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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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336 M. A. Boojari et al.
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where F is the relative centrifugal radial force,r is the distance from the center of
rotation [cm], and N is the rotational speed [rpm]. In order to estimate the amount
of separated cells in a simplified procedure, a separation efficiency factor (X)is
implemented. This factor represents the mass fraction of biomass that is removed
from the feed. This method is effectively adopted in multiple industrial cases [64].
The selection of the separation efficiency factor is guided by literature research.
The value ranges from X = 90% to X = 98%; for the present case, the selected
factor is X =95%. Considering that it is required to obtain a biomass concentration
below the threshold of 30 mg/l, a single centrifuge is not sufficient. It has been
assumed that the centrifugation process influences only the biomass concentration;
moreover, consistent with the other process units, mixture density is considered as
constant. For simplicity, only the balances of the first centrifuge are reported below;
the balances of the second one are similar.
Q
− Q5= 0 (25)
6
1 −X)· Q
(
Q
T
Q
T
Q
T
· C
· C
· C
6
LO,6
LA,6
N,6
· C
X,6
− Q5· C
− Q5· C
− Q5· C
− Q5· C
LO,5
LA,5
N,5
= 0 (26)
X,5
= 0 (27)
= 0 (28)
= 0 (29)
7.2 Buffer Tank: P8/DCS10
Buffer tanks are the most straightforward solution in continuous manufacturing if
it is required to reduce the flow rate and concentration fluctuations. The capacity
of the tank is proportional to the degree of smoothing it can offer. Installation
of the tank before the nanofiltration process has been necessary also to allow
the membrane cleaning or replacement without shutting down the entire process.
The buffer tank volume capacity has to be large enough in order to smoothen
flow rate and concentration deviations, but on the other hand, the residence time
is constrained by the biological degradation of the product molecules. Lovastatin
biological degradation time is correlated to pH, temperature, by-products, and other
parameters [61] that in the current work have not been examined. In the absence
of experimental data, a reference liquid volume of 1000 L has been considered.
The total volumetric flow rate and component mass balances are expressed in Eqs.
(30)–(33).
dV
dt
= Q
8
− Q
7
(30)
Dynamic Modeling and Control of a Continuous Biopharmaceutical... 337
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Fig. 2 Generic concentration
profile of the solute in
nanofiltration membrane
processing
d(CLA.V
d(C
d(C
dt
LO
dt
dt
)
= C
.V
)
= C
.V
)
N
= C
LA,8
LO,8
N,8
· Q8− C
· Q8− C
· Q8− C
LA,7
L0,7
N,7
· Q
· Q
· Q
N
N
N
(31)
(32)
(33)
7.3 Nanofiltration: P11/NF10
It is necessary to concentrate as much as possible the API solution in order to reduce
the costs and the required capacity of the chromatography section. The API is a
minor constituent dissolved in a large water volume. Nanofiltration is the general
choice to concentrate the fermentation broth, especially in the pharmaceutical
industry [65]. The typical operating pressure is between 5 and 40 bar, and the
membrane pore size varies between 0.5 and 2 nm for a molecular weight cutoff
value of approximately 300–500 g/mol (lovastatin molecular weight is around
404.5 g/mol) [59]. The mass transfer process plays a key role in understanding the
mechanism of filtration, and two parameters characterize this process: the rejection
factor (Rej) of the solute and the permeance (Perm) of the solvent. The rejection
factor for a specific compound is given by
C
Rej
= 1 −
i
i,P E
C
i,f
(34)
where C
is the concentration of the solute in the permeate stream and C
i,p
concentration of the solute on the retentive side of the membrane, as reported in
Fig. 2.
i,f
is the
338 M. A. Boojari et al.
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Considering that the separation of components in this application is caused by
a steric exclusion mechanism, the model is based on a set of experimental Rej
values of comparable components on a commercial membrane—DOW NF90—with
a molecular weight cutoff of 300–500 g/mol. A strong assumption adopted in the
refence work is to maintain constant rejection factors in different operating conditions; for a more specific implementation, it would be necessary to retrieve different
Rej values depending on inlet stream concentration and operating conditions. In this
case, a pressure-dependent rejection factor is implemented for lovastatin, based on
experimental performance curves from literature [66, 67]. It is useful to introduce
the permeate volumetric flux Jw, which is generally estimated through the HagenPoiseuille equation adapted for membrane parameters [67]:
J
W
=
μ ·(Rm+ R
1
· ΔP (35)
)
CP
where μ is the feed’s dynamic viscosity, Rm is the membrane resistance, and RCP
is the concentration polarization boundary resistance. The membrane resistance Rm
is dependent on the effective pore radius, the effective membrane thickness, and the
effective porosity of the membrane.
In this case, the membrane selection is based on the work of Košuti´cetal.
