Quality-by-control of intensified continuous filtration-drying of active pharmaceutical ingredients

Abstract

Continuous manufacturing and closed-loop quality control are emerging technologies that are pivotal for next-generation pharmaceutical modernization. We develop a process control framework for a continuous carousel for integrated filtration-drying of crystallization slurries. The proposed control system includes model-based monitoring and control routines, such as state estimation and real-time optimization, implemented in a hierarchical, three-layer quality-by-control (QbC) framework. We implement the control system in ContCarSim, a publicly available carousel simulator. We benchmark the proposed control system against simpler methods, comprising a reduced subset of the elements of the overall control system, and against open-loop operation (the current standard in pharmaceutical manufacturing). The proposed control system demonstrates superior performance in terms of higher consistency in product quality and increased productivity, proving the benefits of closed-loop control and of model-based techniques in pharmaceutical manufacturing. This study represents a step forward toward end-to-end continuous pharmaceutical processing, and in the evolution of quality-by-design toward quality-by-control.

1 |. INTRODUCTION

1.1 |. Pharmaceutical regulatory background: toward quality-by-control

In recent years, pharmaceutical development and manufacturing have been undergoing a modernization trend, aimed at increasing the economic efficiency and the capability to attain and consistently maintain product quality. These efforts have been promoted by regulators such as the United States Food and Drug Administration (FDA) and the European Medicines Agency, through the process analytical technology¹ and quality-by-design^2–5 (QbD) initiatives. Under a QbD approach, quality is inherently built into the product, by designing and operating the manufacturing process with a science- and risk-based approach. QbD represents a major improvement with respect to the traditional quality-by-testing approach (QbT), based on extensive testing on the end-product. Following a typical QbD pharmaceutical development workflow,⁵ the product critical quality attributes (CQAs) are first identified from the quality target product profile. Subsequently, the critical process parameters (CPPs) and the raw materials properties (critical material attributes, CMAs) that are critical for product CQAs attainment are established through a quality risk assessment procedure. Then, the design space (DS) is optionally determined as the multivariable region of CMAs and CPPs that allow ensuring the target product CQAs. Finally, a control strategy is designed as a set of actions, routines, and controls to maintain the process in a state of control and guarantee the product quality. A guidance by FDA distinguishes among three levels of control strategies^6,7 (Figure 1). A Level 3 control strategy, corresponding to the QbT approach, consists in operating the process under tight intervals of CMAs and CPPs, which have been proven to yield the target product CQAs in the regulatory approval application. In a more advanced Level 2 control strategy, a DS is established, and the process is operated therein. Level 2 control strategies guarantee more flexibility on the operating conditions than Level 3 strategies, and allow reducing the reliance on end-product testing. In a Level 2 control strategy, the product CQAs are still controlled at open-loop, by moving the CPPs within the DS in response to registered changes in the CMAs. Level 2 and Level 3 control strategies, both controlling quality at open-loop, are the current standard in pharmaceutical manufacturing.⁷

FIGURE 1.

Three levels of control strategies for a pharmaceutical process. A Level 1 control strategy is made up of multiple layers. The figure is taken from Destro and Barolo⁷

In a Level 1 control strategy, instead, the product quality is actively controlled by making use of process control techniques. The CPPs are automatically adjusted in response to measured changes of the CMAs (feedforward control) or of the CQAs (feedback control). This is achieved through proportional-integral-derivative (PID) control or more advanced techniques, such as model predictive control and real time optimization (RTO).⁸

Although both Level 1 and Level 2 control strategies include QbD elements, the ultimate aim of QbD is to reach widespread establishment of Level 1 control strategies, from which both pharmaceutical companies and patients can benefit.^9,10 Active process control (also known as closed-loop process control in standard engineering term) on CQAs, or measurements directly related to this, can reduce the occurrence of shortages and recalls, and can provide more consistent quality attainment than traditional open-loop operation.¹¹ McKinsey¹² even estimated that introducing an advanced quality control system for process development and manufacturing could reduce the product launch time by up to 30%, with a very strong impact on profit of pharmaceutical companies.¹⁰

Even though the adoption of Level 1 control strategies is still in embryonic phase, many recent academic publications^7,13–15 demonstrated the advantages of closed-loop quality control in different pharmaceutical processes, including plant-wide applications.¹⁶ The recent interest toward control strategies comprising elements of active process control has led to the establishment of a novel paradigm in pharmaceutical development and manufacturing, which has been named quality-by-control (QbC).¹⁷ QbC should be seen as an evolution of QbD, rather than a QbD-independent initiative, in which active process control represents the core feature of the control strategy. Upon QbC implementation, feedback control strategies are employed to obtain the DS under closed-loop conditions, rather than simply used to maintain the process operation within the DS determined a priori via open-loop design of experiments, as in the QbD approach. A hierarchical structure, following the ISA-95 Enterprise-Control System Integration Standard, has recently been proposed to support the development of QbC control systems.^17,18 The hierarchical structure includes three layers (Figure 1): (i) Layer 0, consisting of the built-in control systems of the equipment that controls some of the CPPs, (ii) Layer 1, that uses PAT tools to close the loops on CQAs, typically using simple feedback control strategies, such as PID control, and (iii) Layer 2, which features advanced model-based process control and process monitoring techniques.

In addition to active quality control, the transition to a more continuous production mode is another pharmaceutical emerging technology promoted by the QbD initiative.^19,20 Continuous processing, when feasible, is preferred to the traditional batch production mode, due to the many advantages it offers.^10,21,22 As for active process control, the benefits of continuous processing are for both manufacturers and patients, and include reduced manufacturing time and cost, greater product quality consistency, and potential to reduce recalls and shortages. Furthermore, process control is usually easier in a continuous plant, and has been explored more in the literature for continuous processes than for their batch counterparts.

A strong inter-dependence exists between the two most important pharmaceutical emerging technologies: QbC is gaining much relevance under the increasing popularity of continuous processes, while the transition to more continuous operation modes is boosted by the QbC tools for systematic control system design. Several contributions on continuous processing and QbC have been published around the topics of reacting systems,^23,24 crystallization,^25–27 and of solid-dosage form manufacturing lines.^28,29 However, end-to-end continuous pharmaceutical processes with active quality control have been somewhat overlooked.^13,16 Consider for example filtration, washing and drying of crystallization slurries. They are pivotal unit operations for connecting the drug substance and drug product manufacturing sections of a pharmaceutical process into an end-to-end continuous integrated system. However, although they can constitute a bottleneck in the implementation of plant-wide continuous automated pharmaceutical processes, they have been scarcely studied from a continuous integrated processing and QbC perspective.

1.2 |. Continuous carousel for integrated filtration and drying

In this study, we make a step forward toward the implementation of end-to-end continuous pharmaceutical manufacturing with closed-loop quality control, by presenting a QbC framework for a novel continuous carousel for integrated filtration and drying of crystallization slurries.^25,30,31 The carousel technology, developed by Alconbury Weston Ltd (AWL; UK), features a primary cylindrical body containing multiple processing stations. The slurry coming from an upstream crystallizer is loaded into the first station and is eventually discharged as a dry crystals cake from the last station after being processed in all the stations of the carousel. During every processing cycle, all stations operate simultaneously batch-wise, each one processing a different batch of material throughout the whole cycle duration, which is an operating variable of the unit. At the end of a processing cycle, the content of the last station (dry crystals cake) is discharged. Then, the main body of the carousel rotates and transfers each batch of material to the following station, thus enabling continuous operation.

Previous studies involved experimental and digital design of the carousel technology and its operation,^30,32–34 and experimental and modeling results that successfully integrated carousels, in filtration-only mode, with upstream continuous crystallization systems.^25,31,35 Recently, a comprehensive benchmark simulator (ContCarSim³⁶) has been proposed for the digital design and testing of control strategies for this type of units. ContCarSim, implemented in MATLAB, reproduces the operation of physical carousels and their sensors and actuators network,³³ and it allows the implementation of control loops by the user.

1.3 |. Objectives of this study

In this study, we develop a three-layer QbC control strategy for the carousel technology, and we test its performance on the ContCarSim simulator. Layer 0 consists of the built-in controllers of the equipment, such as those needed for manipulating the CPPs and handling the carousel rotation/feeding routines. In Layer 1, the control strategy features an end-point controller for automatically triggering a carousel rotation when the target product CQAs are met in the final station of the carousel. In Layer 2, an RTO algorithm is used for optimizing the slurry volume fed to the carousel at each cycle. A soft-sensing (SS) framework including parameter and state estimation routine is also put in place. The framework monitors the product CQAs and other key process variables, to support the end-point controller at Layer 1 and to promote real time release testing. The proposed QbC control strategy is benchmarked against a traditional QbD approach and other simpler QbC control strategies, which include only selected components of the complete QbC control system presented in this work. We consider different scenarios, such as the presence of a set of variability sources occurring under normal operating conditions (e.g., mesh fouling and solid concentration fluctuations in the fed slurry), but also the occurrence of abnormal events. The control system is implemented through MATLAB functions that follow the standard ContCarSim template. The relevant code is made publicly available in the ContCarSim repository (https://github.com/CryPTSys/ContCarSim).

The remainder of the manuscript is organized as follows. In Section 2, a description of the process under investigation is provided, and the case study on which the proposed control system will be tested are presented. Section 3 outlines the development of the control system, while in Section 4 its performance on a case study is thoroughly discussed and benchmarked against simpler control systems, consisting of a reduced subset of the elements of the overall control system, and against traditional open-loop operation. The concluding section follows.

2 |. PROCESS DESCRIPTION AND CASE STUDY DEFINITION

2.1 |. Process description

The ContCarSim simulator³⁶ accurately mimics the operation of a carousel continuously isolating paracetamol from a paracetamol/ethanol slurry through filtration and drying. A detailed description of the process and of the simulator (including the input/output structure) is provided in the article accompanying the simulator release,³⁶ and only a summary of the simulator features is given here.

A schematic of the unit is reported in Figure 2. Details on the controllers and sensors network are provided in Table 1, whereas the list of measurements coming from the plant and their symbols are summarized in Table 2. The carousel is made up of a main cylindrical body embedding five cylindrical ports, aligned to five processing stations (V101–V105). It alternates processing cycles, during which the content of every port is processed batch-wise, to carousel rotations, which enable continuous operation by moving each port and its content from one processing station to the following one. At the beginning of any processing cycle, a given amount of slurry is fed to the port aligned with V101. Then, the content of every port is processed for a given cycle duration. In V101–V103, slurry filtration and cake deliquoring occur. Deliquoring consists of the mechanical drying of the liquid contained in the cake pores by the action of a pressure gradient between the top and the bottom of the cake.

FIGURE 2.

ContCarSim: schematic drawing of the carousel for continuous integrated filtration-drying of crystallization slurries mimicked by the benchmark simulator. Filter meshes are placed at the bottom of Stations V101–V104. Station V105, instead, is open for cake discharge. The legend of the unit operations and of the ancillary equipment is reported in Table 1. The drawing is adapted from ContCarSim’s User Manual.³⁶

TABLE 1.

ContCarSim: unit operations, controller, and sensors.

