volume, mass or size of the particle, is sufficient to describe the particle property (readers can refer to [
24] for further details). However, a single parameter is not always sufficient to describe various physical systems. For example, fragment mass distribution obtained by crushing gypsum or glass depends on the initial geometry of the particles. On the other hand, the degradation of polyelectrolyte may depend upon both their mass and excitation (or kinetic) energy. Therefore, the fragmentation dynamics need to be represented by including additional variables to the mathematical model. These variables are equivalently classified as the degrees of freedom of the dynamical system and hence, the multidimensional formulation of the fragmentation equations becomes necessary to represent such cases. The purpose of this article is to take in account more than one particle property and present an efficient numerical model which estimates them with high accuracy. In particular, we present the mathematical representations of two-dimensional and three-dimensional volume conservative linear fragmentation equations. Further extension of the mathematical formulation can be done in a similar manner.
For the population balance models, the moment functions of the particle property distribution play a major role as some of them describe a significant physical property of the system. Therefore, before we proceed further, let us first gather some important information about the moment functions in a generalized multidimensional setup.
Moment Functions
Let
, with
s representing different particle properties like, mass, entropy, moisture content, shape factor, etc. and thus the function
denotes the distribution of particle property
at some instance
t. The formal definition of the moment functions for a general
n-dimensional population balance problem is written as follows:
where the integrations are defined as
In Equation (
6),
are nonnegative integers. As mentioned earlier, the moment functions play an important role to define various physical properties of the system. Like the zeroth moment,
defines the total number of particles present in the system. The first-order moment
(1 is the
rth position) denotes the total content of the
th component in the system, which can be equivalently represented as the total volume of
th property. Hence, for a multidimensional system, the volume conservation of the system can be defined as the total conservation of all the first-order moments taken together. Thus, defining
, the volume conservation law for the
n-dimensional system is expressed as
Furthermore, the
-th order cross moment is defined by
and it represents the particle geometry or hypervolume. Therefore, to preserve the initial geometry of the particles, we need to preserve the cross moments; hence, the hypervolume preservation law is written as
where
.
Similarly, other higher order moments can be defined using the formulation (
6), and depending upon the problem they may correlate to some physical properties of the system. For example, in a pipeline flow for the transport of natural gas from seabed, the breakage of hydrate particles often takes place. In this event, if the first moment
is proportional to the mean radius of the hydrate particle, then the corresponding second order moment
and third order moment
are proportional to the total area and the volume concentration of the hydrate particles, respectively. In general, only the zeroth, first-order and the cross-moments bear the same meaning for any population balance models.However, it should not be misunderstood that higher order moments should always correspond to certain physical characteristics.
In the literature, a limited number of articles are dedicated to the numerical study of multidimensional fragmentation events, and therefore several aspects of study still remain unexplored. The articles of [
25,
26,
27,
28,
29] discuss the development of different numerical schemes to approximate the fragmentation problems. To note that, unlike the methods, e.g., cell average technique, fixed pivot techniques, method of moments, etc., the finite volume methods have gained popularity because the latter are robust to be applied on a multidimensional setup. Moreover, the underlying stencil of the finite volume scheme is simpler, and easy to compute (the readers can refer to the articles of [
23,
29] for further details on the computational advantage of finite volume schemes).
The article is organized in the following manner. In the next section, we present the mathematical representations of the continuous two- and three-dimensional equations. In this regard, the three-dimensional model is represented using vector notation, which will also provide an outline to extend the equations into further higher dimensions. In
Section 3, step-by-step derivation of the numerical schemes are presented. An interesting outcome of this presentation includes the multidimensional extension of the finite volume scheme presented in [
23].
Section 4 contains the numerical validation of the proposed models over some standard empirical test problems. Finally some conclusions and a summary of the work are presented.
4. Results
In this section, we validate the efficiency of the newly proposed finite volume models with the standard finite volume scheme over several test problems. Since the new models are defined with the help of a weight factor, we call it weighted finite volume scheme (WFVS). On the other hand, Remark 2 indicates that the standard forms of the schemes which are directly derived from the continuous equations were initially proposed by [
23] for fragmentation models with one degree of freedom. Therefore, for future reference, we call the standard models the existing finite volume schemes (EFVS). However, we need to mention that the two-dimensional extension of EFVS is not available in the literature to date, and this article not only proposes an improved model, but also extends the existing finite volume schemes for multidimensional fragmentation events.
For one-dimensional fragmentation problems, Refs. [
30,
31] has obtained the exact solutions for a certain class of fragmentation kinetics. However, exact solutions in closed forms are very rare in the multidimensional setup. In order to validate the accuracy of the proposed schemes, we choose four test problems with two degrees of freedom, and two test problems with three degrees of freedom. For all the test problems, exact solutions in closed form are not always available in the literature. However, the zeroth and the first-order moments can be computed exactly, which is sufficient to validate the accuracy of the new schemes. Therefore, in the following section we discuss the efficiency of the WFV scheme to predict the different physically important moment functions over the EFV scheme. Our study builds up on both qualitative and quantitative assessments. The qualitative accuracy is represented through graphical representation of the different entities whereas, the qualitative analysis is performed by computing relative errors of the moment functions over different grid points.
