Abstract
In this study, an efficient systemwide controlsystem has been designed for the integrated continuous purification and processing of the active pharmaceutical ingredient (API). The control strategy is based on the regulatory PID controller which is most widely used in the manufacturing industry because of its simplicity and robustness. The designed control system consists of single and cascade (nested) control loops. The control system has been simulated in gPROMS^{TM} (Process System Enterprise). The ability of the control system to track the specified set point changes as well as to reject disturbances has been evaluated. Results demonstrate that the model shows an enhanced performance in the presence of random disturbances under closedloop control compared to an openloop operation. The control system is also able to track the set point changes effectively. This proves that closedloop feedback control can be used in improving pharmaceutical manufacturing operations based on the Quality by Design (QbD) paradigm.
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Acknowledgments
This work is supported by the National Science Foundation Engineering Research Center on Structured Organic Particulate Systems, through Grant NSFECC 0540855.
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Appendix and Mathematical Model
Appendix and Mathematical Model
The equations used for developing the mathematical model for each unit operation of the flowsheet have been listed.
Crystallizer
In this study, a 2D population balance model considering growth in two directions has been implemented for the crystallization process, as shown below.
The growth and the nucleation terms can be written as follows:
Another internal coordinate (Liq(L _{ eq })) is added to the PBE which tracks the amount of liquid present in the crystals based on the works of Miki et al. [42]. Considering the width of the crystal to be L _{1}, the equivalent length of the crystal can be expressed as follows:
The mass balance equation can be written as shown in the equation below [43]:
Since this is a case of cooling crystallization, the temperature cooling schedule can be expressed as a function of time. C _{ sat } has been expressed as a function of temperature. The expression has been obtained by fitting experimental data as obtained from BristolMyers Squibb Co., NJ.
The heat transfer between the coolant and crystallizer can be represented as follows:
The controller equations for the crystallization process are given below:

Master Controller:
$$\begin{array}{@{}rcl@{}} &&T(t)=K_{c}(Set Point(t)C_{sat}(t)) \notag \\ &&+\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)C_{sat}(t))dt \\ &&+K_{c}\tau_{D}\frac{d(Set Point(t)C_{sat}(t))}{dt}+c_{s} \notag \\ \end{array} $$(12) 
Slave Controller:
$$\begin{array}{@{}rcl@{}} &&T_{c}(t)=K_{c}(Set Point(t)T(t)) \notag \\ &&+\frac{K_{c}}{\tau_{I}}\int^{t}_{0}(Set Point(t)T(t))dt \\ &&+K_{c}\tau_{D}\frac{d(Set Point(t)T(t))}{dt}+c_s \end{array} $$(13)
Filtration Process
The main design equations of the cake filter is as follows (adapted from McCabe et al. [44]):
c and α can be found as follows:
Assuming that there are no solid particles present in the filtrate, the mass of wet cake deposited on the septum is given as follows:
Drying Process
For the drying process, a model has been developed where the liquid is being evaporated from the solid surface (adapted from Mezhericher et al. [45]).
The change in particle diameter with time can be represented as follows:
The temperature profile of particle can be given as shown below:
The evaporation rate can be calculated as follows:
The heat and mass transfer coefficients are given below:
such that Nusselts number is given as follows:
and Sherwood number is given as shown in Eq. 26.
The outflow from dryer is given as shown below:
The heat transfer between the air and superheated steam can be represented as follows:
The controller equation for drying can be given as follows:
Mixer
The mixing model has been assumed to be independent of size change based on previous work [46]. Hence the internal coordinates have been dropped from the population balance model. The PBM for the mixer can be written as follows:
The above equation can be written in multidimensional form as shown below:
The mass balance of a single component can be simplified according to the equation given below:
The properties of the final blend from the mixer output have been presented in terms of mean API composition (y _{ API }) relative standard deviation (RSD).