Name	Description
Unit ID
P101	Compressor
E101	Drying air electrical heater
V101-V105	Carousel station 1–5 (respectively)
V106	Filtrate collector
V107	Slurry tank
Controllers and sensors
AI-101	Slurry concentration indicator
FQC-101	Fed slurry volume controller
FI-101	Flowmeter for drying air entering carousel ports
KIC-101	Carousel rotation controller
LI-101	Camera system measuring volume of fed slurry and cake height
PC-101	Air pressure controller
PI-101	Compressor delivery pressure indicator
PI-102	Filtrate pressure indicator
TC-101	Drying air inlet temperature controller
TI-101	Thermocouple for drying air inlet temperature
TI-102	Thermocouple for drying air outlet temperature
WI-101	Scale for inferring filtrate flow rate

Source: Adapted from the literature 36.

TABLE 2.

ContCarSim: measurements generated by the carousel simulator.

Symbol	Variable name	Units	Sensor
$y_{c_{\text{slury }}}$	Slurry concentration	kg/m3	AI-101
$y_{H_{\text{cake }}}$	Cake height	m	LI-101
$y_{M_{\text{filt }}}$	Filtrate mass	kg	WI-101
$y_{P_{\text{in }}}$	Compressor delivery pressure (gauge)	Pag	PI-101
$y_{P_{\text{out }}}$	Filtrate pressure (gauge)	Pag	PI-102
$y_{T_{\text{in }}}$	Drying air inlet temperature	K	TI-101
$y_{T_{\text{out }}}$	Drying air outlet temperature	K	TI-102
$y_{{\dot{V}}_{\text{in},\text{g}}}$	Air flow rate	NL/min	FI-101
$y_{V_{\text{slurry }}}$	Fed slurry volume	m	LI-101

Deliquoring immediately begins at the end of filtration, namely when the slurry hold-up on top of a cake being formed has been completely filtered out. The equilibrium amount of liquid in cake pores cannot be eliminated through deliquoring, but only through thermal drying. For this purpose, in V104 cake drying is carried out through a hot air flow. Compressor P101 provides the pressure gradient necessary for filtration, deliquoring, and convective drying operations. Dry crystals cakes are eventually discharged from the carousel through a piston action in V105.

Stations V101–V104 present a filter mesh at the bottom. When a threshold mesh fouling is reached, a cleaning-in-place (CIP) routine is automatically triggered. During the following three processing cycles, no additional slurry is loaded into the carousel, and only the material that is already inside the unit is processed. At the onset of the fourth cycle after CIP triggering, all ports are empty, and they are automatically cleaned through an ethanol flow. Then, normal carousel operation resumes, loading fresh slurry into V101.

Table 3 summarizes CMAs, CPPs, non-CPPs, and CQAs. The objective of the process is obtaining cakes meeting the target quality, namely an ethanol content in cakes discharged by the carousel (w_final) below 0.5 wt%. Although ContCarSim makes w_dry(t,z), that is, the dynamic axial profile of ethanol content in the cake being dried, available to the user, the ethanol content in cakes being processed inside the carousel is not an available process measurement (Table 2). Within ContCarSim, w_final is calculated as the value at the cake discharge of the cake-averaged ethanol content in the cake $w_{\mathrm{d}\mathrm{r}\mathrm{y}}^{\text{avg}}\left(t\right)$:

$$w_{\mathrm{d}\mathrm{r}\mathrm{y}}^{\mathrm{a}\mathrm{v}\mathrm{g}}\left(t\right)=\frac{1}{H_{\text{cake}}}{\int }_0^{H_{\text{cake}}} w_{\mathrm{d}\mathrm{r}\mathrm{y}}(t,z)\mathrm{d}z,$$ (1)

where H_cake is the cake height. During the operation, four operating variables can be adjusted to meet the target product CQA (Table 3): the compressor delivery pressure (P_compr), the drying air inlet temperature (T_dry), the fed slurry volume (V_slurry), and the cycle duration (Δt_cycle). In the actual process, the operating variables are adjusted through lower-level controllers (Figure 2). Upon setup of a ContCarSim simulation, the user can decide whether to keep the setpoints of the lower-level controllers for the operating variables constant throughout all the simulation, or to specify a control law for an upper-level controller that, based on the process measurements (Table 2), updates the set-points of the lower-level controllers to control the product quality. In the remainder of this manuscript, we refer to the former scenario as an “open-loop” operation and to the latter one as a “closed-loop” operation, where the loop under consideration is the one that controls the product quality. All lower-level controllers are assumed to be perfect, namely the actual responses perfectly track the relevant setpoints, except for FQC-101; namely, we assume that V_slurry is affected by Gaussian fluctuations around the set-point, to mimic phenomena occurring in real carousel, as further outlined in Section 2.3. Following the QbD jargon,⁵ set-points $V_{\text{slurry}}^{\text{sp}}$ and $\Delta t_{\text{cycle}}^{\text{sp}}$ are the CPPs, while $P_{\text{compr}}^{\text{sp}}$ and $T_{\text{dry}}^{\text{sp}}$, instead, are non-CPPs that impact the CQA less significantly.

TABLE 3.

ContCarSim: CMAs, CQAs, CPPs, and non-critical process parameters.

Symbol	Variable type	Variable name
$c_{\text{slurry }}$	Critical material attribute	Slurry concentration
$w_{\text{final }}$	Critical quality attribute	Ethanol content in cake discharged by the carousel
$V_{\text{slurry }}^{\text{sp }}$	Critical process parameter	Set-point of fed slurry volume
$\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}^{\mathrm{s}\mathrm{p}}$	Critical process parameter	Set-point of cycle duration
$T_{\text{dry }}^{\text{sp }}$	Non-critical process parameter	Set-point of drying air inlet temperature
$P_{\text{compr }}^{\text{sp }}$	Non-critical process parameter	Set-point of compressor delivery pressure

2.2 |. Case study definition

We develop an in silico case study with ContCarSim, based on challenges 1, 2, 4, and 5 proposed in the article accompanying the release of the simulator.³⁶ We design and implement a closed-loop control system, acting on the CPPs ($V_{\text{slurry}}^{\text{sp}}$ and ^{$\Delta t_{\text{cycle}}^{\text{sp}}$}) to maximize the production of cakes meeting the target quality discharged by the carousel in a given time window. We compare the control system performance with open-loop operation and with simpler control systems, including only a subset of the elements of the complete approach, under the reference scenario proposed in challenge #5 (Table 4). The inputs of ContCarSim reported in Table 4 have been used in all the simulations presented in this study. The only exceptions are the simulations performed for DS description, as will be discussed in Section 3.2. The case study benchmarks the control systems on the three default disturbance scenarios implemented in ContCarSim, as defined by different patterns of process variability. Further information on process variability and disturbance scenarios is provided in the next subsection.

TABLE 4.

Case study: summary of inputs to ContCarSim

Input	Value/Type	Units
Control interval	1	s
Inter-cycle idle time	0	s
Mesh cleaning idle time	0	s
Nominal slurry concentration	250.0	kg/m3
Nominal set-point of drying air inlet temperature	323.0	K
Nominal set-point of fed slurry volume	Depends on control strategy (Table 6)	m3
Nominal set-point of compressor delivery pressure	1.0 ×105	Pag
Nominal set-point of cycle duration	Depends on control strategy (Table 6)	s
Disturbance scenario	0, 1, 2 (control strategies tested in different disturbance scenarios)	-
Sampling time	0.025	s
Simulation duration	3600	s

Note: The indicated nominal set-points of the operating variables are the values of the set-points at the operation onset.

2.3 |. Additional details on process variability and disturbance scenarios

A set of 10 sources of process variability (variability sources #1–10) typically occurring in real carousels are implemented in ContCarSim (Table 5). The variability sources are simulated as changes in some of the model parameters, which are kept constant during a given processing cycle, but updated at every carousel rotation. Variability sources #1–4 refer to the resistance of the filter meshes of, respectively, carousel stations 1–4 (R_m,i, for i = 1, 2, 3, 4), and mimic mesh fouling and cleaning as they occur during process operation. Variability sources #5–10 are implemented through a parameter (γ^dist) that represents a deviation with respect to a nominal condition:

$$\gamma =\bar{\gamma }{\gamma }^{\text{dist}}$$ (2)

where $\bar{\gamma }$ is a nominal value, and γ is a value under process variability. Variability sources #5–10 represent variability in the process inputs (fed slurry concentration, $c_{\text{slurry}}^{\text{dist}}$, fed slurry volume, $V_{\text{slurry}}^{\text{dist}}$, specific cake resistance, α^dist, and cake porosity, ϵ^dist), and intrinsic process variability (drying kinetic constant, $h_M^{\text{dist }}$ , and heat transfer coefficient between cake and air during drying, $h_T^{\text{dist}}$).

TABLE 5.

ContCarSim: variability sources

Variability source #	Name of parameter causing the variability	Symbol of parameter causing the variability	Units
1–4	Mesh resistance in station i, for i = 1, 2, 3, 4	$R_{m,i}$	1/m
5	Slurry concentration variability source	$c_{\text{slurry }}^{\text{dist }}$	—
6	Drying kinetic constant variability source	$h_M^{\text{dist }}$	—
7	Cake/gas heat transfer coefficient variability source	$h_T^{\text{dist }}$	—
8	Fed slurry volume variability source	$V_{\text{slurry }}^{\text{dist }}$	—
9	Cake specific resistance variability source	${\alpha }^{\text{dist }}$	—
10	Cake porosity variability source	${ϵ}^{\text{dist }}$	—

The inter-cycle profile of the variability sources depends on the disturbance scenario selected by the user for a simulation. The three default disturbance scenarios implemented in ContCarSim are:

• Disturbance scenario #0: normal operating conditions. The variability sources reproduce the intrinsic inputs and process variability occurring in normal operating conditions;

• Disturbance scenario #1: slurry concentration ramp change. All variability sources behave as in normal operating conditions, except for the inlet slurry concentration, which undergoes an increase of 2% of the nominal value every minute. The ramp change starts 5 min after the operation onset, and stops 25 min afterward, with the inlet slurry concentration stabilizing at 140% of the nominal value;

• Disturbance scenario #2: specific cake resistance step change. All variability sources behave as in normal operating conditions, except for the specific cake resistance, which undergoes a 100% step increase from its nominal value, starting 5 min after the operation onset.

A thorough description of variability sources, their variation profile and of default disturbance scenarios is given in the article accompanying the release of ContCarSim.

3 |. CONTROL SYSTEM DEVELOPMENT

3.1 |. Overview

We design a three-layer (Layer 0, 1, and 2) control system (Figure 3) for the carousel, based on the recently proposed QbC framework (Figure 1).¹⁷ Layer 0 is made up by the built-in controls of the carousel setup (Figure 2). Layer 1 consists in an end-point controller, that automatically triggers a carousel rotation when the control system estimates that the target residual ethanol content has been reached in the cake being dried in V104. Layer 2 features a SS framework consisting in parameter estimation and state estimation routines, and an RTO component for assessing at each carousel cycle, which is the optimal slurry volume to be fed to the unit. The SS framework of Layer 2 estimates, among other key variables of the process, the residual ethanol content in the cake being dried in V104, which is, in turn, used by the end-point controller of Layer 1 as an indicator for triggering carousel rotations.