The computational domain
is considered for all the two-dimensional test problems. The domain
V is discretized into
nonuniform subintervals bearing the geometric relation
where
is the geometric ratio. Additionally, all the test problems are supported by a mono-dispersed initial condition
Similar extensions of the above data are considered for the three dimensional models. The semi-discrete schemes (
7) and (
8) are solved using
Matlab-2019b software in a standard PC with i5-7500 CPU processor @ 3.41 GHz and 8 GB RAM.
4.1. Examples in Two-Dimensions
4.1.1. Volume Conservation Problems
In the first instance, we consider test problems with constant particle selection rate, that is
and two different daughter distribution functions
The first breakage function
is a size-independent function of its arguments and physically represents random scission of particles. On the other hand, the second breakage function
represents size-dependent distribution of daughter fragments, choosing the daughter-particles exactly half the size of the parent particle. The exact solution in closed form for the above set of fragmentation kernels are not available in the literature. However, we can calculate the zeroth
, first
and the cross moments
exactly for both the problems (calculations of the exact moments are given in
Appendix B). In this regard, the exact moments are given in the
Table 1.
Figure 1 and
Figure 2 represent the numerical moments obtained from EFVS and WFVS against their exact values with breakage functions
and
, respectively. More precisely,
Figure 1a and
Figure 2a present the comparison of zeroth and first-order moment functions, and
Figure 1b and
Figure 2b presents the first-order cross moment function
. In order to obtain a clear visibility of different markers, we plot
Figure 1a and
Figure 2a on a semilogarithmic scale with respect to the
y-axis. In both the cases, it is observed that WFVS estimates the zeroth-order, first-order and the cross moments with high accuracy, whereas the EFVS conserves the total volume but fails to produce a good estimate of the other moments.
In the
Table 2, the relative error of the moment functions for
and
are calculated at
over three different grid sizes. The geometric ratios to generate
,
and
grids are 7.9433, 3.9811 and 2.8184, respectively. The discrete
-error norm
is used to calculate the errors. Similarly, the relative error acquired while computing the moments for
and
over different grid points are represented in
Table 3.
In the second instance, we choose two problems by setting the size-dependent selection function
, along with the previously chosen particle daughter distribution functions
and
. In this case, also the exact solutions are not available in the literature, however only the zeroth- and first-order moment functions can be calculated exactly. In both the cases, the moment functions are given as
The following
Figure 3a,b represent the efficiency of the WFV scheme over the EFV scheme to estimate the zeroth- and the first-order moments. It is observed that both the schemes obey the volume conservation laws with high accuracy, but the WFV scheme is highly accurate to predict the evolution of total number of particles.
Table 4 and
Table 5 represent the relative errors over different grid points at time
.
4.1.2. Hypervolume Conservation Problems
In this part, the considered breakage functions
b should satisfy the hypervolume conservation rule (
8). In this regard, we choose the following breakage functions
The breakage function is independent of the daughter-particle size, whereas represents the particle breakage along the longer side of the rectangular structure. Similar to the examples of volume conservation models, we choose two types of selection functions: (i) size-independent kernels , and (ii) size-dependent kernels .
Like before, the exact solutions are not available in the literature, however we can calculate three moment functions exactly, and they are given in
Table 6.
In
Figure 4a, we plot the zeroth- and the cross-moment functions and observe that the new WFV scheme predicts the corresponding moments with high accuracy. On the other hand, we plot the first-order moment in
Figure 4b. In this case also, the weighted scheme exhibits high accuracy to estimate the moment compared to the standard model. In
Figure 4b, we take the the axes in loglog scale for a distinct visibility of the plots. In this problem,
Figure 4c represents the comparison of hypervolume distribution functions with the numerical values obtained from the two schemes. We follow the flat pictorial representation to plot the hypervolume distribution as presented in [
32]. For the other problems, only the exact moment functions can be calculated for comparison with the numerical models.
The relative errors over different grid points are presented at
Table 7.
In the second instance, we consider the size-dependent selection function
and
as the daughter distribution function. The exact solution is not available for this problem, but we can evaluate the zeroth and cross moments exactly. From the
Figure 5 and
Table 8, we can see that the newly proposed WFV scheme predicts the moments with high accuracy.
Next, we consider two problems with size-dependent selection function
and the breakage functions
and
. Only the zeroth- and first-order moment functions can be calculated in closed form, and are given in the
Table 9.
Numerical evaluation of the moments using the WFV and EFV schemes are presented in
Figure 6 and
Table 10 and
Table 11. The improved accuracy of the new scheme is observed.
4.2. Three-Dimensional Examples
Volume conservative problems: In this section, we consider two test problems with size-dependent selection function
. The three-dimensional extension of the above-mentioned breakage functions
and
, that is,
are considered here. Like before, we can only calculate the zeroth and first moments exactly and they are given in the following
Table 12.
From
Figure 7 and
Table 13 and
Table 14, we can observe that the WFV scheme estimates the moment functions more accurately as compared to the EFV scheme.
Hypervolume conservation: In this instance, we consider the size-independent daughter distribution function
along with the constant selection
and size-dependent selection
. The exact moments are calculated in the
Table 15.
Figure 8 and
Table 16 exhibit the improved accuracy obtained from the WFV scheme over the standard scheme to predict the above mentioned three moments.
On the other hand,
Figure 9 and
Table 17 represent the comparison of the above moments as predicted by the WFV and EFV schemes.