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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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90 A. Das and J. Kumar
https://t.me/medicina_free
The collision probability of volume pair (x, y) is the same as the collision
probability of volume pair (y, x). For this, we considered the fraction
1
in the right-
2
hand side of Eq. (20).
On the contrary, the rate of change of total number of particles in the nonlinear
breakage PBE can be computed by integrating Eq. (14) with respect to x and y from
0to∞.
∞
dN
(t)
p
= K
(t)
dt
0
∞
[ν(x;y) − 1]K∗(x,y)ψ(x,y,t)n(x,t)n(y,t)dx dy
0
0
(21)
Comparing Eqs. (20) and (21), we get
f
(t) Np(t)
K
(t) =
0
∞
∞
c
(22)
K∗(x, y) n(x, t ) n(y, t ) dx dy
0
0
Then, using Eqs. (17), (18), and (22), we have the volume and time-dependent
collision kernel for nonlinear breakage process as
∗
(x, y)
K(x,y,t) = f
(x,y,t)Np(t)
s
∞
∞
K
(23)
K∗(x, y) n(x, t ) n(y, t ) dx dy
0
0
Monte Carlo Simulation Details
In this case also, particle collision is the only possible event which takes place in the
MC simulation. For each collision event, we use the probability function of Eq. (19)
to choose two different particles. Then, we break the colliding particles according
to the breakage behavior and update the system accordingly.
Verification of the Model
Similar to the case of linear breakage process, we will use the MC simulation
results to verify the developed model of collisional breakage kernel. The collisional
breakage PBE (Eq. (14)) was solved numerically using the developed model (23),
and the results were compared against the results obtained from the MC simulations.
The weighted finite volume scheme to solve the PBE can be found in Das et al. [20].
To test the authenticity of the model, we have considered a collisional breakage
process where, in cases of successful collision events, both the colliding particles
break into two smaller fragments with volumes 60% and 40% of the parent particles,
respectively. Then, in this case, the breakage distribution function can be written as
[20]
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 91
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b(x, y; z) = δ(x −0.4 y) + δ(x − 0.6 y) (24)
Furthermore, we considered that the collisions are independent of their volumes,
∗
i.e., K
(x, y) = 1. Other simulation parameters that were kept constant during the
simulations are the same as the case of linear breakage process (see Table 1). The
initial size distribution of particlesis also providedin Fig. 2. The verification process
was conducted for the following cases of ψ:
• ψ
(x,y,t)= 1, i.e., every collision event between particles results in a breakage
1
event.
• ψ
(x,y,t) =
2
0ifv ≤ 200
1ifv>200
, where v =
2xy
is the characteristic volume
x +y
of the colliding particles, and breakage event occurs only if the characteristic
volume of the colliding particles is greater than 200.
To verify the developed model accuracy, the evolution of normalized total
number of particles and PSDs obtained from solving the collisional breakage PBE
and from CNMC simulations were plotted against each other in Figs. 4 and 5.From
the figures, it is clear that the model predicted the CNMC results meticulously for
both the considered cases. Since the particles were breaking in 60% and 40% of
the volume and initially particles were distributed normally around the normalized
volume 400, we can observe some discrete normally distributed peaks around some
particular sizes, for example, 240, 160, etc. Also, for the case ψ
= 1, i.e., when
1
every collision resultsin a breakageevent, Fig. 5a illustratedthat a greaternumber of
particles are concentrated in the smaller particle zone compared to the other case. In
the case of ψ
, breakage events were occurring only when the characteristic volume
2
of the colliding particles is higher than or equal to 200. For this, a more prominent
peak can be observed around the normalized volume 160 in Fig. 5b, since once a
particle breaks into this zone, then it is unlikely to break again.
Fig. 4 Evolution of normalized total number of particles
92 A. Das and J. Kumar
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Fig. 5 Comparison of particle size distributions obtained from PBE and MC results after 30 s
The developed model was further verified for several other test cases. For the
detailed discussion of the verification process, the readers are referred to see the
work of Das et al. [20].
3.3 Sonofragmentation of Rectangular Plate-Like Crystals
The ultrasound assisted sonofragmentation process is well known in chemical and
pharmaceutical industries. The use of ultrasound creates cavitation bubbles, and
the implosion of the bubbles creates shock waves, which lead to crystal breakage.