FIGURE 3.

Proposed QbC control system for the carousel. Thick arrows represent process inputs and outputs, while thin arrows represent process measurements. The drying air outlet temperature measurement is exploited (by Layer 1) only when Layer 2 is not activated.

To highlight the advantages of each component of the control system, in Section 4 we compare the product quality attainment performance and the throughput achieved with control strategies featuring only selected parts of the complete control system:

• L0 control strategy: includes only Layer 0 of the control system;

• L1 control strategy: includes only Layers 0–1 of the control system. Since the estimation of the residual ethanol content in the cake being dried in V104 from Layer 2 is not available, Layer 1 directly infers the drying end-point from the drying air outlet temperature measurement (Table 2);

• L2-SS control strategy: features Layers 0–1 and the SS framework (i.e., parameter and state estimation routines) of Layer 2;

• L2-SS-RTO control strategy: complete control system featuring Layers 0–2 implementation, including the RTO at Layer 2.

To understand how much the accuracy of the SS framework affects the performance of the L2-SS and L2-SS-RTO control strategies, we also consider an additional control strategy that can be implemented in ContCarSim, but that cannot be achieved in physical systems:

• L2-perfectSS control strategy: features Layers 0–1 and an ideal SS framework, capable of providing perfect estimations of all the system states. Such perfect SS framework is implemented in this study by directly accessing the unmeasured system states provided by ContCarSim, but that are not available in real carousels.

The differences among the control strategies consist of the way by which they manipulate the CPPs ($V_{\text{slurry}}^{\text{sp}}$ and $\Delta t_{\text{cycle }}^{\text{sp}}$). Such differences are presented in the remainder of this paragraph, which outlines the implementation of the layers of the proposed control system. The code of ContCarSim contains blocks where the user can implement suitable control laws or parameter and state estimation routines to be tested on the simulator. The control strategies described in this section are directly implemented in these dedicated blocks of ContCarSim as MATLAB functions for developing the case studies of Section 4.

3.2 |. Layer 0

Layer 0 of the control system consists of the built-in controls of the carousel and of the ancillary equipment implemented in the simulator (Table 1). Conducting carousel operation with only Layer 0 in place (L0 control strategy) is equivalent to adopting a traditional open-loop QbD approach, in which the CPPs are kept at a fixed point within the DS. Hence, for designing the L0 control strategy, it is necessary to first describe the DS of the process, and then to select an operating point within it.

3.2.1 |. DS description

The DS corresponds to the combination of CMA (nominal slurry concentration,$\left.{\bar{c}}_{\text{slurry}}\right)$, and of CPPs ($V_{\text{slurry }}^{\text{sp }}$ and $\left.\Delta t_{\text{cycle}}^{\text{sp}}\right)$ that result in a product meeting the target quality. We determine the DS of the process by following a probabilistic approach proposed in the literature³⁴ for the carousel technology. The approach requires the development of a model of the process that, for a given cake, given a set of values for the CMA, the CPPs and the uncertain parameters ϕ of the model, calculates the corresponding values of the cake CQA, namely:

$$w_{\text{final}}=w_{\text{final}}\left({\bar{c}}_{\text{slurry}},V_{\text{slurry}}^{\text{sp}},\Delta t_{\text{cycle}}^{\text{sp}},\mathbf{\varphi }\right),$$ (3)

where the same cycle duration $\Delta t_{\text{cycle}}^{\mathrm{s}\mathrm{p}}$ is considered for all the four processing cycles in which the cake is processed within the carousel. We denote the model of Equation (3) as the input/output carousel model. We develop and implement in MATLAB the input/output carousel model to calculate the DS by assembling together filtration, deliquoring, and drying models to reproduce the train of operations occurring in the carousel (Figure 4). More details on the internal structure of the input/output carousel model are provided in the literature.³⁴ The filtration, deliquoring, and drying models, detailed in Appendix A are the same ones implemented in ContCarSim. The main difference between the input–output carousel model and ContCarSim is that the latter simulates the actual carousel operation, consisting in the simultaneous processing of slurry and cakes in all the five carousel stations, while the input/output carousel model simulates the processing of a single cake, directly providing w_final as only output (Figure 4). Instead, ContCarSim provides an extended set of outputs (including real time process measurements) and offers the possibility to modify many simulation settings and to implement a control system.

FIGURE 4.

Structure of the input/output carousel model used for DS description and for RTO.

In this study, ContCarSim is used as the “plant” to test the control strategies, while the input/output carousel model is used for DS description. Hence, the actual values of variability sources #1–10 are not known when running a simulation with the input–output model, and they are the uncertain parameters that originate process/model mismatch:

$$\mathbf{\varphi }=\left[R_{m,1}{R}_{m,2}{R}_{m,3}{R}_{m,4}{c}_{\text{slurry}}^{\text{dist}}{h}_M^{\text{dist}}{h}_T^{\text{dist}}{V}_{\text{slurry}}^{\text{dist}}{\alpha }^{\text{dist}}{ϵ}^{\text{dist}}\right].$$ (4)

For DS description, we build a three-dimensional grid whose axes are: $V_{\text{slurry }}^{\text{sp }}$ (varying from 0.5 ml to 10 ml, step 0.5 ml), $\Delta t_{\text{cycle }}^{\text{sp}}$ (varying from 5–300 s, step 5 s), and ${\bar{c}}_{\text{slurry}}$ (varying from 50 kg/m³ to 250 kg/m³, step 50 kg/m³). For each point of the grid, we carry out a Monte Carlo simulation with 400 realizations. For each realization, the elements of ϕ are sampled from the respective probability distributions implemented in ContCarsim.³⁶ A single value is sampled for all the filter mesh resistances (R_m,1, R_m,2, R_m,3, and R_m,4), since the carousel rotation mechanism (Section 2) causes a fouling lag among stations; hence, a given cake will encounter filters of approximately the same resistance while progressing in the carousel stations. Finally, w_final (i.e., the product CQA) is calculated for the given set of grid point conditions and disturbances using the input/output carousel model. The probability of attaining the target quality in each grid point corresponds to the percentage of realizations satisfying the target CQA (i.e., w_final < 0.5 wt%). The DS (Figure 5; green triangles) corresponds to the region of the grid where such probability is greater than 90%.

FIGURE 5.

Probabilistic DS with a control system consisting of only Layer 0 (open-loop with respect to quality). The symbols indicate the probability of attaining the target product quality at a given set of CMA and CPPs. Green triangles: probability ≥ 90%, yellow circles: 80% ≤ probability < 90%, orange squares: 60% ≤ probability < 80%, and red diamonds: probability < 60%.

3.2.2 |. L0 control strategy

In the reference scenario (Table 4) selected for the case study, a nominal slurry concentration equal to 250 kg/m³ arrives from upstream. Figure 6A shows the section of the DS of Figure 5 corresponding to this inlet concentration. Under the L0 control strategy, the CPPs are assigned offline and maintained constant during process operation, unless significant variations in the slurry concentration are registered. For operation under the L0 control strategy, we select the combination of $V_{\text{slurry }}^{\text{sp}}$ and $\Delta t_{\text{cycle }}^{\text{sp }}$ within the DS that maximize the carousel throughput for each cycle (T_cycle):

$$T_{\text{cycle}}=\frac{V_{\text{slurry}}^{\text{sp}}{c}_{\text{slurry}}^{\text{nom}}}{\Delta t_{\text{cycle}}^{\text{sp}}}.$$ (5)

FIGURE 6.

(A) Probabilistic DS at nominal slurry concentration equal to 250 kg/m³ with a control system consisting of only Layer 0 (open-loop with respect to quality). (B) Slurry throughput within the DS. The selected operating conditions for carousel open-loop operation are highlighted by a gray circle.

The maximum T_cycle in the DS is achieved at the DS boundary (Figure 6B). However, in order to minimize the risk of obtaining an out-of-specification product, we select an operating point (gray circle in Figure 6B) close to the maximum value of T_cycle, but slightly more inside the DS, namely $V_{\text{slurry }}^{\text{sp }}$ = 6 ml and $\Delta t_{\text{cycle }}^{\text{sp }}$ = 80 s. In control strategy L0, these values of $V_{\text{slurry }}^{\mathrm{s}\mathrm{p}}$ and $\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}^{\mathrm{s}\mathrm{p}}$ are set at the operation onset, and never changed during the process.

3.3 |. Layer 1

In Layer 1 of the control system, an end-point controller is implemented to close the loop on the CQA. Since no real-time measurements are available in the carousel for the ethanol content in the cakes, the cycle termination time is inferred indirectly. In L2 control strategies, the SS framework uses state estimation to estimate w_dry(t,z) (Section 3.4). In the L1 control strategy, instead, the drying end-point is determined indirectly from the measured drying air outlet temperature $y_{T_{\text{out }}}$.

Figure 7 shows the relation between $w_{\text{dry}}^{\text{avg }}\left(t\right)$ and $y_{T_{\text{out }}}$ during drying for three batches of slurry, processed with filter meshes that present increasing resistances (3 × 10⁹ m⁻¹, 9 × 10⁹ m⁻¹, and 15 × 10⁹ m⁻¹). The selected values of mesh resistance correspond to the range of variability encountered during carousel operation, in between two CIP procedures. Data are generated with ContCarSim under normal operating conditions (disturbance scenario 0), with the settings of Table 4. In all the three batches, the target quality value is reached by $w_{\text{dry }}^{\text{avg }}$ after $y_{T_{\text{out }}}$ has reached a minimum, although for slightly different values of $y_{T_{\text{out }}}$ in the increasing branch of the curve (Figure 7). Based on these results, the L1 control strategy is designed as outlined in the following sub-section.

FIGURE 7.

Average ethanol content in the cake $\left(w_{\text{dry }}^{\text{avg }}\right)$ vs outlet gas temperature measurement $\left(y_{T_{\text{out }}}\right.$) during drying for filter mesh resistances equal to (3 × 10⁹ m⁻¹, 9 × 10⁹ m⁻¹, and 15 × 10⁹ m⁻¹). Temperature end-point for the Layer 1 end-point controller is also reported. Data generated with ContCarSim in normal operating conditions with the settings of Table 4.

3.3.1 |. L1 control strategy

In the L1 control strategy, during the first three cycles after the operation onset or after mesh cleaning, no cakes are dried in V104, hence the end-point controller cannot be used for triggering the carousel rotation. During these cycles, only filtration and deliquoring occur, which are faster processes than drying. Hence, $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ is set to the fixed value of 30 s. From the fourth cycle after the operation onset or after mesh cleaning, based on the results of Figure 7, $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ is set as the point of time in which a value of $y_{T_{\text{out }}}$ equal to 18.7°C is recorded after the temperature inversion. This is a conservative choice, as for clean filter meshes (e.g., resistance equal to 3E9 1/m), the cycle could be terminated earlier. Through additional simulations (not reported), we assessed that this temperature-inferred end-point is not robust to significant changes of cake height or other cake parameters (e.g., porosity and specific resistance). Hence, in the L1 control strategy, $V_{\text{slurry }}^{\text{sp }}$ is operated at open-loop and kept at the nominal value of 6 ml; clearly, the L1 control strategy can be used only with fixed ${\bar{c}}_{\text{slurry }}$, and in the absence of significant disturbances to the fed slurry properties.

