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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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Linearized Parameter Estimation Methods for Modeled Crystallization... 59
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1.2
y = 0.272 x +1.10
r = 8.57
) [-]
m
0.8
(ΔT
10
log
0.75*10^9
r = 12.9
1.00*10^9
r = 17.1
1.25*10^9
r = 21.4
1.50*10^9
r = 25.7
1.75*10^9
Approx. line
0.4
-2.0 -1.6 -1.2 -0.8
log10(Rlnr) [-]
Fig. 2 Regression analysis for estimation of kinetic parameters of secondary nucleation. (Reproduced from Ref. [6])
⇐⇒ μ
0,avg
r
μ
=
ln r
0s
⇐⇒ μ
3,avg
r
μ
=
ln r
3s
(14)
Equations (13) and (14) are substituted into Eq. (4) and organized, and the
following equation can be obtained:
B
= k
2
b2
⇐⇒ log(ΔT
(ΔT)
)
m
b2
=
μ
3,avg
1
b2 +1
= k
(ΔT)
b2
log(R ln r)+
b2
r
ln r
b2 +1
dμ
=
log
dt
0
μ
0s
μ
3s
1
= R
b2 +1
(
kb2μ
dμ
d(ΔT
)
3s
0
)
(15)
Therefore, secondary nucleation parameters can be estimated by the slope and
the intercept of log(T
) vs. log(Rlnr). This type of parameter estimation was
m
originally reported by Unno et al. [7], and a concrete example is shown in Fig.
2, which was originally reported by Unno and Hirasawa [6]. Here, target substance
was Arg. A saturated amount of Arg anhydrate was dissolved in 300 mL of water
at an initial temperature higher than the saturated temperature. Then, the solution
was linearly cooled down to the final temperature and stirred with a four-blade
agitator (ϕ40) at the rotation speed of 400 rpm. In cooling, an adequate quantity
of seed crystals was added when the solution temperature reached the saturated
temperature. The value of secondary nucleation order was estimated to be 2.67 and
that of secondary nucleation coefficient 1.65 × 10
10s−1m−3K−2.67
by this linear
fitting.
60 I. Hirasawa et al.
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1.6 Stochastic Primary Nucleation Kinetics
In general, secondary nucleation may occur so frequently that it can be regarded
as deterministic, which means that the same result can be obtained by repeating
the same operation. However, primary nucleation may not occur frequently but
rather stochastically, which means that the results will vary with some trend
even if the same operation is repeated. Therefore, it is difficult to control the
crystallization processes which involve primary nucleation. For predicting this
type of crystallization processes, the PBE and MBE including stochastic primary
nucleation need to be solved. The probability that the number of crystals newly
nucleated from t to t + τ will equal k follows the Poisson distribution with the
parameter of the expected nuclei number and is denoted as follows:
P(t,τ;k)=Pr
t+τ
t
WsB∗dτ=k=
t+τ
t
WsB∗dτ
k!
k
exp−
t+τ
WsB∗dτ
t
(16)
t+τ
WsB∗dτ∼ Po
t
t+τ
WsBdτ
t
(17)
Here, the superscript “*” emphasizes that the value is a stochastic variable, and
Eq. (17) means that the left side of the equation follows the Poisson distribution.
Then, τ is replaced by a reasonably short time t to derive an approximate equation
as follows:
B
∗
(t) ∼
Po[W
s
Ws(t)Δt
]
(18)
(t)B(t )Δt
In Eq. (18), t should be so short a time that the change in the expectation of
the nucleation rate and the increase in the crystal size can be ignored during t.The
stochastic nucleation rate expressed by Eq. (18) is substituted into Eqs. (1) and (2)to
derive stochastic differential equations (SDEs), which is solved once to obtain one
of sample paths. The SDEs should be solved so many times that one can estimate the
statistical properties of them. Maggioni and Mazzotti [8] have recently investigated
the modeling of stochastic primary nucleation and reviewed the relevant literature.
