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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5395_Библиотеки_им_академика_М_И_Перельмана
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2 Mathematical Modeling of API Synthesis, Crystallization,
Filtration, and Drying
In this section, we present four subsections describing the mathematical modeling
employed for each step: synthesis, crystallization, filtration, and drying. Further
information on the modeling environments used for each model is discussed with
the case studies in Sect. 3.
2.1 Synthesis Modeling
For the first case study, a plug flow reactor (PFR) for the synthesis of paracetamol
is modeled under the following assumptions: (1) no radial gradients, (2) negligible
axial dispersion, and (3) constant heat transfer fluid temperature T
represented below as Eq. (1):
. The model is
ht
n
comp
%
j=1
CjC
p,j
∂T
∂t
∂C
j
∂t
=−.F
=−˙F
∂C
j
+ νj· rj= A,B, C, D,
in
∂V
n
comp
%
CjC
in
j=1
p,j
∂T
∂V
+ ΔH
(T ) · r −Ua(T − T
rxn
,
)
ht
(1)
where U is the heat transfer coefficient (W m
2m−3
(m
), and C
Heat of reaction H
is the heat capacity of component j (J mol−1K−1), given by
p, j
(T ) = Aj+ BjT + CjT2+ ... (2)
C
p,j
(J mol−1) was computed from data taken from literature
rxn
−2K−1
), a is the heat transfer area
[15, 16]. For simulation purposes, the model is translated to a set of ordinary
differential equations (ODEs) via an upwind discretization in the V dimension (Eq.
(3)):
dC
n
j
=−˙F
dt
C
in
p,vol
C
j
dT
n
−C
ΔV
n
dt
n−1
j
+ νj· rC
=−.FinC
p,vol
n
A
n−Tn−1
T
ΔV
,C
B
n
,T
+ ΔH
n
j = A, B, C, D, n = 1,...,n
n
T
rxn
· r − UaT
discr
n
− T
,
ht
(3)
with boundary and initial conditions for concentrations:
,
256 D. J. Laky et al.
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0
C
= C
j
V,0)= 0 j = A,B, C,
C
(
j
C
D
j = A, B, C, D
j,in
V,0)= 55 mol L
(
(4)
-
1
,
and
0
T
= Tin,
(5)
T(V,0)= 298.15 K
for temperature. Volumetric heat capacity C
n
comp
C
p,vol
=
C
j=1
(J m−3K−1) is given by
p, vol
p,j
n
T
n
· C
j
. (6)
2.2 Crystallization Modeling
For continuous crystallization, a one-dimensional population balance with sizeindependent growth is used to represent the crystal phase, as shown in Eq. (7).
∂f(t,x
∂t
where f represents the crystal size distribution (CSD) of the solids present in
suspension (# μm
−1m−3
to a system of ODEs by discretizing the spatial coordinate x using a high-resolution
method ([17, 18]). The change in crystal size distribution (CSD) can then be written
as the following semi-discretized equation system (n = 1, ···, n
∂f(t,x
)
+ G
)
∂x
= B ·δ(x −x
−˙F
)
0
− f), (7)
f
(
in
in
). The crystallization model shown in Eq. (7) is converted
):
discr
F
df
n
=
dt
n+1/2
− F
Δx
n−1/2
, (8)
where fluxes F are given by
F
n−1/2
= Gf
n−1
+
F
= B,
1
(
2
+
1−1/2
− f
f
n
n−1
.
f
F
in
n,in
F
n
discr
ϕ
)
n−1
,n= 2,...,n
− f
n
= 0.
