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As process modernization and computerization continue, adoption of such digital
frameworks for new and improved manufacturing schemes is a key part to quantitatively optimal and qualitatively feasible operation.
References
1. FDA (2004) ‘Pharmaceutical cgmps for the 21 st century-a risk-based approach final report’.
2. Su, Q., Ganesh, S., Moreno, M., Bommireddy, Y., Gonzalez, M., Reklaitis, G. V., et al. (2019)
‘A perspective on Quality-by-Control (QbC) in pharmaceutical continuous manufacturing’,
Computers and Chemical Engineering. Elsevier Ltd, 125, pp. 216–231. doi: https://doi.org/
10.1016/j.compchemeng.2019.03.001.
3. FDA (2009) ‘Guidance for Industry Q8(R2) Pharmaceutical Development’. Available at: http:/
/www.fda.gov/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/default.htm.
4. Rojko, A. (2017) ‘Industry 4.0 concept: Background and overview’, International Journal of
Interactive Mobile Technologies. International Association of Online Engineering, 11(5), pp.
77–90. doi: https://doi.org/10.3991/ijim.v11i5.7072.
5. García-Muñoz, S. et al. (2015) ‘Definition of Design Spaces Using Mechanistic Models and
Geometric Projections of Probability Maps’, Organic Process Research and Development,
19(8), pp. 1012–1023. doi: https://doi.org/10.1021/acs.oprd.5b00158.
6. García-Muñoz, S., Dolph, S. and Ward, H. W. (2010) ‘Handling uncertainty in the
establishment of a design space for the manufacture of a pharmaceutical product’, Com-
puters and Chemical Engineering, 34(7), pp. 1098–1107. doi: https://doi.org/10.1016/
j.compchemeng.2010.02.027.
7. Ochoa, M. P. et al. (2019) ‘Flexibility Analysis For Design Space Definition’, in Computer
Aided Chemical Engineering. Elsevier Masson SAS, pp. 323–328. doi: https://doi.org/10.1016/
B978-0-12-818597-1.50051-5.
8. Boukouvala, F., Muzzio, F. J. and Ierapetritou, M. G. (2010) ‘Design space of pharmaceutical
processes using data-driven-based methods’, Journal of Pharmaceutical Innovation, 5(3), pp.
119–137. doi: https://doi.org/10.1007/s12247-010-9086-y.
9. Wang, Z., Escotet-Espinoza, M. S. and Ierapetritou, M. (2017) ‘Process analysis and
optimization of continuous pharmaceutical manufacturing using flowsheet models’, Comput-
ers and Chemical Engineering. Elsevier Ltd, 107, pp. 77–91. doi: https://doi.org/10.1016/
j.compchemeng.2017.02.030.
10. Bano, G. et al. (2019) ‘Design space maintenance by online model adaptation in pharmaceutical manufacturing’, Computers and Chemical Engineering, 127, pp. 254–271. doi: https://
doi.org/10.1016/j.compchemeng.2019.05.019.
11. Laky, D. et al. (2019) ‘An optimization-based framework to define the probabilistic design
space of pharmaceutical processes with model uncertainty’, Processes, 7(2). doi: https://
doi.org/10.3390/pr7020096.
12. Process Systems Enterprise (2020) Process Systems Enterprise, gPROMS. Available at:
www.psenterprise.com/products/gproms (Accessed: 24 December 2020).
13. Casas-Orozco, D. et al. (2021) ‘PharmaPy: an object-oriented tool for the development of
hybrid pharmaceutical processes’, In Review.
14. Yu,L. X. et al. (2014) ‘Understanding pharmaceutical qualityby design’, AAPS Journal, 16(4),
pp. 771–783. doi: https://doi.org/10.1208/s12248-014-9598-3.
15. National Institute of Standards and Technology (2015) Libro del Web de Química del NIST
Available at: http://webbook.nist.gov/chemistry/ (Accessed: 20 May 2015).
16. Picciochi, R., Diogo, H. P. and Minas Da Piedade, M. E. (2010) ‘Thermochemistry of
paracetamol’, Journal of Thermal Analysis and Calorimetry, 100(2), pp. 391–401. doi: https:/
/doi.org/10.1007/s10973-009-0634-y.
.
286 D. J. Laky et al.
https://t.me/medicina_free
17. Gunawan, R., Fusman, I. and Braatz, R. D. (2004) ‘High resolution algorithms for multidimensional population balance equations’, AIChE Journal, 50(11), pp. 2738–2749. doi: https://
doi.org/10.1002/aic.10228.