3.4 |. Layer 2

In Layer 2 of the control system, advanced model-based techniques are implemented for process monitoring and control; namely, we use: online parameter estimation and state estimation for implementing a SS framework, and RTO for assessing the optimal slurry volume to be fed to the carousel at every processing cycle.

3.4.1 |. SS framework

The SS framework implemented at Layer 2 is sketched in Figure 8. For every cake processed in the carousel, the framework provides the estimated specific cake resistance $\widehat{\alpha }$, cake porosity $\widehat{ϵ}$, filtration duration $\Delta {\widehat{t}}_{\text{filt }}$, and dynamic axial profile of ethanol content in the cake during drying ${\widehat{w}}_{\text{dry }}(t,z)$, together with its associated estimation error in the form of standard deviation ${\sigma }_{{\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}}(t,z)$. These estimated variables are used online for monitoring the process, and are then stored for offline analysis. The estimated resistances of the filter mesh of Station $1\left({\widehat{R}}_{m,1}\right)$ and Station $4\left({\widehat{R}}_{m,4}\right)$ are also provided by the SS framework, and are used for monitoring and inferring the evolution of the filter mesh fouling.

FIGURE 8.

Soft-sensing framework implemented at Layer 2 for monitoring key variables and parameters of every cake processed in the carousel.

At the end of every processing cycle during which a batch of slurry has been processed in V101, online parameter estimation is carried out for estimating the current ${\widehat{R}}_{m,1}$, and the $\widehat{\alpha },\widehat{ϵ}$, and $\Delta {\widehat{t}}_{\text{filt }}$ of the cake that has just formed. First, porosity is estimated from a mass balance:

$$\widehat{ϵ}=1-\frac{y_{V_{\text{slurry}}}{y}_{c_{\text{slurry}}}}{{\rho }_s{y}_{H_{\text{cake}}}A},$$ (6)

where ρ_s is the paracetamol crystals density and A is the filter cross-section (both parameters are taken as in ContCarSim³⁶). Then, maximum likelihood estimation³⁷ is resorted to for estimating $\widehat{\alpha }$ and ${\widehat{R}}_{m,1}$ from the collected $y_{M_{\text{fit }}}$ profile, by solving the following optimization problem:

$$\widehat{\alpha },{\widehat{R}}_{m,1}=\underset{\widehat{\alpha },{\widehat{R}}_{m,1}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\frac{N_{\text{exp}}}{2}\mathrm{l}\mathrm{o}\mathrm{g}\left({\sum }_{i=0}^{i=N_{\text{exp}}} {\left(M_{\text{filt}}\left(t_i\right)-y_{M_{\text{filt}}}\left(t_i\right)\right)}^2\right),$$ (7)

where N_exp is the number of experimental points $y_{M_{\text{filt }}}\left(t_i\right)$ collected during the processing cycle just finished, and the corresponding $M_{\text{filt }}\left(t_i\right)$, depending on $\widehat{\alpha }$ and ${\widehat{R}}_{m,1}$, are the points of the filtrate mass profile calculated with a filtration model (Appendix A). We use the sequential quadratic programming (SQP) algorithm implemented in the MATLAB fmincon optimizer (from MATLAB Optimization Toolbox, version 9.1) for achieving the solution of the optimization problem of Equation (7). The estimation uncertainty is obtained through the calculation of the variance–covariance matrix at the solution.³⁷ Note that, although $y_{M_{\text{filti }}}$ is the sum of the weight of the filtrate collected from V101 to V104, in the formulation of the optimization problem of Equation (7) we assume it to be equal to the filtrate from V101. This is a reasonable assumption, considering that, with the operating conditions used in this study, filtration always ends in V101, and the amount of filtrate collected from the deliquoring in the other stations is negligible compared to the one from the filtration and the deliquoring occurring in V101. From the obtained ${\widehat{R}}_{m,1},\widehat{\alpha },\widehat{ϵ}$, and measurements $y_{c_{\text{slurry }}}$ and $y_{V_{\text{slurry }}},\Delta {\widehat{t}}_{\text{filt }}$ is estimated through a filtration model (Appendix A).

Equation (7) and the filtration model of Appendix A (used for the estimation of ${\widehat{R}}_{m,1},\widehat{\alpha },\widehat{ϵ}$, and $\Delta {\widehat{t}}_{\text{filt }}$) present no mismatch with respect to the corresponding models implemented in ContCarSim. However, the achieved estimations present an error with respect to the real value of the corresponding variables, because of the noise in the process measurements, which is factored into the equations.

The obtained values of ${\widehat{R}}_{m,1},\widehat{\alpha },\widehat{ϵ}$, and $\Delta {\widehat{t}}_{\text{filt }}$ for a given cake are stored and used in a deliquoring model (Appendix A), together with $y_{H_{\text{cake }}}$, to estimate ${\widehat{w}}_{\text{dry }}(0,z)$, namely the ethanol content profile of the cake when it enters V104 for drying. Given the duration of the processing cycles that the considered cake has undergone, respectively, in Stations 1–3 ($\left(\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e},\mathit{i}}^{\mathrm{s}\mathrm{p}}\right.$, for i = 1–3), the duration of the step to be simulated for computing the deliquoring that the cake has undergone from the end of filtration in V101 until the onset of drying in V104 is calculated through:

$$\Delta {\widehat{t}}_{\text{deliquoring}}=\Delta t_{\text{cycle},1}^{\mathrm{s}\mathrm{p}}+\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e},2}^{\mathrm{s}\mathrm{p}}+\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e},3}^{\mathrm{s}\mathrm{p}}-\Delta {\widehat{\mathrm{t}}}_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}},$$ (8)

where $\Delta {\widehat{t}}_{\text{deliquoring }}$ is the estimated deliquoring duration. The filter mesh resistances encountered by a given cake when entering both Stations 2 and 3 are imposed equal to ${\widehat{R}}_{m,1}$ obtained for the same cake, due to the intrinsic mesh fouling lag among stations caused by the carousel mechanism. The deliquoring model (Appendix A) used for obtaining ${\widehat{w}}_{\text{dry }}(0,z)$ is the same model implemented in ContCarSim.³⁶ However, a coarse discretization grid of 10 discretization nodes is implemented when solving the deliquoring model within the SS framework, instead of the finer grid implemented in ContCarSim, which automatically adapts based on the cake height. Further error in the estimation of ${\widehat{w}}_{\text{dry }}(0,z)$ comes from the propagation of the error of the estimations $\left({\widehat{R}}_{m,1},\widehat{\alpha },\widehat{ϵ}\right.$ and $\left.\Delta {\widehat{t}}_{\text{filt }}\right)$ and of the measurements $\left(y_{H_{\text{cake }}}\right)$ that are used as inputs to the deliquoring model of Figure 7.

After the drying onset, a state estimator (extended Kalman filter;³⁸ EKF) is run for estimating ${\widehat{w}}_{\text{dry }}(t,z)$, from the initial condition ${\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}(0,z)$ The state estimator also provides ${\sigma }_{{\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}}(t,z)$, namely the standard deviation of the estimation error of ${\widehat{w}}_{\text{dry }}(t,z)$. In the EKF, the filter mesh resistance of station 4 during drying of a given cake is considered equal to ${\widehat{R}}_{m,1}$, obtained for the same cake upon processing in station 1. This approach follows the previously presented rationale on the fouling lag among stations. At the end of drying, the Darcy equation (same form implemented in ContCarSim) is used for estimating ${\widehat{R}}_{m,4}$ from $y_{{\dot{V}}_{\text{dryer }}}$:

$$y_{{\dot{V}}_{\text{dryer}}}=\frac{A}{\widehat{\alpha }{\text{ρ}}_s(1-ϵ){\mu }_g}\frac{\Delta P}{Y_{H_{\text{cake}}}}$$ (9)

The ${\widehat{R}}_{m,1}$ used for state estimation is then validated against the obtained ${\widehat{R}}_{m,4}$.

Details on the EKF algorithm and of its implementation in Layer 2 are reported in Appendix B. The drying model used in the state estimator (Appendix A) is the same as the one implemented in ContCarSim. However, the following sources of process-model mismatch are identified. For state estimation, the grid used for the spatial discretization of the partial differential equations of the model has 10 points, independently from the cake height, whereas in ContCarSim, the equations are discretized with a finer grid (of spacing 0.3 mm), originating process-model mismatch. Additional process-model mismatch arises from the propagation of the estimation error of the estimated parameters ($\widehat{\alpha }$ and $\widehat{ϵ}$) and of the noise of the measurements $\left(y_{H_{\text{cake }}},y_{{\dot{V}}_{\text{dryer }}},y_{T_{\text{in }}}\right.$ and $\left.y_{T_{\text{out }}}\right)$ used in the state estimator (Figure 7). Furthermore, in the drying model implemented in the state estimator, variability sources #6–7 ($h_M^{\text{dist }}$ and $h_T^{\text{dist }}$) are unknown and hence set to 1 [–], instead than assuming the local stochastic value that they have in the process. Finally, potential misalignment between ${\widehat{R}}_{m,1}$, used in the state estimator for approximating $R_{m,4}$, and the actual value of $R_{m,4}$ also contributes to process-model mismatch.

3.4.2 |. Real-time optimization

While $\Delta t_{\text{cycle }}^{\text{sp }}$ is the only operating variable adjusted by the controllers presented so far, the RTO routine of Layer 2, presented in this subsection, is meant to optimize $V_{\text{slurry }}^{\text{sp }}$ at every cycle in which a batch of slurry is fed to the carousel. This is done by solving the following optimization problem, before the beginning of every cycle:

$$\underset{T_{\text{cycle}},V_{\text{slurry}}^{\text{sp}}}{\mathrm{m}\mathrm{a}\mathrm{x}} T_{\text{cycle}}$$ (10)

$$\text{Subject to :}{T}_{\text{cycle}}=\frac{V_{\text{slurry}}^{\text{sp}}{c}_{\text{slurry}}^{\text{nom}}}{\Delta t_{\text{cycle}}^{\text{sp}}}$$ (11a)

$$W_{\text{final}}=W_{\text{final}}\left({\bar{c}}_{\text{slurry}},V_{\text{slurry}}^{\text{sp}},\Delta t_{\text{cycle}}^{\text{sp}},\mathbf{\varphi }\right)<0.5\mathrm{w}\mathrm{t}\%$$ (11b)

$$0\mathrm{m}\mathrm{l}<V_{\text{slurry}}^{\text{sp}}<10\mathrm{m}\mathrm{l},$$ (11c)

$$T_{\text{cycle}}\ge 0$$ (11d)

The optimization problem of Equations (10) and (11) maximizes the throughput T_cycle per cycle (Equation (11a); cfr Equation (5) in Section 3.2.2) by adjusting $\Delta t_{\text{cycle }}^{\text{sp }}$ and $V_{\text{slurry }}^{\text{sp }}$, under the constraint of meeting the target product quality (Equation 11b). In Equation (11b), $w_{\text{final }}=w_{\text{final }}\left({\bar{c}}_{\text{slurry }},V_{\text{slurry }}^{\text{sp }},\Delta t_{\text{cycle }}^{\text{sp }},\mathit{\varphi }\right)$ is the input/output carousel model discussed in Section 3.2.1, to which the following simplifying assumptions are introduced to increase the computational speed of the RTO: (i) the deliquoring model of Appendix A is substituted by a simplified design charts approach,^35,39 and (ii) a fixed grid of 10 discretization points is used for solving the drying model (as in the state estimator; Section 3.4.1), instead of the finer grid implemented in ContCarSim.