Unno and Hirasawa [6] reported the concrete method for the parameter estimation which involves stochastic nucleation. At first, for thedata acquisition, the means
and standard derivations of the waiting times were measured in the same system as
the concrete examples in Sects. 1.4 and 1.5 until the total crystal number reached
5
6 ×10
and 1.5 ×106in unseeded crystallization. Next, parameter space was made
by simulation using stochastic primary nucleation rates represented as follows:
−2
)
(19)
B
1
= EB
∗
= k
1
exp−b1(ln S
b1
Linearized Parameter Estimation Methods for Modeled Crystallization... 61
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Fig. 3 Simulated and observed cumulative distribution functions of waiting times until the total
crystal number of 6 × 10
5
. (Reproduced from Ref. [6])
Finally, the primary nucleation parameters were estimated so that the means and
standard derivations corresponded with experimental values. The value of primary
nucleation order was estimated to be 2.28 and that of primary nucleation coefficient
69.7 kg-solvent
the simulated waiting times until the total crystal number of 6 × 10
−1s−1
by this parameter estimation. Using estimated parameters,
5
are compared
to the observed ones. The relation between the waiting time and the cumulative
distribution function is illustrated in Fig. 3. The number of observations was 25
while that of simulations 500.
1.7 Breakage Kinetics
As mentioned in Sect. 1.1, breakage has a birth term Bband a death term Db, which
are represented as follows:
∞
B
L, t)=
(
b
D
L
L, t)= k
(
b
f
bre
L|λ)k
(
bre
bre
(L)n(L, t
λ, t)dλ (20)
(λ)n(
)
(21)
Here, k
is the breakage kernel and f
bre
(20),asinglecrystalwithasizeofλ breaks to yield two crystals, one of which has a
size of L. The mathematical expressions for k
by Marchisio et al. [9], Hasseine et al. [10], Bari and Pandit [11], and Li and Yang
[12] are listed in Tables 1 and 2, respectively. The kinetic parameters of breakage
is the fragment distribution function. In Eq.
bre
bre
and f
of various models presented
bre
62 I. Hirasawa et al.
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Table 1 Functions of
breakage kernel
Mechanism Function
Constant q
Power law q1L
1
q
2
Exponential q1exp (q2L3)
Table 2 Fragment distribution functions
Mechanism Fragment distribution function
1/3
3
+ δL − λ
8
72L
−
9
λ
Erosion
L
is minimum size
1
Constant ratio fragmentation
Mass ratio 1:r
m
Symmetric fragmentation at rm= 1
Parabolic distribution
Uniform at C = 2
δ(L − L
δL − λ
2
3CL
+1 −
3
λ
+ δL −λ
)
1
1
1+r
m
C
2
− L
72L
1
3
3
1
1/3
r
m
1+r
m
5
6
λ
2
18L
+
3
λ
in these models are estimated using the PBE. For example, “power law” model and
“mass ratio” model are applied to the breakage kinetics, and PBE is expressed as
follows if only breakage happens in the crystallizer:
∂(W
n
)
s
∂t
= W
− D
B
(
s
)
a
a
(22)
Applying the MOM to Eq. (22) derives the following equation:
Here, R
is represented as follows for “mass ratio” model:
k
In Eqs. (23) and (24), breakage parameters are q
ODE only about breakage. The mass conservation during breakage can be utilized
to derive the following relation and to estimate q
which satisfies the relation:
d(W
d(W
sμα
dt
dt
R
k
)
)
sμk
1 +r
=
1 +r
(
= W
W
sμ3
⇐⇒ q
= W
sRkq1μk+q
k/3
m
k/3
)
m
sRαq1μα+q
2
− 1 (24)
, and rm.Eq.(23)isthe
1,q2
based on the moment order α,
2
= const.
2
= const.
= 3 − α (25)
2
(23)
Linearized Parameter Estimation Methods for Modeled Crystallization... 63
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0
)
2
μ
)/Δt/(W
μ
Δ(W
-2
k+q
s
-4
/s]
2
q
/m
-6
9
k
[10
s
-8
-10
y = 287 x
rm= 4.63ʹ10
-3
-0.04 -0.03 -0.02 -0.01 0
Rk[-]
Fig. 4 Parameter estimation of breakage kinetics
In addition, finite differences in Eq. (23) are computed for some moment orders
k to derive the following equation and to estimate q
ΔW
μ
s
k
/W
μ
s
Δt
k+q
and rm:
1
= R
kq1
2
(26)
Here, both numerator and denominator of the left side are vectors for a time
horizon, and the right side is a scalar. If R
parameter(s), the other parameters excluding q
is represented by a linear function of
k
can be estimated by linear least-
2
squares method. Otherwise, as in Eq. (24), they can be estimated only by non-linear
least-squares method. As a concrete example, the parameter estimation of breakage
kinetics was performed in the same system as in Sects. 1.4 and 1.5. The resulting
relation between (W
parameter estimation, at first q
estimated at 2.87 × 10
)/t/(Wsμ
sμk
11m−4.0s−1
2
) and Rkis depicted in Fig. 4.Inthis
k + q2
was determined at 4.0, and then q1and rmwere
and 4.63 × 10−3, respectively.