+1/2
+ Df
1
+
n
2
f
(
discr
n−1
− f
,
ϕ
)
n
n
(9)
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Van Leer flux limiters ϕnwere used to incorporate the high-resolution component
of the discretization method:
|
|
θ
+θ
n
1+|θ
fn−f
f
n+1
−f
n
n−1
n
,
|
(10)
.
n
ϕ
=
n
=
θ
n
Material balance on the continuous phase is represented by
dC
˙
j
dt
F
1
in
=
ε
ε
inCj,in
V
tr = 3 k
ε = 1 −k
− εC
vρsμ2
j
vμ3
− trδ
G,
,
j,tg
C
j
−
,
ρ
s
(11)
where ρ
species j and the target crystallizing component, indexed as tg (j = 1..., n
is the crystal density (kg m−3) and δ
s
δ
=
j,tg
j, tg
1 j = tg
0 j = tg
is the Kronecker delta that relates
):
comp
(12)
The nth moment of the CSD, f, is given by
∞
=
μ
n
xnf(x,t)dx, (13)
0
which can be computed numerically from the model solution given a specified size
grid in the x dimension.
Regarding batch and semibatch operation, solid-phase population balances are
−1
computed for the CSD expressed as total particle number˜f (# μm
). On the
other hand, liquid-phase material balances for batch and semibatch operation can
be obtained from Eqs. (7) and (11) by appropriate manipulation, which requires
including an expression for the change in liquid volume (dV
/dt), as described
L
elsewhere [19, 20].
Furthermore, energy balances for the different operation modes analyzed for
crystallizers are given by
mC
mC
p
sl
− ΔH
p
sl
dT
dt
=
=˙F
cry
ερ
in
l
tr − UA(T − T
⎧
C
Vερ
⎨
⎩
V
p,l
l
ρ
L
lCp,l
hl+(1 −ε)ρsh
dm
−
)
ht
dt
+(1 −ε)ρsC
1−ε
(
)
+
ρsC
p,s
ε
−ερlhl+(1 −ε)ρsh
s
in
h(T ),
,(MSMPR
p,s
)
.(Batch and semibatch
s
(14)
)
where C
(J kg
p, l
−1K−1
and C
p, s
), and H
are the heat capacity of the liquid and solid phase, respectively
is the latent heat of fusion of the target compound (J kg−1).
cry
258 D. J. Laky et al.
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Table 1 Kinetic constants
for CR01
Constant Va l ue Unit
k
b
b 6.23 –
k
g
g 1.54 –
k
d
d 1.54 –
16.034 # s−1(kg m3)
6.56E-03 μms−1(kg m3)
6.56E-03 μms−1(kg m3)
−b
−g
−d
Operation of batch and mixed suspension-mixed product removal (MSMPR) crystallizers is assumed at constant mass dm/dt = 0, whereas for semibatch crystallizers,
dm
dt
=˙F
ερ
in
+(1 −ε)ρ
l,in
. (15)
s,in
Crystallization kinetics are represented as power law expressions for nucleation,
growth, and dissolution depending on absolute supersaturation (Eq. 16):
S = C −C
sat
(16)
Nucleation rate is expressed as the sum of primary and secondary nucleation
rates [21]:
B = B
B
p
B
s
+ Bs,
p
= kpexp−Ep/RTSp,
= ksexp(−Es/RT)S
s
1
s
2
vμ3
,
)
k
(
(17)
B = max(B,0).
Growth and dissolution rates are expressed similarly:
D = mink
The operators max and min in Eqs. (17)–(19) dictate which crystallization
kinetics are active during the growth regime (growth and nucleation) and dissolution
regime (only dissolution). Kinetic constants used in Sect. 3 were adapted from Nagy
et al. [22] (Table 1).
Paracetamol solubility in a mixture of water and acetic anhydride + acetic acid
was modeled according to a polynomial expansion [23]:
C
= 4.442 × 103− 30.86 T + 5.368 ×10−2T2. (20)
sat
G = maxk
d
exp−Eg/RTSg, 0
g
exp(−Ed/RT)S|S
d−1
|
, 0. (19)
(18)
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Finally, the influence of chemical species other than paracetamol on growth
kinetics was modeled using a simplified multi-impurity adsorption model (MIAM),
which can be written as [24]
K
jCj
p
imp
= 1 − α
1 +
%
, (21)
KiC
i
i
where K
(m3kg−1) is the adsorption constant of species j and α is the effectiveness
j
factor [25].