18. LeVeque, R. J. (2002) Finite Volume Methods for Hyperbolic Problems. New York: Cambridge
University Press.
19. Rachah, A. et al. (2016) ‘A mathematical model for continuous crystallization’, Mathemat-
ical Methods in the Applied Sciences, 39(5), pp. 1101–1120. doi: https://doi.org/10.1002/
mma.3553.
20. Rawlings, J. B., Miller, S. M. and Witkowski, W. R. (1993) ‘Model Identification and Control
of Solution Crystallization Processes: A Review’, Industrial and Engineering Chemistry
Research, 32(7), pp. 1275–1296. doi: https://doi.org/10.1021/ie00019a002.
21. Szilagyi, B., Majumder, A. andNagy, Z. K. (2020) ‘Fundamentals of PopulationBalance Based
Crystallization Process Modeling’, in The Handbook of Continuous Crystallization.Cambridge: Royal Society of Chemistry, pp. 51–101. doi: https://doi.org/10.1039/9781788013581-
00051.
22. Nagy, Z. K. et al. (2008) ‘Determination of the kinetic parameters for the crystallization of
paracetamol from water using metastable zone width experiments’, Industrial and Engineering
Chemistry Research, 47(4), pp. 1245–1252. doi: https://doi.org/10.1021/ie060637c.
23. Lee, T., Lin, H. Y. and Lee, H. L. (2013) ‘Engineering reaction and crystallization and
the impact on filtration, drying, and dissolution behaviors: The study of acetaminophen
(paracetamol) by in-process controls’, Organic Process Research and Development, 17(9), pp.
1168–1178. doi: https://doi.org/10.1021/op400129n.
24. Borsos, A., Majumder, A. and Nagy, Z. K. (2016) ‘Multi-Impurity Adsorption Model for Modeling Crystal Purity and Shape Evolution during Crystallization Processes in Impure Media’,
Crystal Growth & Design, 16(2), pp. 555–568. doi: https://doi.org/10.1021/acs.cgd.5b00320.
25. Kubota, N. and Mullin, J. W. (1995) ‘A kinetic model for crystal growth from aqueous solution
in the presence of impurity’, Journal of Crystal Growth, 152(3), pp. 203–208. doi: https://
doi.org/10.1016/0022-0248(95)00128-X.
26. Yu, A. B., Zou, R. P. and Standish, N. (1996) ‘Modifying the linear packing model for predicting the porosity of nonspherical particle mixtures’, Industrial and Engineering Chemistry
Research, 35(10), pp. 3730–3741. doi: https://doi.org/10.1021/ie950616a.
27. Zou, R. P. and Yu, A. B. (1996) ‘Evaluation of the packing characteristics of mono-sized nonspherical particles’, Powder Technology, 88(1), pp. 71–79. doi: https://doi.org/10.1016/0032-
5910(96)03106-3.
28. Bourcier, D. et al. (2016) ‘Influence of particle size and shape properties on cake resistance
and compressibility during pressure filtration’, Chemical Engineering Science. Elsevier, 144,
pp. 176–187. doi: https://doi.org/10.1016/j.ces.2016.01.023.
29. Huggins, S., Cosbie, A. and Gaertner, J. (2019) ‘Filtration case studies’, Chemical Engineering
in the Pharmaceutical Industry: Active Pharmaceutical Ingredients. Wiley Online Library, pp.
833–845.
30. Muskat, M. and Meres, M. W. (1936) ‘Theflow of heterogeneous fluids through porous media’,
Physics. American Institute of Physics, 7(9), pp. 346–363.
31. Wakeman, R. J. (1976) ‘Vacuum dewatering and residual saturation of incompressible filter
cakes’, International Journal of Mineral Processing. Elsevier, 3(3), pp. 193–206.
32. Wakeman, R. J. (1979) ‘Low-pressure dewatering kinetics of incompressible filter cakes,
I. Variable total pressure loss or low-capacity systems’,
Processing, 5(4), pp. 379–393. doi: https://doi.org/10.1016/0301-7516(79)90046-2.
33. Van Leer, B. (1974) ‘Towards the ultimate conservative difference scheme. II. Monotonicity
and conservation combined in a second-order scheme’, Journal of computational physics.
Academic Press, 14(4), pp. 361–370.
34. Burgschweiger, J. and Tsotsas, E. (2002) ‘Experimental investigation and modelling of continuous fluidized bed drying under steady-state and dynamic conditions’, Chemical Engineering
Science, 57(24), pp. 5021–5038. doi: https://doi.org/10.1016/S0009-2509(02)00424-4.