Within the inputs of the input/output carousel model (Equation 11b), for RTO ${\bar{c}}_{\text{slurry }}$ is equal to 250 kg/m³ (Table 4), $V_{\text{slurry }}^{\text{sp }}$ is an optimization variable, and $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ is calculated through Equation (11a) from the optimization variable T_cycle (within the optimization problem, the same $\Delta t_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}^{\mathrm{s}\mathrm{p}}$ is considered for all the four cycles during which a given cake is processed in the carousel). The remaining set of inputs of the input/output carousel model are the elements of vector ϕ (Equation 4), namely the variability sources of the process (Table 5), which are generally unknown. Considering the fouling schedule implemented in the simulator, for a generic cycle, variability sources #1–4 in ϕ are imposed equal to the ${\widehat{R}}_{m,1}$ estimated from the previous processing cycle, to which a fouling correction R_f (=2 × 10⁹) is applied as:

$$R_{m,i}={\widehat{R}}_{m,1}+R_f$$ (12)

In the specific case of cycles initiated immediately after a CIP routine, R_m,i (for i = 1, 2, 3, 4) is directly approximated to 3E9 1/m, the average filter mesh resistance after cleaning in ContCarSim. We fix variability sources #5–10 in ϕ to 1 [–], the mean value of their probability distributions.

The RTO problem of Equations (10) and (11a–d) is solved with the SQP algorithm of the fmincon optimizer implemented in MATLAB. To enhance the robustness of the RTO solution, at every cycle we solve the optimization problem four times, varying the initial guess provided to the optimizer among a set of four initial points belonging to the DS. The (feasible) solution leading to the largest T_cycle is implemented in the process. Note that, since the end-point detection based on the temperature measurement depends non-linearly on the cake mass, an RTO (or any other) procedure for $V_{\text{slurry }}^{\text{sp }}$ adaptation can be used only when a state estimator has been put in place for supporting end-point detection.

3.4.3 |. L2-SS control strategy

In the L2-SS control strategy, as for L0 and L1 control strategies, $V_{\text{slurry }}^{\text{sp }}$ is operated at the fixed value of 6 ml. For the first three cycles after CIP, cake drying is not carried out, and $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ is fixed to 30 s. From the fourth cycle on, cake drying is carried out, and $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ is automatically adjusted to terminate the cycle when the average mass fraction of ethanol in the cake being dried in V104 reaches the target quality threshold. The ${\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}(t,z)$ and ${\sigma }_{{\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}}(t,z)$ obtained from the EKF are exploited at Layer 1 for detecting the drying end-point. The profiles ${\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}(t,z)$ and ${\sigma }_{{\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}}(t,z)$ are averaged with respect to space, yielding, ${\widehat{w}}_{\text{dry }}^{\text{avg }}\left(t\right)$ and ${\sigma }_{{\widehat{w}}_{\text{dry }}^{\text{avg }}}\left(t\right)$, respectively. Following the risk-based approach promoted by the QbD paradigm, the estimation uncertainty is accounted by the end-point controller of Layer 1, which triggers the carousel rotation only when:

$${\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}^{\mathrm{a}\mathrm{v}\mathrm{g}}+2{\sigma }_{\mathrm{d}\mathrm{r}\mathrm{y}}^{\mathrm{a}\mathrm{v}\mathrm{g}}<0.5\mathrm{w}\mathrm{t}\%.$$ (13)

The L2-perfectSS control strategy is implemented as the L2-SS control strategy, but the actual value of $w_{\text{dry }}^{\text{avg }}$, obtained from ContCarSim but inaccessible in physical systems, is factored into Equation (13), and ${\sigma }_{{\hat{w}}_{\text{avg }}^{dry}}$ dryavg is imposed equal to zero.

3.4.4 |. L2-SS-RTO control strategy

When operating under the L2-SS-RTO control strategy, the solution of the RTO of Equations (10) and (11) yields the optimal $V_{\text{slurry }}^{\text{sp }}$ to feed at the incoming processing cycle, and the optimal $\Delta t_{\text{cycle }}^{\text{sp }}$ to use for processing the current batch of slurry. For the first three cycles after a mesh cleaning operation, when no cake is being dried in V104, the optimal $\Delta t_{\text{cycle }}^{\mathrm{s}\mathrm{p}}$ found at the end of the cleaning (namely, a state of clean meshes) is actually used. However, when a cake is being dried in V104, the actual $\Delta t_{\text{cycle }}^{\text{sp }}$ used in the process is adjusted by the Layer 1 end-point controller based on Equation (13), as for the L2-SS control strategy.

4 |. CASE STUDY: RESULTS AND DISCUSSION

We compare the performances of L0, L1, L2-SS, L2-SS-RTO, and L2-perfectSS control strategies on ContCarSim for the case study presented in Section 2.2 (Table 4). Table 6 summarizes the control laws implemented for the CPPs in the different control strategies. All other simulation settings and operating conditions are the same across all control strategies, and are reported in Table 4. The total production of cakes (in cumulative mass terms) meeting the target quality is benchmarked in all the three default disturbance scenarios implemented in ContCarSim (Section 2.3): normal operating conditions (disturbance scenario 0) and abnormal operating conditions scenarios (disturbance scenario 1: slurry concentration ramp increase; disturbance scenario 2: specific cake resistance step).

TABLE 6.

Control laws for the CPPs (set-point of fed slurry volume and set-point of cycle duration) under the different control strategies tested in the case study.

	Set-point of fed slurry volume		Set-point of cycle duration
Control strategy	Nominal value	During operation	Nominal value	During operation
L0	6 ml	Kept fixed	80 s	Kept fixed
L1	6 ml	Kept fixed	30 s	Adjusted by end-point controller, based on measurement of drying air outlet temperature
L2-SS	6 ml	Kept fixed	30 s	Adjusted by end-point controller, based on estimation by SS framework of ethanol content in cake being dried
L2-SSperfect	6 ml	Kept fixed	30 s	Adjusted by end-point controller, based on estimation by a (n ideal) perfect SS of ethanol content in cake being dried
L2-SS-RTO	From RTO	Adjusted by RTO	From RTO	Adjusted by end-point controller, based on estimation by SS framework of ethanol content in cake being dried

Note: The CPPs are initiated at the respective nominal values at the operation onset, and then adjusted (or kept fixed) during process operation as indicated in the table. For the first three cycles following mesh cleaning, the CPPs are restored to their respective nominal values.

All the simulations are carried out with MATLAB R2021b on a laptop computer with an Intel^® CoreTM i7-8565U CPU @1.80 GHz processor and total memory of 16.0 GB RAM. The computational time of the simulations show that the proposed control system is compatible with real time implementation, including the EKF and RTO, the most computationally-demanding features.

4.1 |. Normal operating conditions

Figure 9 shows that all control strategies under investigation allow achieving the target product quality in all the cakes discharged from the carousel under normal operating conditions. However, the cumulative mass of the cakes meeting the target quality (Table 7; output directly provided by ContCarSim) varies significantly based on the control strategy. The L0 control strategy is the most conservative approach, consisting in drying the cakes much more than needed (Figure 9) to compensate for the lack of feedback control. Since drying is the slowest processing step in the carousel under the process settings considered in this study (Table 3), drying longer than needed implies achieving a sub-optimal throughput. Hence, the L0 control strategy leads to the poorest production performance (Table 7). The throughput increase due to implementation of end-point controllers (even simple temperature-based ones) can be very large, due to the benefits of adjusting $\Delta t_{\text{cycle }}^{\text{sp }}$ based on the inferred drying duration, as opposite to keeping $\Delta t_{\text{cycle }}^{\text{sp }}$ fixed. The introduction of the Layer 1 end-point controller based on the temperature measurement (L1 control strategy) leads, under normal operating conditions, to a throughput increase of about 24% compared to the L0 control strategy. However, the L1 control strategy in general over-dries the cakes (Figure 9), since a conservative approach has been adopted for the end-point selection (Figure 7), due to the lack of robustness of drying end-point determination based on the temperature measurement.

FIGURE 9.

Disturbance scenario 0: residual ethanol in discharged cakes (CQA) conditions under different control strategies.

TABLE 7.

Disturbance scenario 0: cumulative mass of discharged cakes meeting the target quality, production increase with respect to L0 production, and cake rejection rate under different control strategies.

Control strategy	Production (mg)	Production increase wrt L0 (%)	% rejected cakes
L0	45.2	0	0
L1	55.9	23.6	0
L2-SS	60.4	33.6	0
L2-SSperfect	65.0	43.8	0
L2-SS-RTO	62.1	37.3	0

The L2 control strategies can better estimate the drying end-point, as these feature the SS framework that directly estimates the ethanol content in cakes being dried. Hence, these allow for an additional production increase. Under the L2-SS control strategy, an increase of production of about 34% with respect to the L0control strategy is registered; when also the RTO routine for selecting $V_{\text{slurry }}^{\text{sp }}$ at each cycle is activated (L2-SS-RTO control strategy), the production increase is of about 37%. If an (ideal) perfect state estimator was available for monitoring the residual ethanol in the cakes being dried with no estimation uncertainty (L2-perfectSS control strategy), a throughput increase with respect to Layer 0-only operation of about 44% could even be achieved (Table 7), by immediately triggering a carousel rotation when the acceptable quality is met in the cakes being dried. Since the production increase is larger than for the L2-SS-RTO control strategy, it appears that, for the considered processing conditions, a prompt drying end-point detection is crucial for throughput maximization. However, the RTO implementation of Equations (10) and (11) optimizes the operating conditions for one batch of slurry at a time. More advanced RTO implementations could therefore outperform the production achieved by the L2-perfectSS control strategy, because RTO enables the adjustment of two CPPs ($\Delta t_{\text{cycle }}^{\text{sp }}$ and $\left.V_{\text{slurry }}^{\text{sp }}\right)$, and not only one $\Delta t_{\text{cycle }}^{\text{sp }}$. For example, one could optimize the operating conditions for a series of consecutive batches, and factor the triggering of the CIP procedure inside the optimization problem. However, this is beyond the scope of this study, which aims at introducing the benefits of active and model-based process control for the carousel operation.