The use of ultrasound creates lesser impurities compared to other fragmentation
techniques. There exist several studies in the literature related to sonofragmentation
experiments of one-dimensional crystal particles. However, this is not the case
for sonofragmentation of multi-dimensional crystals, as it is mostly unexplored
regarding experiments and their mathematical modeling. In 2019, Bhoi et al. [26]
performed sonofragmentation experiments on rectangular shaped pyrazinamide
crystals and studied the effects of sonication period and sonication power. The
authors also developed an MC algorithm to understand the breakage behavior of
crystals. Later, Das et al. [19] developed the corresponding bivariate PBM. In this
section, we will discuss the abovementioned studies.
Experimental Section
Sonofragmentation experiments were carried out on the δ-form of pyrazinamide
crystals in a toluene medium. Experiments were conducted for different sonication
periods (30, 60, and 90 s), keeping the ultrasonic amplitude value fixed at 30%.
After each experiment, images of filtered and dried crystal fragments were captured
using an optical microscope (see Fig. 6). The initial size distribution of crystals
along length and width axes is shown in Fig. 7. The key observations from the
experimental setup are following:
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 93
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Fig. 6 Optical microscope image of δ-form pyrazinamide crystals (taken from Das et al. [19])
0.08
Initial crystal length distribution
Initial crystal width distribution
0.06
0.04
0.02
Number fraction
0
0 100 200 300 400 500 600
Crystal dimensions (mm)
Fig. 7 Initial crystal size distributions along length and width
1. δ-form of pyrazinamide crystals is considered as thin rectangular plate-type
particles with constant thickness and the particles break into only two fragments.
2. Due to the rectangular shape of crystals, they mostly break across their width.
Further binary breakage of crystals was observed.
3. Crystal particles with length lesser than 20 μm do not break further into smaller
fragments.
4. The total frequency of stressing events for the whole system stays constant
throughout the experiment (f
amplitude.
(t) = constant), for a constant ultrasonic
ev,tot
94 A. Das and J. Kumar
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Monte Carlo Simulation Details
In the MC algorithm, crystals were represented with a two-dimensional array,
and the elements of that array were the characteristic length and width of crystal
particles. The trial and error method was used to fix the value of f
ev,tot
(i.e., the
total frequency of events) for the fixed ultrasonic amplitude value at 30%. The fixed
value of f
was 3.15 × 105[19]. Furthermore, to find the breaking behavior of
ev,tot
crystals, several breakage techniques were tested and compared with experimental
results. However, the best possible match was observed when crystals break only
across the width axis, and the fracture occurs at any random point between 30%
and 70% of its length (see Figs. 6 and 8)[26]. These observations are used in the
modeling of PBE kernels. Furthermore, the MC simulation results will be discussed
in the validation part.
Modeling of Bivariate Breakage PBE Kernels
Since the length of the sides of crystal particles is the only property of concern and
the fragmentation mechanism is of linear type, we use the bivariate linear breakage
PBE. The continuous bivariate linear breakage PBE is
∞
∂n(x
1,x2
∂t
,t)
=
∞
S(y1,y2,t)b(x1,x2|y1,y2)n(y1,y2,t)dy1dy
x
x
1
2
! "
birth of crystal particles with dimensions (x1,x2)
2
Fragment-1
Fig. 8 Schematic diagram of the crystal breakage mechanism, in which rectangular crystals break
into two fragments across the minor dimension and the breakup mechanism starts at any random
point along the major dimension between 30% and 70% of its length. The dashed line in the figure
represents the breakup position on the crystal
Break up region
Fragment-2
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 95
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where n(x
,t) is the number density function, S(x1,x2,t) is the bivariate break-
1,x2
age selection function, and b(x
− S(x1,x2,t)n(x1,x2,t)
1,x2|y1,y2
death of crystal particles with dimensions (x1,x2)
) is the breakage distribution function,
! "
(25)
respectively. To develop the population balance model accurately, we have to
formulate the two kernel functions S(x
The bivariate linear breakage selection function S(x
,t) and b(x1,x2|y1,y2).