[68] in which an analogous case of filtration for a similar component is developed
with a DOW NF90 membrane. From the same study, an experimental value for the
permeate volumetric flux Jw = 57.90 L/m
2
/h has been adopted as the steady-state
reference value, corresponding to a pressure gradient of P = 8 bar. The retentive
flow rate is typically 10–15% of the incoming flux [31]; with this target value, the
membrane surface (AM) is set to 0.398 m
viscosity μ and a membrane resistance R
2
. Assuming a constant feed dynamic
not dependent on a pressure gradient, it
m
is possible to define a resistance factor RF:
RF =(μ.Rm
−1
)
(36)
Peeva et al. formulated a model for continuous nanofiltration operations based on
a set of linear algebraic equations (pore flow mechanism) [69]. The model is applied
to the present study to simulate the behavior of lactose, nitrogen, and lovastatin
concentrations in the retentive and permeate streams.
The nanofiltration model equations reported below consist of an overall volumetric balance (Eq. 37), three mass component balances (Eqs. 38–40), the membrane
permeate equation (Eq. 41), and three concentration balances derived from the
rejection factor definition:
Q
· C
9
Q
· C
9
LO,9
LA,9
Q
− QPE− Q10= 0 (37)
9
− QPE· C
− QPE· C
LO,P E
LA,P E
− Q10· C
− Q10· C
= 0 (38)
LO,10
= 0 (39)
LA,10
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Q9· C
− QPE· C
N,9
Q
= AM· J
PE
N,PE
− Q10· C
W
= 0 (40)
N,10
(41)
C
LA,P E
= C
LA,10
·1 −Rej
LA
(42)
C
N,PE
= C
N,10
·1 −Rej
N
(43)
C
LO,P E
= C
LO,10
·1 −Rej
LO
(44)
7.4 Buffer Tank: PC13/DCS10
This tank accumulates the volume that is the fed in a discontinuous operation to
the chromatography section. It fills up constantly and it discharges a specific liquid
volume at regular intervals. It is a crucial element considering the high residence
time of the chromatography operation. The tank also allows to mitigate small
variations in the concentration of the feed, before the injection. The precautions
described for the T-01 tank are still valid. The total volumetric flow rate and
component mass balances for the tank are expressed as follows:
dV
= Q
d(C
d(C
d(C
LA
dt
LO
dt
dt
− Q
10
dt
.V
)
= C
LA,10
.V
)
= C
LO,10
.V
)
N
= C
N,10
11
· Q10− C
· Q10− C
· Q10− C
LA,11
L0,11
N10
· Q
· Q
· Q
H
H
H
(45)
(46)
(47)
(48)
7.5 Chromatography Columns – P16/C10
The concentrated stream produced from nanofiltration is stored in a buffer tank,
ready to be processed by chromatographic columns. This represents the last stage of
downstream processing and is referred to as the purification step, in which lovastatin
is separated from the other compounds. As lovastatin is a non-polar compound, the
reversed-phase HPLC (RP-HPLC)method was adopted for the presentcase study. In
order to obtain industrial-scale production, it is required to use multiple RP-HPLC
columns in parallel.
The implemented species mass balance is
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∂
∂t
x,t)=˙m
m
(
acc,i
x
conv,i
x,t)−˙m
(
x+dx
conv,i
x,t)+˙m
(
x
disp,i
x,t)+˙m
(
x+dx
disp,i
x,t)−˙m
(
x
mt,i
(
x,t
)
(49)
where m
m
mt,i
is a mass accumulation term, m
acc
is convective mass flow rate, and
conv,i
is mass transfer rate into particles. It is assumed that the total mass transfer
into the particle surface is equal to the accumulation of the ith component.
where m ¯m
acc, ads,i
∂
m
acc,ads,i
∂t
is the overall accumulation of the ith component in the
=˙m
mt,i
(50)
stationary phase.
The design and the count of the columns were developed with the following
constraints. It is assumed that the injected solvent flow rate is pumped at constant
Q
=20I/h, and the solvent injection time is retrieved as
S
V
inj
=
Q
S
[l] is injected solvent volume. Considering
inj
(51)
where t
[h] is injection time and V
inj
t
inj
a column with a volume capacity of 15 L, the working time of the developed
chromatography model takes around 12 h to achieve compound separation. Thus,
the aim of the sizing process is that the columns process the amount of liquid that is
expected to enter the buffer tank in 12 h. Each injection provides an overall volume
of 30.8 L that has to be distributed equally among the chromatography columns.
Utilizing columns with a capacity volume of 15 L, at least six HPLC columns are
required to process the injected liquid volume. Therefore, considering six HPLC
columns working in parallel, the steady-state injection time is around 0.26 h for
each column, and the overall injection time lasts around 1.54 h. The mobile phase
flow velocity u is given by the ratio between the solvent flow rate and the crosssection area A of the column:
where A
[m2] is the section area of HPLC column and u[m/h] is the mobile
HPLC
phase flow velocity.