Figure 10 reports the profiles of the CPPs for all the cycles carried out under normal operating conditions with all proposed control strategies. Focusing on the fed slurry volume set-point first (Figure 10A), it can be argued that all the control strategies, except for L2-SS-RTO, always use a constant value for $V_{\text{slurry }}^{\text{sp }}$ (Table 6). The three consecutive cycles during which the fed slurry is null (e.g., cycle #7–9), appearing periodically with all control strategies, correspond to the cycles during which no slurry is fed to the carousel so that it can be emptied for mesh CIP. When the RTO routine is turned on, the optimizer selects values of $V_{\text{slurry }}^{\text{sp }}$ that are initially smaller than 6 ml for all cycles, but that increase with the filter mesh fouling. Considering a set of process cycles among two different CIP routines (e.g., process cycles #10–15), the optimal $V_{\text{slurry }}^{\text{sp }}$ from the RTO becomes larger from cycle #10 (clean meshes conditions) up to cycle #15, the last one before mesh cleaning. Even though the increasing trend of the optimal $V_{\text{slurry }}^{\text{sp }}$ in between two CIP procedures is consistent during carousel operation, slightly different optimal $V_{\text{slurry }}^{\text{sp }}$ are found for different cycles occurring after a given number of cycles following the CIP (e.g., cycles #10, #19, and #28). This is because the optimal $V_{\text{slurry }}^{\text{sp }}$ calculated by the RTO depends on several other factors (Equations 10 and 11), beside the local fouling conditions.

FIGURE 10.

Disturbance scenario 0: CPPs under different control strategies: (A) fed slurry volume set-points, and (B) cycle duration set-points.

The profiles of cycle duration under the different control strategies (Figure 10B) further elucidate the rationale of the different control strategies. All control strategies adjust $\Delta t_{\text{cycle }}^{\text{sp }}$ during operation, except for the L0 one. During the first three cycles after a CIP procedure, the L1 and all L2 control strategies use a lower $\Delta t_{\text{cycle }}^{\text{sp }}$ than for other cycles (Table 6) since only filtration and deliquoring are carried out. The L2-SS-RTO control strategy selects a growing $\Delta t_{\text{cycle }}^{\text{sp }}$ with the passing of cycles following CIP, consistently with the increasing $V_{\text{slurry }}^{\text{sp }}$ profile found from the solutions of the RTO problem at increasing fouling mesh conditions (Figure 10A). All other control strategies do not present clear $\Delta t_{\text{cycle }}^{\text{sp }}$ patterns. Considering the cycles during which a cake is being dried in the carousel, the $\Delta t_{\text{cycle }}^{\text{sp }}$ of the L1 control strategy is always larger than the one of the L2-SS control strategy, which is, in turn, always larger than the one for the L2-perfectSS controller. Since under these control strategies $V_{\text{slurry }}^{\text{sp }}$ is always the same for a given cycle, this result is consistent with a lower production (Table 7) and smaller residual ethanol content (Figure 9) found for the L1 control strategy with respect to the L2-SS one, and for the latter with respect to the L2-perfectSS one.

Figures 11 and 12 further demonstrate the importance of the SS framework within L2 control strategies. Figures 11A shows the estimations $\left({\widehat{R}}_{m,1}\right)$ and the actual values of the filter mesh resistance of Station 1 during the simulation performed under normal operating conditions for the L2-SS control strategy. The estimations are in good agreement with the actual resistances. Satisfying estimation is also observed for the specific resistance of the cakes processed during the same simulation (Figure 11B). The estimated filter mesh and specific cake resistances and the other estimated parameters (Section 3.4.1), not reported here for conciseness, are important for process monitoring as shown in Section 4.2. Moreover, they are also pivotal to achieving an accurate estimation of the ethanol content at the drying onset $\left({\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}(0,z)\right)$.

FIGURE 11.

Disturbance scenario 0, L2-SS control strategy: estimated vs. actual (A) filter mesh resistance in Station 1 during every processing cycle, (B) specific cake resistance of every processed cake, and (C) ethanol content axial profile at the drying onset of the first cake processed in the carousel.

FIGURE 12.

Disturbance scenario 0, L2-SS control strategy: residual ethanol during drying of the first cake. Confidence limits (CL) for the estimations are also provided by the state estimator.

Figure 11c shows the estimated and actual ethanol content axial profile in the first cake processed in the carousel in normal operating conditions under the L2-SS control strategy. The estimated profile presents a coarser grid with respect to the actual profile generated by ContCarSim, to fasten the computations of the SS framework (Section 3.4.1). The estimated profile is in good agreement with the actual one, despite the process-model mismatch and the error propagation from the parameters coming from upstream in the estimation train of the SS framework (Figure 8). The obtained ${\widehat{w}}_{\text{dry }}(0,z)$ is fed to the state estimator as initial point for estimating the ethanol content profile during drying: ${\widehat{w}}_{\text{dry }}(t,z)$.

Figure 12 comparesb ${\widehat{w}}_{\text{dry }}^{\text{avg }}\left(t\right)$, obtained averaging ${\widehat{w}}_{\text{dry }}(t,z)$ coming from the state estimator, with the actual profile, for the first cake dried in the carousel under normal operating conditions for the L2-SS control strategy. The estimation is good, despite the process-model mismatch (Section 3.4.1). Figure 12 also elucidates the end-point control mechanism for the control strategies in which the SS framework of Layer 2 is activated. The actual ethanol content reaches the acceptable value about 65 s from the drying onset. However, the end-point controller triggers a carousel rotation only about 10 s later, due to the estimation error and to the implemented conservative approach that accounts for the estimation uncertainty to detect the drying end-point (Equation 13). For this reason, with the L2-perfectSS control strategy, the throughput is much larger than when using the EKF (Table 7), thus remarking the importance of achieving high accuracy parameter and state estimation for improving the control system performance in physical system.

4.2 |. Abnormal operating conditions

Tables 8 and 9 report the production and cake rejection rate achieved using the proposed control strategies upon occurrence of, respectively, a slurry concentration ramp increase (disturbance scenario 1), and a specific cake resistance step increase (disturbance scenario 2).

TABLE 8.

Disturbance scenario 1: cumulative mass of discharged cakes meeting the target quality, production increase with respect to L0 production, and cake rejection rate under different control strategies.

Control strategy	Production (mg)	Production increase wrt L0 (%)	% rejected cakes
L0	16.3	0	67
L1	48.3	196.3	0
L2-SS	58.3	257.7	0
L2-SSperfect	64.3	294.5	0
L2-SS-RTO	62.1	281.0	0

TABLE 9.

Disturbance scenario 2: cumulative mass of discharged cakes meeting the target quality, production increase with respect to L0 production, and cake rejection rate under different control strategies.

Control strategy	Production (mg)	Production increase wrt L0 (%)	% rejected cakes
L0	5.9	0	87
L1	31.9	440.7	0
L2-SS	39.2	564.4	0
L2-SSperfect	43.8	642.4	0
L2-SS-RTO	41.2	598.3	0

The L0 control strategy leads to a smaller production rate in both scenarios, as most of the cakes do not meet the target quality, and must therefore be rejected. Since the slurry concentration is a measured variable, in the scenario in which it abnormally changes, the DS of Figure 5 can be used for adjusting the CPPs at open-loop to meet the target product quality. However, in the occurrence of the (unmeasured) specific cake resistance change, it is not possible to achieve the target product quality operating at open-loop. All the other control strategies always lead to discharging cakes of acceptable quality from the carousel, thus demonstrating that not only does closed-loop control lead to greater throughput under normal operating conditions, but it is also more robust to abnormal conditions.

The L1 control strategy, despite being based on a simple end-point controller, is robust to the considered abnormal operation scenarios. However, the production gap with respect to the L2-SS control strategy (Tables 8 and 9) is significantly larger than under normal operating conditions (Table 7). Although, for this particular abnormal scenario, the L1 control strategy proved effective, it must be used with caution when operating in conditions different from those at which the end-point based on the temperature was measured, as discussed in Section 3.3.1. Introduction of RTO for optimizing $V_{\text{slurry }}^{\text{sp }}$ at each cycle leads to a slight increase in throughput also under abnormal operating conditions. The largest throughput is, however, achieved with the L2-perfectSS control strategy, confirming the importance of a prompt drying end-point detection.

As under normal operating conditions, the SS framework is key to process monitoring and control during the occurrence of abnormal conditions. Let us consider the abnormal scenario involving the specific cake resistance step change occurrence, which is not measured and is, therefore, more difficult to be handled compared to the slurry concentration change. Figure 13A compares the estimated and the actual specific cake resistance of the cake discharged from the carousel under the L2-SS control strategy. The online parameter estimation routine can effectively track the change of specific resistance that occurs from the seventh cake processed in the carousel on. The abnormal event occurrence is promptly detected using control limits (Figure 13A), calculated on the specific cake resistances estimated in normal operating conditions (Figure 11B). A larger cycle duration is automatically enforced by the control system for cycles in which cakes are dried from processing cycle 13, when the seventh cake enters the dryer, on (Figure 13B). Exploiting the accurate estimations of specific cake resistance obtained from parameter estimation, the state estimator can accurately track the ethanol content during drying, and therefore it appropriately sets the cycle duration, as shown in Figure 14 for the first and the seventh (i.e., the first presenting the abnormal specific resistance) cakes processed in the carousel.

FIGURE 13.

Disturbance scenario 2, L2-SS control strategy: (A) estimated vs. actual specific cake resistance, and (B) cycle duration set-points. Control limits at 95% calculated in normal operating conditions for the estimated specific cake resistance are also reported.

FIGURE 14.

Disturbance scenario 2, L2-SS control strategy: residual ethanol during drying of the first and of the seventh cake. Confidence limits (CL) for the estimations are also provided by the state estimator.

5 |. CONCLUSIONS

This article presents an active control system for a continuous integrated filtration-drying carousel for pharmaceutical manufacturing. The control system features three layers. Layer 0 is made up of the built-in control systems of the equipment. Layer 1 consists of an end-point controller that automatically adjusts the carousel processing cycles duration to meet the target product quality. Layer 2 features model-based techniques, namely a SS framework that estimates key parameters and properties of the product in real time and an RTO component, which optimizes the slurry volume to feed to the unit at every processing cycle. We tested the developed control system on ContCarSim, a benchmark simulator of a carousel for continuous filtration-drying of paracetamol/ethanol slurries. Compared to traditional open-loop operation within the design space, closed-loop operation increased the production of product of acceptable quality by up to about 40% in normal operating conditions, and of up to about 600% in the considered abnormal operating conditions. To further appreciate the significance of each of the control layers, we benchmarked the proposed control system with simpler control strategies, including only selected components of the overall control system. Although all active control strategies allowed to always obtaining product satisfying the target quality specifications, the complete three-layered control system achieved the highest throughput. The model-based techniques implemented at Layer 2 of the control system led to a production increase, compared to closed-loop operation with only Layers 0 and 1 activated, of about 10% in normal operating conditions, and of about 30% in the considered abnormal operating conditions.