1.8 Agglomeration Kinetics
As mentioned in Sect. 1.1, agglomeration has a birth term Baand a death term Da,
which are represented as follows:
B
L, t)=
(
a
3
λ,L
k
agg
L
2
L
2
0
− λ
3
L
− λ
1
3
3
n
2
3
3
L3− λ
1
3
3
,tn(λ, t)dλ (27)
64 I. Hirasawa et al.
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Table 3 Basis functions of aggregation kernel
Mechanism Basis function
ConstantSize independent 1
Size independent
Brownian motion
Sum L3+ λ
Hydrodynamic (L + λ)
(
L+λ
Lλ
2
)
3
3
Laminar shear
Isotropic turbulence
Differential forceTurbulent inertiaDifferential sedimentation (L + λ)2|L2− λ2|
Turbulent inertia
Differential sedimentation
Table 4 Coefficients of aggregation kernel
Mechanism Coefficient Symbols used
Constant and so on a
1
Size independent and so on a1G
Brownian motion
Laminar shear
2kBT
3μ
4γ
3
Isotropic turbulence
Turbulent inertia
Differential sedimentation
1.27(ρp−ρ
0.7g(ρp−ρ
8πε
15ν
a
a
1G
1ε
ε
)
f
μ
)
f
μ
a1= constant [equation-dependent]
G = growth rate [m/s]
ε =turbulent dissipation rate [m
a
= exponent of G [−]
1G
a
= exponent of ε [−]
1ε
kB= Boltzmann constant [J/K]
T = thermodynamic temperature [K]
μ = viscosity [pa s]
γ = velocity gradient [s−1]
ν = kinematic viscosity [m2/s]
3
4
ε
ρp= particle density [kg/m3]
ν
= fluid density [kg/m3]
ρ
f
g = gravitational acceleration [m/s2]
2/s3
]
D
(
a
Here k
is the agglomeration kernel. In Eq. (27), two crystals, one of which has a
agg
size of λ, agglomerate to yield a single crystal with a size of L, and in Eq. (28), the
reverse relation is established. The mathematical expressions of the basis functions
and the coefficients for k
et al. [9], Laloue et al. [14], Ó’Ciardhá et al. [15], and Gencaslan et al. [16]are
listed in Tables 3 and 4, respectively. In Tables 3 and 4, a basis function multiplied
by a corresponding coefficient is an agglomeration kernel. When agglomeration is
dependent on supersaturation, nucleation and growth must occur at the same time in
the crystallizer. Thus, it is difficult to prepare the experimental system where only
agglomeration may stand out. In this case, we have no choice but to substitute the
other parameters into Eq. (1).
∞
L, t)= n(L, t
of various models presented by Vanni [13], Marchisio
agg
k
)
λ, L)n(λ, t)dλ (28)
(
agg
0
Linearized Parameter Estimation Methods for Modeled Crystallization... 65
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2 Application of Modeling to Optimization
Mathematical models for crystallization are usually described with the differential
equations, which may include the parameters depending on materials and have the
initial, boundary, and other conditions determined by procedures. The quantities
resulting from crystallization or related to its process can be simulated by solving
the differential equations. In mathematical modeling, the parameters are estimated
to minimize the error between the simulated quantities and the observed ones. On
the other hand, in optimization after mathematical modeling, some procedures are
determined to minimize the difference between the simulated quantities and the
desired ones. Therefore, both modeling and optimization can be regarded as the
minimization problems via the differential equations, and in optimization similar
algorithms can be applied as in modeling.