1,x2
,t) defines the rate
1,x2
of successful stressing events, i.e., those events where crystals break due to the
implosion of cavitation bubbles. To model the bivariate linear breakage selection
function S(x
,t), Das et al. [19] proposed the following partition as
1,x2
S(x
,t) = S∗(x1,x2)S0(t) ψ(x1,x2,t) (26)
1,x2
Then, performing similar calculations as the monovariate linear breakage process,
one can get the following expressions [19]:
f
S
(t) =
0
∞
∞
S∗(x1,x2)n(x1,x2,t)dx1dx
0
0
ev,tot
(27)
2
and
S(x
1,x2
,t) = f
ev,tot
S∗(x1,x2)ψ(x1,x2,t)
∞
∞
S∗(x1,x2)n(x1,x2,t)dx1dx
0
0
(28)
2
The breakage distribution function b(x
The breakage distribution function satisfies the following properties:
1.
0
where ν(y
) is the number of smaller fragments created due to the breakage
1,y2
of a crystal with dimensions (y
2. Since the breakage of the rectangular crystals conserves total area, we have
0
In addition to this, from the experimental observations and MC simulation results,
we know that the crystals break across the width into two smaller fragments and the
fracture mechanism starts at any random point in the confined region (30%, 70%)
1,x2|y1,y2
) by breakage ofcrystals with dimension (y1,y2).
1,x2
y
y
1
2
b(x1,x2|y1,y2) dx1dx2= ν(y1,y2) (29)
0
).
1,y2
y
y
1
2
x1x2b(x1,x2|y1,y2) dx1dx2= y1y
0
) defines the rate of formation
2
(30)
96 A. Das and J. Kumar
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along the length of the crystals (see Fig. 8). Das et al. [19] proposed the following
model for the breakage distribution function which follows the abovementioned
constraints as
b(x
1,x2|y1,y2
) =
× δ(x
+
× δ(x
=
#
2
0.4y
1
− y2)H(y1− y2)
2
2
0.4y
2
− y1)H(y2− y1)
1
2
0.4y1y
2
H(x
− 0.3y1) − H(x1− 0.7y1)
1
#
− 0.3y2) − H(x2− 0.7y2)
H(x
2
y
δ(x1− y1)H(y2− y1)
1
$
$
$
×#H(x
+ y
− 0.3y2) − H(x2− 0.7y2)
2
δ(x2− y2)H(y1− y2)#H(x1− 0.3y1) − H(x1− 0.7y1)
2
(31)
where δ and H are the Kronecker delta function and Heaviside step function,
respectively. In formulation (31), the terms H(y
− y2) and H(y2− y1) help
1
to determine the major dimension (length) of the breaking crystal, and the terms
δ(x
− y1) and δ(x2− y2) help to keep the length of the minor dimensions (width)
1
intact inthe progeny crystal particles. Further, the terms {H(x−0.3y)−H(x−0.7y)}
make sure that the breakup region is the restricted interval (0.3y, 0.7y). One can
easily verify the correctness of the formulation (31) by satisfying the fundamental
properties of any breakage distribution function, i.e., Eqs. (29) and (30). The
detailed development of this model is available in Das et al. [19].
$
Validation of the Model
In this section, we will validate the accuracy and efficiency of the developed
models of bivariate breakage selection function (Eq. (28)) and bivariate breakage
distribution function (Eq. (31)). The bivariate breakage PBE was solved numerically
using the weighted finite volume scheme of Saha et al. [14]. Then, the numerical
results were compared against the experimental observations and MC simulation
results. The experiments were conducted for a fixed ultrasonic amplitude value
(30%), and the sonication period was varied for 30, 60, and 90 s. The initial crystal
length and width distributions aredepicted in Fig. 7. The following expressions were
used while solving the PBE:
∗
S
(x1,x2) = x1x2, (32)
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 97
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0.12
0.1
0.08
0.06
0.04
Number fraction
0.02
0
0 100 200 300 400 500 600
Length of particles (mm)
0.14
0.12
0.1
0.08
0.06
0.04
Number fraction
0.02
0
0 100 200 300 400 500 600
Length of particles (mm)
0.15
0.1
0.05
Number fraction
0
0 100 200 300 400 500 600
Length of particles (mm)
EXP
PBE
MC
EXP
PBE
MC
EXP
PBE
MC
0.15
0.1
0.05
Number fraction
0
0 50 100 150 200 250 300 350
(a)
0.2
0.15
0.1
0.05
Number fraction
0
0 50 100 150 200 250 300 350
(b)
0.25
0.2
0.15
0.1
Number fraction
0.05
0
0 50 100 150 200 250 300 350
Width of particles (mm)
Width of particles (mm)
Width of particles (mm)
(c)
EXP
PBE
MC
EXP
PBE
MC
EXP
PBE
MC
Fig. 9 Comparison of PBE, MC, and experimental results along length and width axes at time (a)
30 s, (b)60s,and(c) 90 s. Ultrasonic amplitude of this experiment is set at 30%
and ψ(x1,x2,t) =
The comparisons of crystal length and width distributions obtained from solving
the PBE, MC simulations, and experimental observations at instances 30, 60, and
90 s are illustrated in Fig. 9. From the figures, it is clear that the PBM results
predicted both the experimental results and MC simulation results satisfactorily
at all instances. From the figures, it is clear that the average length and width of
crystal particles are reducing gradually, and the span of the distributions are getting
narrower with higher peak values with increase in sonication time.