Abbreviations:

CIP

cleaning‐in‐place

CMA

critical material attribute

CPP

critical process parameter

CQA

critical quality attribute

DS

design space

FDA

Food & Drug Administration

PID

proportional‐integral‐derivative

QbC

quality‐by‐control

QbD

quality‐by‐design

QbT

quality‐by‐testing

RTO

real time optimization

SS

soft‐sensing

APPENDIX A: FILTRATION, DELIQUORING AND DRYING MODEL FOR PROCESS CONTROL

In this Appendix, we outline the filtration, deliquoring and drying models used in the input/output model, exploited for DS description (Section 3.2.1) and RTO (Section 3.4.4), and within the SS framework (Section 3.4.1) for parameter and state estimation. Unless where differently specified, the equations and the relevant parameters are taken as in ContCarSim.³⁶

A.1. |. Filtration model

A.1.1. |. Equations

The filtration model is used for calculating the filtrate mass profile $M_{\text{filt }}\left(t\right)$ and the filtration duration $\Delta t_{\text{filtration }}$. From a mass balance, $M_{\text{filt }}\left(t\right)$ is:

$$M_{\text{filt}}\left(t\right)=\frac{-b_{\text{filt}}+\sqrt{b_{\text{filt}}^2-4a_{\text{filt}}{c}_{\text{filt}}}}{2a_{\text{filt}}}{\rho }_l,$$ (A1)

where ρ_l is the filtrate density, and:

$$a_{\text{filt}}=\frac{\alpha {\mu }_l}{2A^2}\frac{V_{\text{slurry}}{c}_{\text{slurry}}}{V_{\text{filt,final}}}$$ (A2)

$$b_{\text{filt}}=\frac{R_m{\mu }_l}{A}$$ (A3)

$$c_{\text{filt}}=-\Delta P$$ (A4)

where α is the specific cake resistance, μ_l is the liquid viscosity, V_slurry is the fed slurry volume, c_slurry is the fed slurry concentration, A is the filter cross-section, R_m is the filter mesh resistance, ΔP is the pressure gradient applied to the system and V_filt,final is the volume of filtrate at the end of filtration. From a mass balance:

$$V_{\text{filt,final}}=V_{\text{slurry}}\left(1-\frac{c_{\text{slurry}}}{{\text{ρ}}_{\text{s}}}\left(1+\frac{ϵ}{1+ϵ}\right)\right),$$ (A5)

where ϵ is the cake porosity and ρ_s is the solid density. Finally, Δt_filtration is calculated as:

$$\Delta t_{\text{filtration}}=\frac{{\mu }_l\alpha V_{\text{slurry}}{c}_{\text{slurry}}{V}_{\text{filt,final}}}{2A^2\Delta P}+\frac{{\mu }_l{R}_m{V}_{\text{filt,final}}}{A\Delta P}$$ (A6)

A.1.2. |. Model parameters and inputs

In Equations (A1)–(A6), parameters ρ_l, μ_l, A, and ρ_s are always taken as in ContCarSim. ΔP is fixed to 10⁵ Pa_g (Table 4). The inputs of the model are c_slurry, R_m, V_slurry, α, and ϵ, and their origin depends on the application of the model, as further detailed in the main text. In the input/output model, c_slurry, V_slurry, α, and ϵ are calculated as product between their nominal values (${\bar{c}}_{\text{slurry }},V_{\text{slurry }}^{\text{sp }},\bar{\alpha }$, and $\left.\overline{ϵ}\right)$ and the respective variability sources $\left(c_{\text{slurry }}^{\text{dist }},V_{\text{slurry }}^{\text{dist }},{\alpha }^{\text{nomdist }}\right.$ and $\left.{ϵ}^{\text{dist }}\right)$, which are all inputs of the input/output model. While ${\bar{c}}_{\text{slurry }}$ and $V_{\text{slurry }}^{\text{sp }}$ are inputs of the input/output model, $\bar{\alpha }$ and $\overline{ϵ}$ are fixed parameters, taken as in ContCarSim.³⁶ Within the SS framework, instead, the available measurements of c_slurry and V_slurry (respectively, $y_{c_{\text{slurry }}}$ and $y_{V_{\text{slurry }}}$) and the estimation of α and ϵ (respectively, $\widehat{\alpha }$ and $\widehat{ϵ}$) are directly used.

A.2. |. Deliquoring model

A.2.1. |. Equations

The deliquoring model is used for calculating the axial profile of ethanol content in the cake at the end of deliquoring. The model consists of one partial differential equation, the dynamic liquid phase total mass balance with respect to the axial coordinate of the cake z:

$$\frac{\partial S}{\partial t}=-\frac{1}{ϵ}\frac{\partial u_l}{\partial z},$$ (A7)

where S is the cake saturation (ratio between pores volume occupied by liquid and total pores volume of the cake) and u_l is the local liquid velocity, calculated through:

$${\mathrm{u}}_{\mathrm{l}}=-\frac{S_R^{2+3{\lambda }_p}}{\alpha {\mathrm{\rho }}_{\mathrm{s}}(1-ϵ){\mu }_l}\frac{\mathrm{d}{\mathrm{P}}_l}{\mathrm{d}\mathrm{z}}$$ (A8)

$$S_R=\frac{S-S_{\infty }}{1-S_{\infty }}={\left(\frac{P_b}{P_g-P_l}\right)}^{{\lambda }_p},$$ (A9)

where λp is the pore-size distribution parameter, S_R is the local reduced saturation (defined by Equation A9), P_b is the minimum pressure to apply to the cake displace the liquid in the pores, and P_g and P_l are the local pressure of, respectively, the gas and of the liquid phases. P_l is obtained from Equation (A9), with total gas pressure drop through the cake corresponding to ΔP_cake:

$$\Delta P_{\text{cake}}=\Delta P\left(1-\frac{R_m}{\alpha H_{\text{cake}}{\mathrm{\rho }}_{\mathrm{s}}(1-ϵ)+R_m}\right),$$ (A10)

where H_cake is the cake height. The ethanol content within the cake in mass fraction terms is immediately obtained from the cake saturation through a mass balance.

To simulate deliquoring, Equation (A7) is semi-discretized with a high-resolution finite volume approach⁴⁰ along z. The obtained ODEs are integrated with MATLAB’s ode23s solver. The initial condition for deliquoring necessary for integrating Equation (A7) corresponds to full saturation conditions, which are the conditions achieved at the end of filtration. The boundary conditions are:

$$S(t,z=0)=S_{\infty },\forall t>0$$ (A11)

The deliquoring model is implemented in dimensionless form⁴¹ for the purposes of this work, to increase numerical robustness.

A.2.2. |. Model parameters and inputs

Parameters P_b, S_∞, λ_p, and ρ_s are always taken as in ContCarSim. ΔP is fixed to 10⁵ Pa_g (Table 4). The inputs of the model are H_cake, R_m, and ϵ, and they are provided to the model based on the application, as further detailed in the main text. In the input/output model, H_cake is calculated with a mass balance from the inputs of the input/output carousel model ${\bar{c}}_{\text{slurry }},c_{\text{slurry }}^{\text{dist }},V_{\text{slurry }}^{\text{sp }}$, and $V_{\text{slurry }}^{\text{dist }}\cdot R_m$. is an input, and ϵ is given by the product of $\overline{ϵ}$ (taken as in ContCarSim) by ϵ^dist (input of the input/output carousel model). When the deliquoring model is used within the SS framework, the available measurement of $H_{\text{cake }}\left(y_{H_{\text{cake }}}\right)$ is directly used, while R_m and ϵ are estimated.

A.3. |. Drying model

A.3.1. |. Equations

The drying model is used for calculating the axial profile of ethanol content in the cake during cake drying. The model consists of four partial differential equations, namely one-dimensional dynamic conservation balances developed along z:

$$\frac{\partial }{\partial t}{W}_{\text{dry}}=-\frac{{\dot{m}}^{L\to G}}{{\mathrm{\rho }}_{\text{cake}}}$$ (A12)

$${\mathrm{\rho }}_gϵ(1-S)\frac{\partial }{\partial t}{W}_g=-{\mathrm{\rho }}_g{u}_g\frac{\partial }{\partial z}{W}_g+{\dot{m}}^{L\to G}$$ (A13)

$$\left({\rho }_s{c}_{p,s}(1-ϵ)+{\mathrm{\rho }}_l{c}_{p,l}ϵS\right)\frac{\partial T_{\text{cake}}}{\partial t}=h_{T}a\left(T_g-T_{\text{cake}}\right)+{\dot{m}}^{L\to G}\lambda$$ (A14)

$$\left({\mathrm{\rho }}_g{c}_{p,g}\in (1-S)\right)\frac{\partial T_g}{\partial t}=-h_{T}a\left(T_g-T_{\text{cake}}\right)-u_g{c}_{p,g}{\rho }_g\frac{\partial T_g}{\partial z},$$ (A15)

where ${\rho }_{\text{cake }}\left(={\rho }_s(1-ϵ)+{\rho }_lϵS\right)$ is the cake density, ${\dot{m}}^{L\to G}$ is the specific drying rate of ethanol, ρ_g is the gas density, w_g is the mass fraction of ethanol in the gas phase, u_g is the superficial gas velocity, c_p,s is the solid phase specific heat, c_p,l is the liquid phase specific heat, T_cake is the temperature of the cake, a is the cake specific surface, T_g is the gas temperature, λ is the latent heat of vaporization of ethanol and c_p,g is the gas phase specific heat. The Darcy law for mono-phase gas flow in a porous medium⁴² is used for calculating u_g:

$$u_g=\frac{1}{\alpha {\mathrm{\rho }}_s(1-ϵ){\mu }_g}\frac{\Delta P}{H_{\text{cake }}},$$ (A16)

The specific drying rate of ethanol is calculated as:

$${\dot{m}}^{L\to G}=\left\{\begin{array}{ll}{h}_{M}a\left(P_{\text{sat}}-P_g\right)\eta & \text{if}{P}_{\text{sat}}>P_g\\ 0& \text{if}{P}_{\text{sat}}\le P_g\end{array}\right.,$$ (A17)

where h_M is the drying kinetic constant, P_sat is the local saturation pressure and η is the effectiveness factor (implemented as in ContCarSim). We calculate h_M and h_T by multiplying, respectively, $h_M^{\text{dist }}$ and $h_T^{\text{dist }}$ (inputs of the model) by the relevant nominal values ${\bar{h}}_M$ and ${\bar{h}}_T$, obtained through³⁶:

$${\bar{h}}_M=\frac{{\mathrm{\rho }}_g{u}_g}{a\xi d_{32}{P}_g}$$ (A18)

$${\bar{h}}_T=\frac{c_{p,g}{\mathrm{\rho }}_g{u}_g}{a\xi d_{32}},$$ (A19)

where ξ is the channeling parameter and d₃₂ the Sauter mean diameter of the crystal-size distribution of the slurry. For achieving the solution of the drying model, partial differential Equations (A12)–(A14) are semi-discretized with respect to space,⁴⁰ and then integrated with MATLAB solver ode15s. In the carousel, the drying air entering the dryer undergoes a temperature drop from $T_{\text{dry }}^{\text{sp }}$ before reaching the top of the cake. This temperature drop needs to be computed for setting the boundary conditions. For this purpose, we use the heat loss component of the drying model implemented in ContCarSim.