In seeded batch cooling crystallization, the seeding method and the cooling
method may be optimized for improving the quantities related to the crystal product
quality. In Sect. 2, several seeding policies classified based on seed loading quantity
are discussed at first, and among them partial seeding is focused on as a good
choice for pharmaceutical processes. Secondly, in partially seeded crystallization of
a model substance, the seeding and cooling methods are simultaneously optimized
for the best product quality. Thirdly, the crystallization process designed optimally
for the model substance is postulated to be implemented. Fourthly, we show the case
study of partially seeded crystallization of Arg and the optimization of the seeding
method. Finally, we make a few comments on the effect of stochastic nucleation
mentioned above on quality stability or fluctuation of the crystallization products.
2.1 Seeding Policies
In the batch crystallization, the surface area of the suspended crystals and the
supersaturation of the solution need to be controlled for obtaining the product
crystals with the desired quality. The initial surface area of the suspended crystals
can be controlled with the seeding method, while the supersaturation may be
controlled with the cooling method in the cooling crystallization. The seeding and
cooling policies have been studied by many researchers since the pioneering work
of Griffiths [17] on industrial crystallization.
In manufacturing pharmaceuticals, large and uniform crystals are desired for
high efficiency of downstream processes, such as filtering and drying. The seeding
methods have been developed for the optimization of the crystal size. The seeding
policies can be roughly classified into internal seeding and external seeding.
In internal seeding, seed crystals are not added to the solution in cooling, and
primary nuclei and secondary ones following them grow up to yield the product
crystals. In this process, the resulting products are not contaminated by the external
seed crystals, and hence the internal seeding attracts attention as a good method
66 I. Hirasawa et al.
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for manufacturing the added-value crystal products such as pharmaceuticals. The
applications of internal seeding have been reported by Doki et al. [18], Kim et
al. [19], Lenka and Sarkar [20], and so on. However, in internal seeding the extra
nuclei need to be dissolved by temperature cycling in order to obtain large and
uniform crystals. Finally, a reasonable number of nuclei survive and are treated
as seed crystals in typical seeding. The heater and the cooler may expend much
time and energy on this temperature cycling in large industrial plants. In addition,
the process containing internal seeding may be strongly affected by the stochastic
nature of primary nucleation, and hence the batch time or the product quality can
fluctuate under a certain controlling condition.
In external seeding, seed crystals are added to the solution for some purpose. The
external seeding may be classified into full seeding and partial seeding, based on the
purpose of seeding.
In fullseeding, a large quantity ofseed crystals isadded to consumeand lower the
supersaturation by the crystal growth and to suppress nucleation. The full seeding
was first investigated with the statistical method by Doki et al. [21], and then has
been widely studied from laboratory scale to industrial plant. The full seeding is
so robust a method that it may yield the crystal products of very high quality and
be hardly affected by the cooling method. Meanwhile, in full seeding, it may be
preferable to add a sufficient quantity of small and uniform seed crystals because
the product crystal quality strongly depends on the seed one. However, a large
quantity of seed crystals is avoided in the production of added-value crystals due to
the impurities from filtrated, dried, milled, and sieved seed crystals, and to the risk
of exposure to their dust. Moreover, unlike the crystals nucleated in the solution, the
dry seed ones may not necessarily grow successfully.
On the other hand, in partial seeding, a small quantity of seed crystals is added
to induce secondary nucleation. And then, the nuclei grow to yield the products of
moderate quality. Roughly speaking, partial seeding has been applied unintentionally to the actual plants so far, which means that some seeded crystallizers may
not be fully seeded for inhibiting nucleation practically. Lee et al. [22] referred
to the partial seeding as a reasonable method which might contribute to a high
reproducibility.In partial seeding, the seed crystalsmay not act as children whichare
going to grow up to be the product crystals but as catalysts for secondary nucleation,
and hence the quality of the seed crystals does not necessarily affect the product
quality. The simulated results supporting this claim, which were originally reported
by Unno and Hirasawa [23], are shown in Sect. 2.2. Therefore, the seed crystals may
not need milling and sieving, and they can be replaced with the seed slurry, which
may contribute to a significant reduction in impurities and dust.
The concept of classification of seeding policies is outlined in Fig. 5.From
the above, partial seeding can be considered to be a relatively good choice for
the seeding policy for pharmaceutical processes. However, the process containing
partial seeding can be strongly affected by the cooling method, which is attributed to
the dependency of secondary nucleation kinetics on supersaturation. For the optimal
control of partially seeded crystallization, a moderate number of secondary nuclei
should be induced at an early stage, and then they should grow to be the product
Linearized Parameter Estimation Methods for Modeled Crystallization... 67
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Are crystals added
from outside?
