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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5637_Библиотеки_им_академика_М_И_Перельмана
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80 A. Das and J. Kumar
https://t.me/medicina_free
macroscopic physically motivated PBE kernels. Over the years, the researchers have
used empirical or semi-empirical kernels, which generally use fitted constants to
predict the experimental outcomes for some particular set of operating conditions.
Furthermore, although the PBE kernels are dependent of material properties and
time simultaneously, most of the studies available in the literature omitted either the
time dependency or the dependency on material properties. It is still a challenging
task to incorporate the physics of the particulate process (e.g., process parameters,
material properties, etc.) and propose the PBE kernels that can track the variations in
the material properties under concern and time simultaneously. In recent years, Das
et al. [18–20] have developed PBE kernels for some processes, which are dependent
on particle dimensions and process time simultaneously. The focus of this chapter
will be to discuss the development and verification of those PBE kernels.
On the other hand, the Monte Carlo (MC) technique became popular as a
replication tool to experimental particulate systems. In a series of studies, TerrazasVelarde et al. [21–23] developed a constant volume MC algorithm to simulate the
aggregation mechanism in a spray fluidized bed granulator. The simulation results
predicted the lab-scale experimental results qualitatively for porous and non-porous
particles. In later years, Dernedde et al. [24, 25] developed an efficient concept
of positions and sectors on the agglomerate surfaces to model the spray fluidized
bed aggregation process and to determine the moisture content on agglomerate
surfaces. In 2019, Bhoi et al. [26] developed a constant number MC algorithm which
predicted the sonofragmentation experimental observations accurately. Recently,
Singh and Tsotsas [27, 28] developed a tunable constant volume MC model of
fluidized bed granulation process using different morphological descriptors. Instead
of working with experimental results, working with the MC technique may be
advantageous for the development of PBE kernels. For instance, access to many
properties, which are challenging or impossible tomeasure in anexperimental setup,
is easily available. Additionally, the user achieves the control to switch on and off
any specific event in the MC simulation in order to evaluate its impact on the whole
simulation [26, 29]. Due to these reasons, in this chapter, we have used the MC
technique as a replacement of experimental setup.
In this chapter, at first, we will discuss the MC algorithm in detail and then will
discuss the development of some breakage PBE kernels which depend on particle
dimensions and process time simultaneously. Finally, the validation of the discussed
models will be done with the help of MC simulation results.
2 Monte Carlo Algorithm
The Monte Carlo (MC) method is a probabilistic approach which uses the generation of random numbers to estimate the values of entities under consideration.
Metropolis and Ulam [30] introduced the MC method in 1949. The uncomplicated
concepts and the discrete nature of Monte Carlo help to naturally adapt itself in
dynamic processes [30, 31]. The MC algorithm works with a representative sample
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 81
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of the whole system, which is considered to have identical properties as the original
system, and after simulation, a scale-up factor is used to predict the original results
[26, 32].
The MC algorithms are divided into two categories given the time step length,
which are “time-driven MC” and “event-driven MC.” In the time-driven approach,
at first, we assign a time step and then implement all possible events within the time
step. However, in the event-driven approach, first, an event is selected to occur, and
then the time is advanced accordingly. The event-driven MC has an upper hand over
the time-driven MC, as an event is guaranteed in each time step of the event-driven
MC, which is not the case for the time-driven MC [31].
Another critical aspect of the MC method is about the size of the simulation box,
i.e., the considered total number of particles in the MC simulation box. The number
of particles in the simulation box changes according to the nature of the process.
In the case of aggregation process, the number of particles decreases with time
and eventually reduces to only one after prolonged simulation. This compromises
the accuracy of the process. On the contrary, the number of particles increases
with time in case of breakage processes. This situation increases the computational
cost of the MC simulation process. To counter this scenario and to regulate the
number of particles in the simulation, MC methods can also be divided into two
types in view of the size of the simulation box, which are “constant number MC”
(CNMC) and “constant volume MC” (CVMC). In CVMC, the total volume of the
MC simulation box is kept fixed until some drastic change occurs in the total number
of particles. By convention, the CVMC method doubles or halves the simulation
box size, respectively, when its size has reduced (in case of aggregation process) or
expanded (in case of breakage process) by a factor of two. This method balances
the accuracy and computational efficiency of the simulation. On the other hand, in
the CNMC method, we fix the total number of particles in the simulation box at
each time step by either duplicating one particle of the simulation box (in case of
aggregation) or deleting one particle randomly from the simulation box (in case of
breakage process) [15–17, 33].
Zhao et al. [31] suggested that the CVMC method works better in case of
aggregation processes, whilst CNMC performs better in case of breakage processes.
That is why we have used the event-driven CNMC algorithm to replicate different
breakage mechanisms. The general flowchart of a CNMC algorithm is depicted in
Fig. 1. At the start, we initialize the process conditions and simulation parameters.