The time evolution of the total number of crystal particles in the system was not
observed in the experimental setup. However, the time evolution of total number
of particles was tracked from solving the PBE and from MC simulations, and the
1ifmax(x
0 elsewhere
) ≥ 20 μm,
1,x2
(33)
98 A. Das and J. Kumar
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6
PBE
5
4
3
2
Normalized number of particles
1
0 102030405060708090
MC
Time (s)
Fig. 10 Comparison of the PBM and MC generated time evolution of normalized number of
particles in the system when the ultrasonic amplitude is set at 30 AMP
)/N
,x
N(x
p
2
1
0.012
0.01
0.008
0.006
0.004
0.002
0
0
200
0
x
1
200
100
x
2
(a)
)/N
,x
N(x
p
2
1
0.02
0.015
0.01
0.005
0.02
p
0.015
)/N
2
,x
0.01
1
N(x
0.005
0
0
200
0
x
1
200
100
x
2
(b)
0
0
200
0
x
1
200
100
x
2
(c)
Fig. 11 Population balance model generated crystal size distribution (CSD) of the 30 AMP
experimental system at instances (a)30s,(b)60s,and(c)90s
obtained results are illustrated in Fig. 10, which showed meticulous agreement
between MC and PBE predictions.
Furthermore, to predict the outcome of the sonofragmentation process in a
more accurate manner, crystal size distribution (CSD) of the system was computed
by solving the breakage PBE using the developed model of bivariate selection
function and breakage distribution function. The computed CSD of the system at
instances 30, 60, and 90 s is illustrated in Fig. 11. The time evolution of the crystal
size distribution clearly shows how the particles are broken to finer dimensions
with increasing sonication time and distribution evolves toward a narrower size
distribution.
Mathematical Modelingof Different Breakage PBE KernelsUsing Monte Carlo... 99
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Table 2 Comparison of the computational efficiency
Sonication time (s) MC simulation time (s) PBM simulation time (s)
30 53.98 2.98
60 80.55 3.66
90 138.40 7.65
Furthermore, to check the efficiency of the developed PBM, computation
times was calculated for both PBM and MC simulations. The comparison of the
computational times is provided in Table 2. From the table, it is clear that the PBM
technique is computationally very efficient than the MC simulation technique. In
fact, for all the three cases, the PBM simulation time was only 5% (at most) of the
MC simulation time.
4 Conclusion
This chapter discusses the development and verification of some breakage PBE kernels, which are dependent on particle dimensions and process time simultaneously.
Mathematical models of the volume and time-dependent linear breakage selection
function andbinary collisional breakagekernel were presented.Furthermore, a comprehensive population balance model of the ultrasound assisted sonofragmentation
experiments was developed, which includes the modeling of bivariate breakage
selection function, and breakage distribution functions were discussed.
On the other hand, constant number MC simulation algorithms were developed
to verify the accuracy of the developed PBMs. The MC simulations verified
the accuracy of the developed models (linear breakage selection function and
nonlinear collisional breakage kernel). In case of sonofragmentation experiments,
MC simulations were first used to understand the breakage mechanism correctly,
i.e., by fixing the frequency of stressing events and by fixing the breakup region.
Consequently, the developed bivariate PBM and MC simulation results predicted
the experimental observations meticulously. Also, it was shown that the population
balance modeling techniques are computationally very efficient compared to MC
simulations. In conclusion, this chapter promotes the use of population balance
equations and the Monte Carlo method to model different breakage processes
accurately.
References
1. D. Ramkrishna, Population balances: Theory and applications to particulate systems in
engineering, Academic Press, 2000.
2. J. Kumar, G. Warnecke, Convergence analysis of sectional methods for solving breakage
population balance equations-II: The cell average technique, Numerische Mathematik 110
(2008) 539–559.
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