A.3.2. |. Model parameters and inputs

When using the drying model, parameters a, c_p,l, c_p,s, d₃₂, λ, ρ_l, ρ_s, and ξ are always taken as in ContCarSim. $T_{\text{dry }}^{\text{sp }}$ and ΔP are fixed to the conditions of Table 4. The inputs of the model are H_cake, $H_{\text{cake }},h_M^{\text{dist }},h_T^{\text{dist }},R_{m,}{w}_{\text{dry }}(0,z)$, α, and ϵ, and their value depends on the application of the model, as further detailed in the main text. When the drying model is used within the input/output model, H_cake is calculated with a mass balance from the inputs of the input/output carousel model, while $h_M^{\text{dist }},h_T^{\text{dist }},R_m$, and R_m are inputs. $w_{\text{dry }}(0,z)$ is the output of the deliquoring model, while α and ϵ are given by the product of the respective nominal values (taken as in ContCarSim) by the respective variability sources (inputs of the input/output model). Furthermore, within the SS framework, u_g is directly computed from the air flow rate measurement $y_{{\dot{V}}_{\text{ing, }}}$, instead of resorting to Equation (A16).

APPENDIX B: IMPLEMENTATION OF THE STATE ESTIMATOR OF LAYER 2

Let us consider the nonlinear process model of Equation (B1), represented by a set of ordinary differential equations:

$$\dot{\mathbf{x}}\left(t\right)=\mathbf{f}\left(\mathbf{x}\left(t\right),\mathbf{u}\left(t\right),t\right)+\mathbf{w}\left(t\right),$$ (B1)

where x is the state vector, u is the input vector, f is a vector of nonlinear functions, and the process noise w is considered to follow an $N(0,\mathbf{Q}(t\left)\right)$ distribution, where Q is the model error variance. Let us define a measurement model h, relating the vector y of measurements from a plant, available at finite sampling times t_k:

$$\mathbf{y}\left(t\right)=\mathbf{h}\left(\mathbf{x}\left(t\right),\mathbf{u}\left(t\right),t\right)+\mathbf{v}\left(t\right),$$ (B2)

where the measurement noise v is assumed to follow an N (0, R) distribution, and R is the measurement error variance. The discrete-time data EKF algorithm,³⁸ given the models of Equations (B1) and (B2), provides $\widehat{\mathbf{x}}$, estimation of the state vector, and P, the estimation error covariance, through a series of subsequent prediction and updates steps. First, the EKF is initialized through the initial state estimation ${\widehat{\mathbf{x}}}_0$ and the initial estimation error covariance P₀. A prediction step follows. For a general time interval in between two sampling times $t_{k-1}$ and t_k, prediction steps are carried out, by integrating Equations (B3) and (B4) from, respectively, the updated estimations of states $\widehat{\mathbf{x}}\left(t_{k-1}\mid t_{k-1}\right)$ and estimation error covariance $\mathbf{P}\left(t_{k-1}\mid t_{k-1}\right)$ at $t_{k-1}$, to yield the predictions of the states $\widehat{\mathbf{x}}\left(t_k\mid t_{k-1}\right)$ and of the estimation error covariance $\mathbf{P}\left(t_k\mid t_{k-1}\right)$ at t_k:

$$\dot{\widehat{\mathbf{x}}}\left(t\right)=\mathbf{f}\left(\widehat{\mathbf{x}}\right(t),\mathbf{u}(t),t)$$ (B3)

$$\dot{\mathbf{P}}\left(t\right)=\mathbf{F}\mathbf{P}+{\mathbf{P}\mathbf{F}}^{\mathrm{\top }}+\mathbf{Q}\left(t\right),$$ (B4)

where F is the Jacobian matrix of the process model:

$$\mathbf{F}={\left(\frac{\partial \mathbf{f}}{\partial \mathbf{x}}\right)}_{\widehat{\mathbf{x}}\left(t\right),\mathbf{u}\left(t\right),t}$$ (B5)

At each sampling time t_k, the estimations are updated through:

$$\widehat{\mathbf{x}}\left(t_k\mid t_k\right)=\widehat{\mathbf{x}}\left(t_k\mid t_{k-1}\right)+\mathbf{K}\left(t_k\right)\left[\mathbf{y}\left(t_k\right)-\mathbf{h}\left(\widehat{\mathbf{x}}\left(t_k\mid t_{k-1}\right),\mathbf{u}\left(t_k\right),t_k\right)\right]$$ (B6)

$$\mathbf{P}\left(t_k\mid t_k\right)=\mathbf{P}\left(t_k\mid t_{k-1}\right)-\mathbf{K}\left(t_k\right){\mathbf{H}}^{\mathrm{\top }}\mathbf{P}\left(t_k\mid t_{k-1}\right),$$ (B7)

where the Kalman gain K and the Jacobian matrix H, both at t_k, are respectively calculated following:

$$\mathbf{K}\left(t_k\right)=\mathbf{P}\left(t_k\mid t_{k-1}\right){\mathbf{H}}^{\mathbf{\top }}{\left[\mathbf{H}\mathbf{P}\left(t_k\mid t_{k-1}\right){\mathbf{H}}^{\mathrm{\top }}+\mathbf{R}\right]}^{-1}$$ (B8)

$$\mathbf{H}={\left(\frac{\partial \mathbf{h}}{\partial \mathbf{x}}\right)}_{\widehat{\mathbf{x}}\left(t_k\mid t_{k-1}\right),\mathbf{u}\left(t_k\right),t_k}$$ (B9)

The implementation of Equations (B1)–(B9) for state estimation in Layer 2 of the carousel control system (Section 3.4.1) is performed as follow. The drying model presented in Appendix A is used for state estimation within the EKF. The differential states corresponding to the four partial differential equations of the drying model are the content of ethanol in the cake being dried $w_{\text{dry }}(t,z)$, the ethanol content in the drying air $w_{\text{air }}(t,z)$, the cake temperature $T_{\text{cake }}(t,z)$, and the air temperature $T_{\text{air }}(t,z)$. The equations are discretized with respect to space in a 10-points grid, and integrated with MATLAB solver ode15s during the prediction step of the EKF.

The corresponding states vector of the EKF $\mathbf{x}\in {\mathbb{R}}^{40}$ is:

$$\mathbf{x}=\left[w_{\text{dry}}\left(t,z_1\right)w_{\text{dry}}\left(t,z_2\right)\dots w_{\text{dry}}\left(t,z_{10}\right)w_{\text{air}}\left(t,z_1\right)w_{\text{air}}\left(t,z_2\right)\dots w_{\text{air}}\left(t,z_{10}\right)\left.T_{\text{cake}}\left(t,z_1\right)T_{\text{cake}}\left(t,z_2\right)\dots T_{\text{cake}}\left(t,z_{10}\right)T_{\text{air}}\left(t,z_1\right)T_{\text{air}}\left(t,z_2\right)\dots T_{\text{air}}\left(t,z_{10}\right)\right]\right.,$$ (B10)

where z₁, z₂, …, z₁₀ are the coordinates of the 10 nodes of the discretization grid.

The inputs from the process used by the EKF are:

$$\mathbf{u}=\left[y_{{\dot{V}}_{\text{dryer}}}{y}_{H_{\text{cake}}}{y}_{T_{\text{in}}}\right]$$ (B11)

The output vector used within the EKF is $\mathbf{y}={\mathrm{y}}_{{\mathrm{T}}_{\text{out }}}$, and the corresponding measurement model is:

$$\mathbf{h}=T_g\left(z=z_{10}\right).$$ (B12)

In the initialization of the EKF, x₀ is built with the structure of Equation (B11). For every grid point $z_i:w_{\text{dry}}\left(0,z_i\right)$ is obtained through a mass balance from ${\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}\left(0,z_i\right),w_{\text{air }}\left(0,z_i\right)=0$, and $T_{\text{cake }}\left(t,z_i\right)=T_{\text{air }}\left(t,z_i\right)=295\mathrm{K}$ (i.e., room temperature). The only uncertain elements of x₀ are $w_{\text{dry}}\left(0,z_i\right)$, for each z_i node of the discretization grid. From considerations on estimation error propagation and based on the results of simulations carried out for this purpose, the estimation variance of ${\widehat{w}}_{\text{dry }}\left(0,z_i\right)$ is approximated to 3 × 10⁻⁵. Hence, ${\mathrm{P}}_0\in {\mathbb{R}}^{40\times 40}$ is defined as a diagonal matrix, whose only non-null entries are the first 10 elements of the diagonal, set to 3 × 10⁻⁵. Based on the measurement noise of TI102: R = 0.01.

$\mathbf{Q}\left(t_k\right)$ is calculated as a time-variant matrix, updated at every t_k, through⁴³:

$$\mathbf{Q}\left(t_k\right)={\mathbf{J}}_{p,\text{nom}}\left(t_k\right){\mathbf{C}}_p{\mathbf{J}}_{p,\text{nom}}^{\mathrm{\top }}\left(t_k\right),$$ (B13)

where C_p is the covariance matrix of the uncertain parameters p, and $J_{p,\text{ nom }}\left(t_k\right)$ is the Jacobian of f with respect to p:

$${\mathbf{J}}_{p,\text{nom}}\left(t_k\right)={\left(\frac{\partial \mathbf{f}}{\partial \mathbf{p}}\right)}_{\widehat{\mathbf{x}}\left(t_k\mid t_{k-1}\right),\mathbf{u}\left(t_k\right),t_k}$$ (B14)

Vector p contains the parameters of the drying model that present most mismatch with respect to those implemented in ContCarSim (i.e., the process):$p=\left[u_g{h}_M{h}_Tϵ\right]$. As suggested in the literature⁴³ for systems affected only by parametric mismatch (neglecting the structural mismatch introduced by the approximation in the discretization grid), C_p is built as a diagonal matrix, presenting the variance of the parametric fluctuations as diagonal elements: ${\mathbf{C}}_p=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left[4\times {10}^{-6}1.6\times {10}^{-21}2.5\times {10}^{-5}4.9\times {10}^{-5}\right]\right)$. Note that, Equations (B13) and (B14) inherently yield the local variance of the model error, and provide a reliable⁴⁴ estimation of $\mathbf{Q}\left(t_k\right)$. Hence, the designed Q and R matrices allow to effectively propagate the uncertainty P₀ in the initial state estimate x₀ into the estimation error covariance P of the estimated states x.

The EKF update step is called every 0.25 s during the process, even though the measurements are sampled every 0.025 s (Table 3). The states ${\widehat{w}}_{\text{dry }}\left(t,z_i\right)$ estimated by the EKF for every grid point z_i, are sent to the control system for monitoring and control purposes. The associated estimation standard deviations ${\sigma }_{{\widehat{w}}_{\mathrm{d}\mathrm{r}\mathrm{y}}}\left(t,z_i\right)$ are also sent to the control system for end-point detection (Equation B13).

As all the code implemented for the purposes of this work, the EKF presented in this Appendix B is implemented in the MATLAB environment. We use the System Identification Toolbox (version 9.15) for the implementation of the EKF, and the Adaptive Robust Numerical Differentiation Toolbox⁴⁵ (version 1.6) for computing the Jacobians of Equation (B14).

DATA AVAILABILITY STATEMENT

The code and the data that support the findings of this study are openly available in the ContCarSim repository at https://github.com/CryPTSys/ContCarSim; The repository contains the code of the carousel simulator used in the case study, and the MATLAB functions that resulted from the implementation of the proposed control system.