We also fix the size of the simulation box, i.e., the number of particles in the
simulation box (N
), which has identical properties to the original system. We
MC
further represent the concerned particle properties in terms of arrays in the MC
system. If the process has I number of events, which are occurring simultaneously,
we calculate the frequency of each individual events (λ
). Then, the total number of
i
events per unit time is given as
λ =
I
i=1
λ
i
(1)
82 A. Das and J. Kumar
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Initialize the process conditions and simulation
parameters
Calculate the individual frequency of all events
Compute:
1.
Total event rate
2.
Time step length (Δt)
Choose the next event according to their
probability of occurrence
Randomly choose the entities associated with
the chosen event
Update time and other system variables
(keeping the size of simulation box constant)
NO
Check if stopping criteria is fulfilled?
YES
End simulation and predict result
Fig. 1 Constant number Monte Carlo (CNMC) flowchart
Then, we calculate the time step length of the next event as
−1
Δt =
ln(1 −ζ) (2)
λ
where ζ is a uniformly distributed random number between 0 and 1. To select
the next event depending on their event rates, we first calculate the cumulative
frequencies using Eq. (3).
i
R
=
i
λk, for i = 1, 2,...,5(3)
k=1
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 83
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Then, we select the ith state for the next event if the below equation is satisfied.
R
i−1
<rλ≤ R
i
(4)
where r is another uniformly distributed random number in (0, 1). Furthermore,
using a similar technique, the corresponding entities associated with the selected
event were chosen. Then, we execute the event and update the system accordingly.
Since no particulate processes have been discussed here, we have not discussed
about theevents and their frequencies.We will discuss this inthe respective sections.
3 Mathematical Modeling of PBE Kernels
This section will focus on developing and verifying the PBE kernels which depend
on particle dimensions and process time simultaneously. The considered particulate
processes are linear breakage, nonlinear collisional breakage, and sonofragmentation of rectangular crystals. Particle breakage can occur in various ways. However,
in view of the process conditions, the breakage process can be broadly classified into
two major classes: linear and nonlinear breakage. Linear breakage process occurs
only due to some internal stresses of particles or some process-specific conditions,
for example, thermal or mechanical conditions of the particulate process. In addition
to that, if the breaking behavior of particles is also influenced due to collisions (or,
interactions) between particles, then we classify it as the nonlinear breakage. A
particular class of nonlinear breakage process is binary collisional breakage, where
the fragmentation occurs solely through instantaneous binary collisions among
particles. Sonofragmentation is one special type of linear breakage process, which
is very popular in crystallization process. It uses the ultrasound to break crystal
structures. In this chapter, we will discuss about the PBE kernels corresponding to
the abovementioned fragmentation mechanisms.
3.1 Linear Breakage
The continuous one-dimensional linear breakage PBE is the following [18]:
∂n(x,t)
∂t
=
with the given initial data
∞
b(x, y)S(y,t)n(y,t) dy
x
n(x, 0) = n
! "
birth of particle x
(x), x ∈ R
0
− S(x, t)n(x,t)
+
! "
death of particle x
(5)
(6)
84 A. Das and J. Kumar
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where n(x, t ) is the number density function, S(x, t) is the breakage selection func-
tion, and b(x,y) is the breakage distribution function. This breakage distribution
function b(x, y) illustrates the breakage rate for formation of particles of volume x
from a particle of volume y and satisfies the following two properties:
y
1.
b(x, y) dx = ν(y), where ν(y) is the number of daughter particles produced
0
due to breakage of the particle of volume y.
y
xb(x,y)dx = y ,∀y>0. This property is called the mass conservation
2.
0
property.
Modeling of Linear Breakage Selection Function
The rate of successful linear breakage events is expressed by the selection function
S(x,t). In the literature [34, 35], S(x,t) is usually partitioned into a product
of volume-dependent and volume independent (i.e., time-dependent) components.