The controller equation for drying can be given as follows:
Nomenclature
Symbol
A _{ c } B _{0} b C C _{ sat } C _{ pw } D _{ p } F G _{1} G _{2} g _{1} g _{2} k _{ g1} k _{ g2} k _{ b } L _{1} L _{2} L _{3} M _{ w } T T _{ c } T _{ in } U 𝜖 ρ _{ avg } ρ _{ s } ΔP μ α A C _{ F } c ᅟ m _{ F } m _{ c } m _{ v } N _{ a } R _{ m } V V _{ p } A _{ s } c _{ ps } C _{ psteam } k h _{ fg } h T _{ s } T _{ p } T _{ g } T _{ in − steam } mass _{ out } M _{ s } Nu Pr Re Sh U x _{ p } x _{ eql } n n _{ max } t V _{ f } V _{ b } V _{ r } x x _{ max } y y _{ max } y _{ API } y _{ avg } y _{ i } 
Description
Area of heat transfer Primary nucleation term Kinetic parameter for crystallization Solute concentration in crystallization Saturation concentration of solute Specific heat constant for water Crytal diameter Particle density Growth rate Growth rate Kinetic parameter for crystallization Kinetic parameter for crystallization Kinetic parameter for crystallization Kinetic parameter for crystallization Kinetic parameter for crystallization Internal coordinate for length of solid Internal coordinate for length of solid Internal coordinate for length of liquid Cooling water flowrate Temperature (cooling schedule) Temperature of cooling water Inlet temperature of water Overall heat transfer coefficient porosity of cake Average density of wet particles Density of solid Filter pressure difference Fluid viscosity Specific cake resistance Filter surface area concentration of solutes in slurry Mass of solute deposited on filter per unit volume of filtrate Mass of wet cake Mass of dry cake Rate of evaporation during drying Avogadro number Filter medium resistance Filtrate volume Particle volume Area of heat transfer Specific heat capacity Specific heat constant for steam Mass transfer coefficient Specific heat of evaporation Heat transfer coefficient Temperature of steam Temperature of particle Drying gas temperature Inlet temperature of steam Outlet flowrate of API crystals from dryer Steam flowrate Nusselts Number Prandtl Number Reynolds Number Sherwood Number Overall heat transfer coefficient Liquid content of solid particle Liquid content of solid particle at equillibrium Counter for number of components Maximum number of components Time Forward axial velocity Backward axial velocity Radial velocity Spatial coordinate in axial direction Maximum number of axial compartments Spatial coordinate in axial direction Maximum number of radial compartments Fractional API composition at mixer outlet Average spatial composition of component A Composition of component A in ith compartment 
Units
m ^{2} particles/m ^{3}/s Dimensionless moles/m ^{3} moles/m ^{3} J/K μm particles/m ^{3} m/s m/s Dimensionless Dimensionless m/s m/s particles/m ^{3}/s m m m kg/s Kelvin Kelvin Kelvin W/m ^{2} − K Dimensionless kg/m ^{3} kg/m ^{3} kPa kg/m − s m/kg m ^{2} moles/m ^{3} kg/m ^{3} ᅟ kg kg kg/s Dimensionless 1/m m ^{3} m ^{3} m ^{2} J/kgK J/K m/s J/kg W/M ^{2} K Kelvin Kelvin Kelvin Kelvin kg/s kg/s Dimensionless Dimensionless Dimensionless Dimensionless W/m ^{2} − K Dimensionless Dimensionless Dimensionless [−] s m/s m/s m/s [−] [−] [−] [−] Dimensionless moles/m ^{3} moles/m ^{3} 
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Sen, M., Singh, R. & Ramachandran, R. SimulationBased Design of an Efficient Control System for the Continuous Purification and Processing of Active Pharmaceutical Ingredients. J Pharm Innov 9, 65–81 (2014). https://doi.org/10.1007/s1224701491736
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Keywords
 Process control
 Continuous processing
 Flowsheet simulation
 Powder mixing
 Pharmaceutical manufacturing
 Crystallization