The Holding tank operation model depends on the presence/absence of crystals
in the incoming material. For material coming from a crystallizer, the holding tank
acts as an adiabatic, semibatch crystallizer, for which material and energy balances
can be adapted, as explained above. On the other hand, for streams composed only
by liquid, the balances shown in Eq. (22) are used:
.
where w
flow (kg s
represents the mass fraction of component j, ˙minis the entering mass
j
−1
enthalpy (J kg
dw
dt
dm
dT
dt
), and m
tot
−1
) is calculated as
m
j
in
=
tot
dt
=
w
m
tot
=.min(t),
.
m
in
%
m
C
tot
j
j,in
p,jwj
is the instantaneous accumulated liquid mass (kg). Specific
n
comp
h(T ) =
w
j=1
j
− w
h
(
,
j
(22)
T
− h(T )),
(
)
in
in
T
ref
C
dT. (23)
p,j
T
2.3 Filtration Modeling
We predict ε and α from the crystal size and shape distribution of the slurry being
filtered according to the following models. The porosity model is based on the work
of Yu, Zou, and Standish [26], who proposed and validated a modified linear packing
model for predicting the porosity of nonspherical particle mixtures. Given the CSD
expressed as percentage volume distribution f (d
the modified linear packing model assumes that one bin of size d
component, determining the porosity of the whole mixture. Under this assumption,
the cake specific volume V (with ε = 1 − 1/V) is given by
V = maxV
), with respect to particle size di,
i
is the controlling
i
C
i
(24)
260 D. J. Laky et al.
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where V
the CSD is the controlling one. Arranging the bins in decreasing size order, V
C
is the specific volume calculated assuming that the ith component of
i
C
i
obtained with the following set of equations, for every ith component:
V
C
i
= Vi+
i−1
−Vj− 1g(r) − V
V
j
j=1
x
i
v,j
n
j=i+1
V
− Vjf(r) − V
j
+
x
,
i
v,j
(25)
⎧
d
p,i
⎨
if j<i,
d
p,j
d
p,j
⎩
if j>i,
d
p,i
3.3
+ 2.8 r(1 −r
)
2
+ 0.4 r(1 −r
)
2.7
, (27)
)
3.7
, (28)
)
where V
r =
f(r)=(1 −r
g(r) =(1 −r
is the specific volume of a cake composed only by particles of size di, g(r)
i
and f (r) are interaction functions between two components of size ratio r, x
volumetric fraction of the ith component in the mixture (calculated from f (d
d
is the equivalent packing diameter of the ith component (calculated followingYu
p,i
is the
v,i
)), and
i
(26)
et al. [26]). The effect of the shape distribution on ε is accounted for by calculating
V
with the relations proposed by Zou and Yu [27].
i
We calculate α following a resistance additivity hypothesis [28], considering the
contribution of every component of the particle mixture:
is
where ρ
is the solid mass density and iis the sphericity of the ith component of
s
the particle mixture, calculated as
in which k
crystals of size d
) and kS(di) are the surface and the volume shape factors of the
V(di
, respectively. We neglect the cake compressibility, as the cakes
i
processed in the carousel are not particularly subject to this phenomenon due to
small cake size and relatively low applied P. Suitable power law relations [29]
can be introduced for compressible systems. The cake permeability k is then defined
from α as
α =f(d
Φ
=
i
1 − ε
180
)
i
1
3
π
(
6 k
k
3
ε
d
(
V
d
(
)
S
i
1
, (29)
2
2
φ
d
ρ
s
i
i
2
3
))
i
, (30)
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k =
αmρ
1
(
s
, (31)
1 −ε
)
with cake properties established; batch filtration is modeled under the following
rules. At the end of filtration, the cake saturation S (ratio between volume of liquid
in the cake porosity and pore volume) is equal to one in every point of the cake, and
the liquid composition is the same as in the fed slurry. The remaining outputs of the