However, the occurrence of a successful breakage event depends on the particle’s
internal bonding strength. If the applied stress upon the surface of the particle is
higher than the internal bonding strength of the particle, then the particle breaks
into fragments. This indicates that the rate of successful breakage events is also
dependent on the probability of successful events (ψ(x,t)). For this, Das et al. [18]
proposed the following factorization of the breakage selection function:
∗
(x) S0(t) ψ(x, t) (7)
where the pre-factor S
S(x,t) = S
∗
(x) describes the particle volume dependency on breakage
process, i.e., how the particle selection depends on the particle volume. The second
factor S
(t) is the time dependency in particle breakage process. The last factor
0
ψ(x,t) is the probability of successful breakage events and can be defined as the
ratio of the frequency of successful breakage events (f
per particle (f
str
), i.e.,
ψ(x,t) =
f
s
f
str
) to the stressing frequency
s
(8)
In the process, particle selection for the stressing events mainly depends on the
volume-dependent part of the selection function S
∗
(x) and their availability in the
system. Therefore, the selection probability of any particle at time t is
∗
(x) n(x, t )
S
P(x,t) =
∞
(9)
S∗(x) n(x, t ) dx
0
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If Np(t) denotes the total number of particles present in the system, then the
frequency of total number of stressing events is f
(t) Np(t). Furthermore, in the
str
event of a particle of volume x breaking into ν(x) number of smaller fragments, the
increase in the total number of particles is (ν(x) − 1). Then, the rate of change in
the total number of particles (N
dN
(t)
p
dt
∞
=
=
ν(x)− 1)[f
(
0
f
str
∞
(t) Np(t)
S∗(x) n(x, t ) dx
0
) can be written as
p
(t) Np(t)]P(x,t) ψ(x,t)dx
str
∞
ν(x)− 1]S
[
0
∗
(x) ψ(x , t ) n(x, t ) dx (10)
On the other hand, integrating equation (5) with respect to x, we get the rate of
change of total number of particles as
dN
(t)
p
= S
dt
∞
(t)
0
ν(x)− 1]S
[
0
∗
(x) ψ(x , t ) n(x, t ) dx (11)
Now, comparing Eqs. (10) and (11), we get
N
(t)
(t) = f
S
0
str
(t)
p
∞
(12)
S∗(x) n(x, t ) dx
0
Finally, using Eqs. (7), (8), and (12), we get the mathematical formulation of the
volume and time-dependent linear breakage selection function as
∗
(x)
S(x,t) = S
∗
(x) S0(t) ψ(x, t) = fs(x, t) Np(t)
S
∞
(13)
S∗(x)f (x, t ) dx
0
Monte Carlo Simulation Details
To verify the accuracy of the discussed model, a CNMC algorithm is used to
replicate a simple linear breakage process. Since the whole algorithm is discussed
before, here we will only discuss the requisites of the simulation. The selection of
particles for stressing events is the only event that takes place in this MC simulation.
In each of the stressing event, we choose one particle using the probability function
in Eq. (9). After selection of the particle, we use the predefined ψ(x,t) to check
whether the particle will break or not.
86 A. Das and J. Kumar
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Verification of the Model
For the verification of the developed model of linear breakage selection function,
the breakage PBE (Eq. (5)) was numerically solved using the developed model (13),
and the numerical results were compared with the results obtained from the CNMC
simulations. A detailed discussion on the model verification can be found in Das et
al. [18]. For this chapter, the verification has been conducted for the following two
simple test cases:
• Case 1: ψ
(x, t) = 1, i.e., every stressing event on particles results in breakage
1
event.
• Case 2: ψ
with the volume of the chosen particle, but it is independent of time t . For this
case, we consider ψ
(x, t) ∝ x, i.e., the probability of successful stressing event varies
2
(x, t) =
2
x
x
max
.
The simulation parameters and material properties that were kept constant for
the simulations are given in Table 1. For simplicity, we have considered a fixed
value of the stressingfrequency per particle (f
). Furthermore, we have considered
str
that particles get selected for the stressing events depending on their volume (i.e.,
∗
S
(x) = x), and in case of breakage events, particles randomly break into two
smaller fragments (i.e., b(x, y) =
2
).
y
To start any MC simulation or to solve any PBE, we need to have the initial size
distribution of particles beforehand. For this linear breakage process, a normally
distributed particle size distribution (PSD) was considered with a mean 400v
variance 20 v
(see Fig. 2). Here, vppdenotes the volume of a primary particle
pp
pp
and
(monomer). Forthe verification of the model, the comparisons of the total number of
particles and the final PSD are considered. The comparisons for both the considered
cases are illustrated in Fig. 3. The evolution of normalized total number of particles
for both the cases is presented in Fig. 3a. Furthermore, the comparison of PSDs
is illustrated in Fig. 3b. From the figures, it is clear that the results obtained from
solving the PBE predicted the MC simulation results with good agreement. For the
case ψ
= 1, every stressing event results in breakage. That is why we can observe
1
Table 1 Process parameters and material properties for the linear breakage system
Parameter/Property Symbol Va l ue Unit
Bed mass M
Primary particle (monomer) diameter d
Particle density ρ
Stressing frequency per particle f
Number of particles in MC simulation box N
Total process time t
Volume-dependent part of the selection function S∗(x) x –
Breakage distribution function b(x, y)
bed
pp
p
str
MC
process
1 kg
−4
4 ×10
2400 kg/m
0.0125 s
50,000 –
100 s
2
y
m
−1
–
3
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0.02
0.015
0.01
0.005
Number fraction
0
0 50 100 150 200 250 300 350 400 450 500
Normalized volume
Fig. 2 Normally distributed initial size distribution of particles with normalized mean volume 400
and variance 20
Fig. 3 Comparison of (a) normalized total number of particles and (b) particle size distributions
obtained from PBE and MC results after 100 s
rapid increase in the number of particles compared to the other case in Fig. 3a. Due
to this, we can observe that more particles have accumulated in the smaller volume
region in this case (Fig. 3b).