International Journal of Mineral
Integrated Synthesis, Crystallization, Filtration, and Drying of Active... 287
https://t.me/medicina_free
35. Bird, R. B., Lightfoot, E. N. and Stewart, W. E. (1960) Transport Phenomena. John Wiley &
Sons.
36. Ellis, F. (2002) ‘Paracetamol – a curriculum resource’ Royal Society of Chemistry
37. Bates, D. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications.New
York: John Wiley & Sons.
38. Bilardello, P. et al. (1993) ‘A general strategy for parameter estimation in differential—
algebraic systems’, Computers & Chemical Engineering, 17(5–6), pp. 517–525. doi: https://
doi.org/10.1016/0098-1354(93)80040-T.
39. Bard, Y. (1974) Nonlinear Parameter Estimation, Nonlinear Parameter Estimation. London.:
Academic Press.
40. Casas-Orozco, D. et al. (2018) ‘Dynamic parameter estimation and identifiability analysis
for heterogeneously-catalyzed reactions: Catalytic synthesis of nopol’, Chemical Engineering
Research and Design. Institution of Chemical Engineers, 134,pp. 226–237. doi:https://doi.org/
10.1016/j.cherd.2018.04.002.
41. Wächter, A. and Biegler, L. T. (2006) ‘On the implementation of an interior-point filter linesearch algorithm for large-scale nonlinear programming’, Mathematical Programming, 106(1),
pp. 25–57. doi: https://doi.org/10.1007/s10107-004-0559-y.
42. Rapin, J. and Teytaud, O. (2018) ‘Nevergrad - A gradient-free optimization platform’, GitHub
repository.GitHub.
43. Destro, F., I. Hur, V. Wang, M. Abdi, X. Feng, E. Wood, S. Coleman, P. Firth, A. Barton,
M. Barolo, Z. K. Nagy (2021). Mathematical modeling and digital design of an intensified
filtration-washing-drying unit for pharmaceutical continuous manufacturing. Chemical Engineering Science, 244, 116803
44. Liu, Y. C. et al. (2019) ‘Development of Continuous Filtration in a Novel Continuous
Filtration Carousel Integrated with Continuous Crystallization’,Organic Process Research and
Development, 23(12), pp. 2655–2665. doi: https://doi.org/10.1021/acs.oprd.9b00342.
45. Destro, F., I. Hur, V. Wang, M. Abdi, X. Feng, E. Wood, M. Barolo, Z. K. Nagy (2020). Digital
design of an intensified filtration-drying unit for pharmaceutical upstream manufacturing.
Presented at: the 2020 Virtual AIChE Annual Meeting, November 16-20.
Fast Model Predictive Control of
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Modular Systems for Continuous
Manufacturing of Pharmaceuticals
Anastasia Nikolakopoulou, Matthias von Andrian, and Richard D. Braatz
1 Introduction
Pharmaceuticals have been traditionally manufactured using batch processing. The
potential for reducing drug costs, production times, waste material, and product
quality variations while providing the ability to respond to abrupt changes in
demand has motivated research efforts in academia and industry over the last decade
to develop continuous-flow processes for pharmaceutical manufacturing [1–3]. Of
special interest is end-to-end synthesis in compact modular reconfigurable systems
for on-demand continuous-flow manufacturing, which refers to the integration of
multiple molecular synthesis and separation steps in series, starting with simple
inexpensive molecules and continuously going through all of the manufacturing
steps of the product. Such modular systems can substantially decrease manufacturing times, while reducing the potential for supply chain disruptions by enabling
spatially localized on-demand production [4].
A compact modular continuous manufacturing platform has been developed
at MIT where both synthesis and final drug formulation are combined [4]. This
refrigerator-sized system allows for multistep synthesis, in-line purifications, semibatch crystallization, and real-time process monitoring. It has lower level regulatory
control systems that are designed to maintain local state variables at specified
setpoint values. An in-line attenuated total reflection (ATR) Fourier transform
infrared (FTIR) system was used to monitor the formation of the active pharmaceutical ingredients (APIs) in real-time. Other process parameters monitored through
sensors, a data acquisition device (DAQ), and LabVIEW (National Instruments)
were pressure, reactor temperature, and flowrates. More recently, artificial intelli-
A. Nikolakopoulou · M. von Andrian · R. D. Braatz ()
Massachusetts Institute of Technology, Cambridge, MA, USA
e-mail: anikol@mit.edu; matthias.von.andrian@gmail.com; braatz@mit.edu
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
A. Fytopoulos et al. (eds.), Optimization of Pharmaceutical Processes, Springer
Optimization and Its Applications 189, https://doi.org/10.1007/978-3-030-90924-6_11
289
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Fig. 1 Hierarchy of plant-wide optimization and control. In (a), bottom: a small-scale upstream
atropine synthesis continuous manufacturing plant has sensors operating in the measurement level,
middle: a PID control scheme is used at the regulatory level, top: MPC acts on the supervisory
control level. In (b), optimization and control of processes follow a hierarchy based on information
flow [6]
gence and robotics have been employed to design synthetic routes by generalizing
previously published chemical reactions and consequently execute chemical recipe
files in a robotically reconfigurable flow chemistry platform [5].