filtration model to be calculated are the filtration duration t
height H
, and the dynamic profile of filtrate V
cake
, which are obtained according
filt
filtration
, the final cake
to the following discussion. The driving force for filtration P is equal to the sum
of the pressure drops through the cake P
filter mesh P
filter
:
ΔP = ΔP
cake
(t) +ΔP
and the pressure drops through the
cake
(t) (32)
filter
Factoring the Darcy law [30] into Eq. (32) and rearranging, the instantaneous
filtrate flow rate is
dV
filt
=
αμ
l
2
A
where μ
dt
is the liquid viscosity, A is the filter cross-section, V
l
volume loaded in the carousel at every cycle, c
the slurry, R
is the filter mesh resistance, and V
m
ΔP
V
slurrycslurry
V
filt, final
, (33)
Rmμ
V +
slurry
filt, final
l
A
slurry
is the crystal concentration in
is the volume of filtrate at
is the slurry
the end of filtration, which is calculated with a mass balance:
V
filt, final
The integration of Eq. (33) assuming constant P yields the quadratic law for
V
(t):
filt
V
αμ
2 A
Imposing V
Δt
slurrycslurry
l
2
V
filtration
filt, final
equal to V
filt
μ
l
=
At the end of filtration and during all the subsequent carousel processing, H
and P
correspond to Eqs. (37) and (38), respectively,
cake
= V
slurry
2
V
(t) +
filt
filt, final
αV
slurrycslurryVfilt, final
1 −
R
mμl
A
in Eq. (35), t
2 A2ΔP
c
slurry
V
1 +
ρ
s
(t) − ΔP t = 0 (35)
filt
is obtained as
filtration
+
ε
1 +ε
μ
lRmVfiltrate,final
. (34)
AΔP
.
(36)
cake
262 D. J. Laky et al.
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V
H
cake
slurrycslurry
=
ρ
s
1 −ε)A
(
, (37)
. (38)
m
ΔP
cake
t ≥ Δt
filtration
= ΔP1 −
αH
cakeρs
R
m
1 − ε)+ R
(
At the end of filtration, cake deliquoring immediately starts, consisting of the
mechanical removal of the liquid retained in the pores under the action of P. Due
to capillary forces, there is a minimum pressure threshold P
to be applied to the
b
cake to initiate liquid removal. Because of capillarity, there is also an equilibrium
saturation of the cake S
can be carried out only through thermal drying. For this study, we calculate P
S
with literature equations [31] for cakes of mono-sized particles, to which we
∞
, at which deliquoring stops, and further cake desaturation
∞
and
b
introduce an additive hypothesis to account for the CSD in the cake as follows:
P
=f(d
S
∞
=
b
)
i
f(d
where σ is the liquid surface tension and N
2
3
ε
d
i
=
N
cap
4.6(1 − ε)σ
)
i
εd
i
0.1551 +0.031 N
is the capillary number, calculated as
cap
ρ
(
1 −ε
(
l
gH
+ ΔP
cake
2
H
)
cake
cake
σ
, (39)
−0.49
, (40)
cap
)
. (41)
The local velocity of the liquid u
[30]:
in which P
is the local pressure of the liquid and krlis the liquid relative cake
l
permeability. We calculate k
is given by the Darcy law for multiphase flow
l
dP
kk
rl
l
u
=−
l
μ
with the following relations [32]:
rl
= kS
k
rl
S −S
1 −S
∞
=
∞
S
=
R
, (42)
dz
l
2+3λ
, (43)
R
P
b
Pg− P
λ
, (44)
l
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where λ is the pore size distribution parameter (usually assumed equal to 5), S
is the local reduced saturation, and Pgis the local gas pressure. In Eq. (42), we
calculate P
during the process, with total gas pressure drop through the cake equal to P
through Eq. (44), assuming linear and constant gas pressure gradient
l
cake
(Eq. 32). With this assumption, we neglect the initial deliquoring transient for a fully
saturated cake, during which there is no gas at the outlet of the bed, to avoid solving
the gas mass balance and the Darcy law for the gas phase. This initial transient is
relatively fast compared to process dynamics and has little impact on the process.