8 Control Strategy
Every chemical process is a dynamic system that changes over time. To handle such
variations, the study and application of an effective control system is crucial. A
well-functioning control strategy is vital for the automated operation and monitoring
u =
A
Q
S
HPLC
(52)
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of complex technical processes. An effective process control strategy can improve
process safety and profitability.
In this section, a plant-wide control strategy has been developed for a continuous
biopharmaceutical manufacturing case. Continuous processing is gaining significant
ground in pharmaceutical drug development, as the cost benefits in many cases
outweigh the practical challenges. Moving to continuous manufacturing generally
requires more process knowledge, advanced monitoring, and control technologies
than batch processes. On the other hand, the potential benefits are increased
productivity, higher timeefficiency, and a reduction of both energy needsand overall
amount of waste.
As already stated, the basis of the current work is a steady-state model taken
from the literature of a pharmaceutical bioprocess designed for the production of
lovastatin [52], in which a plant-wide process synthesis has been undertaken and a
computer simulation has been developed in the MATLAB/Simulink programming
environment (Fig. 3). The challenge now is to design and implement an effective
multivariable feedback control strategy in this reference model, which should optimize the performance of the plant as a whole instead of isolated unit operations. The
questions to be answered concern what are the variables to measure, manipulate, or
regulate and how to achieve a robust control system for the pharmaceutical process
that correctly copes with external disturbances, uncertainties, and implementation
errors. A comprehensive control structure has been developed and implemented
into a computational testing model at a later stage in the next section. The control
structure strategy adopted in the current study relies on feedback control principles.
Fig. 3 Continuous open-loop benchmark simulation in Simulink
342 M. A. Boojari et al.
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Table 1 The values of the
stationary base case scenario
for the upstream process
(concentrations and
bioreactor volume)
Table 2 The values of the
stationary base case scenario
for the upstream process
(flow rates)
Concentration Biomass Lactose Adenine Lovastatin
CI[g/l] 0 20 4 0
C2[g/l] 36.71 7.35 1.49 0.1
C3[g/l] 106.35 14.81 2.52 1.2
C5[g/l] 10.64 14.81 2.52 1.2
CR[g/l] 170.16 14.81 2.52 1.2
CPE[g/l] 0 7.79 1.83 0.11
C00[g/l] 0 77.91 8.7 11
CSTR volume, L 5000
Q1[l/h] 35.60
Q2[l/h] 64.02
Q5[l/h] 25.63
QC[l/h] 38.42
QR[l/h] 9.96
QP[l/h] 28.50
QPE[l/h] 23.07
Q10[l/h] 2.56
8.1 Upstream Control Structure
Based on the values of the key parameters suggested in the optimal design
parameters evaluation carried out in the literature reference study [52], which
represents the starting point of the current work, an optimal steady-state base case
for the upstream process has been determined using the computational model. In
other words, a steady-state base case scenario has been achieved by setting the flow
rates Q
values, according to the reference study. The stationary values obtained for each
concentration and flow rate are listed in Tables 1 and 2, respectively.
and QPto the proper value to obtain the best possible key parameter
1,Q4,
Control Design and Tuning
The selection and implementation of a suitable control strategy for the upstream
process was pursued. The control system design has been preceded by degrees of
freedom (DOF) analysis of the upstream process in order to identify the variables
suitable tobe manipulated. The DOF is defined as the number of controlledvariables
that can be regulated by the control system [70]. In this framework, the number of
degrees of freedom thus corresponds strictly to the number of manipulated variables
that may be utilized in control loops.The DOF quantityis evaluated as thedifference
between the total number of variables and the number of chemical and physical
equations:
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Table 3 Independent
variables of the continuous
section
Stream Va ri a b l es
1 Q
2 Q2,C
3 Q3,C
5 Q5,C
C C
P Q
R Q
1
X,2,CLA,2,CN,2,CLO,2
LA,3,CN,3,CN,2,CLO,3
X,5
X,3
P
R
DOF = number of variables of the system–number of equations of the system.
All the upstream process units such as mixer, bioreactor, and hydro-cyclone rely
on one overall volumetric balance and four species balance equations. When these
equations are added together with the purge volumetric balance (Eq. 20), the total
number of equations results in 16 (from Eq. 5 to Eq. 20). Considering hydro-cyclone
dynamics – hydro-cyclone split ratio a=0.6 and hydro-cyclone separation factor
β=1.9 – two degrees of freedom are saturated. β value is proportional to the amount
of biomass that is present in the reactor at any time, therefore increasing the total
production of the molecule. Another effect is that increasing β reduces the amount
of biomass sent to the downstream section, easing the separation procedure.