Traditionally, in the chemical industry, the regulatory control layer is part of
a hierarchical scheme with higher level controllers and optimization layers above
[6] (see Fig.1). The regulatory layer involves proportional-integral-derivative (PID)
controllers tuned using methods such as internal model control [7] and operating
at fast time scales. The aim of the supervisory control level is to realize plant-wide
objectives determined by a top level optimization. At the supervisory control level,
more sophisticated control technologies such as model predictive control (MPC)
that can handle input, state, and output constraints are often adopted [8]. MPC
describes a class of algorithms that use a process model to predict the future plant
behavior given past, present, and future control actions. The algorithm optimizes
the future plant behavior subject to the sequence of future control actions at every
control interval. The first control action of the sequence is implemented in the plant,
and the optimization is solved again for the subsequent control interval.
MPC is the natural framework for the automation and control of continuous
pharmaceutical processes. An advantage of using MPC is the ability to impose
constraints in the manipulated variables (i.e., pump magnitudes and rate of change
limitations) and the controlled variables (i.e., impurity content [9]). The inputs
to the MPC layer are the continuous pharmaceutical plant’s measured variables
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and overall plant-wide control objectives. Using plant-wide economic objectives
such as the overall yield or production rate as the MPC controlled variables
for pharmaceutical manufacturing plants has been previously explored [10]. The
outputs of the MPC layer are the manipulated variables which are often mass
flowrates. These flowrates constitute setpoints to lower level regulatory control
systems. Together, the regulatory control layer and MPC layer form the plant-wide
control system (e.g., [11]).
The MPC technology that is most widely used in the manufacturing industries is
dynamic matrix control (DMC) [12], which is an algorithm that solves an online
optimization at each control interval subject to operational constraints based on
an input-output (IO) model for the process (e.g., [13–16]). DMC employs a finite
step response model to make the predictions needed for computing the optimization
objective. This approach has the advantage of making the online computational cost
a function of the number of inputs of the manufacturing system as opposed to the
number of states, which can easily be in the hundreds to thousands or more for an
end-to-end manufacturing plant. Quadratic dynamic matrix control (QDMC) is a
variation of the DMC algorithm that uses a quadratic rather than a linear control
objective [17], is easier to tune and has been used in closed-loop simulation studies
of continuous pharmaceutical manufacturing plants [9, 10, 18]. The wide usage of
DMC in the chemical industry facilitates its use in the pharmaceutical industry,
especially for small-molecule drugs, which are made by chemical synthesis.
A step response model can be constructed via system identification, specifically
by perturbingthe manipulated variables and recording the controlled variables (plant
outputs) behavior. A plant-wide model can be used in place of the actual plant to
enable step response model construction before experimental data is available, e.g.,
shortly after the manufacturing plant starts up. The parameters in the first-principles
unit operations models can be identified in a much smaller scale system offline (e.g.,
a droplet-based system [19]). For the larger scale compact modular system, the same
first-principles process model can be used but with larger geometric parameters and
flowrates associated with the higher production rates. Using the same first-principles
model equations for the different scales are enabled by the relatively small volumes
at both scales; atthese scales, the unit operations have fluid flows that are predictable
and nearly ideal. Even with the simplified fluid flows, the first-principles model
for the entire manufacturing plant has hundreds to thousands of states, due to the
number of components and the numerics of unit operations (i.e., discretization of
tubular reactors).