3.2 Nonlinear Collisional Breakage
If particle fragmentation occurs only due to the impacts from particle collision, we
term it as nonlinear collisional breakage. Cheng and Redner [36] mathematically
formulated the one-dimensional binary collisional breakage in terms of an integrodifferential equation as
88 A. Das and J. Kumar
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∞
∂n(x,t)
∂t
=
∞
K(y,z,t)b(x,y;z)n(y, t)n(z, t )dydz
0
x
∞
−n(x, t)
0
death of particle of volume x
birth of particle of volume x
! "
K(x,y,t)n(y,t)dy
! "
(14)
along the initial distribution,
n(x, 0) = n
(x), x ∈ R
0
+
(15)
where K(x, y,t) is the collisional breakage kernel and denotes the rate of successful
collisions between particles of volume x and y at time t. Here, a successful collision
means those collisions where at least one of the colliding particles breaks into
smaller fragments. In practice, the collision between a particle pair of volumes x
and y is equivalent to the collision between the pair of volumes y and x. Therefore,
the collision kernel is assumed to be symmetric in its last two arguments, i.e.,
K(x,y,t) = K(y, x,t), for all x,y ∈ R
+
and t>0 (16)
Also, b(x, y; z) is the breakage distribution function, which illustrates the rate of
formation of particles of volume x by breakage of particle of volume y , due to
collision between y and z. The breakage distribution function b(x, y; z) satisfies the
following two properties:
y
1.
b(x, y; z) dx = ν(y;z), where ν(y;z) is the number of daughter particles
0
produced due to breakage of the particle of volume y after its collision with a
particle of volume z.
y
xb(x,y;z) dx = y, ∀y>0,z > 0, and x ≤ y. This property is called
2.
0
the mass conservation property. This condition confirms that the total volume of
daughter particles generated due to the breakup process is equal to the volume of
the mother particle.
Modeling of Nonlinear Collisional Breakage Kernel
Following a similar line to the previous model, Das et al. [20] proposed the
following factorization of the collisional breakage kernel:
K(x,y,t) = K
∗
(x, y) K0(t) ψ(x, y, t) (17)
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 89
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where K∗(x, y) is the volume dependency in particle collisions, K0(t) is the time
dependency in collisional breakage process, and the last factor ψ(x,y, t) is the
probability of successful collisions and can be defined as the ratio of the frequency
of successful breakage events (f
) to the collision frequency per particle (fc), i.e.,
s
f
ψ(x,y,t) =
s
f
c
(18)
Generally, in a collision event, two particles of the system get selected according
to their volume dependency and collide with each other. At the time of collision,
particles exert force on each other, and if the collision energy is more than the
particle’s internal bonding strength (for example, dynamic-yield strength, shear
strength, etc.), then one or both of the colliding particles break into two or more
fragments of smaller volume depending upon the breakage distribution function.
The total number of collision events per second is
the total number of particles of the system at time t . The factor
1
(t)Np(t), where Np(t) is
f
c
2
1
is considered to
2
avoid the double counting of collisions. Also, particles in the system are selected
for collision events depending on the volume-dependent part of collision kernel
∗
K
(x, y) and the availability of particles of those particular volumes in the system.
Then, the collision probability of particles of volume x and y at any instance t can
be written as
∗
(x, y) n(x, t ) n(y, t )
P(x,y, t) =
1
2
K
∞
∞
K∗(x, y) n(x, t ) n(y, t ) dx dy
0
0
(19)
The fraction
1
in the denominator is considered to avoid the double counting of
2
particles. If the breakage criterion satisfies, then the colliding particles break into
smaller fragments. If the breakup events of volume x and y result in ν(x;y) and
ν(y;x) number of daughter particles, respectively, then the increase in the total
number of particles due to this collisional breakage event is(ν(x;y) + ν(y;x) − 2).
Consequently, the rate of change of total number of particles can be represented by
the following equation:
dNp(t)
dt
1
=
2
×P(x,y, t) ψ(x,y, t)dx dy
(t) Np(t)
= f
c
×
∞
∞
ν(x;y) + ν(y; x) − 2
(
0
0
∞
∞
ν(x;y) − 1]K
[
0
0
∞
∞
K∗(x, y) n(x , t) n(y, t ) dx dy
0
0
1
f
)
c
2
∗
(x, y) ψ (x, y , t) n(x , t) n(y, t ) dxdy
(t)Np(t)
(20)
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