Rearranging and modifying the equations by applying all the simplifications,
three hydro-cyclone balances turn into null identities. The number of equations is
therefore reduced to 13.
The total number of independent variables (reported in Table 3) considering the
bioreactor volume V is 16. Ultimately, the final DOF count is three, meaning that
there are three variables suitable to be manipulated (Q
controlled variables (V, C
, and RF) to be selected.
LOV,4
, and Q1) and as many
4,QP
The bioreactor has been equipped with a level controller, whose aim is to
maintain a stable amount of liquid in the vessel by manipulating the outgoing flow
rate. A Relative Gain Array (RGA) and the Niederlinski index (NI) [71] have then
been computed to obtain an appropriate pairing of the controlled and manipulated
variables for the design of two other control loops. Given the process transfer
functions, the RGA is used to obtain a tentative loop pairing for the decentralized
control system. For a 2 × 2 control system, the RGA reads
RGA =
λ
λ21λ
11λ12
,
22
where the columns and the rows correspond to the manipulated and controlled
variables, respectively. Pairing of a controlled variable with a manipulated variable
must be avoided if their corresponding relative gain λ
criterion favors pairing of a controlled and a manipulated variable with the relative
gain as close to 1 as possible in order to minimize the effects of loop interactions
is negative. The RGA
ij
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Table 4 Index notation for
the upstream manipulated and
controlled variables
Table 5 RGA and NI results
of the control loop pairing
configurations
Fig. 4 Complete control system design of the upstream process
Pairing RGA NI
Diagonal (u1-y1,u2-y2) λ11= λ22=−0.57 −1.75
Off-diagonal (u1-y2,u2-y1) λ12= λ21= 1.57 0.64
Manipulated variables Controlled variables
Q1Q
u1u
P
2
C
y
LOV,4
1
RF
y
2
on the control performance. The NI is then used to ascertain the stability of the
closed-loop system using the recommended RGA pairing. A positive NI provides
a sufficient condition for stability. The index notation used for these manipulated
and controlled variables is shown in Table 4, and the results of the two pairing
combinations are reported in Table 5.
Thus, off-diagonal pairing (u
1-y2,u2-y1
the upstream process. This means that product concentration C
by manipulating the purge flow rate Q
by manipulating the mixer inlet flow rate Q
) has been selected to be implemented in
is controlled
LOV,4
and the recirculation factor RF is controlled
P
. The comprehensive control structure
1
is represented in Fig. 4.
The parameters of each controller have been selected by relying on analytical
tuning rules (the SIMC rule [65] is used here) and through an integrated Simulink
tuning tool named “PID Tuner.” The devised process units and control structure
have been implemented into a computational simulation model, developed in the
MATLAB/Simulink environment. For implementing “PID Controller” blocks in the
computational model, controller parameters are specified as shown in Table 6.
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Table 6 Upstream controllers’ tuned parameters
Controlled Manipulated Controller parameters
variable variable Tuning method K
Bioreactor
level, V
Lovastatin
concentration,
C
LOV,4
Recirculation
factor, RF
Bioreactor
effluent rate, Q
Purge flow rate,
Q
P
Mixer inlet flow
rate, Q
1
SIMC −12.5 0.8 –
4
PID tuner −8.211 406.08 92.72(τ
PID tuner 0.047 0.1546 –
c
τI[h] τD[h]
D = 2.41
)
Table 7 The values of the
stationary base case scenario
for the downstream process
(concentrations and volume
capacities of buffer tanks)
Table 8 The values of the
stationary base case scenario
for the downstream process
(flow rates)
Concentration Cells Lactose Nitrogen Lovastatin
C5[g/l] 10.64 14.81 2.52 1.2
C6[g/l] 0.53 14.81 2.52 1.2
C7[g/l] 0.03 14.81 2.52 1.2
C9[g/l] – 14.81 2.52 1.2
C10[g/l] – 14.81 8.70 11
CPE[g/l] – 7.79 1.83 0.11
C11[g/l] – 77.91 8.7 11
Buffer tank “A” volume, L 1000
Buffer tank “B” volume, L 100
Q5[l/h] 25.63
Q6[l/h] 25.63
Q7[l/h] 25.63
Q8[l/h] 25.63
Q10[l/h] 2.57
QPE[l/h] 23.07
Q11[l/h] 20 fort ≤ t
inj
8.2 Downstream Control Structure
Starting from the upstream inputs, an optimal steady-state base case for the downstream process has been determined based on the considerations and assumptions
made throughout the downstream equipment’s modeling. The steady-state values
for the concentrations, flow rates, and buffer tank volumes are listed in Tables 7
and 8.
Control Design and Tuning
The design and implementation of a suitable control strategy for downstream
processing have been dealt with differently compared to the upstream process. Due
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