No
External seeding
To trigger
secondary nucleation
Internal seeding Full seeding
Fig. 5 Concept of classification of seeding policies
Yes
The purpose
of addition is:
To grow
seed crystals
Partial seeding
crystals withextra nucleation inhibited. Very few studies on this type of optimization
problem have been reported, and the optimization of the cooling method is still
debatable. Simulation examples of the optimal process design are shown in Sect.
2.3.
2.2 Optimization of Partial Seeding
In Sect. 2.2, the simulated results of seeded batch cooling crystallization in the
model system, which were originally reported by Unno and Hirasawa [23], are
discussed to characterize partial seeding.
At first, an aqueous solution of potassium sulfate is selected as a target substance,
and the following preconditions are assumed to be established: a suspension in
the crystallizer is well-mixed, there is no crystal breakage nor agglomeration,
crystal growth follows the McCabe L law, which means that neither growth
rate dispersion nor size-dependent crystal growth occurs, primary nucleation and
secondary nucleation are considered to be deterministic processes, the rate of each
phenomenon is expressed as the power function of the supercooling degree, and the
secondary nucleation rate is proportional to the magma density, or the third moment
of CSD. The physical properties and the kinetic parameters are listed in Table 5.
Next, the experimental procedure is considered as follows. Seed crystals are
added to saturated potassium sulfate aqueous solution at 50
is cooled down to 30
◦
C. Then, the solution is kept at 30◦C for an hour.
Finally, the PBE and MBE, which contain differential equations, are solved
to deduce several lower-order moments of the product CSD. Using the resulting
moments, the mean size and the CV are calculated to evaluate the product quality.
Here the mean size should be defined or weighted according to the purpose of
optimization. The mean size L
, which will be the size of the crystal with the mean
i,j
◦
C, and the solution
68 I. Hirasawa et al.
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Table 5 Physical properties and kinetic parameters of potassium sulfate
Name Symbol Unit Va l ue
Density of crystal ρ
Volumetric shape factor k
Constant of solubility curve α
Constant of solubility curve β
Constant of solubility curve γ
Primary nucleation coefficient k
Primary nucleation order b1 [−] 5.96
Secondary nucleation coefficient k
Secondary nucleation order b2 [−] 3.00
Linear growth coefficient k
Linear growth order g [−] 0.9
Nucleated crystal size L
Solubility curve is represented by C/(g/g-solvent) = α
is the concentration of saturated solution and θ is the solution temperature.
Reproduced from Ref. [24]
c
v
sat
sat
sat
b1
b2
g
0
[kg/m3] 2662
[−] 1.5
[−] 6.29 × 10
[−] 2.46 × 10
[−] −7.14 × 10
[#/(s kg-solvent Kb1)] 1.0 × 10
[#/(s Kb2m3)] 1.0 × 10
[m/(s Kg)] 1.0 ×10
[μm] 1.00
+ β
sat
(θ /◦C) + γ
sat
sat
−6
6
−7
(θ/◦C)2,whereC
−2
−3
−6
value of the quantity proportional to Liif j = 0orbetheLj-weighted mean size if
i – j = 1, is represented using the moments as follows:
1
i−j
μ
L
i,j
i
=
μ
j
(29)
For example, mean volume size, which is the size of the crystal with the mean
volume, is denoted by L
mean size, by L
. The mean volume size is often used when the full seeding is
4,3
and volume mean size, which is the volume-weighted
3,0
investigated because only the total volume of crystals is increased without changing
the total number in successful full seeding. The CV may also be weighted by L
j
and
is given as follows:
μjμ
=
CV
j
For example, the volume-weighted CV is denoted by CV
j+2
− 1 (30)
2
μ
j+1
. The CV originally
3
means the ratio of the standard deviation in CSD to the mean size, and hence
the CV of the size distribution of large and uniform crystals is small. Therefore,
the minimization of the CV can correspond to the optimization of partial seeding.
These statistics defined or weighted are deduced from the histogram in which the
horizontal axis indicates the quantity proportional to L
that proportional to L
j
. In the typical histogram, i is set to 1 and j to 0. Moreover,
i – j
and the vertical axis does
in simulations, one can distinguish the origins of product crystals. In Sect. 2, based
on the origin, the product crystals are classified as seed-grown, seed-originated, or
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