First-principles models for modular systems usually involve many tightly coupled partial and ordinary differential-algebraic equations (PDAEs/DAEs). For
example, the widely applied numerical method of lines [20] involves spatial
discretization of the PDAEs, resulting in a sparse system of DAEs with hundreds
to tens of thousands of state variables. These models are commonly referred to
as singular systems or descriptor systems in the control literature [21, 22]. Building first-principles plant-wide models introduces opportunities for optimizing the
pharmaceutical plant operation. Many continuous pharmaceutical manufacturing
processes are designed to operate over short operation times, which makes their
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operation more strongly impacted by unproductive dynamical plant operations (i.e.,
startup and shutdown). Therefore, methodologies that address the computation of
optimal dynamical operations need to be explored.
Dynamic optimization (DO) is a widely known methodology for obtaining optimal trajectories for a dynamical system by optimizing a cost objective with respect
to certain operating conditions (inputs, states) and constrained by equations that
describe the dynamical operation (such as ODEs, DAEs, PDEs) [23]. This type of
description is general enough to encompass a vast range of problem characteristics
(integer-valued decision variables, multi-point constraints, uncertainty) and it is
for the same reason that ubiquitous software solutions are difficult to implement
[23]. The DOs are infinite-dimensional problems in their original formulation
and there are many approaches for numerically solving these infinite-dimensional
problems. Dynamic programming [24, 25] provides a mathematical formulation for
computing the global optimum, but is very computationally expensive. Pontryagin’s
maximum principle [26] is a less expensive indirect approach, but may produce a
solution that is only locally optimal. Limitations in these two approaches, related to
treating problems with high state dimension for the former and treating inequality
constraints for the latter, gave rise to the development of the so-called direct
methods. In direct methods, the trajectory is parametrized and a nonlinear program
(NLP) is formulated. Direct methods usually adopt one of three classes of numerical
approaches: (1) the sequential approach, in which the time-varying input vector is
parameterized in terms of a finite number of parameters to produce an optimization
in which the process model appears as a constraint in the form of a DAE system,
(2) the simultaneous approach, in which both the input and state variables are
discretized to produce a nonlinear algebraic optimization [27], and (3) the direct
multiple shooting method which is a hybrid approach where the state trajectory is
partially eliminated from the NLP [28].
For the DO of startup of a continuous pharmaceutical manufacturing plant,
the sequential method which deals with the DAE constraints by direct numerical
simulation has been previously adopted resulting in a nonlinear program in which
the only optimization variables are parameters that specify the input vector [18].
This approach is commonly used for this type of application [29–31].
When disturbances and model uncertainties are neglected, DO is a computationally tractable approach for DAE models with tens of thousands of states, since
it can be solved offline and then have its optimal trajectory implemented online.
During the first startup, the effects of model uncertainties can be significant when
modeling modular systems for continuous-flow pharmaceutical manufacturing, as
experimental data from the system are not yet available. In such a situation, feedback
control is needed to reduce the effects of model uncertainties. Modular systems
are highly nonlinear, which motivates the use of nonlinear model predictive control
(NMPC) to suppress the effects of model uncertainties on operations and product
quality by using nonlinear model predictions and real-time measurements to identify
the optimal control inputs and then re-solving the arising nonlinear program at
discrete time instances. NMPC has been implemented in some cases in fairly large
scale systems [32].
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The application of NMPC based on first-principles models during the startup
of continuous-flow pharmaceutical manufacturing is challenging, however, as the
models typically have highly nonlinear dynamics and thousands to tenths of
thousands of states. Then the arising NLPs are usually nonconvex and have too
high of a computational cost to be used in real-time implementation. On the other
hand, implementing an optimal policy in an open-loop scheme has the disadvantage
that the process disturbances and model uncertainties will result in a suboptimal
operation and potentially off-spec product. As such, the closed-loop control of
dynamical operations must be addressed, to suppress the effects of disturbances and
model uncertainties on product quality. Government regulatory agencies are more
likely toapprove an industrial drug production facilitythat uses control software that
has undergone extensive testing and implementation. This regulatory consideration
results in a strong preference in the pharmaceutical industry to use linear MPC
(LMPC) over NMPC. NMPC software solutions have so far been implemented in
applications limited in size and scope compared to LMPC, partly due to the much
higher computational complexity of NMPC and partly because LMPC provides
adequate closed-loop performance in most supervisory control applications [8].
This chapter discusses how informed construction of IO models for QDMC can be
leveraged to successfully control dynamical regions of the operation. LMPC such as
QDMC can be combined with polynomial chaos theory to give fast stochastic MPC
formulations to rigorously address parametric uncertainty [33, 34].
The rest of this chapter describes methodologies for dynamic optimization
and QDMC implementation for modular systems with a high state dimension
and significant nonlinear behavior. These methodologies are demonstrated in a
computational case study for a compact modular plant for the continuous upstream
manufacturing of atropine [35].