We develop a one-dimensional dynamic model for deliquoring, along the cake
axial coordinate z. The liquid phase mass balance reads
=−
1εdu
l
. (45)
dz
∂S
∂t
We account for initial gradients in the liquid composition by including in the
model theliquid phase species mass balances, assuming absence of species diffusion
in the liquid:
∂c
i,l
∂t
=−
u
l
εS
∂c
i,l
, for i = 1,...,N
∂z
. (46)
L
The boundary conditions are
&
S(t,z = 0)= 0, ∀t>0,
∂c
t,z=0
(
)
i,l
∂z
= 0, ∀t>0.
(47)
R
The model of Eqs. (39)–(46)presents1+ N
(PDEs), which we semi-discretize with a high-resolution finite volume approach
[33] along the z dimension.
2.4 Thermal Drying Modeling
A one-dimensional dynamic model is developed for dead-end convective cake
drying, based on the following equations. The local drying rate ˙m
for the N
where h
volatile species still present in the liquid phase is [34]
L, vol
L→G
= h
˙m
i
is the mass transfer coefficient, calculated with correlations or from
M, i
M,i
experimental data; a is the cake specific surface, either computed as a = 6/d
d
is the Sauter diameter from the CSD) or measured; and for species i, P
p
aP
i,sat
− P
η
, for i = 1,...,N
i,g
i
partial differential equations
L
L→G
[kg/(m3s)]
i
, (48)
L,vol
(where
p
is the
i, sat
264 D. J. Laky et al.
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saturation pressure (from the Antoine equation), P
is the partial pressure, and ηiis
i, g
a factor accounting for mass transfer limitations occurring, mostly due to capillarity,
when the mass fraction of i in the cake (w
crit
(falling rate period). ηiis instead equal to one when w
w
i,cake
crit
w
(constant rate period). In the falling rate period, ηiis typically linearly or
i,cake
quadratically dependent on w
, and it should be estimated with experiments.
i, cake
) becomes lower than a critical value
i, cake
is greater than
i, cake
More than one falling rate period (and the corresponding critical solvent content)
can be identified for certain systems. The species mass balance with respect to the
concentration of i in the cake c
∂c
i,cake
=−˙m
∂t
i, cake
is
L→G
, for i = 1,...,N
i
. (49)
L,vol
Equation (49) is not solved for non-volatile species, as their concentration in
the cake does not vary during drying. In addition, in the equation, both c
L→G
˙m
are functions of time and z. The local cake saturation is related to the species
i
i, cake
and
concentrations in the cake with
N
L
%
c
i,cake/ρi,l
i=1
where ρ
S =
is the liquid density of pure i. The species mass balances in the gas phase
i, l
ε
for the volatile solvents and impurities read, in terms of mass fraction w
, for i = 1,...,N
, (50)
L
,
i,g
N
L
ρgε(1 −S
∂w
i,g
=−ρ
)
∂t
gug
∂w
∂z
i,g
L→G
+˙m
i
− w
L→G
˙m
i,g
i=1
, for i = 1,...,N
i
L,vol
(51)
where ρ
is the density of the gas and ugis the gas velocity, calculated with the
g
Darcy law for mono-phase gas flow in a porous medium [30].
Inter-phase energy transfer is very fast for the process, as found from experiments
on the carousel and literature correlations [35]. Hence, we assume local thermal
equilibrium among phases for the development of the differential energy balance:
ρ
scp,s
1 −ε)+ ρ
(
lcp,l
εS + ρgc
p,g
ε(1 −S
∂T
=−
∂t
i
)
L→G
˙m
i
− u
λ
i
gcp,gρg
∂T
∂z
+˙Q,
(52)
where ρ
the liquid phase specific heat, c
temperature, λ
is the liquid phase density, c
l
is the solid phase specific heat, T is the local
p, s
is the latent heat of vaporization of species i, and˙Q is the heat
i
is the gas phase specific heat, c
p, g
p, l
exchange with the environment, namely, the heat loss through the dryer walls
,
is
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