2 Modeling, Control, and Optimization of Modular Systems
This section describes a plant-wide model developed for a compact modular
reconfigurable system for continuous-flow pharmaceutical manufacturing, discusses
the supervisory control design methodology, and presents the dynamic optimization
formulation to determine optimal dynamical operations such as startup.
2.1 Process Flowsheet
The plant-wide model is constructed in the context of a case study for the
upstream synthesis of atropine, a central nervous system depressant, which can treat
certain types of nerve agents, pesticide poisonings, certain types of slow heart rate
conditions, and can decrease saliva production during surgery [35].
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Fig. 2 Upstream atropine synthesis flowsheet showing three mixers, three tubular reactors, and
one liquid-liquid separator. The mass flowrates of the six feed streams u
variables, and the T
are reactor temperatures controlled to a setpoint by resistive heating [18]
i
are the manipulated
i
The process flowsheet in Fig.2 shows the unit operations, which are three mixers,
three tubular reactors, and a liquid-liquid separator, as well as the manipulated
volumetric flowrates (u
in the first two reactors (T
), and potential disturbances which are the temperatures
i
). The flowrates were manipulated through positive
i
displacement pumps controlled in LabVIEW. The temperatures of the reactors 1
and 2 (Fig. 2) are held constant by keeping a jacket at a constant temperature using
resistances.
The inputs to the atropine synthesis simulation are the volumetric flowrates of
six feed streams:
• Tropine in dimethylformamide (u
• Phenylacetyl chloride (u
• Formaldehyde (u
3
• Sodium hydroxide (u
• Buffer solution (u
5
• Organic solvent (u
)
2
)
)
4
)
).
6
)
1
Each of the streams contains one or more of the fourteen species in the simulation.
Streams 1 and 2 are mixed in the first mixer, whose outlet flow connects to the
first tubular reactor to produce a reaction intermediate in solution. That reactor outlet
flow is then mixed with Streams 3 and 4 in the second mixer and directed to a second
tubular reactor. The outlet flow of that chemical reactor is mixed with streams 5 and
6 in a third mixer, and subsequently sent to a packed bed. In the last unit operation
a liquid-liquid separation is used to extract atropine in the aqueous phase.
A plug-and-play software module was developed for each unit operation.
Modules can be automatically selected and interconnected to construct a dynamic
model for the entire plant from the process flowsheet (Fig. 2). Each software module
contains first-principles model equations for the associated unit operation, where
“first-principles” refers to mass conservation equations, reaction stoichiometry, and
reaction kinetics. Each tubular reactor was held at uniform, constant temperature
by a heated jacked. Due to the high surface-to-volume ratio of the tubular reactors,
the inlet flows to the reactors quickly reach the jacket temperature, so distributed
parameter models for the energy balances were not needed and the temperature of
each reactor was considered essentially equal to its jacket temperature. The temperature dynamics for the jackets were too slow to be used in the regulatory controls,
so the temperature setpoints were determined by the higher level optimization.
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2.2 Mathematical Description of Modules
Below is a description of the mathematical models for each module.
Mixer
Each mixer in the system is in-line, has very small volume, and is designed to
have intensive mixing. The mixing of multiple streams is described by a mass
conservation equation without accumulation for each chemical species,
n
s
=
˙m
˙m
out,i
k=1
, for i = 1,...,nc,
in,i,k
(1)
where ˙m
is the mixer outlet mass flowrate of species i , ˙m
out,i
flowrate of species i in the inlet flow k, n
mixer, and n
is the number of species. Some associated equations that describe
c
is the number of streams directed to the
s
is the inlet mass
in,i,k
variables of interest within each mixer unit are the total outlet mass flowrate,
n
c
˙m
out, tot
=
˙m
, (2)
out,i
i=1
and the molar concentration of species i,
=
c
out,i
˙m
out, totMi
where ρ is the solution volumetric mass density and M
, for i = 1,...,nc, (3)
is the molar mass of species
i
ρ
˙m
out,i
i. The solution volumetric mass density is calculated assuming additive volumes,
where x
ρ =
is the mass fraction of species i in the stream of interest, and ρithe
i
n
i=1
−1
c
x
i
, (4)
ρ
i
volumetric mass density of species i. This assumption is accurate for ideal solutions
or for completely immiscible nonreacting mixtures. To simplify the nomenclature,
the integer index that refers to each mixer is not shown. The above equations are
replicated for every mixer, with the respective input and output variables.
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